J. Phys. Chem. C 2007, 111, 6113-6123
6113
Multiple Mode Exciton-Phonon Coupling: Applications to Photoluminescence in Oligothiophene Thin Films Zhen Zhao and Frank C. Spano* Department of Chemistry, Temple UniVersity, Philadelphia, PennsylVania 19122 ReceiVed: NoVember 29, 2006
The impact of exciton-phonon (EP) coupling including two or more vibrational modes on emission in herringbone aggregates with N molecules is investigated theoretically. When the total nuclear relaxation due to all modes is much smaller than the exciton band curvature at the band minimum, the effects of several modes are additive, with each mode contributing a single sideband to the emission spectrum in the limit of large N. Outside the weak coupling regime synergistic intermode effects arise. In particular, the presence of a strongly coupled mode enhances the sideband intensity of a weakly coupled mode. Applications are made to photoluminescence spectra of oligothiophene thin films.
I. Introduction For the past several years many groups have vigorously investigated the fundamental electronic excitations in nanoaggregates, thin films, and crystals of the unsubstituted oligothiophenes (OTn) and oligophenylene vinylenes (OPVn), where n refers to the number of thiophene and phenyl rings, respectively.1-14 These molecules and their associated polymers are finding utility as active materials in a range of electronic and optoelectronic devices, including field effect transistors,15,16 solar cells,17 and light-emitting diodes.16,18,19 In OPVn and OTn thin films and crystals, electronic excitations are delocalized primarily within herringbone (HB) layers where intermolecular coupling is strongest. The excitations are dressed with phonons originating mainly from high-frequency intramolecular vibrational modes. Much of our prior work20-26 dealt with exciton-phonon (EP) coupling involving the dominant ∼1400 cm-1 ring-breathing/bond-stretching mode common to virtually all molecules with extended π-conjugation and responsible for pronounced vibronic progressions in singlemolecule solution spectra.27,28 In the current work we expand our investigation to include EP coupling to two or more intramolecular modes in order to better understand highresolution emission spectra of OTn thin films. However, multiple-mode EP coupling has a far broader significance and is important in understanding charge and energy transport in an array of condensed-phase organic systems, from artificial materials such as FETs and solar cells17 to naturally occurring systems such as photosynthetic antenna arrays.29 A secondary theme of the current work is nonlinear effects arising from the intermode interactions. We are particularly interested in conditions under which the effects of EP coupling to several modes are no longer additive. EP coupling involving the 1400 cm-1 mode results in vibrationally “dressed” excitons with center-of-mass wave vector K, subsequently referred to as (neutral) polarons. An HB aggregate supports a distribution of polaron effective masses.24,30 Heavy K ) 0 polarons near the band bottom are largely responsible for the weak and featureless A1 and A2 absorption bands observed in nanoaggregates.9,10,14 By contrast, weakly dressed “light” polaronssessentially free excitonssare responsible for the strongly blue-shifted main absorption peak or H-band (relative
to the monomer band in solution) consistent with a Davydov splitting of approximately one eV in OTn and OPV3.11,12,14,22 The A1 band in high-quality films and crystals of OTn and OPV3 at low temperature exhibits additional fine structure due to low-frequency intramolecular modes.5,6,11-13 Birnbaum et al.31 have identified at least five symmetric modes from solid solution excitation and fluorescence spectra of OT4 ranging from approximately 165 cm-1 up to 1550 cm-1. OPV3 also has an array of modes covering a comparable spectral range beginning at approximately 150 cm-1.5,9,27 Recent theoretical treatments underscored the importance of an aggregate version of HerzbergTeller coupling to account for the A1 fine structure.32,33 Excited states near the exciton band bottom were found to be primarily products of heavy polarons due to dressing with high-frequency modes, and phonons due to the low-frequency modes (< n| + j)R,β,... ∑ λj2ωj
j)R,β,... n † ωjλj (bn,j j)R,β,... n
∑
(6)
where p ) 1. Zero energy corresponds to the ground state of † the aggregate with no electronic or vibrational excitations. bn,j and bn,j are creation and anihilation operators, respectively, for intramolecular vibrations of mode j on molecule n relative to the ground-state (unshifted) potential. The complete Hamiltonian including all modes can be represented as a simple generalization of eq 6 by expanding the summations over j to include also the weakly coupled a,b,... modes. However, we have chosen to separate out the weakly coupled modes in order to anticipate their treatment via perturbation theory. We therefore introduce the zero-order Hamiltonian, H(0) a,b,.. for modes a,b,... as (0) ) Ha,b,....
† † ωj b˜ q)0, ˜ q)0, j + ∑ ∑ ωjbq, ∑ jb jbq, j + j)a,b,... j)a,b,...q*0
( )∑ 1-
1
N
λj2ωj (7)
j)a,b,...
(0) is written in the (k,q) representation with the where Ha,b,.... (Einstein) phonon operator given by
bq,j ≡
1
xN
∑n exp(-iq‚n)bn, j
qx,qy ) 0, (2π/M, ..., π (8)
The totally symmetric q ) 0 phonon for any mode j ∈{a,b,...} does not participate in EP coupling and is exactly treatable using the shift operator
b˜ q)0,j ≡ bq)0,j +
λj
xN
(9)
6116 J. Phys. Chem. C, Vol. 111, No. 16, 2007
Zhao and Spano
Finally, H′ represents the coupling of the weak nonsymmetric (qj * 0) modes
H′ )
∑ j)a,b,...
ωjλj
xN
∑k q*0 ∑ (bq,j† |k〉〈k + q| + bq,j|k〉〈k - q|)
III. All Weakly Coupled Modes We begin our analysis of low-temperature emission by considering an aggregate with several weakly coupled modes (a,b,...), each obeying the inequality in inequality (4). In this case HR,β,... in eq 6 reduces to the free excitonic Hamiltonian, (0) Hex in eq 1. The eigenstates of Hex + Ha,b,... can be factored into an exciton part and a phonon part, which are then mixed by H′ in eq 10. We assume, as will be justified shortly, that the mixing, though H′ is weak, is under the collective influence of all modes. Because the aggregates of Figure 1 are essentially Haggregates the lowest energy-emitting state is an exciton with wave vector k ) (π,π), as depicted in Figure 2. Correct to first order in H′, the lowest-energy exciton is expressed as
|ψem〉 ) |k ) (π,π)〉 X |Vaca,b,...〉 ωjλj † |k〉 X b(π,π)k,j|Vaca,b,...〉 (11) k(*(π,π)) j)a,b,... x N (Wk + ωj)
∑ ∑
with energy
∑j)a,b,... λ2j ωj
in the large N limit. In eq 11, the vibrational vacuum state is represented in a phonon number basis as
|Vaca,b,...〉 )
j {|n˜q)0 ) 0〉 X ∏ |nqj ) 0〉} ∏ j)a,b,...
(12)
q(* 0)
The symmetric (q ) 0) mode in eq 12 causes a shift λj/xN along the vibrational coordinate j in all molecules, leading to emission sidebands that vanish with N as is shown in the following section. The energy difference Wk in eq 11 is defined as
Wk ≡ J˜k - J˜(π,π)
∑ ∑
(10)
The division of the Hamiltonian in eq 5 is exact. Although this suggests that several modes satisfying (4) can be treated collectively perturbatively (i.e. to low order in H′), this is not generally true, as is shown in the following section. Moreover, synergistic effects may arise between two or more modes. For example, the presence of a strongly coupled mode further reduces the curvature of the band bottom below 2Jc, potentially moving a weakly coupled mode out of the perturbative regime. Such nonlinear cases are most interesting and are discussed in greater detail in section IV.
ωem ≈ ω0-0 + D - 4J0 + 4J1 +
of the mixing coefficients is small, i.e., when
(13)
The free exciton Davydov splitting W is therefore also denoted Wk)0. To zero order the exciton (11) is primarily a BO state consisting of a product of the free k ) (π,π) exciton and the phonon vacuum state. Nonadiabatic admixtures of higher k states are made possible by phonons with wave vectors q ) (π,π) k. The BO approximation is valid for all modes when the admixture is weaksthis happens when the sum of the squares
j)a,b,... k
λ2j ω2j N(Wk + ωj)2
,1
(14)
Converting the sum over k to an integral (assuming large N) and utilizing the parabolic approximation25 in the vicinity of the band minimum converts the inequality to
∑j λ2j ωj , 4πJc
(15)
Condition 15 requires the total nuclear relaxation to be smaller than 2π times the exciton curvature at the band bottom in order for the BO approximation in eq 11 to remain valid. Note that a derivation along the lines of McRae39 yields condition 15 but with the rhs replaced by W. However, ref 39 uses an energyindependent density of states function which is valid only when J1 ) 0. Figure 2b clearly shows the density of states is focused toward the band bottom as J1 increases, thereby justifying conditions 4 and 15. The exciton (eq 11) gives rise to the very weak origin or J-peak in the absorption spectrum of HB aggregates. The transition is b-polarized due to the out-of-phase oscillation of the two sublattices. At low temperature the state 11 is also the origin of intrinsic exciton emission. Such emission terminates on the electronic ground state of the aggregate containing any number of phonons corresponding to combinations involving one or more intramolecular vibrational modes. Hence, when the temperature is low enough to ignore the thermally activated emission from higher excited states, the emission spectrum consists of all transitions
|ψem〉 f |G;
nqj 〉 ∏ ∏ q
(16)
j)a,b,...
where the terminal state is the product of a pure electronic ground state G (with all molecules electronically unexcited) and a phonon number state with njq ()0,1,2,...) phonons of mode j with wave vector q. The intramolecular vibrational excitations from which the phonons are built correspond to the unshifted nuclear potential in the electronic ground state. We further define Vjt to be the total number of mode-j phonons in the terminal state, Vjt ≡ ∑qnjq. Then, neglecting phonon dispersion (as for Einstein phonons), emission terminating on the electronic ground state with Vat mode-a phonons, Vbt mode-b phonons, etc., corresponds to the creation of a photon with energy, ωem - ∑j a b t Vt Vjtωj with a F-polarized oscillator strength denoted by I0-V . F The resulting emission spectrum consists of vibronic progressions involving multiple intramolecular vibrational modes
SF(ω) )
∑
t Vt ... I 0-V Wh(ω - ωem + F a b
Vat ,Vbt ,...
∑j Vtjωj)
(17)
The spectrum 17 is simplified since we have omitted the cubic frequency dependence as well as refractive index dependencies in order to focus entirely on the line-strength dependence. Wh is a normalized homogeneous line shape function, taken here to be a Gaussian. The dimensionless F-polarized 0 - Vat Vbt ... line strength is generally given by
IF0-Vt Vt ... ) a b
1 2
µ
∑
{naq,nbq,...}
′||〈 ψem|M ˆ F|G;
njq〉||2 ∏ j,q
(18)
Multiple Mode Exciton-Phonon Coupling
J. Phys. Chem. C, Vol. 111, No. 16, 2007 6117
(µ⊥ cos φ)2
)N
Ib0-00...
µ
2
exp(-
∑j λ2j /N)
(21)
shows the cooperative N-fold enhancement arising from a constructive interference of N tdm’s, characteristic of superradiance.23 In the thermodynamic limit, N in eq 21 is replaced by the thermal coherence size, NT.25 By contrast, the b-polarized line strength corresponding to the first replica of a given mode, say a, is
I0-100... b
(µ⊥ cos φ)2
)
µ
Figure 3. Polarized emission spectrum for a pinwheel (a) and 4 × 4 aggregate (b) including two weakly coupled vibrational modes with frequencies ωa ) 150 cm-1 and ωb ) 250 cm-1 with λ2a ) λ2b ) 1. In addition, J0 ) 700 cm-1 and J1 ) 0.4J0. Solid lines are obtained from eqs 19 and 23. Dotted curves are obtained using the hybrid basis set. c-Polarized spectra are vertically offset for clarity.
The matrix element contains the transition dipole moment (tdm) operator, M ˆ F ) ∑nµn,F|n〉〈G| + hc, where µn,F is the F-component of the molecular tdm at site n (see Figure 1). The sum in eq 18 is over all sets of quantum numbers {naq,nbq,...} under the constraints indicated by the prime, ∑qnjq ) Vjt, for all j. Inserting the emitting exciton (eq 11) into eq 18 gives the emission line strengths. In b-polarized emission only the zeroorder term in eq 11 contributes, giving
Ib0-Vt Vt ... ) a b
N
(µ⊥ cos φ)2 µ2
{ } (λ2j /N)Vt
j
∑j
exp( -
λ2j /N)
∏j
Vjt!
(19)
where all vibrational excitations in the terminal state originate from the totally symmetric phonons with q ) (0,0). Equation 19 utilizes the size-enhanced tdm matrix element, 〈G| M ˆ b|k ) (π,π)〉 ) xN µ⊥ cos φ, and the Franck-Condon factors:
( )
-λ2j j 2 nq)0 exp λj N j j |n˜ q)0 ) 0〉|2 ) |〈nq)0 j N (nq)0 )!
()
According to eq 19 only the origin (Vjt ) 0, all j) and first replica (Vjt ) 1, Vj′(*j) ) 0) for any given mode j appear in the t b-polarized spectrum in the limit of large N. All higher replicas and combinations bands diminish at least as strong as N-1. The highest-energy emission peak (origin) corresponds to emission to the vibrationless ground state. The associated line strength
λ2a exp(-
∑j λ2j /N)
(22)
and is size independent due to a cancellation of the factor of N in the excitonic tdm by the factor of 1/N arising from the FC factor (eq 20). Interestingly, the sidebands (eq 22) are also independent of mode frequency as well as the strength of the intermolecular interactions. For the ac-polarized spectrum, inserting the wave function (eq 11) into the line strength (eq 18) shows that sideband emission is activated by the admixed states |k ) 0〉 X + bq)(π,π),j |Vac〉, where the dominant c-polarized component derives from 〈G|M ˆ c|k ) 0〉 ) xN µ| Such emission requires termination on the ground electronic state having a single q ) (π,π) phonon in mode j, and any number of totally symmetric phonons in any of the modes (including j). The general expression for the c-polarized line strengths is the sum over all such terms a b Ic0-Vt Vt ,...
)
µ2| µ
2
∑j λ2j /N) ×
exp(-
∑j
′
{
ω2j λ2j
(λ2j /N)Vt -1 j
(W + ωj)2 (Vtj - 1)!
∏ i*j
(λ2i /N)Vt (Vti)!
}
i
(23)
where the prime indicates that the jth term in the sum is zero unless Vjt g1. Finally, the a-polarized line strengths are given by the rhs of eq 23 multiplied by (µ2⊥/µ2| ) sin2 φ. Equation 23 shows that no cooperative enhancement is present in the ac-polarized emission spectrum. The first sideband of mode a arises from intensity borrowing (ala Herzberg-Teller) + from |k ) 0〉 X bq)(π,π),a |Vac〉 . The higher-order ac-polarized a replicas (Vt g 2, Vj(*a) ) 0) involve Vat - 1 additional totally t symmetric q ) (0,0) phonons. According to eq 23 all higher replicas, as well as combination bands built from additional modes, vanish in the limit of large N, leaving only the first replica for each mode. For mode a the strength of the first sideband is
Ic0-100... ) j ) a,b,... (20)
2
µ|2
ω2aλ2a
µ (W + ωa) 2
2
∑j λ2j /N)
exp( -
(24)
which, like its b-polarized partner, lacks coherent (N) enhancement. The single-mode version of eq 24 has been derived previously.25,40 To demonstrate the emission properties outlined above, we present in Figure 3 the polarized emission spectrum involving two vibrational modes, a and b, with frequencies ωa ) 150 cm-1 and ωb ) 250 cm-1 and with HR factors, λ2a ) λ2b ) 1, coupled to excitons in a pinwheel aggregate and a larger 4 × 4
6118 J. Phys. Chem. C, Vol. 111, No. 16, 2007
Zhao and Spano
aggregate. Using J0 ) 700 cm-1 and J1 ) 0.4J0 the inequality (15) becomes
∑j λ2j ωj , 1800 cm-1
(25)
which is satisfied for the two modes. Calculations were conducted using the perturbative expressions for the line strengths (eqs 19 and 23) in the emission spectrum (eq 17). We also show numerical calculations using the hybrid basis set, as discussed in the following section, which are practically indistinguishable from the analytical results. In Figure 3 peaks are labeled as 0 - Vat Vbt with the number of quanta in the lowest-frequency mode reported first, a convention maintained throughout this paper. As the aggregate size increases from N ) 4 to 16, the b-polarized origin is enhanced 4-fold, in accord with the factor of N scaling in eq 21. By contrast, the b-polarized 0-1 intensities remain virtually unchanged (ala eq 22), while the ac-polarized 0-1 peaks are diminished by approximately a factor of 2.5 due primarily to a increase in the DS (see eq 24); from W ) 2800 cm-1 in the pinwheel to 5600 cm-1 in the 4 × 4 aggregate. Moreover, all higher-order sidebands including the 0-11 combination band and second replicas (0-20 and 0-02) are strongly attenuated when going from the pinwheel (N ) 4) to the 4 × 4 aggregate (N ) 16). In summary, for large aggregates (N g 16) in the weak coupling regime defined by condition 15 the low-temperature emission spectrum is greatly simplified, consisting of a sizeenhanced b-polarized emission origin, followed by a series of b- and ac-polarized first sidebands, one for each mode. In this regime, the effects of multiple modes are additive. IV. Polaron-Phonons In the case of OPVn and OTn aggregates, thin films, or crystals, the several vibrational modes coupled to the optical gap electronic transition are divided among the weak and intermediate EP coupling regimes, so that the perturbation theory of the previous section cannot be universally applied. The more strongly coupled modes increase the effective mass of the emitting exciton, altering the band curvature and thereby potentially moving weakly coupled modes out of the perturbation regime. In order to clearly demonstrate such intermode nonadditive effects we consider just two vibrational modes, one in each of the aforementioned regimes. Maintaining the same J0 and J1 as in the previous section we take the weakly coupled mode (see eq 25) to have ωa ) 150 cm-1 and λ2a ) 1, comparable to the low-energy symmetric modes found in OPV35,9,27 and OT4.31 The intermediate mode is the symmetric ring-breathing/-stretching mode with ωβ ) 1400 cm-1 and λ2β ) 1. (To avoid possible notational confusion we use β instead of R.) Hence, λ2βωβ is approximately 4πJc ()1800 cm-1) but is much smaller than W. The emission spectrum is calculated numerically using the hybrid basis set introduced in ref 33, which coincides with the Hamiltonian subdivision in eq 5. Figure 4 shows emission spectra for a pinwheel and 4 × 4 aggregate calculated using an energy cutoff of 3000 cm-1 and phonon cap of two, details of which are found in ref 33. Also shown is the two-mode perturbation theory result obtained using eqs 19 and 23. The exact spectrum is blue-shifted compared to the perturbation theory result due to band renormalization. Although perturbation theory only slightly overestimates the origin and first sideband of mode β (0-01) it fails completely to capture the substantial intensity of the higher mode-β replicas (0-02, 0-03, ...,).
Figure 4. Polarized emission spectrum for (a) a pinwheel and (b) a 4 × 4 aggregate including coupling to two vibrational modes: an intermediately coupled mode with ωβ ) 1400 cm-1 and a weakly coupled mode with ωa ) 150 cm-1. Both modes have HR factors of unity. J0 and J1 are the same as in Figure 3. Dotted line is the perturbation theory result (see text). b-Polarized spectra are vertically displaced for clarity. Note the different vertical scales in (a) and (b).
As in Figure 3, the b-polarized emission origin (0-00) in Figure 4 scales with the number of coherently connected molecules, N. A striking feature in Figure 4 is the strong suppression of the ac-polarized component of the first sideband of the low-frequency mode a (0-10) compared to the corresponding sideband of mode β (0-01) despite identical HR factors. The disparity is most obvious in comparing the monomer spectrum to the pinwheel spectrum as is done in Figure 5. This suppression is entirely consistent with what is observed in the emission spectra of both OPV35 and OTn1,4 thin films. The integrated spectrum, which is proportional to the radiative decay rate (and therefore the quantum yield) is also significantly reduced in the aggregatesan example of aggregation-induced quenching.41 The primary reason for the biased suppression of the lowfrequency c-polarized sidebands is roughly contained in the perturbation expression 24, which shows the peak intensity to scale as the square of the mode frequency. Application of expression 24 to both modes yields:
Ic0-10 Ic0-01
≈
λ2aω2a(W + ωβ)2 λ2βω2β(W + ωa)2
For the 4 × 4 aggregate this yields a factor of 0.017 compared to a factor of 0.024 obtained numerically from Figure 4. The increase over the perturbation theory result stems primarily from an amplification of the low-frequency sideband in the presence of the more strongly coupled mode β. The extra enhancement of the 0-10 sideband in the acpolarized spectrum is a synergistic effect due to the dressing of
Multiple Mode Exciton-Phonon Coupling
J. Phys. Chem. C, Vol. 111, No. 16, 2007 6119 denoted |ψrK〉, is a neutral polaron with the central vibronic excitation almost completely relaxed in the upper-state nuclear potential, surrounded by a vibrational distortion field in neighboring molecules. Under the two-particle approximation |ψrK〉 is expressed as21,24,30,42
|ψrK 〉 )
cVK;r ˜ ;sV 〉 ∑V˜ cVK;r˜ |KV˜ 〉 + V˜∑ ˜ ;s,V|KV ;s,V
(26)
with energy denoted rK. In contrast to the case of a free exciton where each exciton has a unique K, there are generally many polarons with a given wave vector K. We take r ) 1, 2, ..., to label such states in order of increasing energy. Hence, the r)1 lowest-energy (emitting) state is |ψK)(π,π) 〉. In eq 26 the states |KV˜ 〉 are one-particle (vibronic) excitons,43 Figure 5. Calculated unpolarized monomer vs pinwheel emission spectra including two vibrational modes with frequencies ωβ ) 1400 cm-1 and ωa ) 150 cm-1 and HR factors of unity for both modes. Aggregate parameters are the same as in Figure 4a.
|KV˜ 〉 ≡
1
xN
∑n exp(iK‚n)|nV˜ 〉
(27)
where |nV˜ 〉 denotes a vibronic excitation at n, with V˜ vibrational quanta in the excited state (shifted) nuclear potential. |KV˜ ;sV〉 are two-particle excitons30,42 consisting of delocalized vibronic/ vibrational pair states
|KV˜ ;sV 〉 ≡
1
xN
∑n exp(iK‚n)|nV˜ ;n + s,V 〉
(28)
|nV˜ ;n + s,V〉 denotes the state with a vibronic excitation at n and a vibrational excitation with V quanta in the unshifted (ground state) potential at n + s. The expansion coefficients in eq 26 are determined numerically as described previously21,24 and are normalized according to
Figure 6. c-Polarized mode-a sidebands near the emission origin calculated in a 4 × 4 aggregate for several values of the mode-β frequency, ωβ ) nω0 (n ) 1, 2, 5) with ω0 ) 1400 cm-1. Peaks have been shifted spectrally into alignment with the n ) 1, λ2β ) 1 spectrum for clarity. Inset shows the effect of increasing λ2β while holding ωβ ) ω0. All other parameters are the same as in Figure 4b.
the emitting state by coupling to mode β. In Figure 6 the mode-a sideband (0-10) is plotted for several values of ωβ (holding λ2β ) 1) and several values of λ2β (holding ωβ ) 1400 cm-1; see Figure 6 inset). The figure clearly shows that the extra enhancement relative to the case when mode β is absent originates from increased EP coupling in mode β. Moreover, the second sideband (0-20) gains prominence compared to the 0-10 peak as λ2βωβ increases, with I0-20 /I0-10 ranging from c c 0.11 when mode β is absent to 0.18 when ωβ ) 5ω0, quite smaller than the value of 0.5 expected in the single-molecule limit for λ2a ) 1 (but larger than the perturbative result of 1/16). The polaron enhancement of the ac-polarized 0-10 peak can be approached perturbatively by starting with a zero-order K ) (π,π) polaron state in contrast to the free exciton used in eq 11. The enhancement can then be traced to intensity borrowing from a range of low-energy K ) 0 polarons. In several previous works21-26,30 we have considered the nature of the excited states when free excitons couple to 1400 cm-1 intramolecular vibrationssthe eigenstates of Hβ in eq 6. The rth such eigenstate with center-of-mass wave vector K,
2 |cVK;r ∑V˜ |cVK;r˜ |2 + V˜∑ ˜ ;s;V| ) 1 ;s,V
(29)
With the lowest-energy polaron in hand, we proceed to write a general expression for the emitting state in the presence of a weakly coupled mode a. Based on the Hamiltonian subdivision in eq 5 , first-order perturbation theory gives r)1 〉 X |Vaca〉 |ψem〉 ) |ψK)(π,π) ωaλaSK,r
∑ ∑r
K(*(π,π))
xN(WrK + ωa)
† | Vaca〉 (30) |ψrK〉 X b(π,π)-K,a
The expansion is in polaron-phonon product states, analogous to the exciton-phonon product states of eq 11. The second term contains a polaron overlap factor SK,r involving states |ψrK〉 and r)1 |ψK)(π,π) 〉
SK,r ≡
K)(π,π),r)1 + ∑ cVK;r ∑V˜ cVK;r˜ cVK)(π,π),r)1 ˜ ˜ ;sVcV˜ ;sV V˜ ,s,V
(31)
r)1 . Note that as well as the energy differential, WrK ≡ rK - K)(π,π) the factor SK,r is an “overlap” without consideration of the r)1 〉 is, of K-dependent phase. The “true” overlap, 〈ψrK|ψK)(π,π) course, just δK,(π,π)δr,1 since eigenstates 26 are orthonormal. Due to the coupling to mode β, the band bottom curvature is further reduced from 2Jc as the effective mass of the emitting exciton increases. In ref 25 it was shown that the curvature 2ωc
6120 J. Phys. Chem. C, Vol. 111, No. 16, 2007
Zhao and Spano
is given by, 2ωc ≈ 2FJc where F is a generalized FC factor
F≡|
)1 〈0 | V˜ 〉|2 ∑V˜ cVK)(π,π),r ˜
(32)
with the molecular vibrational overlap, 〈0 | V˜ 〉 ) (λβ)V˜ exp( - λ2β/2)/xV˜ !. F measures the strength of the EP coupling, ranging from exp(-λ2β) in the strong EP coupling regime to exp(-λ2β/N) in the weak coupling regime. With the present choice of parameters numerical calculations give F ≈ 0.61, reflecting intermediate coupling. The renormalized curvature further restricts the regime over which mode a can be treated perturbatively. Under the influence of mode β, the inequality 4 is replaced by
λ2aωa , 4πωc
(µ⊥ cos φ)2 µ2
F exp
( ) -λ2a N
(34)
which remains superradiantly enhanced through the factor of N, as in the previous section. Continuing with the b-polarized spectrum we find a progression built on the origin (eq 34) from the weakly coupled mode a. As in the previous section, only the first replica survives the large N limit with intensity:
Ib0-10 )
(µ⊥ cos φ)2 µ2
λ2aF exp
( ) -λ2a N
(35)
Division of eq 34 by eq 35 shows that I0-00 /I0-10 ) N/λ2a, b b which is also obtained in the perturbative regime of the previous section. This simple ratio of b-polarized line strengths does not depend on the nature of the zero-order emitting state; it may be either a free exciton or a polaron, making the ratio a robust measure of the exciton coherence size, N. The c-polarized contributions to the weakly coupled mode-a replicas are somewhat more involved but follow readily from the general line strength expression in eq S1 (SI). As in the previous section, only the first replica (Vat ) 1) survives large N:
Ic0-10
)
µ|2 µ
2
( ) -λ2a
exp
N
ω2aλ2a |
∑r
SK)0,r r (WK)0
E)
(33)
The general line strength expression based on the hybrid basis appears in the Supporting Information (SI). Inserting the wave function in eq 30 into eq S1 gives the b-polarized emission origin:
Ib0-00 ) N
polarized 0-10 emission line strength, whereas the A2-band and strongly blue-shifted H-band contribute about 20% and 40%, respectively. Overall the enhancement caused by the 1400 cm-1 mode is modest (∼1.2). However, the enhancement increases with the EP coupling strength (see Figure 6). The maximum enhancement is achieved in the limit of very strong EP coupling, λ2βωβ . W, where the emitting state lies as the bottom of the V˜ ) 0 vibronic band of width W exp(-λ2β).25 In this case, the 0-10 line strength is driven by intensity borrowing from the nearby κ ) 0 upper band state, higher in energy by only [W exp(-λ2β)]. As long as ωa , W exp(-λ2β) the enhancement, E, is
+ ωa )
〈0|V˜ 〉|2 ∑V˜ cVK)0,r ˜ (36)
and is similarly independent of N. Equation 36 shows that the intensity of the weakly coupled mode replica is borrowed from many K ) 0 polarons as indicated by the sum over r with the lending efficiency of the rth polaron given by the overlap factor, SK)0,r , and the energy-dependent factor, (WK)0,r + ωa)-1, the latter of which is enhanced for low-energy polarons. The energies and oscillator strengths of all K ) 0 polarons are revealed in the mode-β absorption spectrum, shown in Figure S1 plotted alongside the free-exciton spectrum. Low-energy heavy polarons give rise to the ubiquitous low-energy A1 and A2 peaks observed in OPVn and OTn aggregates, thin films, and crystals. Our analysis shows that the intensity borrowing from the nearby A1 peak contributes about 40% to the ac-
exp(-λ2β)
(W + ωa)2 ((W exp(-λ2β) + ωa)2
(37)
For W ) 5600 cm-1, ωa ) 150 cm-1, and λ2β ) 1, we obtain E ) 2.5, slightly greater than the largest enhancement observed in Figure 6. Increasing λ2βto 2 increases E to about 5.4. From eq 37 the optimal HR factor, λ2β ) ln(W/ωa), is 3.6, but the associated factor of 10 enhancement is not realized because the condition, ωa , W exp(-λ2β), necessary to justify perturbation theory is violated. As is shown in Figure 4, in contrast to the low-frequency mode (a), replicas for the high-frequency mode (β) survive at large N to form a well-defined progression. The c-polarized intensity of the first sideband can be expressed in the multiparticle basis set as was done in ref 24 where one- and twoparticle interferences were shown to be responsible for the reduction in the sideband intensity due to aggregation. A far more useful form for the first replica intensity for mode β can be obtained by observing that, since ωβ , W, the admixture of the product state |k ) 0〉 X |q ) (π,π)〉 (which supplies the 0-1 oscillator strength) with energy W + ωβ in the emitting exciton (eq 11) can still be accounted for perturbatively. Hence, the intensity is closely approximated by the weak coupling from
Ic0-01 )
µ|2 µ2
exp
[
]
-(λ2a + λ2β) λ2βω2β N (W + ωβ)2
(38)
as demonstrated in Figure 4. In ref 25 we showed that eq 38 is valid as long as W is greater than approximately 3ωβ. The current example shows that, although the emitting polaron is heavily dressed by excitons near the band bottom (since ωβ J 4πJc) , the coupling to the much higher states (which contain the sideband oscillator strength) remains treatable by perturbation theory whenever ωβ , W. Expressions for the higher mode-β replicas with Vβt g 2 are more involved but have appeared elsewhere.21,24 As Figure 4 shows, the ratio of the intensities of these higher replicas to that of the first sideband is very similar to the single-molecule result and can be estimated by using single-molecule FC factors: β
Ic0-0Vt
Ic0-01
(λ2β)Vt exp(-λ2β)/(Vβt )! β
≈
λ2β
exp(-λ2β)
(λ2β)Vt -1 β
)
(Vβt )!
Vβt g 2
(39)
Hence, the second (third) replica is approximately one-half (onesixth) the first replica intensity when λ2β ) 1 in excellent agreement with the numerical results in Figure 4. Finally, Figure 4 shows that the b-polarized component of the 1400 cm-1 sideband is negligible in comparison to the acpolarized component primarily because the off-axis component
Multiple Mode Exciton-Phonon Coupling
Figure 7. (a) Experimental emission spectrum for high-quality crystalline films of OT4 and OT5 from ref 1 at T ) 5 K. (b) The calculated spectrum for a 4 × 4 aggregate including only two vibrational modes: the high-frequency dominant mode with ωβ ) 1470 cm-1 and λ2β ) 1; and the lowest-frequency mode, with ωa ) 165 cm-1 and λ2a ) 1. The exciton coupling parameters are J0 ) 825 cm-1 and J1 ) 388 cm-1 (see text). The off-axis dipole moment for the OT4 simulation is µ⊥ ) 0.05 µ and φ ) 55°.
of the molecular dipole moment is much smaller than the longaxis component (see Figure 1). Furthermore, the b-polarized components of both the 1400 cm-1 and 150 cm-1 sidebands are practically identical. This surprising result can be appreciated from the weak mode expression (eq 35) which shows the b-polarized component to be independent of the mode frequency. V. Comparison to Experiment We compare our calculations with the emission spectra of high-quality crystalline samples of OT4 and OT5 as obtained by Meinardi et al.1 and reproduced in Figure 7. Very similar spectra for OT4 films have more recently been obtained by Gebauer, et al.4 The OT5 spectrum lacks the 0-0 origin which is clearly defined in OT4 films at low temperature. This arises because OTn’s with odd n have C2V symmetry forcing the transition dipole moment to align exactly along the long molecular axis. Hence, µ⊥ ) 0, and the lower Davydov component is optically forbidden (see eq 34).1,11 Although the film spectra in Figure 7a are unpolarized, crystal spectra of OT6 show the 0-0 emission line to be entirely b-polarized1,3 in agreement with the current model. Both spectra in Figure 7 show several low-energy sidebands due to weakly coupled modes (a,b,...) with energies ∼165 cm-1, 320 cm-1, etc., which have also been been identified in solid solution spectra.31 The β mode (or collection of closely spaced modes) with energy 1470 cm-1 dominates the emission spectrum with its first replica at least
J. Phys. Chem. C, Vol. 111, No. 16, 2007 6121 an order of magnitude larger than the low-energy (first) sidebands in both films. This is in sharp contrast to the solid solution (isolated molecule) spectrum where the 1470 and 165 cm-1 sidebands are approximately equal and therefore have similar HR factors.31 OT6 crystal spectra show the 1470 cm-1 sideband to be primarily ac-polarized in contrast to the emission origin.3 Since µ⊥ ) 0 in OT5, its aggregate spectrum should be primarily ac-polarized, thereby simplifying the analysis. The weak highest-energy peak from Figure 7a is a false origin, being the first replica of mode a (ωa ) 165 cm-1). The peak is approximately 20 times less intense than the mode-β (1470 cm-1) sideband. In OT4 the mode-a sideband is relatively more intense, most likely due to an added b-polarized component. To evaluate the ratio of the mode-a and mode-β sideband intensities theoretically requires estimates of W and Jc from which the exciton couplings J0 and J1 readily follow (see section II). On the basis of the measured DS of Wexp ≈ 1 eV for OT4 and OT512 and the relationship Wexp ≈ W + λ2βωβ (see ref 24 and Figure S1), we obtain W ) 6600 cm-1 (or J0 ) 825 cm-1) using λ2β ) 1 and ωβ )1470 cm-1. The renormalized band curvature in the presence of the 1470 cm-1 mode has been calculated previously25 on the basis of the complete set of extended excitonic couplings in a 5 × 5 aggregate22 yielding 2ωc ) 50 cm-1 and F ) 0.55.25,36 Using the relationship ωc ≈ FJc 25 gives Jc ≈ 50 cm-1 (from which J1 ) 0.47J0). Thus, 4πJc ≈ 600 cm-1, placing the 165 and 330 cm-1 modes approximately within the weak EP coupling regime, whereas the 1470 cm-1 is clearly in the intermediate-to-strong coupling regime. With the interaction parameters in hand, we then evaluated the OT5 spectrum (µ⊥ ) 0) for a 4 × 4 aggregate appearing at the bottom of Figure 7b. The solid curves show exact (numerical) calculations including only the 165 and 1470 cm-1 modes with HR factors of unity,28 as additional modes led to an enormous increase in computational effort. Moreover, we inserted the usual cubic frequency dependence in the emission spectrum (eq 17) which attenuates the 1470 cm-1 sideband by a factor of 0.8 relative to the 165 cm-1 sideband. In the calculated spectrum the ratio, Rβa, of the 1470 to 165 cm-1 sideband intensities is ∼30, in very good agreement with the measured value of ∼20 and not dramatically different from the value Rβa ) 45 calculated using the crude perturbation expression in eq 26 (including the cubic frequency dependence.) We also verified the insensitivity of Rβa to the band curvature by holding J0 fixed and varying J1 from 0 to 0.5J0, the latter value corresponding to a flat band, and noting a change in Rβa of only 3%. To evaluate the OT4 spectrum we recalculated the spectrum by setting µ⊥/µ to 0.05, as was determined for OT4 using MRDCI methods,22 and φ ) 55°.35 We also replaced the factor of N in the (0-0) emission origin with the thermal coherence size NT. As was shown in ref 25, in the thermodynamic limit and for minimal static disorder, the 0-0 line strength scales not with N but with NT; I0-0 ≈ NT(Fµ2⊥ cos2 φ/µ2) with NT ≡ 1 + b 4πpωc/kbTh. As pointed out by Wu et al.,5 the temperature Th following optical excitation well above the band minimum may be considerably higher than the cryostat temperature (5 K in Figure 7a) due to the heat generated during relaxation to the band bottom. Hence, we cannot accurately determine NT from ωc and Th. In ref 25 we showed that NT could be determined 0-1 from the line strength ratio, R0β ≡ I0-0 b /Ic , since the mode-β 0-1 sideband intensity, Ic , is independent of NT. From Figure 7a we obtain R0β ≈ 0.2, taking into account that line strengths are
6122 J. Phys. Chem. C, Vol. 111, No. 16, 2007
Zhao and Spano
proportional to the spectrally integrated intensities and the line width of the mode-β sideband cluster is about twice that of the ≈ 0.02 from the calculated OT5 origin line width. Taking I0-1 c spectrum we obtain, NT ≈ 0.02R0β/(Fµ2⊥cos2 φ/µ2). Using the parameters previously defined, we finally determined NT ≈ 10. After replacing N with 10 in the origin intensity and activating µ⊥, we obtained the spectrum shown at the top of Figure 7b. In the calculated spectra, the mode-a sideband intensity increases in going from OT5 to OT4 due to the addition of a significant b-polarized component arising from µ⊥. The increase also occurs in the experimental spectra. In both the calculated and measured OT4 spectra the total intensity of the mode-a sideband is approximately 4 times smaller than the origin intensity, providing an excellent check on our procedure. In Figure 7b the first sidebands of both the 165 and 1470 cm-1 modes are within 20% of the perturbation theory predictions based on eq 24. Assuming that the 320 and 688 cm-1 modes behave similarly, we added in Figure 7b the perturbation theory results for the first sidebands of both of these modes. We utilized HR factors obtained from the ratio of the first sidebands in the solid solution PL spectra in ref 31; i.e. λ2 ) 0.5 and 0.4 for the 320 and 688 cm-1 modes, respectively. The agreement with the measured spectra is quite satisfactory. Finally, we introduce a superior method for obtaining the exciton coherence volume from the polarized emission spectrum. As we have seen, determining NT from the ratio R0β requires knowledge of the off-axis dipole moment, the Davydov splitting, and the generalized FC factor. The method also assumes that disorder is minimalsif disorder disrupts the orientation of the off-axis dipole moments, as is the case for point orientational defects and stacking faults, the origin line strength becomes sensitive to ηNT, where η (e1) accounts for disorder.26,36 Since the mainly c-polarized mode-β sidebands are η-independent (they depend on µ| and not µ⊥), the ratio R0β includes η, which is difficult to estimate. By contrast, the b-polarized sidebands originate from µ⊥ and are therefore also proportional to η. Taking the ratio of the origin line strength to the b-polarized sideband line strength of a weakly coupled mode, say a (i.e., the 165 cm-1 mode) cancels the η-dependence, giving
Ib0-00 Ib0-10
)
NT λ2a
(40)
(see eqs 34 and 35). Equation 40 side-steps the need for accurate values of µ⊥, W, F, and most importantly, η, in determining NT. The tradeoff is an accurate measurement of the polarized emission spectra of thin films, which is a formidable experiment. Using eq 40 on our simulations in Figure 7b gives NT ) 10, identical to what we obtained earlier using R0β since we assumed no disorder (η ) 1). Hence, the first sideband of a weakly coupled mode serves as an efficient probe into the exciton coherence volume. VI. Conclusion and Discussion In this study, we investigated the influence of EP coupling involving several intramolecular vibrational modes on the emission spectra of HB molecular aggregates, successfully accounting for the strong suppression in the low-frequency sidebands observed in OTn and OPV3 thin films and crystals. Generally, weakly coupled modes contribute only a single sideband in the emission spectrum through an aggregate version of Herzberg-Teller coupling. If only weakly coupled modes with frequency ω are present, the ac-polarized sidebands are
attenuated by a factor of approximately (ω/W),2 whereas the b-polarized component is virtually frequency independent. Coupling to multiple vibrational modes can lead to nonlinearities, where the effects of several modes are not additive. We showed here that the presence of a strongly coupled mode can significantly boost the sideband intensity of a weakly coupled mode, an enhancement driven primarily by band renormalization from the strongly coupled mode. In OTn the boosting mode with frequency 1400 cm-1 is only intermediately coupled, but still provides a modest ∼20% enhancement to the low-frequency 165 cm-1 mode. We are continuing to investigate other possible synergistic intermode effects. The present study underscores the essential role played by the band curvature near the band minimum in determining the impact of EP coupling through polaron formation. ωc reflects the strength of the excitonic interactions in the vicinity of the emitting exciton and is proportional to the bandwidth of the commonly referred to au exciton. ωc plays a central role in defining the overall nature of excited states: ∆dis/ωc determines the coherence localization due to a static disorder distribution of width ∆dis,36 and kbT/ωc determines the impact of temperature (dynamic lattice-phonon scattering) on the exciton coherence size.25 Hence, the much lower origin line strength measured in OPV3 crystals at low temperature5,6 compared to that measured in OT4 films1,2 may be due to a significantly smaller ωc in OPV3. Currently, we are pursuing this hypothesis further. We are also considering multimode EP coupling in other architectures, such as achiral and chiral π-stacks of alkylpolythiophenes,44 molecules which are finding important applications in FETs15,16 and organic photovoltaic devices.17 Acknowledgment. F.C.S. is supported by the National Science Foundation, Grant DMR-0606028. Supporting Information Available: Additional calculations; an absorption spectrum of a free exciton and polaron. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Meinardi, F.; Cerminara, M.; Sassella, A.; Borghesi, A.; Spearman, P.; Bongiovanni, G.; Mura, A.; Tubino, R. Phys. ReV. Lett. 2002, 89, 157403. (2) Meinardi, F.; Cerminara, M.; Blumstengel, S.; Sassella, A.; Borghesi, A.; Turbino, R. Phys. ReV. B 2003, 67, 184205. (3) Muccini, M.; Lunedei, E.; Bree, A.; Horowitz, G.; Garnier, F.; Taliani, C. J. Chem. Phys. 1998, 108, 7327. (4) Gebauer, W.; Langner, A.; Schneider, M.; Sokolowski, M.; Umbach, E. Phys. ReV. B 2004, 69, 1254201. (5) Wu, C. C.; Ehrenfreund, E.; Gutierrez, J. J.; Ferraris, J. P.; Vardeny, Z. V. Phys. ReV. B 2005, 71, 081201. (6) Wu, C. C.; Korovyanko, O. J.; Delong, M. C.; Vardeny, Z. V.; Ferris, J. P. Synth. Met. 2003, 139, 735. (7) Wu, C. C.; Delong, M. C.; Vardeny, Z. V.; Ferraris, J. P. Synth. Met. 2003, 137, 939. (8) Gierschner, J.; Egelhaaf, H.-J.; Oelkrug, D. Synth. Met. 1997, 84, 529. (9) Gierschner, J.; Oelkrug, D. Optical properties of oligophenylenevinylenes. In Encyclopedia of Nanoscience and Nanotechnology; Nalwa, H., Ed.; American Scientific Publishers: Stevenson Ranch, CA, 2004; Vol. 8, p 219. (10) Lim, S.-H.; Bjorklund, T. G.; Bardeen, C. J. J. Phys. Chem. B 2004, 108, 4289. (11) Kouki, F.; Spearman, P.; Valet, P.; Horowitz, G.; Garnier, F. J. Chem. Phys. 2000, 113, 385. (12) Tavazzi, S.; Campione, M.; Laicini, M.; Raimondo, L.; Borghesi, A.; Spearman, P. J. Chem. Phys. 2006, 124, 194710. (13) Fichou, D.; Horowitz, G.; Xu, V.; Garnier, F. Synth. Met. 1992, 48, 167. (14) Oelkrug, D.; Egelhaaf, H.-J.; Gierschner, J.; Tompert, A. Synth. Met. 1996, 76, 249. (15) Garnier, F. Acc. Chem. Res. 1999, 32, 209.
Multiple Mode Exciton-Phonon Coupling (16) Malliaras, G.; Friend, R. H. Phys. Today 2005, 58, 53. (17) Brabec, C. J.; Dyakonov, V.; Parisi, J.; Sariciftci, N. S. Organic PhotoVoltaics: Concepts and Realization; Springer: Heidelberg, 2003. (18) Perepichka, I. F.; Perepichka, D. F.; Meng, H.; Wudl, F. AdV. Mater. 2005, 17, 2281. (19) Friend, R. H.; Gymer, R. W.; Holmes, A. B.; Burroughes, J. H.; Marks, R. N.; Taliani, C.; Bradley, D. D. C.; Santos, D. A. D.; Bredas, J. L.; Logdlund, M.; Salaneck, W. R. Nature 1999, 397, 121. (20) Spano, F. C. Annu. ReV. Phys. Chem. 2006, 57, 217. (21) Spano, F. C. J. Chem. Phys. 2002, 116, 5877. (22) Sun, X. H.; Zhao, Z.; Spano, F. C.; Beljonne, D.; Cornil, J.; Shuai, Z.; Bredas, J.-L. AdV. Mat. 2003, 15, 818. (23) Spano, F. C. Chem. Phys. Lett. 2000, 331, 7. (24) Spano, F. C. J. Chem. Phys. 2003, 118, 981. (25) Spano, F. C. J. Chem. Phys. 2004, 120, 7643. (26) Spano, F. C. J. Lumin. 2005, 112, 395. (27) Gierschner, J.; Mack, H.-G.; Luer, L.; Oelkrug, D. J. Chem. Phys. 2002, 116, 8596. (28) Cornil, J.; Beljonne, D.; Heller, C. M.; Campbell, I. H.; Laurich, B. K.; Smith, D. L.; Bradley, D. D. C.; Mullen, K.; Bredas, J. L. Chem. Phys. Lett. 1997, 278, 139. (29) van Grondelle, R.; Novoderezhkin, V. I. Phys. Chem. Chem. Phys. 2006, 8, 793. (30) Spano, F. C. Ann. ReV. Phys. Chem. 2006. (31) Birnbaum, D.; Fichou, D.; Kohler, B. E. J. Chem. Phys. 1992, 96, 165.
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