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Strong Photophysical Similarities between Conjugated Polymers and J‑aggregates Hajime Yamagata and Frank C. Spano* Department of Chemistry, Temple University, 1901 North 13th Street, Philadelphia, Pennsylvania 19122, United States S Supporting Information *

ABSTRACT: The photophysical properties of emissive conjugated polymer (CP) chains are compared to those of linear J-aggregates. The two systems share many properties in common, including a red-shifted absorption spectrum with increasing chain/aggregate length, enhanced radiative decay rates (superradiance) relative to a single monomer/molecule, and several vibronic signatures involving the vinylstretching mode common to many conjugated molecules. In particular, the scaling of the 0−0/0−1 photoluminescence ratio and radiative decay rate with the inverse square root of temperature in red-phase polydiacetylene is also characteristic of linear, disorder-free J-aggregates. The strong photophysical resemblance is traced to the excitonic band structure; in one-dimensional direct band gap semiconductors as well as J-aggregates, the exciton band curvature is positive at the gamma point (k = 0).

O

riginally discovered independently by Scheibe1 and Jelley,2 J-aggregates constitute an important class of organic molecular aggregates in which the main UV−Vis absorption peak is narrowed and red-shifted relative to the solvated (unaggregated) chromophore. Such aggregates are also characterized by small Stokes shifts and enhanced radiative decay rates or superradiance.3−5 The collective behavior displayed by J-aggregates is derived from the delocalized nature of the electronic excited states (or Frenkel excitons) caused by Coulombic intermolecular coupling. Many molecules are known to form J-aggregates, including cyanine dyes, perylene diimides, and porphyrins, as detailed in a recent review.6

The intrinsic nature of the exciton behavior in CP chains is exposed only when disorder is minimized or eﬀectively eliminated. Although the prospect of creating such a polymer sounds improbable, it has nevertheless been achieved by Schott and co-workers for several derivatives of polydiacetylene (PDA).9,13−17 In situ polymerization via UV activation of the monomer crystal results in micrometers-long isolated chains, which, because of the steric hindrance imposed by the surrounding, unpolymerized monomer, are practically devoid of torsional defects. Such chains have been shown to support enormous coherence lengths on the order of a micrometer, roughly 2 orders of magnitude larger than typical diﬀusion lengths measured in CP ﬁlms.18 Despite the obvious diﬀerences between a linear J-aggregate of (through-space) coupled molecules and a single CP chain of (through-bond) coupled monomers, the photophysical properties of both systems are surprisingly similar; for example, superradiance has been observed in isolated chains of red-phase PDA derivatives14 (see below), and superradiant phosphorescence has been observed in a Pt-containing CP.19 In what follows, we explore the intimate relationship between linear Jaggregates and emissive CPs, showing that ultimately it is the positive band curvature at the gamma point (k = 0) common to both systems that establishes similar photophysics. The Jaggregate behavior of single CP chains provides the basis for understanding higher-order structures such as π-stacked aggregates, where the side-by-side orientations of polymer segments induce H-like coupling, resulting in overall hybrid “HJ” behavior.20 Before beginning, we point out that CPs are emissive only when the one-photon allowed 1Bu exciton lies below the two-photon allowed 2Ag state.8,21 This is usually the

Despite the obvious diﬀerences between a linear J-aggregate of (through-space) coupled molecules and a single conjugated polymer chain of (through-bond) coupled monomers, the photophysical properties of both systems are surprisingly similar. Excited states in a single conjugated polymer (CP) chain are also commonly understood in terms of excitons,7,8 which have been described as one-dimensional (1D) Wannier-like in the sense that the electron and hole are not bound to the same monomeric unit.9 Due to the inherent disorder present in such macromolecules, the coherent exciton range is limited to only a small portion of the chain, typically less than 10 repeat units, with Forster-like hopping between such segments. How the socalled conjugation length is related to the microscopic nature of disorder, vibronic coupling, and bath ﬂuctuations is not entirely clear and has been the subject of several investigations.10−12 © 2014 American Chemical Society

Received: November 13, 2013 Accepted: January 21, 2014 Published: January 29, 2014 622

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excitations surrounded by a nuclear distortion ﬁeld.27,28 Such excitons are eigenstates of a Holstein-style Hamiltonian, originally applied to J-aggregates by Philpott29 and Fischer and Scherer.30 In addition to the term representing excitation transfer mediated by Jnn, the Hamiltonian contains local exciton−vibrational coupling, assuming harmonic ground(S0) and excited-state (S1) nuclear potentials of identical curvature but shifted relative to each other along the nuclear coordinate (see Figure 1), so that S1 has more quinoidal character. The nuclear relaxation subsequent to vertical excitation is λ2ℏωvib, where, λ2 is the Huang−Rhys (HR) factor. All of the parameters deﬁning the Hamiltonian in a vibronically coupled linear J-aggregate are shown in Figure 1. A CP has a very similar topology except that the chromophores are monomeric units coupled by through-bond interactions. As in the Kasha model, when disorder is absent, excitons are completely delocalized and characterized by a wave vector k. However, vibronic coupling leads to far more complex energy dispersion bands, as demonstrated in Figure 2 for a linear J-

case for polymers like polythiophene, which have nondegenerate ground states. When the reverse energetic ordering occurs, as in polyacetylene or blue-phase PDA, the molecule is weakly ﬂuorescent, and the J-aggregate analogy breaks down. In conventional J-aggregates, excited-state delocalization is induced by intermolecular Coulombic coupling. In the original analysis by Kasha and co-workers,22−24 it was shown that Jaggregates derive their unique properties from the band structure of the delocalized Frenkel excitons. The simplest model assumes only nearest-neighbor (nn) dipole−dipole interactions, Jnn. When adjacent molecules are oriented in a “head-to-tail” fashion, as depicted in Figure 1, Jnn is negative,25

Figure 1. A linear J-aggregate consisting of N (=5) chromophores with nn coupling, Jnn () to the k = 0 excitons in the ﬁrst three vibronic bands. Red arrows indicate emission pathways from the k = 0 exciton (center) and from thermally activated excitons (left) to the electronic ground state with up to two vibrational quanta.

aggregate with intermediate excitonic coupling. The dispersion curves are obtained via numerical diagonalization of the molecular Holstein Hamiltonian using the two-particle approximation, as described in greater detail in the Supporting Information (SI). The basic level structure consists of a series of main vibronic bands (ν = 0, 1, 2, ...), each of which contains several sub-bands (except when ν = 0) containing avoided crossings between dispersive one-particle states and largely nondispersive two-particle states. The oscillator strength is concentrated in the k = 0 exciton, which, in a J-aggregate (Jnn < 0), is located at the bottom of each vibronic band. 623

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The ν = 0 band is especially important for photophysics because by Kasha’s rule, emission takes place from the lowestenergy excitons subsequent to rapid intraband relaxation from the initially excited level. In the limit where the number of chromophores, N, is large, k becomes a continuous variable. In this limit the dispersion of the ν = 0 band, E(k), can be approximated by a parabola near k = 0

emitting excitons as well as an ensemble average over static disorder conﬁgurations. The coherence function in eq 4 is similar to that studied by Mukamel and co-workers41 and Kuhn and Sundstrom.42 Ncoh is deﬁned as the spatial extent of the coherence function envelope and is deﬁned by38,43 Ncoh ≡ C̅(0)−1 ∑ |C̅(r )|

(1)

The coherence number determined from eqs 4 and 5 is quite general; for a conventional J-aggregate, the chromophore identiﬁed by n is an individual molecule, while for a CP, the chromophore is a given monomeric unit. Ncoh has an enormous impact on photophysical properties; increasing Ncoh by increasing W, reducing disorder and/or lowering temperature, leads to superradiant emission,3−5 enhanced nonlinear optical responses,44 and more rapid “ballistic” exciton transport.45 The important excitonic properties (curvature and coherence length) outlined above are directly obtainable from the absorption and PL spectral line shapes of J-aggregates and CPs. This arises because the interplay between excitonic coupling and vibronic coupling profoundly impacts the absorption and steady-state PL vibronic progressions. Consider an isolated chromophore, which results when all couplings Jnn vanish. In this case, the absorption and PL spectra consist of a simple vibronic progression, where the intensity of the 0−n peak (n = 0, 1, 2, ...) is proportional to the Franck−Condon (FC) factor, exp(−λ2)λ2n/n!. When the electronic couplings are turned on, the vibronic progressions are distorted in a way that reveals the exciton bandwidth and coherence length. Moreover, the distortion depends on the sign of Jnn, thereby giving rise to additional spectral signatures with which to distinguish J- and H-aggregation.27 Figure 2 shows the various transitions giving rise to the absorption and PL vibronic progressions in linear Jaggregates. The profound impact of excitonic coupling on the optical spectra is most easily demonstrated in a molecular J-aggregate in which the HR factor is unity, so that the ﬁrst two vibronic peaks (0−0 and 0−1) in the vibronic progression are of equal intensities when Jnn = 0 (i.e., for isolated chromophores). Figure 3 shows the eﬀect of increasing Jnn on the absorption and PL spectra for aggregates with N = 10 chromophores. The spectra were evaluated by the methods outlined in the SI. The absorption spectra in Figure 3a are due to excitations from the ground state to the k = 0 excitons in the various bands shown in Figure 2. As the excitonic bandwidth W increases, the spectra undergo a signiﬁcant red shift, as expected for J-aggregates. In addition, the oscillator strength of the ﬁrst peak (0−0) increases relative to that of the second peak (0−1). This is also a signature of J-aggregation, as has recently been demonstrated by Pochas et al.46 for a series of linear perylene diimide complexes. The eﬀect is readily appreciated in the perturbative limit (W ≲ ℏωvib), where the line strength ratio becomes27

where k is dimensionless (in units of d−1, where d is the nn distance) and with the curvature, ℏωc, deﬁned by ℏωc ≡

1 d 2E(k) 2 dk 2

(2)

k=0

The curvature directly impacts the exciton’s optical response as well as its coherent and incoherent (diﬀusive) transport as it is inversely proportional to the exciton’s eﬀective mass. Under the nn-only approximation, the curvature is directly related to the exciton bandwidth

ℏωc ≈ −FJnn =

FW 4

(3) 2

where F is a generalized FC factor ranging from exp(−λ ) in the weak excitonic coupling limit (W ≪ ℏλ2ωvib) to exp(−λ2/ N) in the strong coupling limit (W ≫ ℏλ2ωvib). Equation 3 shows that the curvature is positive for J-aggregates because Jnn < 0. As we show below, the curvature is also positive for excitons within a CP chain but due to an entirely diﬀerent mechanism. Instead of Coulombic coupling, the positive curvature in direct band gap semiconductors is derived from the electron- and hole-transfer integrals having the same sign. The positive curvature, whatever its origin, is ultimately responsible for J-aggregate behavior. When static or dynamic disorder is present, excitons in Jaggregates (or CPs) are no longer delocalized over the entire chain; the number of coherently connected chromophores, Ncoh, becomes smaller than N, and the associated coherence length, Lcoh ≡ (Ncoh − 1)d, is smaller than the chain or aggregate length. Essentially, Lcoh is the length over which the exciton retains its wave-like properties. The nature of exciton coherence is a hotly debated topic especially regarding its role in light harvesting31−34 and charge generation in bulk heterojunction solar cells.35,36 Coherence has recently been analyzed in detail in MEH-PPV solutions and nanoparticles using ultrafast polarization decay, where it was shown that coherence is primarily intrachain in nature.37 In P3HT πstacked aggregates the coherence length along the polymer backbone was shown to approach 2 nm in high molecular weight samples, while the interchain coherence length is somewhat smaller, < 1 nm.38,39 For the thermal distribution of low-energy excitons, the exciton coherence number is determined from the Boltzmannaveraged thermal coherence function, given by38,40 C̅(r ) ≡ ⟨⟨Ψ

(em)

|∑ n

Bn†Bn + r |Ψ(em)⟩⟩T , C

(5)

r

E(k) ≈ E(0) + ℏωck 2

I A0−0 I A0−1

=

(1 + 0.24W /ℏωvib)2 (1 − 0.073W /ℏωvib)2

λ2 = 1 (6)

Equation 6 is strictly valid under the nn approximation and for λ2 = 1 but is easily generalized to include extended interactions and arbitrary HR factors.27 Equation 6 resembles the result for an H-aggregate27,47 except that the signs in the numerator and denominator are reversed in the latter case. Hence, the ratio decreases with W in an H-aggregate, exactly opposite to what

(4)

where B†n ≡ |n;vac⟩⟨g;vac| creates an exciton on the nth chromophore with no vibrational quanta (vac) relative to the ground-state unshifted potential well (S0). Here, ⟨...⟩T,C represents a thermal average over a Boltzmann distribution of 624

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=

N λ2

k bT

4πℏωc/N2, where the sum in the partition function (see the SI) can be converted into an integral. The latter can be evaluated analytically using the quadratic form for E(k) in eq 1. Under such conditions, the PL ratio in a disorder-free linear J-aggregate takes on a simple form 0− 1 ⟨IPL ⟩T

=

1 λ2

4π ℏωc k bT

ℏωc > k bT >

4π ℏωc N2

ℏωc > k bT >

4π ℏωc N2

(10)

In our initial work, we added a “1” to the right-hand side of eq 10 in order to obtain an expression that is approximately valid over the entire temperature range. In this way, the thermal coherence length approaches unity in the high-temperature limit where all excitons in the lowest vibronic band are equally populated. Although this high-T “ﬁx” worked well in 2D herringbone aggregates of ref 49, where the coherence number scales as the inverse temperature, it is overly approximate in one dimension for the physically realistic temperature range in Figure 4b. Finally, using eq 10 the inequality expressing the thermodynamic limit can be recast simply as N > Ncoh. The enhanced coherence number at low temperatures for Jaggregates or 1D direct band gap semiconductors leads to cooperative emission or superradiance, a phenomenon in which the radiative decay rate of an aggregate exceeds that of an individual chromophore by the number of coherently coupled chromophores, Ncoh.3,4 For vibronically coupled J-aggregates with a dominant 0−0 emission peak, the radiative decay is given by50

Figure 4. (a) Calculated PL spectra for disorder-free N = 10 Jaggregates for several temperatures. Spectra are calculated using the Holstein Hamiltonian, parametrized as in Figure 2. (b) The thermally averaged 0−0/0−1 PL ratio (solid dots) as a function of T−1/2 for N = 30 aggregates (λ2 = 1) and for several values of Jnn. Lines are evaluated using eq 8, with the band curvature calculated numerically. The asterisks represent Ncoh/λ2, with Ncoh calculated using eq 5. Asterisks are practically superimposed on the solid circles, showing the strong agreement between the PL ratio and Ncoh/λ2. The inset shows a magniﬁcation of the high-temperature region.

0− 0 ⟨IPL ⟩T

4π ℏωc k bT

k rad = FNcoh k mon

(8)

where the left inequality ensures the validity of the parabolic approximation. Figure 4b shows how the PL ratio (solid circles) scales with the inverse square root of temperature, with the solid lines representing eq 8. At low temperatures, where ﬁnite size eﬀects dominate, the ratio is held steadfast at N/λ2, as predicted from eq 7. Increasing the temperature into the thermodynamic limit leads to the T−1/2 behavior of eq 8, which is maintained as long as the parabolic approximation remains valid. Hence, eq 8 allows one to extract the exciton curvature (or eﬀective mass) directly from a plot of the PL ratio as a function of temperature. The inverse square-root dependence of the PL ratio on temperature is a deﬁning signature of ordered linear Jaggregates.43 As our derivation depended only the curvature through the parabolic approximation, the result is also valid for any 1D exciton exhibiting positive curvature: a J-aggregate, a

(11a)

where k mon ≡

3 2 n3ωem μ0

3πε0ℏc 3

(11b)

is the radiative decay rate of a single chromophore (but with the emission frequency ωem matching that of the aggregate), n is the refractive index of the medium, and μ0 is the monomer transition dipole moment. Equation 11a shows that, like the PL ratio, the radiative decay rate is directly proportional to the number of coherently connected chromophores. In the limit of no disorder, ﬁnite size eﬀects dominate at low temperatures (kbT < 4πℏωc/N2), and Ncoh = N. In this case, the radiative decay rate in eq 11a is fully enhanced by FN. Increasing temperature into the thermodynamic limit results in 626

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ℏωc > k bT >

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therefore also present in single chains of red-phase PDA. Moreover, the 0−0/0−1 peak ratio in the polymer absorption spectrum for red- (and blue-) phase PDA16 is about 5 times larger than that found in small oligomers,53 where the ratio is approximately one, consistent with J-aggregate behavior.46 The increase in both the absorption and PL ratios with chain length has also been observed in a series of oligomeric phenylene vinylenes54 and oligothiophenes,55 although in absorption, the relative 0−0 enhancement is somewhat obscured by the presence of torsional disorder. In order to better understand the association between an emissive CP chain and a linear J-aggregate, we modeled the former as a 1D semiconductor including vibronic coupling.50 Essentially, the HOMO and LUMO for each polymer repeat unit are retained with the coupling te (th) between neighboring LUMOs (HOMOs), leading to electron (hole) transport along the chain. Here, an electron refers to a single electron in the LUMO, while a hole refers to the absence of an electron in the otherwise ﬁlled HOMO. Hence, there are four possible electronic conﬁgurations for each monomer unit, a “ground” state consisting of no electrons and no holes (double-ﬁlled HOMO), a Frenkel-like excitation consisting of a single electron and a single hole, and two ionic states, a cation (hole) and an anion (electron and doubly ﬁlled HOMO). Each of the four states has a corresponding nuclear potential for the repeat unit vibration (i.e., the vinyl stretching mode), all of identical curvature but with varying displacements. Relative to the minimum of the ground-state potential, the HR factors λ20, λ2+, and λ2− correspond to the Frenkel-like state, the cation, and the anion, respectively. Finally, binding between an electron and hole at a separation of r units is given by the truncated Coulomb potential, V(r) = −Uδr,0 − (1 − δr,0)V1/r. Further details of the Hamiltonian and the nature of the multiparticle basis set used to express it can be found in ref 50. Numerical analysis of the 1D semiconductor Hamiltonian, parametrized to best describe red-phase PDA, gives the energy dispersion curves shown in Figure 6. The lowest-energy band has positive curvature and is associated with the 1Bu exciton, which gives rise to the prominent 0−0 absorption (and emission) feature.50 There are also dispersion curves for the higher-energy vibronic satellites. The 1Bu exciton has been described as a 1D Wannier exciton56 with a Bohr radius estimated at 12 Å and a binding energy of 0.6 eV,9 in good agreement with our calculated values. Figure 6 shows that beyond the 1Bu exciton binding energy lies a continuum of free (unbound) electron and hole polarons, a feature not shared by J-aggregates (see Figure 2) because in Frenkel excitons the electron and hole are bound to the same chromophore. The most important property of the 1D semiconductor for establishing J-like behavior is the positive curvature of the 1Bu exciton. The sign of the curvature ultimately depends on the relative signs of te and th; when the signs are the same, as in a direct band gap semiconductor, the curvature of the lowestenergy exciton band is positive at k = 0 (see Figure 6), making the k = 0 exciton the lowest in energy, as in a conventional Jaggregate. This can be shown analytically using perturbation theory when |teth| ≪ U − V1. In this limit, the transfer of a Frenkel-like excitation between adjacent monomers is accomplished via a two-step charge-transfer process proceeding through a higher-energy “virtual” intermediate charge-separated (CS) state, as shown in Figure 7. A second-order pertubative treatment reveals an eﬀective Frenkel-like coupling between adjacent unit cells20,43,57

4π ℏωc N2

(12)

where we have substituted eq 10 into eq 11a. Hence, the radiative decay rate follows an inverse square-root temperature dependence, decreasing with increasing temperature as the coherence size diminishes. The inverse-square-root temperature dependence of krad was originally derived by Citrin51 for a 1D semiconductor (without vibronic coupling) and has been observed in single-chain PDA14 (see below) as well as also in single-walled carbon nanotubes.52 Having analyzed the temperature dependence of the PL spectrum in ideal linear J-aggregates, we now turn to the PDA derivative 3BCMU, studied extensively by Schott and coworkers.9,13−17 Virtually defect-free straight polymers or “quantum wires” of the red and blue phases of 3BCMU were prepared in situ, through UV activation of the monomer (diacetylene) crystal. The red and blue phases are believed to be diﬀerent conformations of the polymer; in the blue phase, the dipole-forbidden 2Ag state likely falls below the optically allowed 1B u state, rendering the blue phase weakly ﬂuorescent.21 By contrast, the red phase is strongly emissive, indicative of the 1Bu exciton9 lying below the 2Ag state. The low-temperature PL spectrum of red-phase 3BCMU is shown in Figure 5. The spectrum is characterized by a dominant 0−0

Figure 5. The steady-state PL spectrum of isolated chains of red-phase 3BCMU at T = 15K from ref 14. The sideband (0−1) peaks corresponding to the double (D) and triple (T) bond symmetric stretching modes have been ampliﬁed by a factor of 60. The inset shows the ratio of ID/I0−0, where ID (I0−0) is the area under the D (0− 0) peak. The ratio I0−0/ID therefore scales as the inverse square root of temperature. Adapted with permission from ref 14. Copyright (2014) by the American Physical Society.

peak at 2.28 eV and much smaller vibronic side bands approximately 0.18 and 0.25 eV below the 0−0 peak due to the double-bond (“D”) and triple-bond (“T”) stretching modes, respectively.14 The lack of disorder is evident in the extremely narrow spectral line widths; each vibronic line is only ∼20 cm−1 wide, almost 2 orders of magnitude narrower than what is typically measured in P3HT and MEH-PPV ﬁlms at similar temperatures. As shown in the ﬁgure, the 0−0 peak dwarfs the much smaller side bands by a factor of ∼100 at T = 15 K. As shown in the inset, increasing temperature leads to a reduction in the ratio of the 0−0 to 0−1 (D) spectral intensities (I0−0/ID), which scales as the inverse square root of temperature. Schott and co-workers also showed that the radiative decay rate scales as T−1/2. The vibronic signatures of J-aggregate emission are 627

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The ability to realize large coherence lengths in CPs may revolutionize organic solar cells by exploiting rapid coherent energy transport to acceptor sites where charge separation occurs. Jintra = −

2teth U − V1

(13)

J-aggregate behavior results when Jintra is negative, which occurs when te and th have the same sign because U > V1. (Here the sign convention is based on the translation operator − see ref 65). The implications of J-like behavior from a negative Jintra have also been noted by Barford.58 One can further show that the relationships for the PL ratio in a (1D) direct band gap semiconductor are essentially the same as those for a linear J-aggregate. For example, when the temperature is low so that ﬁnite size eﬀects prevail, the 0−0/0− 1 PL ratio is given by50

Figure 6. Energy bands corresponding to the 1D semiconductor Hamiltonian of ref 50 with N = 60 repeat units, parametrized for PDA using te = th = −1.15 eV, U = 2V1 = 2 eV, ℏωvib = 0.175 eV, and the HR factors, λ20 = 2λ2+ = 2λ2− = 1. The lowest band corresponds to the 1Bu exciton and is responsible for 0−0 absorption (and emission). The next higher band is the 0−1 vibronic satellite. The parabolic dispersion in red is evaluated from eq 1 using the numerically obtained curvature, ℏωc = 0.46 eV. The blue curve outlines the band edge of the (quasi)continuum consisting of unbound electrons and hole polarons. The exciton binding energy, Eb, is 0.8 eV, in good agreement with published values.9 Note the break in the vertical axis.

0− 0 IPL 0− 1 IPL

=

κN λ 02

k bT

0) . Using second-order perturbation theory, the eﬀective Frenkel-like transfer (bottom path) is represented by the coupling, Jintra. The latter is negative, inducing J-like behavior, when the electron (te) and hole (th) transfer integrals have the same sign as in a direct band gap semiconductor.

0− 1 ⟨IPL ⟩T

=

4π ℏωc k bT

κ λ 02

ℏωc > k bT >

4π ℏωc N2

(15)

again diﬀering from the linear J-aggregate result by the additional factor of κ . Equation 15 also shows that the coherence number remains the same as that for a J-aggregate in eq 10. Finally, the radiative decay rate for the 1D semiconductor is20,50 k rad 4π ℏωc = Fϕ0 k mon k bT

(16)

Table 1. Comparison of the PL Ratio and Radiative Decay Rates in Disorder-Free Linear J-Aggregates and 1D Direct Band Gap Semiconductors low temperature, kbT < 4πℏωc/N2 0−1 I0−0 PL /IPL

linear J-aggregate

1D semiconductor

N/λ2 κN/λ02

krad/kmon FN

Fϕ0N

thermodynamic limit, kbT > 4πℏωc/N Ncoh N

N

628

0−1 ⟨I0−0 PL ⟩T/⟨IPL ⟩T

krad/kmon

Ncoh

4π ℏωc k bT

4π ℏωc k bT

1 λ2

4π ℏωc k bT

F

κ λ02

4π ℏωc k bT

Fϕ0

4π ℏωc k bT

4π ℏωc k bT

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where ϕ0 is the probability for the electron and hole to be on the same repeat unit. The latter is unity in the weak coupling (Frenkel-like) limit where eq 12 is recovered. A detailed comparison of the PL properties of linear J-aggregates versus 1D direct band gap semiconductors is made in Table 1. Figure 8 shows measured and theoretical values for the PL ratio and radiative lifetime τrad (=krad−1) of red-phase 3BCMU

energy transport to acceptor sites where charge separation occurs. Evidence for such a process at early times (≪ 1 ps) before the exciton is localized by disorder, has recently been reported for various bulk heterojunction solar cells,36 and the 59 0−1 associated drop in the I0−0 may be PL /IPL ratio with time evidence of coherence collapse in accordance with eq 9. Composite materials consisting of coherent PDA chains surrounded by PCBM could perhaps make far more eﬃcient solar cells than current realizations based on polymer ﬁlms with small coherence lengths (