Polaron Delocalization in Conjugated Polymer Films - The Journal of

Article pubs.acs.org/JPCC

Polaron Delocalization in Conjugated Polymer Films Raja Ghosh, Christopher M. Pochas, and Frank C. Spano* Department of Chemistry Temple University Philadelphia, Pennsylvania 19122, United States ABSTRACT: A theory describing the spatial coherence length of polarons in disordered conjugated polymer films is presented, revealing a simple relationship between oscillator strength of the mid-IR absorption band and the polaron coherence function. Applications are made to positively charged polarons or holes in poly(3-hexylthiophene) π-stacks where polaron delocalization occurs in essentially two dimensions: along the polymer axis (xaxis) and along the interchain stacking axis (y-axis). Based on a lattice model consisting of a 2D array of thiophene units linked electronically along the polymer axis (tintra) and between adjacent chains (tinter), a disordered 2D Holstein-style Hamiltonian is developed. The Hamiltonian includes vibronic coupling involving the polaron-forming vibrational mode responsible for aromatic/quinoidal oscillations. The hole absorption spectrum for light polarized along each axis is calculated along with the corresponding hole coherence function, CP(r). For diagonal disorder arising from a random distribution of hole site energies, the band oscillator strength for light polarized along x (y) scales directly as the product of tintra (tinter) and the coherence function, CP(r), for a separation r equal to one unit along x (one unit along y). For off-diagonal disorder arising from a distribution of electronic couplings, the relationship becomes approximate with tintra (tinter) replaced by its mean value. Calculated spectral line shapes are in good agreement with previously published experimental charge-modulated spectra. Further investigation reveals that the polaron is delocalized isotropically over about six thiophene units in each direction for high molecular weight (MW) films (270 kDa), In the low MW films (15 kDa), the coherence length becomes anisotropic, with delocalization numbers of only three along the polymer axis and two along the interchain axis. The drop in coherence lengths tracks a similar drop in the calculated band oscillator strengths, consistent with experimental observations.

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INTRODUCTION Coherence enhancement of exciton and charge transport is a topic attracting a great deal of attention in the fields of materials science and biology, especially with regards to organic photovoltaic devices1−7 and the light harvesting antenna systems involved in photosynthesis.8−15 However, the unambiguous measurement of electronic coherences has been a subject of considerable controversy: room-temperature temporal coherences have ultrashort (sub-picosecond) lifetimes and the ranges over which spatial coherences are established are extremely small, on the order of a few nanometers or less, due to the inherent disorder characteristic of many organic materials like conjugated polymer films. Moreover, some of the methods used to detect coherences are challenging to interpret; for example, electronic (vibronic) coherences present in the cross-peaks of an electronic two-dimensional spectrum interfere with vibrational coherences.16,17 In the following, we develop a theory describing the spatial coherence of polarons (holes) in conjugated polymer films and show how information about coherence can be obtained from the hole infrared absorption spectrum. The polaron coherence length can be thought of as the range over which the hole’s motion can be considered wave-like; large coherence lengths are associated with more ordered packing and higher charge mobilities, ideal conditions for device applications such as solar cells and field-effect transistors. Here, we seek to understand © 2016 American Chemical Society

how the generally anisotropic hole coherence length depends on morphology in thin films of poly(3-hexylthiophene) or P3HT, the polymer of choice for solar cell applications.18,19 P3HT films exhibit lamellar packing in which the thiophene backbones form π-stacks separated by alkyl-rich layers containing the hexyl side chains.20,21 In the semicrystalline phase of P3HT, the π-stacking distance is only 0.38 nm leading to large interchain interactions, which substantially impact polaron and exciton transport. Indeed, convincing evidence of two-dimensional polaron delocalization in P3HT (along the polymer backbone and along the π-stacking axis) was obtained by Vardeny and co-workers using transient pump−probe spectroscopy22,23 and by Sirringhaus and co-workers using charge modulation spectroscopy (CMS).24−26 For excitons, the coherence function was derived from a careful analysis of the vibronic progressions present in the UV−vis and PL spectra.27,28 Due to the large amount of disorder present in typical P3HT films, the exciton coherence length is quite small, less than 1 or 2 nm along the polymer backbone and aggregate axes.27,28 Although excitons and holes are similar in some respects, they are affected quite differently by disorder, and the question as to how extensive hole delocalization is in P3HT Received: March 21, 2016 Revised: April 29, 2016 Published: May 19, 2016 11394

DOI: 10.1021/acs.jpcc.6b02917 J. Phys. Chem. C 2016, 120, 11394−11406

Article

The Journal of Physical Chemistry C

the large value of ωvib (0.17 eV) and throws into question the very meaning of the polaron self-trapping. Several theories have been advanced to account for the effect of interchain coupling on polarons in conjugated polymers films.26,29,41,44−47 The general consensus of the early works was that disorder is needed to localize the polaron wave function to the point that self-trapping is possible.41,44 The more recent works of Beljonne et al.29 and Chang et al.26 investigated the infrared spectral signatures of interchain coupling. In ref 29, the polaron absorption spectrum was evaluated for a cationic PPV dimer using quantum chemical techniques revealing the presence of a low-energy peak polarized along the interchain axis with an absorption cross-section proportional to the peak absorption energy. A nonadiabatic analysis based on charge transfer between a single donor chain and a single acceptor chain was later employed by Chang et al.26 Their approach mirrors that of Piepho et al.48−50 used to investigate mixedvalence Creutz−Taube complexes.51 Chang et al. correlated the higher mobilities found in the higher MW P3HT films with stronger interchain interactions and smaller polaron relaxation energies. Although the theories of refs 29 and 26 shed important light on the more general features of polaron absorption, they were unable to capture the vibronic structure present in the measured spectra. Moreover, neither work considered the profound influence of disorder. For example, disorder arising from changes in the π-stacking distance, often referred to as paracrystallinity,33,45,46 plays a major role in limiting hole mobility and has been shown to depend strongly on the polymer molecular weight.33 By contrast, the approach used in ref 30 yielded nearly quantitative reproductions of the measured infrared line shapes by adapting the Holstein model to lattices that include both diagonal and off-diagonal disorder. Diagonal disorder is manifest as a distribution of hole site energies, while offdiagonal disorder arises from a distribution of intrachain and interchain couplings, paracrystallinity being an example of the latter. In the present work, we employ the theory of ref 30 to investigate the nature of hole coherence in P3HT films. We continue to utilize the multiparticle basis set52−54 truncated at the two-particle level, which allows us to account for both local and nonlocal nuclear distortions surrounding the hole. The basis set is analogous to that used to successfully account for the photophysics of excitons in P3HT π-stacks.27,55−58 We focus on the case of a single hole confined to a two-dimensional lattice with electronic coupling between adjacent donor and acceptor HOMOs along and across chains. One of the major advantages of our approach to charge transfer over the more conventional theories of Hush59 and Piepho et al.48−50 is that the latter are limited to just two coupled chromophores, that is, a donor−acceptor pair. With current computational facilities, we can investigate lattices as large as 6 × 6, that is, 36 coupled chromophores, all of which participate in nearest-neighbor hole hopping mediated by vibronic coupling. The lattices are large enough to achieve complete spectral convergence for sufficiently large disorder. We investigate diagonal and offdiagonal disorder with variable spatial correlation lengths defined both along the polymer axis and along the stacking axis and perform configurational averaging over thousands of realizations of disorder in order to accurately obtain the mid-IR polaron absorption spectrum and the associated polaron coherence function.

films naturally arises. The answer to this question is the main goal of the present paper. As has been recognized by Beljonne et al.,29 Chang et al.26 and Pochas and Spano,30 the intensity of light-induced polaron transitions depends on the extent of the hole delocalization, which, in turn, depends on film morphology and molecular weight (MW). Chang et al.26 measured an order of magnitude increase in the intensity of the CM spectrum in going from low molecular weight films (15 kDa) with a large amount of disorder to more ordered, high molecular weight films (270 kDa). The change was accompanied by a 30-fold increase in hole mobility. The latter is likely due to the biphasic morphology of high MW P3HT films, where highly ordered aggregates are connected through intervening amorphous regions via tie molecules.31−36 What is the associated change in the generally anisotropic polaron coherence lengths with MW? As we show below, the spatial coherence length of charged polarons along the polymer axis and along the stacking axis can be directly obtained from the broad features of the mid-IR line shapes, with holes generated either by photoexcitation22,23 or through the application of a voltage bias as in CMS.24−26,37 Although the mid-IR line shapes are quite complex, one can crudely describe them as possessing a lowenergy peak in the range 500−1000 cm−1 due to the aforementioned interchain transitions (labeled DP122,23 or CT24−26,37) followed by a much broader region about 1000− 3000 cm−1 higher in energy, presumably dominated by intrachain transitions (labeled P122,23 or C124−26,37). Superimposed on the broad polaron features are very sharp antiresonance lines derived from infrared active vibrational modes (IRAVs),22,23 which are not considered in this work. In our analysis of polaron coherence, we adapt the formalism presented in ref 30, which was able to successfully reproduce the (unpolarized) mid-IR line shapes described above. The theory is based on the Holstein molecular crystal Hamiltonian,38 modified to include spatially correlated diagonal and off-diagonal disorder. Recently, the Hamiltonian was used to investigate Anderson localization on polymer chains.39 In our work, a polymer π-stack is modeled as a “square” lattice of thiophene units composed of a stack of N polymer chains, with each chain containing N thiophene units. The thiophene units are coupled electronically via interchain and intrachain charge transfer (“hole hopping”) integrals. In P3HT, the nuclear relaxation responding to cation formation occurs mainly along the symmetric aromatic-quinoidal stretching coordinate, with an associated energy of approximately, ωvib ≈ 0.17 eV/ℏ. In the Holstein approach, the vibronic coupling involving the symmetric mode is treated nonadiabatically: both the nuclear kinetic and potential energy are treated f ully quantum mechanically. This is in contrast to many approaches, which are based on the adiabatic or Born−Oppenheimer approximations.29,40−43 For example, the conventional quantum chemical understanding of polaron formation starts with the minimal-energy molecular geometry determined for an oxidized chain containing a central hole; the hole-induced chain deformation to the bond order leads to the creation of two midgap molecular orbitals, one slightly above the valence band and one slightly below the conduction band. Transitions involving the midgap orbitals gives rise to the additional midand near-infrared peaks commonly referred to as P1 and P2, respectively, in the absorption spectrum.40 However, such a scenario neglects the nuclear vibrational kinetic energy associated with the hole. This energy is not negligible, given 11395


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MODEL A model P3HT π-stack is shown in Figure 1. The polymer axis is taken to lie along the x-axis while the stacking axis is along y.

in H0 account for local vibronic coupling involving the symmetric aromatic/quinoidal vibration with energy, ℏωvib ≈ † 0.17 eV. The operators bm,n and bm,n, respectively, create and annihilate vibrational quanta on the mth thiophene unit on the nth chain within the ground (S0) potential well. The Huang− Rhys (HR) factor λ2 represents the geometric relaxation energy experienced by a single thiophene unit upon oxidation (in units of ℏωvib). The HR factor is nonzero whenever the ground and cationic potential wells are shifted relative to each other. The estimated excitonic (HOMO−LUMO) HR factor for thiophene in ref 27 was found to be in the range of 1.5−2.0. Typically, ionic HR factors are roughly one-half of the excitonic HR factor. Hence, in all calculations to follow we use λ2 = 1 for cationic thiophene. Finally, the Hamiltonian omits a constant term representing the zero-point vibrational energy as well as the on-site polaron energy. Because we are interested only in transitions between hole eigenstates such terms are unimportant. Disorder. Disorder in polymer π-stacks includes random changes in the site energy of the hole as well as changes in the hole transfer integrals.60,61 Inhomogeneity can be manifest as variations in the nearest-neighbor π-stacking distance, as occurs in paracrystalline films33,34,45,46 accompanied by variations in the intramolecular torsional angles within the polymer.33,45,46 Inhomogeneity may also arise from spatially varying local electric fields. In our model, all such sources of disorder are accounted for phenomenologically. The various forms of disorder induce random changes in the hole energy, Δεm,n, for a hole located at position (m,n), as well as variations of the transfer integrals: Δtm,n inter denotes the random deviation in the interchain coupling between monomer (m,n) and (m,n+1), while Δtm,n intra denotes the random deviation in the intrachain coupling between (m,n) and (m+1,n). The complete Hamiltonian for a disordered N × N πstack is given by

Figure 1. A 4 × 3 P3HT π-stack. The nearest−neighbor distances between thiophene units along the polymer (dintra) and along the stacking axis (dinter) are approximately equal to d = 0.4 nm.

Each thiophene unit is identified by the coordinate (m,n) indicating the mth unit on the nth chain. As has been demonstrated in previous works on excitons27,58 and polarons,30 individual chains can be successfully treated in a coursegrained approximation; a hole on a given thiophene unit is taken to be a missing electron in its local HOMO. The neighboring HOMOs for adjacent units along the polymer backbone are coupled through the hole transfer integral, tintra. In a π-stack of such chains, there is also wave function overlap between the HOMOs on neighboring units on adjacent chains. The associated electronic coupling is represented by the interchain hole transfer integral, tinter. Hence, electronic coupling induces two-dimensional delocalization of the hole.22,24 In order to account for the nuclear relaxation accompanying the formation of a hole on a particular thiophene unit, we consider coupling to the aromatic/quinoidal stretching mode, with frequency at ∼0.17 eV/ℏ, which is responsible for pronounced vibronic progressions in the absorption and emission spectra of aggregated P3HT.27,37,57 To do so, we employ a site based Holstein Hamiltonian, where, in the simplest scheme, the nuclear potentials for molecular vibrations in the ground (S0) and cationic (S+) states of a given thiophene unit are shifted harmonic wells of identical curvature. In the vector subspace containing a single hole within an N × N square π-stack, the Hamiltonian is

inter intra H = H0 + Hdiag + Hoff ‐ diag + Hoff ‐ diag

where H0 was introduced in eq 1. The diagonal and off-diagonal terms are given by N

Hdiag =

∑∑

H0 =

and N inter Hoff ‐ diag =

+ h.c.}

N intra Hoff ‐ diag =

+ ℏωvib ∑

N

+λ

N

}dm† , ndm , n

(4a)

N−1

m,n {dm† + 1, ndm , n + dm† , ndm + 1, n} ∑ ∑ Δt intra m=1 n=1

(4b)

The disorder can be spatially correlated to varying degrees along the polymer and stacking axes. The extreme limits are referred to as long-range (infinite spatial correlation) and shortrange (no spatial correlation). As detailed in ref 30, short- and long-range broadening along each of the two directions defines four general cases, denoted as LL (isotropic long-range), LS (long-range along the polymer axis, short-range along the stacking axis), SL (short-range along the polymer axis, longrange along the stacking axis), and SS (isotropic short-range). Figure 2 shows several examples that are of particular relevance to the current work.

N

∑ bm† ,nbm,n + ℏωvib ∑ ∑ {λ(bm† ,n + bm,n)

m=1 n=1 2

N−1

m,n {dm† , n + 1dm , n + dm† , ndm , n + 1} ∑ ∑ Δt inter m=1 n=1

∑ ∑ t inter{dm† ,n+ 1dm,n + h.c.} m=1 n=1 N

(3)

m=1 n=1

m=1 n=1 N N−1

+

N

∑ ∑ Δεm,ndm† ,ndm,n

N−1 N

t intra{dm† + 1, ndm , n

(2)

m=1 n=1

(1)

where h.c. means hermittian conjugate. The first term † represents the local hole energy. The operators, dm,n and dm,n, respectively, create and annihilate a hole on the mth thiophene unit of the nth chain. These operators obey the usual anticommutation relations for Fermions. The last two terms 11396



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INFRARED ABSORPTION AND HOLE COHERENCE In what follows, we focus our attention entirely on the far-tomid-IR transitions between the polaron ground state, the lowest eigenfunction of the Hamiltonian in eq 2, and all excited polaron states. As was done in ref 26, we exclude the purely infrared transitions, which give rise to the sharp infrared-active vibrational transitions (IRAVs) and antiresonant peaks observed in the absorption spectrum of oxidized polythiophene,22,23 as well as other conjugated polymers.62,63 We concentrate on the broad absorption bands spanning the spectral range from several hundred to several thousand wave numbers. We further neglect the higher-energy transitions in the near-IR, which involve excitations into the polymer conduction band and therefore require the inclusion of the thiophene LUMO levels. Our previous work concerning polymer π-stacks showed that interchain (intrachain) disorder selectively impacts the corresponding polarization component of the hole absorption spectrum.30 As the disorder increases in a particular direction (along the polymer backbone or along the aggregate axis), the corresponding component of the absorption spectrum diminishes. Since increasing disorder leads to localization, the spatial coherence of the polaron should also diminish with increasing disorder. Here, we develop a direct relationship between the CM spectrum and the (ground-state) hole coherence function, establishing a means of obtaining the latter from the former. We begin with the expression for the spectrum polarized along the direction j

Figure 2. Various disorder models considered in this work. Here, the circle radii are proportional to the site energy offsets (sign indicated by color). The off-diagonal disorder in tinter arises from a distribution of interchain distances.

Under the two-particle approximation, the single-hole basis set used to represent H in eq 2 is truncated to include one- and two-particle states, analogous to the one- and two-particle states used in treating neutral excitons.55,56 In a single-particle state, denoted as |m,n,ν̃⟩, a hole resides on the mth thiophene unit of the nth chain with ν̃ vibrational quanta in its shif ted (S+) potential well. The remaining N2 − 1 monomers are in their vibrationless ground states (filled HOMO with no vibrations in the S0 well). In a two-particle state, denoted |m,n,ν̃;m′,n′,ν′⟩, monomer (m,n) is ionized with ṽ vibrational quanta in S+, while monomer (m′,n′) is electronically neutral with ν′ > 0 vibrational quanta in the unshifted S0 potential. The remaining N2 − 2 monomers are in their vibrationless ground states. In all calculations to follow, we cap the total number of vibrational quanta at four, sufficient to obtain convergence. Three- and higher particle states with three or more monomer excitations (electronic plus vibrational) can also be included, but their impact is negligible on the calculated infrared spectra for the vibronic coupling parameters used here. When inter- and intrachain electronic couplings are neglected the multiparticle states are eigenstates of H in eq 2. Electronic coupling induces mixing among the one- and twoparticle states. In the general case, the αth eigenstate of H in eq 2 can be written as, |Ψα⟩ =

∑ ∑

Aj (ω) ≡ ⟨∑ f jex WLS[ℏω − (Eex − EG)]⟩C

(6)

where the sum is over all polaron excited states and ⟨···⟩C represents an average over an ensemble of disorder configurations. The oscillator strength involving the transition between the polaron ground state, |ΨG⟩ and an excited state |Ψex⟩ (both of which are generally dependent upon the configuration of disorder) is given by the oscillator strength, 2me

f jex ≡

2 2

3e ℏ

(Eex − EG)|⟨ΨG|μĵ |Ψex⟩|2

(7)

where the vector operator μ̂ in eq 7, given by, μ̂ ≡ e ∑ rm , ndm† , ndm , n (8)

m,n

arises from the interaction of the infrared electric field E and the hole, through Hint = −μ̂ ·Ε. Finally, WLS in eq 6 is the homogeneous line shape function. In what follows, we take WLS to be an area-normalized Gaussian with standard deviation Γ. The total oscillator strength corresponding to the jth polarization component is obtained by spectrally integrating over the j-polarized component of the absorption spectrum:

cmα , n , ν|̃ m , n , ν⟩̃

∑ ∑ ∑ ∑

(j = x , y )

ex

m , n ν ̃= 0,1,...

+

Article

cmα , n , ν ;̃ m ′ , n ′ , ν ′|m , n , ν ;̃ m′, n′, ν′⟩

m , n ν ̃= 0,1,... m ′ , n ′ v ′= 1,2,...

(5)

f jtot =

The one- and two-particle expansion coefficients can be readily obtained numerically for all eigenstates. The ground state polaron, indicated by |ΨG⟩, is taken to be the lowest energy eigenstate of the Hamiltonian in eq 2. This represents a major departure with exciton theory where the ground state is simply the state in which all chromophores are unexcited. As we show below, it is the coherence properties of |ΨG⟩ that determine infrared absorption.

=

∫ Aj(ω) dω

2me 3e 2ℏ2

⟨∑ (Eex − EG)|⟨ΨG|μĵ |Ψex⟩|2 ⟩C ex

(9)

which, using H from eq 2 can be rewritten as f jtot = 11397

2me 3e 2ℏ2

⟨ ΨG|[μĵ , H ]μĵ |ΨG ⟩C

(10)


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Figure 3. (a) Hole absorption spectrum in the mid-infrared for a 6 × 6 π-stack with dinter = dintra = d = 0.4 nm and with no disorder. The integrated oscillator strength of each band is shown in the inset. (b) The associated coherence function as a function of the number of thiophene units along the x (intrachain) and y (interchain) directions. eqs 11a and 11b are exactly satisfied using the oscillator strengths in panel a and the coherence points indicated by the red dots in panel b. The parameters defining the Hamiltonian in eq 1 are tinter= −0.15 eV, tintra = −0.3 eV, ωvib = 0.174 eV, and λ2 = 1.

Since the diagonal disorder component of H in eq 3 commutes with the dipole operator in eq 8, the expression in eq 10 can be further simplified in the case where only diagonal disorder exists. After several steps, we obtain the exact expressions, tot f intra =

tot f inter =

2me 3ℏ2

2me 3ℏ2

d intra 2|t intra|C P[r = (1, 0)]

d inter 2|t inter|C P[r = (0, 1)]

begin to appreciate the width of the coherence function in each direction using eqs 11 and 13. These equations therefore provide a direct link between the polarized components of the infrared absorption spectrum and the spatial coherence lengths in the intrachain (x) and interchain (y) directions, respectively. In the limit of highly mobile holes along x (y), where CP[(1,0)] tot (CP[(0,1)]) approaches unity, f tot x (f y ) is largest, while in the opposite extreme of completely trapped holes, CP(r) = CP[(0,0)]δr,(0,0), and the oscillator strength vanishes in either direction. This makes sense, because the polaron transitions induced by infrared excitation involve moving charge. If a polaron is trapped on a site with sufficiently low energy, it cannot respond to the electric field and the oscillator strength vanishes. Interestingly, the total oscillator strength for polaron absorption is not conserved with changing disorder, in marked contrast to exciton absorption, where a loss of oscillator strength in one spectral region is compensated by an increase in oscillator strength in another.

(11a)

(11b)

where dintra (dinter) is the nearest neighbor distance along the intrachain (interchain) direction and CP(r) is the twodimensional coherence function describing hole delocalization within the ground state28,64,65 C P(r) ≡ ⟨⟨ΨG| ∑ dR† dR + r|ΨG⟩⟩C R

(12)

■

Here, R = (m,n) is a dimensionless position vector. When r = 0, the coherence reduces to unity, CP[(0,0)] = 1, since ∑Rd†RdR is the operator for the total number of holes, which is taken to be unity. We point out that the form of the hole coherence function defined in eq 12 is consistent with that defined in refs 14, 66, and 67 to investigate electronic coherence but differs from that used for exciton coherence27,28 derived directly from the photoluminescence 0−0/0−1 ratio.25,26 The exciton coherence operator in refs 25 and 26 contains an electronic component like the sum in eq 12 but also a projection onto the ground-state nuclear coordinates. When off-diagonal disorder (which does not commute with dipole operator) is also present, the relationship between the oscillator strength and the coherence functions becomes approximate, tot f intra ≈

tot f inter ≈

2me 2

3ℏ

2me 3ℏ2

d intra 2|⟨t intra⟩|C P[r = (1, 0)] d inter 2|⟨t inter⟩|C P[r = (0, 1)]

RESULTS Figure 3 shows the calculated absorption spectrum and corresponding coherence function for a hole in a disorderfree 6 × 6 lattice described by H0 in eq 1 parametrized with the values shown in the caption. The component of the CM spectrum polarized along the stacking axis (shown in red) arises from interchain charge transfer; the first peak is referred to as the CT peak by Sirringhaus and co-workers24 and DP1 by Vardeny and co-workers.22 The spectral component polarized along the polymer axis (shown in blue) arises from intrachain polaron transitions, with the dominant peak labeled C1 (also referred to as P122). Figure 3b shows that despite the total absence of disorder, the coherence function is not uniform over all thiophene units. This arises because we have taken open boundary conditions. Confinement limits coherences with values of r approaching the sample dimensions, much like the wave functions for the particle-in-a-box. The total number of coherently connected thiophene units, Ncoh, is derived from the coherence function via65

(13a)

(13b)

Here, the mean intrachain and interchain couplings are given by ⟨tintra⟩ and ⟨tinter⟩, respectively. Equations 11 and 13 show that the total oscillator strength for light polarized along the direction j (=x,y) scales as the polaron coherence for a separation of one unit along direction j. Since CP[r=(0,0)] is unity independent of disorder, we can

Ncoh =

∑ |CP(r)| r

(14)

which reduces to unity in the limit of a strongly localized or trapped polaron. For the mobile polaron in Figure 3b, eq 14 gives Ncoh ≈ 28 somewhat less than the total number of units 11398


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The Journal of Physical Chemistry C (36). Note that if periodic boundary conditions were employed then Ncoh would be equal to the total number of units (=36) in a disorder-free lattice. From the coherence function in eq 14, one can also determine the coherence lengths along the polymer chain direction (Lintra) and along the π-stacking axis (Linter). Following ref 27, we have L intra /d intra = {

∑

|C P(r)|} − 1

r ∈ chain

L inter /d inter = {

∑

The set of offsets, {Δεm,n}, defines a particular configuration (or “realization”) of disorder within a large ensemble. The Hamiltonian H0 + Hdiag from eqs 1 and 3 can readily be numerically diagonalized for each configuration, from which an infrared spectrum can be assembled after averaging over many such configuration-specific spectra. In what follows, we typically employ 5000 configurations. In the limit of short-range disorder (i.e. no spatial correlation) each detuning Δεm,n within a given configuration is chosen independently of all others, while in the opposite extreme of long-range broadening, all detunings within a given configuration are identical and equal to ΔεC (although ΔεC is selected randomly from the Gaussian distribution of width σ). As indicated in Figure 2, short- and long-range broadening can be directional. In the SL distribution, for example, the detunings along the polymer backbone are uncorrelated, while the detunings between polymer chains are fully correlated. As pointed out in ref 30, a disorder-free lattice is spectrally equivalent to a lattice with long-range disorder in each direction, that is, an LL lattice. Hence the spectrum in Figure 3a is identical to that corresponding to an ensemble of LL lattices with configuration detunings ΔεC chosen from the distribution in eq 16. The absence of any change from the disorder-free spectrum arises because the polaron ground state and all excited states are uniformly shifted by ΔεC so that the transition energy is unaffected, that is, is identical for each configuration despite differing values of ΔεC. This is quite different from the case of exciton absorption where long-range disorder leads to a distribution of transition frequencies resulting in a line shape broadened by σ. The impact on the coherence function for LL broadening in a polaron lattice is also the same as for a disorder-free lattice. Hence, LL broadening has no ef fect on the polaron coherence. Polarons are more robust against site disorder than are excitons. The situation is quite different for SS broadening. In Figure 5, we demonstrate the impact of increasing SS disorder on the absorption spectrum and coherence function for a polaron confined to 6 × 6 polymer stack with H0 parametrized as in Figure 3. The disorder width σ increases from 0.2 to 0.6 eV in going from the top to bottom panels. Unlike LL broadening, the effect of SS broadening is profound. The most striking observation in Figure 5 is a dramatic reduction in the width of the coherence function with increasing disorder, consistent with efficient disorder-induced polaron trapping. The localization is faithfully tracked by the oscillator strengths of the polarized components, which also precipitously drop with increasing disorder. (Note the different vertical scales in Figure 5a−c.) The relationship between the oscillator strength and coherence is entirely consistent with eqs 11a and 11b,which show that throughout the increase in disorder the quantities

(15a)

|C P(r)|} − 1

r ∈ chain normal

(15b)

Note that the term r = (0,0) is included in the sums appearing in eqs 15a and 15b. Both coherence lengths properly reduce to zero in the limit of strong localization. When disorder is absent, as in Figure 3a, the coherence lengths approach their maximum values. Using eqs 15a and 15b, the coherence lengths in Figure 3b are approximately 4.2 units along each direction, smaller than the maximum separation of 5 units along each direction due to the open boundary conditions. The highly delocalized polaron responsible for the absorption spectrum in Figure 3 is not stable, in the sense that its properties are strongly size dependent. This is demonstrated in Figure 4, which shows that the (unpolarized)

Figure 4. Hole absorption spectrum for various size π-stacks with no disorder. Parameters defining the Hamiltonian in eq 1 are the same as in Figure 3.

spectra are not convergent with increasing aggregate size; the spectra continue to shift to the red as the polaron coherence expands with size, reflecting an increase in the density of energy levels responsible for absorption.30 This behavior is also similar to what is expected for a particle-in-a-box. Polaron stability requires disorder so that the polaron coherence size eventually stabilizes with increasing size. As we will see, this can be accomplished with short-range site disorder or with paracrystallinity disorder. Diagonal (Site) Disorder. To account for diagonal (site) disorder, we assume a Gaussian distribution of thiophene unit energy shifts (detunings) with zero mean. Each of the N2 detunings, Δεm,n, within an N × N π-stack is selected from a Gaussian distribution with standard deviation σ: P(Δεm , n) = (2πσ 2)−1/2 exp(−Δεm , n 2 /(2σ 2))

tot f intra /C P[r = (1, 0)] tot f inter /C P[r = (0, 1)]

remain constant, that is, the oscillator strength tracks coherence and vice versa. Unlike in the LL limit, in the SS limit polarons are stable if the aggregate dimensions sufficiently exceed the coherence lengths. Under these conditions, the polaron no longer depends on the total number of thiophene units in the π-stack. This is demonstrated in Figure 6, which shows CM spectra for SS

(16) 11399


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The Journal of Physical Chemistry C Nx − 1 ≳ 2L intra /d intra

(17a)

Ny − 1 ≳ 2L inter /d inter

(17b)

As shown in ref 30 anisotropic disorder distributions (i.e., SL and LS) are characterized by a selective attenuation of the absorption component polarized along the direction containing the short-range disorder. The selective attenuation is also reflected in the coherence lengths in a manner consistent with eqs 11a and 11b. Hence, in an SL distribution, for example, the component of the absorption spectrum polarized along the polymer chain as well as the associated coherence length are selectively attenuated. Figure 7 demonstrates the effect in a SLdisordered 6 × 6 π-stack with σ = 0.6 eV. Compared with the disorder-free lattice in Figure 3, the x-polarized (intrachain) spectrum is significantly reduced with f tot intra decreasing from 0.37 to 0.21, while practically no change occurs in f tot inter. The coherence length responds similarly, with CP[r=(1,0)] dropping from 0.87 to 0.50, while CP[r=(0,1)] is only mildly affected, dropping from 0.86 to 0.83. Interestingly, the coherence lengths are far more sensitive to SL disorder, with Lintra decreasing from 4.2 to 1.7 units, while Linter is hardly affected. Moreover, the x-polarized peak absorption intensity is also extremely sensitive to SL disorder, diminishing by about an order of magnitude compared with the disorder-free value in Figure 3a, as the line width increases substantially reflecting the increased disorder. Conversely, the entire line shape corresponding to the interchain component is essentially unaffected by SL disorder. Off-Diagonal Disorder. X-ray diffraction studies by Salleo and co-workers33,46 have revealed the presence of paracrystallinity in poly(2,5-bis(3-tetradecylthiophen-2-yl)thieno[3,2,-b]thiophene) (PBTTT)46 and P3HT33 films, in which deviations exist in the interchain π-stacking distances. In a perfectly ordered crystal, all such distances are equal to d, which is approximately 0.4 nm in P3HT π-stacks. In treating paracrystallinity, the deviation, Δdm,n, in the separation between the mth unit on chain n and the mth unit on chain n+1 can be considered as a normal random variable, chosen from a Gaussian distribution with standard deviation σd. The unitless parameter g, defined as g ≡ σ d /d, is a measure of paracrystallinity. A film with a g > 0.1−0.15 is considered amorphous.33,46 A deviation in interunit separation leads naturally to a change in the corresponding coupling. The interchain coupling between the repeat unit at (m,n) and one at (m,n+1) is exponentially sensitive to the distance and is given by33,46

Figure 5. (a−c) Hole absorption spectrum in the mid-infrared for a 6 × 6 π-stack with d = 0.4 nm and with short-range site disorder in each direction (SS). The integrated oscillator strength of each band is shown in the inset. (d−f) The associated coherence functions. The parameters defining the Hamiltonian in eq 1 are tinter= −0.15 eV, tintra = −0.3 eV, ωvib = 0.174 eV, and λ2 = 1. An average over 5000 configurations of disorder was performed in each panel.

mn t inter = t inter exp( −β Δdm , n)

(18)

where d + Δdm,n is the distance between the two monomers and tinter is the coupling when Δdm,n = 0. The factor β dictates how rapidly the interchain orbital overlap diminishes with increasing separation and was evaluated to be β = 2.35 for P3HT using DFT in ref 46. The deviation in the interchain coupling appearing in eq 4a is thus

Figure 6. Hole absorption spectrum for various size π-stacks with SS site-disorder and σ = 0.4 eV. Remaining parameters defining the Hamiltonian in eq 2 are the same as in Figure 5.

disordered N × N π-stacks for various values of N. Here, the disorder width is σ = 0.4 eV, as in Figure 4b. The spectra are convergent by approximately N = 5, so that r in eq 12 is bounded by 4 units in either direction, roughly double the coherence lengths (see Figure 4b, inset). Hence, polarons are stable in an Nx × Ny π-stack if the following conditions are satisfied:

mn mn Δt inter ≡ t inter − t inter

(19)

Note that because of the exponential dependence in the coupling strength the mean coupling, ⟨tmn inter⟩, which is important in determining oscillator strength (see eq 13b) is not simply tinter but is somewhat greater since for a given |Δdm,n|, Δtmn inter is larger when Δdm,n is negative vs positive. In our first order 11400


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Figure 7. (a) Hole absorption spectrum in the mid-infrared for a 6 × 6 π-stack with d = 0.4 nm and with SL disorder (σ = 0.6 eV). The integrated oscillator strength of each band is shown in the inset. (b) The associated coherence function. The parameters defining the Hamiltonian in eq 1 and eq 3 are tinter= −0.15 eV, tintra = −0.3 eV, ωvib = 0.174 eV, and λ2 = 1.

Figure 8. (a) Hole absorption spectrum in the mid-infrared for a 6 × 6 π-stack with off-diagonal disorder in tinter. The top two panels present the xpolarized (intrachain) and y-polarized (interchain) spectra for LL disorder. The bottom two panels represent SS disorder. Several values of the crystallinity parameter g are shown. Insets show the oscillator strengths and coherence lengths in units of d = 0.4 nm. The parameters defining the Hamiltonian in eq 1 are tinter= −0.15 eV, tintra = −0.3 eV, ωvib = 0.174 eV, and λ2 = 1.

absorption spectra for 6 × 6 π-stacks with LL and SS distributions of off-diagonal disorder. As in the case of site disorder, in the LL limit, the coherence function is entirely unchanged from the disorder-free aggregate shown in Figure 3. The insensitivity of CP(r) to the presence of LL off-diagonal disorder is readily understood: each configuration in the LL distribution is perfectly ordered with uniform values of Δtmn inter,C, although its particular value varies between members of the disordered ensemble. Figure 8 shows that for LL off-diagonal disorder the component of the absorption spectrum polarized along the polymer axis is essentially unchanged as g varies from zero to 0.06. This is consistent with an unvarying tintra, since we have assumed that paracrystallinity affects only tinter. The constant oscillator strength is entirely consistent with eq 13a, which is

approach to understanding paracrystallinity, we neglect changes in tintra. In general, the deviations, Δdm,n, and therefore Δtmn inter, are functions of both m and n and can therefore be distributed with varying degrees of spatial correlation along the two directions defining a π-stack. The four extreme cases, LL, SL, LS, and SS, are defined in a manner similar to their site-disorder analogues. For example, for SS off-diagonal disorder depicted in Figure 2, all deviations Δdm,n within a given configuration are chosen independently from a Gaussian distribution of width σd, while in LL disorder, all deviations Δdm,n within a given configuration are identical and equal to ΔdC, (while ΔdC is chosen from a Gaussian distribution of width σd.) To demonstrate the impact of paracrystallinity in the extreme limits, we evaluated the coherence functions and the hole 11401


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Figure 9. (a, b) Comparison of simulated CM spectra to the measured spectra of Chang et al.26 for two different molecular weight P3HT polymer films (37 and 15 kDa) spin-cast from trichlorobenzene solution. Calculated spectra (dashed lines) were obtained using 6 × 6 π-stacks with d = 0.4 nm. For the high MW films, the computed spectrum represents an average over 5000 randomly generated configurations, characterized by SL diagonal disorder with σ = 0.28 eV and SS paracrystallinity with g = 0.065. The homogeneous line width Γ was set to 0.018 eV (standard deviation) and tintra and tinter were, respectively, set to −0.40 eV and −0.15 eV. By contrast, the low MW films, were characterized by increased SS diagonal disorder (σ = 0.37 eV), increased homogeneous broadening (Γ = 0.03 eV), and g = 0.05. The values of tintra and tinter were taken to be −0.28 and −0.15 eV, respectively. The vibrational parameters ωvib = 0.174 eV and λ2 = 1 were taken in both cases. The associated coherence functions for the high and low MW films are shown in panels c and d, respectively.

spectra with the published spectra of Chang et al.26 for P3HT films prepared with polymers of varying molecular weights, spin-cast from a trichlorobenzene solution. For our analysis, we choose the 37 and 15 kDa films, referred to hereon as the high and low MW films, respectively. Our calculations are significantly improved over our those presented in ref 30 due to the expansion to larger lattices (6 × 6 vs 5 × 5), and our implementation of the more realistic SS form of paracrystallinity (see Figure 2), which was not considered in ref 30. Figure 9a,b shows the experimental CM spectra from ref 26. The spectra are considered to be independent of incident field polarization because the samples were not oriented. The spectra are quite different for the two MWs. Note first the marked change in ordinate units between panels a and b in Figure 9, which masks a precipitous drop in peak intensity by about a factor of 4 in going from the high to low MW film. There is also an increase in the overall line broadening, indicative of increased disorder in the lower MW films. Both high and low MW spectra also display sharp anti-resonances due to infrared active vibrations (IRAVs).22,23 We have not accounted for purely vibrational transitions in our theory. Rather, we are concerned with the overall spectral envelope, which arises from all transitions starting from the ground state of the hole, |ΨG⟩, to all of the generally delocalized hole states, which are eigenstates of the Hamiltonian in eq 2. Although the line shapes of the both high and low MW spectra are very different, they basically consist of two broad peaks, the first of which (labeled peak A) contains most of the sharp anti-resonances and has been attributed to the interchain charge transfer by several groups.22−26 The second peak (peak

exact for the intrachain component. By contrast, the interchain component undergoes significant attenuation, although the oscillator strength, which scales as the area of the band, actually slightly increases with g (see figure inset). This unusual behavior is readily appreciated from eq 13b. Because of the exponential dependence of the interchain coupling on separation, the mean coupling ⟨tmn inter⟩ is larger than tinter and the disparity increases with increasing g. Since the coherence function is entirely unchanged with g, the oscillator strength according to eq 13b increases as observed in Figure 8 insets. The case for SS paracrystallinity is quite different. Our calculations reveal that the coherence function narrows with increasing g in both directions, and this is reflected in the diminishing coherence lengths with increasing g shown in the insets of Figure 8. This behavior is again analogous to the case of SS site disorder, where the coherence also narrows isotropically with increasing disorder. Figure 8 further shows an attenuation of the main absorption peak in both polarization components. In the case of the intrachain component, the oscillator strength shown in the inset also diminishes in accordance with eq 13a, and faithfully tracks the diminishing value of CP[r = (1,0)]. By contrast, the interchain component of the oscillator strength increases, due to the increasing mean value of the interchain coupling, which overcompensates the diminishing coherence factor.

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COMPARISON TO EXPERIMENT Based on the approach described in the previous sections, we can now determine the extent to which holes are delocalized in P3HT. We accomplish this by comparing our calculated CM 11402


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diagonal disorder significantly to σ = 0.37 eV and change its nature from SL to SS. To account for paracrystallinity, the value of g was slightly decreased to g = 0.05, in agreement with X-ray diffraction measurements by Koch et al.,34 who showed a 10− 15% decrease in g in going from 40 to 15 kDa P3HT films. In addition to increasing the diagonal disorder, we also reduced the value of |tintra| to 0.28 eV, consistent with enhanced intrachain (torsional) disorder. As shown in ref 27, torsional disorder is larger in the lower MW films, where the paraffinic phase is prominent. We emphasize that the simulated spectra in Figure 9 are not individually normalized. The spectral attenuation by about a factor of 4 observed by Chang et al. in going from the 37 to 15 kDa films is well-captured by our simulations and justifies the validity of our model. The attenuation is linked to a drop in band oscillator strength, which is reported in the Figure 9 inset. Peak A in the simulated spectrum of Figure 9b is slightly larger than its experimental counterpart, but the remaining line shape is very well captured, especially the appearance of the broad, high-energy shoulder centered around three vibrational quanta, which is also observed in Figure 5 with increasing diagonal disorder and is due primarily to the intrachain component. Note that in this regime of strong disorder the two polarized components more closely resemble each other. Of particular interest is the composition of peak A. In marked contrast to the high MW spectra, the main contributor to peak A is now the intrachain component, making the association of peak A exclusively with interchain transitions questionable. We now turn to polaron delocalization in the high and low MW films. In Figure 9c,d, the calculated coherence functions for the high and low MW films are displayed, with the coherence lengths (in units of d) reported in the insets. Consistent with the increased disorder in going from the high to low MW film is a pronounced drop in the spatial coherence lengths: (Lintra, Linter) decreases from (2.94d, 2.81d) for the 37 kDa films to only (1.83d, 1.14d) in the 15 kDa films. Hence, in the high MW film, the polaron is delocalized over roughly four thiophene units in each direction, decreasing to only three and two in the intrachain and interchain directions, respectively, for the low MW films. (The number of coherently connected thiophene units in direction j is one greater than the corresponding coherence length, Lj/d). Interestingly, the drops in Lintra and Linter track similar drops in the polarized band oscillator strengths, which are reported in the insets of Figure 9a,b; f intra and f inter diminish by factors of 1.9 and 2.3, respectively, compared with the factors of 1.6 and 2.5 for Lintra and Linter. The decrease in the oscillator strengths and coherence lengths with decreasing MW is primarily due to an increase in SS diagonal disorder as well as the associated decrease in the |tintra| (0.40 to 0.28 eV) used in our simulations. Interestingly, because intrachain disorder is diagonal, eq 13a is exact and can be used to quantitatively account for the decrease in f intra by a factor of 1.9 in going from the high MW to the low MW film.

B) has been loosely associated with the intrachain polaron transition, that is, peak C1 or P1 in refs 22−24 and 26. Interestingly, in both spectra peaks A and B are separated by a significant dip, exacerbated by the presence of sharp antiresonances, which lies close in energy to the prominent vibrational mode at approximately 0.17 eV (1400 cm−1). There are also additional features that appear to be vibronic in origin: in the high MW films peak B displays a splitting (indicated by the arrows) that is slightly more difficult to discern in the lower MW film presumably due to increased disorder-induced line width. The lower MW film, however, develops a pronounced high energy band centered at approximately 3 vibrational quanta (or ∼0.5 eV), which can be tentatively associated with P1 or C1. Figure 9a also shows the simulated spectrum consisting of the sum of the x- and y-polarized components based on the Hamiltonian in eq 2 for 6 × 6 lattices. In order to make the simulations tractable, we focus only on site disorder and paracrystallinity, relegating the impact of off-diagonal disorder along the chains to an ef fective value of tintra. DFT band structure calculations by Northrup68 give |tintra| ≈ 0.45 eV and | tinter| ≈ 0.15 eV for planarized chains. For the high MW films, we reduced the intrachain value to 0.40 eV to account for torsions but left the interchain value intact since paracrystallinity is treated directly. The vibrational parameters are the same as in previous figures and are consistent with what is known for P3HT.27 Although we use λ2 = 1, we have verified (not shown) that the spectra are not that sensitive to changes of λ2 within the range 0.7−1.3. Within this range, all peak positions (including the dip) and the high-energy tail remain stable. In order to obtain the best agreement with experiment, we incorporated SL diagonal broadening (σ = 0.28 eV) and SS paracrystallinity (g = 0.065) and averaged over 5000 configurations of disorder to obtain converged spectra. The substantial value of g is in good agreement with X-ray diffraction measurements of Salleo and co-workers for high MW P3HT films33,34 and may arise from the stress induced by chain folding.31,32 The SL nature of the diagonal (site) disorder emphasizes intrachain disorder and is necessary to maintain the dominance of the first broad peak,30 peak A in the figure. As shown in previous sections, SS broadening effectively attenuates both peaks (A and B).30 Overall, the agreement with experiment is quite good. The calculated polarized components in Figure 9a show that peak A is predominantly due to interchain transitions in agreement with the interpretations of refs 22−26. However, there is also a significant contribution to peak A from the intrachain transitions.30 The broad asymmetric nature of peak B is also well-reproduced, including the enhanced oscillator strength in the vicinity of the split peaks, which are mainly due to transitions along the polymer backbone. Our calculations also account for the dip observed between the peaks A and B, which occurs at an energy slightly greater than a vibrational quantum. The dip position (in units of vibrational quanta) is remarkably insensitive to changes in the vibrational frequency as well as the HR factor. For example, changing ωvib or λ2 by ±30% results in a shift in the dip position by less than ±3%. Furthermore, the dip only arises when two-particle states are included. We are further investigating the nature of the dip to see if is related to the antiresonances observed in refs 22 and 23. In order to capture the increased broadening and dramatic drop in intensity displayed by CM spectrum of the low MW film displayed in Figure 9b, it was necessary to increase the

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SUMMARY/CONCLUSION Based on a nonadiabatic Holstein-style model put forth by Pochas and Spano,30 we have analyzed in detail the nature of hole delocalization in conjugated polymer films. The model is highly successful in quantitatively reproducing most of the salient features present in the mid-IR absorption spectrum of P3HT films, including the pronounced dip observed at an energy approximately equal to one quantum of vibrational 11403


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when each lattice dimension exceeds roughly twice the corresponding coherence length, see eq 17. In other words the polaron’s wave function extends beyond the values of Lintra and Linter defined in eqs 15a and 15b by roughly a factor of 2. We can therefore also claim a delocalization number of 10−11 along the polymer chain in the very high MW (270 kDa) films. Our definition of Lintra (Linter) is but one way to characterize the width of the coherence function along x (y). A much more accurate determination of the dimensionality of polaron delocalization requires oriented samples, from which the spectral components for IR excitation polarized along the polymer axis and along the stacking axis can be derived. With the two spectra in hand for each MW, one can exploit the relationships between the x-polarized and y-polarized band oscillator strengths and the coherence points, CP[r=(1,0)] and CP[r=(0,1)], respectively, expressed by eqs 13a and 13b to extract approximate coherence lengths. As absolute oscillator strengths are difficult to measure, an alternative method is to vary the nature and magnitude of disorder until the simulated x- and y-polarized spectral line shapes agree with experiment, as was done for the unpolarized spectra in this work. Oriented samples and the ability to resolve the mid-IR spectrum along the x and y axes will also provide additional insight into the nature of peak A, also referred to as the CT peak24 and DP122 in previous works. These works utilized standard polaron theory,29,40 to show that the peak is entirely sourced by interchain transitions. However, as shown here, peak A has contributions from both interchain and intrachain transitions (see Figure 9a,b). The interchain origin of peak A appears to dominate in the higher MW films with SL diagonal disorder. As shown in ref 30, short-range disorder along a given axis selectively attenuates the polarized spectrum along that same axis. This is most dramatically demonstrated in the model calculations shown in Figure 7, where the x-polarized (intrachain) spectrum is severely attenuated under SL disorder, leaving behind an intense low-energy peak (A) polarized entirely along the interchain axis. However, when significant interchain short-range disorder is added, as in going from the simulated spectrum in Figure 9a (SL) to that in Figure 9b (SS), the situation is reversed: peak A becomes dominated by intrachain transitions, because the interchain component is selectively attenuated. CM spectra from oriented samples will allow one to verify the nature of peak A, as well as the intrachain nature of the split peak, which appears in the broad region of peak B. Hopefully this work will stimulate future experimental work along these lines.

energy involving the aromatic/quinoidal stretching mode, the additional split pair of peaks at slightly higher energies, and the broad high-energy shoulder at ∼0.5 eV, which is most pronounced in the more disordered, lower MW films. Such fine structure cannot be accounted for in the standard polaron theory, which is based on the adiabatic approximation.29,40 With the excellent agreement between the theory and experiment (see Figure 9) in hand, we proceeded to derive the generally anisotropic hole coherence lengths in P3HT thin films, where polymer molecular weight is a major factor in determining thin film morphology.33−35 For the intermediate MW films (37 kDa) of ref 26, we found the hole to be delocalized over approximately four thiophene units in each direction. Hence, the total number of thiophene units within the coherence “area” is approximately Ncoh = 16, which can also be obtained using eq 14. Reducing the molecular weight to 15 kDa results in significant line broadening due to increased disorder and a 4-fold reduction in peak intensity. The polaron delocalization number diminishes to three units along the polymer axis and only 2 units along the interchain axis, reducing Ncoh to about 6−7. Chang et al.26 also investigated a much higher MW film (270 kDa) showing a roughly 30% increase in oscillator strength compared with the 37 kDa films, but with a similar line width. The change is likely linked to an increase in ⟨tintra⟩ in the 270 kDa films due to enhanced chain planarization, which according to eq 13a would result in an increased oscillator strength through the quantity CP[(1,0)]⟨tintra⟩. One would therefore expect an increase in the intrachain delocalization number to roughly 5−6 units, which is too large for an accurate reproduction of the measured spectrum using our 6 × 6 lattices (see eq 17) . Interestingly, the charge mobility was also shown to increase by almost 2 orders of magnitude in going from the 15 to 270 kDa films,26 although it is not likely that the increase is due entirely to the increase in Ncoh. Hole mobility is sensitive to disorder on various length scales and, in particular, to the presence of “tie” polymers, which connect ordered aggregates in the presence of intervening amorphous regions in high MW films.31−36 A previous work by Niklas et al.69 based on a combination of EPR spectroscopy and DFT calculations estimated the delocalization number to be about 15 thiophene units in highly regioregular P3HT/fullerene film blends. At first glance, this number agrees well with our estimates for the 37 kDa films; however, the MW was not reported in ref 69. Moreover, the authors of ref 69 conceded that their analysis could not determine the dimensionality of the delocalization, for example, whether the polaron is delocalized over 15 units on a single chain or over 7−8 units on two neighboring chains within a πstack. However, because the film EPR spectrum was very similar to the spectrum corresponding to a frozen P3HT/ toluene solution, they concluded that delocalization was probably intramolecular. Takeda and Miller70 estimated the intramolecular delocalization number to be significantly smaller, about 8−9 units for poly(3-decylthiophene) dissolved in chloroform, where holes were generated using pulsed radiolysis and the delocalization number was deduced from the magnitude of the bleach signal in a transient absorption measurement. To fairly compare all of the aforementioned delocalization lengths depends on a consistent definition of the polaron delocalization number. We do not know how Ncoh from eq 14 exactly relates to the numbers deduced from the analyses in refs 69 and 70. In the discussion surrounding Figure 6, we pointed out that the hole absorption spectrum converges

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the National Science Foundation, Grant DMR 1533954. This research was also supported in part by the National Science Foundation through major research instrumentation Grant Number CNS-09-58854.

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REFERENCES

(1) Gelinas, S.; Rao, A.; Kumar, A.; Smith, S. L.; Chin, A. W.; Clark, J.; van der Poll, T. S.; Bazan, G. C.; Friend, R. H. Ultrafast Long-Range

11404


Article

The Journal of Physical Chemistry C Charge Separation in Organic Semiconductor Photovoltaic Diodes. Science 2014, 343, 512−516. (2) Kaake, L. G.; Moses, D.; Heeger, A. J. Coherence and Uncertainty in Nanostructured Organic Photovoltaics. J. Phys. Chem. Lett. 2013, 4, 2264−2268. (3) Kaake, L. G.; Jasieniak, J. J.; Bakus, R. C.; Welch, G. C.; Moses, D.; Bazan, G. C.; Heeger, A. J. Photoinduced Charge Generation in a Molecular Bulk Heterojunction Material. J. Am. Chem. Soc. 2012, 134, 19828−19838. (4) Banerji, N. Sub-Picosecond Delocalization in the Excited State of Conjugated Homopolymers and Donor−Acceptor Copolymers. J. Mater. Chem. C 2013, 1, 3052−3066. (5) Mukamel, S. Comment on ″Coherence and Uncertainty in Nanostructured Organic Photovoltaics″. J. Phys. Chem. A 2013, 117, 10563−10564. (6) Song, Y.; Clafton, S. N.; Pensack, R. D.; Kee, T. W.; Scholes, G. D. Vibrational Coherence Probes the Mechanism of Ultrafast Electron Transfer in Polymer−Fullerene Blends. Nat. Commun. 2014, 5, 4933. (7) Bittner, E. R.; Silva, C. Noise-Induced Quantum Coherence Drives Photo-Carrier Generation Dynamics at Polymeric Semiconductor Heterojunctions. Nat. Commun. 2014, 5, 3119. (8) Kassal, I.; Yuen-Zhou, J.; Rahimi-Keshari, S. Does Coherence Enhance Transport in Photosynthesis? J. Phys. Chem. Lett. 2013, 4, 362−367. (9) Strümpfer, J.; Şener, M.; Schulten, K. How Quantum Coherence Assists Photosynthetic Light-Harvesting. J. Phys. Chem. Lett. 2012, 3, 536−542. (10) Hwang, I.; Scholes, G. D. Electronic Energy Transfer and Quantum-Coherence in Pi-Conjugated Polymers. Chem. Mater. 2011, 23, 610−620. (11) Donehue, J. E.; Varnavski, O. P.; Cemborski, R.; Iyoda, M.; Goodson, I. T. Probing Coherence in Synthetic Cyclic LightHarvesting Pigments. J. Am. Chem. Soc. 2011, 133, 4819−4828. (12) Panitchayangkoon, G.; Hayes, D.; Fransted, K. A.; Caram, J. R.; Harel, E.; Wen, J. Z.; Blankenship, R. E.; Engel, G. S. Long-Lived Quantum Coherence in Photosynthetic Complexes at Physiological Temperature. Proc. Natl. Acad. Sci. U. S. A. 2010, 107, 12766−12770. (13) Engel, G. S.; Calhoun, T. R.; Read, E. L.; Ahn, T. K.; Mancal, T.; Cheng, Y. C.; Blankenship, R. E.; Fleming, G. R. Evidence for Wavelike Energy Transfer through Quantum Coherence in Photosynthetic Systems. Nature 2007, 446, 782−786. (14) Dahlbom, M.; Pullerits, T.; Mukamel, S.; Sundstrom, V. Exciton Delocalization in the B850 Light-Harvesting Complex: Comparison of Different Measures. J. Phys. Chem. B 2001, 105, 5515−5524. (15) Chachisvilis, M.; Kuhn, O.; Pullerits, T.; Sundstrom, V. Excitons in Photosynthetic Purple Bacteria: Wavelike Motion or Incoherent Hopping? J. Phys. Chem. B 1997, 101, 7275−7283. (16) Halpin, A.; Johnson, P. J. M.; Tempelaar, R.; Murphy, R. S.; Knoester, J.; Jansen, T. L. C.; Miller, R. J. D. Two-Dimensional Spectroscopy of a Molecular Dimer Unveils the Effects of Vibronic Coupling on Exciton Coherences. Nat. Chem. 2014, 6, 196−201. (17) Tempelaar, R.; Jansen, T. L. C.; Knoester, J. Vibrational Beatings Conceal Evidence of Electronic Coherence in the Fmo LightHarvesting Complex. J. Phys. Chem. B 2014, 118, 12865−12872. (18) Kim, Y.; Cook, S.; Tuladhar, S. M.; Choulis, S. A.; Nelson, J.; Durrant, J. R.; Bradley, D. D. C.; Giles, M.; McCulloch, I.; Ha, C. S.; Ree, M. A Strong Regioregularity Effect in Self-Organizing Conjugated Polymer Films and High-Efficiency Polythiophene: Fullerene Solar Cells. Nat. Mater. 2006, 5, 197−203. (19) Mihailetchi, V. D.; Xie, H. X.; de Boer, B.; Koster, L. J. A.; Blom, P. W. M. Charge Transport and Photocurrent Generation in Poly (3Hexylthiophene): Methanofullerene Bulk-Heterojunction Solar Cells. Adv. Funct. Mater. 2006, 16, 699−708. (20) Prosa, T. J.; Winokur, M. J.; McCullough, R. D. Evidence of a Novel Side Chain Structure in Regioregular Poly(3-Alkylthiophenes). Macromolecules 1996, 29, 3654−3656. (21) Prosa, T. J.; Winokur, M. J.; Moulton, J.; Smith, P.; Heeger, A. J. X-Ray Structural Studies of Poly(3-Alkylthiophenes): An Example of an Inverse Comb. Macromolecules 1992, 25, 4364−4372.

(22) Osterbacka, R.; An, C. P.; Jiang, X. M.; Vardeny, Z. V. TwoDimensional Electronic Excitations in Self-Assembled Conjugated Polymer Nanocrystals. Science 2000, 287, 839−842. (23) Jiang, X. M.; Osterbacka, R.; Korovyanko, O.; An, C. P.; Horovitz, B.; Janssen, R. A. J.; Vardeny, Z. V. Spectroscopic Studies of Photoexcitations in Regioregular and Regiorandom Polythiophene Films. Adv. Funct. Mater. 2002, 12, 587−597. (24) Sirringhaus, H.; Brown, P. J.; Friend, R. H.; Nielsen, M. M.; Bechgaard, K.; Langeveld-Voss, B. M. W.; Spiering, A. J. H.; Janssen, R. A. J.; Meijer, E. W.; Herwig, P.; de Leeuw, D. M. Two-Dimensional Charge Transport in Self-Organized, High-Mobility Conjugated Polymers. Nature 1999, 401, 685. (25) Chang, J. F.; Clark, J.; Zhao, N.; Sirringhaus, H.; Breiby, D. W.; Andreasen, J. W.; Nielsen, M. M.; Giles, M.; Heeney, M.; McCulloch, I. Molecular-Weight Dependence of Interchain Polaron Delocalization and Exciton Bandwidth in High-Mobility Conjugated Polymers. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 74, 115318. (26) Chang, J.-F.; Sirringhaus, H.; Giles, M.; Heeney, M.; McCulloch, I. Relative Importance of Polaron Activation and Disorder on Charge Transport in High-Mobility Conjugated Polymer Field-Effect Transistors. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 205204. (27) Paquin, F.; Yamagata, H.; Hestand, N. J.; Sakowicz, M.; Bérubé, N.; Côté, M.; Reynolds, L. X.; Haque, S. A.; Stingelin, N.; Spano, F. C.; Silva, C. Two-Dimensional Spatial Coherence of Excitons in Semicrystalline Polymeric Semiconductors: Effect of Molecular Weight. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 155202. (28) Spano, F. C.; Clark, J.; Silva, C.; Friend, R. H. Determining Exciton Coherence from the Photoluminescence Spectral Line Shape in Poly(3-Hexylthiophene) Thin Films. J. Chem. Phys. 2009, 130, 074904. (29) Beljonne, D.; Cornil, J.; Sirringhaus, H.; Brown, P. J.; Shkunov, M.; Friend, R. H.; Bredas, J. L. Optical Signature of Delocalized Polarons in Conjugated Polymers. Adv. Funct. Mater. 2001, 11, 229− 234. (30) Pochas, C. M.; Spano, F. C. New Insights on the Nature of Two-Dimensional Polarons in Semiconducting Polymers: Infrared Absorption in Poly(3-Hexylthiophene). J. Chem. Phys. 2014, 140, 244902. (31) Brinkmann, M.; Rannou, P. Molecular Weight Dependence of Chain Packing and Semicrystalline Structure in Oriented Films of Regioregular Poly(3-Hexylthiophene) Revealed by High-Resolution Transmission Electron Microscopy. Macromolecules 2009, 42, 1125− 1130. (32) Brinkmann, M.; Rannou, P. Effect of Molecular Weight on the Structure and Morphology of Oriented Thin Films of Regioregular Poly(3-Hexylthiophene) Grown by Directional Epitaxial Solidification. Adv. Funct. Mater. 2007, 17, 101−108. (33) Noriega, R.; Rivnay, J.; Vandewal, K.; Koch, F. P. V.; Stingelin, N.; Smith, P.; Toney, M. F.; Salleo, A. A General Relationship between Disorder, Aggregation and Charge Transport in Conjugated Polymers. Nat. Mater. 2013, 12, 1038−1044. (34) Koch, F. P. V.; Rivnay, J.; Foster, S.; Muller, C.; Downing, J. M.; Buchaca-Domingo, E.; Westacott, P.; Yu, L. Y.; Yuan, M. J.; Baklar, M.; Fei, Z. P.; Luscombe, C.; McLachlan, M. A.; Heeney, M.; Rumbles, G.; Silva, C.; Salleo, A.; Nelson, J.; Smith, P.; Stingelin, N. The Impact of Molecular Weight on Microstructure and Charge Transport in Semicrystalline Polymer Semiconductors Poly(3-Hexylthiophene), a Model Study. Prog. Polym. Sci. 2013, 38, 1978−1989. (35) Botiz, I.; Stingelin, N. Influence of Molecular Conformations and Microstructure on the Optoelectronic Properties of Conjugated Polymers. Materials 2014, 7, 2273−2300. (36) O’Connor, B. T.; Reid, O. G.; Zhang, X. R.; Kline, R. J.; Richter, L. J.; Gundlach, D. J.; DeLongchamp, D. M.; Toney, M. F.; Kopidakis, N.; Rumbles, G. Morphological Origin of Charge Transport Anisotropy in Aligned Polythiophene Thin Films. Adv. Funct. Mater. 2014, 24, 3422−3431. (37) Brown, P. J.; Thomas, S. D.; Kohler, A.; Wilson, J. S.; Kim, J.-S.; Ramsdale, C. M.; Sirringhaus, H.; Friend, R. H. Effect of Interchain 11405


Article

The Journal of Physical Chemistry C Interactions on the Absorption and Emission of Poly(3-Hexylthiophene). Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 67, 064203. (38) Holstein, T. Polaron Motion. I. Molecular-Crystal Model. Ann. Phys. 1959, 8, 325−342. (39) Barford, W.; Marcus, M.; Tozer, O. R. Polarons in ΠConjugated Polymers: Anderson or Landau? J. Phys. Chem. A 2016, 120, 615−620. (40) Bredas, J. L.; Street, G. B. Polarons, Bipolarons and Solitons in Conducting Polymers. Acc. Chem. Res. 1985, 18, 309−315. (41) Vogl, P.; Campbell, D. K. Three-Dimensional Structure and Intrinsic Defects in Trans-Polyacetylene. Phys. Rev. Lett. 1989, 62, 2012−2015. (42) Mizes, H. A.; Conwell, E. Stability of Polarons in Conducting Polymers. Phys. Rev. Lett. 1993, 70, 1505−1508. (43) Gomes da Costa, P.; Dandrea, R. G.; Conwell, E. FirstPrinciples Calculation of the Three-Dimensional Band Structure of Ppv. Phys. Rev. B: Condens. Matter Mater. Phys. 1993, 47, 1800−1810. (44) Gomes da Costa, P.; Dandrea, R. G.; Conwell, E. M. FirstPrinciples Calculation of the Three-Dimensional Band Structure of Poly(Phenylene Vinylene). Phys. Rev. B: Condens. Matter Mater. Phys. 1993, 47, 1800−1810. (45) McMahon, D. P.; Cheung, D. L.; Goris, L.; Dacuña, J.; Salleo, A.; Troisi, A. Relation between Microstructure and Charge Transport in Polymers of Different Regioregularity. J. Phys. Chem. C 2011, 115, 19386−19393. (46) Rivnay, J.; Noriega, R.; Northrup, J. E.; Kline, R. J.; Toney, M. F.; Salleo, A. Structural Origin of Gap States in Semicrystalline Polymers and the Implications for Charge Transport. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 121306. (47) Poelking, C.; Andrienko, D. Effect of Polymorphism, Regioregularity and Paracrystallinity on Charge Transport in Poly(3Hexylthiophene) P3ht Nanofibers. Macromolecules 2013, 46, 8941− 8956. (48) Piepho, S. B. Vibronic Coupling Model for the Calculation of Mixed-Valence Line-Shapes - a New Look at the Creutz Taube Ion. J. Am. Chem. Soc. 1990, 112, 4197−4206. (49) Piepho, S. B. Vibronic Couplng Model for the Calculation of Mixed-Valence Line-Shapes - the Interdependence of Vibronic and Mo Effects. J. Am. Chem. Soc. 1988, 110, 6319−6326. (50) Piepho, S. B.; Krausz, E. R.; Schatz, P. N. Vibronic Coupling Model for Calculation of Mixed-Valence Absorption Profiles. J. Am. Chem. Soc. 1978, 100, 2996−3005. (51) Creutz, C.; Taube, H. A Direct Approach to Measuring FranckCondon Barrier to Electron Transfer between Metal Ions. J. Am. Chem. Soc. 1969, 91, 3988−3989. (52) Philpott, M. R. Theory of Coupling of Electronic and Vibrational Excitations in Molecular Crystals and Helical Polymers. J. Chem. Phys. 1971, 55, 2039−2054. (53) Spano, F. C. Absorption and Emission in Oligo-Phenylene Vinylene Nanoaggregates: The Role of Disorder and Structural Defects. J. Chem. Phys. 2002, 116, 5877−5891. (54) Stradomska, A.; Petelenz, P. Intermediate Vibronic Coupling in Sexithiophene Single Crystals. Ii. Three-Particle Contributions. J. Chem. Phys. 2009, 131, 044507. (55) Spano, F. C. Modeling Disorder in Polymer Aggregates: The Optical Spectroscopy of Regioregular Poly(3-Hexylthiophene) Thin Films. J. Chem. Phys. 2005, 122, 234701. (56) Spano, F. C. Absorption in Regioregular Poly(3-Hexylthiophene) Thin Films: Fermi Resonances, Interband Coupling and Disorder. Chem. Phys. 2006, 325, 22−35. (57) Clark, J.; Silva, C.; Friend, R. H.; Spano, F. C. Role of Intermolecular Coupling in the Photophysics of Disordered Organic Semiconductors: Aggregate Emission in Regioregular Polythiophene. Phys. Rev. Lett. 2007, 98, 206406. (58) Yamagata, H.; Spano, F. C. Interplay between Intrachain and Interchain Interactions in Semiconducting Polymer Assemblies: The Hj-Aggregate Model. J. Chem. Phys. 2012, 136, 184901. (59) Hush, N. S. Distance Dependence of Electron Transfer Rates. Coord. Chem. Rev. 1985, 64, 135−157.

(60) Hoffmann, S. T.; Bassler, H.; Kohler, A. What Determines Inhomogeneous Broadening of Electronic Transitions in Conjugated Polymers? J. Phys. Chem. B 2010, 114, 17037−17048. (61) Bassler, H. Localized States and Electronic Transport in Single Component Organic Solids with Diagonal Disorder. Phys. Status Solidi B 1981, 107, 9. (62) Blanchet, G. B.; Fincher, C. R.; Chung, T. C.; Heeger, A. J. Photoexcitations in Trans - (Ch)X a Fourier-Transform Infrared Study. Phys. Rev. Lett. 1983, 50, 1938−1941. (63) Tian, B.; Zerbi, G. Lattice Dynamics and Vibrational Spectra of Pristine and Doped Polypyrrole: Effective Conjugation Coordinate. J. Chem. Phys. 1990, 92, 3892−3898. (64) Spano, F. C.; Meskers, S. C. J.; Hennebicq, E.; Beljonne, D. Probing Excitation Delocalization in Supramolecular Chiral Stacks by Means of Circularly Polarized Light: Experiment and Modeling. J. Am. Chem. Soc. 2007, 129, 7044−7054. (65) Spano, F. C.; Yamagata, H. Vibronic Coupling in J-Aggregates and Beyond: A Direct Means of Determining the Exciton Coherence Length from the Photoluminescence Spectrum. J. Phys. Chem. B 2011, 115, 5133−5143. (66) Meier, T.; Zhao, Y.; Chernyak, V.; Mukamel, S. Polarons, Localization, and Excitonic Coherence in Superradiance of Biological Antenna Complexes. J. Chem. Phys. 1997, 107, 3876−3893. (67) Meier, T.; Chernyak, V.; Mukamel, S. Multiple Exciton Coherence Sizes in Photosynthetic Antenna Complexes Viewed by Pump-Probe Spectroscopy. J. Phys. Chem. B 1997, 101, 7332−7342. (68) Northrup, J. E. Atomic and Electronic Structure of Polymer Organic Semiconductors: P3ht, Pqt, and Pbttt. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 245202. (69) Niklas, J.; Mardis, K. L.; Banks, B. P.; Grooms, G. M.; Sperlich, A.; Dyakonov, V.; Beaupre, S.; Leclerc, M.; Xu, T.; Yu, L. P.; Poluektov, O. G. Highly-Efficient Charge Separation and Polaron Delocalization in Polymer-Fullerene Bulk-Heterojunctions: A Comparative Multi-Frequency Epr and Dft Study. Phys. Chem. Chem. Phys. 2013, 15, 9562−9574. (70) Takeda, N.; Miller, J. R. Poly(3-Decylthiophene) Radical Anions and Cations in Solution: Single and Multiple Polarons and Their Delocalization Lengths in Conjugated Polymers. J. Phys. Chem. B 2012, 116, 14715−14723.

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