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Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Spectral Signatures and Spatial Coherence of Bound and Unbound Polarons in P3HT Films: Theory Versus Experiment Raja Ghosh,† Annabel R. Chew,‡ Jonathan Onorato,§ Viktoria Pakhnyuk,∥ Christine K. Luscombe,§ Alberto Salleo,‡ and Frank C. Spano*,† †

Department of Chemistry, Temple University, Philadelphia Pennsylvania 19122, United States Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, United States § Materials Science and Engineering Department, University of Washington, Seattle, Washington 98195-2120, United States ∥ Department of Chemistry, University of Washington, Seattle, Washington 98195-1700, United States

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‡

S Supporting Information *

ABSTRACT: The origin of the mid-infrared (IR) spectral features for hole absorption in doped poly(3-hexylthiophene) (P3HT) films is investigated theoretically and experimentally. Using a Holstein-style Hamiltonian to treat vibronic coupling involving the prominent vinyl stretching mode, the low-energy peak (A) occurring in the spectral interval 0.1−0.15 eV is found to contain a substantial (sometimes dominant) intrachain-polarized component, in contrast to the predictions of the more conventional treatments based on self-trapped, midgap polaron states where peak A is entirely interchain in origin. A higher-energy peak (B) located between 0.35 and 0.7 eV and largely intrachain-polarized is also obtained and associated with the conventional P1 polaron transition. Spectral signatures for polaron coherence based on peaks A and B are identified and used to analyze the molecular weight dependence of the IR spectral line shapes of P3HT doped with 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane. Generally, the polaron coherence lengths along the chain and stacking axes increase with molecular weight, which is consistent with similar studies conducted on hole absorption in undoped P3HT. been identified such as Coulomb trapping by ionized dopants,5 disorder in the doped polymer energetic landscape,28 and carrier localization near-ionized dopants.8 The situation is further complicated by the occurrence of different processes as a function of the dopant concentration. At low dopant concentrations, for instance, Coulomb traps5 and carrier localization in amorphous polymer regions9 leading to a reduction in mobility were observed. At higher concentrations, it was found that the interaction with the dopant straightens the P3HT chains, leading to an increase in mobility.8 There is therefore considerable interest in providing a fundamental understanding of the nature of the charges generated by doping. In this manuscript, we present a theoretical and experimental investigation of the infrared (IR) spectral line shape derived from holes in doped P3HT films, emphasizing the important role played by polaron coherence. The latter defines the spatial extent over which the polaron wave function is wave-like and hence figures prominently in the hole mobility. The theory is based on a Holstein-style Hamiltonian for mobile holes, which differs markedly from the conventional interpretation in terms

1. INTRODUCTION Chemical doping of semiconducting conjugated polymers is a key enabling technology to bring these materials to market and has therefore been actively studied for several decades.1−21 The dopant molecules act as efficient charge injectors and can introduce either electrons, or, more often, holes into the polymer matrix. The addition of holes into the conjugated polymer poly(3-hexylthiophene) or P3HT has been shown to dramatically increase conductivity,16,21 motivating several recent works to investigate the important relationship between the polymer/dopant morphology and the method used to introduce the dopant.8,15−17,20 While it has been shown that the ionization efficiency of dopants in P3HT can be as high as 70−80%,5,10 the fraction of mobile charges remains low, on the order of 5%,5 leading to overall low functional doping efficiencies and therefore requiring high dopant concentrations to significantly increase conductivity. The high concentration of dopant molecules can induce dramatic microstructural changes, such as the appearance of new co-crystallized phases.14,22 These microstructural instabilities pose problems in cases where high conductivities are required, such as thermoelectric materials23 or contact/channel doping in transistors.24−27 The origin of such low efficiency of mobile carrier generation is still being investigated. A few mechanisms have © XXXX American Chemical Society

Received: April 24, 2018 Revised: July 8, 2018

A

DOI: 10.1021/acs.jpcc.8b03873 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C of self-trapped, mid-gap polarons.29 The theory described herein accounts for peak positions and line widths and resolves several curious features, most notably, the simultaneous presence of a low-energy, narrow peak (A) followed by a higher-energy, broad peak (B) separated by a pronounced spectral dip at approximately one vibrational quantum of the aromatic-quinoidal mode. We show that the low-energy peak (“A”) derives from a polaron version of Herzberg−Teller (HT) coupling,30 in part borrowing oscillator strength from peak B via both intrachain and interchain interactions. Importantly, the spatial coherence of the dopant-induced polarons can be deduced from spectral signatures, such as the relative intensity of peak A or the absorption maximum of peak B. The predicted spectra are in excellent agreement with the measured spectra for 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane (F4TCNQ)-doped regioregular P3HT samples with different molecular weights (MWs). Spectral analysis supports the presence of two polymorphs in the low-MW films, with the intrachain and interchain polaron coherence lengths generally increasing with MW, which is consistent with previous studies of undoped films.31 Our methodology allows one to understand how polymer microstructure, MW, and processing affect the delocalization of the dopant-induced polarons, information that is important in the design of more efficient processing routes or new dopants.

induced backbone distortion.29,32,33,35−37 From here on, we refer to the broad peak in Figure 1 as peak B. The IR spectrum of doped P3HT also shows a lower-energy peak (labeled A in Figure 1) at approximately 0.150 eV s,8,15 which is narrower and attenuated compared to its higher energy partner (peak B). As shown in the figure, peak A spectrally overlaps a cluster of much narrower peaks derived from IR-active vibrational modes or IRAVs.32,33 These are Raman active modes made IR-active by hole-induced symmetry breaking. In doped films, simulations have shown that peak A red-shifts and substantially increases in intensity as the anion moves further away from the polymer chain(s) because of an increase in the polaron delocalization length.8,15 This behavior is consistent with observations involving (unbound) holes in the absence of dopants: peak A is relatively more intense and appears at lower energies in undoped P3HT with optically-32−34 or electrically31,35,36generated holes. Moreover, peak A also increases with MW as the films become more ordered, leading to substantial increases in hole mobility.31 Vardeny and co-workers32,33 referred to the peak as DP1 (delocalized polaron)a transition among polarons delocalized over several chains because of significant interchain coupling. The view was also shared by Sirringhaus and co-workers who labeled the peak as CT (charge transfer).31,35−37 In later works, however, numerical simulations based on a Holstein-style Hamiltonian showed that the low-energy peak also contains a significant intrachain contribution, which can surpass the interchain contribution in the presence of uncorrelated disorder between chains,38,39 that is, the low-energy peak cannot be solely identified with interchain CT, although CT can greatly amplify the peak intensity. Indeed, recent measurements on oriented P3HT films doped with F4TCNQ have shown peak A to contain a substantial component polarized along the chain axis.20 Moreover, an attenuated peak (A) due to photoinduced holes is discernible in unaggregated regiorandom P3HT.32 In what follows, we establish the origins of peak A and its higher-energy partner, peak B. The latter has been associated with absorption to the low-energy gap state (P1) since the early days of polyacetylene.40 According to the conventional theory, peak A arises from interchain charge transfer, which causes the gap-polaron responsible for P1 to split into two states, the transition between them being responsible for a low-energy peak polarized along the interchain direction.32,35,37 As such, the conventional interpretation fails to account for the appearance of peak A in single chains or in π-stacks with the electric field polarized along the chain direction.20 In what follows, we show that in chemically doped P3HT films, peak A is primarily of intrachain origin. The dopant anion leads to hole localization on a single chain, with peak A deriving oscillator strength from a mechanism akin to HT coupling. Interchain interactions contribute to peak A, but only in undoped films where the unbound holes, generated either optically or electrically, are delocalized over two or more chains.

2. IR SPECTRA OF POLARONS IN P3HT: AN OVERVIEW OF THE CURRENT STATUS There exists a vast collection of experimental literature describing the spectral features that arise in the IR absorption spectrum of doped P3HT.2−21 As exemplified by the spectra taken from ref 15 and displayed in Figure 1, the IR spectrum of

Figure 1. Measured IR spectra of sequentially-processed P3HT films cast from chloroform (blue curve), o-dichlorobenzene (green curve), and 100% RR P3HT (red curve) doped with 1 mg/mL F4TCNQ. Reprinted with permission from ref 15. Copyright 2017, John Wiley and Sons.

the dopant-induced hole is dominated by a featureless, broad peak, with a maximum occurring in the range of 0.35−0.60 eV, depending on sample morphology. As shown in ref 15, the peak position is sensitive to crystallinity, with the most redshifted spectrum belonging to the most ordered form of P3HT. The broad peak also figures prominently in undoped samples with holes induced either optically32−34 or electrically,31,35,36 where it has been referred to as P132−34 and C131,35−37 and has generally been attributed to intrachain absorption to the midgap polaron state which splits off the valence band due to hole-

3. MODEL HAMILTONIAN In our analysis of the mid-IR polaron absorption spectrum of P3HT π-stacks in the presence (or absence) of F4TCNQ dopants, we continue to utilize the Holstein molecular crystal Hamiltonian41 adapted to polarons in polymers in our previous works.38,39 The polymer backbone is taken to be along the xaxis, while the chains are π-stacked along the y-axis. A B


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The Journal of Physical Chemistry C simplified N × M π-stack consists of M chains, each containing N thiophene units, as depicted by the 4 × 3 stack shown in Figure 2. A thiophene unit in the two dimensional aggregate is

where we have set ℏ = 1. The first two terms in eq 2 account for the hole’s kinetic energy with tintra and tinter taken to be −0.40 and −0.15 eV, respectively, which is consistent with density functional theory (DFT) calculations of Northrup.44 The interchain value is also consistent with the results obtained in ref 45 assuming eclipsed OT chains. The remaining terms account for the local vibronic coupling involving the symmetric aromatic/quinoidal vibrational stretching mode with energy, ωvib, equal to 0.17 eV. The operators b†m,n (bm,n) create (annihilate) a vibrational quantum on the mth thiophene unit of the nth chain. The Huang Rhys (HR) factor, λ2, is a measure of the relaxation energy, λ2ωvib, experienced by a thiophene unit upon photoionization. The cationic HR factor is set to λ2 = 1, which is in agreement with calculated relaxation energies based on the coupled cluster methodology.46 The cationic HR factor is significantly less than the HR factor of 1.5−2 corresponding to the neutral (S1) excited state (in individual thiophene units) used in analyzing the optical properties derived from excitons.47 Energetic disorder in polymer films arises from variations in nearest neighbor π-stacking distances, changes in the intramolecular torsional angles, and the presence of spatially fluctuating electric fields. As in earlier works,38,39 disorder is represented as random changes in the hole’s energy, Δm,n, when located on the (m, n) thiophene unit, with

Figure 2. 4 × 3 P3HT π-stack showing inter- and intrachain interactions and the indexing scheme used in this work. C6 chains have been omitted for clarity.

indexed by (m, n), indicating the mth thiophene unit in the nth chain. The local highest occupied molecular orbital (HOMO) of each thiophene unit is electronically coupled to its neighboring thiophene HOMOs within a given polymer chain through the hole transfer integral tintra, allowing the missing electron (“hole”) to hop along the chain. Similarly, interchain coupling, which arises from wave function overlap between the HOMOs of neighboring thiophene units on adjacent chains, is governed by tinter. To account for the nuclear relaxation energy accompanying the formation of a hole on a thiophene unit, we consider coupling to the aromatic/ quinoidal stretching mode with energy 0.17 eV. This mode is responsible for the pronounced vibronic progression in the UV−vis absorption and emission spectrum in P3HT aggregates.42,43 The Hamiltonian also includes spatially uncorrelated (short-range) site-energy deviations due to structural/torsional disorder, chemical defects, inhomogeneous electric fields, and the presence of dopants. Within the subspace containing a single hole residing within a polymer πstack consisting of M, N-mers interacting coulombically with a dopant anion, the Hamiltonian can be written as H = H0 + Hdis + HC

N

Hdis =

where the deviations Δm,n are randomly selected from a Gaussian distribution with standard deviation σ P(Δm , n) =

N

(1)

HC = − ∑

∑ ∑ tinter{dm† ,n+ 1dm, n + h.c.} n=1 m=1 M N

∑ bm† ,nbm, n

n=1 m=1 M N

∑ {λ(bm† ,n + bm,n) + λ 2}dm† ,ndm,n

n=1 m=1

(4)

e2 4πε0ε|rm , n − ranion|

dm† , ndm , n

(5)

where rm,n is the position vector identifying the mth thiophene ring center on the nth chain, while ranion is the position of the dopant anion. In what follows, we assume a nearest neighbor separation of 0.40 nm between thiophene rings along a chain, and a separation between adjacent chains within a π-stack of 0.40 nm. Finally, in eq 5 ε0 is the vacuum permittivity, and ε is (unitless) relative permittivity of the polymer film. Similar to the case of neutral excitons, we represent the Hamiltonian in eq 1 in a basis set consisting of one- and twoparticle states. For holes, a single-particle state, denoted as |m, n , ṽ⟩, consists of a hole residing on the mth thiophene unit of the nth chain with ṽ vibrational quanta in its shifted cationic nuclear potential well, S+. The remaining MN − 1 units in the π-stack remain neutral with no excess vibrations in the S0 well. In a two-particle state, denoted as |m, n, ṽ; m′, n′, v′⟩, a hole exists at the thiophene unit (m, n) with ṽ vibrational quanta (ṽ ≥ 0) in its shifted S+ potential well, while the thiophene unit (m′, n′) remains neutral but is vibrationally excited with v′ ≥ 1

n=1 m=1 M−1 N

+ ωvib∑

M

∑

m=1 n=1

M N−1

+ ωvib∑

ij Δm , n 2 yz 1 zz expjjjj− z j 2σ 2 zz 2π σ k {

The final term in the Hamiltonian in eq 1 represents the Coulomb interaction between the hole and the dopant anion. In the simplest manifestation, the anion is considered to be a negative point charge. Treating the holes also as point charges (located at the thiophene ring centers), the attractive Coulomb coupling is given by

∑ ∑ tintra{dm† + 1,ndm,n + h.c.} +

(3)

m=1 n=1

The first term is the Holstein Hamiltonian, which accounts for the electronic/vibrational coupling within a π-stack, while the second and third terms account for site disorder and the Coulomb binding induced by the dopant anion, respectively. In terms of the operators d†m,n (dm,n) which create (annihilate) a hole on the mth thiophene unit of the nth chain, H0 takes the form H0 =

M

∑ ∑ Δm,ndm† ,ndm,n

(2) C


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The Journal of Physical Chemistry C

5. IR ABSORPTION IN SINGLE POLYMER CHAINS: THEORY Figure 3 shows the calculated IR spectrum corresponding to a disordered ensemble of isolated 10-mer chains, each hosting a

vibrational quanta in the unshifted S0 potential well. The remaining MN − 2 units are neutral with no vibrational excitations. In all our calculations, we cap the total number of vibrations to four and neglect three- and higher-particle states, approximations which manage to yield converged IR spectral line shapes.38 The basis set so described has been very successful in reproducing the mid-IR line shapes of unbound holes as measured by Chang et al.36 and Ö sterbacka et al.32 as detailed in refs 38 and 39.

4. IR ABSORPTION AND HOLE COHERENCE The polarized absorption spectrum, Aj(ω), for a disordered polymer π-stack is evaluated using

∑ f jex WLS[ℏω − (Eex − EG)]

Aj (ω) =

ex

c

(6)

where the index j (=x, y) indicates the polarization direction, with x lying along a polymer chain axis and y lying along the aggregate (interchain) axis. The sum in eq 6 is over all the polaron excited states, and ⟨...⟩c represents an average over many site-energy disorder configurations. W LS is the homogenous line shape function taken here to be a narrow Gaussian with a standard deviation Γ = 0.03 eV, an order of magnitude smaller than the assumed inhomogeneous width, σ = 0.3 eV. The oscillator strength, f ex j in eq 6 pertaining to the jpolarized transition from the polaron ground state, ψG to the higher excited state, ψex is given by 2me

f jex =

3e 2ℏ2

(Eex − EG)|⟨ψG|μĵ |ψex ⟩|2

Figure 3. Calculated IR spectrum of a disordered anion/10-mer complex for various distances between the anion and the chain midpoint. The parameters defining the Hamiltonian in eq 1 are tintra = −0.4 eV, ωvib = 0.17 eV, λ2 = 1, and ε = 1. Additionally, the standard deviation of the Gaussian disorder distribution in eq 4 is σ = 0.3 eV, and the homogeneous line width is set to Γ = 0.03 eV. The full-width at half maximum (fwhm) corresponding to the homogeneous and inhomogeneous broadening are indicated by the horizontal bars. 103 configurations were averaged together to create the spectra.

single hole, with a point-charge anion placed alongside each chain at a distance danion from the chain midpoint (see figure inset). Spectra are shown for several values of danion. The simulations are based on the Hamiltonian in eq 1 and the disorder distribution function in eq 4, characterized by a standard deviation, σ = 0.3 eV. In the figure, the dielectric constant is set to unity so that the Coulomb binding between charges is unscreened. When large π-stacked lattices are considered, the screening due to the induced polarization on each thiophene unit can be approximately included by shielding the Coulomb attraction using a dielectric constant ε closer to three, the bulk value for P3HT.2 More sophisticated methods involve solving for the induced dipole at each site selfconsistently, as described by D’Avino et al.48 Considering the size of the F4TCNQ molecule, the point charge approximation is most accurate for danion values greater than approximately 0.4 nm. The lateral extent of a realistic anion prompted us to also investigate the limit where the point-anion is placed directly across a given thiophene unit (as opposed to the midpoint between two thiophene units). We found that if danion is greater than approximately 0.5 nm, there is little change from the calculated “midpoint” IR spectrum (see the Supporting Information, and in particular Figure S4 for additional details and discussion). Importantly, this regime includes the experimentally relevant regime, where peak B is broadest (see, e.g., the orange curve in Figure 3). In Figure 3, the spectrum for danion = ∞ corresponds to unbound holes, which become increasingly localized as danion decreases. Hence, Figure 3 compares the signatures of dopantinduced (bound) polarons to those of field-induced or optically induced (unbound) polarons. The spectra generally consist of a low-energy narrow peak (A) followed by a broader, higher-energy peak B. Peak B resides in the spectral region associated with conventional polaron absorption in single chains, that is, the peak P1 in refs 32 and 33 and the peak C1 in

(7)

where Eex and EG are the excited and ground-state energies, respectively, and μ̂ = e∑ rm , ndm† , ndm , n (8)

m,n

is the vector dipole moment operator. In what follows, we also evaluate the two-dimensional coherence function, C(r), describing hole delocalization in the ground state39 ⟨ψG|∑ dR† dR + r |ψG⟩

C(r ) =

R

c

(9)

where R = (m, n) is a dimensionless position vector. When r = (0, 0), the coherence function, C(r), reduces to unity because ∑Rd†RdR is the number operator for holes and we assume only a single hole. The total number of coherently connected thiophene units, Ncoh, is obtained from the coherence function by evaluating the sum Ncoh =

∑ |C(r )| r

(10)

To obtain the coherence number along the polymer chain direction, Nintra, one needs to restrict the sum in eq 10 to position vectors r lying along the chain direction. Similarly, Ninter is evaluated by restricting the sum to vectors lying along the interchain (stack) axis. D


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The Journal of Physical Chemistry C refs,31,35,37 whereas peak A resides in the spectral region normally associated with delocalized polaron derived from interchain interactions, that is, the peak DP1 in refs 32 and 33 and the peak CT in refs.31,35,37 The spectra in Figure 3 with peak B centered in the range of 0.4−0.6 eV strongly resemble the measured line shapes in refs 8 and 15, see Figure 1. Consistent with experiment as well as our previous simulations,38,39 the transition A peaks at an energy slightly lower than a vibrational quantum of the aromatic-quinoidal stretching mode with ωvib ≈ 0.17 eV. Interestingly, peak A is unusually narrow; its line width, defined as the fwhm, is a full order of magnitude smaller than the fwhm corresponding to disorder-induced inhomogeneous broadening

broader than A with a peak energy near 0.3 eV. At danion = 1 nm, the peak remains broad, with an fwhm approximately equal to two-thirds of the inhomogeneous line width, σfwhm. As danion is further reduced, the line width narrows, converging to a value only slightly larger than the homogeneous line width, Γfwhm. Curiously, as danion approaches zero, the energy of peak B converges to 0.85 eV, which is slightly larger than 2|tintra| = 0.8 eV (see Figure 4a). Figure 5 shows how the coherence number, Ncoh, in a 10mer chain responds to changes in danion. When the anion is very

σfwhm = 2 2 ln 2 σ ≈ 0.7 eV

As can be appreciated from Figure 3, the width of peak A is only slightly larger than the homogeneous line width, Γfwhm = 2 2 ln 2 Γ = 0.074 eV. The figure further shows that as the anion is brought closer to the chain, peak A slightly blue shifts, narrows further, and strongly attenuates. Figure 4 summarizes how the position and the line width of peak A change with danion. The peak position tends toward ωvib and the line width tends toward Γfwhm with diminishing danion. In contrast, the peak position and line width of transition B is far more sensitive to the proximity of the anion, strongly blue-shifting and narrowing as danion is reduced. When danion is very large so that the anion is effectively absent, peak B is much

Figure 5. Calculated coherence number as a function of danion for a chain containing 10 thiophene units. The parameters defining the chain/anion Hamiltonian are tintra = −0.4 eV, ωvib = 0.17 eV, λ2 = 1, and ε = 1. Additionally, the standard deviation of the Gaussian disorder distribution is σ = 0.3 eV, and the homogeneous line width is set to Γ = 0.03 eV. 103 configurations were averaged together to determine Ncoh.

close to the chain, the coherence number is minimized (Ncoh ≈ 2), mainly because of the localizing effects associated with Coulomb binding, which is consistent with the findings in refs 8 and 15. As danion increases, the coherence number expands to approximately 4.5 but is still significantly smaller than the maximum value of ≈10 because of the additional localizing effects caused by diagonal disorder. When the anion is very close to the chain, Ncoh converges to the value of two because of the energetic isolation of a degenerate pair of dimer levels as indicated in Figure 6. This arises because the anion is equidistant from the two thiophene

Figure 4. (a) Frequency corresponding to the absorption maximum in the A and B transitions as a function of danion. (b) Line widths (fwhms) of A and B as a function of danion. The fwhm of peak A is evaluated by doubling the half width at half maximum obtained from the red side of the line. The peaks A and B begin to merge after danion = 1 nm so that the fwhm of B becomes ill-defined.

Figure 6. Impact of Coulomb binding on the energies of nearby thiophene units for the anion symmetrically situated between the two thiophene units I and II. Here, VI = VII = V(danion,a/2) and Va = Vb = V(danion,3a/2). E


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The Journal of Physical Chemistry C units in the middle of the chainunits 5 and 6 in the case of a 10-mer. The isolated units, labeled I and II in Figure 6, are each a distance a/2 from the chain midpoint, where a is the spacing between adjacent thiophene units along a chain (a ≈ 0.4 nm in P3HT). Hence, the central pair of thiophene units are energetically isolated from the remaining units whenever V (danion , 3a /2) − V (danion , a /2) ≫ |t intra|

(11)

where the Coulomb energy for the thiophene unit at a distance, aM, from the chain midpoint is given by V (danion , aM) =

−e 2 4πε0ε danion 2 + aM 2

(12)

The relative Coulomb energies for the various thiophene units are depicted in Figure 6. Figure S8 shows that the 10-mer with danion = 0 from Figure 3 is spectrally equivalent to a physical dimer with the same degree of site disorder (but with no associated anion). In the case when the anion is directly across from a given thiophene unit, only that particular unit is energetically isolated, as demonstrated in Figure S3. For this geometry, Ncoh converges to unity with decreasing danion. However, as shown in Figure S4, substantial changes in the mid-IR spectrum as the anion slides away from the midpoint are only important for very small and physically irrelevant values for danion.

Figure 7. Energy-level diagram in the limiting case of very small danion values displaying the first-order corrected energies. Note the total cancellation of disorder in both the A and B transitions.

the order of the antisymmetric and symmetric states, but the resulting absorption spectrum remains unchanged, that is, the IR spectrum is invariant to the sign of the electronic coupling, in contrast to the case of Frenkel excitons. The appearance of the low-energy peak A and the unusual sensitivity of peaks A and B to disorder displayed in Figure 3 can be understood by treating the (electronic) off-diagonal parts of the polaron-vibrational coupling as well as the inhomogeneous site disorder perturbatively. One first needs to express the Hamiltonian in eq 1 in terms of the symmetryadapted electronic states in eqs 13 and 14 as well as the symmetric and antisymmetric vibrational modes through the creation operators, b†S ≡ 2−1/2(b†I + b†II) and b†AS ≡ 2−1/2(b†I − b†II), respectively. The result appears in eq S1. In the strong polaron coupling limit, defined by

6. ORIGIN OF THE A AND B POLARON PEAKS To better appreciate the unusual structure of the IR spectrum in Figure 3, we turn to the limit of small danion values, where an energetically isolated (symmetric) dimer pair is created via condition 11 and treat vibronic coupling and disorder perturbatively. Although very small values of danion are unphysical, the symmetric dimer is the simplest geometry which supports the coexistence of both peaks A and B and demonstrates their disparate behaviors. Later in this section, we consider the more physically relevant models which consider larger chain segments, which are relevant for increased values of danion. As indicated in Figure 6, the isolated thiophene units (I and II) are each lowered in energy by the Coulomb interaction, VI, so that the resonant interaction through the intrachain hole transfer integral tintra is maintained. Taking tintra < 0 and neglecting vibronic coupling and disorder in the Hamiltonian of eq 1 results in a symmetric ground state 1 | S⟩ = {|I⟩ + |II⟩} (13) 2

2|t intra| ≫ σ , ωvibλ 2

the vibronic and disorder terms, Hv and HΔ, defined in eqs S10 and S11, which connect symmetric and antisymmetric electronic states, can be treated perturbatively. The zeroorder electronic/vibrational ground state is then given by |G⟩(0) = |S; 0S̃ , 0AS⟩

(16)

where 0̃S and 0AS indicate that both vibrational modes have no quanta. The overstrike bar in the symmetric case indicates an excited-state nuclear potential shifted by λ / 2 . The zeroorder excited states which are relevant to the A and B transitions include the vibrationally excited ground state

with energy VI − |tintra|, and an excited, antisymmetric state 1 |AS⟩ ≡ {|I⟩ − |II⟩ 2

(15)

|A⟩(0) = |S ; 0S̃ , 1AS ⟩

(17)

and the electronically excited state, |B⟩

(14)

|B⟩(0) = |AS; 0S̃ , 0AS⟩

with energy VI + |tintra|. The IR transition between the ground state |G⟩ (=|S⟩), and the electronic excited state |B⟩ (=|AS⟩) is dipole-allowed via the matrix element, ⟨G|μ̃ |B⟩ = ea/2, with transition energy equal to 2|tintra|. Hence, without disorder or vibronic coupling, the associated IR spectrum consists of a single peak with energy 2|tintra|, as indicated by the blue-colored transition in Figure 7, which agrees well with the position of peak B in Figure 3 when danion is less than approximately 0.4 nm. Interestingly, although we have chosen tintra to be negative, a positive value makes no differenceit merely interchanges

(0)

(18)

Figure 7 displays the corresponding zero-order energy levels which are also correct to first order in both disorder and polaron-vibrational coupling because the perturbing Hamiltonians in eqs S10 and S11 are off-diagonal in the electronic basis. The transition to state |A⟩ occurs at an energy ωAG, which is approximately equal to one vibrational quantum, while the transition to state |B⟩ occurs at ωBG ≈ 2|tintra|. The peak positions are in rough agreement with the A and B peaks in Figure 3 in the limit when danion is less than about 0.4 nm. F


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The Journal of Physical Chemistry C The transition, |G⟩ → |A⟩, which is much weaker than the |G⟩ → |B⟩ transition, is allowed by a form of HT coupling where the same-symmetry states in eqs 17 and 18 are mixed together by the vibronic perturbation in eq S10, thereby allowing the A transition to borrow oscillator strength from the B transition. There is also a contribution arising from the mixing of the ground state |G⟩ with |AS; 0̃S, 1AS⟩. Complete details can be found in the Supporting Information. All considered, the square of the transition dipole moment is obtained using state |A⟩ corrected to first order giving (see eq S16) ÄÅ 2 ÉÑ 2 ÅÅ σ ÑÑ 4λωvibt intra e 2a 2 2 ÑÑ ÅÅ O |⟨G|μ̂ |A⟩| = + Ñ ÅÅ ÅÅÅ t intra 2 ÑÑÑÑ (19) 8 (4t intra 2 − ωvib 2) Ö Ç

second-order correction arising from the difference in the energies at sites I and II. The resulting heightened sensitivity of peak B versus peak A to disorder is apparent in Figure 4b. Moreover, eqs 22 and 23 show that both A and B transitions contain a second-order correction arising from the polaronvibrational coupling which is equal in magnitude but opposite in sign for the two transitions. The correction neatly explains the small increase in the transition frequency of peak B above 2|tintra| by ≈0.04 eV observed in Figure 4a as danion approaches zero, and the small decrease in the transition frequency of peak A below ωvib by ≈0.04 eV in the same limit. Finally, to demonstrate the validity of the perturbation theory, Figure S8 compares the numerically evaluated spectrum for the physical dimer (no disorder) with the spectrum obtained using perturbation theory. 6.1. Beyond the Dimer. As the anion is further separated from the polymer chain and the Coulomb well depth decreases, the polaron becomes less bound and is delocalized over an increasing number of thiophene units. As Figure 3 shows, delocalization is characterized mainly by a diminishing peak B transition frequency and a strongly-increasing peak A oscillator strength. Both signatures reflect the increasing resonance among thiophene units, which surprisingly occurs even in the presence of considerable site disorder. The spectral behavior can be qualitatively appreciated by evaluating the IR spectrum for a series of well-defined N-mers with no anion present. Figure 8 shows the results for N increasing from 2 to 10. In the figure, the disorder width σ is taken to be the same as that used in Figure 3, that is, σ = 0.3 eV.

Equation 19 shows that under the strong polaron coupling condition 15, peak A is vibronic in origin, with a line strength increasing with the HR factor, but decreasing with |tintra|. For comparison, the square of the |G⟩ → |B⟩ transition dipole moment corrected to first order is much larger (see eq S17) ÄÅ 2 ÉÑ 2 ÅÅ σ ÑÑ λ 2ωvib 2 e 2a 2 1 2 ÑÑ ÅÅ 1− O |⟨G|μ̂ |B⟩| = + Ñ ÅÅ ÅÅÅ t intra 2 ÑÑÑÑ 4 2 (4t intra 2 − ωvib 2) Ö Ç (20) 2 2

Equation 20 reduces to our earlier result, e a /4, in the absence of disorder and polaron-vibrational coupling. Using the parameters appropriate for P3HT (|tintra| = 0.4 eV, ωvib = 0.17 eV and λ2 = 1), the ratio of the |G⟩ → |A⟩ and |G⟩ → |B⟩ oscillator strengths is ωAG |⟨G|μ̂ |A⟩|2 ≈ 0.02 ωBG |⟨G|μ̂ |B⟩|2

2|t intra| ≫ σ , ωvibλ 2

(21)

which is consistent with the dominance of peak B in the spectra in Figure 3 in the limit of small danion (i.e., the dimer limit). The unusual sensitivities of the A and B transitions to disorder can also be readily accounted for in the dimer and strong coupling limits defined by conditions 11 and 15, respectively. Recall from Figure 4 that peak A is hardly affected by disorder, independent of danion, while peak B is insensitive only when danion is very close to the polymer chain. As can be appreciated from Figure 7, the disorder terms corrected to first order exactly cancel out in both the |G⟩ → |A⟩ and |G⟩ → |B⟩ transition energies, thereby accounting for the narrow bandwidths of both peaks A and B in Figure 3 when danion is smaller than approximately 0.4 nm. The enhanced sensitivity of peak B versus peak A to disorder is revealed only when incorporating the second-order corrections. Expanding the transition energies ωAG and ωBG to second order in both perturbations (disorder and polaron-vibrational coupling) gives ωAG ≈ ωvib − 2

Figure 8. Calculated IR spectrum of a hole delocalized within various disordered N-mers without the presence of an anion. The parameters defining the chain/anion Hamiltonian are tintra = −0.4 eV, ωvib = 0.17 eV, λ2 = 1, and ε = 1. Additionally, the standard deviation of the Gaussian disorder distribution in eq 4 is σ = 0.3 eV, and the homogeneous line width is set to Γ = 0.03 eV. 103 configurations were averaged together to create the spectra.

Figure 8 shows that as the physical chain size varies from N = 2 to 10, the coherence number (reported in the figure inset) varies only from 1.8 to 4.6 because of the localizing influence of disorder. As anticipated, the sizeable red-shift of peak B and the enhancement of the oscillator strength of peak A with increasing N roughly parallels the behavior previously observed with increasing danion in Figure 3. By N = 8, the spectrum is already converged and is essentially identical to the 10-mer spectrum in Figure 3 with danion = infinity. In the sizeconverged spectrum, the maximum of peak B is red-shifted from the dimer value (N = 2) by about 0.6 eV, whereas the

λ 2ωvib 2|t intra| 4t intra 2 − ωvib 2

(22)

and ωBG ≈ 2t intra + 2

λ 2ωvib 2|t intra| 4t intra 2 − ωvib 2

+

(ΔI − ΔII)2 4t intra

(23)

The equations show that transition A entirely lacks a secondorder correction from disorder, while transition B contains a G


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The Journal of Physical Chemistry C oscillator strength of peak A undergoes an ≈15-fold increase, comparable to the increase observed in Figure 3, as danion varies from zero to infinity. The red shift of peak B and the increase in the oscillator strength of peak A can be understood as follows. For N-mers larger than a dimer, the zeroth-order G → B transition corresponds to an excitation from a delocalized ground state polaron with wave vector k = π/(N + 1) (i.e., |G⟩ = |k = π/(N + 1)⟩) to an excited polaron with wave vector k = 2π/(N + 1). Here, we have maintained tintra < 0 and invoked open boundary conditions; see ref 38 for details. The corresponding Ndependent transition frequency, ω(0) BG(N), which is corrected to zeroth order in the vibronic coupling and disorder, is given by38

{

(0) ωBG (N ) = 2|t intra| cos

π 2π − cos N+1 N+1

Subsequent works by Pochas and Spano38 and Ghosh et al.39 employing a theory based on the Holstein Hamiltonian, however, showed that peak A can also carry a significant intrachain component, which, as shown in the prior section, derives from a HT mechanism operative in single polymer chains. The theory is consistent with recent measurements on oriented P3HT films doped with F4TCNQ, where the polarization of peak A was shown to mainly oriented along the chain.20 To appreciate how the IR spectrum is modified in the presence of interchain interactions, we recalculated the IR spectrum of Figure 3 (with no anion) but in a π-stack consisting of five, 10-mer chains, and resolved the absorption in both intra- and interchain directions. The result is shown in Figure 9a. For the calculated spectra, we maintained a spatially

}

(0) ωBG ≫ λ 2ωvib , σ

(24)

where the inequality defines the strong coupling limit. Equation 24 shows that ω(0) BG(N) diminishes with increasing N, making it increasingly difficult to justify strong coupling; for a tetramer (N = 4), ω(0) BG is exactly one-half of the value for the dimer, that is, ω(0) BG(4) = |tintra|. When N ≫ 1, the transition frequency vanishes as 3π2|tintra|/(N + 1)2. For polarons defined by a coherence number Ncoh < N (due to a nearby anion and/ or the presence of disorder), ω(0) BG can be approximately obtained by replacing N with Ncoh in eq 24, that is, ω(0) BG(Ncoh). For example, the converged value for the peak frequency of B in Figure 8 (N = 10 spectrum) agrees well with the value ω(0) BG(4.6) = 0.33 eV, using the coherence number reported in the inset. Furthermore, the associated increase in the oscillator strength of peak A arises from the scaling ωAG⟨A|μ̂ |G⟩2 ∼ Ncoh

ωvib3λ 2 (0) [ωBG (Ncoh)]2

(0) ωBG ≫ λ 2ωvib

(25)

where the Ncoh-prefactor arises from a combination of a quadratic scaling of ⟨k = 2π/(Ncoh + 1)|μ̃ |G⟩2 with Ncoh together with an inverse Ncoh dependence derived from the polaron-vibrational coupling. The approximately 15-fold enhancement of peak A in going from N = 2 to N = 10 in Figure 8 follows from eq 25 after inserting the 2.5-fold increase in Ncoh (Ncoh ranges from 1.8 to 4.6) and an almost three-fold reduction in ω(0) BG, from 0.85 eV when N = 2 to 0.3 eV when N = 10. Note that eq 25 remains zeroth order with respect to the vibronic perturbation. For P3HT with |tintra| = 0.4−0.5 eV, the strong polaron coupling condition is satisfied for Ncoh ≲ 5; when Ncoh = 4, ωBG is approximately twice as large as ωvib (≈0.17 eV).

Figure 9. Calculated IR spectrum for an (a) unbound hole (no anion) and a (b) bound hole in a disordered lattice consisting of 5, 10-mers using the Hamiltonian in eq 1 with tintra = −0.4 eV, tinter = −0.15 eV, ωvib = 0.17 eV, λ2 = 1, and ε = 1. Additionally, the standard deviation of the Gaussian disorder distribution in eq 4 is σ = 0.3 eV, and the homogeneous line width is set to Γ = 0.03 eV. 103 configurations were averaged together to create the spectra.

7. INTERCHAIN EFFECTS In this section, we briefly consider how interchain interactions impact the IR spectrum, in particular, in the low-energy region occupied by peak A. In the seminal works of Ö sterbacka et al.32 and Sirringhaus,35 interchain interactions and the subsequent delocalization of the polaron over two or more chains within the P3HT stack were believed to be solely responsible for the mid-IR peak (A). Although the polarization of transition A could not be experimentally determined (as the P3HT π-stacks were not oriented), the conventional polaron theory predicts the peak to be polarized in the interchain direction.37

uncorrelated disorder distribution along both axes (the “SS” model in refs 38 and 39) and utilized an interchain interaction, tinter = −0.15 eV, the value derived by Northrup using DFT44 and consistent with the values supported by Ö sterbacka et al.32 As can be immediately appreciated from the figure, the intrachain component (blue) is significantly larger than the interchain component (red). The polarized components of peak A also appear at roughly the same frequency; a more detailed analysis38,39 shows that in the absence of disorder, interchain and intrachain transitions occur to distinct polaron excited states, which are subsequently mixed by disorder to H


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The Journal of Physical Chemistry C create peak A.38,39 As also shown in refs 38 and 39, the integrated band intensity of the inter- or intrachain component is approximately proportional to the corresponding coherence number. Hence, the interchain component can be enhanced by allowing the polaron coherence to expand along the interchain direction, for example, by reducing disorder along the interchain direction. Indeed, in the so-called “SL” disorder distribution, where disorder is assumed to be short-range (S) along the polymer chain, but long-range (L) between chains, the interchain component dominates peak A.39 As shown in Figure 9b, the additional localization induced by the presence of a nearby dopant anion displaced along the stacking (b) axis leads to a sharp reduction in peak A relative to peak B, and a dominance of the intrachain component, as the polaron is essentially localized on a single chain.8,15 Figure S7 shows that placing the anion in the lamellar region gives similar results to the “side-on” geometry of Figure 9b for similar values of danion. To summarize, the polaron theory outlined here predicts the presence of two main peaks in the IR spectrum for bound or unbound holes in single and π-stacked conjugated polymer chains; the broad, higher-energy peak B is associated with the “classical” polaron peak (P1), while the narrower, lower-energy peak A arises from HT coupling in the limit of strong polaron coupling, with an intensity that sharply increases with the coherence number. Interchain interactions in π-stacks give rise to a vibronic peak polarized along the stacking axis which spectrally overlaps the intrachain peak. In any case, the intensity of peak A is a direct probe of the exciton coherence number, showing that analysis of the IR spectrum allows one to determine the polaron coherence lengths along the interand intrachain directions.

Figure 10. Measured IR absorption spectrum for F4TCNQ-doped P3HT films with an average polymer Mn of 3.6 kg/mol alongside the theoretical spectrum using a dielectric constant of ε = 1 in (a) and ε = 3 in (b). The anion is situated at a distance danion from the midpoint of the first chain in a stack of five chains as in Figure 9b, each chain containing 10 thiophene units. The anion distances are shown in the insets. In both cases, two distances were needed to simulate the experimental spectrum. Both simulations use tintra = −0.4 eV, tinter = −0.15 eV, ωvib = 0.174 eV, and λ2 = 1. Site disorder was distributed using σ = 0.3 eV, and 103 configurations were averaged together to create the spectrum. The homogeneous line width is set to Γ = 0.03 eV.

8. COMPARISON TO THE EXPERIMENT To test our model, P3HT samples of variable MW but with 100% regioregularity were synthesized according to a literature method, as described briefly in the Supporting Information.49 The measured IR spectrum for F4TCNQ-doped regioregular P3HT films with an average MW of 3.6 kg/mol is shown in Figure 10. The films were created via sequential processing by doping neat P3HT films with a 0.01 mg/mL solution of F4TCNQ in acetonitrile as described elsewhere.8,16 The peaks A and B are clearly discernible along with a cluster of very narrow IRAV peaks in the vicinity of peak A. In conducting the simulations for comparison to the experimental spectra, we considered an anion in association with an entire π-stack, displaced along the stacking (b)-axis as indicated in the inset of Figure 9b. (Figure S7 shows that similar IR spectra are obtained when the anion is in the lamellar region.) The effect of screening on the dopant/hole interaction was included via the dielectric constant. Increasing the dielectric constant from the value of unity (used up to now) to the bulk value of three, typical of nonpolar organic solids,2 reduces the Coulomb well depth and therefore decreases the distance, danion, needed to simulate the measured spectrum. In ref 15, a more accurate spatially dependent dielectric constant was used to interpolate between the vacuum value applicable for small danion to the bulk value at larger distances. To demonstrate the effect of screening in this work, we show in Figure 10 two simulations, one using ε = 1 (10a) and the other using ε = 3 (10b). In both cases, the anion was placed on the “side” of a π-stack consisting of five, 10-mer chains, laterally situated between the 5th and 6th thiophene units. However, as shown in Figure S5,

averaging over the lateral displacements has only a minor effect on the spectral line shape and coherence numbers. To properly reproduce the spectral details in the regions near 0.3 and 0.7 eV, it was necessary to assume a bimodal distribution of anion distances as indicated in the inset of Figure 10. Overall, all of the salient features present in the measured spectrum are captured by the theory except for the narrow IRAVs. Such resonances have been considered by others33 but were not included in our analysis. Although both simulated spectra in Figure 10a,b agree well with experiment, there is a substantial decrease in the danion values in the spectrum with the higher dielectric constant, as expected. The maximum absorption in the measured spectrum near 0.7 eV (peak B) is indicative of anions in relatively close proximity to the P3HT stacks. Interestingly, the anion distance required for the ε = 3 simulation (Figure 10b) is unphysically small, only 0.2 nm. Even in the most intimate arrangement in which the anion is situated within a π-stack, danion would be approximately 0.4 nm, twice the value needed to be in agreement with the experiment using ε = 3. The value danion = 0.5 nm needed to reproduce the main absorption peak for ε = 1 (Figure 10a) is more realistic and more in line with the simulations of ref 15 where the spatially dependent dielectric I


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The Journal of Physical Chemistry C constant is biased toward ε = 1 for danion less than approximately 0.5 nm. More importantly, because the spectral features such as the position of the peak maximum (B) and the relative oscillator strength of peak A are directly correlated to the polaron coherence lengths, we expect the latter to be very similar in the simulations in Figure 10a,b, despite the difference in danion. To show that this is indeed the case, the coherence numbers for both cases are reported in Table 1. Independent Table 1. Coherence Numbers Corresponding to the Simulated Spectra in Figures 10 and 11 Nintra

3.6 kg/mol (ε = 1) danion = 0.50 nm danion = 0.85 nm (ε = 3) danion = 0.20 nm danion = 0.38 nm 30 kg/mol Nintra (avg) ε=1 ε=3

4.04 3.92

Ninter

Ncoh

3.11 1.27 4.27 1.64 3.11 1.34 4.06 1.75 Ninter (avg)

4.09 7.60 4.44 7.94 (avg)

1.62 1.69

Ncoh

7.35 7.63

of the dielectric constant, the coherence number along the chain is 3, while across chains the coherence is much smaller, about 1.3, for the conformation with the closest anion, while the coherence numbers are larger (Nintra ≈ 4, Ninter ≈ 1.7) for the conformation with the more distant anion. Hence, in chemically doped P3HT samples with relatively small MW polymers, the polaron is mainly confined to a single chain. This was also the conclusion reached in refs 8 and 15. Interestingly, the high-MW polymers (>30 kg/mol) in ref 8 were simulated using coherence lengths similar to those corresponding to the red-shifted component of the low MW polymers in Figure 10. As shown in Figure 11, increasing the MW to 30 kg/mol has a dramatic effect on the IR line shape. Peak B red-shifts by ≈0.2 eV to a maximum at ≈0.5 eV, while the oscillator strength of peak A increases several-fold. The simulated spectra also rely on (at least) two anion distances, but unlike the case for the lower MW polymers, the two values are more distant, and relatively closer together in value, suggesting a distribution of dopant/polymer distances rather than two distinct polymorphs. A more sophisticated calculation would therefore require an average over a distribution of anion distances. As determined in the previous section, a red shift in peak B and an increase in the oscillator strength of peak A are both signatures of an expanded polaron coherence length, arising from a more distant anion and a shallower Coulomb binding well. This is consistent with the simulations in Figure 11, where, on average, the anion is nearly twice the distance from the nearest polymer chain as compared to the lower MW simulations. Table 1 shows the coherence numbers, obtained by averaging the two values corresponding to the slightly different danion values. The number of coherently coupled thiophene units along a chain (Nintra) is approximately four, up from the value of three for the closest anion configuration in the 3.6 kg/mol films, but in good agreement with Nintra for the more distant anion configuration in the 3.6 kg/mol films. Similar agreement exists for the interchain coherence numbers; the average Ninter for the 30 kg/mol films (1.6) is significantly larger than Ninter for the closest anion configuration in the 3.6 kg/mol films (1.3) but agrees well with the more distant anion value. The analysis suggests that the 30 kg/mol films are simply devoid of the polymorph supporting the closest anion configuration.

Figure 11. Measured IR absorption spectrum for F4TCNQ-doped P3HT films with an average polymer MW of 30 kg/mol alongside the theoretical spectrum using a dielectric constant of ε = 1 in (a) and ε = 3 in (b). The anion is situated at a distance danion from the midpoint of the first chain in a stack of five chains as in Figure 9b, each chain containing 10 thiophene units. The anion distances are shown in the insets. In both cases, two distances were needed to simulate the experimental spectrum. Both simulations use tintra = −0.4 eV, tinter = −0.15 eV, ωvib = 0.174 eV, and λ2 = 1. Site disorder was distributed using σ = 0.3 eV, and 103 configurations were averaged together to create the spectrum. The homogeneous line width is set to Γ = 0.03 eV.

The crystalline structure of P3HT depends on MW,13 with the largest structural differences being observed at the lower MWs where chain-end effects are more apparent. Different crystalline structures are likely to exhibit different chain/anion distances for the F4TCNQ anion. The quantitative theoretical treatment of the IR absorption of the dopant-induced polarons presented above allows us to detect the existence of different dopant locations near the P3HT chain, which can be dopantpolymer polymorphs. Importantly, our analysis shows that dopant-induced charges are more delocalized in high molecular-weight P3HT, where the enhanced order of the polymer keeps the anion further away from the conjugated backbone. This is consistent with the conclusions of ref 15, where the increasing red shift of peak B (and increasing oscillator strength of peak A) with increasing polymer order shown in Figure 1 was attributed mainly to increasing danion values, a scenario supported by GWAXS measurements. F4TCNQ anions were believed to be incorporated into the lamellar and amorphous regions, which is consistent with the findings of other works.16,20 J


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9. CONCLUSIONS In this paper, we developed a theory for the IR spectral line shape of positively charged polarons (holes) in semiconducting conjugated polymers. The theory is robust, providing a quantitative description of the IR line shape for chemically induced (“bound”) holes as well as photoinduced and electrically induced (“unbound”) holes in P3HT. Spectral signatures are identified which can be used to extract the polaron coherence lengths along the chain and stacking directions.38,39 The theory, based on a Holstein-like Hamiltonian including site disorder and the Coulomb binding between the dopant and the hole, treats the nuclear kinetic energy derived from the aromatic-quinoidal stretching mode exactly, thereby differing from conventional Born−Oppenheimer descriptions.29,40 In particular, the theory naturally accounts for the appearance of a low-energy peak (A) occurring at a frequency slightly lower than the quinoidalaromatic vibrational energy (ωvib ≈ 0.18 eV) both in π-stacks as well as isolated chains, as well as a much broader and higherenergy peak B. In the strong polaron coupling regime, ω(0) BG ≫ λ2ωvib, peak A arises from HT coupling, with oscillator strength borrowed from peak B. In P3HT, strong coupling is justified when the number of coherently connected thiophene units, Nintra, is smaller than five. Localization to four or fewer units is driven mainly by the presence of substantial site disorder; in doped films, further localization results from Coulomb binding to the dopant anion. Dominant disorder-induced localization in conjugated polymers (compared with localization due to self-trapping) is in line with the results of Barford and coworkers.50,51 Were it not for disorder- or dopant-induced localization, the polaron coherence would encompass the entire chain length (as well as all chains within a π-stack); in such an unphysical limit, increasing chain length (or number of chains) results in a spectrum dominated by a single peak which increasingly red-shifts below ωvib.38,39 The theoretically evaluated absorption spectra compare very favorably to the measured spectra in F4TCNQ-doped P3HT films with variable MW; increasing the MW from 3.6 to 30 kg/ mol leads to a marked red shift of the peak (B) maximum along with a substantial increase in the oscillator strength of peak A, both characteristics of increasing polaron delocalization. Such trends with increasing MW have also been observed for unbound holes generated electrically in P3HT films by Chang et al.31 and accounted for in the theory presented in refs 38 and 39. Extending the theory to treat bound holes8,15 shows that the low-MW films (3.6 kg/mol) allow for a more intimate contact between the dopants and polymer chains. Moreover, the presence of two distinct “B” peaks in the IR line shape indicates a bimodal distribution of anion distances with a roughly 3 to 1 ratio of close contacts (≲0.5 nm) to more distant ones (≲0.9 nm). The bimodal distribution in the 3.6 kg/mol films may indicate the coexistence of the two known polymorphs of P3HT, referred to as form I and form II.52 The different packing of the alkyl chains in the lamellar regions in these two forms leads to significantly different lamellar spacings (12 and 16 Å)52 which may be responsible for the bimodal anion distribution (assuming that the anion is situated in the lamellar regionsee ref 20 and Figure S7), but further work needs to be conducted to clarify this. Interestingly, the coexistence of the two polymorphs has been found to be more likely in P3HT films with lower MWs.53 The spectral features in the films consisting of longer P3HT chains (30 kg/mol)

indicate, on average, more distant anion/P3HT contacts, which appears to be consistent with the elimination of the close-anion phase found in the lower MW films. In contrast to the conventional polaron theory, the theory presented herein shows that peak A in chemically doped samples is not related to a self-trapped, mid-gap polaron state derived solely from interchain coupling. In the presence of the dopant anion, holes are mainly localized on single chains where peak A is polarized along the chain. For unbound holes generated optically or electrically, interchain interactions are responsible for a second transition polarized in the stacking direction with an energy similar to that of the intrachain peak A.38,39 Peak A is therefore composed of spectrally overlapping intrachain and interchain components mixed by disorder. Peak A is enhanced and red-shifted as the polaron is delocalized in either direction, providing a direct measure of the corresponding coherence length. The overall picture presented here is consistent with the experiment: peak A is much weaker than peak B, occurring at approximately 0.15 eV in F4TCNQ-doped P3HT samples,8,15 but becomes the dominant peak with a lower energy in undoped samples, as revealed, for example, in the transient absorption measurements of Ö sterbacka et al.,32 where peak A occurs at approximately 0.08 eV (≈ωvib/2). These spectral changes correspond to a total polaron coherence number Ncoh (roughly equal to the product of Ninter and Nintra) of approximately 4−7 for bound holes and ≈16−18 for unbound holes in P3HT.39 The relative intensity of peak A (compared to peak B) is a key piece of information for characterizing how polymer processing and microstructure affect polaron delocalization and ultimately charge transport. In future work, we intend to investigate oriented P3HT samples to extract the components of the IR spectrum polarized along the chain direction and along the stacking direction. Such information will allow us to more accurately test the polaron theory and will enable more accurate measures of the anisotropic coherence lengths.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b03873.

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Detailed derivation of the spectral characteristics of peaks A and B using perturbation theory in the dimer limit, additional information about how the lateral position of the anion impacts the IR spectrum, and the synthesis and characterization of the P3HT films (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Annabel R. Chew: 0000-0001-9617-3528 Christine K. Luscombe: 0000-0001-7456-1343 Frank C. Spano: 0000-0003-3044-6727 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation (DMREF1533954, 1533987 and 1533372). K


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