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Polarons, bipolarons, and absorption spectroscopy of PEDOT Igor V. Zozoulenko, Amritpal Singh, Sandeep Kumar Singh, Viktor Gueskine, Xavier Crispin, and Magnus Berggren ACS Appl. Polym. Mater., Just Accepted Manuscript • DOI: 10.1021/acsapm.8b00061 • Publication Date (Web): 11 Dec 2018 Downloaded from http://pubs.acs.org on December 18, 2018

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ACS Applied Polymer Materials

POLARONS, BIPOLARONS, and ABSORPTION SPECTROSCOPY OF PEDOT

Igor Zozoulenko*, Amritpal Singh, Sandeep Kumar Singh†, Viktor Gueskine, Xavier Crispin and Magnus Berggren Laboratory of Organic Electronics, ITN, Linköping University, 60174 Norrköping, Sweden

AUTHOR INFORMATION Corresponding Author: *E-mail: [email protected] Present address: †Department of Physics, Chalmers University of Technology, 41296 Göteborg, Sweden

ORCID: Igor Zozoulenko: 0000-0002-6078-3006 Xavier Crispin: 0000-0001-8845-6296 Magnus Berggren: 0000-0001-5154-0291

KEYWORDS: Polarons, bipolarons, conducting polymers, PEDOT, Vis/NIR absorbance spectroscopy, electron paramagnetic resonance spectroscopy

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ABSTRACT Electronic structure and optical absorption spectra of poly(3,4-ethylenedioxythiophene) (PEDOT) for different oxidation levels were studied using density functional theory (DFT) and time-dependent DFT. It is shown that the DFT-based predictions for the polaronic and bipolaronic states and the nature of corresponding optical transitions are qualitatively different from the widely used traditional picture based on semi-empirical pre-DFT approaches that still dominate the current literature. Based on the results of our calculations, the experimental Vis/NIR absorbance spectroscopy and the electron paramagnetic resonance spectroscopy are re-examined and a new interpretation of the measured spectra and the spin signal which is qualitatively different from the traditional interpretation is provided. The findings and conclusions concerning the nature of polaronic and bipolaronic states, band structure and absorption spectra presented for PEDOT, are generic for a wide class of conducting polymers (such as polythiophenes and their derivatives) that have similar structure of monomer units.


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I. Introduction Since the discovery of conducting polymers their electronic structure and optical properties have attracted massive attention, both experimentally and theoretically1. The electronic and optical properties of conducting polymers are related to the conjugated nature of their backbones and to the formation of polarons and bipolarons that represent singly and doubly charged quasiparticles, respectively, that are localized along the polymer chains due to a strong electron-phonon coupling. One of the most important conducting polymers is poly(3,4-ethylenedioxythiophene) (best known as PEDOT)2 that find its application in variety of fields including energy storage3-4 and conversion5, bioelectronics,6 photovoltaics,7 sensing and electrochemical devices, such as electrochemical transistors,8-10 and electrochromic displays. The electronic structure and optical properties of PEDOT are similar to the ones of polythiophene (PT) and its derivates11, since both polymers have a similar monomer unit comprising the thiophene ring. PEDOT and PTs are typically p-doped, and the polarons and bipolarons in these materials represent therefore positively charged “hole” quasiparticles. (In organic chemistry, polarons and bipolarons are usually referred to as radical cations and dications, respectively). In order to understand the electronic properties and optical transitions of conjugated polymers, in particular those of PTs and its derivatives, a number of semi-empirical models were developed in the eighties and early nighties. These studies included approaches based on the Su-Schrieffer-Heeger Hamiltonian,12 semiempirical MNDO method (Modified Neglect of Diatomic overlap),13 semiempirical Hartree-Fock INDO method (Intermediate Neglect of Differential Overlap) combined with configuration interaction technique,14-16 and others.17 In the past decades, the density functional theory (DFT) has emerged as the prime method of choice for the description of the electronic and optical properties of most material systems including conducting polymers and macromolecules. This is because of the computational efficiency of the DFT technique and its ability to provide a good agreement with more accurate high-level ab initio calculations and with the experiments.18-19 Apparently, DFT, as any other approximate method, has its own limitations and yet unsolved issues arising from the approximations in the treatment of the exchange-correlation functionals. (For the perspective and insights into limitations of DFT in the theoretical chemistry see e.g.

20-22).

Despite these limitations the DFT approach has de-facto become one of the most utilized

techniques in all branches of theoretical chemistry. The DFT and the time-dependent (TD-) DFT approach have been used to study various electronic and optical properties of PEDOT and PTs23-28. However, it has been realized that DFT and TD-DFT20-22 predict the electronic structure and optical transitions that are not just quantitatively, but also qualitatively different as compared to above-mentioned earlier studies based of the pre-DFT semiempirical approaches.18-19, 29-30 This is illustrated in Figure 1 which shows band diagrams along with corresponding optical transitions for polarons and bipolarons for PTs predicted by different pre-DFT 3 ACS Paragon Plus Environment


Figure 1. Polarons and bipolarons in conjugated polymers: a brief overview. Band diagrams and optical transitions for polaronic and bipolaronic states for p-type doping reported in the literature and used for interpretation of experimental data, (a) ref. 12, (b) ref. 13, (c) ref. 16, (d) ref. 17, (e) ref. 31, (f) ref. 2, (g) ref.32, (h) ref. 33, (i) ref. 29, (j) ref. 30. Note that not only optical transitions predicted by different theories differ qualitatively, the band diagrams themselves can be qualitatively different. Moreover, in some of the experimental interpretations the assignment of the optical transitions (such as (f) and (g)) seems to be quite arbitrary and not related to particular theoretical predictions.

approaches 12-17 (Figure 1a-e) and compared to those obtained from the DFT methods18-19, 29-30 (Figure 1h-j). (Note that results based on the pre-DFT approaches were termed as “polaron model” in18-19, and a “traditional view” in29). The distinctive feature of “pre-DFT” band structure is the appearance of a pair of the spin-degenerate level(s) in the gap (polaronic and bipolaronic levels, respectively, for chains charged +1 or +2), see Figure 1a-e. The bipolaronic levels are empty, whereas the lowest polaronic level is occupied by one electron. For the case of the crystal this single occupancy leads to a half-filled polaronic band, which should be contrasted with the empty bipolaronic band (see Figure 1b). The optical transitions (indicated by blue lines) reflect this band structure (it should be noted however that even within this band structure the predicted optical transitions are different for different approaches, c.f. Figure 1a-e). A qualitatively different band structure and optical transitions emerge from the DFT calculations of PTs and related polymers19, 29-30, 33. For the case of the polaron, the spin degeneracy is lifted and the band 4 ACS Paragon Plus Environment

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structure of the spin-up and spin-down states is totally different: while one of the spin species (say, spindown ↓) shows an empty level in the gap (corresponding to the polaronic state), the spin-up species ↑ do not exhibit any polaronic level, see Figure 1h-j. It should be emphasized that the nature of the polaronic states is different between the pre-DFT and the DFT approaches; there are two spin-degenerate polaronic states where the lowest one is half-filled in the pre-DFT approach, while there is only one empty spin-resolved state in the gap predicted by the DFT approach. This in turn, leads to the qualitative difference in the polaronic optical transitions: because the polaronic level is empty in the DFT approach, no optical transitions are possible from this level. In contrast, optical transitions are possible from the polaronic level in the pre-DFT approach because this level is occupied. Also, no transitions are possible into the second polaronic level because it is absent in the DFT approach, whereas such transitions are possible in the pre-DFT approach where this level is present. For the case of bipolaronic state, the DFT approach predicts one empty spin degenerate level in the gap (that should be contrasted by two empty spin degenerate levels obtained in the pre-DFT approaches), which, in turn also leads to different predictions concerning the optical transitions. The striking difference between the pre-DFT models and the DFT predictions can be traced back to a number of simplified assumptions adopted in the former approaches, most importantly, to neglecting to lift the spin degeneracy for polaronic states. Earlier experimental studies utilized the pre-DFT predictions to interpret the absorption spectra of conducting polymers, simply because no DFT calculations were available at that time,17, 31 see Figure 1d-g. It is however remarkable that even with the availability of the modern DFT calculations, the traditional semi-empirical pre-DFT model still remains the dominant picture in the current literature; it is still widely used in the interpretations of the experimental data

32, 34-41

derived from experiments

including PTs and PEDOT2, 11, 42 as the active material. (Note that DFT approaches have been used in the past to calculate the electronic structure of the crystalline PEDOT.25-26,

43

However, it is well

established, both theoretically and experimentally, that PEDOT represents a highly amorphous material with very limited degree of crystallinity41, 44-46). Note also that the predictions based on the traditional picture were used for interpretation of the transport measurements, where the different behaviour of the conductivity and the Seebeck coefficient was attributed to the difference in the nature of charge carriers and the corresponding band structures (polarons vs bipolarons).47 Despite the vast interest to the optical and electrochemical properties of PEDOT, a detailed analysis of the polaronic and bipolaronic band structure and an interpretation of the corresponding optical transitions based on the modern DFT-based approaches are not available. One of the aims of this paper is to fill this gap and to provide a corresponding theoretical treatment of polarons and bipolarons in PEDOT. It is also important to stress that PEDOT is distinctly different from other polymers including PTs and their derivatives, since it is heavily doped in the pristine state (i.e. as polymerized). Therefore, 5 ACS Paragon Plus Environment


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the understanding of its band structure and corresponding optical transitions, at high oxidation levels, is of the outmost importance. This understanding is also of a vital importance for the interpretation and modelling of the transport properties of the PEDOT thin films,46-48 because the polarons/bipolarons are the charge carriers of the system. Thus, a proper description of the transport properties requires a detailed knowledge of the structure of the polaronic/bipolaronic states and the corresponding density of states. To our knowledge a corresponding theoretical treatment is missing for these properties of the PEDOT system. The main goal of the current paper is to provide a DFT-based description of the polaronic and bipolaronic states, and the optical transitions in PEDOT at all oxidation levels. Based on the results of our

calculations

we

re-examine

the

available

experimental

results,

in

particular,

on

spectroelectrochemistry and the electron paramagnetic resonance (EPR) spectroscopy of PEDOT, and provide an alternative explanation which is distinctly different from the traditional view relying on the pre-DFT results, typically achieved in the eighties. Finally, we discuss and correct some widely spread misconceptions, reported in current literature, that originate from pre-DFT studies including the notion of a half-filled polaronic band, the assignment of the spin 1/2 to the polaronic level in the gap. The latter aspect can have important implications for the interpretations of the experiments probing the nature of the states (i.e. polarons vs bipolarons) in polymeric films.

II. Results and discussions (II.a) Electronic structure and optical transitions

Figure 2. Spin multiplicity of the ground state. The total energies of different spin states in PEDOT chains of the length (a) N=12, and (b) N=18 for different oxidation levels (i.e. different number of positive charges q = +1,…,+6 residing on a chain). Numbers in parentheses indicate the spin multiplicity M=2S+1, with S being the total spin. For convenience of comparison between different oxidation levels, all ground state energies are shifted to zero.


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We start with the determination of the ground state of the PEDOT chains at different oxidation levels. Here and hereafter we will call states with q = +1e and +2e as polarons and bipolarons, and, generally, states with odd numbers of positive charges (q = +1e, +3e, +5e) as polaronic states, and those with even numbers (q = +2e, +4e, +6e) as bipolaronic states. Note that the oxidation level of a pristine (as polymerized) PEDOT chains is cox ≈ 33%, i.e. one positive charge per three monomer units2 and it can be subsequently reduced or increased by electrochemical means. Note also that the chain length of PEDOT is not known exactly, but here estimated to consist of 5-20 monomer units

2, 49-50).

Figure 2

shows the calculated total energies of the states of different spin multiplicities M=2S+1 for the chains of the length N=12 and N=18, with S being the total spin. For the case of a single polaron in the chain (q = +1), there is one unpaired electron with a spin S=½ and hence the only possible state is a doublet (M=2). For the bipolaronic states with q = +2, the total spin can be S=0, 1 and thus singlet (M=1) and triplet (M=3) states are possible. The calculations show that the triplet state has a lower energy in comparison to the singlet one. This result might look unexpected because in the pre-DFT approaches the bipolaron states were postulated to be spinless (S = 0). There is however no physical reason why this should be the case. In fact, the triplet character of the ground state for the bipolaron can be considered as a manifestation of the Hund’s rule which favors the single filling of available degenerate orbitals, leading to the lowering of the total energy as the exchange energy is gained due to the increased total spin. It is noteworthy that the Hund’s rule is also observed in the semiconductor quantum dots (often called “artificial atoms”) when they are filled by excess electrons.51-52 The ground states for the case of polaronic states for higher oxidation levels (q = +3, +5) are also doublets for both chain lengths N=12 and N=18. In contrast, the character of the bipolaronic states for higher oxidation levels (q = +4, +6) is different for N=12 and N=18. Namely, the ground state for q = +4, +6 is singlet for N=12 and triplet for N=18. It should be noted however that for the former case the energy difference between the triplet and singlet states are rather small and for the case of q = +6 it is even comparable to the room temperature thermal energy. We speculate that a difference in the character of the ground state (singlet vs triplet) for different N can be a result of a subtle interplay of various factors including the chain length, oxidation level, the exchange-correlation interaction in the chosen DFT functional, leading to different electrostatic interaction for different q, and different wave function localization for different spins. Figure 3 shows the band structure of a PEDOT chain of the length of N=12 monomer units when it is consequently doped (oxidized) from a neutral state (Q=0) to the oxidized one, Q = +6e. The corresponding calculated optical spectra and schematic diagrams indicating the dominant transitions contributing to the absorption peaks are presented in Figure 4. (Here and hereafter we focus on the case N=12. The band diagrams and optical spectra for the case of N=18 exhibit very similar trends and features, and they are shown and discussed in the Supporting Information, see Figure S1 and Figure S2.



Figure 3. Electronic structure of doped PEDOT chains. Positively charged polaron and bipolaron states in a PEDOT chain N=12 in the ground state configuration for different number of positive charges q = +1,…,+6. (The spin S of the ground state is indicated). (a)-(b) Bond length alternation (red curves) and the electron density distribution in a PEDOT chain with a polaron +e (corresponding to the band diagram (f)). (c)-(d) the same as (a)-(b), but for a bipolaron +2e in (f). The black lines in (a) and (c) correspond to the bond length alternation for a neutral chain. (e)-(k) The band diagram for the polaron and bipolaron states as the PEDOT chain is consequently charged from 0 to +6e. All band diagrams are aligned by setting EHOMO = 0. Occupied electronic levels (in the valence band) are shown in blue, where electrons filling the levels are represented by arrows. The empty electronic levels (in the conduction band) are shown in red. Dashed red lines are unoccupied polaronic/bipolaronic levels. (l) and (m) are the electron density distribution for the polaronic (+5e) and bipolaronic (+6e) states (j) and (k), along with the corresponding bond length alternation. Arrows in the conduction band describe the occupied spin-up and spin-down electron levels. 8 ACS Paragon Plus Environment

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Figure 4. Vis/NIR absorption spectra. Calculated absorption spectra of a PEDOT chain N=12 using the TD DFT approach for different oxidation levels cox (i.e. number of charges per monomer). (a)-(g) correspond to a different number of charges per chain, 0e … +6e as indicated in the figure. Schematic diagrams indicate the dominant transitions contributing to the absorption peaks corresponding to the largest configuration interaction (CI) coefficients as given by the DT DFT calculations.

The absorption spectrum of a neutral chain calculated using TD DFT exhibits a single peak with a wavelength λabs = 481 nm, see Figure 4a. The analysis of the configuration interaction coefficients shows that this peak, as expected, is primarily due to the transition from HOMO to LUMO, see Figure 4a. Let us now consider a polaron state of a charge +e. Because only one electron is removed from the chain, the spin degeneracy of otherwise doubly occupied states is lifted and spin-up and spin-down orbitals show a qualitatively different structure. Namely, a new unoccupied level corresponding to one of the spin species (say, spin-down) appears in the gap. This level is defined as a polaronic level. In contrast, the spin-up electrons do not have levels in the gap. The optical transitions corresponding to the case of the polaron give rise to two absorption peaks, λ = 1772 nm and λ = 799 nm, marked as P1 and P2 in Figure 4b. The peak P1 (λ = 1772nm) corresponds to the transition from HOMO to the polaronic level and therefore includes a contribution of electrons of one spin species only, whereas the peak P2 (λ = 799nm) corresponds to the transitions from the HOMO to the levels in the conduction band and includes 9 ACS Paragon Plus Environment


a contribution of both spin species (It should be noted that the exact position of the peaks and their oscillator strength depends on the parameter ω in the exchange-correlation functional ωB97XD, as well as on the number of excited states included in the TD DFT calculations. We did not make any systematic attempts to adjust the value of ω as in this study because we focus on the generic properties of the spectra and optical transitions). Consider now a bipolaron state of a charge +2e (i.e. two removed electrons). The ground state is the spin-polarized (see Figure 2) and, as a result, the band diagram and the optical transitions show similar features with the polaronic state (which is spin-polarized too). Namely, two new unoccupied levels corresponding to the same spin species (say, spin-down) appears in the gap, see Figure 3g, whereas there are no levels in the gap for the spin-up electrons. The optical transitions also give rise to two absorption peaks, λ = 1300 nm, and λ = 791 nm marked as B1 and B2 in Figure 4c. The peak B1 (λ = 1300nm) corresponds to the transition from HOMO to the bipolaronic levels and therefore includes a contribution of electrons of one spin species only, whereas the peak B2 (λ = 791nm) corresponds mostly to the transitions from the HOMO to the levels in the conduction band and includes a contribution of both spin species. Figures 3a,b and c,d show the bond length alternation in the PEDOT chain and the wave function for the polaron and bipolaron levels. The wave functions of both of these levels are strongly localized over 6-7 monomer units (A detailed discussion of the quasiparticle localization in PEDOT in terms of the inverse participation ratio can be found in 53). In the regions of the PEDOT chains, where the wave function is localized, the character of the bond length alternations changes from aromatic to quinoid. The calculated band structure of the polaron states in PEDOT is qualitatively similar to those calculated using the DFT approaches for PTs,18-19, 33 ETE-S,30 and PPP 29 (which has a similar aromatic ring). It is important to stress that, as mentioned in the introduction, these results are qualitatively different from the pre-DFT results (cf Figs. 3f and 1a-d). The reason for this is that pre-DFT approaches treat the polaronic states as spin-degenerate, and the lifting of the spin-degeneracy is essential for the case when an odd number of electrons are removed from the chain. Also, the pre-DFT approaches treat the lowest polaron level as occupied which is hardly reconcilable with the fact that the electron is removed from the chain. With this respect, the pre-DFT predictions are not mutually consistent for polaron and bipolaron states: in both case electrons are removed from the system, but the polaron level is considered as occupied, whereas the bipolaron level is empty (see Figs. 1a-e). It should be mentioned that for the case of bipolarons (e = +2), our predictions are not only qualitatively different from the pre-DFT results, but they are somehow different from the DFT-based calculations for PTs,18-19, 33 and ETE-S30 (note that only polaron states were studies in 29 for PPP). The difference with the previous DFT-based calculations is that for the case of PEDOT the ground state for the bipolaron is


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triplet, which apparently lead to a different band structure as compared to the singlet states considered in Refs. 18-19, 33 and 30. It is noteworthy that there exists a misconception that the energy difference ΔELUMO-HOMO between the LUMO and HOMO as calculated by the DFT approach would give the energy of the absorption peak Eabs . In fact, the wavelength for the optical transition corresponds to the total energy difference ΔEtot between the ground and corresponding excited state of the system as obtained from the TD DFT calculations. These two (i.e ΔELUMO-HOMO and ΔEtot) are distinct because the total energy of the Nelectron system is not just a sum of the energies of Kohn-Sham orbitals, but also includes contributions related to the Hartree and exchange terms. For example, for a neutral chain Eabs ≈ 2.57 eV, (corresponding to λabs ≈ 481 nm), whereas Eg ≈ 5.43 eV, is more than twice of this value. Also for a bipolaron shown in Figure 3g and Figure 4c, the absorption wavelength naively expected from the orbital energy difference is λ= hc/(Ebipolaron – EHOMO) ≈ 348 nm, which is much smaller than the TD DFT calculated wavelength λ ≈ 1300 nm. (Note that an interesting discussion concerning definitions and differences between different energy gaps such as fundamental gap, optical gap, transport gap, electrochemical gap can be found in refs. 19 and 54). Let us now consider higher oxidation levels, see Figure 3h-k. For a chosen chain length N=12, states +3e, +4e, +5e, +6e correspond to the oxidation levels cox = 25%, 33%, 42% and 50% (It is noteworthy that cox = 33% corresponds to a pristine (i.e. as polymerized) PEDOT; higher oxidation levels exceeding 50% can be achieved by electrochemical means41). For N=12 the bipolaronic states +4e and +6e are spin degenerated singlets with the total spin S=0. The polaronic states +3e and +5e are spin-polarized doublets with the total spin S=½. Even though the band structure changes when the PEDOT chain is consequently charged and thus undergoes a transition between the polaronic and bipolaronic states of different spin nature, the main features of the band structure remain very similar for the bipolaronic and polaronic states. Indeed, the band diagrams show a band of unoccupied levels (polaronic/bipolaronic levels) separated by gaps from both the conduction band (with empty electronic levels) and the valence band (with occupied electronic levels). It is important to stress that polaronic and bipolaronic bands correspond to empty electronic levels, with the number of levels (counting the two-fold degeneracy for the bipolaron states for the cases of +4e and +6e) being equal to the number of removed electrons. Also, in contrast to the pre-DFT approaches,13 the DFT approach does not predict a half-filled polaronic band (c.f. Figure 1b and Figure 3h,j). Note that within the DFT approach a number of polarons/bipolarons levels always equals to the number of electrons removed from the chain. This does not hold for the case of pre-DFT semiempirical approach. For example, for the case of one removed electron the later gives four polaronic levels (one occupied and three unoccupied, see Figure 1a,c-e) which apparently does not match the number of removed electrons. 11 ACS Paragon Plus Environment


Figure 3l,m shows the bond length alternation in the PEDOT chain and the wave function for the polaron and bipolaron levels for higher oxidation levels (+5e and +6e respectively). As expected, the rearrangement of the lattice from the aromatic to quinoid form becomes even more pronounced as compared to the polaron and bipolaron levels +e and +2e, (c.f. Figure 3a-d and Figure 3l-m). Figure 4d,f shows the optical transitions for polaronic states for higher oxidation level. For polaronic state +3e the spectrum exhibits two absorption peaks, λ = 958nm (peak P′1) and λ = 1282nm (peak P1). As in the case of the single polaron, peak P1 originates from the transition from the levels in the valence band to the polaronic levels; peak P′1 also represents transitions from the valence band to the polaronic levels but with some admixture of the transitions to the conduction band. Note that each of these peaks corresponds to the contribution of different spin species. For the case of the polaron state +5e the absorption spectrum exhibits only one peak at λ = 1266nm (P1) originating from the transitions from the valence band into the polaronic band. Both spin-up and spin-down spin species contribute to this peak. The absorption spectra for the bipolaronic states at higher oxidation levels are shown in Figure 4e,g. They exhibit a single peak (B1) (λ = 915nm and λ = 933nm for +4e and +6e respectively) which originates from the transition from the valence band to empty (spin-degenerate) bipolaronic levels in the gap. For higher oxidation levels (+4e and +6e) the peak is blue-shifted in comparison to the case of a single bipolaron +2e. We conclude this section by elaborating a definition of the polaronic and bipolaronic states that we use, namely, addressing a question of bipolarons vs polaron pairs that has been extensively discussed in the literature previously.33, 55-57 As discussed above, the polaronic state +Q corresponds to one unpaired electron, which results in spin S = ½, whereas the bipolaronic state +2Q can have spin S = 0 or 1. For the case S = 0 the solution is spin-degenerated and describes a state called in the literature as “bipolaron”. For the case S = 1 the spin degeneracy is lifted, and the solution corresponds to a triplet state which in literature is referred to as “a polaron pair” (“biradical” in the chemical nomenclature). (Note that the polaron pair can theoretically correspond to the singlet state in the spin-unrestricted case; we however do not consider this case, see Method section for a detailed explanation). Thus, in the present case of the chain N = 12 the calculated ground state for Q = +2 (corresponding to S = 1) would be called the polaron pair in the notations adopted in the literature. While the distinction of bipolarons and polaron pairs is clear for the case of two electron removed from the chain, the notion of the polaron pair becomes rather ambiguous for higher oxidation levels with more than two electrons being removed from the chain. In order to use the same notations for all oxidation levels, in this paper we do not use a separate term (i.e. a term “polaron pair”) for the case of two removed electrons. Instead, for all oxidation levels we refer to the states with even number of removed electrons as bipolaronic ones with a corresponding multiplicity. For example, for Q = +2 we call a ground state as a bipolaronic one in a triplet state (S = 1). 12 ACS Paragon Plus Environment

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II.c misconceptions in the field There are several common misconceptions in the field that originate from the predictions of the traditional pre-DFT approaches and which are not supported by the modern DFT calculations. Because of the complete dominance of the traditional pre-DFT approaches over last decades these misconceptions are rather common and can be found in many recent papers and reviews.2, 11 In this section we address these misconception, including (i) half-filled polaronic band, and (ii) positive polarons as spin ½ quasiparticles occupying the polaronic level in the gap. (i) half-filled polaronic band. A traditional pre-DFT approach13 (based on the semiempirical MNDO (modified neglect of diatomic overlap)) predicts a half-filled polaronic band in contrast to the empty bipolaron band, see Figure 1b. It has been argued in the literature that this difference would lead to different transport properties of the thin films depending on whether electronic conductance is mediated by polarons or bipolarons.47 However, as shown above in Sec. IIa, the DFT approach does not predict the half-filed polaronic band. In fact, the character of the electronic structure and optical transitions for polaronic and bipolaronic states are rather similar, especially at higher oxidation levels, see Figs. 3 and 4. The similarity of the band structures for the cases of polarons and bipolarons, as well as the similarity in charge localization and the character of the wave function strongly suggest that the character and the features of electron transport are the same for polarons and bipolarons. (ii) positive polaron as a spin ½ quasiparticle occupying the polaronic level in the gap. Traditional preDFT approaches predict that a positive polaron occupies a level in the gap and therefore the positive polaron possess a spin ½. In contrast, according to the DFT predictions, the polarons corresponds to empty levels in the gaps, see Figure 3. It is electrons that possess the spin, not the empty levels. Note that according to the DFT calculation the ground state of the oligomer with a polaron state is doublet (i.e. spin = ½), which means that such the oligomer would give rise to a spin count into the EPR signal. The essential difference from the traditional interpretation is that this spin comes from the electrons that fill in the levels in the valence band, not from the empty polaron level in the gap.

(II.c) Vis/NIR absorbance spectroscopy and EPS spectroscopy of PEDOT thin films: reinterpretation of experimental results.

(i) Vis/NIR absorbance spectroscopy



Figure 5. Vis/NIR absorbance spectroscopy and EPS spectroscopy: re-interpretation of experimental results. (a)-(b). Representative absorption spectrum of PEDOT (reproduced with permission from ref 35. Copyright 2014 Royal Society of Chemistry), and its interpretation based on (a) the traditional pre-DFT picture and (b) our DFT calculations. In (a),(b), the curves labelled a through e correspond to different oxidation levels with a corresponding to fully oxidized PEDOT and e to fully reduced one. Peaks P1, B1 and P2, B2 and corresponding transitions are defined in Figure 4. (c)-(d) A schematic figure illustrating a typical EPR spin signal recorded as the oxidation level of the PEDOT film is varied from a fully reduced state to a fully oxidized one using standard cyclic voltammetry scans. (c) the traditional pre-DFT interpretation and (d) the DFT-based interpretation.

A powerful method for investigation of the electronic structure of conducting polymers at different oxidation levels is the Vis/NIR absorbance spectroscopy, where the optical absorption is recorded while varying the oxidation level of the polymer from an oxidized to a reduced state. This is done using the standard electrochemical setup when ions from the electrolyte are injected into the polymer film and therefore attracting (or forcing out) charges in the polymer chains into metallic contacts,39-41 or by treating the film by reducing agents. 35 A representative Vis/NIR absorption measurement of PEDOT (reproduced from 35) is shown in Figure 5. (It should be stressed that similar results for PEDOT absorption were reported in many other studies, e.g. Ref. 5, 39-41 . Note also that the absorption spectra of PTs and their derivatives such as P3HT exhibit similar features34, 36-38). All these measurements show similar behavior of the absorption spectra as the oxidation level is varied. In a pristine (oxidized) state the absorption shows a broad signal for λ ≳ 9001000nm, Figure 5a, curve a. Within the traditional interpretation this is attributed to the bipolaron absorption, see schematic diagrams in Figure 1a,c,d. As the oxidation level is reduced, the intensity of the broad signal at λ ≳ 900-1000nm is reduced, and a new broad peak centered at λ ≈ 800 nm starts to 14 ACS Paragon Plus Environment

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develop, Figure 5a, curves b,c,d. Within the traditional interpretation this peak is attributed to the polaron absorption, which becomes dominant in the spectrum when the number of charge carriers is reduced. Finally, in the reduced state, the absorption shows a peak at λ ≈ 600 nm, Figure 5a, (curve e) which is attributed to the HOMO-LUMO transition in neutral chains. The results based on the DFT approach provide a completely different interpretation of Vis/NIR absorption spectra. The broad signal λ ≳ 900-1000 nm at high oxidation levels is now attributed to both polaronic- and bipolaronic states, namely, to the peaks P1 and B1 due to transitions from the occupied HOMO levels to the empty polaronic and bipolaronic levels as shown in Figure 4d-g. The peak at λ ≈ 800 nm at low oxidation level is also due to both polaronic and bipolaronic states. It corresponds to peaks P2 and B2 in Figure 4b,c arising due to the transitions from the valence band to the conduction band. Interesting to note that the calculated peaks P2 and B2 are much lower for the case of PEDOT chains of the length N=18, see Figure S2 in Supporting Information. This might be an indirect indication that PEDOT chains are shorter that N=18. However, an alternative explanation might be that due to the inelastic scattering, the coherence length for the polaronic/bipolaronic wave functions is shorter than the chain length. It is noteworthy that even for the low oxidation level the experimental intensity of the broad signal λ ≳ 900-1000 nm is higher than the one for the peak at λ ≈ 800 nm, which is consistent with the relative values of the oscillator strength for different peaks obtained within the DFT calculations. Finally, for the neutral PEDOT the absorption peak is due to transitions from the valence band to the conduction band as shown in Figure 4a. One important comment is in order. The calculated absorption spectra in the NIR region consists of single polaron/bipolaron peaks, whereas the measured spectra show a broad signal with λ ≳ 900-1000 nm. What is the reason for this discrepancy? Above, we discussed the absorption spectra of PEDOT films based on the electronic structure of a single chain. This is apparently a simplification, because the PEDOT thin film represents a rather amorphous structure composed of small crystallites consisting of several π-π stacked chains surrounded by less ordered regions as well as counterions. (For discussion of morphology of PEDOT films see e.g. Refs. 41, 44-45, 58-59). This model nevertheless represents a reasonable starting ground for the interpretation of the absorption spectra because the optical properties of the polymer are primarily determined by the electronic states in the individual chains, and to a much lesser extent by the inter-chain coupling because the latter is much weaker than the former. However, the interchain coupling leads to a transformation of the polaronic and bipolaronic states into corresponding bands that eventually merges with the conduction band at the high oxidation levels, see Figure 1 in Ref. 60, as well as Ref. 53. Note that the counterions affect the position of the levels in the band due to the Coulomb interaction leading to the further band broadening. All these would affect the calculated spectra, causing, in turn, a significant line broadening. The most pronounced effect of the above factors is expected to be 15 ACS Paragon Plus Environment


on the peaks in the NIR region, because these peaks are due to the transitions to the polaronic/bipolaronic levels, that transforms in a broad band that merges with the conduction bands as discussed above.53, 60 It is noteworthy that the measurement of the absorption spectra of individual polymeric oligomers was performed by van Haare et al. 17 Unfortunately, such the measurements can not be performed for PEDOT where isolation of individual oligomers is not possible because of the insolubility of the material.

(ii) EPR spectroscopy Electron paramagnetic resonance (EPR) spectroscopy is widely used to study the spin nature of charge carriers in PEDOT and related polymers. The EPR signal provides a spin count that can be used to determine the concentration of the spin carriers in the system under investigation. A number of groups reported electrochemical EPR studies, where the spin signal was recorded as the oxidation level of the PEDOT film was varied in the standard cyclic voltammetry scans.47, 61-65 A typical spin signal as a function of the applied potential is schematically depicted in Figure 5c,d. The spin signal first increases and then slowly decreases as the film undergoes a transition from the fully reduced to the fully oxidized state when the voltage is increased. A traditional interpretation of this behavior is that the polymer is first doped with polarons (which contribute to the spin signal because of their spin nature). As the doping level is increased the concentration of bipolarons starts to rise, which leads to reduction of the spin signal because the bipolarons are spinless, see Figure 5c. It is worth to point out that the applicability of this interpretation at higher doping levels (for fully oxidized films) is questionable, because in this case all chains contain more than one or two charges, such that a tradition picture of one or two carrier per chain (polarons and bipolarons) does not apply. Our DFT-based calculations provide an alternative interpretation of the EPR signal of PEDOT films, see Figure 5d. For low doping levels the polymer chains remain neutral or contain no more than one or two charges (polarons or bipolarons). According to the DFT predictions both polarons (+e) and bipolarons (+2e) contribute to the spin signal. As voltage increases the number of charged polymeric chains increases, which apparently causes the increase of the spin signal. (It is worth stressing again that, as discussed in Sec. IIb, it is electrons in valence band, not empty polaron/bipolaron levels in the gap that give rise to the spin signal). The spin signal will increase until most of the chains are oxidized, and after that it is expected to somehow decrease. Indeed, with further increase of the voltage (and therefore the oxidation level), more charges will be induced in each chains (+3e, +4e, +5e,…). Chains with the odd number of charges (polaronic states) correspond to the doublet states (S=½) regardless of the charge, and therefore the spin signal from the polaronic states would not increase any longer. On the contrary, chains with the even number of charges (bipolaronic states) gives rise to the spin signal only for the bipolarons (+2e), whereas +4e and +6e bipolaronic states are spinless singlets (see Figure 2). As


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a result, the spin signal from the chains with bipolaronic states at high oxidation level is expected to decrease, which matches very well the reported experimental data. The interpretation of the spin signal outline above corresponds to the chain length N=12. For the case of N=18, PEDOT chains containing +4e and +6e bipolaronic states are triplets (see Figure 2). However, even in this case the dependence of the spin signal on the applied voltage would remain qualitatively similar, exhibiting first an increase of the spin signal due to an increased concentration of polarons (+e) and bipolarons (+2e), and then showing a saturation when all chains become highly oxidized and contain whether +3e or +5e polaronic states that are doublets, or +4e or +6e bipolaronic states that are triplets.

III. Conclusions

Figure 6. A graphical summary of the present work. (a),(b) Schematics of energy band diagrams and main optical transitions for (a) low oxidation level, (b) high oxidation level; (c),(d) Interpretation of the observed Vis/NIR absorption spectra the PEDOT thin film indicating contribution of the optical transitions shown in (a),(b). (e) Interpretation of the EPR spin signal for high and low oxidation levels.



The predictions based on traditional pre-DFT semi-empirical approaches developed in eighties are still commonly used for discussion of the band structure and interpretation of the optical absorption in conducting polymers, in particular PTs and PEDOT.12-17 It has been recently realized that modern approaches based on DFT and TD DFT predict the electronic structure and optical transitions in conducting polymers are qualitatively different in comparison to the above-mentioned earlier studies.1819, 29-30, 33

Despite of the vast interest to the optical, transport and electrochemical properties of PEDOT,

a detailed analysis of its polaronic and bipolaronic band structure and interpretation of the corresponding optical transition based on the modern DFT-based approaches are not available. In the present paper we use DFT and TD DFT to calculate the band structure of PEDOT and its absorption spectrum at different oxidation levels. A particular attention is given to the case of high oxidation levels because PEDOT is heavily doped in its pristine (as polymerized) state. We discuss the evolution of the band structure as the oxidation changes from the neutral state cox=0 to the pristine state cox=33%. A graphical summary of our results is presented in Figure 6 and outlined below as follows. First, we determine the ground state of the PEDOT chains at different oxidation levels. For the case of the polaron (+e) the spin degeneracy is lifted, and the ground state is doublet (spin S=1/2). The electronic structure of the polaron state shows one empty level in the gap for one of spin species, and no levels in the gap for the opposite spin. These finding for PEDOT are in agreement with corresponding DFT studies for related polymers

18-19, 29-30, 33

and are qualitatively different from the traditional pre-DFT

predictions.12-17 The main reason for the above discrepancy is that the traditional pre-DFT approaches treated the polaronic states as spin-degenerate, disregarding the lifting of the spin-degeneracy. The optical spectrum for the case of polaron exhibit two absorption peaks due to the transition: valence band  the polaronic level (peak P1), and valence band  conduction band (peak P2), see Figure 6a. For the case of the bipolaron (+2e) the ground state is triplet (spin S=1), which is in a stark contrast to the traditional results assuming the spineless ground state S=0. The triplet character of the ground state for the bipolaron can be considered as a manifestation of the Hund’s rule which favors the single filling of available degenerate orbitals. Both the band diagram and the optical transitions for the bipolaron show similar features with the polaronic state. Namely, two new unoccupied levels corresponding to one of spin species appear in the gap whereas electrons of the opposite spin do not have levels in the gap. The spectrum also shows two peaks due to the transitions: valence band  the bipolaron levels (peak B1), and valence band  conduction band (peak B2), see Figure 6a. For the high oxidation level, the ground states for the polaronic states (+3e, +5e) are doublets (S=1/2), whereas the ones for bipolaronic states (+4e, +6e) are singlet (S=0). Despite of that the main features of the band structure are very similar for bipolaronic and polaronic states. Namely, the band diagrams show a band of polaronic/bipolaronic states of comparable width separated from both the conduction band (with empty electronic states) and the valence band (with occupied electronic states) as outlined in 18 ACS Paragon Plus Environment

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Figure 6b. Both polaronic and bipolaronic states give rise to a peak in optical absorption caused by the transitions: valence band  polaronic/bipolaronic levels as indicated in Figure 6b. The similarity of the band structures for the cases of polarons and bipolarons, as well as the similarity in charge localization and the character of the wave function strongly suggest that the character and the features of electron transport are the same for polarons and bipolarons. Hence, the interpretation of the transport experiments47 attributing different transport behaviour to the difference in the nature of charge carriers (polarons vs bipolarons) and the corresponding band structures following from the traditional picture should be revisited. Based on the results of our calculations we re-examine the experimental Vis/NIR absorbance spectroscopy and the EPR spectroscopy, where the respectively optical absorption and the spin signal are recorded while varying the oxidation level of the polymer from an oxidized to a reduced state. We provide a new interpretation of the measured Vis/NIR absorbance spectra and the EPR spin signal (summarized respectively in Figure 6c,d and Figure 6e), which is qualitatively different from the traditional interpretation. Finally, we discuss and correct two common misconceptions in the field, including (i) half-filled polaronic band, and (ii) treating positive polarons as spin ½ quasiparticles occupying the polaronic level in the gap. Even though our calculations have been done for PEDOT, we believe that all our findings and conclusions are generic for a wide class of conducting polymers (such as PTs and their derivatives, P3HT, PPPs and others) that have similar structure of monomer units.

Method The geometry optimizations for this study were accomplished using Gaussian 09 package

66

without

imposing any constraints on initial structures. Neutral PEDOT (Q = 0) and p-doped PEDOT (Q = +1e, +2e, +4e, +6e) were optimized at ωB97XD/6-31G(d) level of DFT. Vis/NIR absorption spectra were simulated by using TD-DFT at same level of theory i.e. ωB97XD/6-31G(d). Range separated hybrid functional ωB97XD accounts for 22% Hartree-Fock (HF) exact exchange at a short range and 100% HF exact exchange at long range, with Grimme’s D2 dispersion effects.67-68 Restricted (close-shell) spin was used in the calculations for a neutral PEDOT chains (Q = 0) and chains with even numbers of electrons for the singlet state S=0 (with the spin multiplicity M=2S+1=1). For all other cases the unrestricted (open-shell) spin calculations were performed. Note that we also performed unrestricted spin calculation for the even number of electrons for the singlet case S=0. In some cases, these 19 ACS Paragon Plus Environment


calculations converge to the spin-degenerate solution (identical to the results obtained for the case of the restricted spin calculations for S=0), and in some cases they do not converge at all. For the case when the solution is converged and distinct from the spin-degenerated one, it always has the unacceptable high spin contamination. The similar problem with the spin-contamination for this case was noticed before,

33, 55, 57

and the obtained results were interpreted in

33

as mixtures of singlet and triplet states.

Note that the condition S=0 refers in the single-determinant spin-unrestricted ansatz to spin projection Sz, rather than to the total spin, so the solution can be a mixture of the Sz=0 components of various multiplicities, including open-shell singlets. Because of this we treat the spin-contaminated solution for the spin-unrestricted case S=0 as an artefact, and therefore always use the restricted spin calculations for this case. It is noteworthy that unrestricted spin calculations for all other cases (i.e. S≠0) show none or very little spin contamination thus providing correct solutions for the maximum Sz components of higher spin multiplicities. Evasive singlet polaron-pairs (open-shell singlets) are at least two-determinant states and as such are inaccessible for any single-reference method including DFT. However, previous studies (ref.

57

and references therein), suggest that singlet and triplet polaron pairs are quasi-degenerate in

energy when the two polarons in them are almost non-overlapping. Note that in the current literature there are many studies comparing the performance of different functionals including ωB97XD, B3LYP, CAM-B3LYP, LC-wPBE, BNL, MO6.69-70 In our study we use the range separated hybrid functional ωB97XD because it was shown to be highly suitable for the description of π-conjugated oligomers giving an accurate prediction of ionization energies, electron affinities, and excitation energies of neutral and oxidized polyenes, thiophene, and furan oligomers.69 it is also important that this functional overcomes the problem of the charge delocalization which is inherent to the popular functional B3LYP. Note that we performed comparative calculations with the hybrid functionals M06, B3LYP and we found that they give quantitatively same electronic structure compared to ωB97XD. Finally, we note that we performed a normal mode analysis for the cases of a neutral chain as well as for a highly oxidized chain (cox = 50%) to verify the stationary points as minima (zero imaginary frequency). Note that an analysis of vibrational frequencies of PEDOT using the same functional was reported in Ref. 71.

ASSOCIATED CONTENT Supporting Information


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The Supporting Information is available free of charge on the ACS Publications website at DOI: XXX. Section SI. The electronic structure and optical transitions in PEDOT chain of the length N=18 (PDF).

Funding Sources: This work was supported by Swedish Research Council (2017-04474 and 201605990), and the Knut and Alice Wallenberg Foundation through the project The Tail of the Sun. IZ, CX, and MB thank the Advanced Functional Material Center at Linköping University for support. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at NSC and HPC2N.

References

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