Exciton Diffusion in Highly-Ordered One Dimensional Conjugated Polymers: Effects of Back-Bone Torsion, Electronic Symmetry, Phonons and Annihilation

Many optoelectronic devices based on organic materials require rapid and long-range singlet exciton transport. Key factors controlling exciton transport include material structure, exciton–phonon coupling and electronic state symmetry. Here, we employ femtosecond transient absorption microscopy to study the influence of these parameters on exciton transport in one-dimensional conjugated polymers. We find that excitons with 21Ag– symmetry and a planar backbone exhibit a significantly higher diffusion coefficient (34 ± 10 cm2 s–1) compared to excitons with 11Bu+ symmetry (7 ± 6 cm2 s–1) with a twisted backbone. We also find that exciton transport in the 21Ag– state occurs without exciton–exciton annihilation. Both 21Ag– and 11Bu+ states are found to exhibit subdiffusive behavior. Ab initio GW-BSE calculations reveal that this is due to the comparable strengths of the exciton–phonon interaction and exciton coupling. Our results demonstrate the link between electronic state symmetry, backbone torsion and phonons in exciton transport in π-conjugated polymers.

S2 S1: Optical micrographs and photothermal deflection spectra of 'blue' and 'red' PDA Figure S1: a-b. Optical micrographs of 'blue' and 'red' PDA. When the incident light is polarised perpendicular to the long-axis of the chains, there is full transmission (left side image). When the light is polarised parallel to the chain long-axis incident light is extinguished (right side image). Regions of the samples with high-degrees of polymer alignment are identified and masked before performing fs-TAM measurements. White dashes indicate line along which intensity is measured as a function of analyser angle in Figure S1i. Yellow  A nalyser Angle (°) Figure S1i: Intensity of light along 2 um segment (5 sample locations; see Figure S1i) as analyser is rotated. Uniform extinction of the light is observed suggesting little bending and linear alignment of the chains.   Measurements are taken at the same sample location and pump fluence with only the probe wavelength adjusted. As can be seen from the traces moving to bluer (shorter) probe wavelengths within the broad 2 1 Agand 1 1 Bu + PIAs does not alter the MSD. Due to experimental limitations wavelengths longer than 800 nm could not be studied.
To ensure there is no contribution from the 'hot' GS population to the diffusion of S1 excitons fs-TAM measurements were repeated at 710 nm and 860 nm (same sample location). As shown in Figure S2i the 710 nm MSD trace shows markedly different evolution to that at 860 nm and 800 nm. In a previous study 2 the signal at 710 nm was shown to be dominated by a 'hot' GS population of excitons. The spectral overlap between the PIA associated with this state and S1 is almost negligible (~12 %) at the probe wavelength of 800 nm used in our experiments. Hence we do not believe this state contributes significantly to the rapid expansion observed. Understanding the dynamics of the 'hot' ground state is beyond the scope of this work and reserved for future studies. We also note that in the previous works it was shown that 'hot' ground state evolves from S 1 after ~2 ps hence is unlikely to influence the early time behaviour we study here. S9 S3: Extraction of diffusion coefficients from fs-TAM data for 'blue' and 'red' PDA For 'blue' PDA the diffusion coefficient was extracted as follows: (i) fs-TAM images were fit with a 2D Gaussian function.
(ii) The Gaussian standard deviation (σ) was extracted for each image along the wire longand short (principle) axes.
(iv) The MSD was fit using equation 2 of the main text.
(v) A differential diffusion equation was calculated at the diffusion coefficient extracted from the value at 2500 fs.
For 'red' PDA exciton-exciton annihilation effects are present and the above method cannot be used to determine the diffusion coefficient. Instead the fs-TAM data was fit to equation 1 of the main text as follows: (i) Derive initial parameters.
(ii) Numerically solve equation 1 of main text to data.
(iii) Fit Gaussian to solution at each time point.
(iv) Compute sum of squares of residuals between solution's Gaussian-fitted variance and data's Gaussian-fitted variance at each time point until error is sufficiently low.
(v) Extract MSD and annihilation coefficients.
All fitting was performed using the Mathematica software package.     Figure S3ii. The fit shows that the diffraction ring is reproduced faithfully. This implies that the deviation from a true two dimensional spatial profile in our measurements due to the diffraction ring has no effect on the retrieved spatial standard deviation. The slight asymmetry in our point spread function arises from imperfections in our imaging objective as noted in previous studies [3][4][5] . In the case of 'blue' PDA there is no dependence of the kinetics or MSD on the carrier density, whereas in 'red' PDA the decay quickens and diffusion coefficient increases at higher pump fluences. In this latter case exciton-exciton annihilation effects can be considered to be important.
To assess the role of defects fs-TAM measurements were repeated on thicker PDA crystals which have been shown to have lower energetic disorder 2 and less intrinsic defects. There is a negligible difference in the MSD traces between the two samples for both 'red' and 'blue' PDA as shown in Figure S4i (measurements performed under identical conditions). Fitting the traces gives diffusion coefficients of: blue Dthick = 27 ± 3 cm 2 s -1 , Dthin = 25 ± 2 cm 2 s -1 ; red Dthick = 8 ± 0.8 cm 2 s -1 , Dthin = 7.6 ± 0.8 cm 2 s -1 .
The difference in the electronic disorder between thick and thin crystals (~10%) is similar in magnitude to the difference in diffusion coefficient between these different materials. Tentatively this suggests that S14 disorder may be responsible for the observed differences, however further work is required to fully confirm this. The separation between excitons is tabulated below for each of the fluences in our measurements. The calculation is performed as follows: 1. The excitation density, n, can then be calculated from × where φ is the photon flux, OD the optical density of film at the peak excitation wavelength and d is the film thickness.
2. The volume (V) of the chain is then calculated from the minimal separation between chains, representing the chain space.
3. n/V will give the excitations per chain (n0), dividing this by the chain length ~2000-5000 nm will give the excitations per unit length (nm).  Although Figure S5 captures the spread in annihilation and diffusion coefficients at a single fluence, performing a fluence series at multiple sample locations allows us to further quantify the heterogeneity as shown in Figure S5i.

Exciton-phonon coupling and reorganization energy
In this section, we study the model PDA structure which is described in section S6. Following a geometry optimisation within density functional theory using the Quantum Espresso software package 10 and the PBE functional, we perform a phonon calculation using finite differences 11 .
We then calculate the exciton-phonon coupling elements for each mode as where a small displacement of the equilibrium structure. Hence for every vibrational mode, two GW-BSE calculations are necessary in order to obtain , . The Huang-Rhys factors are obtained as 12 with being the exciton coherence length, i.e. the number of diacetylene monomers that the exciton can travel over coherently, before being scattered by a phonon = • .
Here the exciton group velocity computed in section S6, the length of a single monomer of diacetylene, and the characteristic scattering timescale. The calculation of , reveals that the double-bond stretching motion has the steepest excited state surface, i.e. the largest , /ℏ value, and will dominate the scattering. We thus approximate = 23 fs, which is the period of this mode, and we find = 15 monomers. The values of , and are summarized alongside the frequencies of the respective vibrations in Table S2. From these, the reorganization energy is approximated as 13 = 2 ∑ ℏ = 0.43 eV.  20 . This corresponds to approximately 70 repeat units for the exciton coherence length. Therefore, our value of 15 is between the two extremes seems to be a reasonable estimate. We hence suggest the large coherence length in PDA reported to be reasonable and a consequence of the low disorder 21 and strong inter-monomer interactions 2 .

Mode frequency
From Table S2 we see that the carbon-carbon double-bond stretch (1488 cm -1 ) and the triple-bond stretch (2115 cm -1 ) are the vibrational modes most strongly coupled to the exciton. This can intuitively be understood from the fact that these motions allow the polymer chains to transiently explore configurations closer to the limit of a structure without Peierls distortion and hence to the metallic limit 22 . Another way of understanding this is that compared to other vibrational motions, these phonon modes lead to strong changes to the excitonic wavefunction once they are displaced, as visualized in Figure S6 for the case of the double-bond stretch. The electronic density of the exciton wavefunction is plotted in red for a hole localized at the blue dot in each case. We compare the case of the undistorted geometry, to that of displacing the structure along the weakly coupled lowest frequency mode, and to that of displacing along the double-bond stretching motion. The latter evidently leads to a significant change in the excitonic wavefunction, altering the bonding pattern of the electronic density. For visualization purposes, here we have exaggerated the displacement of the phonons compared to its values at room temperature. Figure S7: Visualization of exciton wavefunctions. The electronic density is given in red for a hole localized at the position in blue for a. the undistorted geometry b. displacement along the 389 cm -1 phonon, c. displacement along the 1488 cm -1 phonon (double-bond stretch).

Phonon-induced renormalization of excitonic observables
Observable quantities often undergo a renormalization at a given finite temperature due to the presence of phonons. In particular, when accounting for phonons, the value of some observable at temperature is 23 : where , the frequency and displacement of phonon mode with momentum , and the Bose-Einstein distribution. This quadratic approximation is accurate up to third order in , and we employ it in order to obtain the exciton energy, the excitonic coupling , and the magnitude of the transition dipole moment | | at 300 K. The obtained value becomes increasingly accurate with the inclusion of morepoints. For the coupling to excitons, phonons at the Γ ( = (0,0,0)) and X ( = ( 2 , 0,0)) points dominate the interaction 24 , and we hence restrict the calculation to these two points in order to keep the computational cost at a minimum (the evaluation of the second derivative for each mode requires two GW-BSE calculations, in the + and − directions).
The computed static exciton energy in the planar PDA is found to be 1.88 eV, with a renormalization of +14 meV due to phonons at 300 K. A strong static coupling of 0.54 eV is found (see section S6), with only a small renormalization of −5 meV. Finally, the absolute magnitude of the transition dipole moment is found to have a value of 1.82 a.u. at the ground state geometry, and accounting for phonons is found to lead to a significant renormalization of its value by −0.52 a.u.. While the quadratic approximation can be less accurate compared to a Monte Carlo sampling of the phonon-induced renormalization of the observable 25 , it has the advantage that it provides mode-resolved information.
Therefore, in Table S3 we report the specific contribution of the individual modes to the total renormalization reported here. For the transition dipole moment, which is the only quantity among the a b c S24 three studied ones with a non-negligible renormalization, the modes with frequencies 1488 cm -1 and 2115 cm -1 are found to dominate, as was also the case for exciton-phonon coupling.  Table S3. Renormalization of the exciton energy, excitonic coupling and absolute magnitude of the transition dipole moment due to individual phonon modes at 300 K.