Polaron Delocalization in Conjugated Polymer Films

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