How Many Parameters Actually Affect the Mobility of Conjugated Polymers?

We describe charge transport along a polymer chain with a generic theoretical model depending in principle on tens of parameters, reflecting the chemistry of the material. The charge carrier states are obtained from a model Hamiltonian that incorporates different types of disorder and electronic structure (e.g., the difference between homo- and copolymer). The hopping rate between these states is described with a general rate expression, which contains the rates most used in the literature as special cases. We demonstrate that the steady state charge mobility in the limit of low charge density and low field ultimately depends on only two parameters: an effective structural disorder and an effective electron-phonon coupling, weighted by the size of the monomer. The results support the experimental observation [N. I. Craciun, J. Wildeman, and P. W. M. Blom, Phys. Rev. Lett. 100, 056601 (2008)] that the mobility in a broad range of (polymeric) semiconductors follows a universal behavior, insensitive to the chemical detail.

Figure 1

Electronic energy landscape for the lowest energy eigenstates of a disordered polymer chain of 1000 monomers with periodic boundary conditions and parameters of model 5. Each eigenstate is represented by a horizontal segment centered on the site where $c_{ni}^2$ is maximum, whose length is the localization length, defined as $2{(⟨n^2⟩-{⟨n⟩}^2)}^{1/2}$. The transitions between eigenstates that contribute most to the steady state mobility are represented by arrows. They are selected to make up 90% of the total particle velocity (definition given in Supplemental Material [18]). One should note the increased delocalization at higher energy and the importance of delocalized states in promoting long range displacement of the carrier.

Figure 2

Mobility as a function of temperature computed using different values for the reorganization energy of the single site ${\lambda }_1$. Results are shown for disorder models 5 and 12 (but are similar for any disorder model). They are computed with model (a) (see the caption of Fig. 3) for the inducing modes electron-phonon coupling.

Figure 3

Mobility as a function of temperature for disorder models 5 and 12 computed using four different models for the inducing modes. The inducing mode frequencies $\hslash {\omega }_k^I$ and coupling strengths $M_k^2$ have been set as follows. Model (a), one low energy mode (6.2 meV); model (b), four modes from low to intermediate energy (6.2, 12.4, 37.2, 49.6 meV); model (c), five modes from low to high energy (6.2, 12.4, 37.2, 49.6, 186 meV); model (d), one high energy mode (186 meV). The values of $M_k^2$ have been chosen to be identical for all inducing modes with strength such that ${\sum }_k{M}_k^2=4.0\times {10}^{-8}{\mathrm{eV}}^2$ to reproduce the experimental range of mobilities.

Figure 4

Top: experimental hole mobility vs $1/T$ for a range of organic semiconductors, adapted from Ref. [4] and augmented with additional data points [28]. Bottom: mobility vs $1/T$ from various models of chains of 1000 monomers with a variety of disorder parameters (see Table 1), including an alternating copolymer (model 14). The lines are obtained from a least squares linear fitting of log $\mu$ vs $1000/T$.