Conductivity as a function of conjugation length: Theory and experiment for conducting polymer complexes

The dependence of electronic conductivity upon the length (L) of conjugated segments is calculated for conducting polymers for which (1) interruptions in conjugation on neighboring chains are uncorrelated, (2) all chains are equivalent, and (3) intersegment hopping controls chain-direction conductivity. If interchain carrier hopping dominates, the calculated ratio of conductivities parallel (${\sigma }_1$) and perpendicular (${\sigma }_2$) to the chain is ${\sigma }_1$/${\sigma }_2$=$L^2$/6$d^2$F, where d is the interchain separation in the hopping direction and F is a geometric factor which is near unity for directions in which ${\sigma }_2$ is maximum. If intrachain hopping between conjugated segments dominates transport, then this calculated anisotropy ratio is linearly proportional to conjugation length. Derived temperature, pressure, dopant level, and electric field dependencies for the anisotropy ratio are generally near zero only for the case where interchain hopping dominates both ${\sigma }_1$ and ${\sigma }_2$. The calculated dependence of conductivity on conjugation length is in good agreement with observations for iodine-doped polyacetylene, potassium-doped poly(p-phenylene), and iodine-doped polythiophene. Relationships are derived between the conjugation length and the temperature dependencies of conductivity components, which are also supported by experimental results. For conducting polymers having short conjugation lengths, conductivity is predicted to be approximately proportional to exp[-(T/$T_0$ ${\left)\right]}^{\mathrm{-}1/3}$, where $T_0$ is inversely proportional to conjugation length. This expression, which can also be obtained for variable-range hopping, is here derived for nearest-neighbor hopping with a distribution of activation energies stemming from a distribution of conjugation lengths.