Statistics for electrons, solitons, and polarons intrans-polyacetylene and consequences for conductivity

The equations that govern the temperature variation, in thermal equilibrium, of electron populations in soliton (kink) and polaron levels and the conduction band are set up for trans-polyacetylene. These are the electrical neutrality equation, the relations between chemical potentials of electrons, holes, solitons, and polarons, and the relations between the chemical potentials and the concentrations of the different excitations. It is pointed out that the lifetime predicted for a conduction electron against dropping into a polaron level, short though it is, is comparable to scattering times for conduction electrons in some typical three-dimensional (3D) semiconductors; it therefore does not preclude treatment of electrons in the conduction band of polyacetylene as quasiparticles. From the relations derived for the concentrations of solitons and polarons it is deduced that in trans-(CH${)}_x$ polaron concentration is negligible compared with soliton concentration in thermal equilibrium within the range of doping where both can exist, i.e., up to ∼5% or 6%. The 300-K conduction-electron concentration in trans-(CH${)}_x$ with 5% doping is found to be ∼${10}^{18}$/${\mathrm{cm}}^3$. On theoretical grounds, and according to x-ray and other types of evidence, dopant ions are, however, generally not random in their distribution as was assumed in the calculation of the thermal-equilibrium level populations. As a result a 5% dopant concentration, say, and correspondingly high electron concentration might be achieved in some local regions when the average doping is much less than 5%. With much lower doping in the surrounding regions, however, a local high electron concentration would not have much effect on dc conductivity. It could, however, contribute to high-frequency conductivity. Recent measurements of Genzel et al. may show this contribution.