Theory of bipolaron lattices in quasi-one-dimensional conducting polymers

We present a theory of the bipolaron lattice in the Peierls-distorted system with nondegenerate ground states. First, we obtain an exact solution for the order parameter of a bipolaron lattice in terms of the Weierstrass ζ function for the continuum model of Brazovskii and Kirova. Making use of the exact solution, we determine the complete band structure of a bipolaron lattice. In addition to the conduction and valence bands, we find two bipolaron bands symmetrically located about the middle of the Peierls energy gap. We also calculate the electronic density of states and the energy of formation of a bipolaron lattice as a function of the bipolaron density $n_b$. The chemical potential of a bipolaron determines the domain of stability of the lattice as a function of the confinement parameter γ. In the weak-coupling limit (infinite momentum cutoff Λ) the bipolaron-bipolaron interaction decays as a repulsive exponential while for finite Λ it decays inversely proportional to the distance and is attractive. This yields the possibility of phase separation in doped quasi-one-dimensional conducting polymers with nondegenerate ground states. For higher dopant concentrations we describe a possible first-order phase transition to a metallic phase consisting of an array of polaronlike distortions from a lattice of charged bipolarons. As well, the optical absorption spectrum of a bipolaron lattice is briefly discussed. Our results apply to cis-polyacetylene as well as to a class of heterocyclic compound polymers, namely, polypyrroles, polythiophenes, and their derivatives. Finally, we make contact with recent experiments which study the evolution of ordered phases during the electrochemical doping of polyparaphenylene by alkali atoms and indicate the possibility of a bipolaron lattice.