Periodic superstructures in tetrahedrally bonded homopolymers

Using a lattice sum over single bipolaron potentials displaced by periodicity d, we have analytically obtained a solution for a bipolaron lattice for a tetrahedrally bonded homopolymer within the continuum model of Rice and Phillpot. This solution is used to derive the band structure, which consists of two bipolaron bands symmetrically located about the middle of the band gap in addition to the conduction and valence bands. The electronic density of states, chemical potential, and the energy of formation of a bipolaron lattice are also calculated as a function of the bipolaron density b. The bipolaron chemical potential lies between the conduction-band edge and the upper edge of the upper bipolaron band, indicating that the bipolaron lattice is energetically the most favorable charge configuration at low b. In the strict weak-coupling limit (infinite momentum cutoff ) the bipolaron-bipolaron interaction is found to be repulsive and varies with bipolaron density as (1/b)exp(-2/bp), p being the bipolaron characteristic length. Thus, the bipolaron lattice is stable only in the range 0b<2/p. This suggests the possibility of a phase separation of a doped homopolymer into conducting bipolaron droplets at b=2/p, while the rest of the system is insulating. Our results apply to polysilylenes, polygermylenes, and their derivatives, as well as to a wide class of carbon-based polymers.