LOW-DIMENSIONAL AND DISORDERED SYSTEMS Linear diatomic crystal: single-electron states and large-radius excitons

the quasi-one-dimensional large-radius exciton problem can be reduced to a 1D system of two quasi-particles with a potential having a Coulomb attraction tail. Due to the parity of the interaction potential the exciton states should split into odd and even series. In Ref. 10 we show that for the bare and screened Coulomb interaction potentials the binding energy of even excitons in the ground state well exceeds the energy gap in the same work we also discuss the factors which prevent the collapse of single-electron states in isolated semiconducting single-walled carbon nanotubes SWCNTs. But the electron-hole e-h interaction potential and thus the corresponding exciton binding energies may depend noticeably on the electron and hole charge distributions. It is therefore worthwhile to ascertain whether the effect of seeming instability of single-electron states near the gap is inherent to the all quasi-one-dimensional semiconductors in vacuum or maybe takes place only in SWCNTs for the specific localization of electrons holes at their surface and weak screening by the bound electrons. That is why we consider here the simplest model of a quasi-one-dimensional semiconductor with cylindrical symmetry, namely the linear crystal with two atoms in the unit cell. The electrons holes in this crystal are simply localized at its axis. The aim of this work is only a qualitative analysis of the effect mentioned. For study of electron structure of the 1D crystal concerned, we apply here the zero-range potentials ZRP method 11,12 see Sec. II. The matter is that results on the band structure and single-electron states, obtained by this method for SWCNTs in Refs. 13 and 14, appeared to be in good accordance with the experimental data and results of ab initio calculations related to the band states. For specificity we use the linear crystal parameters the electron bare mass, lattice parameters taken from works on nanotubes. 13,14 In Sec. III we obtain the e-h bare interaction potential and that screened by the crystal band electrons, and then the largeradius exciton spectrum for the linear crystal in vacuum. All these data are used in Sec. IV, where we present results of calculations for crystals with different lattice periods which also means different band structures. As it turns out, the binding energy of even excitons in the ground state well exceeds 2 – 5 times the energy gap for the linear crystal in vacuum, and the screening by the crystal band electrons is negligible. Note that this result was obtained within the framework of exactly solvable ZRP model with feasible parameters. Therefore, the instability effect mentioned may take place not only for the considered simplest case but, most likely, also for other quasi-onedimensional isolated semiconductors in vacuum.