Band structures of atomic chains of group IV, III–V, and II–VI elements

The relativistic band structures of AN and ANB8–N chains have been calculated by the linear augmented-cylindrical-wave method, which is an extension of the augmented-plane-wave method for cylindrical polyatomic systems. The band structures of covalent monatomic chains of Group IV elements are characterized by σ(s), π+, and π–, and σ(pz)* bands. The C, Si, Ge, and Sn chains are metallic. There is a considerable difference between the relativistic and nonrelativistic band structures. Because of the cylindrical symmetry of chains in the nonrelativistic model, the π bands crossing the Fermi level are orbitally doubly degenerate (i.e., the π+ and π– band energies are exactly the same). The spin and orbital motion of electrons are coupled in the chains to split π bands, but each π+ and π- band is doubly spin-degenerate. The spin–orbit splitting energy for C and Sn chains varies from 1.7 meV to 0.67 eV. The mass–velocity correction reduces all valence band levels: the level shifts are 2–5 meV for C and up to 2.2 eV for Sn. The Darwin corrections are several-fold lower than the mass–velocity contributions. A sharp change in the band structure is observed in going from covalent to partially ionic chains. The carbon chain has a metallic band structure with a zero gap in the center of the Brillouin zone, and a boron nitride chain is an insulator with an optical gap of 8 eV and optical transitions between the occupied π and vacant π* states at the edge of the Brillouin zone (this is explained by the existence of the antisymmetric component of the electron potential in the BN wire, which mixes even bonding and odd antibonding π states). Going from the BN chain to the AlP, GaAs, and InSb chains is accompanied by a decrease in the chemical bond ionicity, which leads to a gradual decrease in the π–π* gaps.