{\it Ab initio} computations of structural and electronic properties of doped and undoped Ge nanowires

We report results of structural and electronic properties of hydrogen passivated doped and undoped Ge nanowires along [100], [110] and [111] growth directions using density functional theory in the local density approximation (LDA). Cross-sections of nanowires with diameters $\ $ d $>$ 2.0 nm are facetted reflecting the crystal symmetry about their axis. Nanowires along [100] direction with d below (above) 1.5 nm are found to be direct (indirect) band gap (E$_g$) semiconductors. Nanowires along [110] have direct E$_g$ for d $>$ 1.0 nm. Nanowires along [111] have indirect E$_g$ for d $>$ 1.0 nm. The magnitude of E$_g$ increases as the wire diameter decreases with values as high as 4.3 eV for a [100] wire with d $=$ 0.41 nm. For a fixed diameter E$_{g}^{[100]}$ $>$ E$_{g}^{[111]}$ $>$ E$_{g}^ {[110]}$. Doping with P or B did not have a significant effect on the valence and conduction band dispersions.