Theory of defects in one-dimensional systems: the case of Al in Si nanowires

The energetic cost of creating a defect within a host material is given by the formation energy. Here we present a formulation allowing the calculation of formation energies in one-dimensional nanostructures, which overcomes the difficulties involved in defining the chemical potential of the constituent species and the possible passivation of the surface. We also develop a formula for the Madelung correction for general dielectric tensors and computational cell shapes. We apply this formalism to the formation energies of charged Al impurities in silicon nanowires, obtaining concentrations significantly larger than in their bulk counterparts.