Theory of defects in one-dimensional systems: application to Al-catalyzed 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 applying the bulk formalism and the possible passivation of the surface. We also develop a formula for the Madelung correction for general dielectric tensors. We apply this formalism to the technologically important case of Al-nanoparticle-catalyzed Si nanowires, obtaining Al concentrations significantly larger than in their bulk counterparts and predicting the fast consumption of the nanoparticles when the wires are grown on n-type substrates.