Multi-scale modeling of edge effect on band gap offset in polygonal cross-section Silicon nanowires

The band gap offset is an effect of coordination numbers (CNs) of atom reduction at the edge of transversal cross-section of Silicon nanowires (SiNWs). In this paper, a hierarchical multi-scale technique is developed to model the edge effect on the band gap shift of SiNWs since the geometric effect is dominant in the energy gap due to the appearance of strain in the self-equilibrium state. The multi-scale model is performed based on the molecular dynamics approach and finite element method for the micro- (atomistic) and macro-scale levels, respectively. The Cauchy-Born (CB) hypothesis is used to relate the atomic positions to the continuum field through the deformation gradient. Finally, the applicability of proposed multi-scale model is illustrated in numerical simulations of four SiNWs cross-sections, i.e. the circular, hexagonal, rectangular and triangular, and the results are compared with the fully atomistic model, experimental data and analytical solution.