Quantum size effect in core-shell structured silicon-germanium nanowires

First-principles density functional theory has been employed to study the composition dependent quantum size effect in a series of silicon-germanium core-shell structured nanowires with the diameters ranging from $0.5\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}3.2\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$. Analysis of the calculated band gap energies in Si-core/Ge-shell and Ge-core/Si-shell structured nanowires shows a nonlinear composition dependence for nanowires with fixed diameter (fixed total number $N={N}_{\mathit{\text{Core}}}+{N}_{\mathit{\text{Shell}}}$ of Si and Ge atoms in the unit cell). In contrast, for nanowires with fixed core size and varying shell thickness, our calculation results reveal a striking linear blueshift of the direct band gap with composition. The obtained linear composition effect implies an inverse square relation between the energy of the fundamental band gap and the size of nanowire, in agreement with experimental observations. Our results provide useful guidelines for experimental gap engineering in the core-shell structured nanowire heterostructures.