Quantum confinement effect in thin quantum wires

The electronic states and optical transition properties of three semiconductor wires Si, GaAs, and ZnSe are studied by the empirical pseudopotential homojunction model. The energy levels, wave functions, optical transition matrix elements, and lifetimes are obtained for wires of square cross section with width from 2 to 5 ($\sqrt{2}$a/2), where a is the lattice constant. It is found that these three kinds of wires have different quantum confinement properties. For Si wires, the energy gap is pseudodirect, and the wave function of the electronic ground state consists mainly of four bulk Δ states. The optical transition matrix elements are much smaller than that of a direct transition, and increase with decreasing wire width. Where the width of wire is 7.7 Å, the Si wire changes from an indirect energy gap to a direct energy gap due to mixing of the bulk ${\mathrm{\Gamma }}_{15}$ state. For GaAs wires, the energy gap is also pseudodirect in the width range considered, but the optical transition matrix elements are larger than those of Si wires by two orders of magnitude for the same width. However, there is no transfer to a direct energy gap as the wire width decreases. For ZnSe wires, the energy gap is always direct, and the optical transition matrix elements are comparable to those of the direct energy gap bulk semiconductors. They decrease with decreasing wire width due to mixing of the bulk ${\mathrm{\Gamma }}_1$ state with other states. All quantum confinement properties are discussed and explained by our theoretical model and the semiconductor energy band structures derived. The calculated lifetimes of the Si wire, and the positions of photoluminescence peaks, are in good agreement with experimental results.