Electronic structure and optical transition of semiconductor nanocrystallites

The electronic states and optical transition properties of three semiconductor nanocrystallites, Si, GaAs, and ZnSe, are studied using the empirical pseudopotential homojunction model. The energy levels, wave functions, optical transition matrix elements, and lifetimes are obtained for quadratic prisms with widths from 11 to 27 Å. It is found that the three kinds of prism have different quantum confinement properties. For Si prisms, the energy gaps vary with the equivalent diameter d as d^-1.37, in agreement with previous theoretical calculations. For the same d the energy gaps are slightly different for different shapes: large for the prism with large aspect ratio; small for the prism with small aspect ratio. The exponent of d depends on the boundary barrier height, i.e. the extent of penetration of the wave function into the vacuum. The wave function of the LUMO states consists mainly of bulk X states. The optical transition matrix elements are much smaller than those of direct transition, and increase with decreasing width. The corresponding lifetimes decrease from the millisecond range to the microsecond range, and the change is abrupt depending on the symmetry and composition of the wave function of the LUMO and HOMO states. For GaAs prisms, the energy gap is also pseudo-direct, but the optical transition matrix elements are larger than those of Si prisms by two orders of magnitude for the same width. For ZnSe prisms, the energy gap is always direct, and the optical transition matrix elements are comparable with those of direct energy gap bulk semiconductors. In some cases the symmetry of the HOMO state changes, resulting in an abrupt decrease of the transition matrix element. The calculated lifetimes of the Si prism and the positions of PL peaks are in agreement with experimental results for porous Si.