Number and density of states in quantum semiconductor structures

Pushed by the semiconductor industry to achieve greater speed and functionality, device dimensions are becoming sufficiently small to exhibit prominent quantum mechanical effects. In addition, devices are now being developed that utilize these quantum effects. The number and density of states are fundamentally important in the operation of any quantum device. Traditionally in a classroom setting, one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) continuum approximations are presented to analyze the quantum wire, well, and box, respectively. As shown in this paper, the exact number and density of states can be straightforwardly calculated by students for real semiconductor quantum structures. These results clearly illustrate the overall true 3-D form of each of these structures. These correct calculations also reveal an overestimation in the number of states when using the continuum approximations.