In this paper, classical thermionic emission model is employed to explore the formulas and their evolution for derivation of thermionic emission current in semiconductor Schottky junctions under three-dimensional, two-dimensional, and one-dimensional scenarios. These extended formulas can be widely applied to current research hotspot of interface barrier extraction and analysis in low-dimensional semiconductors, and should further contribute to enhancing students' understanding of semiconductor knowledge and the integrated application of related scientific research and analytical methods.