Towards the Analytical Generalization of the Transcendental Energy Equation, Group Velocity, and Effective Mass in One-Dimensional Periodic Potential Wells with a Computational Application to Common Coupled Potentials

Featured Application: The findings presented in this work will allow the calculation of the electronic density states as well as the electrical and optical properties of complex one-dimensional coupled periodic potential wells without using expensive licensed software. The analytical generalization for N periodic potential wells coupled to a probe rectangular-like potential and a zero potential is extremely important in the study of one-dimensional periodic potentials in solid state physics, e.g., in the calculation of transport, optical, and magnetic properties. These findings raise the possibility of calculating equations for the generalization of N arbitrary potentials related to any potential (Formula presented.) using special functions as a solution. In this work, a novel analytical generalization of the transcendental energy equation, group velocity, and effective mass for N-coupled potentials to a probe one-dimensional potential (Formula presented.) was proposed. Initially, two well-known linear periodic potentials (Formula presented.) were employed to obtain analytical solutions for rectangular-like and Dirac-delta potentials. Python libraries were used to easily represent the equations for one or two rectangular-like potentials coupled with an arbitrary potential, highlighting the transcendental energy, group velocity, and effective mass. The results showed that the group velocity behavior changed its orientation due to the sign of the potential, whereas the width of the potential (Formula presented.) strongly influenced the group velocity behavior. The effective mass was also modified by the potential shapes, and their combinations, both effective mass and group velocity, exhibited similar physical behaviors to those found in ordinary rectangular-like potentials.