Approximate construction of new conservative physical magnitudes through the fractional derivative of polynomial-type functions: A particular case in semiconductors of type Al_xGa_1-xAs

The fractional calculus has a very large diversification as it relates to applications from physical interpretations to experimental facts to the modeling of new problems in the natural sciences. Within the framework of a recently published article, we obtained the fractional derivative of the variable concentration x(z), the effective mass of the electron dependent on the position m(z) and the potential energy V (z), produced by the confinement of the electron in a semiconductor of type Al_xGa_1-xAs, with which we can intuit a possible geometric and physical interpretation. As a consequence, it is proposed the existence of three physical and geometric conservative quantities approximate character, associated with each of these parameters of the semiconductor, which add to the many physical magnitudes that already exist in the literature within the context of fractional variation rates. Likewise, we find that the fractional derivatives of these magnitudes, apart from having a common critical point, manifest self-similar behavior, which could characterize them as a type of fractal associated with the type of semiconductor structures under study.