Ballistic electron transport described by a generalized Schrödinger equation

We propose a Schrödinger equation (SE) of arbitrary order for modeling charge transport in semiconductors operating in the ballistic regime. This formulation incorporates non-parabolic effects through the Kane dispersion relation, thereby extending beyond the conventional effective mass approximation. Building upon the framework introduced in Aliffi et al (2025 Z. Angew. Math. Phys. 76 155), we derive a hierarchy of models, each governed by a SE of increasing order. As in the standard second-order formulation, the problem is formulated on a bounded spatial domain with suitable transparent boundary conditions. These conditions are designed to simulate charge transport in a quantum coupler where an active region - representing the electron device - is connected to leads acting as reservoirs. We investigate several analytical properties of the proposed models and derive a generalized expression for the current, valid for any order. This formula includes additional terms that account for interference effects arising from the richer wave structure inherent in higher-order SEs, which are not captured by the effective mass approximation. Numerical simulations of a resonant tunneling diode illustrate the key features of the solutions and highlight the impact of the generalized formulation on device behavior.