A fourth-order Schrödinger equation for charge transport in semiconductors operating in

the ballistic regime is introduced, incorporating non-parabolic effects in the dispersion

relation and thereby going beyond the simple effective-mass approximation. As in the

standard second-order formulation, the problem is confined to a finite spatial domain

and equipped with transparent boundary conditions to simulate charge transport in a

quantum coupler, where an active device region is connected to leads acting as reservoirs.

Several analytical properties of the model are established, and a new expression

for the current is derived. Numerical results highlight the main qualitative features of

the solutions. In particular, interference effects appear due to the richer wave structure

induced by the fourth-order formulation—effects absent in the effective-mass approximation.

Building on this approach, a hierarchy of models is further developed, each governed

by a Schr¨odinger equation of progressively higher order. Several analytical properties of

these generalized models are analyzed and a unified current formula valid for any order

is derived. Numerical simulations of a resonant tunneling diode (RTD) demonstrate the

impact of the generalized formulation on device behavior.