Ballistic electron transport described by a fourth-order Schrödinger equation

A fourth-order Schrödinger equation for the description of charge transport in semiconductors in the ballistic regime is proposed with the inclusion of non-parabolic effects in the dispersion relation in order to go beyond the simple effective mass approximation. Similarly to the standard (second order) Schrödinger equation, the problem is reduced to a finite spatial domain with appropriate transparent boundary conditions to simulate charge transport in a quantum coupler (Lent and Kirkner in J Appl Phys 67:6353, 1990; Ben Abdallah et al. in ZAMP 48:135–155, 1997; Ben Abdallah in J. Math. Phys. 41:4241–4261, 2000), where an active region representing an electron device is coupled to leads which take the role of reservoirs. Some analytical properties are investigated, and a generalized formula for the current is obtained. Numerical results show the main features of the solutions of the new model. In particular, an effect of interference appears due to a richer wave structure than that arising for the second-order Schrödinger equation in the effective mass approximation.