Nonlinear quantum motions in 2D nano-channels part I: Complex potential and quantum trajectories

As the size of electronic devices is narrowed down to the nanoscale, quantum effects become so prominent that the conventional Newton mechanics is no longer able to provide an accurate description for the electrons moving in nanostructures. On the other hand, the probabilistic description provided by quantum mechanics requires a large enough ensemble of electrons to yield the representative mean motion consistent with probability prediction. However, in some nanoelectronic devices, moving electrons may be so few that they do not averagely exhibit the mean motion predicted from quantum mechanics. Under such a circumstance, we need a new method that can describe not only the particle behavior of an individual electron, but also the wave behavior of an ensemble of electrons. In this paper, the complex-valued Newton mechanics is shown to possess the desired ability of manifesting the wave-particle duality of electrons moving in nanostructures. The first part of this paper establishes the nonlinear quantum dynamics characterizing the motion of individual electrons and finds their quantum trajectories in the presence of channelized quantum potential. The second part is devoted to showing how the collective motion of electrons yields the phenomenon of conductance quantization and the various wave behaviors such as tunneling, transmission and reflection within narrow-channel nanostructures.