Boundary conditions for electron tunneling in complex two- and three-dimensional structures

We present boundary conditions given in integro-differential form for the single-particle two-dimensional (2D) or 3D Schrodinger equation, which allows for a treatment of nontrivial geometries, and an arbitrary number of input and output channels. The formalism is easy to implement using standard finite element packages. We consider a resonant dot structure and transport through a ringlike waveguide without barriers. The current in the dot is focused on an ellipsoid dot via a tunneling tip. The current-voltage characteristic is calculated for this system at the temperature 4.2 K. Our results show that the current maxima appear close to the eigenstates of the quantum dot. We show, however, that only those modes which obey certain symmetry properties give rise to resonance in the dot, and current maxima are absent for antisymmetric modes at low temperatures. The current in the waveguide is shown to be a resonant function of the voltage, and the system exhibits current feedback and turbulence. Finally, we extend the formalism to other types of channels and equations other than the Schrodinger equation and we discuss some possible applications for these systems.