R-Matrix Theory in a Semiconductor Quantum Device: Weak Formulation and Current Conserving Approximations

In a series of previous publications an R-matrix approach was developed to describe transport in N-pole quantum devices in the Landauer–Büttiker formalism. Central quantities in this formalism are the transmission coefficients occurring in coherent scattering functions ranging throughout the device. Here we develop the weak formulation version of this approach. It allows to introduce systematically approximations that reduce the exact problem to suitable matrix equations that can be solved on the computer. As a major advantage of our method any approximation found in the weak formulation approach to the R-matrix is current conserving by construction. Here the essential step is the representation of the current S-matrix by a Cayley transform. Restricting us to the one-dimensional case we find that the R-matrix in weak formulation generates a real symmetric Cayley transform by construction. From general theory it follows immediately that the current is conserved.