Calculation method for the conductance of mesoscopic systems: assembly of Tight Binding Hamiltonians with spin effects-revisited.

We discretize the Schrödinger equation in the approximation of the effective mass for the two-dimensional electron gas of GaAs, without magnetic field and on the other hand, with magnetic field. This discretization leads naturally to Tight Binding (TB) Hamiltonians in the approximation of the effective mass. An analysis of this discretization allows us to gain insight into the role of site and hopping energies, which allows us to model the TB Hamiltonian assembly with spin: Zeeman and spin-orbit coupling effects, especially the case Rashba. With this tool we can assemble Hamiltonians of quantum boxes, Aharanov-Bohm interferometers, anti-dots lattices and effects of imperfections, as well as disorder in the system. The extension to mount quantum billiards is natural. We also explain here how to adapt the recursive equations of Green's functions for the case of spin modes, apart from transverse modes, for the calculation of conductance in these mesoscopic systems. The assembled Hamiltonians allow to identify the matrix elements (depending on the different parameters of the system) associated with splitting or spin flipping, which gives a starting point to model specific systems of interest, manipulating certain parameters. In general, the approach of this work allows us to clearly see the relationship between the wave and matrix description of quantum mechanics. We discuss here also, the extension of the method for 1D and 3D systems, for the extension apart from the first neighbors and for the inclusion of other types of interaction. The way we approach the method, has the objective of showing how specifically the site and hopping energies change in the presence of new interactions. This is very important in the case of spin interactions, because by looking at the matrix elements (site or hopping) we can directly identify the conditions that can lead to splitting, flipping or a mixture of these effects. Which is essential for the design of devices based on spintronics. Finally, we discuss spin-conductance modulation (Rashba spin precession) for the states of an open quantum dot (resonant states). Unlike the case of a quantum wire, the spin-flipping observed in the conductance is not perfectly sinusoidal, there is an envelope that modulates the sinusoidal component, which depends on the discrete-continuous coupling of the resonant states.