Capacitance in open quantum structures

R-matrix Formalism for Electron Scattering in Two Dimensions with Applications to Nanostructures with Quantum Dots

“…The transversal component of the density current j ⊥ (x ⊥ , x , x 3 ) is zero in leads, because φ n (x ⊥ ) are real functions. The component x 3 of the incident density current is also zero, either due to the confinement in the third direction, like in Cartesian geometry [47], or due to the symmetry reasons like for the cylindrical geometry [31]. What remains is the longitudinal component of the particle density current j (x ⊥ , x , x 3 ), which provides after the integration over the cross section of the lead with the corresponding measure, dΩ, the very well-known relations for the transmission and reflection probabilities.…”

“…These effects can be incorporated in the formalism, but within this paper we neglect them for the simplicity of the exposure. For the Cartesian geometry [47], in comparison with Eq. (4), we have…”

“…We have to mention that for the Cartesian coordinates [47], the measures are dΩ = dy, dx = dx, while for the cylindrical geometry [31] they are dΩ = rdr, dx = dz.…”

“…The eigenfunctions φ n (x ⊥ ) depend on the geometry (Cartesian or cylindrical) and on the confinement potential. In the case of a hard wall confinement, the transversal modes are given for the Cartesian geometry by sine functions [47], while for the cylindrical geometry they are expressed in terms of the Bessel functions of the first kind [31]. The transversal modes form an orthonormal and complete system of functions.…”

“…as a sum of an incoming component on the channel (sn) and a linear combination of outgoing components on each scattering channel. One can write the scattering wave functions in a compact form [47] …”