Abstract
The Schr\"odinger equation for an electron in a mesoscopic ring, in the presence of the Rashba and linear Dresselhaus terms of the spin-orbit interaction (SOI) and of a magnetic field $B$, is solved exactly. The effective electric fields of these terms as well as $B$ have perpendicular and radial components. The interplay between them and $B$ and their influence on the spectrum is studied. The transmission through such a ring, with two leads connected to it, is evaluated as a function of the SOI strengths and of the orientations of these fields. The Rashba and Dresselhaus terms affect the transmission in different ways. The transmission through a series of rings with different radii and with SOI in both arms of the rings or only in one of them is also evaluated. For weak magnetic fields $B\ensuremath{\leqslant}1\phantom{\rule{0.3em}{0ex}}\mathrm{T}$ the influence of the Zeeman term on the transmission, assessed by perturbation theory, is negligible.
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