  We investigate theoretically electron spin states in one dimensional (1D) and
two dimensional (2D) hard-wall mesoscopic rings in the presence of both the
Rashba spin-orbit interaction (RSOI) and the Dresselhaus spin-orbit interaction
(DSOI) in a perpendicular magnetic field. The Hamiltonian of the RSOI alone is
mathematically equivalent to that of the DSOI alone using an SU(2) spin
rotation transformation. Our theoretical results show that the interplay
between the RSOI and DSOI results in an effective periodic potential, which
consequently leads to gaps in the energy spectrum. This periodic potential also
weakens and smoothens the oscillations of the persistent charge current (CC)
and spin current (SC) and results in the localization of electrons. For a 2D
ring with a finite width, higher radial modes destroy the periodic oscillations
of persistent currents.
