  Circulating orbital currents, originating from the spin-orbit interaction,
are calculated for semiconductor nanostructures in the shape of spheres, disks,
spherical shells and rings for the electron ground state with spin oriented
along a symmetry axis. The currents and resulting orbital and spin magnetic
moments, which combine to yield the effective electron g factor, are calculated
using a recently introduced formalism that allows the relative contributions of
different regions of the nanostructure to be identified. For all these
spherically or cylindrically symmetric hollow or solid nanostructures,
independent of material composition and whether the boundary conditions are
hard or soft, the dominant orbital current originates from intermixing of
valence band states in the electron ground state, circulates within the
nanostructure, and peaks approximately halfway between the center and edge of
the nanostructure in the plane perpendicular to the spin orientation. For a
specific material composition and confinement character, the confinement energy
and orbital moment are determined by a single size-dependent parameter for
spherically symmetrical nanostructures, whereas they can be independently tuned
for cylindrically symmetric nanostructures.
