  In superconductors that lack inversion symmetry, the flow of supercurrent can
induce a non-vanishing magnetization, a phenomenon which is at the heart of
non-dissipative magneto-electric effects, also known as Edelstein effects. For
electrons carrying spin and orbital moments a question of fundamental relevance
deals with the orbital nature of magneto-electric effects in conventional
spin-singlet superconductors with Rashba coupling. Remarkably, we find that the
supercurrent-induced orbital magnetization is more than one order of magnitude
greater than that due to the spin, giving rise to a colossal magneto-electric
effect. The induced orbital magnetization is shown to be sign tunable, with the
sign change occurring for the Fermi level lying in proximity of avoiding
crossing points in the Brillouin zone and in the presence of superconducting
phase inhomogeneities, yielding domains with opposite orbital moment
orientation. The orbital-dominated magneto-electric phenomena, hence, have
clear-cut marks for detection both in the bulk and at the edge of the system
and are expected to be a general feature of multi-orbital superconductors
without inversion symmetry breaking.
