  Superconducting magnetoelectric effect, which is the current-induced
magnetization in a superconductor, mainly focused on the spin magnetization in
previous studies, but ignore the effect of the orbital magnetic moments carried
by the paired Bloch electrons. In this work, we show that orbital magnetic
moments in superconductors can induce large orbital magnetization in the
presence of a current. We constructed a unified description for the
current-induced spin and orbital magnetization across the superconductivity
normal metal phase transition. We find that in a superconductor with uniform
pairing, the current-induced magnetization at a given current density is the
same as that in its normal metal state, while with the nonuniform
superconducting pairing, the current-induced magnetization exhibits an abrupt
change in magnitude near the superconductivity normal metal phase transition.
Importantly, our theory predicts the orbital magnetoelectric effect in
superconducting twisted bilayer graphene which has paired Bloch electrons with
large orbital magnetic moments and negligible spin-orbit coupling. We propose
that the measurement of the current-induced orbital magnetoelectric effect can
be used to detect the possible nonuniform pairings in twisted bilayer graphene
and other newly discovered superconductors with non-trivial Berry curvatures.
