  In noncentrosymmetric superconductors, previous studies on the
supercurrent-induced magnetization mainly focused on the spin magnetization
related to spin-orbit coupling (called the spin magnetoelectric effect or the
Edelstein effect). However, the Berry curvature effects of the normal state
band structure is ignored. In this work, we show that Berry curvatures in
superconductors can induce large orbital magnetization in the presence of a
supercurrent. We constructed a unified description for the current-induced spin
and orbital magnetization effect 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 phase, while with the nonuniform superconducting pairing, the
current-induced magnetization can have an abrupt change in magnitude near the
superconductor-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 measurement of the
current-induced orbital magnetoelectric effect can be used to detect the
possible non-uniform pairings in twisted-bilayer graphene and other newly
discovered superconductors with non-trivial Berry curvatures.
