The single-site anisotropy vanishes for the spin-1/2 as a consequence of Kramers degeneracy. We argue that similar property holds for the magnetically induced electric polarization P, which should depend only on the relative orientation of spins in the bonds but not on the direction of each individual spin. Thus, for insulating multiferroic compounds, P can be decomposed in terms of pairwise isotropic, antisymmetric, and anisotropic contributions, which can be rigorously derived in the framework of the superexchange (SE) theory, in an analogy with the spin Hamiltonian. The SE theory also allows us to identify the microscopic mechanism, which stands behind each contribution. The most controversial and intriguing one is antisymmetric or spin-current mechanism. In this work, we propose that the disputed magnetoelectric (ME) properties of Ba2CuGe2O7 can be explained solely by the spin-current mechanism, while other contributions are either small or forbidden by symmetry. First we explicitly show how the cycloidal spin order induces the experimentally observed P in the direction perpendicular to the xy plane, which can be naturally explained by the spin-current mechanism operating in the out-of-plane bonds. Then, we unveil previously overlooked ME effect, where the application of the magnetic field perpendicular to the plane not only causes the incommensurate-commensurate transition, but also flips P into the plane due to the spin-current mechanism operating in the neighboring bonds within this plane. In both cases, the magnitude and direction of P can be controlled by rotating the spin pattern in the xy plane. Our analysis is based on a realistic spin model, which was rigorously derived from the first-principles calculations and supplemented with the new algorithm for the construction of localized Wannier functions obeying the crystallographic symmetry of Ba2CuGe2O7.
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