On the magnetoelectric coupling in (Ni, Cu)B$_2$O$_4$

The mechanism of the magnetoelectric coupling of spins of Ni$^{2+}$ and Cu$^{2+}$ ions with the applied electric field in (Ni, Cu)B$_2$O$_4$, caused by the coupling of $3d$ electrons with the electric field and by the spin–orbit coupling, is analyzed. It is shown that the ordering of spins in the ab crystallographic plane of the crystal induces the electric polarization along the c axis of the crystal, and this polarization is mainly related to nickel ions. To obtain the electric polarization in the CuB$_2$O$_4$ antiferromagnet, it turns out to be necessary to apply the magnetic field in the ab plane.