  The honeycomb antiferromagnet Co4Nb2O9 is known to exhibit an interesting
magnetoelectric effect that the electric polarization rotates at the twice
speed in the opposite direction relative to the rotation of the external
magnetic field applied in the basal ab-plane. The spin-dependent electric
dipole can be an origin of the magnetoelectric effect. It is described by the
product of spin operators at different sites (type-I theory) or at the same
site (type-II theory). We examine the electric polarization for the two cases
on the basis of the symmetry analysis of the crystal structure of Co4Nb2O9, and
conclude that the latter is the origin of the observed result. This paper also
gives a general description of the field-induced electric polarization on
honeycomb lattices with the C3 point group symmetry on the basis of the type-I
theory.
