Magnetization plateaus in weakly coupled dimer spin system

I study a spin system consisting of strongly coupled dimers which are, in turn, weakly coupled in a plane by zigzag interactions. The model can be viewed as the strong-coupling limit of a two-dimensional zigzag chain structure typical, e.g., for the $\mathrm{}\left(\mathrm{ac}\right)\mathrm{}$ planes of ${\mathrm{KCuCl}}_3.$ It is shown that the magnetization curve in this model has plateaus at $\frac{1}{3}$ and $\frac{2}{3}$ of the saturation magnetization, and an additional plateau at $\frac{1}{2};$ the critical fields are calculated perturbatively. It is argued that for the three-dimensional lattice structure of the ${\mathrm{KCuCl}}_3$ family the plateaus at $\frac{1}{4}$ and $\frac{3}{4}$ of the saturation can be favored in a similar way, which might be relevant to the recent experiments on ${\mathrm{NH}}_4{\mathrm{}\mathrm{CuCl}}_3$ by Shiramura et al., J. Phys. Soc. Jpn. 67, 1548 (1998).