  The olivine compound Mn2GeO4 is shown to feature both a ferroelectric
polarization and a ferromagnetic magnetization that are directly coupled and
point along the same direction. We show that a spin spiral generates
ferroelectricity (FE), and a canted commensurate order leads to weak
ferromagnetism (FM). Symmetry suggests that the direct coupling between the FM
and FE is mediated by Dzyaloshinskii-Moriya interactions that exist only in the
ferroelectric phase, controlling both the sense of the spiral rotation and the
canting of the commensurate structure. Our study demonstrates how
multi-component magnetic structures found in magnetically-frustrated materials
like Mn2GeO4 provide a new route towards functional materials that exhibit
coupled FM and FE.
