Ferroelectricity in Incommensurate Magnets
  We review the phenomenology of coupled magnetic and electric order parameters for systems in which ferroelectric and incommensurate magnetic order occur simultaneously. We discuss the role that such materials might play in fabricating novel magnetoelectric devices. Then we briefly review the mean-field description of ferroelectricity and modulated magnetic ordering as a preliminary to analyzing the symmetry of the interaction between the spontaneous polarization and the order parameters describing long-range modulated magnetic ordering. As illustration we show how this formulation provides a phenomenological explanation for the observed phase transitions in Ni$_3$V$_2$O$_8$ and TbMnO$_3$ in which ferroelectric and magnetic order parameters simultaneously become nonzero at a single phase transition. In addition, this approach explains the fact that the spontaneous polarization only appears along a specific crystallographic direction. We analyze the symmetry of the strain dependence of the exchange tensor and show that it is consistent with the macroscopic symmetry analysis. We conclude with a brief discussion of how our approach might be relevant in understanding other systems with coupled magnetic and ferroelectric order, and more importantly, how these principles relate to the search for materials with larger magnetoelectric couplings at room temperature. 
A.B. Harris and G. Lawes