Multiferroicity due to charge ordering

In this contribution to the special issue on multiferroics we focus on multiferroicity driven by different forms of charge ordering. We will present the generic mechanisms by which charge ordering can induce ferroelectricity in magnetic systems. There is a number of specific classes of materials for which this is relevant. We will discuss in some detail $(i)$ perovskite manganites of the type (PrCa)MnO$_3$, $(ii)$ the complex and interesting situation in magnetite Fe$_3$O$_4$, $(iii)$ strongly ferroelectric frustrated LuFe$_2$O$_4$, $(iv)$ an example of a quasi one-dimensional organic system. All these are "type-I" multiferroics, in which ferroelectricity and magnetism have different origin and occur at different temperatures. In the second part of this article we discuss "type-II" multiferroics, in which ferroelectricity is completely {\it due to} magnetism, but with charge ordering playing important role, such as $(v)$ the newly-discovered multiferroic Ca$_3$CoMnO$_6$, $(vi)$ possible ferroelectricity in rare earth perovskite nickelates of the type RNiO$_3$, $(vii)$ multiferroic properties of manganites of the type RMn$_2$O$_5$, $(viii)$ of perovskite manganites with magnetic E-type ordering and $(ix)$ of bilayer manganites.

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