The invention relates to heterostructures including a layer of a two-dimensional material placed on a multiferroic layer. An ordered array of differing polarization domains in the multiferroic layer produces corresponding domains having differing properties in the two-dimensional material. When the multiferroic layer is ferroelectric, the ferroelectric polarization domains in the layer produce local electric fields that penetrate the two-dimensional material. The local electric fields can influence properties of the two-dimensional material, including carrier density, transport properties, optical properties, surface chemistry, piezoelectric-induced strain, magnetic properties, and interlayer spacing. Methods for producing the heterostructures are provided. Devices incorporating the heterostructures are also provided, including tunable sensors, optical emitters, and programmable logic gates.