Most multiferroic materials with coexisting ferroelectric and magnetic order exhibit cycloidal antiferromagnetism with wavelength of several nanometers. The prototypical example is bismuth ferrite (BiFeO$_3$ or BFO), a room-temperature multiferroic considered for a number of technological applications. While most applications require small sizes such as nanoparticles, little is known about the state of these materials when their sizes are comparable to the cycloid wavelength. This work describes a microscopic theory of cycloidal magnetism in nanoparticles based on Hamiltonian calculations. It is demonstrated that magnetic anisotropy close to the surface has a huge impact on the multiferroic ground state. For certain nanoparticle sizes the modulus of the ferromagnetic and ferroelectric moments are bistable, an effect that may be used in the design of ideal memory bits that can be switched electrically and read out magnetically.
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