Atomistic treatment of depolarizing energy and field in ferroelectric nanostructures

An {\it atomistic} approach allowing an accurate and efficient treatment of depolarizing energy and field in {\it any} low-dimensional ferroelectric structure is developed. Application of this approach demonstrates the limits of the widely used continuum model (even) for simple test cases. Moreover, implementation of this approach within a first-principles-based model reveals an unusual phase transition -- from a state exhibiting a spontaneous polarization to a phase associated with a toroid moment of polarization -- in a ferroelectric nanodot for a critical value of the depolarizing field.