Magnetic states in multiply-connected flat nano-elements

Flat magnetic nano-elements are an essential component of current and future spintronic devices. By shaping an element it is possible to select and stabilize chosen metastable magnetic states, control its magnetization dynamics. Here, using a recent significant development in mathematics of conformal mapping, complex variable based approach to the description of magnetic states in planar nano-elements is extended to the case when elements are multiply-connected (that is, contain holes or magnetic anti-dots). We show that presence of holes implies a certain restriction on the set of magnetic states of nano-element.