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In numerous functional materials, such as steels, ferroelectrics and magnets,new functionalities can be achieved through the engineering of the domainstructures, which are associated with the ordering of certain parameters withinthe material. The recent progress in imaging techniques with atomic-scalespatial resolution transformed our understanding of domain topology revealingthat, along with simple stripe-like or irregularly shaped domains, intriguingvortex-type topological domain configurations also exist. In this Review, wepresent a new classification scheme of "ZmxZn domains with Zl vortices" fortwo-dimensional macroscopic domain structures with m directional variants and ntranslational antiphases. This classification, together with the concept oftopological protection and topological charge conservation, can be applied to awide range of materials, such as multiferroics, improper ferroelectrics,layered transition-metal dichalcogenides and magnetic superconductors, as wediscuss through selected examples. The resulting topological considerationsprovide a new basis for the understanding of the formation, kinetics,manipulation and property optimization of domains and domain boundaries infunctional materials.

Aperiodic topological order in the domain configurations of functional materials