Ordered and Disordered Skyrmion States on a Square Substrate

We examine the ordering of skyrmions interacting with a square substrate created from a modulation of anisotropy using atomistic simulations. We consider fillings of {\it f} = 1.5, 2.0, 2.5, 3.0, and 3.5 skyrmions per potential minimum as a function of magnetic field and sample size. For a filling of {\it f} = 2.0, we find various dimer orderings, such as tilted dimer states, as well as antiferromagnetic ordering that is similar to the colloidal dimer ordering seen on square substrates. The ability of the skyrmions to change shape or annihilate produces additional states that do not occur in the colloidal systems. For certain parameters at {\it f} = 2.0, half of the skyrmions can annihilate to form a square lattice, or a superlattice of trimers and monomers containing skyrmions of different sizes can form. At lower fields, ordered stretched skyrmion states can appear, and for zero field, there can be ordered stripe states. For {\it f} = 3.0, we find ferromagnetic ordered trimers, tilted lattices, columnar lattices, and stretched phases. For fillings of {\it f} = 1.5 and 2.5, we find bipartite lattices, different ordered and disordered states, and several extended disordered regions produced by frustration effects.