Interaction between magnetic vortex cores in a pair of nonidentical nanodisks

Interacting magnetic nanoobjects constitute one key component for many proposed spintronic devices, from microwave nano-oscillators to magnetic memory elements. For this reason, the mechanism of this interaction and its dependence with distance $d$ between these nanoobjects has been the subject of several recent studies. In the present work, the problem of the interaction between magnetic nanodisks with magnetic vortex structures is treated both analytically and through micromagnetic simulation. The coupling of two magnetic nanodisks is obtained, in the case of nanodisks with different gyrotropic frequencies. The motion of the two vortex cores, and consequently the time dependence of the total magnetization $M(t)$ is derived using Thiele's equation and by simulation. The magnetic coupling strength as a function of the distance between the disks of each pair is computed with a recently developed method, using the magnetic vortex echoes, analogous to the NMR spin echoes. The results of the two approaches differ by approximately $10\%$ and can be fitted with a multipolar expansion of odd negative powers of $d$; using one single term, a dependence with distance found is broadly in agreement with studies employing other techniques. The coupling frequencies as a function of distance for the pair of nonidentical disks are also obtained.