Modeling the spatial resolution of magnetic solitons in magnetic force microscopy and the effect on their sizes

In this work, we explored theoretically the spatial resolution of magnetic solitons and the variations of their sizes when subjected to a magnetic force microscopy (MFM) measurement. Next to tip-sample separation, we considered reversal in the magnetization direction of the tip, showing that the magnetic soliton size measurement can be strongly affected by the magnetization direction of the tip. In addition to previous studies that only consider thermal fluctuations, we developed a theoretical method to obtain the minimum observable length of a magnetic soliton and its length variation due to the influence of the MFM tip by minimizing the soliton's magnetic energy. We show that a simple spherical model for the MFM tip can capture most of the physics underlying tip-sample interactions, with the key requirement being an estimate of the magnetization field within the sample. Our model uses analytical and numerical calculations and prevents overestimating the characteristic length scales from MFM images. We compared our method with available data from MFM measurements of domain wall widths, and we performed micromagnetic simulations of a skyrmion-tip system, finding a good agreement for both attractive and repulsive domain wall profile signals and for the skyrmion diameter in the presence of the magnetic tip. In addition, the theoretically calculated frequency shift presents good qualitative agreement with experimental measurements. Our results provide significant insights for a better interpretation of MFM measurements of different magnetic solitons and will be helpful in the design of potential reading devices based on magnetic solitons as information carriers.