Topological Hall effect in diffusive ferromagnetic thin films with spin-flip scattering

So far, most electrical measurements of magnetic skyrmions have been based on the interpretation that the topological Hall (TH) resistivity is proportional to the number of skyrmions multiplied by the magnetic flux quantum, which only applies to bulk systems in the strong exchange-coupling and nondiffusive regimes. In this work, we theoretically studied the TH effect in diffusive ferromagnetic metal thin films by solving a Boltzmann transport equation in the presence of spin-flip scattering. We found that, even in the strong exchange limit, the TH effect itself is not topologically protected in the presence of spin diffusion, owing to the spin accumulation built up in the vicinity of the skyrmions. A more general formula for the TH resistivity was derived, which establishes the relation between the TH resistivity and the ratio of the spin diffusion length to the skyrmion radius and would be increasingly important for extracting information about skyrmion density from experimental data when the size of room temperature skyrmions is further reduced to tens of nanometers, close to the spin diffusion lengths of most transition metal ferromagnetic thin films.