Surface Roughness Dominated Pinning Mechanism of Magnetic Vortices in Soft Ferromagnetic Films

Although pinning of domain walls in ferromagnets is ubiquitous, the absence of an appropriate characterization tool has limited the ability to correlate the physical and magnetic microstructures of ferromagnetic films with specific pinning mechanisms. Here, we show that the pinning of a magnetic vortex, the simplest possible domain structure in soft ferromagnets, is strongly correlated with surface roughness, and we make a quantitative comparison of the pinning energy and spatial range in films of various thickness. The results demonstrate that thickness fluctuations on the lateral length scale of the vortex core diameter, i.e., an effective roughness at a specific length scale, provides the dominant pinning mechanism. We argue that this mechanism will be important in virtually any soft ferromagnetic film.

Figure 1

Contour maps of the gyrotropic frequency $f_G$ as a function of the in-plane static field for disk thicknesses of (a) 20, (b) 35, (c) 50, (d) 65, (e) 80, (f) 100, and (g) 130 nm. High frequency areas in the contour maps correspond to pinning sites, as indicated by the dashed circle in (c) for example. $\Delta H$ corresponds to the radius of a pinning site as defined in the text.

Figure 2

Thickness dependence of (a) the pinned ($f_{\mathrm{pin}}$) and the unpinned ($f_u$) gyrotropic frequency, (b) the averaged depinning field $\Delta H$, (c) the pinning energy $E_{\mathrm{pin}}$, and (d) the pinning range $D_{\mathrm{pin}}$. The solid line in (d) is the size of core obtained from the micromagnetic simulation. The inset shows a schematic of how the total potential energy changes in an applied field.

Figure 3

(a) Roughness power spectral density (PSD) as a function of wave vector (lines), obtained from the Fourier transforms of atomic force micrographs (AFM). The solid Gaussian curve is a weighting function, which is determined from a fit of the probability distribution of $\pi /D_{\mathrm{pin}}$ in (b). (b) Probability distribution of the pinning range as a function of $\pi /D_{\mathrm{pin}}$. Data are obtained from the pinning sites in Fig. 1. This distribution is used to determine the normalized weighting function shown by the solid Gaussian curve in (a). (c) Effective roughness ${\sigma }_{\mathrm{eff}}$ versus the disk thickness $L$. The two points shown in open symbols are additional 50-nm thick films prepared with different roughness characteristics.

Figure 4

Pinning energy versus (a) the rms roughness, and (b) the effective roughness (on the length scale of the core diameter). The solid line is a linear fit. (c) Pinning energy normalized by the effective roughness versus disk thickness. Data obtained on the 50 nm thick sample #2 are indicated by open circles. Data obtained on the 50 nm thick sample #3 are indicated by open triangles. The dashed line indicates the average value of $E_{\mathrm{pin}}/{\sigma }_{\mathrm{eff}}$.