Closure magnetization configuration around a single hole in a magnetic film

The basic problem of a single hole in a magnetic film with uniaxial anisotropy has been analyzed in detail by magnetic force microscopy, micromagnetic simulations, and an analytical model. The closure magnetization configuration can be described by two $-1/2$ half vortices located at the hole edge along the easy anisotropy axis and confined within a distance $L$ that is determined by the minimization of magnetostatic and anisotropy energies constrained by the magnetic charge conservation within the system.

Figure 1

(Color online) [(a)–(c)] MFM images of the domain configuration around several patterned holes in a Co-Zr thin film. Double arrows indicate the uniaxial anisotropy easy axis in each image. (a1), (b1), and (b2) MFM signal along the indicated profiles taken across the different walls [indicated by solid lines in panels (a) and (b)].

Figure 2

(Color online) MFM image of one of the walls emanating from the patterned hole. Top inset: MFM signal along a profile taken across the wall (indicated by solid line). Bottom inset: evolution along the wall of the MFM signal at the maximum and minimum contrast points (dark and clear lines defining the wall).

Figure 3

(Color online) OOMMF simulations of the magnetic configuration around a hole in a Co-Zr magnetic film. (a) Map of the magnetization divergence around the hole. Inset shows the divergence profile taken across one of the walls (solid line) revealing an asymmetric charge configuration. (b) Same as in (a) but with an enhanced contrast by a factor of 3. Arrows indicate the direction of the local magnetization.

Figure 4

(Color online) Size of the closure structure around a hole normalized by the hole radius $R$ (a) as a function of exchange constant $A$ ($K=8\times {10}^2\text{J}/{\text{m}}^3$, $M_s=2\times {10}^5\text{A}/\text{m}$); (b) as a function of uniaxial anisotropy $K$ $\left(M_s=2\times {10}^5\text{A}/\text{m},A=1\times {10}^{-11}\text{J}/\text{m}\right)$; and (c) as a function of saturation magnetization $M_s$ $\left(K=8\times {10}^2\text{J}/{\text{m}}^3,A=1\times {10}^{-11}\text{J}/\text{m}\right)$: circles along the easy axis $\left(L_x/R\right)$ and triangles perpendicular to the easy axis $\left(L_y/R\right)$.

Figure 5

(Color online) (a) Magnetization lines around a nonmagnetic hole derived from $\psi \left(\rho ,\theta \right)$. (b) Magnetization divergence (i.e., magnetic charges) that is associated to the magnetization distribution in (a) upon imposing the condition of constant modulus $\left|\mathbf{M}\right|=M_s$ everywhere (note the mixture between positive and negative poles around the hole).

Figure 6

(Color online) Sketch of the domain structure around the hole after introducing two pairs of straight domain walls emanating from the singularity points at $\left(\pm 1,0\right)$. Also indicated are the area ${\Omega }_1$, the wall $C$, and the point $\left({\xi }_w,{\eta }_w\right)$ that define the area of integration for the application of the Gauss theorem.

Figure 7

(Color online) (a) Closure domain structure around the hole with walls described by Eq. (6) with $a=1$. (b) Linear density of magnetic charges $\lambda$ along the wall calculated for $a=1$. Arrow indicates the point ${\xi }_w$ at which Gauss theorem is satisfied.

Figure 8

(Color online) (a) Normalized magnetostatic energy ${\epsilon }_m$ (circles) and normalized anisotropy energy ${\epsilon }_k$ (squares) as a function of domain-wall parameter $a$. Dashed line is a fit to $1/a$ dependence and solid line is a linear fit. (b) Total system energy vs $a$ for different values of $\kappa$. Note that the minima in the curves are displaced to larger values of $a$ as $\kappa$ decreases.

Figure 9

(Color online) $\kappa$ dependence of the size of the closure structure along the hard axis obtained from micromagnetic simulations: triangles varying $M_s$ ($K=8\times {10}^2\text{J}/{\text{m}}^3$, $A=1\times {10}^{-11}\text{J}/\text{m}$) and squares varying $K$ ($M_s=2\times {10}^5\text{A}/\text{m}$, $A=1\times {10}^{-11}\text{J}/\text{m}$). Solid line corresponds to a $L_y\sim {\kappa }^{-0.4}$ dependence.