Magnetization Patterns in Ferromagnetic Nanoelements as Functions of Complex Variable

The assumption of a certain hierarchy of soft ferromagnet energy terms, realized in small enough flat nanoelements, allows us to obtain explicit expressions for their magnetization distributions. By minimizing the energy terms sequentially, from the most to the least important, magnetization distributions are expressed as solutions of the Riemann-Hilbert boundary value problem for a function of complex variable. A number of free parameters, corresponding to positions of vortices and antivortices, still remain in the expression. Thus, the presented approach is a factory of realistic Ritz functions for analytical (or numerical) micromagnetic calculations. Examples are given for multivortex magnetization distributions in a circular cylinder, and for two-dimensional domain walls in thin magnetic strips.

Figure 1

A cylinder with Cartesian coordinate system axes.

Figure 2

A multivortex magnetization distribution in a circular cylinder, comparable to the numerical simulations of Ref. 20. It is Eqs. (2, 5, 7, 8) with $h=4$, $\alpha =\pi /4$, $e_1=0.01$, $e_2=400$, $(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},\overrightarrow{d})=(\{ı,-4+4ı,6+7ı\},\{3\},\{2ı,6+2ı\},\{\})$. There are three vortices, two antivortices, and two side-bound Skyrmions (their centers, where the magnetization vector is vertical, are marked with dots).

Figure 3

Some domain wall configurations in thin strip: (a) the vortex wall (comparable to the numerical simulation of Ref. 13); (b) antivortex wall; (c) and (d) two different arrangements of a pair of two vortex walls. As before, dots mark the singularities in the meron. It is Eqs. (2, 5, 7, 9) with $e_1=0.01$, $e_2=400$, and ($\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$, $\overrightarrow{d}$): (a) $(\{I\},\{0\},\{\},\{5,-1/5\})$; (b) $(\{\},\{0,5,-1/5\},\{I\},\{\})$; (c) $(\{90ı,ı/90\},\{0\},\{\},\{360,1/360,-20,-1/20\})$; (d) $(\{90ı,ı/90\},\{0\},\{\},\{360,-1/360,-20,1/20\})$.