Theory of Vortex-Like Structures in Perforated Magnetic Films Accounting Demagnetizing Fields

Abstract

Vortex-like inhomogeneities that can appear in ferromagnetic films with strong easy-plane uniaxial anisotropy in the presence of antidotes in the form of artificial nanoscale holes or nonmagnetic inclusions are investigated. Peculiarities of the structure of such type of nanoobjects are considered for various geometries of holes, and the effect of demagnetizing fields on vortex-like inhomogeneities localized near one, two, or four holes is analyzed. It is shown that the analytic estimates obtained here are in all cases in good agreement with the results of numerical simulation.

APPENDIX A. CALCULATION OF ENERGY OF THE STATE LOCALIZED ON TWO HOLES

To begin with, we note that vector ∇ϕ in polar coordinate system (r, ϕ) is equal in magnitude to vector r/r² and forms a constant angle of π/2 with it. Taking this into account and substituting relation (2) into (1), we obtain

$$\begin{gathered} E = {{k}^{2}}Ah{{\int {\left( {\frac{{{{{\mathbf{r}}}_{1}}}}{{r_{1}^{2}}} - \frac{{{{{\mathbf{r}}}_{2}}}}{{r_{2}^{2}}}} \right)} }^{2}}dS \\ = {{k}^{2}}Ah\int {\frac{{{{a}^{2}}}}{{r_{1}^{2}r_{2}^{2}}}} dS = {{k}^{2}}Ah({{G}_{1}} + {{G}_{2}}), \\ \end{gathered} $$

where

$${{G}_{1}} = \int {\frac{{{{a}^{2}}}}{{r_{1}^{2}(r_{1}^{2} + r_{2}^{2})}}dS,\quad } {{G}_{2}} = \int {\frac{{{{a}^{2}}}}{{r_{2}^{2}(r_{1}^{2} + r_{2}^{2})}}dS.} $$

The integration domain for G₁ is the entire plane except circles r₁ < R₁ and r₂ < R₂. However, the integrand of the expression for G₁ in region r₂ < R₂ is close to a^–2; consequently, circle r₂ < R₂ can be included into the integration domain without the loss of accuracy of the result. Then

$${{G}_{1}} = \int\limits_0^{2\pi } {\int\limits_{{{R}_{1}}}^\infty {\frac{{{{a}^{2}}}}{{r_{1}^{2}(r_{1}^{2} + r_{2}^{2})}}{{r}_{1}}d{{r}_{1}}d{{\phi }_{1}}.} } $$

Evaluating this integral considering that $r_{2}^{2}$ = $r_{1}^{2}$ + a² + 2r₁acosϕ₁ and neglecting terms of order (R₁/a)⁴, we obtain G₁ = 2πln(a/R₁). Analogously, G₂ = 2πln(a/R₂), which gives expression (3).

1.1 APPENDIX B. CALCULATION OF ENERGY OF THE STATE LOCALIZED ON FOUR HOLES

The magnetization distribution corresponding to the third inhomogeneity in Fig. 3 is described by the following relation analogous to expression (2):

$$\theta = {{\phi }_{1}} - {{\phi }_{2}} + {{\phi }_{3}} - {{\phi }_{4}},$$

where the holes are numbered clockwise. Analogously to Appendix A, we obtain

$$E = Ah\int {{{{\left( {\frac{{{{{\mathbf{r}}}_{1}}}}{{r_{1}^{2}}} - \frac{{{{{\mathbf{r}}}_{2}}}}{{r_{2}^{2}}} + \frac{{{{{\mathbf{r}}}_{3}}}}{{r_{3}^{2}}} - \frac{{{{{\mathbf{r}}}_{4}}}}{{r_{4}^{2}}}} \right)}}^{2}}dS.} $$

Using identity

$$\begin{gathered} {{(a - b + c - d)}^{2}} = {{(a - b)}^{2}} + {{(b - c)}^{2}} \\ \, + {{(c - d)}^{2}} + {{(d - a)}^{2}} - {{(a - c)}^{2}} - {{(n - d)}^{2}} \\ \end{gathered} $$

and taking into account the system symmetry, we obtain E = Ah(4M₁ – 2M₂), where

$${{M}_{1}} = \int {{{{\left( {\frac{{{{{\mathbf{r}}}_{1}}}}{{r_{1}^{2}}} - \frac{{{{{\mathbf{r}}}_{2}}}}{{r_{2}^{2}}}} \right)}}^{2}}dS,\quad {{M}_{2}} = \int {{{{\left( {\frac{{{{{\mathbf{r}}}_{1}}}}{{r_{1}^{2}}} - \frac{{{{{\mathbf{r}}}_{3}}}}{{r_{3}^{2}}}} \right)}}^{2}}dS.} } $$

Disregarding the fact that the circles corresponding to holes 3 and 4 are not contained in integration domain of M₁ and repeating the arguments used in Appendix A, we obtain M₁ = 4πln(a/R). Integral M₂ differs from M₁ only in the distance between the centers of the circles, which equals a$\sqrt 2$ in the case of circles 1 and 3. Consequently, M₂ = 2πln(a$\sqrt 2$/R), which finally gives E = 8πAhln(a/R$\sqrt 2$).