2020-01-15

Computes the skyrmion profile $$m_z(r)$$ and the skyrmion radius $$R_\text{Sky}$$ of an axially symmetric Neel-type skyrmion tube in an infinite ferromagnetic film of finite thickness $$t$$ with perpendicular magnetocrystalline anisotropy $$K$$ and constant modulus of the magnetization density $$M_s$$, described by the spin-stiffness constant $$A$$ and the Dzyaloshinskii-Moriya interaction (DMI) $$D$$ including the magnetostatic self-energy $$E_\text{mag}$$ due to dipole-dipole interactions under an external magnetic field $$B$$ assuming a constant magnetization profile along the tube.

We consider the micromagnetic energy functional for the continuous magnetisation direction $$\mathbf{m}(\mathbf{r})$$:

$E(\mathbf{m}) = E_\text{exc} + E_\text{dmi} + E_\text{anis} + E_\text{zeeman} + E_\text{mag}, \text{ where}$

$E_\text{exc} = A \int (\nabla \mathbf{m})^2 \mathrm{d}^3\mathbf{r},$

$E_\text{dmi} = D \int \left[(\mathbf{m} \cdot \nabla)m_z – m_z(\nabla \cdot \mathbf{m})\right] \mathrm{d}^3\mathbf{r},$

$E_\text{anis} = K \int m_z^2 \mathrm{d}^3\mathbf{r},$

$E_\text{zeeman} = B M_s \int m_z \mathrm{d}^3\mathbf{r},$

and we have the magnetostatic energy $$E_\text{mag}$$.

To simplify the discussion, we introduce cylindrical coordinates for the real space vector $\mathbf{r} = \left(\begin{array}{c} x = r \cos(\phi) \\ y = r \sin(\phi) \\ z = z \end{array}\right)$ and spherical coordinates for the magnetisation direction $\mathbf{m}(\mathbf{r}) = \left(\begin{array}{c} m_x = \sin(\theta) \cos(\psi) \\ m_y = \sin(\theta)\sin(\psi) \\ m_z = \cos(\theta)\end{array}\right).$

And, finally, we invoke the circular domain wall approximation by setting$\theta(r) = \sum_{\pm} \arcsin \left( \tanh \left( -\frac{r \pm c}{w /2}\right) \right) + \pi$ and $$\psi = \phi$$.

Note that this approximation neglects variations of the profile under change of the $$z$$ coordinate and that we have two free parameters $$c$$ and $$w$$. By fixing the skyrmion phase $$\psi=\phi$$ we restrict ourselves to Neel type skyrmions – however, an extension to arbitrary skyrmion phases is straight forward.

By evaluating the energy functional for our profile and performing a numerical optimization it is now possible to find the energy-minimizing values of $$c$$ and $$w$$. From these the skyrmion radius $$R_{Sk}$$ is found – defined as the value of $$r$$ for which $$m_z = 0$$.

Within the approximation of the profile, the dipolar interactions are taken into full account. They introduce an additional dependence on the film thickness $$t$$.

To arrive at universal results, independent of any superfluous scale, a reduced unit system is employed.

• Distances are scaled by the characterisic length $$L_D = \frac{A}{|D|}$$.
• The external field is scaled by $$H_D = \frac{A}{\mu_0 M_s L_D^2}$$, leading to $$h = \frac{B}{\mu_0 H_D}$$.
• The magneto-crystalline anisotropy is scaled by $$K_D = \frac{A}{L_D^2}$$, leading to $$k = \frac{K}{K_D}$$.
• We further have the energy scale $$E_0 = t A$$ and the strength of dipolar interactions $$dip = \mu_0 \frac{M_s^2}{2K_D}$$.

Values taken from https://www.nature.com/articles/nnano.2015.313, “Methods”.

Values taken from https://www.nature.com/articles/s42005-018-0029-0, “Discussion”.

Values taken from https://www.nature.com/articles/s41565-018-0255-3, Figure 4b.

Citation

“Semi-analytical approaches for the radius of skyrmions in thin magnetic films”, B. Zimmermann, F. Lux, M. Sallermann, S. Bluegel, in preparation