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 \(dip = \mu_0 \frac{M_s^2}{2K_D}\).
