9.2.1 The Analytical Rigid Vortex Model

The rigid vortex model assumes a ``rigid vortex'', which does not change its shape in an external field. Together with a certain magnetization distribution it gives an approximation for the magnetization distribution of a curling state (vortex state) in a fine cylindrical particle. An analytical model for the magnetization distribution $\boldsymbol{M}(\boldsymbol{x})$ in zero field has been developed using a variational principle by Usov and coworkers [113,114]. It is split into two parts (cf. Fig. 9.2)

Figure 9.2: Geometry of a flat cylindrical nanodot.

The first part describes the magnetization in the core of the vortex ($r \le a$, $a$ is the vortex core radius), which is defined by $M_z \neq 0$:

$$M_x = -\frac{2ar}{a^2+r^2} \sin \varphi$$ (9.1)

$$M_y = \frac{2ar}{a^2+r^2} \cos \varphi$$ (9.2)

$$M_z = \sqrt{1-\left( M_x^2+M_y^2 \right)}$$

$$= \sqrt{1-\frac{2ar}{a^2+r^2}}$$ (9.3)

where $r$, $\varphi$ are the polar coordinates. The other part describes the magnetization outside the core ($r>a$):

$$M_x = -\sin \varphi$$ (9.4)

$$M_y = -\cos \varphi$$ (9.5)

$$M_z = 0$$ (9.6)

$a$ denotes the radius of the core and it is given by

$$a = \left( \frac{l_\mathrm{ex}^2 R}{12 \kappa g} \right)^{1/3} \quad,$$

where $l_\mathrm{ex}$ (the exchange length) is given by

$$l_\mathrm{ex} = \sqrt{\frac{A}{\frac{1}{2}\mu_0 M_{\mathrm{s}}^2}} \quad,$$

$R$ is the radius of the nanodot, $\kappa$ is a numerical constant (cf. [113]) and $g$ is the ratio $R/L$, where $L$ is the height of the nanodot. For permalloy we find $l_\mathrm{ex}=5.7 \mathrm{nm}$ and with $R=100 \mathrm{nm}$ and $L=20 \mathrm{nm}$ we get $a \approx 11 \mathrm{nm}$. The core radius is obtained from the minimization of the total energy (exchange and magnetostatic energy).

There are some typical properties of the rigid vortex model with Usov's magnetization distribution:

• In equilibrium in zero field there are surface charges only on the top and bottom surface within the vortex core.
• For shifted vortices surface charges are induced on the circumference of the nanodot.
• There are no volume charges in the model.