9.1 Introduction

The recent advances in microfabrication techniques [103] have stimulated interest in the properties of submicron sized patterned magnetic elements [104,105]. Promising applications include magnetic random access memory, high-density magnetic recording media, and magnetic sensors [106]. However, in order to exploit the special behavior of magnetic nanoelements it is necessary to study and understand their fundamental properties. We have studied the static properties of cylindrical magnetic nanodots of different sizes and aspect ratios with analytical models and numerical finite element (FE) simulations, especially magnetic vortex states.

Direct experimental evidence for the existence of these magnetic vortex states has been found by the method of magnetic force microscopy. Shinjo and coworkers [107] have used magnetic force microscopy (MFM) to characterize magnetic nanodots of permalloy (Ni$_{80}$Fe$_{20}$) with a thickness of 50 nm and a radius between 300 nm and 1 $\mu$m, for example. An MFM image is given in Fig. 9.1. It shows the magnetic contrast of nanodots with different radii. The dark spots in the center of the nanodots indicate the position of the vortex core, where the strongest stray field is sensed by the MFM tip. However, the lateral resolution is not high enough to estimate the diameter of the vortex core. In addition, the the MFM tip is sensitive only to the out-of-plane component of the stray field gradient, and the interaction between the magnetization of the nanodot and the MFM tip plays an important role for the contrast. These problems can be overcome using spin-polarized scanning tunneling microscopy and the direct observation of the magnetization distribution in nanoscale iron islands with magnetic vortex cores have been reported [108]. Lorentz transmission electron microscopy allows in situ magnetizing experiments with thin samples and it has been used to characterize the magnetization distribution in individual circular and elliptical particles [109].

Figure 9.1: MFM image of nanodots with 50 nm thickness and different diameters (0.3 to 1 $\mu$m) [107]. The dark spots in the center of the dots indicate the magnetic vortex core, where the MFM detects the stray field caused by the perpendicular magnetization.

The hysteresis loops of magnetic nanodots have been measured by vibrating sample magnetometer [105] and magneto-optical methods [110,111]. Single domain and vortex states have been successfully identified. Furthermore, these magnetic vortex states are an interesting object for high frequency magnetization dynamics [112] experiments, which are important for high-density magnetic recording media, where high-frequency field pulses of the magnetic write head store the information by reversing the magnetization.

In most of the simulations the material parameters given in Tab. 9.1, which are typical of permalloy (Ni$_{80}$Fe$_{20}$), have been used.

Table 9.1: Typical material parameters of permalloy (Ni$_{80}$Fe$_{20}$).

Saturation magnetization	$M_{\mathrm{s}}$	$8\times 10^{5} \mathrm{A/m}=8\times 10^{2} \mathrm{G}$
Saturation polarization	$J_{\mathrm{s}}= \mu_0 M_{\mathrm{s}}$	$\approx 1 \mathrm{T}$
Exchange constant	$A$	$13\times 10^{-12} \mathrm{J/m}=$
		$1.3\times 10^{6} \mathrm{erg/cm}$
Exchange stiffness constant	$C=2A$	$26\times 10^{-12} \mathrm{J/m}$
Anisotropy has been neglected.