1. Introduction

The physical properties of magnetic materials have been utilized since the ability of loadstone (Fe$_2$O$_3$) to attract iron and to align in the earth's magnetic field was discovered. About 1000 years ago compasses were used by Chinese sailors [1]. In the 13th century their use became known in Europe.

Until the 19th century electric and magnetic effects were seen as two independent physical occurrences. In 1820 Oersted proved, that electric currents can influence the needle of a compass. Ampère and Faraday explained that behaviour and laid the foundation for the unified theory of electrodynamics, which was elaborated by James Clerk Maxwell.

Since then, permanent magnets have found many applications in energy transforming and storage devices. In microphones and electric generators they transform the mechanical energy of a membrane or a rotor into electric energy. In loudspeakers and motors the electric energy is transformed back into mechanical energy.

Around 1900 a new application has been found: magnetic recording [2]. Poulsen was the first who recorded acoustic signals on a ferromagnetic wire. In 1927 the magnetic tape, a paper tape coated with dried ferrimagnetic liquid, was invented in the USA and Germany, where a tape containing iron powder was used. The magnetic material was no longer a bulk magnet, but consisted of small magnetic particles. In the 1940's oxide tapes were developed and soon after the appearance of audio recording devices also video signals were stored on magnetic tape.

With the invention of digital computers there was need to store data on reliable and fast, yet easy to handle media. Magnetic tapes and later floppy disks and hard disks proved to be suitable storage devices. Storage capacity and density, access time and data transfer rate have improved constantly due to continuous research and development in magnetic materials. However, there is one parameter which can be hardly influenced: temperature. Products, which are not designed for a few highly specialized applications but for widespread use in a mass market must not require special environment conditions. They should operate at room temperature and normal pressure. As the space required by a piece of information in a magnetic storage device (the bit size) shrinks, the effects of temperature (thermal activation, thermal perturbations) become increasingly important. These can cause spontaneous switching of the magnetization, which modifies the stored data and results in data loss.

For future magnetic storage devices there are designs of ``quantum magnetic disks'' and magnetic random access memory devices (MRAMs). These consist of small magnetic elements (with typical dimensions of a few nanometres), where each element stores a single bit. The storage density is given by the size of and distance between the elements. The data writing speed is limited by the magnetization switching time. Projected data storage systems with a frequency greater than 250 MHz [3] will be a challenge for both head and media design, as the desired switching rate leads to writing times, which approach the intrinsic relaxation time of the media. Therefore, a precise understanding of the magnetization process is crucial for the optimal design of magnetic recording media. Micromagnetic simulations can provide information, which is experimentally not accessible. It is possible to study the magnetization switching process [4], optimize the shape [5], and the magnetization reversal time [6,7] of magnetic nano-elements.

It is the aim of this thesis to investigate the implementation and effects of thermal perturbations in micromagnetic simulations. The micromagnetic formalism is extended by a temperature dependent fluctuation field. The resulting stochastic differential equation is studied by a finite difference simulation program. Simple geometries like a rigid magnetic moment and small magnetic cubes are simulated. Then the algorithms are implemented in a finite element simulation package, which has been developed for deterministic problems and improved over the years [8]. It provides a reliable and flexible tool for micromagnetic simulations and my contribution is the implementation of time integration schemes for the solution of the Langevin type equation of motion for the magnetization.

In chapter  the basic theory of micromagnetism is explained and the Landau-Lifshitz equation of motion for the magnetization derived. Its solution by the finite difference and finite element method, and an outline of these methods for numerical computer simulations are described in chapters  and . Then we extend the theory to take into account thermal perturbations and find a stochastic differential equation in chapter . In chapter  stochastic calculus is summarized and we find the quantitative properties of the thermal field. For the numerical solution of our Langevin equation we develop suitable numerical integration schemes in chapter . The implementation of the finite difference and the finite element model as well as the numerical time integration schemes are explained in chapter . Then we study the behaviour of a rigid magnetic moment in chapter  before we go on to cubic and spherical particles in chapter , which are discretized into smaller computational cells. Furthermore we study the interaction of spherical particles. Finally, a review of experimental results is given.