1 Motivation and introduction

The temperature dependence of magnetocrystalline anisotropy of pure ferromagnets has been well known for decades, following the Callen-Callen theory [106], who established a scalling law of the anisotropy K with magnetization M. Examples of the systems following this law and classed as pure ferromagnets in the anisotropic sense include Gd with uniaxial anisotropy ( $ K\propto M^{3}$) and Fe with cubic anisotropy ( $ K\propto M^{10}$), in contrast there are other systems such as magnetic transition metal alloys that exhibit more complex temperature dependence. For example, the anisotropy of FePt alloy has been demonstrated to follow $ K\propto M^{2,09}$ law [107] or CoPt alloy that shows a $ K\propto M^{2}$ dependence [108].

Recently, the high temperature behavior of magnetic anisotropy has become important due to the applications in heat-assisted magnetic recording (HAMR) [109,110,111]. The idea of HAMR is based on the heating of the recording media to decrease the writing field of the high anisotropy media (such as FePt) to values compatible with the writing fields provided by conventional recording heads. Since the writing field is proportional to the anisotropy field $ H_{\mathrm{k}}=2K(T)/M(T)$, the knowledge of the scaling behavior of the anisotropy $ K$ with the magnetization $ M$ has become a paramount consideration for HAMR[112]. It should be noted that even in relatively simple systems, a simple scaling behavior predicted by the Callen-Callen theory is only valid at temperatures far from the Curie temperature. The systems proposed for HAMR applications can also include more complex composite media such as soft/hard bilayers [113], FePt/FeRh with metamagnetic phase transition [114,115], or exchange-bias systems [7].

The evaluation of the temperature dependence of magnetic anisotropy is also important for the modeling of the laser-induced demagnetization processes. The thermal decrease of the anisotropy during the laser-induced demagnetization has been shown to be responsible for the optically-induced magnetization precession [116]. Thus the ability to evaluate the temperature dependence of the anisotropy in complex systems at arbitrary temperatures is highly desired from the fundamental and applied perspectives.

In this sense magnetic thin films with surface anisotropy are a representative example of this more complicated situation. Since the surface anisotropy has a different temperature dependence from the bulk, multiple experiments on thin films have demonstrated the occurrence of the spin re-orientation transition from an out-of-plane to in-plane magnetization as a function of the temperature and the thin film thickness [63,117,118,54,66,119,56,120]. The possibility to engineer the re-orientation transition also requires the capability to evaluate the temperature dependence of the surface anisotropy independently from the bulk.

In this chapter we introduce the constrained Monte Carlo method (in the following CMC) for the calculation of the macroscopic temperature dependence of the magnetic anisotropy. We will show the possibility to calculate the temperature-dependent anisotropies in principle with this method. When CMC is combined with detailed magnetic information, such as that available from ab-initio methods (see Ref. [121] and also chapter 6), forms a very powerful method of engineering the temperature dependent properties of a magnetic system. The flexibility of the constrained Monte Carlo method allows for a thorough investigation of the temperature dependence of the Néel surface anisotropy. Since the constrained Monte Carlo is a new method first we test the method and later we investigate the temperature behavior of the surface anisotropy in thin films.

Thin films have attracted a lot of research interest over the past 50 years and so a large body of experimental data exists [122,118,66]. Nevertheless, achieving good experimental data on the temperature dependence of surface anisotropy requires the creation of very thin films with very sharp interfaces, which has only been technologically feasible within the last decade. This is because the influence of surface anisotropy is usually determined by varying the thickness of the magnetic layer, so that volume and surface contributions can be separated. For thick films the volume component strongly dominates the overall anisotropy, leading to a large degree of uncertainty in the strength of the surface contribution.

The present chapter is divided into the following sections:

In section 4.2 we describe the constrained Monte Carlo algorithm and perform several tests.
In the section 4.3 we present the results of the temperature dependence of the effective magnetic anisotropy in thin films with Néel surface anisotropy.
And finally, in section 4.4 we summarize the conclusions of these studies.

Rocio Yanes