# Magnetic Properties of Gd-Doped Fe\({}_{3}\)O\({}_{4}\) Nanoparticles
## 1 Introduction  
Nanoparticles (NPs) have received a lot of attention in the last decade due to their unique properties and wide range of applications. The magnetic behavior of magnetic NPs greatly depends on their size, shape, purity and crystal structure. Magnetic Fe\({}_{3}\)O\({}_{4}\) is one of the most widely used nanomaterials for biomedical applications. [PERSON] et al. [1] performed self-consistent ab initio calculations and investigated the magnetic properties of bulk Fe\({}_{3}\)O\({}_{4}\). The magnetic behavior of spherical and cubic magnetite NPs are studied by [PERSON] et al. [2], [PERSON] et al. [3], [PERSON] et al. [4], [PERSON] et al. [5], [PERSON] et al. [6]. With decreasing particle size, the magnetic properties begin to change. Specifically, the lattice parameters are slightly enhanced [7]. Decreasing NP size leads to reduced ferrimagnetic behavior [8]. The saturation magnetization and the Curie temperature \(T_{\rm C}\) of Fe\({}_{3}\)O\({}_{4}\) NPs are lower than that of the bulk one [7, 9]. High-saturation magnetization in small Fe\({}_{3}\)O\({}_{4}\) NPs is observed by [PERSON] et al. [10] and [PERSON] et al. [11].  
The magnetic properties of magnetite Fe\({}_{3}\)O\({}_{4}\) can be achieved by ion doping. From a theoretical point of view, doping is a method by which the structural parameters of the crystal lattice can be drastically changed, causing shrinkage and stretching processes and changes in the nature of magnetic arrangement. Doping can significantly renormalize the magnetic phase transition temperature, magnetization, magnetic anisotropy and coercivity. The influence of rare earth (RE) doping on the magnetic properties of Fe\({}_{3}\)O\({}_{4}\) NPs is investigated by [PERSON] et al. [12]. They observed a reduction of the magnetization and increase of the band gap. The influence of Gd doping on structural and magnetic properties of Fe\({}_{3}\)O\({}_{4}\) NPs is studied by [PERSON] et al. [13]. [PERSON] et al. [14] also considered the magnetic properties of Gd-doped magnetite NPs.  
The aim of the present paper is to investigate theoretically for the first time the magnetic and optical properties of Fe\({}_{3}\)O\({}_{4}\) NPs using a microscopic model and the [PERSON]'s function theory.  
It should be noted that the most theoretical papers consider the magnetic properties of bulk Fe\({}_{3}\)O\({}_{4}\) using the density functional theory (DFT) [1, 15]. The DFT is a very powerful tool in investigation of many body problems. However, DFT is mostly concerned with ground-state properties at zero temperature. In our approach, we are able to cover thewhole temperature regime. It is a finite temperature analysis including the entire excitation spectrum. In particular, the method allows us to study the total phase diagram, which is based on the different excitation energies realized in the system. The disadvantage of our approach consists of the consideration of collective properties from the beginning. Our basic quantities are not the \"naked\" electrons, but the effective spins of the underlying quasi-particles. Whereas, within DFT, all of the parameters of the system can be--at least in principle--calculated, we are forced to use additional models to find out those parameters. We are convinced that both approaches, DFT and [PERSON]'s function method, are appropriate and, to a certain extent, are an alternative in describing many body systems.