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The dipolar 4-state Potts model : Infinite 1D chain

Now that the dipolar energy behavior in the case of two spins has been determined, it is possible to extend this study in the case of an infinite 1D chain of Potts 4-state spins. Obviously an infinite 1D chain leads to an infinite of configurations. In order to do a simple analytic analysis, we will essentially restrict ourselves to configurations with a period of two spins. And so the configurations selected in this part are those directly given by the 6 configurations determined in the case of two spins. Indeed to form the infinite 1D chain, it is selected a “unit mesh” composed of two spins and this unit mesh is repeated an infinite number of times. An example of this process is exposed in the figure 1-5, which represents the 1D chain with a unit mesh composed of two perpendicular spins. Considering 6 possible configurations between two Potts 4-state spins, it will also be the case for the infinite 1D chain. Note that this study does not give the fundamental state of the chain but it provides an indication of its behavior according to the alpha angle (angle between spins and chain axis) for the 6 energy levels corresponding to configurations with a period of 2.

Simple and 2 spins periodic configurations

The main purpose of this section is to determine the behavior of the ground state for an infinite lattice according to the alpha angle. As it is not possible to determine directly this ground state, a way is to observe firstly the behavior of simple configurations. Towards this goal, a possibility to realize these simple configurations is to define (as for the infinite chain) a unit mesh composed by 2×2 spins and then to repeat this one an infinite number of times. Obviously “manually” it is not possible, for a reasonable time, to calculate all the energies related to the 256 possible configurations for the unit mesh. Thus 4 simple configurations are selected and these configurations are chosen due to the type of coupling present in the unit mesh. Indeed thanks to the two spins study it appears that according to the alpha angle, some configurations that we know (see section 1.2) are lowest in energy. And so in observing the possible configurations for 2×2 spins, 4 configurations seem to be advantageous in term of energy, and these 4 configurations are represented in the figure 1-13.

Micromagnetism introduction: contribution of Brown free energy

As mentioned previously the support of this chapter is the micromagnetism. Indeed micromagnetic simulations are based on a theory developed in the 1940’s by W.F.Brown: the micromagnetism. This theory describes the properties of ferromagnetic environment [36] in taking the magnetization and also the different internal fields as continuous thermodynamic variables. In order to understand the different processes taking part in ferromagnetic environments as well as the difficulty to obtain a nanomagnet which can be considered as a macrospin (uniform magnetization in all the volume), it is essential to describe the different energies involved in the magnetization behavior for magnetic materials.
The magnetization behavior in magnetic material is fixed by the Brown free-energy minimization ETot. This energy is given in the following equation: 𝐸𝑇𝑜𝑡=𝐸𝐸𝑥𝑐ℎ𝑎𝑛𝑔𝑒+𝐸𝑍𝑒𝑒𝑚𝑎𝑛+𝐸𝐷𝑖𝑝𝑜𝑙𝑎𝑟+𝐸𝐴𝑛𝑖𝑠𝑜𝑡𝑟𝑜𝑝𝑦.
Where it appears 4 energies which are exchange energy, Zeeman energy, dipolar energy and anisotropy energy.
Thus in this section, the different energies reported in the equation (2-1) are described and we discuss the required compromises between these energies in order to minimize the Brown free-energy.

Micromagnetism as pathway to design artificial spin

In order to carry out micromagnetic simulations, the processing of the different equations governing the dynamic of the magnetization, impose to split the magnetic nanostructures in cells which can take several shapes (tetrahedral, orthorhombic, cubic…). In each cell, the parameters like magnetization, energy or effective field are fixed. According to the cells uniformity, two main micromagnetic models allow to describe the magnetization inside nanomagnets. The first is based on the finite elements method [42, 43, 44] and the second is based on the finite differences method [45, 46, 47, 48].
As part of this thesis, the software used is an open-source-GPU accelerated micromagnetic simulation program: Mumax3 [49]. This program is based on a finite elements space discretization, as it is the case for a lot of micromagnetic simulation programs like “The object Oriented MicroMagnetic Framework” (OOMF). The main advantage related to Mumax3 is the use of GPU in order to make the calculations, which allows a time calculation shorter than the others softwares. In the aim to use the finite differences method, the space is discretized in a structured grid (2D or 3D) composed of orthorhombic cells. Thus the volume quantities, like magnetization or effective field, are defined in the center of each cell while the coupling, like exchange, are defined at the interfaces between two cells. Moreover at each cells is associated a region with a value ranging from 0 to 256. These regions are independent and for each number related to one region, it can correspond a different material (with different parameters). This software allows also to define a lot of shapes for the nanomagnets. In this aim the geometry is defined as a function f(x, y, z), which gives true if (x, y, z) are inside geometric shape and false in the contrary case. In order to determine the magnetization dynamic, Mumax3 calculates the evolution of the reduced magnetization 𝑚⃗⃗ (𝑟 ,𝑡), where this reduced magnetization can presented a time and space dependence but where the amplitude is kept constant. Thus to determine the time and space dependence of the reduced magnetization, the program calculates the time derivative of the reduced magnetization which represents the torque 𝜏 =𝜕𝑚⃗⃗⃗ 𝜕𝑡, where 𝜏 possesses three contributions which are: the Landau-Lifshitz torque, the spin transfer torque of Zhang-Li and the spin transfer torque of Slonczewski. As part of this thesis, only the Landau-Lifshitz torque is relevant, the two other being related to electron transport phenomena.
To make micromagnetic simulations, the first point is to fix the different parameters use for the magnetic material, like exchange stiffness, anisotropy constant, saturation magnetization and Landau-Lifshitz damping constant. In this aim all the simulations presented in this study have been realized with iron parameters with the values shown in the following table.

The stability diagram of the monodomain state

As exposed in the beginning of this part, the selected material for the micromagnetic simulations presented in this section is iron with cubic anisotropy. Generally in the nanostructures, the equilibrium state is not necessarily monodomain but can be more complex (Landau structures, vortices…). Up to now in the artificial spin systems, the spin model is related to Ising spins and in order to design this spin, one find in the literature that a magnet with typical size around a hundred nanometers combined with an uniaxial shape anisotropy (elongated shape) is a good candidate [11, 12, 16]. In our case, where the aim is to design a spin with an uniform magnetization presenting 4 preferential directions, this elongated shape is not adapted, however one can assume that the nanometric scale for the magnet remains an appropriate choice. Moreover as it is required 4 preferential orientations for the magnetization in the nanostructure, the elongated shape seen in the literature is replaced by a shape presenting a cubic symmetry. Thus the cubic symmetry of the shape combined with the cubic anisotropy of Fe should allow the realization of an artificial spin which matches the spin used in the dipolar 4-state Potts model. Consequently, three shapes are selected for the simulations: a disc and two squares named square 0° and square 45° after the relative orientations of their borders with respect to the magnetic anisotropy axes (see figure 2-3). As exposed in the literature [50], it exists several magnetic configurations for magnet. Depending of the size and thickness of nanomagnet, different magnetic configurations arise like single domain uniform, several uniform domains separate by domain walls [51] or vortex structure where the magnetization continuously curls around the center with the magnetization in-plane and in the center of the core the magnetization is perpendicular to the plane [52]. Micromagnetic simulations have been used in order to determine the stable configuration of thin nanomagnet as a function of different parameters (thickness, size or magneto crystalline anisotropy). An example is shown in the figure 2-4 (extracted of the Ref [53]) where it is studied the single domain to flux closure (vortex) transitions in thin ferromagnetic disks of Co, according to a variable uniaxial magneto crystalline anisotropy.

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Validity of the dipolar approximation

Now that the best condition to realize experimentally the spin is determined, it is useful to study the coupling between these two spins. Indeed in the first part of this work, the theoretical study concerned dipolar interactions between spins (magnetic dipole) possessing 4 states. But experimentally even if the magnetization in nanostructure is monodomain, the magnetic configuration is not really a magnetic dipole but it has a spatial extension and some deviation from the strictly uniform state. So in this section, it is shown how the micromagnetism changes the interaction behavior between two spins possessing 4 states. In this aim, it’s interesting to study the energy levels given by the micromagnetic simulations and to compare with the energy levels expected in the case of two magnetic dipoles.
For these simulations, the energy of a system composed by two squares with a length of 300 nanometers is determined. The magnetic material is a 2 nanometers layer of iron with cubic anisotropy (to stabilize monodomain state) and these anisotropy axes are aligned with diagonals squares. The initial magnetization is always uniform along a diagonal square and the magnetic configurations in a square can take 4 preferential directions which are the four directions along diagonals square. For these simulations the system probes the 16 possible configurations between the magnetic configurations squares and give the total energy for each equilibrium state. Moreover with the theoretical study, it is determined that the energy levels of a system composed by two spins with four states depend on the angle between the two spins. So in these simulations two angle are studied: 0 and 45 degrees. As squares stay in the same position against the anisotropy axes, to define the angle, it’s the position between two squares which is modified. Indeed the angle alpha defined in the chapter 1 is represented in these simulations as the angle between mondomain direction in a square and the axis between two centers squares. On the figure 2-7, it is represented the configurations for an angle of 0 degree and 45 degrees.

Magnetometric study

Our sample composition includes two interfaces V/Fe. Yet, contrary to the interfaces MgO/Fe and Au/Fe, we should not find a significant value of the interface anisotropy for V/Fe interface. Although this interface has been widely studied [29, 69-71], we could not find any study providing a clear thickness dependence allowing to precisely determine this interface anisotropy value. Besides it is crucial to determine if with this composition, the sample obtained present effectively a cubic anisotropy and if it does not appear a supplementary anisotropy in the material due for example to a problem during the growth (crystallographic arrangement) or due to the interface anisotropies. This is why a detailed analysis of the magnetic properties is needed, and for this study the magnetization curves are measured using rotating sample vibrating sample magnetometer (VSM) and SQUID-VSM.
In order to obtain the interface anisotropies and all the magnetic properties of the sample, we have realized a magnetometry study on epitaxial V/Fe (t)/V trilayer, for different Fe thicknesses. And so the Fe layers of thickness t ranging from 0.7 nm (5 atomic layers) to 5 nm (35 atomic layers) were grown on V (20 nm) buffer and capped with V (5 nm)/Au (5 nm).

Fe volumic anisotropy

First, the volume (“bulk”) magnetization of a V/Fe (2 nm)/V film has been probed under both for in-plane and out-plane magnetic field in order to determine the magnetic easy and hard axes directions. The two hysteresis loops are presented in the figure 3-3.
Figure 3-3_ (a) Normalized magnetization versus field loop for in-plane along (100) Fe direction (black solid squares) and out-of-plane (open red circles) field respectively for a V/Fe (2 nm)/V stack. (b) Zoom of the main figure around zero field show square hysteresis cycle.
The figure 3-3 shows normalized magnetization versus field loops for in-plane (black squares) magnetic field applied along Fe (100) direction and for out-of-plane (red circles) magnetic field. Thus it appears that the out-of-plane direction corresponds to a hard axis direction for the magnetization in the Fe layer. Moreover the figure 3-3 (b) shows clearly that the Fe (100) direction correspond to an easy axis direction. Indeed the black curve represent a square loop with full magnetization at remanence which is characteristic of magnetic easy axis. Thus thanks to the figure 3-3 it can be conclude that, as expected for the Fe bulk, the magnetization in the Fe layer lies preferentially in the film plane. Now let’s confirm the quadratic anisotropy of the Fe layer. In this aim the normalized remanent magnetization extracted from hysteresis loops obtained for in-plane applied field, where the field is applied in all sample directions from 0 to 360 degrees with a step of 1 degree, is plotted in the figure 3-4 (c).

Table of contents :

1.1 The Potts model
1.2 The dipolar 4-state Potts model
1.3 The dipolar 4-state Potts model : Infinite 1D chain
1.4 The dipolar 4-state Potts model : 2D square lattice
1.4.1 Numerical issues
1.4.2 Simple and 2 spins periodic configurations
1.4.3 Monte Carlo simulations
1.5 Finite lattice
1.6 Summary
2.1 Micromagnetism introduction: contributions of Brown free energy
2.2 Micromagnetism as pathway to design artificial spin
2.2.1 The program code
2.2.2 The stability diagram of the monodomain state
2.2.3 Internal magnetic configurations
2.3 Validity of the dipolar approximation
2.4 Summary and perspectives
3.1 Sample preparation
3.1.1 The buffer
3.1.2 The iron deposition
3.1.3 The capping
3.2 Magnetometric study
3.2.1 Fe volumic anisotropy
3.2.2 Thermal stability
3.2.3 Magnetization versus Fe thickness
3.2.4 Interfaces anisotropy
3.3 Samples overview
3.4 Lattices and alpha definition
3.5 Nanofabrication
3.5.1 Ebeam lithography
3.5.2 Aluminum mask
3.5.3 Ionic etching
3.5.4 Dose optimization
3.6 Magnetic characterization and tip influence
3.6.1 Standard tip
3.6.2 Low moment tip
3.6.3 Low moment tip with spacer layer at surface sample Spacer layer of PMMA Spacer layer of Aluminum
3.7 Lattice distortion
3.8 Summary
4.1 Pathway to fundamental state: the demagnetization
4.1.1 AC field demagnetization Protocol Efficiency
4.1.2 Thermal demagnetization Protocol Thermally induced magnetization reversal as a function of square size Efficiency Heating time influence Temperature influence
5.1 Coupled Potts spin lattice: qualitative study
5.1.1 Magnetic configurations measured after field demagnetization
5.1.2 Magnetic configurations measured after thermal demagnetization
5.1.3 Thermal demagnetization performed after field demagnetization
5.2 Discussion: spins repartition and broken symmetry
5.3 Dipolar coupling effects
5.4 Comparison between demagnetization protocols
5.5 Some insights in the demagnetization process
5.6 Summary of the chapters 4 and 5
6.1 Conclusions
6.2 Perspectives


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