Table of contents
1 Introduction2
2 Micromagnetism2
2.1 Domains in thin films with perpendicular anisotropy3
2.2 RT films4
2.3 Thin film stripe domains in magnetic fields4
2.4 MFM and resonant x-ray scattering5
3 X-ray scattering studies of stripe domains 5 3.1 Simulations and MFM studies 6
3.2 Skyrmion7
4 OOMMF8
4.1 Magnetic constants8
4.2 Qualitative effect of parameters8
5 Results9
5.1 Comparison with 1-dimensional model9
5.2 Magnetization loop and vortex structure10
5.3 Magnetization loop: Vxx,Vyy and hysteresis11
5.4 Skyrmion structure13
5.5 Magnetization loop: contrast functions15
5.6 Comparison magnetization curve with 1-d model 17 5.7 Energy 19
6 Discussion23
6.1 First look23
6.2 X-ray diffraction data24
6.3 Nucleation and skyrmion collapse 25
6.4 The 1-dimensional model27
7 Conclusion28
8 Further research28
9 References29
Appendix 30
1 Introduction
Ferromagnetic objects consist of a great number of magnetic domains. These domains explain why objects made from ferromagnetic materials do not always seem to be magnetic from the outside. The study of magnetic domains has been important for electrical engineering, but even more important for data recording. Better understanding of magnetic domains has enabled magnetic storage technology to keep pace with semiconductor technology for the last few decades.
The formation of magnetic domains is a complex process that depends on the properties of the material but also on the size and shape of the object. In this thesis we studied thin-film systems where the magnetic structure becomes a one-dimensional vortex lattice.
We used simulations to study the influence of an magnetic field applied to the film on this structure, which transforms to a skyrmion structure at the coercive field. The skyrmion topology has been observed before in condensed-matter systems, for instance in Bose-Einstein condensates and 2-dimensional Quantum Hall systems and in thin-film chiral magnets 5,6.
The systems we studied have been studied before with resonant x-ray scattering and with magnetic force microscopy. In this thesis we compare the results with micromagnetic simulations and with earlier a theoretical model. We will start with an introduction of the of micromagnetism and of the films we studied. Then we discuss the results of the earlier research and continue with our research; the simulations we have done and the results. In the discussion we compare this to the resonant x-ray scattering studies and to the model. Where we conclude that our simulations confirm both the model and the x-ray diffraction experiments.
2 Micromagnetism
Magnetic domains are regions where the magnetization is uniform. Magnetic domain theory explains why objects made of ferromagnetic materials do not always appear to be magnetic from the outside. This is because the magnetization in the material has different directions in each domain, which reduces the external magnetic field created. The regions in between magnetic domains where the magnetization changes orientation are domain walls. When an external magnetic field is applied the domain walls shift to increase the size of the domains with magnetization parallel to the applied field.
Current domain theory is based on the work of Landau and Lifschitz whichuses the variational principle to find the configuration with the lowest energy.
Magnetic structures on the micro-scale are described by the micromagnetic equations.
The total energy of the system is given by:
The first term is the exchange energy density, given by the exchange constant times the gradient of the local magnetization squared.The second term is the anisotropy energy, associated with an easy axis (in this case the z-axis) to which the magnetic moment tends to align. Ku is the anisotropy constant.The third term is the Zeeman energy originating from an external field. The fourth term is the magnetostatic energy. This is the energy that comes from the magnetic field an object with a certain magnetization would create in free space. This term is dependant on the shape of the object and as Hd depends on M this term makes the problem nonlinear.
Other terms are the magneto-elastic energies, which describe the influence of the magnetic structure on the crystal lattice, but they are negligible in our case.
One of the characteristic lengths in a micromagnetic system is the Bloch wall width.
This is the thickness of the area between two magnetic domains; regions with different directions of magnetization. It ranges between a few nanometers and a hundred nanometers. There are two ways to make the transition from one domain to the other, a Bloch wall and a Neél wall. In a Neél wall the magnetization stays in the plane of the magnetization in the domains and in a Bloch wall the magnetization goes out of this plane. The surface energy of a 180° Bloch wall is
Fig. 1. Schematic picture of a Bloch wall (top) and a Neél wall (bottom).
2.1 Domains in thin films with perpendicular anisotropy
In the thin film samples studied here, the thickness was reduced to the order of the domain wall with, so there is only one domain in the direction perpendicular to the plane. This reduces the domain structure to two dimensions, although the internal structure of the domain walls is still three dimensional.
Thin films can be grown with perpendicular magnetocrystalline anisotropy. This anisotropy aligns the magnetization in the direction perpendicular to the plane. However, if the magnetization would be in this direction throughout the material there would be a large stray field. To reduce this the magnetization must be either turned in the plane or break up in domains. The combination of these effects gives rise to a configuration of alternating up and down domains, as shown by Kittel 7. The size of these domains is influenced by the material parameters, the exchange constant, the anisotropy constant and saturation magnetization, but also by the field history. This phenomenon, where the current magnetic properties depend on magnetic fields applied in the past is called hysteresis. A common source of hysteresis is the fact that moving domain walls costs energy. In our system this effect is small since there are no crystal grains to pin the domain walls. A high value of the exchange constant means that rotating the magnetization is expensive, resulting in wider domains. Large saturation magnetization produces high demagnetizing fields, leading to smaller domains. To reduce the stray field energy, Neél closure domains can form at the surfaces (see fig. 2).
Fig. 2.Example of a Neél closure domain around a Bloch wall
2.2 RT films
Films made of rare earth transition metal alloys are particularly suited for the study of magnetic domains due to the fact that they are amorphous. This means there are no crystal boundaries on which domain walls can pin. They are ferrimagnetic and the material parameters can be tuned by choosing the composition. Due to the amorphous structure, both the Fe and the Gd sublattices have considerable spread in spin direction around the local magnetization.
The materials we studied are GdFe9 and GdFe5. They are amorphous,ferrimagnetic materials and the magnetization of the sublattices are antiparallel. The layers are about 40 nm thick. During their growth, an out-of-plane magneto-crystalline anisotropy emerges due to the anisotropy of the surface. This anisotropy aligns the magnetization in the direction perpendicular to the plane. The value of the anisotropy constant depends on the growth parameters.
Fig. 3.MFM image of the GdFe5 sample, image size is 5×5 . Red and yellow represent up- and downward magnetization.
2.3 Thin film stripe domains in magnetic fields
The grown films display a magnetic stripe domain structure with a period that is the result between the tendency to reduce the demagnetizing field, which favors small magnetic domains, and the cost of forming domain walls.
The stripe domains respond to applied magnetic fields. Their size changes and they align with the magnetic field. A strong magnetic field can magnetize the material uniformly, completely destroying the domains. We studied the influence of an in-plane magnetic field on the magnetic structure. The magnetic structure in the sample plane is hard to measure with most techniques because many of them only measure the magnetization at the surface and cannot be used in high magnetic fields. One cannot simply cut the sample and measure the magnetization since cutting will influence the magnetization.
2.4MFM and resonant x-ray scattering
A technique to study magnetic domain structures is magnetic force microscopy (MFM). It is a scanning technique closely related to atomic force microscopy (AFM). One records the atomic force between a sharp tip and a surface as the tip scans the surface. This returns a topographic map of the surface. For magnetic force microscopy the tip has some magnetic material ( usually some cobalt or nickel ) attached to it to measure the magnetic field produced by the sample. This makes it hard to use MFM in high magnetic fields, since the external magnetic field can dominate the interaction between the sample and the tip. First the surface is scanned at a short distance with atomic force microscopy and then the process is repeated with the needle at a larger distance from the surface to measure the magnetic field.
MFM is best suited for very flat surfaces but the magnetic structure must not be to flux-closed; the field produced by the sample may not be too small. Another problem are tip-domain interactions where the magnetic tip influences the magnetic domains it is scanning.The spatial resolution of MFM is in the order of a few tens of nanometers. MFM can only measure the magnetization at the surface. This means that to determine the internal structure you need to use other techniques or numerical methods.
The best technique to measure the internal structure is X-ray resonant scattering. It has a spatial resolution in the order of ten nanometer and can be used in any external magnetic field. A disadvantage of this technique, however, is that it cannot solve the magnetic structure topographically like MFM. For a detailed explanation of this technique see Resonant soft x-ray scattering studies of the magnetic nanostructure of stripe domains, by Joost Frederik Peters2.
3 X-ray scattering studies of the magnetic nanostructure of stripe domains
In 2003 J.F. Peters studied the evolution of the stripe domains in 40 nm thick amorphous GdFe9 and GdFe5 films with resonant x-ray scattering 2. A magnetic field that saturated the sample was applied in the in-plane direction (we call this the x-direction). The field was then reduced and later applied in the minus x-direction. From the diffraction peaks the stripe period, magnetization and the squared components of the magnetization were resolved. This enabled Peters to deduce the magnetic structure of the domains and its evolution as the field changed (see appendix). The results were also compared to a 1-dimensional model by Marty et al4.
Fig. 4. Schematic view of the internal stripe structure. The direction of H is displayed on the right of each picture.
It was found that when the in-plane magnetization changed from the positive to the negative x-direction the large jumps in the magnetization at the coercive field is accompanied by a huge internal reorganization of the magnetic structure while the domain period stayed the same.. Two ways in which this could occur were proposed: either the magnetization would start to turn at the edges, leaving the magnetization of the core in the original direction until a sudden collapse E or the transition could be smooth and some domains go from C to F directly.
Although the measurements were successful they only provided indirect evidence of themagneticstructure and no conclusion on how the magnetization reversed. For a more detailed understanding finite element numerical simulations were used.
3.1 Simulations and MFM studies
In 2009 we8 studiedthese systems with OOMMF9, a micromagnetic simulation program. We also checked the domain period evolution found by Peters with magnetic force microscopy.
The simulations showed that Peters was largely correct in his depiction of the magnetic structure. They showed that the domains have a vortex structure in alternating orientations around the Bloch walls ( which are more like vortex cores than Bloch walls).
Fig. 5.Image of the simulated structure. The thickness is 42 nm. Red regions represent magnetization pointing inward ( x ). The vertical direction is z and the y-axis lies horizontal.
The simulations also showed the transition happens in the first way Peter proposed. When the applied field already points in the negative x-direction, the magnetization of the vortex core remains in the positive x-direction while the region on the outside twists in the negative x-direction. The mx magnetization is trapped by the circulating magnetization mz and my. This state is stable until it collapses when the field becomes too large.
3.2 Skyrmion
During this phase the vortex has become a skyrmion. This skyrmion is a stable topological defect; there is no continuous transition to a state where the magnetization is in the minus x-direction. Skyrmions were originally proposedby Peter Skyrme in a model for bosons in high energy physics. Skyrmionsalso appear in Bose-Einstein condensates and in quantum hall systems.In nanomagnetism they have been observed in thin film chiral ferromagnets5.
At low temperatures ( 0 to 50 Kelvin ) a ( meta- ) stable skyrmion lattice can form when a magnetic field is applied perpendicular to the film.
Fig. 6.Example of a skyrmion in FeCoSi2 ( left ). The image on the right is the skyrmion lattice that forms. Images taken from Yu5
The diameter of these skyrmions is in the same order of magnitude as the domain size in GdFe5 and GdFe9. The length of the skyrmions in stripe domains could be a lot larger than in these skrymions since in these systems the maximum length is approximately the thickness of the film. A more important difference is the formation of the skyrmions. In the case of Yu and Mühlbauer10the skyrmion lattice is a static solution that exists in a particular space in the B-T phase diagram. In our system the skyrmions are dynamically formed;by a changing magnetic field.
In this study we further investigate the skyrmion and the transition that occurs when the magnetization changes direction by using OOMMF. We compare our results to the research of J.F. Peters and to the 1-dimensional model by Marty4
4 OOMMF
OOMMF is a finite element micromagnetic simulation program that solves the micromagnetic equations iteratively. The physical input parameters are the exchange constant, the uniaxial anisotropy constant and the saturation magnetization. OOMMF is extensible, meaning that anyone can contribute by adding extensions for instance to calculate certain energy terms or add periodic boundary conditions. We used an extension with periodic boundary conditions in the x-direction, where we have translation symmetry.
Other input parameters are the domain sizes in x,y and z- directions and the cell size. The number of cells has a strong influence on the simulation time and it is necessary to find a balance between precision and simulation time.
The simulation starts with a randomly generated vectorfield. The vectors represent the local magnetization. Each iteration the vectorfield changes to reduce the energy until a stable solution has been found. Then the applied magnetic field changes and the simulation enters the next stage, until the solution is stable again. The number of stages, the magnetic field during each stage and the accuracy are also user controlled parameters. An example of an input file can be found in the appendix.
4.1 Magnetic constants
The properties A, Ms and Kuwere taken fromStaticand dynamic x-ray resonant scattering studies on magnetic domains, by Jorge Miguel Soriano3. They were estimated from the nucleation field and the stripe period, measured with vibrating sample magnetometry.
These parameters can only be measured indirectly so there is a lot of uncertainty in the value of these parameters, especially in the value of the exchange constant A. We can even argue that, if our simulations match the X-ray data perfectly the values we assumed for the simulation are more accurate than the values obtained by Peters2 and Soriano3.
4.2 Qualitative effect of parameters
To qualitatively check the simulations and to find out how sensitive the results of the simulations are to the input parameters we ran simulations with different values for A, Ms, Ku and the thickness. First we did a reference simulation and we doubled each of the parameters individually. In all simulations the sample was initially fully saturated with a field in the x-direction. That field was then reduced to a small value. The resulting domain structures are displayed in the appendix.
They show that the effect of tweaking the parameters in OOMMF is qualitatively correct. Reducing Ku means that there will be no domain formation. Increasing A results in larger domains, reducing it results in smaller domains. They also show that the domain structure is very sensitive to the values of the physical parameters. A factor of two can make the difference between domain formation and no domain formation at all.
5 Results
5.1 Comparison with 1-dimensional model
J.F. Peters compared his results to a 1-dimensional model of the magnetization by A.Marty4and so will we. This model assumes the magnetization to be in the xz plane and it varies along the y-direction. The mz magnetization as function of y is given by the Jacobi sine function. Θ0 is the maximum angle between the magnetization and the plane, but my is ignored in this model.Θ0, the reduced stripe width w and shape parameter σ parameterize mz(y) according to , where sn(y) is the Jacobi sine function.