Energy in ferromagnetic materials

In order to understand the magnetization processes we have to consider the minimum total energy of magnetic materials, which determines their magnetization behavior on different scales [44]. This energy is expressed as an integral of different energy densities over the volume of the specimen:

E= Z

V

(_ex+_an+_demag+_ext)dV (2.49) The integrated sum of the energies includes: exexchange,ananisotropy,demag mag-netostatic,extexternal field andDM Dzyaloshinskii Moriya interaction (DMI) energies.

In the following all energy components involved in equation 2.49 will be introduced in details.

• Exchange energy_ex. On the quantum level exchange interaction is based on the Pauli principle that forbids electrons with the same spin to occupy the same or-bit. The situation when electrons with the same spin orientation are located on the different orbitals is energetically more favorable. It explains the tendency of ferro-magnetic materials to form extended regions with uniform magnetization, so called,

magnetic domains. The exchange energy is a short range interaction that includes only neighboring spins. In material with numbered spins it is expressed as:

i,j=−X

i,j

Ji,jS~iS~j, (2.50) whereJis an exchange constant that represents the energy difference between par-allel and anti-parpar-allel spin orientations. For example, for ferromagnetic materials J >0and for antiferromagneticJ <0.S~_iandS~_j are the associated spins. In the continuous magnetic material where spins are not perfectly collinear the exchange interaction can be expressed:

ex=A(∇m)~ ² (2.51)

Ais an exchange stiffness of the material and defined as:

A=J S²n

a , (2.52)

wherenis the number of atoms per cell unit, S - the spin of the lattice,J - the exchange integral, anda- the lattice constant of the material. The exchange stiff-ness can be measured by the methods which probe the magnetic excitations, such as spin-wave resonance experiments, using temperature dependence of the sponta-neous magnetization, Brillouin Light Scattering (BLS) or inelastic neutron scatter-ing (NS). Exchange stiffness is difficult for reliable estimation since its value may locally vary depending on defects in the material. As it is seen from the equation 2.51 exchange energy depends on the angle between neighboring magnetic mo-ments, but not on their relative orientation.

• Anisotropy (magneto crystalline) energyan. The main source of magnetocrys-talline anisotropy is spin-orbit interaction. It results in existence of preferred mag-netization directions with respect to the crystal lattice, i.e. easy axis, or vice versa unfavorable directions, hard axis, which require higher energy for magnetization. In case when only one axis direction is energetically degenerated the uniaxial anisotropy occurs. Therefore anisotropy energy can be expressed as a Tailor series with the first term:

an=K1sin²θ, (2.53)

whereK1 is the anisotropy constant of the first order andθ is an angle between magnetizationM~ and direction of the easy axis. Often magnetic moments in mate-rial biased towards many particular directions which yield the existence of multiple easy axes. For example, Fe and Ni possess cubic crystal anisotropy that is given by first two orders of Tailor series:

_an=K₁(α²₁α²₂+α₂²α²₃+α²₃α²₁) +K₂α²₁α²₂α²₃, (2.54)

whereα_iis a direction cosines of magnetization vector in respect to crystallographic axes,K₁andK₂are anisotropy constants, which depend on temperature and spe-cific for different materials. In case if second term can be neglected forK₁ >0 easy axes are located along the body edges (100) andK1 <0the easy axes exist along the diagonals (111). IfK2 6= 0along with hard and easy axis intermediate magnetization axis can be distinguished. For example, Ni with anisotropy constants K1 = −5.7·10³J/m³andK2 =−2.0·10³J/m³[45] possesses easy axis along diagonal (111), intermediate and hard axis along (110) and (100), respectively.

Ferromagnetic thin films, single- and polycrystalline, as well as alloys, mostly have uniaxial anisotropy. Magnetocrystalline anisotropy can be modified by stress in-duced changes in crystal lattice of ferromagnetic material. The tensile or compres-sive strains change the distances between ions in crystal that modifies spin-orbit coupling in the material. Also anisotropy can be induced by symmetry breaking of the crystal structure at the interfaces and surfaces in multilayer structures.

• Magnetostatic energydemag. The magnetization of ferromagnetic material also can interact with magnetostatic field generated by the sample itself. This magnetic fieldHdis demagnetizing or stray field and acts to reduce total magnetization. The energy density is given as:

demag=−1

2µ0MsHd·m, (2.55)

whereMsis the saturation magnetization. The estimation of stray fieldHdcan be done using some fundamental relations based on Maxwell equations. In the absence of currents the magnetic scalar potentialφcan be introduced as:

Hd=−∇φ (2.56)

The magnetic potentialφcan be found as a solution of Poisson’s equation that can be found in [46].

• External field energy_ext, or Zeeman energy. This interaction refers to the energy occurring in the ferromagnetic material when it is placed in an external magnetic field. The external field energy correlates to the angle between the material magne-tizationM and applied fieldH_a:

ext=−µ0Ha·M, (2.57)

whereµ₀is the vacuum permeability.

The basic magnetic parameters of Ni with face centered cubic (FCC) crystalline structure are listed in table 2.2.

Parameters Ni

Saturation magnetizationMs 4.9·10⁵A/m

Curie temperatureTC 627 K

Damping parameterα 0.05

Exchange stiffnessA 9·10⁻¹²J/m

Anisotropic constant of the1^storderK1 −5.7·10³J/m³

Exchange lengthlexch 9.9 nm

Table 2.2: Magnetic parameters of Ni [47].

Dzyaloshinskii Moriya interaction (DMI) . DMI is an additional magnetic interaction which is observed in the systems with a lack of inversion symmetry. DMI is an antisym-metric exchange between two neighboring magnetic spins, which favors a spin canting of (anti)parallel aligned magnetic moments. If we look at the simplified case involving only two neighboring atomic spinsS~1andS~2interfacial DMI is expressed as following:

_DM =D~₁₂·(S~₁×S~₂), (2.58) whereD~12 is a Dzyaloshinskii vector, which sign determines left or right magnetization rotation. Initially starting from the ferromagnetic state vectorsS~1andS~2are parallel, the DMI deflects the spins by rotation aroundD~12. The interaction of two atomic spins with an atom having large spin orbit coupling (SOC) results inD~12pointing outwards from the plane of the atoms (figure 2.16).

Figure 2.16: DMI interaction imaged at the atomic scales, where blue are ferromagnetic ions and red is an ion of SOC metal.S~1andS~2are two coupled spins andD~12is a vector of Dzyaloshinskii Moriya interaction.D~12is oriented perpendicular towards the triangle plane of involved ions.

The domain patterns of the samples contain information about DMI, since the domain width is determined by the balance between magnetostatic and wall energies. So

mea-surement of the spaces between domains allows to calculate the surface energy density, σ_DW:

σ_DW = 4p

AK_{ef f} −π|D|, (2.59)

whereDis the DMI constant,Ais exchange stiffness, andK_{ef f} is the effective uniaxial anisotropy constant that includes bulk and surface magnetic anisotropies.

Shape anisotropy The existence of preferred magnetization direction occurs not only due to crystal structure of the ferromagnetic material but also because of the shape of the structure itself. There are two main magnetization orientations in thin films: in-plane and out-of-plane as it is shown in figure 2.17 a. Out-of-plane orientation produces high surface charge(~m·~n)because all the magnetic moments aligned normally to the surface.

Therefore this configuration is energetically less favorable since it requires to produce higher stray field than in-plane magnetization orientation.

Figure 2.17: Magnetization configuration in materials with: a) shape anisotropy magnetization in-plane and out-of-in-plane configuration; b) in single domain and two-domain states of magnetization.

Magnetization of the ferromagnetic film can be in multi-domain state or single do-main state as it is shown in figure 2.17 b. In multidodo-main state dodo-mains with different magnetization direction are separated by domain wall.

Domain walls The interface separating domains with differently oriented magnetiza-tion is a magnetic domain wall. Domain walls are formed in order to reduce the energy

Figure 2.18: Two types of domain walls: a) Bloch wall and b)N´eelwall. Reproduced from [47].

between magnetic domains. Magnetic moments in the walls gradually change their orien-tation forming N´eel or Bloch walls as it is shown in figure 2.18. The Bloch wall has the magnetization direction laying in the plane of domain wall, whileN´eelwall has magneti-zation reorientation pointing perpendicular to it.

The width of domain walls depends on the equilibrium of exchange energy and the magnetic anisotropy. It can be estimated as following:

∆ =π q

A/Kef f, (2.60)

whereAis exchange stiffness, andK_{ef f} is effective anisotropy. From the perspective of exchange energy domain walls are an energetically suboptimal state. The alignment of magnetic moments in the same direction make the magnetic walls wider. On the other hand, magnetic anisotropy energy increases when spins are not oriented in the direction of the easy axis and tends to reduce the width of the domain wall. Therefore the larger exchange interaction results in wider walls, when higher anisotropy yields thinner walls.

Additionally, the multiple domains formation is beneficial since it significantly reduces stray field.

Chapter 3

X-ray Imaging

3.1 Background

Microscopy is a basic and essential technique with a very wide application in natural sci-ences and technology. The history of microscopy starts with invention of optical light mi-croscope in the late 16th century. Modern advancement in optics allows to produce optical microscopes with resolution limited by the wavelenght of visible light (400-700 nm) or even bypass diffraction limitations using laser stimulated emission depletion microscopy (STED) that provides resolution of 20 nm. Nowadays the most advanced microscopy in-struments, transmission electron (TEM) or scanning tunneling microscopy (STM), have resolution in sub-nm scale allowing atomic level imaging [1, 2]. Despite all the advantages of high resolving power these methods have their own limitations. For example, TEM has small probing area, needs a fabrication of very thin specimens due to strong scattering of the electrons in the samples volume, also induces high radiation damage and requires sample cooling. STM and atomic force microscope (AFM) are applicable only for surface scanning and can’t provide bulk information [48].

X-rays can be used for imaging, as a logical extension of optical microscopy, providing wavelength limited resolution in a range of few nm. The potential of the combination of spectroscopic method with high penetration depth of X-rays, chemical and magnetic sensitivity advantage it over other techniques. With the development of X-ray focusing optics, i.e. refractive lenses, Kirkpatrick-Baez mirrors and Fresnel Zone Plates, and new highly coherent and powerful X-ray sources provided by modern storage rings and XFELs, X-ray microscopy became a substantial tool covering a wide range of possible applications in materials science, biology, magnetism and time resolved imaging [49]. Moreover, new lensless diffractive imaging methods significantly improved the spatial resolution of X-ray microscopy to less then 10 nm [50] providing the possibility to retrieve phase contrast

information.