16 Anisotropy in magnetic films

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Contents

 

1.      Introduction to anisotropy in ferromagnetic materials

2.      Origin of the magnetic anisotropy in thin films

(a)   Magnetic dipolar anisotropy (shape anisotropy)

(b)   Magneto-crystalline anisotropy

(c)    Magneto-elastic anisotropy

3.      Experimental techniques for the determination of magnetic anisotropy

(a) Magnetometric Measurements

(b) Magneto-optical Kerr effect (MOKE) measurements

 

Introduction to anisotropy in ferromagnetic materials

 

Magnetic anisotropy is recognized as the directional dependence of the magnetic properties in a magnetic material. It is considered as one of the most crucial parameter of ferromagnetic materials, which decides the extent of magnetization that can be achieved in a material. Basically, single crystal ferromagnetic materials possess directional magnetization properties i.e. ‘easy’ and ‘hard’ directions of magnetization, which is attributed on the basis of the energy needed to magnetizing a material in the direction of externally applied field with respect to the crystal axes. Depending upon the type of application, materials with a high, medium or low magnitude of magnetic anisotropy are employed, for respective applications like permanent magnets, magnetic recording heads, information storage media or magnetic cores in transformers. It has been observed that the physical basis which dictate the preferred orientation of magnetic moments in magnetic thin films and multilayer structures can be fairly different from the factors that are responsible for the easy-axis alignment along the symmetry direction in a bulk material. The presence of significant symmetry-breaking elements in films such as planar interfaces and surfaces modify the basis of magnetization in bulks and films. The thicknesses of individual layers in a multi-layered system can be varied to modify the magnetic anisotropy of the system.

 

The effective magnetic anisotropy energy K (J m−3), short termed as MAE, can be written as a sum of the volume contribution i.e. (J m−3) and contribution from the interfaces (J m−2), and can be expressed as follows:

This equation gives an effective magnetic anisotropy energy value taking into consideration the of the interface atoms and the inner atoms of a magnetic thin film structure of thickness t. Equation (1) is written under the consideration that ??/d (where d represents the thickness of a monolayer) is the difference between the anisotropy of the interfacial atoms with respect to the bulk or inner atoms. Secondly, the thin film structure is considered to be bounded on both sides by two identical interfaces, therefore a pre-factor of 2 has been added.
Example: Magnetic anisotropy determination in Co/Pd thin film multilayers
Equation (1) is generally employed in experimental studies on magnetic systems, and the determination of volume and surface contributions i,e. and can be obtained by plotting the product of and t versus t. Figure 1 demonstrates a typical plot for Co/Pd multilayers.

Figure 1: Plot for the determination of volume and surface contributions in MAE. The intercept on vertical axis represents twice the interface anisotropy, whereas the volume contribution is given by the slope. Source: Broeder et al. (1991)

 

A positive value of ???f denotes a preferred direction of the magnetization normal to the plane of thin film. A negative slope signifies a negative volume anisotropy ?v , which favours an in-plane magnetization, whereas the intercept at zero thickness (of Co) shows a positive interface anisotropy , which favours perpendicular magnetization. Lower than a certain thickness t⊥ (= −2 ks⁄ kv  ), the effect of interface anisotropy overshadows the volume contribution, thus giving rise to a perpendicularly magnetized system.

 

Origin of the magnetic anisotropy in thin films

 

Magnetic anisotropy in magnetic thin films arises primarily due to the magnetic dipolar interactions and the spin–orbit interactions. The dipolar interaction (long range) in a system is strongly dependent on the shape/dimensions of the sample, and makes a significant contribution in defining the anisotropy of a system. It is of certain importance in case of thin films, and is also responsible for the in-plane magnetization. In the absence of any spin–orbit and dipolar interactions, the total energy of the electron–spin system has no dependence on the direction of the magnetization. The spins are pictured to be coupled via spin–orbit interaction with the orbits which, in turn, are influenced by the crystal lattice of the material. Therefore, the total energy of the magnetic systems depends on the orientation of the magnetization with respect to the crystalline axes, and which reflects the symmetry of the crystal. This is called the magneto-crystalline contribution to the anisotropy.

 

In concurrence with the overlapping wavefunctions of neighbouring atoms, the effect of spin– orbit interaction too contributes towards the magneto-elastic or magneto-strictive anisotropy observed in a strained magnetic system. Such a situation is generally found in multi-layered thin film systems due to the lattice mismatch between the alternate layers.

 

(a) Magnetic dipolar anisotropy (Shape anisotropy)

 

Long range magnetic dipolar interaction, which is predominant over the outer boundaries of the sample, is one of the most important sources of the magnetic anisotropy in magnetic thin film samples. Overlooking the discrete nature of matter, the dipolar interaction in ellipsoidal ferromagnetic samples due to their shapes can be described using an anisotropic demagnetizing field, Hd, expressed as Hd= −NM   Where, M represents the magnetization vector and N is a demagnetizing tensor, which is shape-dependent.

 

In the case of a thin film, all the tensor elements are zero except for the direction normal to the surface layer, therefore, ?⊥ = 1. The magnetostatic energy can be given by the expression:

where, ?0 represents the permeability of vacuum. The anisotropy energy contribution per unit volume V of the system is given by

 

In the above expression, the magnetization has been presumed to be uniform with a magnitude equal to the saturation magnetization ?? , and subtends an angle θ with respect to the film normal.

 

(b)   Magneto-crystalline anisotropy

 

The microscopic origin of the magneto-crystalline in a magnetic material is the spin– orbit interaction. Technically, the exchange interaction and the dipolar interaction could also contribute towards the magneto-crystalline anisotropy in magnetic materials. However, the exchange interaction cannot contribute to the anisotropy since it is directly proportional to the scalar product of the spin vectors and is thus independent of the angle between the spins and the crystal axes. On the other hand, the dipolar interaction energy, does depend on the orientation of the magnetization relative to the crystal axes.

 

(c)  Magneto-elastic anisotropy

 

Strain developed in a ferromagnetic material changes the magneto-crystalline anisotropy and may thus alter the magnetization direction. It is the ‘inverse effect’ of magnetostriction, where the sample dimensions get altered if the magnetization direction is changed. For an elastically isotropic medium with isotropic magnetostriction, the energy per unit volume associated with this effect can, be expressed as

with

Here σ represents the stress developed in the material, which is in turn related to the strain,? , via the elastic modulus E, expresses as σ = E?. The magnetostriction constant λ depends on the orientation of magnetization and can be either negative or positive. The angle θ denotes the direction of the magnetization with respect to the direction of uniform stress in the film. If the strain in the film is finite, the effective anisotropy is primarily due to the magneto-elastic coupling.

 

Strain in thin films can be induced due various reasons. Amongst the known causes, thermal strain induces due to the difference in coefficient of thermal expansion, intrinsic strain brought about by the nature of the deposition process and strain due to non-matching lattice parameters of adjacent layers.

 

Experimental techniques for the determination of magnetic anisotropy

 

Magnetic anisotropy in films/multilayers can be determined either from the static or dynamical response of the thin films/multi-layered systems magnetic systems. The dynamical response of the magnetic thin films/multilayers can be studied using Ferromagnetic Resonance (FMR) and Brillouin light scattering (BLS). However, most of the experimental data on magnetic samples has been obtained using static measurements. Particularly, magnetization and torque measurements are frequently used to estimate the magnetic anisotropy in samples. Two of the most commonly employed techniques for magnetic anisotropy measurements are briefly described, as follows:

 

1. Magnetometric Measurements

 

The magnetic anisotropy in samples is most commonly determined from magnetometric measurement tools, like VSM or SQUID, across two orthogonal directions of the magnetic field with respect to the sample surface.

 

The angle-dependent part of the energy E of the magnetization of the thin film is expressed as

 

Here, ?i represents all the first-order anisotropy energy contributions (intrinsic) except the shape anisotropy or magnetostatic energy contribution, which is given as 1/2µ0??2 in the case of a saturated film. The last term denotes the interaction between the externally applied field and the resultant magnetization; whereas the angles subtended by the magnetization and field and the film normal are given by θ and φ respectively. The minimization of energy as a function of the applied field H yields the field dependence of the equilibrium angle ??? (H) and the field component of the magnetization, which is expressed as

 

Thus, depending on the ease of magnetization, different cases are possible:

 

(iii) In samples with very large domains, the effect of the interaction amongst the domains is insignificant. At H = 0, although the average perpendicular component of the magnetization is zero, the magnetostatic energy term is correctly described as in equation (6) for both perpendicular and in-plane applied fields. When magnetizing the layer perpendicularly, the magnetostatic energy does not have to be overcome. Minimization of equation (6) therefore gives a perpendicular curve saturating immediately and an in-plane curve saturating at the field 2??/µ0?? − ?? or 2????/µ0?? , see figure 2 (c). As in the previous cases, the area between the in-plane and perpendicular magnetization curve gives the effective MAE,  ????, including the shape anisotropy energy

2.   Magneto-optical Kerr effect (MOKE) measurements

 

Magneto-optical Kerr effect is a well-established experimental tool to study the magnetization in materials. It describes the change in the polarization state of light induced as a result of reflection at the surface of magnetic materials. There are two optical effects: rotation of the plane of linear polarization (i.e. the Kerr rotation) and a change of the ellipticity of the light (i.e. the Kerr ellipticity). Both the effects arise due to the difference in the complex Fresnel reflection coefficients for left and right circularly polarized light. This optical anisotropy can only be observed for non-zero magnetization samples, where its magnitude is determined by the off-diagonal terms of the permitivity tensor, which are odd linear functions of the magnetization. Although no direct information can be obtained regarding the absolute magnitude of the magnetization, the angle of Kerr rotation is directly proportional to the magnetization. In addition, since the magnetization is only monitored in that region of the sample which is illuminated by a (focused) light beam, a localized analysis of magnetic behaviour can be carried out.

 

In polar Kerr effect measurements, the light beam is made incident normal to the sample, and the field is applied perpendicular to the plane of sample. The Kerr rotation and the ellipticity in this geometry are proportional to the perpendicular component of the magnetization. For samples with an in-plane easy axis one obtains a linear increase of the magnetization (and thus of the Kerr signal) up to saturation.

 

Summary

 

The concept of anisotropy in ferromagnetic materials has been introduced, especially for magnetic thin films/multi-layered systems. The various sources of magnetic anisotropy in thin film samples have been briefly discussed. Two characterization techniques (Magneto-metric and MOKE) for experimental determination of magnetic anisotropy in thin films have also been described.

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REFERENCES

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  3. McCaig, Malcolm (1977). Permanent magnets in theory and practice. Pentech press.
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