Magnetic energy terms

Exchange energy

Up until this point, discussion of the interaction between microscopic magnetic moments has been limited to simply whether or not they tend to interact and whether the interac-tion is parallel or antiparallel. The driving force behind the aligning interacinterac-tion is called the exchange coupling interaction. The exchange coupling interaction has its origin with the electrons of the material and their magnetic moments. The magnetic moment of an electron in an atom is comprised of both its intrinsic spin contribution and the orbital angular momentum contribution, both of which are quantum mechanical degrees of free-dom. According to the Pauli principle, two fermions (e.g. electrons) may not occupy the same quantum state, which dictates that two electrons of the same orbital configuration must have oppositely aligned spins and thus approximately zero magnetic moment.

How-ever, states where multiple electrons have parallel spins (and thus a significant magnetic moment) are possible if the electrons have separate orbital angular momentum configura-tions. Electrons with parallel spins in separate orbital states can be favorable due to less coulombic interactions, decreasing the free energy of the system. For further details on the quantum mechanical mechanisms behind such configurations, the reader is referred to Kronm¨uller and F¨ahnle [41].

In quantitative terms, we can write the exchange energy as [4]

Eexch=−µ0MS

where the exchange field,H_exch, is defined by Hexch= 2Aexch

M_S² ∇²m.ˆ (2.7)

In Equation 2.7, the term A_exch denotes the exchange stiffness, a material parameter of units J m⁻¹, related to the magnetic torque exerted by two neighboring magnetic moments.

From Equation 2.6, it is clear that the energy associated with the exchange coupling, Eexch, will be minimized by aligning all microscopic moments in the material. However, an alignment of all microscopic moments in an entire material is not the observed behavior of macroscopic ferromagnets. Recalling the formation of magnetic domains as discussed in Section 2.1.3, it is clear there must be other contributions to the system’s total free energy which does not favor aligningall microscopic magnetic moments.

Magnetostatic energy

In a monodomain ferromagnet, the net magnetization will be significant. Since the mag-netic field must be a divergence free field, the magmag-netic material must sustain a magmag-netic stray field outside the material. The stray field is associated with a high energy cost, and works against the magnetization of the ferromagnet which is why it is often referred to as the demagnetizing field, Hd. The energy associated with the demagnetizing field can be found in a similar fashion to Equation 2.6, by considering the interaction of the material’s magnetization with the field. This gives the relation,

E_demag=−µ₀M_S

and sinceHdand the magnetization direction,mˆ are oppositely aligned, theEdemagterm is minimized only by a small stray field,Hd.

Thus, there is a competing energy term to balance the domain formingE_exchterm, the Edemagterm. Magnetic domain formation is a balance between minimization of both these two terms. The dual competition betweenEexch andEdemagis illustrated in Figure 2.6.

Looking at Figure 2.6 while considering Equation 2.6, it is rather obvious that the

2.2. THE MICROMAGNETIC MODEL 15

N

S

a) b) c)

Figure 2.6: Illustration of domain formation. a) A hypothetical bar magnet with a single magnetic domain. This single domain configuration leads to a minimumEexch. However, a high Edemag makes it an unstable configuration. b) A second domain is introduced, drastically minimizing theEdemagcontribution at the expense of an increase inEexch. c) The introduction of two new domains has seemingly totally minimizedEdemag, although with a further increase in E_exch. Exactly how the domains of a magnet will arrange themselves depend on the exact quantitative magnitudes of the magnetostatic energy and the exchange energy.

increase in exchange energy only occurs at the domain walls, indicated by the dashed lines. Through the bulk of each individual domain, the magnetization is uniform and aligned with Hexch. It is only at the domain borders that an energy increase can be observed. Thus, it is not the amount of distinct domains that contributes to an increase in free energy, but the amount of domain wall.

A domain wall is not a discrete change in magnetization direction, but is a gradual turning of the magnetization over a finite width,δw. The thickness,δw, of a domain wall is determined (mostly) by the exchange stiffness of the materialAexch. Thus, the domain wall is a small, magnetically frustrated volume that is proportional to the boundary areas, and the wall thickness, of each domain. An illustration of the domain wall and its frustrated microscopic moments is provided in Figure 2.7.

The formation of new domains is governed by the free energy balance between min-imizing the energy cost of the domain boundaries and the energy cost of the stray (de-magnetizing) field. As mentioned, the domain walls are not discrete boundaries, but a volume of material where the magnetic moments are changing over a certain width,δw. The domain wall volume will be smaller if the change in magnetic moment can occur over a shorter width, and thus the amount of frustrated microscopic moments will be reduced.

In other words, a thinner domain wall will be energetically favorable as there is less energy cost associated with the exchange energy. The exchange stiffness,Aexch, that determines

δw

Figure 2.7: The microscopic magnetization of a domain wall between two oppositely aligned domains is illustrated in the green inset. Note that type of domain wall illustrated here, where the magnetization orientation turns in plane, is an example of a N´eel wall, typical of magnetic thin films. The other configuration, more common in bulk magnets, is Bloch orientation where the magnetization turns out of plane forming a helix-like shape.

Also indicated is the domain wall widthδw.

the domain wall thickness can thus be expressed as a change in polar angle,δφ, that the material can sustain over a small distance, δl. An illustration of the exchange stiffness and its effect on the width of a domain wall is provided in Figure 2.8.

The magnetostatic energy is also the main driving mechanism behind so-called shape anisotropy. Shape anisotropy is the effect that the geometrical confines (in other words the shape) of a magnetic object will dictate which direction or set of directions that the magnetization will tend to align along. Non-spherically shaped ferromagnets will have preferred magnetization directions where the stray field is minimized. The favorable alignment of magnetic moment along the length of an object can be intuitively understood by considering a long bar magnet. A magnetization direction along the length will align Hdso that it is much smaller than if the magnetization were to be oriented perpendicular to the long axis. A spherical magnet on the other hand, cannot exhibit shape anisotropy due to its complete symmetry.

Applied field energy

There is a magnetic interaction between the externally applied field,Hext, and the mag-netic moments of a material. The energy contribution due to this interaction is simply

Eext=−µ0MS

Z

V

ˆ

m·HextdV. (2.9)

Minimizing this energy contribution is the driving force causing the magnetization to tend to align along the applied field. The energy contribution described in Equation2.9

2.2. THE MICROMAGNETIC MODEL 17

δw

δl δφ

Figure 2.8: The domain wall width is dictated by the exchange stiffness, which may be visualized as the possible change in magnetization angle, δφ, over a finite length, δl, in any given material.

is sometimes called the Zeeman energy [4].

Magnetocrystalline anisotropy

Inherent to some magnetic materials is an anisotropy related to the crystallographic struc-ture of the material. The property of crystal-dependent magnetic anisotropy is called the magnetocrystalline anisotropy, and it describes how the magnetization of the material will favor aligning along certain crystallographic directions. The directions which are fa-vored are termed the easy axes, and the directions which are the least fafa-vored are termed the hard axes. If a system only has a single axis of high symmetry, it is termed a uni-axial anisotropic system. Thin films of complex oxide systems, such as (001) oriented La0.7Sr0.3MnO3 (LSMO) can have two sets of easy axes, making it a biaxial anisotropic system [42].

Magnetocrystalline anisotropy is associated with an energy contribution related to the degree of symmetry. For a certain magnetization direction in a uniaxial system this contribution can be expressed as

E_anis=Z

V

K_u1sin²θdV, (2.10)

where Ku1 is the material (crystal) first order uniaxial magnetocrystalline anisotropy constant, and θ is the angle difference between the magnetization and the easy axis orientations.

An illustration of magnetocrystalline anisotropic (and isotropic) magnetization behav-ior is provided in Figure 2.9. For the iron-nickel alloy named permalloy (Py), the material most studied in this thesis, the magnetocrystalline anisotropy is negligible [43].

[001]

[111]

[110]

[001]

[111]

[110]

a) b)

Eanis Eanis

ˆ

m mˆ

Figure 2.9: Magnetocrystalline anisotropy energies, for a given magnetization direction, ˆ

m, along indicated crystallographic directions. a) No magnetocrystalline anisotropy, the direction of magnetization has no effect on the magnetocrystalline anisotropy energy. b) Magnetocrystalline anisotropy, clearly showing a set of easy axes with energy minima along [111] and hard axes with energy maxima along [110] and [111].

Thermal energy

The last energy contribution is simply the thermal energy, providing a contribution of E_thermal =k_BT, wherek_B is the boltzmann constant. The thermal energy contribution is temperature dependent and can completely change the macroscopic behavior if the temperature is changed sufficiently, as discussed with regards to the Curie temperature in Section 2.1.2. In a regime where the thermal energy is of sufficient magnitude com-pared to the other contributions, it will provide a degree of random orientation in the magnetization.

Total energy

By summing all these contributions it is possible to quantify the total energy of the system. The total energy can be expressed as

E_tot=E_exch+E_demag+E_ext+E_anis. (2.11) By the second law of thermodynamics, any thermodynamical system can be said to follow the principle of minimum energy [44]. The internal energy of a system, such as the one defined by the micromagnetic model by Equation 2.11, will decrease towards a free energy minimum at equilibrium. In other words, the total state of our micromagnetic system is stable if the sum of the free energy contributions is at a minimum. However, it is worth noting that the minimum does not have to be a global minimum, known as a ground state. The system can also reside in local minima, known as stationary states or metastable states.