Chapter 4
Magnetic anisotropy, magnetic domains and superparamagnetism

Suggested Reading

4.1 Introduction

In Lecture 3 we learned that in some crystals electronic spins work in concert to create a spontaneous magnetization that remains in the absence of an external field. The basis of paleomagnetism is that these ferromagnetic particles carry the record of ancient magnetic fields. What allows the magnetic moments to come into equilibrium with the geomagnetic field and then what fixes that equilibrium magnetization into the rock so that we may measure it millions or even billions of years later? We will begin to answer these questions over the next few lectures.

We will start with the second part of the question: what fixes magnetizations in particular directions? The short answer is that certain directions within magnetic crystals are at lower energy than others. To shift the magnetization from one “easy” direction to another requires energy. If the barrier is high enough, the particle will stay magnetized in the same direction for very long periods of time - say billions of years. In this lecture we will address what causes and some of the consequences of these energy barriers.

4.2 The magnetic energy of particles

4.2.1 Magnetic moments and external fields

We know from experience that there are energies associated with magnetic fields. Just as a mass has a potential energy when it is placed in the gravitational field of another mass, a magnetic moment has an energy when it is placed in a magnetic field. We have seen this energy briefly in Lecture 3 . This energy has many names, but here we will call it the “magnetostatic interaction energy density” (E_h):

(4.1)

E_h is at a minimum when the magnetization M is aligned with the field B. It is this energy that drives magnetic compass needles to seek the minimum energy state by aligning themselves with the ambient magnetic field.

4.2.2 Exchange energy

We learned in Lecture 3 that some crystalline states are capable of ferromagnetic behavior because of quantum mechanical considerations. Electrons in neighboring orbitals in certain crystals “know” about each other’s spin states. In order to avoid sharing the same orbital with the same spin (hence having the same quantum numbers - not allowed from Pauli’s exclusion principle), electronic spins in such crystals act in a coordinated fashion. They will be either aligned parallel or antiparallel according to the details of the interaction. This exchange energy density (E_e) is the source of spontaneous magnetization and is given for a pair of spins by:

We define here a parameter that we will use later: the exchange constant A = J_eS²/a where a is the interatomic spacing. A = 1.33 x 10^-11 Jm^-1 for magnetite, a common magnetic mineral.

The 3d electronic orbitals within magnetic crystals are, unlike the s orbitals, anisotropic (recall Lecture 3). They “poke” in certain directions. Hence spins in some directions within crystals will be easier to coordinate than in others. We can illustrate this using the example of magnetite shown in Figure 4.1. Magnetite octahedra (Figure 4.1a), when viewed at the atomic level (Figure 4.1b) are composed of one ferrous (Fe²⁺) cation, two ferric (Fe³⁺) cations and four O^2- anions. Each oxygen anion shares an electron with two neighboring cations in a covalent bond.

In Lecture 3 it was mentioned that in some crystals, spins are aligned anti-parallel, yet there is still a net magnetization, a phenomenon known as “ferrimagnetism”. This can arise from the fact that not all cations have the same number of unpaired spins. Magnetite, with its ferrous (4 m_b) and ferric (5 m_b) states is a good example. There are three iron cations in a magnetite crystal giving a total of 14 m_b to play with. This is HUGE. Magnetite is very magnetic, but not that magnetic! From Figure 4.1b we see that the ferric ions are all sitting on the tetrahedral (A) lattice sites and there are equal numbers of ferrous and ferric ions sitting on the octahedral (B) lattice sites. The A and B sites are aligned anti-parallel to one another because of super exchange (Lecture 3) so we have 9 m_b on the B sites minus 4 m_b on the A sites for a total of 5 m_b per unit cell of magnetite.

4.2.3 Magnetocrystalline anisotropy energy

The energy of moments aligned along different directions in magnetite is shown in Figure 4.1c. The bulges are in directions that have the highest energy ([001, 010, 100]). The lowest energy is along the body diagonal ([111] direction). The energy surface shown in Figure 4.1c represents the magnetocrystalline anisotropy energy, E_a. In a cubic crystal with direction cosines ₁,₂,₃ (the cosines of the angles between the direction and the crystallographic axes [100, 010, 001]; see appendix to Lecture 1 ), the energy density is given by:

(4.2)

where K₁ and K₂ are empirically determined magnetocrystalline anisotropy constants. In the case of (room temperature) magnetite, K₁ is -1.35 x 10⁴ Jm^-3. If you work through the magnetocrystalline equation, you will find that when K₁ is negative, E_a is at a minimum when it is parallel to the [111] direction (the body diagonal).

As a consequence of the magnetocrystalline anisotropy energy, once the magnetization is aligned with an easy direction, work must be done to change it. In Figure 4.1d we show the results of numerical simulations of the magnetization of a cube of magnetite as an applied field is brought from saturation to zero, then changed in sign and increased in the opposite direction. We show the results from two directions in the crystal. The magnetization aligned with the body diagonal [111] (associated with the minimum energy state - see Figure 4.1c) is harder to change than along one of the “hard” directions (e.g. [001]).

A useful parameter in characterizing the stability of a particular particle or assemblage of particles is the field that is required to drive the magnetizations out of the easy directions, over an energy barrier and into another easy direction. This field is called the flipping field, the coercive field or the coercivity (B_c or H_c depending on units), something we will consider in more detail in later lectures.

Cubic symmetry (as in the case of magnetite) is just one of many types of crystal symmetries. One other very important form is the uniaxial symmetry which can arise from crystal shape or structure. The energy density for uniaxial magnetic anisotropy is:

(4.3)

In this equation, when K_u is negative, the magnetization is constrained to lie perpendicular to the axis of symmetry. When K_u > 0, the magnetization lies parallel to it.

An example of a mineral dominated by uniaxial symmetry is hematite. The magnetization of hematite is quite complicated, as we shall learn later, but one source is magnetization lies in the “spin-canting” (see Lecture 3) within the basal plane of a hexagonal crystal. Within the basal plane, the anisotropy constant is very low and the magnetization wanders fairly freely. However, the anisotropy energy away from the basal plane is high, so the magnetization is constrained to lie within the basal plane.

Because electronic interactions depend heavily on inter atomic spacing, magnetocrystalline anisotropy constants are a strong function of temperature (see Figure 4.2). In magnetite, K₁ changes sign at a temperature known as the “isotropic point”. At the isotropic point, there is no large magnetocrystalline anisotropy. The large energy barriers that act to keep the magnetizations parallel to the body diagonal are gone and the spins can wander more freely through the crystal. Below the isotropic point, the energy barriers rise again, but with a different topology in which the crystal axes are the energy minima and the body diagonals are the high energy states.

At room temperature, electrons hop freely between the ferrous and ferric ions on the B lattice sites, so there is no order. Below about 120 K, there is an ordered arrangement of the ferrous and ferric ions. Because of the difference in size between the two, the lattice of the unit cell becomes slightly distorted and becomes monoclinic instead of cubic. This transition is known as the Verwey transition. Although the isotropic point (measured magnetically) and the Verwey transition (measured electrically) are separated in temperature by about 15^o, they are related phenomena (the ordering and electron hopping cause the change in K₁).

The change in magnetocrystalline anisotropy at low temperature can have a profound effect on the magnetization. In Figure 4.3 we show a typical (de)magnetization curve for magnetite taken from the “Rock magnetic bestiary” web site maintained at the Institute for Rock Magnetism: http://www.geo.umn.edu/orgs/irm/bestiary . There is a loss of magnetization at around 100 K. This loss is the basis for “low-temperature demagnetization” (LTD). However, some portion of the magnetization is always recovered after low temperature cycling (called the low temperature memory), so the general utility of LTD is somewhat limited.

4.2.4 Magnetostriction - stress anisotropy

It is intuitively obvious that, because the exchange energy depends strongly on the details of the physical interaction between orbitals in neighboring atoms with respect to one another, changing the positions of these atoms will affect that interaction. Put another way, straining a crystal will alter its magnetic behavior. Similarly, changes in the magnetization can change the shape of the crystal by altering the shapes of the orbitals. This is the phenomenon of magnetostriction. The magnetic energy caused by the application of stress to a crystal be approximated by:

4.2.5 Magnetostatic - or shape anisotropy

There is one more important source of magnetic anisotropy: shape. To understand how crystal shape controls magnetic energy, we need to understand the concept of the internal “demagnetizing field” of a magnetized body. In Figure 4.4a we show the magnetic vectors within a ferromagnetic crystal. These produce a magnetic field external to the crystal that is proportional to the magnetic moment (see Lecture 1 ). This external field is identical to a field produced by a set of “free poles” distributed over the surface of the crystal (Figure 4.4b). The surface poles don’t just produce the external field, they also produce an internal field shown in Figure 4.4c. The internal field is known as the demagnetizing field H_d. H_d is proportional to the magnetization of the body and is sensitive to the shape. For the simple ellipsoid shown in Figure 4.4, the demagnetizing field is given by:

where N is a demagnetizing factor determined by the shape. For a sphere, the surface poles are distributed over the surface such that there are none at the “equator” and most at the “pole” (see Figure 4.4d). The expression for surface pole density is _m = M ^. . By using tricks of potential field theory in which we can pretend that the external field of a uniformly magnetized body is identical to that of a central dipole moment of magnitude m = vM (where v is volume). At the equator of the sphere, H_d = -NM. The external field at the equator (remembering from Lecture 1 ) is given by

r

so H_d = -1 3M, hence N = 1 3.

Different directions within a non-spherical crystal will have different distributions of free poles (see Figures 4.4e,f), so N will depend on direction. In the case of an ellipsoid magnetized parallel to the elongation axis a (Figure 4.4e), the free poles are farther apart than across the grain, hence, intuitively, the demagnetizing field, which depends on 1/r², must be less than in the case of a sphere. Thus, N_a <1 3. Similarly, if the ellipsoid is magnetized along b (Figure 4.4f), the demagnetizing field is stronger or N_b > 1 3. In an ellipsoid there are three axes a,b,c, and N_a + N_b + N_c = 1 (in SI; in cgs units the sum is 4).

Getting back to the anisotropy energy, that arising from the external field of the particle is called magnetostatic energy whose energy density equation is:

N

For a prolate ellipsoid N_c = N_b and a/c = 1.5, N_a -N_c =~ 0.16. Also, N_a = 1 3[1 - 2 5(2 - b a - c a)]. The magnetization of magnetite is 4.8 x 10⁵Am^-1, so K_u 2.3 x 10⁴ Jm³. This is somewhat larger than the absolute value of K₁ for magnetocrystalline anisotropy in magnetite (K₁= -1.35 x 10⁴ Jm^-3), so the magnetization for even slightly elongate grains will be dominated by uniaxial anistropy controlled by shape. Minerals with low saturation magnetizations (like hematite) will not have shape dominated magnetic anisotropy, however.

4.2.6 Thermal energy

We have gone some way to answering some of the questions posed at the beginning of the lecture. It should be clear that it is the anisotropy energy which opposes change in the magnetic direction, preserving the magnetization for posterity. The related question of what allows the magnetization to come into equilibrium with the applied magnetic field, however, requires a little more work. The key is to find some mechanism which allows the moments to “jump over” magnetic anisotropy energy barriers and one answer is thermal energy E_T , given by:

where kT is thermal energy (see Lecture 3).

Imagine a block of material containing a random assemblage of magnetic particles that are for simplicity uniformly magnetized and dominated by uniaxial anisotropy. Suppose that this block has some initial magnetization M_o and is placed in an environment with no ambient magnetic field. Anisotropy energy will tend to keep each tiny magnetic moment in its original direction and the magnetization will not change over time. With some thermal energy, certain grains will have sufficient energy to overcome the anisotropy energy and flip their moments to the other easy axis. Over time, the magnetic moments will become random.

We know from statistical mechanics that the probability of finding a grain with a given thermal energy is P = exp(-E_T /kT). So we may have to wait some time t for a particle to work itself up to having sufficient energy to flip over the energy barrier. Therefore, the magnetization as a function of time in this simple scenario will decay according to this equation:

(4.4)

where t is time and is an empirical constant called the relaxation time which is the time required for the remanence to decay to 1/e of M_o. This equation is the essence of what is called “Néel theory” (see, e.g., Néel, 1955).

The value of is a function of the competition between magnetic anisotropy energy and thermal energy. It is a measure of the probability that a grain will have sufficient thermal energy to overcome the anisotropy energy and switch its moment. Therefore in zero external field:

(4.5)

where C is a frequency factor with a value of something like 10¹⁰ s^-1. The anisotropy energy is given by the dominant anisotropy parameter K (either K_u,K₁, or ) times the grain volume v. It is often convenient to use the relationship K = B_cM_s 2 , which will be derived as a homework assignment.

Thus, the relaxation time is proportional to coercivity, and volume, and is inversely related to temperature. Relaxation time varies rapidly with small changes in v and T. There is a sharp transition between grains with virtually no stability ( is on the order of seconds) and grains with stabilities of millions of years. To see how this works, we can take K = K₁ for magnetite and v = d³ for the grain size of a cube of magnetite (see Figure 4.5). We have only considered equant particles of magnetite to construct Figure 4.5 so it is worth bearing in mind that Kv is a strong function of shape - the more elongate the particle, the higher the stability - a point we will return to at the end of the lecture.

Grains with 10² - 10³ seconds have sufficient thermal energy to overcome the anisotropy energy frequently and are unstable on a laboratory time-scale. In zero field, these grain moments will tend to rapidly become random and in an applied field, they tend to rapidly align with the field. The net magnetization is related to the field by a Langevin function (see Lecture 3). Therefore, this behavior is quite similar to paramagnetism, hence these grains are called superparamagnetic (SP). Such grains can be distinguished from paramagnets, however, because the field required to saturate the moments is typically much less than a tesla, whereas that for paramagnets can exceed hundreds of tesla.

4.3 Magnetic domains

4.3.1 Some theory

So far we have been discussing hypothetical magnetic particles that are uniformly magnetized. In Figure 4.4a we noted that there is an energy associated with the field generated by a magnetic particle. This self energy density is given by:

It is interesting to consider how the self energy of a particle changes with volume. In a sphere with radius r microns (and volume 4 3r³), we get the energy in joules as a function of volume in Figure 4.6 by remembering that M = 4.8 x 10⁵ Am^-1 and _o = 4 x 10^-7 Hm^-1.

Particles with strong magnetizations (like magnetite) have self energies that quickly become quite large. We have been learning about several mechanisms that tend to align magnetic spins. In fact in very small particles, the spins are essentially lined up. The particle is uniformly magnetized and is called single domain (SD). In larger particles (although still pretty small) the self energy exceeds the other exchange and magnetocrystalline energies and crystals have non-uniform states of magnetization.

There are many strategies possible for magnetic particles to reduce self energy. Numerical models (called micromagnetic models) can find internal magnetization configurations that minimize the energies discussed in the preceding sections. Micromagnetic simulations for magnetite particles (e.g. Schabes and Bertram, 1988) allow us to peer into the state of magnetization inside magnetic particles. These simulations give a picture of increasing complexity from so-called “flower” (Figure 4.7a) to “vortex” states (Figure 4.7b) remanent states.

As particles grow even larger, they break into regions of uniform magnetization called magnetic domains separated by narrow zones of rapidly changing spin directions called domain walls. Magnetic domains can take many forms. We illustrate a few in Figure 4.8. The uniform case (single domain) is shown in Figure 4.8a. The external field is very large because the free poles are far apart (at opposite ends of the particle). When the particle organizes itself into two domains (Figure 4.8b), the external field is reduced by about a factor of two. In the case of four lamellar domains (Figure 4.8c), the external field is quite small. The introduction of closure domains as in Figure 4.8d reduces the external field to nothing.

(4.6)

As you might already suspect, domain walls are not “free”. If, as in Figure 4.9a, the spins simply switch from one orientation to the other abruptly, the exchange energy cost would be very high. One way to get around this to spread the change over several hundred atoms, as sketched in Figure 4.9b. The wall width is wider and the exchange energy price is much less. However, there are now spins in unfavorable directions from a magnetocrystalline point of view (they are in “hard” direction). Exchange energy therefore favors wider domain walls while magnetocrystalline anisotropy favors thin walls. With some work (see e.g., Dunlop and Özdemir, 1997, pp. 117-118), it is possible to come up with the following analytical expressions for wall width (_w) and wall energy per unit area (E_w):

In Figure 4.10 we plot the self energy from Figure 4.6 and the wall energy from E_w for spheres of magnetite. We see that the wall energy in particles with radii of a few tenths of a micron is much less than the self energy, yet the width of the walls is also a few tenths of a micron. So the smallest wall is really more like the vortex state and it is only for particles closer to one micron in size that true domains separated by discrete walls are formed.

Finally, it is possible to predict the number of domains (n_d) in a given particle of magnetite. Assuming lamellar domains within cubes of magnetite, Dunlop and Özdemir (1997) derived the following equation:

where Z is a constant incorporating magnetostriction and wall energy and a and b are particle length and width as before. For magnetite, Z 1.1 x 10³. For a 100 m equant grain of magnetite, then, we would expect to find 11 domains.

4.3.2 Some experiments

How can we test the theoretical predictions of domain theory? Do they really exist? Are they the size and shape we expect? Are there as many as we would expect? In order to address these questions we require a way of “seeing” magnetic domains. Bitter (1931) devised a way for doing just that. Magnetic domain walls are regions with large stray fields (as opposed to domains in which the spins are usually parallel to the sides of the crystals to minimize stray fields). In the “Bitter technique” magnetic colloid material is drawn to the regions of high field gradients on highly polished sections allowing the domain walls to be observed.

We show an example of a photomicrograph taken from the interior of a large grain of magnetite (Dunlop and Özdemir, 1997). It appears that if great care is taken, domain walls of about the right size, shape and orientation can be found.

Özdemir and Dunlop (1997) compiled what they considered to be the “best” data on number of domains n_d observed in carefully sized magnetite grains. We replot their data compilation in Figure 4.12. Also shown is the prediction from Equation 4.6. There appear to be “too many” domains for small grain sizes and “too few” for large grain sizes. In a seminal paper, Halgedahl and Fuller (1980) argued that there were far fewer domains than predicted for titanomagnetite, which they explained as arising from the fact that the energy to nucleate a domain wall from nothing had not been taken into account in the theory.

We are now in a position to pull together all the threads we have considered in this lecture and make a plot of what sort of magnetic particles behave as superparamagnets, which should be single domain and which should be multi-domain according to our simple theories. Evans and McElhinny (1969) made a beautiful plot (see Figure 4.13). There is virtually no SD stability field for equant magnetite; they are either SP or MD (multi-domain). As the width to length decreases (the particle gets longer), the stability field for SD magnetite expands. Of course micromagnetic modelling shows that there are several transitional states between uniform magnetization (SD) and MD, i.e. the flower and vortex remaent states, but Figure 4.13 or variations thereof (e.g. Butler and Banerjee, 1975) have enormous predictive power and continue to be used extensively.

Bibliography

 

 

[ Home ]  [ Databases ]  [ Events ]  [ Tools ]  [ Publications ]  [ Links ]  [ Google Site Search ] [ Metadata ]  [ Who's Who ]  [ Browsers and Plugins ]  [ Database Indexes ]
This page was last updated on 04-Apr-2008 Sponsored by NSF EAR 0000998	Supported by the San Diego Supercomputer Center and the Scripps Institution of Oceanography