11.3. Itinerant Electron Theory

11.3. Itinerant Electron Theory

Magentism/David Jiles 11.3. Itinerant Electron Theory

" What happens if the magnetic electrons are not localized at the atomic sites? "

I. Magnetism of Electrons in Energy Bands

" What are the magnetic properties of electrons in energy bands? "

The magnetic moments are due to the angular momentum of unpaired electrons in unfilled shells.

The unpaired electrons are usually outer electrons and so are unlikely to be closely bound to the atoms.

This is true of the 3d transition elements iron, nickel, and cobalt.

We therefore need to find a theoretical description of magnetism due to itinerant electrons in these cases.

Metals such as the alkali metal series lithium, sodium, potassium, rubidium, and caesium all show temperature dependent paramagnetism, for example, which can not be explained by the local moment model.

In this case, the paramagentism is due to the outer electrons, which behave as a free electron gas.

II. Pauli Paramagnetism of Free Electrons

" Can paramagnetism be described simply on the basis of changes in population of nearly free electrons in bands? "

The reason that the paramagnetic susceptibility is so much lower than the expectation based on the localized model is that the electrons are in general not free to rotate into the field direction as required by the classical Langevin model.

This is because, as a result of the Pauli exclusion principle, the electron states needed for reorientation are already occupied by other electrons.

Only those electrons within an energy T of the Fermi level are able to change orientation.

We therefore need to consider an alternative model of paramagnetism due to the band electrons as conceived by Pauli

Let's consider th parabloic distribution of nearly free electrons.

Only the fraction of the total number of electrons can contribute to the magntization.

T is the thermodynamic temperature

is the Fermi temperature, defined such that .

Therefore,

And also, the number density of electrons parallel to the field is

N =

So that,

Equation right above gives the Pauli spin magnetization of the paramagnetic conduction band electrons.

The susceptibility is then

III. Band Theory of Ferromagnetism

" How do electrons in bands behave when there is an exchange interaction, which causes alignment of spins? "

In metallic ferromagnets such as Fe, Co, and Ni, the magnetic properties are due principally to the conduction electrons, and in these cases, an argument can be made in favor of itinerant exchange between these electrons.

This form of exchange interaction is conceptually different from the direct and superexchange theories, which have been developed for localized moments.

The band theory of ferromagnetism is a simple extension of the band theory of paramagnetism by the introduction of an exchange coupling between the electrons.

The simplest case is to consider the electrons to be entirely free, that is, a parabolic energy distribution.

Since magnetic moments can only arise from unpaired electrons, it is immediately clear that a completely filled energy band cannot contribute a magnetic moment since in such an energy band all electron spins will be paired giving L and S = 0.

In a partially filled energy band, it is possible to have an imbalance of spins leading to a net magnetic moment per atom.

This arises because the exchange energy removes the degeneracy of the spin-up and spin-down half bands.

The larger the exchange energy, the greater the difference in energy between two half bands.

When half bands overlap, there will be non-integral number of magnetic moments per atom.

When exchange energy is big enough to completely split the half bands, there will be integral number of magnetic moments per atom

IV. Magnetic Properties of 3d Band Electrons

" Can the band theory provide a satisfactory description of the magnetic properties of the 3d metals? "

In the transition metals such as Fe, Ni, and Co, which are the three ferromagnetic elements with electronic structure for which the band theory should apply, the magnetic properties are due to 3d band electrons.

Of course, there is also a 4s electron band but this contains two paired electrons and so does not affect magnetic properties.

The 3d band can hold up to 10 electrons (5 5), and it is here that we must concentrate attention in order to explain the observed properties.

The exchange interaction is responsible for creating the imbalance in the spin-up and spin-down states.

In the absence of the exchange energy, the spin imbalance would be an excited state, but this does not require too much energy in 3d band because of the high density of states and therefore a positive exchange interaction can be sufficient to cause the alignment, resulting in a spin imbalance and a net magnetic moment per atom.

In this way, the band theory can account for the nonintegral atomic magnetic moments in these metal, a result that is more difficult to justify on the localized moments model.

V. Slater-Pauling Curve

" How well does the itinerant electron model describe the magnetic properties of 3d alloys? "

The Slater-Pauling curve gives the magnetic moments of these 3d metals and their alloys from the premises of the itinerant electron theory.

This is shown below

The metals in these range, Cr, Mn, Fe, Co, Ni, and Cu, have total numbers of electrons ranging from 24 to 29, while the number of 3d electrons ranges from 5 in Cr to 10 in Cu.

The interpretation of the results in figure above is in terms of the rigid band model.

It is considered that the alloy metals share a common 3d band to which both elements contribute electrons.

You can see that the maximum magnetic moment occurs at a point between iron and cobalt.

It appears from tis model that, as expected, the 3d and 4s electrons are responsible for the magnetic properties of these metals and alloys, and that they are relatively free.

Therefore it is a reasonable assumption that they are shared between the ions in a common 3d band.

It has been suggested that the 3d band is broken into two parts:

The upper part : capable of holding 4.8 electrons(2.4 2.4 ) The lower part : capable of holding 5.2 electrons(2.6 2.6 )

This means that as electrons are removed beginning with zinc, for example, the magnetic moment is increased by depletion of spin down electrons in the upper part of the band until a magnetic moment of 2.4 Bohr magneton is reached between iron and cobalt.

Then the moment begins to decrease again toward chromium because the removal of further electrons results in a reduction in spin-up electrons from the upper part of the 3d band.

VI. Critique of the Itinerant Model

" What are the strengths and weaknesses of the itinerant model? "

The drawback of the itinerant electron theory is that it is extremely difficult to make fundamental calculations based on it.

The itinerant electron theory does not provide any simple model from which first principles calculations can be made.

Therefore, although the current opinion is that the itinerant theory is intrinsically closer to reality in most cases, interpretations of magnetic properties are still more often made on the basis of the localized moment model.

VII. Correlation Effects Among Conduction Electrons

" Since 3d magnetic metals seem to have features of both itinerant and localized moment models, how can this explained? "

Ferromagnetic metals, particularly the 3d transition metals, have features that are characteristic of both types of model.

As examples, we can consider the occurrence of spin-wave phenomena in these materials and the strong temperature dependence of the susceptibility. These are indicative or characteristic of a local moment model.

However, the occurrence of magnetic moments with nonintegral numbers of Bohr magnetons per atom is characteristic of an itinerant electron model.

These apparent contradictions have been resolved by the model suggested by Hubbard.

In this treatment, the 3d electrons bands are considered to be narrow enough that with the electron charge density concentrated in the vicinity of the atomic sites, it becomes meaningful to describe an electron as being located on, or at least associated with, a particular atom.

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In real materials, the situation is itself usually somewhere in between the localized and itinerant models.

The conditions of the Hubbard model can be relaxed to represent either localized or bandlike behavior in the appropriate limits.

Thus strong correlation of the electron spins on a particular atom, but weak correlation with other conduction band electrons, tends to produce behavior characteristic of the local moment model.

Weak correlations of electron spins on a particular atom together with itinerant exchange coupling with other conduction electrons tend to produce behavior characteristic of the band model of ferromagnetism.

VIII. Indirect Exchange

" How can the electrons spins be coupled on localized atomic sites if their wave functions do not overlap? "

The polarization of the spins on the conduction electrons as a result of exchange with the unpaired bound electrons could allow ordering of the electron spins on neighboring atomic sites, and could also account for the nonintegral number of Bohr magnetons per atom that are known to occur in these materials.

The later theory of indirect exchange, which became known as the RKKY theory, relies also on the propagation of exchange coupling via the polarized conduction electrons, and can lead to ferromagnetic, antiferromagnetic, sinusoidal, and helical arrangements of the localized magnetic moments under different conditions.

This model therefore takes into account the presence of both conduction and localized electronic magnetic moments in providing an explanation of magnetic ordering in these materials.

IX. Giant Magnetoresistance in Multilayers

" How can the large change in resistance with applied field in magnetic multilayers be explained? "

The giant magnetoresistnace occurs in some multilayers when the relative orientations of the magnetic moments are changed.

When the successive layers are

parallel : the material can have relatively high conductivity.

antiparallel : the material can have relatively high resistivity.

The change in resistance with change in relative orientation of the directions of magnetization in the successive magnetic layers can be attributed to the spin-dependent conduction of the electrons in these itinerant electron ferromagnets.

The band theory of magnetism as described in previous part provides an explanation of this effect.

The majority (e.g., spin up) and the minority (e.g., spin down) electrons have different densities of states and mobilities at any given energy due to the exchange splitting of the energy bands.

Consequently, the conductivities of these two groups of electrons can be very different.

This spin-orientation-dependent conductivity gives the so-called spin valve effect.

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