Ferrimagnetism

A ferrimagnetic material is material that has populations of atoms

with opposing magnetic moments, as in antiferromagnetism. For
ferrimagnetic materials these moments are unequal in magnitude so
a spontaneous magnetization remains.[1] This can for example
happen when the populations consist out of different atoms or ions
(such as Fe2+ and Fe3+).
Ferrimagnetic ordering
Ferrimagnetism has often been confused with ferromagnetism. The
oldest known magnetic substance magnetite (Fe3 O4 ) was
classified as a ferromagnet before Louis Néel discovered ferrimagnetism in 1948.[2] Since the discovery,
numerous uses have been found for ferrimagnetic materials, such as hard drive platters and biomedical
applications.

Contents
History
Physical origin
Derivation
Effects of temperature
Effect of external fields
Properties and uses
Examples
See also
References
External links

History
Until the twentieth century, all naturally magnetic substances were called ferromagnets. In 1936 Louis Néel
published a paper proposing the existence of a new form of cooperative magnetism he called
antiferromagnetism.[3] While working with Mn2 Sb, French physicist Charles Guillaud discovered that the
current theories on magnetism were not adequate to explain the behavior of the material, and made a model
to explain the behavior.[4] In 1948, Néel published a paper about a third type of cooperative magnetism,
based on the assumptions in Guillaud's model. He called it ferrimagnetism. In 1970, Néels was awarded for
his work in magnetism with the Nobel Prize in Physics.[5]

Physical origin
Ferrimagnetism has the same physical origins as ferromagnetism and
antiferromagnetism. In ferrimagnetic materials the magnetization is also
caused by a combination of dipole-dipole interactions and exchange
interactions resulting from the Pauli exclusion principle. The main
difference is that in ferrimagnetic materials there are different types of
atoms in the material's unit cell. An example of this can be seen the
figure on the right. Here the atoms with a smaller magnetic moment
point in the opposite direction of the larger moments. This arrangement
is similar to that present in antiferromagnetic materials, but in ➀ Below the magnetization
ferrimagnetic materials the net moment is nonzero because the opposed compensation point,
moments differ in magnitude. ferrimagnetic material is
magnetic. ➁ At the
Ferrimagnets have a critical temperature above which they become compensation point, the
paramagnetic just as ferromagnets do.[6] At this temperature (called the magnetic components cancel
Curie temperature) there is a second order phase transition[7] and the each other and the total
magnetic moment is zero. ➂
system can no longer maintain a spontaneous magnetization. This is
Above the Curie temperature,
because at higher temperatures the thermal motion is strong enough that
the material loses magnetism.
it exceeds the tendency of the dipoles to align.

Derivation
There are various ways to describe ferrimagnets, the simplest of which is with mean-field theory. In mean-
field theory the field acting on the atoms can be written as:

Where is the applied magnetic field field and is field caused by the interactions between the
atoms. The following assumption then is:

Here is the average magnetization of the lattice and is the molecular field coefficient. When we allow
and to be position and orientation dependent we can then write it in the form:

Here is the field acting on the ith substructure and is the molecular field coefficient between the ith
and the kth substructure. For a diatomic lattice we can designate two types of sites, A and B. We can
designate the number of magnetic ions per unit volume, the fraction of the magnetic ions on the A
sites, and the fraction on the B sites. This then gives:

It can be shown that and that unless the structures are identical. favors a
parallel alignment of and , while favors an anti-parallel alignment. For ferrimagnets,
, so it will be convenient to take as a positive quantity and write the minus sign explicitly in
front of it. For the total fields on A and B this then gives:
Furthermore we will introduce the parameters and which give the ratio between the

strengths of the interactions. At last we will introduce the reduced magnetizations:

with the spin of the ith element. This then gives for the fields:

The solutions to these equations (omitted here) are then given by

where is the Brillouin function. The simplest case to solve now is . Since
. This then gives the following pair of equations:

with and . These equations do not have

a known analytical solution, so they must be solved numerically to find the temperature dependence of .

Effects of temperature
Unlike ferromagnetism, the shapes of the magnetization curves of ferrimagnetism can take many different
shapes depending on the strength of the interactions and the relative abundance of atoms. The most notable
instances of this property are that the direction of magnetization can reverse while heating a ferrimagnetic
material from absolute zero to its critical temperature, and that strength of magnetization can increase while
heating a ferrimagnetic material to the critical temperature, both of which cannot occur for ferromagnetic
materials. These temperature dependencies have also been experimentally observed in NiFe2/5 Cr8/5 O4 [8]
and Li1/2 Fe5/4 Ce5/4 O4 .[9]

A temperature lower than the Curie temperature, but at which the opposing magnetic moments are equal
(resulting in a net magnetic moment of zero) is called a magnetization compensation point. This
compensation point is observed easily in garnets and rare-earth–transition-metal alloys (RE-TM).
Furthermore, ferrimagnets may also have an angular momentum compensation point, at which the net
angular momentum vanishes. This compensation point is a crucial point for achieving high speed
magnetization reversal in magnetic memory devices.

Effect of external fields

When ferrimagnets are exposed to an external magnetic field,
they display what is called magnetic hysteresis, where magnetic
behavior depends on the history of the magnet. They also exhibit
a saturation magnetization ; this magnetization is reached
when the external field is strong enough to make all the moments
align in the same direction. When this point is reached, the
magnetization cannot increase as there are no more moments to
align. When the external field is removed, the magnetization of
the ferrimagnet will not disappear but a nonzero magnetization
will remain. This is effect is often used in applications of
magnets. If an external field in the opposite direction is applied
subsequently, the magnet will demagnetize further until it Theoretical model of magnetization m
eventually reaches a magnetization of . This behavior against magnetic field h. Starting at the
results in what is called a hysteresis loop. [10] origin, the upward curve is the initial
magnetization curve. The downward
curve after saturation, along with the
Properties and uses lower return curve, form the main loop.
The intercepts hc and m rs are the
Ferrimagnetic materials have high resistivity and have coercivity and saturation remanence.
anisotropic properties. The anisotropy is actually induced by an
external applied field. When this applied field aligns with the
magnetic dipoles, it causes a net magnetic dipole moment and causes the magnetic dipoles to precess at a
frequency controlled by the applied field, called Larmor or precession frequency. As a particular example, a
microwave signal circularly polarized in the same direction as this precession strongly interacts with the
magnetic dipole moments; when it is polarized in the opposite direction, the interaction is very low. When
the interaction is strong, the microwave signal can pass through the material. This directional property is
used in the construction of microwave devices like isolators, circulators, and gyrators. Ferrimagnetic
materials are also used to produce optical isolators and circulators. Ferrimagnetic minerals in various rock
types are used to study ancient geomagnetic properties of Earth and other planets. That field of study is
known as paleomagnetism. In addition, it has been shown that ferrimagnets such as magnetite can be used
for thermal energy storage.[11]

Examples
The oldest known magnetic material, magnetite, is a ferrimagnetic substance. The tetrahedral and
octahedral sites of its crystal structure exhibit opposite spin. Other known ferrimagnetic materials include
yttrium iron garnet (YIG); cubic ferrites composed of iron oxides with other elements such as aluminum,
cobalt, nickel, manganese, and zinc; and hexagonal ferrites such as PbFe12 O19 and BaFe12 O19 and
pyrrhotite, Fe1−x S.[12]

Ferrimagnetism can also occur in single-molecule magnets. A classic example is a dodecanuclear

manganese molecule with an effective spin S = 10 derived from antiferromagnetic interaction on Mn(IV)
metal centers with Mn(III) and Mn(II) metal centers.[13]
See also
Anisotropy energy
Orbital magnetization

References
1. Spaldin, Nicola A. (2011). Magnetic materials : fundamentals and applications (2nd ed.).
Cambridge: Cambridge University Press. ISBN 978-0-521-88669-7. OCLC 607986416 (http
s://www.worldcat.org/oclc/607986416).
2. Néel, M. Louis (1948). "Propriétés magnétiques des ferrites ; ferrimagnétisme et
antiferromagnétisme" (https://hal.archives-ouvertes.fr/hal-02888371/file/N%C3%A9el%20-%
201948%20-%20Propri%C3%A9t%C3%A9s%20magn%C3%A9tiques%20des%20ferrite
s%20%3B%20ferrimagn%C3%A9ti.pdf) (PDF). Annales de Physique. 12 (3): 137–198.
doi:10.1051/anphys/194812030137
(https://doi.org/10.1051%2Fanphys%2F194812030137). ISSN 0003-4169 (https://www.worl
dcat.org/issn/0003-4169).
3. Néel, Louis (1936). "Propriétés magnétiques de l'état métallique et énergie d'interaction
entre atomes magnétiques" (https://dx.doi.org/10.1051/anphys/193611050232). Annales de
physique. 11 (5): 232–279. doi:10.1051/anphys/193611050232 (https://doi.org/10.1051%2F
anphys%2F193611050232). ISSN 0003-4169 (https://www.worldcat.org/issn/0003-4169).
4. Smart, J. Samuel (September 1955). "The Néel Theory of Ferrimagnetism" (http://aapt.scitati
on.org/doi/10.1119/1.1934006). American Journal of Physics. 23 (6): 356–370.
doi:10.1119/1.1934006 (https://doi.org/10.1119%2F1.1934006). ISSN 0002-9505 (https://ww
w.worldcat.org/issn/0002-9505).
5. "The Nobel Prize in Physics 1970" (https://www.nobelprize.org/prizes/physics/1970/summar
y/). NobelPrize.org. Retrieved 2021-01-26.
6. Simon, Steven H. (21 June 2013). The Oxford Solid State Basics (First ed.). Oxford.
ISBN 978-0-19-150210-1. OCLC 851099021 (https://www.worldcat.org/oclc/851099021).
7. Blundell, Stephen; Blundell, Katherine M. (2010). Concepts in thermal physics (2nd ed.).
Oxford: Oxford University Press. ISBN 978-0-19-956209-1. OCLC 607907330 (https://www.
worldcat.org/oclc/607907330).
8. Tsushima, Tachiro (August 1963). "Magnetic Properties of Ferrite-Chromite Series of Nickel
and Cobalt" (https://dx.doi.org/10.1143/jpsj.18.1162). Journal of the Physical Society of
Japan. 18 (8): 1162–1166. doi:10.1143/jpsj.18.1162 (https://doi.org/10.1143%2Fjpsj.18.116
2). ISSN 0031-9015 (https://www.worldcat.org/issn/0031-9015).
9. Gorter, E. W.; Schulkes, J. A. (1953-05-01). "Reversal of Spontaneous Magnetization as a
Function of Temperature in LiFeCr Spinels" (https://dx.doi.org/10.1103/physrev.90.487.2).
Physical Review. 90 (3): 487–488. doi:10.1103/physrev.90.487.2 (https://doi.org/10.1103%2
Fphysrev.90.487.2). ISSN 0031-899X (https://www.worldcat.org/issn/0031-899X).
10. Soler, M. A. G.; Paterno, L. G. (2017-01-01), Da Róz, Alessandra L.; Ferreira, Marystela; de
Lima Leite, Fábio; Oliveira, Osvaldo N. (eds.), "6 - Magnetic Nanomaterials" (http://www.scie
ncedirect.com/science/article/pii/B9780323497824000061), Nanostructures, William
Andrew Publishing, pp. 147–186, doi:10.1016/b978-0-323-49782-4.00006-1 (https://doi.org/
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11. Grosu, Yaroslav; Faik, Abdessamad; Ortega-Fernández, Iñigo; D'Aguanno, Bruno (March
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reversible latent heat transition and controlled thermal conductivity" (https://doi.org/10.101
6%2Fj.solmat.2016.12.006). Solar Energy Materials and Solar Cells. 161: 170–176.
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13. Sessoli, Roberta; Tsai, Hui Lien; Schake, Ann R.; Wang, Sheyi; Vincent, John B.; Folting,
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External links
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