Ferrimagnetism
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A ferrimagnetic material is a material that has populations of atoms with opposing magnetic moments, as in antiferromagnetism, but these moments are unequal in magnitude, so a spontaneous magnetization remains. This can for example occur when the populations consist of different
atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s or ions (such as Fe2+ and Fe3+). Like
ferromagnetic Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromagne ...
substances, ferrimagnetic substances are attracted by magnets and can be magnetized to make
permanent magnet A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nickel, c ...
s. The oldest known magnetic substance,
magnetite Magnetite is a mineral and one of the main iron ores, with the chemical formula . It is one of the iron oxide, oxides of iron, and is ferrimagnetism, ferrimagnetic; it is attracted to a magnet and can be magnetization, magnetized to become a ...
(Fe3O4), is ferrimagnetic, but was classified as a ferromagnet before Louis Néel discovered ferrimagnetism in 1948. Since the discovery, numerous uses have been found for ferrimagnetic materials, such as hard-drive platters and biomedical applications.


History

Until the twentieth century, all naturally occurring 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. While working with Mn2Sb, 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. 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, Néel was awarded for his work in magnetism with the Nobel Prize in Physics.


Physical origin

Ferrimagnetism has the same physical origins as
ferromagnetism Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromagne ...
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 in the figure above. 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 ferrimagnetic materials the net moment is nonzero because the opposed moments differ in magnitude. Ferrimagnets have a critical temperature above which they become paramagnetic just as ferromagnets do. At this temperature (called the Curie temperature) there is a second-order phase transition, and the system can no longer maintain a spontaneous magnetization. This is because at higher temperatures the thermal motion is strong enough that 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 : \vec=\vec_0+\vec_m, where \vec_0 is the applied magnetic field, and \vec_m is field caused by the interactions between the atoms. The following assumption then is \vec_m=\gamma\vec. Here, \vec is the average magnetization of the lattice, and \gamma is the molecular field coefficient. When \vec and \gamma is allowed to be position- and orientation-dependent, it can then be written in the form : \vec_i=\vec_0+\sum_^n\gamma_\vec_k, where \vec_i is the field acting on the ''i''-th substructure, and \gamma_ is the molecular field coefficient between the ''i''-th and ''k''-th substructures. For a diatomic lattice, two types of sites can be designated, ''a'' and ''b''. N can be designated the number of magnetic ions per unit volume, \lambda the fraction of the magnetic ions on the ''a'' sites, and \mu=1-\lambda the fraction on the ''b'' sites. This then gives : \vec_ = \gamma_\vec, \quad \vec_ = \gamma_\vec_b, \quad \vec_ = \gamma_\vec_a, \quad \vec_ = \gamma_\vec_b. It can be shown that \gamma_=\gamma_ and that \gamma_\neq\gamma_, unless the structures are identical. \gamma_>0 favors a parallel alignment of \vec_a and \vec_b, while \gamma_<0 favors an anti-parallel alignment. For ferrimagnets, \gamma_<0, so it will be convenient to take \gamma_ 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 : \vec_a=\vec_0+\gamma_\vec_a-\gamma_\vec_b, : \vec_b=\vec_0+\gamma_\vec_b-\gamma_\vec_a. Furthermore, the parameters \alpha=\gamma_/\gamma_ and \beta=\gamma_/\gamma_ will be introduced, which give the ratio between the strengths of the interactions. At last, the reduced magnetizations will be introduced : \vec_a=\vec_a/\lambda Ng\mu_B S_a, : \vec_b=\vec_b/\mu Ng\mu_BS_b with S_i the spin of the ''i''-th element. This then gives for the fields: : \vec_a=\vec_0+Ng\mu_B S_a\gamma_(\lambda\alpha\vec_a-\mu\vec_b), : \vec_b=\vec_0+Ng\mu_BS_b\gamma_(-\lambda\vec_a+\mu\beta\vec_b) The solutions to these equations (omitted here) are then given by : \sigma_a=B_(g\mu_bS_aH_a/k_\textT), : \sigma_b=B_(g\mu_bS_bH_b/k_\textT). where B_J(x) is the Brillouin function. The simplest case to solve now is S_a=S_b=1/2. Since B_(x)=\tanh(x), this then gives the following pair of equations: : \lambda\sigma_a=\frac(\beta\tanh^\sigma_a+\tanh^\sigma_b), : \mu\sigma_b=\frac(\tanh^\sigma_a+\alpha\tanh^\sigma_b) with \tau=T/T_\text and F(\lambda,\alpha,\beta)=\frac\left(\lambda\alpha+\mu\beta+\sqrt\right). These equations do not have a known analytical solution, so they must be solved numerically to find the temperature dependence of \mu.


Effects of temperature

Unlike ferromagnetism, 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 Absolute zero is the lowest possible temperature, a state at which a system's internal energy, and in ideal cases entropy, reach their minimum values. The absolute zero is defined as 0 K on the Kelvin scale, equivalent to −273.15 ° ...
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 NiFe2/5Cr8/5O4 and Li1/2Fe5/4Ce5/4O4. 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 Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
compensation point, at which the net angular momentum vanishes. This compensation point is crucial for achieving fast 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 M_\text; 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 does not disappear, but a nonzero magnetization remains. This 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 eventually reaches a magnetization of -M_\text. This behavior results in what is called a ''hysteresis loop''.


Properties and uses

Ferrimagnetic materials have high resistivity and have
anisotropic Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ver ...
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 Microwave is a form of electromagnetic radiation with wavelengths shorter than other radio waves but longer than infrared waves. Its wavelength ranges from about one meter to one millimeter, corresponding to frequency, frequencies between 300&n ...
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 Magnetite is a mineral and one of the main iron ores, with the chemical formula . It is one of the iron oxide, oxides of iron, and is ferrimagnetism, ferrimagnetic; it is attracted to a magnet and can be magnetization, magnetized to become a ...
can be used for thermal energy storage.


Examples

The oldest known magnetic material,
magnetite Magnetite is a mineral and one of the main iron ores, with the chemical formula . It is one of the iron oxide, oxides of iron, and is ferrimagnetism, ferrimagnetic; it is attracted to a magnet and can be magnetization, magnetized to become a ...
, 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 oxide An iron oxide is a chemical compound composed of iron and oxygen. Several iron oxides are recognized. Often they are non-stoichiometric. Ferric oxyhydroxides are a related class of compounds, perhaps the best known of which is rust. Iron ...
s with other elements such as
aluminum Aluminium (or aluminum in North American English) is a chemical element; it has chemical symbol, symbol Al and atomic number 13. It has a density lower than that of other common metals, about one-third that of steel. Aluminium has ...
, cobalt, nickel,
manganese Manganese is a chemical element; it has Symbol (chemistry), symbol Mn and atomic number 25. It is a hard, brittle, silvery metal, often found in minerals in combination with iron. Manganese was first isolated in the 1770s. It is a transition m ...
, and
zinc Zinc is a chemical element; it has symbol Zn and atomic number 30. It is a slightly brittle metal at room temperature and has a shiny-greyish appearance when oxidation is removed. It is the first element in group 12 (IIB) of the periodic tabl ...
; and hexagonal or spinel type ferrites, including rhenium ferrite, ReFe2O4, PbFe12O19 and BaFe12O19 and
pyrrhotite Pyrrhotite (''Pyrrhus of Epirus, pyrrhos'' in Greek language, Greek meaning "flame-coloured"'')'' is an iron sulfide mineral with the formula Fe(1−x)S (x = 0 to 0.125). It is a nonstoichiometric compound, nonstoichiometric variant of FeS, th ...
, Fe1−''x''S. Ferrimagnetism can also occur in single-molecule magnets. A classic example is a dodecanuclear
manganese Manganese is a chemical element; it has Symbol (chemistry), symbol Mn and atomic number 25. It is a hard, brittle, silvery metal, often found in minerals in combination with iron. Manganese was first isolated in the 1770s. It is a transition m ...
molecule with an effective spin ''S'' = 10 derived from antiferromagnetic interaction on Mn(IV) metal centers with Mn(III) and Mn(II) metal centers.


See also

* *


References


External links

* {{Authority control Magnetic ordering Quantum phases