Ferrimagnetism

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 Mn₂Sb, French physicist Charles Guillaud discoverd that the current theories on magnetism were not adequate to explain the behaviour of the material, and made a model to explain the behaviour.[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]

➀ Below the magnetization compensation point, ferrimagnetic material is magnetic. ➁ At the compensation point, the magnetic components cancel each other and the total magnetic moment is zero. ➂ Above the Curie temperature, the material loses magnetism.

Ferrimagnetism has the same physical origins as Ferromagnetism and Antiferromagnetism. In ferrimagnetic materials the magnetisation is also caused by a combination of dipole-dipole interactions and the exchange interaction caused by the Pauli exclusion principle. The main difference with ferromagnetism and antiferromagnetism is that there are different types of atoms in the 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. So here there is an anti-ferromagnetic ordering but there is still a non-zero magnetic moment. Ferrimagnets have a critical temperature above which they become paramagnetic just as ferromagnets do.[6] At this temperature (called the Curie temperature) there is a second order phase transition[7] and the system can no longer maintain a spontaneous magnetisation. This is because at higher temperatures the thermal motion is big enough so that it outcompetes the tendency of the dipoles to align.

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:

${\displaystyle {\overrightarrow {H}}={\overrightarrow {H}}_{0}+{\overrightarrow {H}}_{m}}$

Where ${\displaystyle {\overrightarrow {H}}_{0}}$ is the Applied magnetic field field and ${\displaystyle {\overrightarrow {H}}_{m}}$ is field caused by the interactions between the atoms. The following assumption then is:${\displaystyle {\overrightarrow {H}}_{m}=\gamma {\overrightarrow {M}}}$ Here ${\textstyle {\overrightarrow {M}}}$ is the average magnetization of the lattice and ${\displaystyle \gamma }$ is the molecular field coefficient. When we allow ${\textstyle {\overrightarrow {M}}}$ and ${\displaystyle \gamma }$ to be position and orientation dependent we can then write it in the form:

${\displaystyle {\overrightarrow {H}}_{i}={\overrightarrow {H}}_{0}+\sum _{k=1}^{n}\gamma _{ik}{\overrightarrow {M}}_{k}}$

Here ${\displaystyle {\overrightarrow {H}}_{i}}$ is the field acting on the i^th substructure and ${\displaystyle \gamma _{ik}}$ is the molecular field coefficient between the i^th and the k^th substructure. For a diatomic lattice we can designate two types of sites, A and B. ${\displaystyle N}$ the number of magnetic ions per unit volume, ${\displaystyle \lambda }$ the fraction of the magnetic ions on the A sites and ${\textstyle \mu =1-\lambda }$ on the B sites. This then gives:

${\displaystyle {\overrightarrow {H}}_{aa}=\gamma _{aa}{\overrightarrow {M}};{\overrightarrow {H}}_{ab}=\gamma _{ab}{\overrightarrow {M}}_{b};{\overrightarrow {H}}_{ba}=\gamma _{ba}{\overrightarrow {M}}_{a};{\overrightarrow {H}}_{bb}=\gamma _{bb}{\overrightarrow {M}}_{b}}$

It can be shown that ${\displaystyle \gamma _{ab}=\gamma _{ba}}$ and that ${\displaystyle \gamma _{aa}\neq \gamma _{bb}}$ unless the structures are identical. ${\displaystyle \gamma _{ab}>0}$ favours a parallel alignment of ${\displaystyle {\overrightarrow {M}}_{a}}$ and ${\displaystyle {\overrightarrow {M}}_{b}}$, while ${\displaystyle \gamma _{ab}<0}$ favours an anti-parallel alignment. For ferrimagnets ${\displaystyle \gamma _{ab}<0}$ so it will be convenient to take ${\displaystyle \gamma _{ab}}$ as a positive quantity and write the minus sing explicitly in front of it. For the total fields on A and B this then gives:

${\displaystyle {\overrightarrow {H}}_{a}={\overrightarrow {H}}_{0}+\gamma _{aa}{\overrightarrow {M}}_{a}-\gamma _{ab}{\overrightarrow {M}}_{b}}$

${\displaystyle {\overrightarrow {H}}_{b}={\overrightarrow {H}}_{0}+\gamma _{bb}{\overrightarrow {M}}_{b}-\gamma _{ab}{\overrightarrow {M}}_{a}}$

Furthermore we will introduce the parameters ${\displaystyle \alpha ={\frac {\gamma _{aa}}{\gamma _{ab}}}}$ and ${\displaystyle \beta ={\frac {\gamma _{bb}}{\gamma _{ab}}}}$ which give the ratio between the strenghts of the interactions. At last we will introduce the reduced magnetization's:

${\displaystyle {\overrightarrow {\sigma }}_{a}={\overrightarrow {M}}_{a}/\lambda Ng\mu _{B}S_{a}}$

${\displaystyle {\overrightarrow {\sigma }}_{b}={\overrightarrow {M}}_{b}/\mu Ng\mu _{B}S_{b}}$

With ${\displaystyle S_{i}}$ the spin of the i^th element. This then gives for the fields:

${\displaystyle {\overrightarrow {H}}_{a}={\overrightarrow {H}}_{0}+Ng\mu _{B}S_{a}\gamma _{ab}(\lambda \alpha {\overrightarrow {\sigma }}_{a}-\mu {\overrightarrow {\sigma }}_{b})}$ ${\displaystyle {\overrightarrow {H}}_{b}={\overrightarrow {H}}_{0}+Ng\mu _{B}S_{b}\gamma _{ab}(-\lambda {\overrightarrow {\sigma }}_{a}+\mu \beta {\overrightarrow {\sigma }}_{b})}$

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

${\displaystyle \sigma _{a}=B_{S_{a}}(g\mu _{b}S_{a}H_{a}/k_{B}T)}$ ${\displaystyle \sigma _{b}=B_{S_{b}}(g\mu _{b}S_{b}H_{b}/k_{B}T)}$

Where ${\displaystyle B_{J}(x)}$ is the Brillouin function. The simplest case to solve now is ${\displaystyle S_{a}=S_{b}={\frac {1}{2}}}$. Since ${\displaystyle B_{\frac {1}{2}}(x)=tanh(x)}$. This then gives the following pair of equations:

${\displaystyle \lambda \sigma _{a}={\frac {\tau F(\lambda ,\alpha ,\beta )}{\alpha \beta -1}}(\beta tanh^{-1}\sigma _{a}+tanh^{-1}\sigma _{b})}$

${\displaystyle \mu \sigma _{b}={\frac {\tau F(\lambda ,\alpha ,\beta )}{\alpha \beta -1}}(tanh^{-1}\sigma _{a}+\alpha tanh^{-1}\sigma _{b})}$

With ${\displaystyle \tau =T/T_{c}}$ and ${\displaystyle F(\lambda ,\alpha ,\beta )={\frac {1}{2}}(\lambda \alpha +\mu \beta +{\sqrt {(\lambda \alpha -\mu \beta )^{2}+4\lambda \mu }})}$. These equations do not have a known analytical solution so these must be solved numerically to find the temperature dependence of ${\displaystyle \mu }$.

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 of which is that the direction of the magnetization can change sign while heating op from Absolute zero to the critical temperature, and that the magnetization increases while increasing the temperature to the critical temperature, which both cannot occur for ferromagnetic materials. These temperature dependencies have also been experimentally observed in NiFe_2/5Cr_8/5O₄[8] and Li_1/2Fe_5/4Ce_5/4O₄.[9]

The point where the net magnetisation is 0 for temperatures below the Curie temperature at which the opposing magnetic moments are equal, resulting in a net magnetic moment of zero is called the 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.

Theoretical model of magnetization m against magnetic field h. Starting at the origin, the upward curve is the initial magnetization curve. The downward curve after saturation, along with the lower return curve, form the main loop. The intercepts h_c and m_rs are the coercivity and saturation remanence.

When ferrimagnets are exposed to an external magnetic field, they display what is called Magnetic hysteresis , the magnetic behaviour depends on the history of the magnet. They also know a saturation magnetisation ${\displaystyle M_{rs}}$, this magnetisation 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 magnetisation cannot increase as there are no more moments to align. Once you decrease the external field back to ${\displaystyle 0}$, the magnetisation of the ferrimagnet will not go back to ${\displaystyle 0}$ but will rather stay magnetised. This is effect is often used in applications of magnets. When the external field is then put in the opposite direction the magnet will demagnetise further until it will eventually reach a magnetisation of ${\displaystyle -M_{rs}}$. This behaviour results in what is called a hysteresis loop.[10]

Ferrimagnetic materials have high resistivity and have 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. Furthermore it has been shown that ferrimagnets such as Magnetite can be used for Thermal energy storage.[11]

The oldest known magnetic material, Magnetite, is a ferrimagnetic substance. The Tetrahedral and Octahedral sites 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 PbFe₁₂O₁₉ and BaFe₁₂O₁₉ and pyrrhotite, Fe_1−xS.[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 centres with Mn(III) and Mn(II) metal centres.[13]

• Anisotropy energy
• Orbital magnetization

External links

• Media related to Ferrimagnetism at Wikimedia Commons