Ferromagnetic materials

1.1 Ferromagnets

In this material, there are domains in which the magnetic fields of the individual atoms align, but the orientation of the magnetic fields of the domains is random, giving rise to no net magnetic field. This is illustrated below.

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A useful property of ferromagnets is that when an external magnetic field is applied to them, the magnetic fields of the individual domains tend to line up in the direction of this external field, due to the nature of the magnetic forces, which causes the external magnetic field to be enhanced. This is illustrated below.

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Historically, the term ferromagnet was used for any material that could exhibit spontaneous magnetization: a net magnetic moment in the absence of an external magnetic field. This general definition is still in common use. More recently, however, different classes of spontaneous magnetization have been identified when there is more than one magnetic ion per primitive cell of the material, leading to a stricter definition of "ferromagnetism" that is often used to distinguish it from ferrimagnetism. In particular, a material is "ferromagnetic" in this narrower sense only if all of its magnetic ions add a positive contribution to the net magnetization. If some of the magnetic ions subtract from the net magnetization (if they are partially anti-aligned), then the material is "ferrimagnetic". If the ions anti-align completely so as to have zero net magnetization, despite the magnetic ordering, then it is an antiferromagnet. All of these alignment effects only occur at temperatures below a certain critical temperature, called the Curie temperature (for ferromagnets and ferrimagnets) or the Néel temperature (for antiferromagnets).

1.2 Origin of magnetization

The spin of an electron, combined with its orbital angular momentum, results in a magnetic dipole moment and creates a magnetic field. In many materials (specifically, those with a filled electron shell), however, the total dipole moment of all the electrons is zero (i.e., the spins are in up/down pairs). Only atoms with partially filled shells can experience a net magnetic moment in the absence of an external field. Ferromagnetic materials contain many atoms with unpaired spins. When these tiny magnetic dipoles are aligned in the same direction, they create a measurable macroscopic field.

These permanent dipoles (often called simply "spins" even though they also generally include orbital angular momentum) tend to align in parallel to an external magnetic field, an effect called paramagnetism. (A related but much weaker effect is diamagnetism, due to the orbital motion induced by an external field, resulting in a dipole moment opposite to the applied field.) Ferromagnetism involves an additional phenomenon, however, the dipoles tend to align spontaneously, without any applied field. This is a purely quantum-mechanical effect.

In a ferromagnet, they tend to align in the same direction because of the Pauli principle: two electrons with the same spin cannot also have the same "position", which effectively reduces the energy of their electrostatic interaction compared to electrons with opposite spin. Mathematically, this is expressed more precisely in terms of the spin-statistics theorem: because electrons are fermions with half-integer spin, their wave functions are antisymmetric under interchange of particle positions. This can be seen in, for example, the Hartree-Fock approximation to lead to a reduction in the electrostatic potential energy. This difference in energy is called the exchange energy.

The exchange interaction is primarily responsible for the ordering of atomic moments occurring in magnetic solids. The aforementioned interaction described by classical electromagnetism usually plays only a marginal role. For instance, in iron (Fe) the exchange interaction between two atoms is about 1000 times stronger than that classical interaction. There is a small number "exotic" ferromagnets in which the exchange interactions are exceptionally weak, and then the classical dipole-dipole interactions may become the dominant ones. However, such system become ferromagnetic only at very low temperatures, usually below 1 K. But if the Curie temperature in a given material is higher than a few Kelvins, then its ferromagnetism is surely produced by exchange interactions. In such systems the classical dipole-dipole interactions may only give rise to secondary effects, e.g., to weak magnetic anisotropy.

1.3 Curie temperature

As we mentioned above, in section 1.2, the very significant dipole alignment in ferromagnetic materials is a magnetic ordering due to quantum effects. Spin ordering is disrupted by thermal energy:

(1)
\begin{equation} E_{thermal}=k_{B}T \end{equation}

where, $k_{B}$ is Boltzmann constant, T is temperature in kelvin.

As the temperature increases, thermal motion, or entropy, competes with the ferromagnetic tendency for dipoles to align. When the temperature rises beyond a certain point, called the Curie temperature Tc, there is a second-order phase transition and the system can no longer maintain a spontaneous magnetization, although it still responds paramagnetically to an external field, i.e. above Tc the spontaneous magnetization due to ferromagnetic ordering is lost, ferromagnetic ordering is destroyed and the material behaves paramagnetically. Below that temperature, there is a spontaneous symmetry breaking and random domains form (in the absence of an external field). The Curie temperature itself is a critical point, where the magnetic susceptibility is theoretically infinite and, although there is no net magnetization, domain-like spin correlations fluctuate at all length scales.

Curie temperatures of some ferromagnetic materials are as follows:

Fe: Tc=1043K
Ni: Tc=627K
Co: Tc=1388K
$FeOFe_{2}O_{3}$: Tc=858K
$NiOFe_{2}O_{3}$: Tc=858K
$CuOFe_{2}O_{3}$: Tc=728K
$MgOFe_{2}O_{3}$: Tc=713K
MnBi: Tc=630K
MnSb: Tc=587K
$MnOFe_{2}O_{3}$: Tc=573K
$Y_{3}Fe_{5}O_{12}$: Tc=560K
$CrO_{2}$: Tc=386K
MnAs: Tc=318K
Gd: Tc=292K

1.4 Domain structure

In ferromagnetic materials, small regions with a particular overall spin orientation are termed domains. This strong (large) spin alignment leads to huge permeabilities:

Material Relative Permeability $\mu_{r}$
Nickel 250
Cobalt 600
Iron (pure) 4,000
compare to paramagnetic metal:
Aluminium ≈ 1

At long distances (after many thousands of ions), the exchange energy advantage is overtaken by the classical tendency of dipoles to anti-align. This is why, in an equilibriated (non-magnetized) ferromagnetic material, the dipoles in the whole material are not aligned. Rather, they organize into magnetic domains (also known as Weiss domains) that are aligned (magnetized) at short range, but at long range adjacent domains are anti-aligned.[3] The boundary between two domains, where the magnetization flips, is called a domain wall (i.e., a Bloch/Néel wall, depending upon whether the magnetization rotates parallel/perpendicular to the domain interface) and is a gradual transition on the atomic scale (covering a distance of about 300 ions for iron).[2]

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1.5 Hysteresis

When a ferromagnetic material is magnetized in one direction, it will not relax back to zero magnetization when the imposed magnetizing field is removed. It must be driven back to zero by a field in the opposite direction. If an alternating magnetic field is applied to the material, its magnetization will trace out a loop called a hysteresis loop. The lack of retraceability of the magnetization curve is the property called hysteresis and it is related to the existence of magnetic domains in the material. Once the magnetic domains are reoriented, it takes some energy to turn them back again. This property of ferrromagnetic materials is useful as a magnetic "memory". Some compositions of ferromagnetic materials will retain an imposed magnetization indefinitely and are useful as "permanent magnets". The magnetic memory aspects of iron and chromium oxides make them useful in audio tape recording and for the magnetic storage of data on computer disks.

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The magnetic field within an unmagnetized piece of steel is zero. As the magnetizing force (H) is increased from zero, the flux density (B) within the part will also increase from zero. The curve from points A to E in Figure above illustrates this behavior. In the region of point E, the flux density increases up to a point and then tends to level off; this condition is called magnetic saturation and for a magnetically saturated ferromagnetic material, the relative permeability (m) is approximately equal to 1. When the magnetizing force is reduced to zero the flux density does not return to zero. Instead, the flux density returns to a value shown at point F in Figure above. This is the amount of residual magnetism resulting from the applied magnetizing force (H) that reached the point E in the hysteresis curve. As the magnetizing force (H) is increased from zero in the opposite direction, the flux density (B) will decrease to zero, as shown at point G in Figure above, and then start to increase to point I. The magnetizing force (H) represented by the distance OG on the H axis in Figure above is called the coercive force. It represents the strength of the magnetizing force (H) required to reduce the flux density (B) to zero in a saturated ferromagnetic material. A further increase in the magnetizing force (H) to the point I results in saturation of the material in a direction opposite to that represented by point E. Reduction of the magnetizing force (H) to zero from point I will reduce the flux density (B) to the value represented by point J. Application of a magnetizing force (H) in the original direction will change the flux density (B) as shown in the portion JK of the hysteresis curve. Increasing the magnetizing force (H) sufficiently will return the material to saturation as illustrated at point E.

1.6 Application

An area where ferromagnetic materials are employed is in magnetic recording devices, such as for cassette tapes, floppy discs for computers, and the magnetic stripe on the back of credit cards. These devices essentially take information in the form of electrical signals and permanently encode it into a magnetic material. The way this is done is illustrated below.

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As an (AC) electrical signal passes through the wire loop, a magnetic field is produced which passes through the ferromagnetic core, which in turn produces a magnetic field in the vicinity of a moving magnetic tape. This magnetic field aligns the magnets of atoms on the tape that happen to be passing by it at that instant. A short time later the direction of the current reverses, which reverses the direction of the magnetic field, which subsequently reverses the orientation of the next atom on the tape which passes by. In this way information stored in the electrical signal is encoded in a particular orientation of the magnetic fields of individual atoms.[4]

There are also many applications, like Magnetic Response Of Ferromagnetic Semiconductor[3] etc.

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Under certain conditions, two-dimensional electron systems in strong magnetic field behave as “quantum Hall ferromagnets”. These unusual ferromagnets have the novel property that the local magnetic vorticity (or “Skyrmion” density) is directly related to the physical charge density. In view of this vorticity/charge equivalence, electrical potentials couple directly to the magnetic vorticity.

Within the present context, a “Skyrmion” is a configuration of a two-dimensional ferromagnet which has a non-trivial global topological structure (this Skyrmion configuration cannot be deformed into the magnetic groundstate, in which all spins are aligned, by a smooth or continuous process). The importance of Skyrmions for quantum Hall systems arises from the discovery (based on both theoretical and experimental evidence) that they describe the lowest-energy charged excitations of quantum Hall ferromagnets, and therefore dominate many of the low-temperature properties of these systems.[5]

1.7 Coupling of Ferromagnetics and Ferroelectrics—Multiferroics

The term multiferroic was first used by H. Schmid in 1994. His definition referred to multiferroics as single phase materials which simultaneously possess two or more primary ferroic properties. [7]

Multiferroic materials simultaneously show ferromagnetism and ferroelectricity. They hold great potential for applications as the multiferroic coupling allows switching of the ferroelectric state with a magnetic field and vice versa. They could lead to a new generation of memory devices that can be electrically written and magnetically read.

The multiferroics offer the possibility of combining the best qualities of ferroelectric random access memories (FeRAMs) and magnetic random access memories (MRAMs): fast low-power electrical write operation, and non-destructive magnetic read operation. At the 256 Mbit level, such memory devices, would be a “disruptive technology” [15]and could eliminate competition such as EEPROMs (electrically erasable programmable read-only memories) for applications including megapixel photo memories for digital cameras or audio memories in devices such as mp3 players.[16]

Typical multiferroics belong to the group of the perovskite transition metal oxides, and include rare-earth manganites and -ferrites (e.g. TbMnO3, HoMn2O5, LuFe2O4). Other examples are the bismuth alloys BiFeO3 and BiMnO3, and non-oxides such as BaNiF4 and spinel chalcogenides, e.g. ZnCr2Se4. These alloys show rich phase diagrams combining different ferroic orders in separate phases. Apart from single phase multiferroics, composites and heterostructures exhibiting more than one ferroic order parameter are studied extensively. Some examples include magnetic thin films on piezoelectric PMN-PT substrates and Metglass/PVDF/Metglass trilayer structures. Besides scientific interest in their physical properties, multiferroics have potential for applications as actuators, switches, magnetic field sensors or new types of electronic memory devices.

For thin-film multiferroics, there are three types of thin-film architectures: single-phase thin films, in which heteroepitaxial strain and crystal chemistry are the key variables in controlling and improving magnetoelectric coupling; horizontal heterostructures, in which the principles of heteroepitaxy at the interfaces can be used to control and initiate magnetoelectric coupling at the atomic scale; and nanoscale ‘vertical heterostructures’, in which coupling occurs through vertical heteroepitaxy.[8]

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(a) Single-component thin-film multiferroics

Perovskite-structure bismuth ferrite is currently the most studied single-component multiferroic, in part because its large
polarization and high Curie temperature (~820 °C) make it appealing for applications in ferroelectric non-volatile memories and hightemperature electronics. Measured polarizations for thin films grown by a variety of techniques[9-12] initially showed a large spread of values. However, the measured values are now converging to ~90 μC cm–2 along the [111] direction of the pseudo-cubic perovskite unit cell, consistent with first-principles calculations[14]. The large difference between the thin-film and bulk values, initially attributed to epitaxial strain, could also result from leakage effects in crystals caused by defect chemistry or the existence of second phases, or mechanical constraints in granular bulk ceramics.

(b) Horizontal multilayer heterostructures

Several interesting interfacial magnetoelectric effects have been proposed theoretically and are particularly exciting areas for future experimental study. First-principles density functional theory is used to demonstrate a magnetoelectric effect in a ferromagnetic/ferroelectric heterostructure that arises from a purely electronic mechanism, not mediated by strain. It is shown that the displacement of atoms at the interface due to the ferroelectric instability changes the overlap between atomic orbitals at the interface, which in turn influences the magnetization of the magnetic layer resulting in a magnetoelectric effect. In addition, first-principles calculations of metal–insulator heterostructures in the presence of a finite electric field show that a polarized dielectric induces an electric polarization in the adjacent metal; if the metal is ferromagnetic, a region of coexisting electrical polarization and magnetization will be produced at the interface.

(c) Vertical heterostructures

Vertical heterostructures have a larger interfacial surface area and are intrinsically heteroepitaxial in three dimensions; this allow for stronger coupling between ferroelectric and magnetic components. In addition, the 2D clamping of the thin film on the substrate, suppress both the piezoelectric response and the magnetoelectric coupling. Recent work has demonstrated very strong magnetoelectric coupling in such nanostructures, through switching of the magnetization on reversal of the ferroelectric polarization[13].

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The figure above is magnetic force microscopy image of thin-film multiferroic nanostructures. The coupling of magnetic and ferroelectric materials make them have many potential applications, such as filters and oscillators, and electrically tuneable microwave. Coupling in the case of magnetic field sensorsis results from the macroscopic mechanical coupling of the two materials at the bonded interface. The crystallographic orientation control play an important role in optimizing the magnetoelectric response.

1.8 References
[1] Richard M. Bozorth, Ferromagnetism, first published 1951, reprinted 1993 by IEEE Press, New York as a "Classic Reissue." ISBN 0-7803-1032-2.
[2] http://en.wikipedia.org/wiki/Ferromagnetism
[3] ScienceDaily (Mar. 11, 2009)
[4] http://theory.uwinnipeg.ca
[5] Information from Theory of Condensed Matter group, University of Cambridge.
[6] http://www.phys.unsw.edu.au
[7] http://en.wikipedia.org/wiki/Multiferroics
[8] R. Ramesh and Nicola A. Spaldin, Nature Materials 6, 21 (2007)
[9] Wang, J. et al. Epitaxial BiFeO3 multiferroic thin fi lm heterostructures. Science 299, 1719 (2003).
[10] Li, J. et al. Dramatically enhanced polarization in (001), (101), and (111) BiFeO3 thin fi lms due to
epitaxial-induced transitions. Appl. Phys. Lett. 84, 5261–5263 (2004).
[11] Qi, X. et al. Epitaxial growth of BiFeO3 thin fi lms by LPE and sol-gel methods. J. Magn. Magn. Mater. 283, 415–421 (2004).
[12] Qi, X., Dho, J., Tomov, R., Blamire, M. G. & MacManus-Dricoll, J. L. Greatly reduced leakage current and conduction mechanism in aliovalent-ion-doped BiFeO3. Appl. Phys. Lett. 86, 062903 (2005).
[13] Zavaliche, F. et al. Electric fi eld-induced magnetization switching in epitaxial columnar nanostructures. Nano Lett. 5, 1793–1796 (2005).
[14] Neaton, J. B., Ederer, C., Waghmare, U. V., Spaldin, N. A. & Rabe, K. M. First-principles study of spontaneous polarization in multiferroic BiFeO3. Phys. Rev. B 71, 014113 (2005).
[15] Christensen, C. M. Th e Innovator’s Dilemma (Harvard Business School Press, Boston, 1997).
[16] J. F. Scott, Nature Materials 6, 256 (2007)

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