1. Fundamentals of Magnetism, Part II
Magnetoelastic Phenomena and Magnetization Processes
Leszek Malkinski
Department of Physics and Advanced Materials Research Institute
University of New Orleans

2. Why Magnetoelastic Properties?
You should always have on your mind that there is a coupling between magnetic and elastic properties while analyzing data or considering applications. Bending, compressing, twisting, stretching, forging, cutting, grinding or polishing materials may alter their magnetic properties. Examples:
You want to measure magnetic permeability of a tape od soft magnetic material. To decrease demagnetization factor you make a ring. Do not ignore the fact that you introduced stresses to bend it!
You polish rough surface of your sample to image domains using magnetooptical Kerr effect. Your domain structure is altered by polishing.
There are interesting applications of magnetoelastic effects for sensing strains or magnetic field, actuation and energy conversion (multiferroics).

3. Brief History
In the first “handbook” on magnetism, published in 1600, William Gilbert experimentally disproved many myths associated with magnetism. But he also described an experiment in which hitting magnets caused deterioration of their magnetization. Also, forging iron oriented North-South was found to magnetize it.
James Prescott Joule observed in 1842 that iron rods elongate while magnetized by weak fields applied along the rods. He also found that elongation is associated with reduction of their transverse dimensions. 5 years later he published additional results on elongation of iron and steel rods under tensile strains. He discovered that at larger strains and higher fields iron rods shrink. Change of dimensions (at constant volume) in the presence of the field is called linear or Joule magnetostriction.
In the meantime, in 1846 Amédée Victor Guillemin reported that bent ferromagnetic rods tend to straighten while magnetized. He ascribed this fact to a change in elasticity triggered by magnetic field.

4. In 1847 Carlo Matteucci observed changes of magnetization due tensile or torsional strains applied to magnetic materials. This phenomenon was re-discovered by Emilio Villari in 1865 and got his name (it is also called inverse Joule effect). Matteucci also showed that current can be induced by twisting iron rods. This effect was named after him although Wertheim discovered it 6 years earlier.
Gustav Heinrich Wiedemann in 1958 observed twisting of a rod with current flowing through it when the rod was placed in a magnetic field. (Matteuchi effect is the inverse Wiedeman effect)
In William Fletcher Barrett discovered negative magnetostriction of Ni and also the volume magnetostriction.
The major discoveries had been done before magnetism in solids was understood.
Other people contributing to discovery of magnetoelastic effects: Sherfold Bidwell, James Alfred Ewing, Carl Barus, Mary Chilton Noeyes, Lord Kelvin, Samuel Barrnet, Albert Einstein and Wander Johannes de Haas. Interestingly, Maria Curie also started her scientific career studying effect of mechanical impact on magnetization change in magnets.

5. Spontaneous Magnetostriction
Ferromagnet
T>TC
T<TC
Antiferromagnet or ferrimagnet
Appearance of magnetic order is accompanied by spontaneous deformation of material. Linear magnetostriction coefficient defined as relative change of dimensions Δl/l is the measure of this effect. Also the volume may change by ΔV/V
λ>0
λ<0

6. Origin of Magnetostriction
Spin
Magnetocrystalline anisotropy
Magnetostrictive strain H
Lattice H
Orbital moment
Interactions between magnetic moments in crystals.
Strong spin-spin coupling is due to exchange interactions. It is responsible for magnetic order.
Electron orbits are strongly coupled to the electric crystal fields. The orbital magnetic moments are quenched and even strong fields cannot change their orientations.
The spins are weakly coupled to the orbital motion of the electrons (spin-orbit coupling). It takes magnetocrystalline anisotropy energy to rotate the spin system with respect to fixed orientation of electron orbits
Note, that we can allow an additional degree of freedom for magnetic atoms to displace. If this helps to minimize total energy the atoms will do this!

7. Link Between Elasticity and Magnetism
All kinds of bonds which form crystals are electrostatic in nature. Magnetic energy of interacting classical magnetic moments is much weaker. gravitational forces are negligible and nuclear forces bind nuclei (very short range) Thus, the whole mechanics on atomic level can be reduced to electricity!
Is spin-spin interaction magnetic or electric in nature? The origin of molecular field which aligns spins in magnetically ordered solids was a mystery until quantum mechanics emerged. The experimental results showed that it can be as large as 1000 T (even today high magnetic field labs cannot produce such high fields). The form of exchange energy in quantum mechanical calculations depends or relative orientation of spins:
The from, which involves a product of spins resembles equations for energy of magnetic moments. But it is deceptive! Let’s make some estimates which demonstrate why exchange energy depends on spin orientation and what the origin of this energy is .
(I will follow reasoning from lectures of Prof. R. Camley from UCCS)

8. Two Magnetic Atoms Let’s consider two electrons on two atoms.
Coulomb’s energy of the system of charged particles is given by: d1 d2 r1 r2
Coulomb repulsion between two electrons
The other terms can be included which involve interaction between the electron at r1 with the nucleus d2, etc.
Pauli principle stipulates that electrons cannot occupy the same quantum state (described by the same quantum numbers and thus the same wavefunctions). Because electrons are indistinguishable the multi-electron state must be described by antisymmetric functions. Wavefunctions can be written in the form:
Wavefunction = (spin part) (space part)
If the spin part is symmetric under interchange of electrons (both spins up for example) than the space part must be antisymmetric. If the spin part is antisymmetric than the space part must be symmetric.
So, the main point is that the spin state affects the spatial distribution of the electrons and this, in turn, changes the Coulomb repulsion term above .

9. Same spin electrons are far apart Coulomb repulsion energy is lowerd
Same spin electrons are far apart
Coulomb repulsion energy is lower
Opposite spin electrons are closer
Coulomb energy is higher
Now, on average, if the two electrons have the same spin direction they spend more time apart from each other.
Numerical estimates
So, again we see the energy depends on the
spin orientation.
Look at
4.8 eV is a reasonable value of energy of the distance between electrons.
Now suppose that the spin orientations are changed so that this distances increases by about 1%. Then the energy difference is ΔE= 0.05 eV
Let's express the energy difference in "thermal units" by kT = DE
We find that T = 580 K
This is the correct order of magnitude by the temperature for a transition from an ordered state (ferromagnetic) to a disordered state (nonmagnetic).
Put the energy in effective field units
Let m be a Bohr magneton and calculate H. We get H = 8.7 x 106 Gauss=870 T
This is a huge effective field and is known as the exchange field. (A typical lab field is about 10T.) Thus, the strength of the coupling which is responsible for the ordering of magnetic materials is much stronger than any external applied field and is electrostatic in nature.

10. Magnetostrictive Strain
As we see the origin of the exchange energy is a part of Coulomb’s energy of interacting electrons who are subject to Pauli’s exclusion principle. Since the same energy is responsible for bonds the change of the exchange energy may affect elastic energy. The only problem is that the exchange energy
depends on relative orientation of the spins, but has no relation to the orientation of the spins with respect to the lattice. Therefore, it may be a source of volume magnetostriction. The isotropic exchange magnetoelastic energy is given by
, where N is a number of atoms in a unit volume and z the number of he nearest neighbors
Volume magnetostriction can be found by minimizing magnetoelastic energy and the elastic energy expressed in terms of compressibility κ
Note that above Tc there is no correlation between spins and average zeroes. This is not the case in the ordered state.

11. Anisotropic Energy
Linear magnetostriction (or Joule magnetostriction) is associated with spin-orbit coupling which is also a source of magnetocrystalline anisotropy. Following Néel approach the energy of two spins can be written as: r ɸ
The first term is associated with the volume magnetostriction and the second term is crucial for understanding the linear magnetostriction. The angular dependence can be expressed for 3-dimensional lattice through direction cosines of the magnetization (α1,α2,α3) and of the bonds (β1,β2, β3) ( or crystal lattice)
It is a tedious task (details in Chikazumi’s book) to express change of bonds in terms of strain tensor components εxx, εyy,εzz, εxy, εyz and εzx. This leads to the expression for magnetoeleastic energy, which for cubic crystals (simple, bcc and fcc) has the form:

12. where the coefficients B1 and B2 depend on the crystal cell type:
for a simple cubic cell
for bcc cell
for fcc cell
The magnetoelastic energy must be counterbalanced by elastic energy (otherwise the crystal would be unstable), which for cubic crystals is:
where C11 C12 and C44 are elastic moduli.

13. Linear Magnetostriction
Minimization of Emagel+Eel with respect to all 6 strains gives relations between the equilibrium strains corresponding to certain orientation of magnetization (given by α1,α2, and α3). This allows formulating general formula for elongation in observed direction given by direction cosines (β1,β2,β3):
Here, two magnetostrictrive coefficients are defined through B1 and B2
They describe elongations in [100] and [111] directions respectively. Finally, for isotropic medium we need only one magnetostriction constant
For polycrystalls, averaging of the strain over different crystal orientations gives effective magnetostriction constant.
Hexagonal crystals may have up to 4 independent magnetostriction coefficients!

14. Quantum Mechanical Picture
Alternative, atomistic approach gives better insight into origin of spontaneous magnetostriction. In the paramagnetic state, electron orbits of ferromagnetic rare–earth compound display 4-fold symmetry. However, below Curie temperature the magnetic symmetry changes to tetragonal, when one of the fourfold axes becomes the easy magnetization direction. The coupling of ellipsoidal orbits promotes expansion in [001] direction and contraction in transverse direction. In this approach it is difficult to give universal formula for magnetostriction.
In highly magnetostrictive materials the anisotropy energy associated with spontaneous magnetostricdtion can be up to 20% of the magnetocrystalline anisotropy energy value.
(Buschow and de Boer)

15. Some Data How large is the magnetostrictive strain?
All magnetic materials respond to applied field by changing dimensions, shape or volume. Some materials (specially designed) have magnetostriciton coefficient as small as 10-9 (materials for shielding) By convention materials can be classified
Nonmagnetostrictive (10-6) Magnetostrictive (10-4) Giant Magnetostrictive
Some examples of magnetostriction of polycrystals at room temperature
Material
Magnetostriction Fe
-7x10-6 Co
62x10-6 Ni
-34x10-6
Tb0.3Dy0.7Fe2 (terfenol-D)
1200x10-6
CoFe2O4
-110x10-6
GaFe (Galfenol)
240x10-6
Amorphous FeCoNiB
<1x10-6
Shape memory alloys may have up to 7% change of dimensions in a magnetic field

16. Real Samples-Domain Theory
Positive magnetostriction
Because the definition of the magnetostrictive strain
it will be the same for domains with magnetization which differ by 180o. Displacements of 180o do not produce any strain.
Therefore only rotations and movements of non-1800domain walls contribute to the magnetostrictive strain. λS H

17. Domain Theory - Negative Magnetostrictionλ

18. Inverse Effect (Magnetoelastic)
How magnetic materials respond to stresses?
The same form of magnetoelastic energy is involved. However we can use relation between elastic strains and stresses to express it in terms of applied stress. For cubic crystals it depends on the magnitude of applied stress, direction cosines of the stress (ϒ1,ϒ2,ϒ3) and of the magnetization direction(α1,α2,α3):
Example:
For isotropic material M
Here ɸ is the angle between the magnetization and the stress axis σ ɸ σ

19. Stress Induced Anisotropy
The magnetoelastic energy expressed in terms of the magnitude of applied stress σ, and the angle ɸ between the stress σ and the magnetization M is for uniaxial stress equivalent of magnetic anisotropy.
For the materials with positive anisotropy, the magnetization tends to align along the direction of tensile stress (direction of stress is an easy axis), but for materials with negative magnetostriciton coefficient the direction of stress is the hard axis. You must be careful using this formula. Note that this anisotropy “constant” changes sign when you apply compressive stresses.
Also, note that the stress anisotropy can affect the domain wall width and domain wall energy
as a part of effective anisotropy Keff=Kmagcrys+Kdemag+Kσ
Therefore, it can also change the pinning of domain wall and consequently corecivity.

20. Stress-Activated Domain Processes
Again, 180o domain walls will not move (but the stresses modify magnetic anisotropy and domain wall energy of these domains if motion is caused by magnetic field). For the domain configuration shown here 90o domain walls move and remaining 180o wall cannot be removed by increasing stress.
You cannot reverse magnetization using uniaxial stress.
It takes field action to magnetize this sample. However, if the material is partially magnetized by the field, the stresses can magnetize it to saturation by moving domain walls.
λS>0 σ σ H H H

21. Extra Strain
When stresses act on the ferromagnetic materials, magnetostrictive strain occurs in addition to elastic strain. This modifies elastic properties!

22. Elastic Constants
Young’s modulus E (or Y) is defined as the ratio of applied stress to the strain of the material in the direction of the stress.
Because of the magnetostrictive strain magnetics do not obey the Hook’s law. Their elastic properties depend on stress and the applied field. Change of Young’s modulus with field or stress level are called ΔE-effect. It can be significant! (over 50%). Magnetostrictive strains are also used to compensate strains due to thermal expansion. This allows designing materials with zero expansion coefficient (INVAR) or constant Young’s modulus (ELINVAR) in certain temperature range.
Strain
Stress

23. Magnetomechanical Damping
Magnetostrictive properties have pronounced effect of damping of vibrations of magnetic materials.
One reason for extra losses of elastic energy is magneto-mechanical hysteresis (When stress is increased and then reduced the strain does not follow the same path)
Another reason is that even for
demagnetized material the domain walls displace due to variable stresses. These displacements and related magnetization changes cause micro-eddy currents, which are converted to a heat.

24. How We Measure Magnetostriction?
Several methods exist to measure magnetoelastic effects, but there are no simgle universal method. The choice of particular method depends on the size and shape of your sample. The methods can be divided into direct (direct measurement of magnetostrictive strain) or indirect (change of magnetization due to stresses). In direct methods elongation can be measured by strain gauge, tunneling current, change of capacitance of a capacitor with movable electrode, optical measurements of deflection of substrate with thin magnetic film deposited, etc.
Dilatometer
Capacitor
Strain Gauge
Thin Film

25. Indirect Methods
Delay line. Magnetoelastic wave is generated, propagates through a material and is detected. The velocity and consequently the time of propagation of the sound wave as well as attenuation depend on elastic magnetostrictive properties.
Also, strain modulated ferromagnetic resonance method is an accurate method to measure stress induced anisotropy and a shift of resonant microwave absorption.
Small Angle Magnetization Rotation
-change of AC permeability due to stress

26. Applications
Ultrasonic transducers (in sonars, drillers, ultrasonic baths)
Magtnetostrictive motors and actuators (High load and high torque linear motors, high frequency range)
Electromechanical filters and microwave filters
Nondestructure Testing –(Magnetoacoustic emission.)
Sensors
Contactless impact power meters
Contactless torque meters
Force sensors
Pressure sensors (hydrophones)
Magnetic field sensors (multiferroic composites)

27. Magnetoelectric Effect
“Marriage” of Piezomagnetism and Piezoelectricity
For small fields and small strains the mutual coupling between magnetic and elastic properties is described by linear equations. In this regime, we are talking about piezomagnetism. In piezoelectric/piezomagnetic composites magnetostrictive strains can be transferred to piezoelectric material and deform it. This deformation changes polarization of the piezoelectric which can be measured as voltage. This provides a mechanism for conversion of magnetic field to electric signal to build highly sensitive magnetic field sensors .
The magnetoelectric coefficients α describe the ability of the magnetic field to change components of polarization (or electric displacement vector)
Piezoelectric shell
Magnetostrictive core
---
+++
Magnetic
Field
Dielectric properties
Piezoelectric properties
Magneto-electric
Magnetomechanical coupling factor k determines how much magnetic energy can be converted into elastic. The best results are about 98%
Magnetoelectric Effect

28. Small Particles and Thin Films
Magnetocrystalline anisotropy is related to the symmetry of crystal fields and a coupling between orbital and spin moments. The number of nearest neighbors and the symmetry conditions are quite different for the two atoms at two different locations. Dramatic change of symmetry at the surface -“broken symmetry”
The atom in the bulk has 8 nearest neighbors which form 4 fold symmetry. The surface atom has only five neighbors on one side. It is much lower symmetry configuration. In the case of interface between two different materials there will be atoms of different kind nest to it but there will be still low symmetry configuration. Surface and interface electronic states differ from those in the bulk which results in different magnetic moments of surface atoms.

29. Surface Anisotropy
Nèel in 1954 pointed out that the reduced symmetry at the surface of a cubic crystal changes the form of the anisotropy which can be written in the form:
where K-s are constants α-s are directional cosines. This anisotropy is a uniaxial anisotropy type and aligns magnetization perpendicular to the surface for Ks1<0 and favors magnetization in plane of the surface in the opposite case. If you grow Fe film its surface anisotropy dominates. Only for thicker films magnetization aligns in the film plane.

30. Multilayers
As we remember magnetocrystalline anisotropy and magnetoastriction are related. A similar effect of broken symmetry and modification of electronic states was found to cause a surface magnetostriction. If you consider just bulk coefficient of magnetostriction you will not be able to fit experimental data of magnetostriction of multialyers or granular nanocomposites without including a surface ansotropy or surface magnetostriction terms. Usually surface effects decrease with thickness d of the film as 1/d. The surface anisotropy can sometimes achieve large values so it is called giant surface anisotropy.
We should also member that change of the arrangement of atoms at the surface modifies the electronic states in the outer atomic layer. We take advantage of the interface anisotropy by increasing number of interfaces in multilayers. For this reason many multilayers (Fe/Ag, Ni/Cu, or Co/Pd) show perpendicular anisotropy. The magnetization is normal to the interface between materials instead of in-plane of the film, as expected from the shape anisotropy. (This is and important effect for perpendicular recoding media)

31. Fine Grains and Anisotropy
Polycrystalline magnetics consisting of large, high quality grains have low coercivity, because there are only a few defects, which can oppose domain wall movements. The coercivity increases with increasing number of grains because grain boundaries and other defects are source of wall pinning. However, below the size of 100 nm a dramatic suppression of coercivity was observed. This fact would be easy to explain in the system of isolated superparamagnetic nanoparticles, because below certain critical size at given temperature magnetization of such
nanoparticles becomes unstable, starts chaotically fluctuating due to thermal excitations and the coercivity vanishes.
However, this explanation is not acceptable for a solid material where exchange coupled grains are in contact and domain walls can cross grain boundaries.

32. Random Anisotropy Model
Think, what happens when the grain size becomes smaller than the domain wall width! Will the domain wall motion be affected by local anisotropies of randomly oriented grains? The answer is NO, the domain will pretty much ignore them!
It was explained by Herzer that when the size of particles decreases below certain value called exchange length the anisotropy of grains averages out. The exchange length represents minimum scale below which the direction of the magnetization cannot vary appreciably.
The random anisotropy model gave correct description of effective anisotropy of nanocrystalline alloys:
Here, D is the nanoparticle size, K1 anisotropy constant of individual grains and vcr the vlomue fraction of the particles. According to this model amoprhous magnets should have no magnetocrystalline anisotropy!

33. Barkhausen Noise
When we look carefully at magnetization loop we may notice that it is not a
smooth curve. The magnetization increases in little steps. It is commonly considered that each jump of magnetization is associated with domain wall motion (or irreversible rotation) which is freed from pinning points and displaces rapidly till trapped by other anchors. The signal detected by a pick-up coil (derivative of magnetic flux) looks like noise. This signal produces an acoustic noise in a loudspeaker.
The Barkhausen noise is not a white noise but deterministic noise described by stochastic processes. You can also ascribe fractal dimension to this signal.

34. Both the probability of distribution of magnitudes of Barkhausen jumps and the frequency characteristics of Barkhausen noise power depend on magnetization rate. At low magnetization rates bursts of Barhausen jumps can be observed but at higher rates the avalanches of jumps cluster.
Barkhausen noise was the first indirect proof of existence of domain structure. It is a useful tool to study irreversible magnetization processes. They contain important information about defects in magnetic materials. Therefore Barkhausen noise is one of appreciable methods of nondestructive materials testing.

35. Mechanical and Acoustic Barkhausen Effect
Moving magnetic domain walls in a magnetized sample interact with defects and generate magnetoelastic waves with acoustic frequency. They can be detected at the surface of the sample by a sensitive microphone or a piezoelectric transducer. This method is called an acoustic Barkhausen effect or magnetoacoustic emission.
On the other hand, as we already discussed, alternating stresses trigger motion of non-180o domain walls. Abrupt changes of magnetization associated with domain wall displacement can be detected by a pick-up coil. This method is called Mechanical Barkhausen effect to distinguish from the Barkhausen effect which uses magnetic field to generate jumps of magnetic flux through a coil.
These two methods provide additional information which allows distinguishing interaction of domain walls with internal stresses and other type of defects (voids or inclusions).
Again, they are useful for nondestructive testing of materials. For example you can evaluate the magnitude of stresses produced by welding of steel elements.
Example of the mechanical Barkhausen
measurements.

36. References
Soshin Chikazumi, Physics of Ferromagnetism, Oxford Science Publications
B.D. Cullity, C. D. Graham, Introduction to Magnetic Materials. D edition John Wiley & Sons
David Jiles, Introduction to Magnetism and Magnetic Materials, 2nd edition, CRC Taylor & Francis
Giorgio Bertotti, Hysteresis in Magnetism, Academic Press
Robert C. O’Handley, Modern Magnetic Materials-Principles and Applications, John Wiley & Sons
K.H.J. Buschow and F.R De Boer, Physics of magnetism and Magnetic Materials
Partick T. Squire, Magnetopmechanical measurments of magnetically soft amorphous materials. Measurements Science and Technology, 5 (1994)
L. Malkinski, Unpublished materials