1. Ferromagnetism

2. Ferromagnetism
The atomic moments in these materials exhibit very strong interactions, resulting in a parallel or antiparallel alignment of atomic moments.
Exchange forces are very large, equivalent to a field on the order of 1000 Tesla, or approximately a 100 million times the strength of the earth's field.
The exchange force is a quantum mechanical phenomenon due to the relative orientation of the spins of two electron.

3. Ferromagnetism
Ferromagnetic materials exhibit parallel alignment of moments resulting in large net magnetization even in the absence of a magnetic field.
The elements Fe, Ni, and Co and many of their alloys are typical ferromagnetic materials.
Two distinct characteristics of ferromagnetic materials are their (1) spontaneous magnetization and the existence of (2) magnetic ordering temperature

4. Ferromagnetism

5. Origin of Ferromagnetism
Hund’s rule
Fe: [Ar]3d64s2

6. Ferromagnetism Ferromagnetic transition metals: Fe, Co, Ni
i) Magnitude of Ms
ii) Way to reach Ms
M (ferro) >> M (para) : 1700 emu/cm3 for Fe >> 10-3 emu/cm3
Hs = 50 Oe

7. Weiss' Assumption
Molecular field is acting in FM not only above Tc but also below Tc and this field is so strong that it could magnetize the substance to saturation even in the absence of an applied field. → spontaneously magnetized (Self-saturating)
Magnetic domain : In demagnetized state, a ferromagnetic material is divided into a number of small regions called domains, each of which is spontaneously magnetized.

8. Magnetization process
a) Unmagnetized specimen for random orientation of domains
b) – c) Single domain process (motion of domain wall)
d) Rotation of the domain along the field
Question: Spontaneous magnetization?
Division into domains?

9. Magnetic Domain
(a) A single-domain sample with a large stray field. (b) A sample split into two domains in order to reduce the magnetostatic energy. (c) A sample divided into four domains. The closure domains at the ends of the sample make the magnetostatic energy zero.

10. Magnetic Domain

12. Domain wall motion
Barkhausen effect

13. Magnetic Order
Are ferromagnets already in an ordered state before a magnetic field is applied or is the order by the field?

14. Explanation of magnetic order in ferromagnets
Weber (1852): The material could already have small atomic magnetic moments within the solid which are randomly aligned in the demagnetized but which became ordered under the action of a magnetic field.
Poisson (1983) : The atomic magnetic moments may not exist at all in the demagnetized state but could be induced by a mangetic field.

15. Explanation of magnetic order in ferromagnets
Ampère (1827): The origin of the atomic moments was suggested that they were due to electrical currents continually circulating within the atom.
Ewing (1893): Followed Weber’s idea and interested in explaining hysteresis.

16. Weiss domain theory
Atomic magnetic moments were in permanent existence (Weber’s hypothesis)
Atomic magnetic moments were ordered even in the demagnetized state. It was the domains only which were randomly aligned in the demagnetized state.
The magnetization process consisting of reorienting the domains so that more domains were aligned with field.

17. Magnetic Domain
In order to minimize its magnetostatic energy, the magnetic material divides up into magnetic domains.
Weiss (1907): concept of magnetic domains. A magnetic material consisted of a number of distinct regions termed ‘domains’ each of which was saturated in a different direction.
The concept of domains is able to explain why ferromagnetic materials can be demagnetized even below their Curie temperature.

18. Weiss Mean Field Theory
What is the origin of the alignment of the atomic magnetic moments?
It is the Weiss mean field (later the “molecular field”, further later exchange coupling from quantum mechanics)

19. Curie-Weiss Law
Curie's law: Individual carrier of magnetic moment (atoms or
molecules) do not interact with one another
Curie-Weiss law:
 Under the consideration of interaction between electrons
 Fictitious internal field Hm (“molecular field”) for interaction
: molecular field constant

20. Molecular field theory
Pierre Weiss introduced molecular field concept.
Interaction between magnetic moments  Fictitious internal filed
: molecular field constant
For  > 0, Hm || M

21. Curie Temperature

22. Curie Temperature
Even though electronic exchange forces in ferromagnets are very large, thermal energy eventually overcomes the exchange and produces a randomizing effect.
This occurs at a particular temperature called the Curie temperature (TC).
Below the Curie temperature, the ferromagnet is ordered and above it, disordered.
The saturation magnetization goes to zero at the Curie temperature.

23. Curie temperature Saturation magnetization of Fe, Co, Ni
as a function of temperature

24. Exchange Energy
Exchange force depends on relative orientation of spins of two electrons due to Pauli's exclusion principle
When two atoms, such as hydrogen atoms, are coming together, there are electrostatic attractive (e-↔p+) and repulsive (e-↔e-, p+↔p+) forces and exchange force.
The internal field is produced by interactions between nearest-neighbor dipole moments.
The interaction arises from the electrostatic electron-electron interaction, and is called the ”exchange interaction” or exchange force.

25. Exchange Energy: Heisenberg Model
Si·Sj: spin angular momentum
Je : a numerical quantity called exchange integral
Relative orientation of two spins determines the energy states.
ra/r3d
Bethe-Slater curve
(1) If Jex is positive, Eex is a minimum when the spins are parallel, leading to ferromagnetism
(2) If Jex is negative, Eex is a minimum when the spins are antiparallel, leading to antiferromagnetism.

26. Band Theory of Ferromagnetism
A simple extension of the band theory of paramagnetism by the introduction of an exchange coupling between the electrons.
Source of magnetic moments: unpaired electrons
In partially filled energy band, an imbalance of spins leads to a net magnetic moment per atom.

27. Band Theory
When N atoms come together to form a solid, each level of the free atom must split into N levels.
In transition metal elements, the outermost electrons are the 3d and 4s; these electron clouds are the first to overlap as the atoms are brought together, and the corresponding levels are the first to split.

28. Density of states

30. Anti-Ferromagnetism

31. Anti-ferromagnetism
If the A and B sublattice moments are exactly equal but opposite, the net moment is zero. This type of magnetic ordering is called antiferromagnetism.
The clue to antiferromagnetism is the behavior of susceptibility above a critical temperature, called the Néel temperature (TN).
Above TN, the susceptibility obeys the Curie-Weiss law for paramagnets but with a negative intercept indicating negative exchange interactions.

32. Wess Model on Anti-ferromagnetism
Two identical sublattices A and B: While the interaction with the moments on other sublattices with a negative coupling coefficient, interaction with the moments on their own sublattice with a positive coupling coefficient
On the basis of nearest-neighbor interactions, with a negative interaction between nearest neighbors, this leads to simple antiferromagnetism

33. Anti-ferromagnetism Electrical Insulator (no free electron)
Molecular field theory
BCC crystal (Cr)

34. Anti-ferromagnetism TN : Néel temperature T < TN : AF state
T > TN : paramagnetic

35. Anti-ferromagnetism

36. Ferrimagnetism

37. Ferrimagnetism
In ferrimagnets, the magnetic moments of the A and B sublattices are not equal and result in a net magnetic moment.
Ferrimagnetism is therefore similar to ferromagnetism. It exhibits all the hallmarks of ferromagnetic behavior- spontaneous magnetization, Curie temperatures, hysteresis, and remanence. However, ferro- and ferrimagnets have very different magnetic ordering.

38. Ferrimagnetism Two groups of ferrites depending on their structure
Cubic :
 General formula : MOFe2O3 where M is a divalent metal ion (Mn, Ni, Fe, Co, Mg, ...)
 CoOFe2O3 is magnetically hard, but all the other cubic ferrites are magnetically soft.
 magnetite : Fe3O4 = FeOFe2O3 : oldest ferrite (lodestone, iron ferrite)
2. Hexagonal :
 Barium ferrite (BaO6Fe2O3) is magnetically hard

39. Cubic ferrites (Spinel structure)
MO·Fe2O3: M = Mn, Ni, Fe, Co, Mg, etc.
 In the unit cell, total 56 ions (8 M2+ ions, 16 Fe3+ ions, 32 O2- ions)
        64 tetrahedral A site / 8 = 8
        32 octahedral B site / 2 = 16
 Normal Spinel : 8 M2+ in A, 16 Fe3+ in B              
 Inverse Spinel : 8 Fe3+ in A, 8 M Fe3+ in B
 Intermediate structure : Nor perfectly normal or inverse structure
        MnO · Fe2O3 (80% on A, 20% on B)
        MgO · Fe2O3 (10% on A, 90% on B)
 Most commercial ferrites : a mixed ferrite like (Ni, Zn)O · Fe2O3

40. Hexagonal Ferrites MO·6Fe2O3(= BaFe12O19) where M = Ba, Sr
Calculated saturation magnetization
= 20μB/molecule (experimental)
 Other oxides
                BaO·2MO·8Fe2O3         W
                2(BaO·2MO·3Fe2O3)     Y
                3BaO·2MO·12Fe2O3)     Z
                 where, M is a divalent ion

41. Other Ferrites γ-Fe2O3 : tetragonal
(calculated net moment/molecule = 2.5μB
↔ 2.39μB experimental)
Garnets : 3M2O3・5Fe2O3 (M = Y or RE)
Alloys : Mn2Sb, Mn3Ga, Mn3Ge2, Mn3In, FeGe2, FeSe,
Cr3As2, CrPt3,
RECo5 (RE: Gd, Tb, Dy, Ho, Eu, or Tm)

42. FeO·Fe2O3 (Iron ferrite)
Crystal structure
FeO·Fe2O3 (Iron ferrite)
Magnetite is a well known ferrimagnetic material. Indeed, magnetite was considered a ferromagnet until Néel in the 1940's, provided the theoretical framework for understanding ferrimagnetism.
Tetrahedral site:Fe ion is surrounded by four oxygens
Octahedral site:Fe ion is surrounded by six oxygens

43. Magnetite Fe3O4
Magnetite (Fe3O4) has a very high Curie temperature (850 °C), but shows complex magnetic behavior. For this reason it seems to be a promising candidate for a high spin polarization degree near 100% even at room temperature.

44. Magnetite Fe3O4

45. Differences with Ferromagnetism
Smaller s/0 than that for Fe
Curie-Weiss behavior above Tc is not obeyed (Non-linear)
NiO •Fe2O3 :
Expected to have 12 B if ferromagnetic
Experiment: 2.3 B (56 emu/g) at 0 K

46. Spontaneous magnetizations
Spontaneous magnetizations of the A and B sublattices and the resultant s

47. Kinds of Magnetism Diamagnetism Paramagnetism
Non-cooperative (statistical) behavior
Ferromagnetism
Antiferromagnetism
Ferrimagnetism
Cooperative behavior

48. Classification of magnetic materials