Ferromagn

etism

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.

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

Ferromagnetism

Hund’s rule

Fe: [Ar]3d64s2

Origin of Ferromagnetism

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

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.

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?

Ms Ms Ms

(a) (b) (c)

Ms Ms Ms

(a) (b) (c)

(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.

Magnetic Domain

Magnetic Domain

Domain wall motion

Barkhausen effect

Magnetic Order

Are ferromagnets

already in an ordered

state before a

magnetic field is

applied or is the order

by the field?

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.

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.

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.

Weiss domain theory

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.

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)

Weiss Mean Field Theory

Curie-Weiss LawCurie'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 constantMHm

mt HHH

T

C

kT

n

H

M

3

2

T

C

MH

M

)(

CT

CHM

T

C

CT

C

H

M

Molecular field theoryPierre Weiss introduced molecular field concept.

Interaction between magnetic moments Fictitious internal filed

T

C

C

MHm

For > 0, Hm || M

: molecular field constant

MHHHH amatot

Curie Temperature

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.

Curie temperature

Saturation magnetization of Fe, Co, Ni

as a function of temperature

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.

Exchange Energy: Heisenberg Model

Si·Sj: spin angular momentum Je : a numerical quantity called exchange integral

cos22 jiexjiexex SSJSSJE

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.

Relative orientation of two spins determines the energy states.

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.

Band TheoryWhen 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.

Density of states

Anti-

Ferromagn

etism

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.

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

Anti-ferromagnetism

BCC crystal (Cr)

Electrical Insulator

(no free electron)

Molecular field theory

Anti-ferromagnetism

TN : Néel temperature

T < TN : AF state

T > TN : paramagnetic

)(

T

C

Anti-ferromagnetism

Ferrimagn

etism

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.

Ferrimagnetism

Ferrimagnetism

Two groups of ferrites depending on their structure

1. Cubic :

General formula : MOFe2O3 where M is a divalent metal ion (Mn, Ni, Fe, Co, Mg, ...)

CoOFe2O3 is magnetically hard, but all the other cubic ferrites are magnetically soft.

magnetite : Fe3O4 = FeOFe2O3 : oldest ferrite (lodestone, iron ferrite)

2. Hexagonal :

Barium ferrite (BaO6Fe2O3) is magnetically hard

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 M2+ + 8 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

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

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)

Crystal structure

Tetrahedral site:Fe ion is surrounded by four oxygens

Octahedral site:Fe ion is surrounded by six oxygens

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.

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.

Magnetite Fe3O4

Magnetite Fe3O4

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

Spontaneous magnetizations

Spontaneous magnetizations of the A and B sublattices and the resultant s

Kinds of Magnetism

DiamagnetismParamagnetism

Non-cooperative (statistical) behavior

FerromagnetismAntiferromagnetismFerrimagnetism

Cooperative behavior

Classification of magnetic materials