Development of Domain Theory• By Ampere in 1827 -The atomic magnetic moments are due to “electrical current continually circulating within the atom” -This was some 75 years before the discovery of electron by J. J. Thomson and no charge separation was known

• By Weber in 1852 -Each atom has a net magnetic moment. The magnetic moments are randomly aligned in the demagnetized state but become ordered by Ha

-Supported by the existence of saturation magnetization and

remanence

• By Poisson in 1893 -No atomic magnetic moments at all in the demagnetized state but could be induced by Ha

•by Ha

• By Ewing in 1893 (similarly to Weber) -Randomly aligned atomic moments in the demagnetized state but

aligned in the magnetized material

• By Weiss in 1906 and 1907 -Suggested the existence of magnetic domains in ferromagnets

-Explained one main problems of the earlier theories – A very

large permeabilities of ferromagnets

• Summary of the theories -Each atom has a permanent magnetic moment

-The atomic moments are aligned (ordered) in the demagnetizied

state

-It is the domains only which are randomly aligned

-Magnetization process consists of reorienting the domains so that

more domains are aligned with Ha

Weiss Mean Field Theory

•What’ 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•Extension of the Lagevin theory for paramagnets by including the Weiss mean field (or molecular field)•The Weiss mean field - Each atomic moment interacts equally with ever other atomic moment - Works well for homogeneous distribution of magnetic moments paramagnets and within domains, for examples)

kT

MHm

kT

HHmx

xxxL

M

M

MHHHH

aeffmaeff

amatot

)()(

1)coth()(

00

0

Weiss Mean Field Theory

• Magnetic structure depending on (ferromagnetic for >0 and antiferromagnetic for <0)

(a) (b)

Weiss Mean Field Theory

Simple ferromagnetism Simple antiferromagnetism Ferrimagnetism

Canted antiferromagnetism Helical spin array

•Equation for the mean fields - The interaction with one moment He=ijmj

- The interaction with all moments He=ijmj

- If the interactions with all moments are

equal(mean field approximation) He=mj= Ms

• How Big is it? – Very big, 850 T for Fe He= Ms= 400 (1.7 !06) A/m =8.5 106 Oe =850 T

Weiss Mean Field Theory

•Energy states of different arrangements of moments -Lower energy for parallel alignment for >0 Ei = -0 mi·He= -0 mi· ijmj - -0 mi · mj

Etot= - 0 mi · mj

Etot= - 0 (6m)(5m)= - 0 30m2

Etot =- 0 (5m3m-m5m)= - 0 10m2

Weiss Mean Field Theory

·

•For an array of 9 magnetic moments, calculate the Curie temperature in the mean field coupling and NN coupling

Weiss Mean Field Theory

20)8)(9( mEtot 2

024 mEtot

kTNkTEthermal )9)(3(3

k

mTC 3

8 20

k

mTC 9

8 20

For mean field coupling

For NN coupling

(3)(9)kT

For mean field coupling

For NN coupling

Domain Observation

•Indirect observation by the Backhausen effect (1919) -Discrete change in magnetic induction, which can be sensed with a search coil•The Bitter method, the first direct observation by Bitter(1931) -Very fine magnetic powders suspended in a carrier liquid Collodial solutions of ferromagnetic particals•Optical methods utilizing the Kerr and Faraday effects -What are the effects?- The axis of polarization of linearly polarized light beam is rotated by the action of a magnetic field

•The Kerr method – A reflection method. Note that the Faraday method is transmission method - The angle of rotation of the axis of polarization is dependent on M(the magnitude and direction of M) at the material surface - The angle of rotation is very small Very little contrast between the different domains

•The Faraday method - Less useful than the Kerr method - Only applicable to thin transparent samples

Domain Observation

• The polar Kerr effect - M is perpendicular to the surface - The largest Kerr angle up to 20 minutes of arc

• The longitudinal Kerr effect - M lies in the incidence plane and also in the surface plane - The smallest Kerr angle up to 4 minutes of arc - The Kerr angle is largest at an incidence angle of 60

•The transverse Kerr effect - M lies in the surface plane but is perpendicular to the incidence - The Kerr angle is similar to that from the Longitudinal Kerr effect

Domain Observation

• Domain Formation as a Result of Energy Minimization(fig.6.6)

• Magnetization Process1. Domain wall motion at low fields2. Incoherent (irreversible) domain

rotation at moderate fields

-The moments within an unfavorably aligned domain overcome the anisotropy energy and suddenly rotate into one of the crystallographic easy axes which is nearest to Ha

-Single domain state in the direction parallel to one of the easy axes

3. Coherent (reversible) domain rotation at high fields -Rotation from one of the easy axes to the Ha direction

-Single domain state in the Ha direction

4. The paraprocess (forced magnetization) at a very high fields

Fig.6.7, fig.6.8

•Magnetization Process

Domain Rotation and MCA

•Domain wall motion is little affected by MCA but (irreversible) incoherent and coherent(reversible) rotation are determined principally by MCA

•Domain rotation can be considered as a competition between the MCA energy and the Zeeman energy

Etot=Ea(, )+Ezeeman=Ea (, )-0Ms.H

Domain Rotation in Cubic Anisotropy

• In a one constant model Ea=K0+K1(cos21cos2 2+cos2 2cos2 3+cos2 3cos2 1

Ea= K0+K1/2(cos2 i cos2 j)

• If K1>0, Ea

Reaches a minimum of Ea = 0 for [100] directions• If K1<0, Ea

Reaches a minimum of Ea = -Kl/3 for [111] directions