Topics in magnetic domains

Domain theoryHysteresis loop

and thermal limit

After reviewing this lecture, you should be familiar with:

1. General features of ferromagnetic hysteresis curves2. Affects of anisotropy3. Affects of domains4. Some physics of single and multi-domain particles

Magnetic dipoles

©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

Origin of magnetic dipoles: (a) The spin of the electron produces a magnetic field with a direction dependent on the quantum number ms. (b) Electrons Electrons orbiting around the nucleus create a magnetic field around the atom.

Classification of Magnetic Materials

Ferromagnetism Ferrimagnetism Diamagnetism Antiferromagnetism Paramagnetism

Diamagnetic, Paramagnetic, Ferromagnetic, Ferrimagnetic Materials

Ferromagnetism - Alignment of the magnetic moments of atoms in the same direction so that a net magnetization remains after the magnetic field is removed.

Ferrimagnetism - Magnetic behavior obtained when ions in a material have their magnetic moments aligned in an antiparallel arrangement such that the moments do not completely cancel out and a net magnetization remains.

Diamagnetism - The effect caused by the magnetic moment due to the orbiting electrons, which produces a slight opposition to the imposed magnetic field.

cont Antiferromagnetism - Arrangement of magnetic

moments such that the magnetic moments of atoms or ions cancel out causing zero net magnetization.

Paramagetism- When a field is applied to them they become magnetized, usually much more weakly than ferromagnetic material. The magnetization depends linearly on the field and always disappears when the field is removed.

Hard magnet - Ferromagnetic or ferrimagnetic material that has a coercivity > 104 A . m-1.

Magnetization, Permeability, and the Magnetic Field

Magnetic permeability - The ratio between inductance or magnetization and magnetic field. It is a measure of the ease with which magnetic flux lines can ‘‘flow’’ through a material. , ( B=H+4M)

Magnetization - The total magnetic moment

per unit volume. Magnetic susceptibility - The ratio between

magnetization and the applied field.

H

B

H

M

BS Saturation flux density/ induction

Br Remanence; flux density remaining after applied field is removed

Hc Coercivity; field required to bring the net flux density to zero.

µ Permeability; = B/H Susceptibility; = M/Hµ0 Permeability of free space; 4π.10-7 henry per meter

µr Relative permeability, = B/µ0H

Ms Saturation magnetization; BS=µ0 Ms

Wh Energy lost per cycle; often the most important parameter for a soft magnetic material.

BHmax Energy product; often the most important parameter for a hard magnetic material.

Theoretical and Actual Saturation Magnetization in Fe

Calculate the maximum, or saturation, magnetization that we expect in iron. The lattice parameter of BCC iron is 2.866 Å . Compare this value with 2.1 tesla (a value of saturation flux density experimentally observed for pure Fe.)

Example: SOLUTION Based on the unpaired electronic spins, we expect each

iron atom to have four electrons that act as magnetic dipoles. The number of atoms per m3 in BCC iron is:

The maximum volume magnetization (Msat) is the total magnetic moment per unit volume:

To convert the value of saturation magnetization M into saturation flux density B in tesla, we need the value of μ0M. In ferromagnetic materials μ0M >> μ0H and therefore, B μ0M.

Saturation induction in tesla = Bsat = μ0Msat.

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The effect of the core material on the flux density. The magnetic moment opposes the field in diamagnetic materials. Progressively stronger moments are present in paramagnetic, ferrimagnetic, and ferromagnetic materials for the same applied field.

Domain theory

Weiss domain theory - Pierre Weiss (1907) postulated that atoms in

ferromagnetic materials had permanent magnetic moments which aligned parallel to one another over extensive regions of the sample.

- The overall magnetization of a block material is the vector sum of the domain magnetization.

Domains - Small regions within a single or polycrystalline material in which all of the magnetization directions are aligned.

Bloch walls - The boundaries between magnetic domains.

Why do domains form? If the material is magnetized, there will be

magnetic field lines in the surrounding area. These field lines, just like the electric field , represent a potential energy (magnetostatic energy)

At the domain boundaries the directions of the magnetic dipole moments change and poles formed at the surface of the sample.

Development of a stable domain configuration The domain structure can be explained by a balance

between four energy terms. - magnetostatic energy - exchange energy - anisotropic energy - magnetoelastic energy ( or magnetostrictive energy)The formation of domains allows a ferromagnetic material to minimize its total magnetic energy.

Energy considerations of a ferromagnet

Domain theory:

EH is the the Zeeman energy ED - is the magneto static ene

rgy, Eλ is the magnetoelastic ene

rgy, Ek is the magnetocrystalline

anisotropy energy, where Eex is the exchange en

ergy

Magneto-static energy: This is a self-energy, due to the interaction of the magnetic field created by the magnetization in some part of the sample on other parts of the same sample (or the potential energy of a magnetized material in its-own demagnetizing field)

Zeeman energy isthe interaction of magnetization in material with the external magnetic field

domain energy (cont) Magnetoelastic energy is a slight change

in the dimensions of the crystal when magne tized.

Anisotropy energy : The crystal lattice is 'e asy' to magnetize in some directions and 'ha

rd' to magnetize in others. Exchange energy:? The energy tends to keep

adjacent magnetic moments parallel to each other,

In ferromagnetic materials, exchange interaction leads to an alignment of atomic spins

However, this leads to a large external and dipolar magnetic fields which will tend to demagnetize the material. Domains are formed to minimize this effect.

Reduction of the magnetostatic energy by domain formation in a ferromagnet

Magnetostatic energy of domain in ferromagnetic materials

Domain wall

No magnetic poles at the surface of the block

Demagnetizing field HD=NDMs

Applie

d fi

eld

Exchange energy in single-multi domain particle.

Domains grow in an applied field When an external

magnetic field is applied the domain wall migrates into domain other domain. The result is that the specimen now acquires net magnetization.

Some physics of single and multi-domain particles When the dimensions of a magnetic particle get close to

the domain wall size, it may be more energetically favorable for the particle to form a single domain.

A permanent magnet creates a magnetic field in space, which carries energy.

The energy can be minimized by creating oppositely facing domains in the materials.

Domain walls

©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.

(a) A qualitative sketch of magnetic domains in a polycrystalline material. The dashed lines show demarcation between different magnetic domains; the dark curves show the grain boundaries. (b) The magnetic moments in adjoining atoms change direction continuously across the boundary between domains.

Domain size and wall size determined by energy cost,dependent on material and geometry.

Domains

Ni thin film

What makes the domain wall motion irreversible? The motion of the walls is

strongly influenced by defects or impurities in the sample

point defect or inclusion

• Domains have different shapes and orientations• Two examples of thin film domain walls:

Domain Walls

Neel wall (rotation in plane)

Bloch wall (rotation out of plane)

Domain wall structure

The series expansion of cosij, ....242

1cos42

Energy is minimized by having a wall of finite width

To calculation the energy and structure of domain wall

Energy cost (exchange)

N spins

Energy cost (exchange + anisotropy)

KNaNa

NJSexani

2

22

K = anisotropy constanta = lattice constantN = number atom per unit cellA =exchange stiffness/exchange constant

2

2 cos2

a

NE

JSE

exex

ijex

N spin pairs in area a2 (the wall surface)

....242

1cos42

A = (JS2/a)

N

Energy is minimized by having a wall of finite width

Domain Walls (180º)

N spins over d, domain wall is Na

NadKa

JSN

Ntotal

3

2

00

For iron, J=2.16x10-21, S=1, K=4.2x104 and a=2.86x10-10

d=42 nm (150 lattice constants)

(domain size will depend on sample geometry)

When an energy is minimized for

21

0

21

3

22

0

)/(

)(

u

u

KAaN

aKJSN

The wall thickness is

Magnetostatic energy and domain structure

Magnetostatic energy of the single-domain(per unit volume)

Nd is demagnetizing factor

The value of Nd for a cube, in a direction parallel to an edge is 4/3(cgs) or1/3(SI)

2

2

1sdms MNE

Magnetostatic energy of the crystal per unit area

LME sms2

6

1 (SI)

Magnetostatic energy of multi-domain crystal

The energy per unit area of the top surface

D<<LTotal energy is the some of

magnetostatic and wall energies Ems+Ewall

DME sms285.0

D

LDME s 285.0

Wall energies= Eex+Eani

Magnetic domains

Domain Structure and the Hysteresis Loop

Saturation magnetization - When all of the dipoles have been aligned by the field, producing the maximum magnetization.

Remanance - The polarization or magnetization that remains in a material after it has been removed from the field.

Hysteresis loop - The loop traced out by magnetization in a ferromagnetic or ferrimagnetic material as the magnetic field is cycled.

Domain model for magnetization of ferromagnetic materials

In ferromagnetic materials, exchange interaction leads to an alignment of atomic spins. When a magnetic field is applied, these spins are reoriented, leading to hysteresis.

M

H

H

H

M=magnetization along direction of H

Features of Hysteresis Curve:

M

H

M=magnetization along direction of H

Saturation magnetization (Ms)

Remnant magnetization (Mr)

Coercivity (Hc)

What determines shape of hysteresis loop?1. Coherent rotation determined mainly by Anisotropy2. Domain formation and domain wall motion

Important principle:

Magnetization will lie in direction which is an energy minimum

Consider a simple example:

H“easy axis”

M

See: http://www.student.uni-kl.de/~mewes/magnet.e.html

H“easy axis”

M

)(sin)cos( 21 KMHE

Simple example:

Zeeman energy Uniaxial anisotropy

Find M () by condition: 0E

(Stoner-Wohlfarth model)

Coherent rotation of magnetization considering only uniaxial anisotropy:

M

HH

M

=00 (along easy axis)

H M

=900 (along hard axis)

Note: Hysteresis shown above is the component of M in the direction of H

Hc=2K1/MsFor 00:

Magnetic Anisotropy

• Anisotropy: preferred (easy axes) and unfavorable (hard axes) directions of magnetization

• Due to coupling of electronic spins to electronic charge density

For this rotation, as long as spins remain parallel, exchange energy does not change, but dipolar and LS coupling energy will change.

Magnetic Anisotropy

• Anisotropy: preferred (easy axes) and unfavorable (hard axes) directions of magnetization

• Due to coupling of electronic spins to electronic charge density

Example: hcp Co

c-axis (hard)

(easy)

hard

M

H (G)8000

easy

Magnetic Anisotropy

Two major types of anisotropy, written in terms of empirical anisotropy coefficients:

Uniaxial:

Cubic:

42

21 sinsin KKEA

23

22

212

23

22

23

21

22

211 )( KKEA

(e.g., Co)

(e.g., Fe, Ni)

Note: cubic lattices can have several easy and hard axes

Domain formation and domain wall motion affects the shape of hysteresis loop:

Domains and Hysteresis

M

H

H

H

Barkhausen noise: Tiny steps of domain walls

Domains and Hysteresis

M

H

Types of medium noise The stationary DC remanent noise

The nonstationary transition noise due to the zig-zag structure at the transition center

The modulation noise due to the surface roughness of the medium

Longitudinal and perpendicular magnetic thin film media noise The longitudinal media

noise comes from distortions in zig-zag pattern of the domain walls separating oppositely magnetized domains

The perpendicular media noise comes from distortion in the semicircular pattern of the domains walls.

Media noise power (modeling) 4

12

2

nl

np

VV

Particle noise of magnetic tape The replay flux is

discontinuous This appears as

noise in the head output

Signal to noise ratio (S/N) is proportional to

* tape/head velocity (v), track width (w), particle/unit volume (n), frequency (f),reproduce signal filter bandwidth f

and film thickness (c)

v

fc

v

fc

e

e

ff

wnv

4

2

1

1

Fine particles Some physical and

mechanical property is found to be depend on size of particle (size effect).

The coercivity Hc shows a marked size effect.

For example, the coercivity of elongated iron particles 150 Å in diameter is some 1014 times that iron in bulk.

How the size is divided in relation to the variation of the coercivity with particle diameter.

Coercivity of fine domains1.Multi-domains

where a, b are constants

2. single-domain (below critical Ds) the particle of size Ds and small change

their magnetization by spin rotation.3. At particle size decreases below Ds the

coercivity decreases (thermal effect)

4. Below a critical diameter DP the coercivity is zero. (thermal effect)

Shape anisotropy& Coercivity The coercivity decreases as

p increases, because particle interactions

An isolated particle (p=0) When p =1,all particles are

everywhere in contact, shape anisotropy is lost, and coercivity becomes zero, if other forms of anisotropy are absent.

Shape anisotropy& Coercivity The coercivity of elongated

particle.

Where Na is the demagnetizing factor along the short axis and NC along the long axis.