MagneticProperties

Magnetic fields are generated by

movement of electric charges

A loop of electric

current generates a

magnetic dipole field

A magnetic dipole

•

Field lines run from the

North pole to the South

pole

•

Field lines indicate the

direction of force that

would be experienced

by a North magnetic

monopole

A bar magnet

A simple bar magnet

behaves like a magnetic

dipole

Far field picture

•

Sometimes the dipoles

are

very

small

compared with their

spatial

field

of

influence

•

An

electron,

for

example

Schematic representation

•

A magnetic dipole is

often

represented

schematically as an

arrow.

•

The head of the arrow

is the North pole.

An electric dipole is composed of 2 electric charges, 1 positive and 1 negative.

The electric field of the electric dipole looks like this.

Dipoles

N S

A magnetic dipole is composed of 2 magnetic “charges” or poles, 1 north pole and 1 south pole.

The magnetic field of the magnetic dipole looks identical to the electric dipole.

E



B



Atoms can exhibit magnetic properties that sometimes mimic this simple bar magnet behavior.

This gives bulk materials magnetic behavior.

Flux density,

B

• Density of flux (or field)

lines determines forces on

magnetic poles

• Direction of flux indicates

direction of force on a

North pole



Flux density,

B

• Higher flux density

exerts more force on

magnetic poles

Magnetic field gradients

• Magnetic field

gradients exist when

flux lines converge of

diverge

Magnetic Moment

• A magnetic dipole in a

field

B

experiences a

torque,



• Magnitude of



depends on B and

magnetic dipole

moment,

m

.



Compass needles

• A magnetic compass

needle has a magnetic

moment

• Needle is oriented in the

Earth’s magnetic field.

• Note that both magnetic

moment and field are

vectors

Flux density,

B

• Density of flux (or field)

lines determines forces on

magnetic poles

• Direction of flux indicates

direction of force on a

North pole



Flux density,

B

• Higher flux density

exerts more force on

magnetic poles

Magnetic field gradients

• Magnetic field

gradients exist when

flux lines converge of

diverge

Magnetic Moment

• A magnetic dipole in a

field

B

experiences a

torque,



• Magnitude of



depends on B and

magnetic dipole

moment,

m

.



Compass needles

• A magnetic compass

needle has a magnetic

moment

• Needle is oriented in the

Earth’s magnetic field.

• Note that both magnetic

moment and field are

vectors

Magnetic Materials -

an Empirical Approach

Magnetization,

M

• Material with a net

magnetic moment is

magnetized

• Magnetization is the

magnetic moment per unit

volume within the

Magnetization depends on……..

• Number density of

magnetic dipole

moments within

material

Magnetization depends on……..

• Magnitude of the

magnetic dipole

moments within the

material

Magnetization depends on……..

• The arrangement of

the magnetic dipoles

within the material

Magnetization in materials arises

from…….

• unpaired electron spins mainly

• the orbital motion of electrons within the

material to a lesser extent

A bulk piece of a magnetic material

Its constituent magnetic atoms (viewed as bar magnets)

Replace the bar magnets with dipole vectors

In this example, all the dipoles are aligned (perhaps by an external magnetic field). In the more general case, the dipoles may be all randomly oriented.

Magnetic Materials

Magnetic materials are formed by collecting a large number of atoms that display this inherent magnetism.

These magnetic dipoles (at the atomic level) are represented by μ_i. Since these are vector quantities, the total magnetic moment μ_total is found by a vector sum of the individual atomic magnetic moments.

Generating a uniform magnetic field

in the laboratory

•

An electric current run through a conducting coil

(solenoid) generates a uniform flux density

Flux density in vacuum (or air)

within coil……..

• Increases in proportion

to the electric current

• Increases in proportion

to the number of turns

per unit length in the

coil

Inserting a specimen into the coil

• Generally, the orbital

and spin magnetic

moments within atoms

respond to an applied

magnetic field

• Flux lines are

Specimen in magnetic field

• If specimen has no

magnetic response,

flux lines are not

perturbed

“Magnetic” materials

• “magnetic” materials tend

to concentrate flux lines

• Examples: materials

containing high

concentrations of

magnetic atoms such as

iron, cobalt

Diamagnetic materials

• Diamagnetic materials

tend to repel flux lines

weakly

• Examples: water,

protein, fat

Flux density

B

within material

determined by both……

• Geometry and current

in solenoid

• Magnetic properties of

the material

• Geometry of material



The

H

Field

•

_H

_{is called the magnetic field strength}

 

is a constant called the permeability of

In the absence of material in the

solenoid……

• There is no

magnetization

M

• So…..



Measuring magnetic moment of

specimen

• Pass specimen thru

small “sensing” coil

• Measure voltage

generated across coil

• Voltage proportional to

Measuring magnetic moment

of specimen

• Use large coil to

apply magnetic

field to specimen

• Use a cryostat or

furnace to vary

temperature of

specimen

Response of material to applied magnetic

field strength

H

• Generally,

M

changes

in magnitude as

H

is

varied.

• Magnitude of response

is called the “magnetic

susceptibility” of the

material

Response of material to applied magnetic

field strength

H

• Diamagnetic materials have a very weak

negative response

• i.e. they have a small negative magnetic

susceptibility

Magnetic susceptibility,



• Magnetic susceptibility is sometimes written

as

• And sometimes as the slope of

M

vs

H







M

_H



The total magnetization of a material is defined as the magnetic dipole density in a material

The total magnetic field in a material with an external field B₀ is given by

The magnetic field intensity in the presence of a magnetic material is given by

The magnetic permeability of a material is given by

where K_m is the relative permeability. The magnetic susceptibility of the material is given by

In a linear material, we have

For paramagnetic & diamagnetic materials, K_m ~ 1. Ferromagnetic materials have very large K_m values. For non-magnetic materials, K_m = 1 and _m = 0.

Magnetic Materials

V

M



_





M

B

B













0







K

1





K



M

B

H













H

M









How does

M

respond to

H

?

• There is a variety of ways that

M

responds to

H

• Response depends on type of material

• Response depends on temperature

• Response can sometimes depend on the previous

history of magnetic field strengths and directions

applied to the material

Non-linear responses

• Generally, the

response of

M

to

H

is

non-linear

• Only at small values

of

H

or high

temperatures is

response sometimes

linear

Non-linear responses

•

_M

_{tends to saturate}

at high fields and

low temperatures

Low field magnetic susceptibility

• For some materials,

low field magnetic

susceptibility is

inversely proportional

to temperature

• Heating a magnetized

material generally

decreases its

magnetization.

• Remnant

magnetization is

reduced to zero above

Curie temperature T

Effect of temperature on remnant

magnetization

Effect of temperature on

remnant magnetization

• Heating a sample

above its Curie

temperature is a

way of

demagnetizing it

• Thermal

The Microscopic Picture of

Magnetic Materials

• We will now revisit the experimentally

observed magnetic behaviours and try to

understand them from a microscopic

Recall the variety of magnetic behaviors that materials & films may exhibit. Diamagnetic--these materials have magnetic susceptibilities that oppose the

application of an external B field. This can be viewed as the opposition of e– _in

their orbitals to the varying B_ext as required by Lenz’s Law. Diamagnetic materials have negative susceptibilities.

All materials are inherently diamagnetic to some degree, but other behavior may dominate. Conductors are strongly diamagnetic in the presence of alternating fields. Superconductors, lacking electrical resistance to current flow, are

perfectly diamagnetic and expel external B fields from their interiors. Material _m (10–5₎

Bismuth –16.6 (–1.66  10–4₎

Mercury – 2.9 Silver – 2.6 Carbon (diamond) – 2.1 Carbon (graphite) – 1.6

Lead – 1.8

Sodium chloride – 1.4 Copper – 1.0

Water – 0.91

H M

Paramagnetic gas

• Imagine a classical gas

of molecules each with

a magnetic dipole

moment

• In zero field the gas

would have zero

Paramagnetic gas

• Applying a magnetic

field would tend to

orient the dipole

moments

• Gas attains a

magnetization

Paramagnetic gas

• Very high fields

would saturate

magnetization

• Heating the gas

would tend to

disorder the

moments and hence

decrease

Paramagnetic--when an external field is applied, these materials are magnetized in the same direction as B. This can be caused by e– _{laying outside closed shells.}

They exhibit a magnetization that is directly proportional to the applied B field. These materials obey Curie’s Law

where C is the Curie constant for the material & T is the absolute temperature.



















T

B

C

T

B

,



Material _m (10–5₎

Iron oxide (FeO) 720 Iron amonium alum 66

Uranium 40

Platinum 26

Tungsten 6.8

Cesium 5.1

Aluminum 2.2

Lithium 1.4

Magnesium 1.2

Sodium 0.72

Oxygen gas 0.19

H M

(T)

T (K) 0

The Curie Law for paramagnetic materials

As the T is lowered, the susceptibility increases inversely with T.

 

T

C

T

~

Ferromagnetism

• Materials that retain a

magnetization in zero field

• Quantum mechanical

exchange interactions

favour parallel alignment

of moments

Ferromagnetism

• Thermal energy can be used

to overcome exchange

interactions

• Curie temp is a measure of

exchange interaction strength

• Note: exchange interactions

much stronger than

dipole-dipole interactions

Magnetic domains

• Ferromagnetic materials

tend to form magnetic

domains

• Each domain is

magnetized in a different

direction

• Domain structure

minimizes energy due to

stray fields

Magnetic domains

• Applying a field

changes domain

structure

• Domains with

magnetization in

direction of field grow

• Other domains shrink

Magnetic domains

• Applying very strong

fields can saturate

magnetization by

Magnetic domains

• Removing the field

does not necessarily

return domain

structure to original

state

• Hence results in

magnetic hysteresis

Magnetic domain walls

Single domain particles

• Particles smaller than

“t” have no domains

Ferromagnetic--some materials exhibit large  ( >>1) in which long-range order causes unpaired e– to line up inside macroscopic regions called domains.

In zero external B, the domains may be randomly oriented with respect to each other. In an external B, the domains will begin to align with each other.

The material will also display hysteretic effects. For ferromagnets, the susceptibility follows the Curie-Weiss Law above T_c.

For T > T_c, the material is paramagnetic. For T < T_c, it is ferromagnetic.





T

T

C

T

B





,



Material T_c (K)

Fe 1043

Co 1388

Ni 627

Gd 293

Dy 85

Cu₂MnAl 630 Fe₂B 1015

The Curie-Weiss Law for ferromagnetic materials

As the T is lowered, the susceptibility increases inversely with T. There is a critical temperature at T_c for the onset of ferromagnetic behavior. The FM is the low temperature phase while the high temperature phase is PM.

(T)

T (K)

0 Tc

Curie-Weiss complex

hysteretic behavior

Hysteresis, Remanence, & Coercivity of Ferromagnetic

Materials

remanent magnetization = M₀ coercivity = H_c

“hard” ferromagnetic material has a large M₀ and large H_c.

“soft” ferromagnetic material has both a small M₀ and H_c.

Magnetoresistance

Magnetoresistance is the variation of a material’s (or film’s) electrical resistance with the applied B field. The resistance can increase or decrease, typically by a few

percent for ordinary materials.

In a semiconductor with a single carrier type, the MR is proportional to (1 + (B)2₎

where  is the carrier mobility (m2_/V-sec).

The Giant Magnetoresistance Effect (GMR) is a quantum mechanical effect observed in thin film structures composed of alternating ferromagnetic and nonmagnetic metal layers. The variation can be large.

In zero field the magnetization of adjacent ferromagnetic layers are antiparallel due to a weak anti-ferromagnetic coupling between layers. This gives rise to a zero-field resistivity.

When a field is applied to the film, a lower resistance appears when the magnetization of the adjacent layers align. The spin of the electrons of the

nonmagnetic metal align parallel or antiparallel with an applied magnetic field in equal numbers. These suffer less magnetic scattering when the M of the