Magnetism

UEEP2024 Solid State Physics

Topic 5 Magnetism

Magnetic properties of solids

• The magnetic moment of a free atom has three principal sources :

1. The spin with which electrons are endowed2. The electron orbital angular momentum about

the nucleus3. The change in the orbital moment induced by an

applied magnetic field

Magnetic properties of solidsMaterials may have intrinsic magnetic dipole moments, or they may have magnetic dipole moments induced in them by an applied external magnetic field of induction. In the presence of a magnetic field of induction, the elementary magnetic dipoles, whether permanent or induced, will act to set up a field of induction of their own that will modify the original field. The magnetic dipole moments are the source of magnetic induction B

MHB oo

Magnetic field strengthMagnetic constant

Magnetization

Magnetic properties of solidsAlternative names for B and H

Bname used bymagnetic flux density electrical engineersmagnetic induction applied mathematicians

electrical engineers

magnetic field physicistsH

name used bymagnetic field intensity electrical engineersmagnetic field strength electrical engineersauxiliary magnetic field physicistsmagnetizing field physicists

Magnetic properties of solidsMagnetization is defined as magnetic moment per unit volume. For certain magnetic materials, it is found empirically that the magnetization M is proportional to magnetic field strength H

HM

Magnetic susceptibility

> 0 -- paramagnetism

< 0 -- diamagnetiism

Atomic Magnets• In classical picture of an atom, the electrons describe a

circular orbit around the nucleus.• Each orbit can therefore be thought of as a loop of electric

current.• From electromagnetic theory, a loop of current produces a

magnetic field, so electrons in an atom also generate a magnetic field.

• Quantum theory predicts that electrons in an atom produce a magnetic field.

• The quantum number n, l, ml and ms label the electrons in an atom.

Atomic Magnets

• The orbital magnetic number ml takes values between –l and +l.

• Spin magnetic number ms takes values of 1/2.• A single electron the component of the spin

magnetic moment, µs, in the direction parallel to the magnetic field is given by

• The quantity is known as the Bohr magneton.

ee

ss m

emem

2

eB m

e2

Atomic Magnets

• The component of orbital magnetic moment is given by • When an atom has more than one electron, for a

particular subshell, Total spin angular momentum Total orbital angular momentum• If a subshell is filled, the values of S and L are both zero.• If a subshell is partially filled, electrons are distributed

between different ml and ms states.

Bll

l mm

em 2

smS

lmL

Hund’s rules

• Two simple rules known as Hund’s rules to determine how theses states are occupied in a partially filled sub band.

1. The states are occupied so that as many electrons as possible have their spin aligned parallel to one another, i.e. so that the values of S as large as possible.

2. When it is determined how the spins are assigned, the electrons occupy states such that the value of L is a maximum.

Total angular momentum J

• The total angular momemtum J is obtained by combining L and S as follows:

If the subshell is less than half filled then J = L-S;If the subshell is more than half filled then J = L + S;If the subshell is exactly half filled then L = 0 and so

J = S.• By applying these rules, the values of S, L and J can

be determined.

Landé splitting factor g

• The maximum component of the magnetic moment of the atom in the direction parallel to the magnetic field is

where g is the Landé splitting factor

• Most atom exhibit a magnetic moment.

Jgm Bj

)1(2)1()1()1(3

JJ

LLSSJJg

Example

Determine the values of S, L and J for Cr3+

which has three electrons in the 3d subshell. (All lower energy subshells are filled)

Solution

• The 3d subshell corresponds to l = 2, and therefore ml takes integer values between -2 and +2. i.e. there are five allowed values of m l. so that the subband has a capacity to hold 10 electrons.

• In Cr3+, the three 3d electrons can therefore all occupy ms = +1/2 and therefore S = 3/2.

• The maximum value of L is obtained if two electrons occupy the state with ml = +2 and one occupies the ml = +1 state. However, the exclusion principle forbids two electrons with the same value of m s from occupying the same ml state.

• Consequently the largest allowed value of L corresponds to having one electron in each of the states m = +2, +1 and 0, and therefore L = 3.

• Since the subband is less than half filled, the value of J is J = L – S = 3 – 3/2 = 3/2.

Which materials have magnetic moment?

• Filled electron subshell in an atom do not affect the magnetic moment because S = L = 0 so J = 0.

• Inert atoms have a magnetic moment of zero.• Ionic materials have a magnetic moment of

zero because the electrons are transferred from one atom to another so that the resulting ions have only filled subshells.


• In covalent material the outer subshell is only partially filled, so these materials have finite magnetic moment.

• However, each covalent bond is formed by a pair of electrons with opposite spin and with a net orbital angular momentum of zero. Covalent solids have a net magnetic moment zero.

• Actually, this is not quite true. The present of a magnetic field also affect the orbital motion of the electrons in an atom in such a way that the atom generate a magnetic field which opposes the external field. This is referred to as diamagnetism.


• Filled electron subshells in an atom do not affect the magnetic momentum of the atom because they have a net angular momentum of zero. (S=L=0, so J = 0).

• This means that inert atoms have a magnetic moment of zero because they have only filled electron subshells.

• Ionic materials have a magnetic moment of zero because the electrons are transferred from one atom type to another so the resulting ions have only filled subshells.


• In covalent materials the outer subshell is only partially filled, and so these materials have a finite magnetic moment.

• However, each covalent bond is formed by a pair of electrons with opposite spin and with net orbital angular momentum of zero, covalent solids have a net magnetic moment of zero.

• Although most atoms have a non-zero magnetic moment, it appears that in majority of solids the effects cancel and the resultant magnetization is zero.


• The presence of a magnetic field affects the orbital motion of the electrons in an atom in such a way that the atom generates a magnetic field which opposes the external field. This is referred to as diamagnetism and occurs in all type of atom.

• It produces a finite magnetic susceptibility in all solids, but the effect is weak.


• It appears that most non-metals display only a small magnetic susceptibility.

• In simple metal, such as sodium and aluminium, the valence electrons are assumed to be localized, in other words they are no longer attached to any particular atom. As a result the metal ions contains only filled electron subshells, and so each has a total angular momentum of zero.

• Although the ions do not contribute to the magnetic moment, the delocalised electrons do produce a non-zero magnetic moment. This is known as Pauli paramagnetism.


• Free electron paramagnetism is a weak effect.• Paramagnetism is caused by the magnetic

dipole moments becoming aligned with the magnetic field, whereas diamagnetism results from an induced magnetic field which opposes the applied field.

• Paramagnetism gives rise to a positive susceptibility, whilst diamagnetism produces a negative susceptibility.


• A group of metallic elements known as the transition metal, and the so called rare earth and actinide elements have more promising characteristics.

• These materials have an electronic structure which is very different to that of the simple metals.

• These elements give rise to Curie paramagnetism.• Under certain circumstances they can form

permanent magnets.

Diamagnetism

• The value of B is smaller in the region of the diamagnetic material than it would be if the material were absent

• The origin of diamagnetism is Lenz’s law

When the flux through an electrical circuit is changed, an induced current is set up in such a direction as to oppose the flux change.

Diamagnetism(Classical theory)

2omRF

Consider an electron in a circular orbit of radius R and angular frequency o

If a magnetic field dB is turned on, and assume it affects angular frequency only

BqvmRF o d 2


BqRmRmR

BqRmRmR

BqvmRmR

oooo

ooo

o

dddd

ddd

d

222

22

22

2

The new angular frequency

Taking only first order form (assume o>>d)

mBqBqRmR oo 2

2 dddd

Classically, the B field will make the electrons revolve slower


dd

22qIqI

Current formed by the loop = charge x revolution/second

For an electron in an atom

mBeI

mBqqI

ddd

d

422

2

The magnetic moment = (curent) x (area of the loop)

mBe

mBe

44

222

2 dd


22

1 4d

mBZeZ

ii

For an atom with Z electrons

If r = radius of the 3-D electron shell

2222 zyxr

For a spherically symmetrical distribution of charge

22222

23 rzyx

is the radius of the electron loops

222 yx


22

6r

mBZe

atomd

The magnetic moment of an atom with Z electrons

Diamagnetic susceptibility per unit volume

BN atomo

d

Classical Langevin equation for diamagnetism

22

6r

mNZeo

Number of atoms per unit volume

Paramagnetism• When atoms possess their own magnetic

moment, paramagnetism will occur

In a paramagnetic material the atoms contain permanent magnetic dipole moments. An externally applied B field will tend to align these dipole moments parallel to the field. The result is an induced field that adds to the applied field so that the susceptibility is positive.

1. Electron spin

2. Orbital motion of the electrons

Paramagnetism

• In atoms with filled subshells, the spin magnetic dipole moments, and separately the orbital magnetic dipole moments, cancel in pairs.

• The inert gas, have closed subshell configurations, so that they do not exhibit paramagnetism and are excellent for diamagnetic studies.

Paramagnetism

• Only unfilled subshells can have unpaired electrons, so that we expect paramagnetism only in material containing atoms whose electronic subshells are partly filled.

• The basic requirement for paramagnetism in solids is that the individual magnetic dipole moments have some degree of isolation

If wave functions of atoms overlap significantly, they will tend to pair up the magnetic dipole moments

Paramagnetism

• Unfilled inner subshells• Isolation of the individual moments results

from the shielding of these inner subshells by the filled outer subshells

Transition elements Rare earth elements

Paramagnetic solids

ParamagnetismTransition elements : an element whose atom has an incomplete d sub-shell, or which can give rise to cations with an incomplete d sub-shell (defined by IUPAC )

Sc 21 Ti 22 V 23 Cr 24 Mn 25 Fe 26 Co 27 Ni 28 Cu 29 Zn 30

Y 39 Zr 40 Nb 41 Mo 42 Tc 43 Ru 44 Rh 45 Pd 46 Ag 47 Cd 48

Lu 71 Hf 72 Ta 73 W 74 Re 75 Os 76 Ir 77 Pt 78 Au 79 Hg 80

Lr 103

Rf 104

Db 105

Sg 106

Bh 107

Hs 108

Mt 109

Ds 110

Rg 111

Uub 112

Rare earth elements : a collection of seventeen chemical elements in the periodic table, namely scandium, yttrium, and the fifteen lanthanoids (defined by IUPAC )

Paramagnetism

Paramagnetism

1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f5d<6p<7s<5f<6d<7p

Paramagnetism

• Magnetic field – line up the magnetic dipole moments

• Thermal motion – make the directions of the magnetic dipoles random

The susceptibility to decrease with increasing temperature

TC

Curie law Positive constant

Paramagnetism

JgJ B The magnetic moment of an atom in free space is

Total angular momentum = sum of the orbital and spin angular momentum, gyromagnetic ratio – the ratio of the magnetic moment to the angular momentum

B, Bohr magneton – is define as eh/4m

g – g factor or spectroscopic splitting factor

Paramagnetism

12

1111

JJLLSSJJg

The g factor comes about during the calculation of the first-order perturbation in the energy of an atom when a weak uniform magnetic field is applied to the system

J : the total electronic angular momentumL : the orbital angular momentumS : the spin angular momentum

The energy levels of the system in an applied magnetic field B are

BE

ParamagnetismWhen there is no field (B=0)

NN

ParamagnetismWhen an external field is applied

Paramagnetism

F

F

E

B

E

B

dEBEDN

dEBEDN

)(21

)(21

The total magnetic moment

BENNNM

F23)(

2

Pauli paramagnetism

Ordered magnetic materials• Interaction between the 3d electrons in these materials

have two effect on the magnetic moments of the ions1. Interaction affects the orbital angular momentum in such a way that the average orbital angular momentum on neighbouring ions cancels. Magnetic moment due to orbital angular momentum become zero.

2. The spins of the 3d electrons interact in such a way that there is a correlation between the spins of 3d electrons on neighbouring ions. This is called the exchange interaction.

Ordered magnetic materials

• If the interaction between the electrons is sufficiently strong, it is energetically favourable for the electrons in the two ions to have the same spin.

• Each ion affects the dipole moment of each of its neighbouring ions, then all of the atomic dipoles in the crystal will be aligned in a common direction.

• Such a material is called a ferromagnet.


• The exchange interaction provides a mechanism for aligning the atomic dipoles without the need for an external field.

• Ferromagnet can have a large magnetization even when no external field is present.


• Some materials retain their magnetization even in the absence of a magnetic field.

• These materials are referred to as permanent magnet.• To explain this type of behaviour, require some

mechanism for causing the magnetic diople on neighbouring ions to become mutually aligned without the aid of an external magnetic field.

• The 3d electrons in the iron group of transistion metals interact strongly with similar electrons on neighbouring ions.

Three simplest type of ordering of atomic magnetic moments .

(a) Ferromagnetic. (adjacent magnetic

moment are aligned)

(b) Antiferromagnetic (adjacent magnetic moment are antiparallel)

(c) Ferrimagnetic (adjacent magnetic moment are

antiparalle and unequal magnitude)

Long range magnetic ordering• Long range magnetic ordering is duo to exchange field from

neighbors• Magnetic ordered states occurs only ay low temperatures (T <

Tc). When T > Tc there will be no ordering and the material has to be paramagnetic

• Three common types of magnetic ordering :

Spontaneous magnetisation


• Another possible way of ordering the dipoles is if the magnetic dipoles in adjacent planes are misaligned in such a way that the dipole form a helix.

• Example, in magnesium dioxide the angle between the dipoles is about 129o.

• These materials are called helimagnets.

In a helimagnet the magnetic dipoles are rotated by a fixed angle

Example

Show that the magnetic susceptibility due to the delocalized electrons in a metal is given by

.)( 2omFm mEg

Solution

• The number of electrons in the shaded region on the left hand side is

• Factor of ½ is because only using one-half of the electron distribution.

• If the total number of valence electrons per unit volume is N, then the number aligned parallel to the field is

• number aligned antiparallel to the field is

omF BmEg )(21

.)(21

2 omF BmEgNN

.)(21

2 omF BmEgNN

Solution

• The difference is therefore• Since each electron contributes a magnetic

moment mm. • The magnetism is

• Susceptibility

.)( onF BmEgNN

omFm BmEgmNNM 2)()(

2)( mFoo

om mEg

BM

Example

the probability that an atomic dipole moment mj is given by where A is a contant as a

function of temperature. Show that the temperature dependence of the magnetic

susceptibility can be written as

where C is a constant.(called Curie constant)

,kTBm oj

Ae

TC

m

Solution

• If there are N magnetic ions per unit volume, then the total number of dipole in this state is

and the magnetic moment of these atom is

Magnetization M of the crystal

kTBm oj

NAe

kTBm

j

oj

NAem

j

j

kTBm

j

oj

NAemM

Solution• Since at room temperature mjBo is very much smaller than

the thermal energy kT,

• Magnetization becomes

where C’ is a constant

kTBme ojkT

Bm oj

1

TBC

mkTNAB

kTBm

NA

kTBm

mNA

kTBm

NAmM

o

j

jj

o

j

j

oj

j

j

ojj

j

j

ojj

2

2

2

1

Solution

• Susceptibility

TC

TC

BM o

o

oB

Example

Calculate the magnetization of magnetite, Fe3O4, assuming that the magnetization is due only to the Fe2+ ions (which have six 3d electrons) and that only the spin angular momentum of the electrons contributes to the magnetic moment of the ions. (The molar volume of Fe3O4 is 4.40×10-5 m-3.)

Solution

• According to the Hund’s rule, the six 3d electrons in each Fe2+ ion are arranged so that the spins of five of the electrons are parallel to one another and the sixth electron is antiparallel.

• This can be represented as ↑↑↑↑↑↓.• Since each electron has a spin magnetic moment

of µB, the net magnetic dipole moment per ions is 4 µB, and so the dipole moment per mole is 4 µBNA where NA is avogadro’s number.

Solution

• The magnetization

.mJT1007.5molm1040.4

)mol1002.6)(T J1027.9)(4(memolar volu

4

315

135

123124

ABNM

Temperature dependence of permanent magnets

• When a permanent magnet is heated it is found that there is a temperature above which the magnetization of the material disappears.

• The transition point is known as the curie temperature.

• Above the curie temperature the material behaves like a paramagnet- the dipoles are randomly aligned unless they are exposed to an external magnetic field.


• The magnetic susceptibility of this magnetic material is given by a modified form of the Curie law, known as Curie-Weiss law,

where c is the Curie temperature.• As the temperature of a ferromagnet is lowered

towards the Curie point, all the atomic dipoles move to the lowest energy state and therefore become aligned along a common direction.

cm T

C

Values of the saturation magnetization at 300 K shown as Ms (JT-1 m-3), µoMs(T) and Curie temperature c(K).

Ms (×105 J T-1 m-3) µoMs (T) c (K)

Iron 17.1 2.15 1043

Cobalt 14.0 1.76 1388

Nickel 4.85 0.61 627

Gadolinium 20.6 2.60 292

CrO2 5.18 0.65 386

Fe3O4 4.80 0.60 858

MnFe2O4 4.10 0.52 573

NiFe2O4 2.70 0.34 858


• In iron, cobalt and nickel the curie temperature is well above room temperature.

• Some material, their Curie points are below room temperature, such materials are not normally used as permanent magnet.

• The existence of the Curie temperature can be explained as follow.

• As the temperature is increased, the thermal vibrations of the ions become so large that the alignment of the magnetic dipoles is destroyed.


• This happens when the thermal energy kT is comparable with the interaction energy between the neighbouring dipoles.

• Since thermal effects are random in nature, we might expect the transition to occur gradually over a temperature range of several degrees, but the Curie temperature is remarkably well defined.

• To understand why this is so, let us consider what happens if we have a ferromagnetic material in the absence of any external magnetic field.


• Suppose that initially the temperature is higher than the Curie temperature.• In this case the magnetic dipoles are orientated in random directions. • If the temperature is now lowered, then as we reach the Curie temperature

we find that the thermal vibrations are weak enough to allow a few neighbouring magnetic dipoles to become aligned.

• As the result these aligned dipoles create a local magnetic field within the sample, so other neighbouring dipoles tend to become aligned in the same direction, which in turn further increases the strength of the internal magnetic field.

• So we have a runaway process which does not stop until all of the magnetic dipoles in the crystal are aligned in the same direction.

• Such process is known as a cooperative transition.• Consequently, the change from a disordered paramagnet to an ordered

magnetic material takes place over a very narrow temperature range.

Application of magnetic materials for information storage

• This includes audio and video tapes, floppy disks and hard disks for computers.

• The basic principle of magnetic storage of information is broadly similar regardless of whether the data stored is digital or analogue.

• The magnetic medium is covered by many small magnetic particles, each of which behaves as a single domian.

• A magnet- known as the write head- is used to record information onto this medium by orientating the magnetic particles along particular direction.

• Reading the information from the tape or disk is simply the reverse of this process- the magnetic field produced by the magnetic particles induces a magnetization in the read head.

Schematic diagram showing the process of recording information onto a magnetic medium

Current in

“1” “1” “1” “1”“0” “0”

B


• The choice of material for magnetic recording medium depends on several factors.

• The coercivity must be low enough so that the orientation of the particles can be altered by the write head, but high enough so that the orientation is not accidentally affected by other external magnetic fields.

• The curie temperature should also be well above the temperature to which the material will normally be exposed.


• Magnetic hard disks are used in computer because they are considerably cheaper than semiconductor memory.

• They are not volatile.• Magnetic system does not require a constant

electric power supply in order to remain in the same memory state.


• In order to keep pace with the advances in semiconductor technology it has been necessary to increase continually the storage density of these magnetic hard disks, from about 2 MB cm-2 in 1980 to 75 MB cm-2 in 1996.

• Firstly the size of magnetic particle is reduced.• The read head are made using magnetoresistive

materials. ( resistance change in the present of magnetic field)

• Align the particle so that they are perpendicular to the surface of the disk.

Orientation of particle perpendicular to the surface to increase its density.

Orientated parallel to the surface

Orientated perpendicular to the surface


• The write process is achieved by using a laser beam to heat a magnetic particle above the Curie temperature as a magnetic field is applied.

• As it cool down the particle retains the magnetization of the applied field.

• To read the information another (weaker) polarized laser beam is directed onto the surface.

• The polarization of the reflected beam is determined by the direction of magnetization of each particle.

Science

Magnetism