3.magnetism

Lecture Notes on Phenomenon at Low Temperatures, P.N.Vishwakarma, Dept. of Physics, NIT-Rkl

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5. Magnetic properties The magnetic property in a material originates from the magnetic moments present in that

material. In general there are three main types of behaviours are observed,

(a) Diamagnetism: When the effective dipole moment m is in a direction opposite to applied

magnetic field H and is proportional to the magnitude of H.

(b) Paramagnetism: When the effective dipole moment m is in the same direction as applied

magnetic field H and is proportional to the magnitude of H.

(c) Ferromagnetism: When m is parallel but not proportional to H; m increases very rapidly as H is

increased and it then saturates at a value which is many orders of magnitude higher than any

which is found in a paramagnetic substance. When the field is reduced to zero some of the

magnetic moment remains, and if this is large enough, the material is said to be a permanent

magnet.


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All the magnetic effects we shall consider are due to the interaction of a magnetic field with the

electrons of the atoms of the substance which is being investigated. This interaction occurs

because the electron, which is both spinning on its own axis and also circulating around the

nucleus, can be considered as a circulating current. Since such a current always has a magnetic

moment associated with it, it follows that the electron also possesses moment which will tend to

orient itself in a magnetic field. The total magnetic moment of an electron can initially be discussed

in terms of two components, lm due to its orbital motion and sm which is associated with its spin.

When a magnetic field is applied to an atom, then for each electron lm and sm will tend to align

themselves with respect to H along one of the special directions which are permitted by quantum

mechanics**. In the ground 1s state of the hydrogen atom the orbital moment is zero, and the magnetic moment is that of the electron spin along with a small induced diamagnetic moment. In

the 1s2 state of helium the spin and orbital moments are both zero, and there is only an induced

moment. Atoms with filled electron shells have zero spin and zero orbital moment: these moments

are associated with unfilled shells.


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** Here we will discuss the magnetic properties in terms of the alignment of the orbit or the spin

with respect to the magnetic field H. This gives an oversimplified picture of the process. In actual

fact when H is applied, the orbital and spin magnetic moments precess with H as the axis of

precession. Thus the components of the orbital and spin moments in the direction of H remain

constant during the precession and it is these which we call lm and sm

The alignment of magnetic moments due to external magnetic field H in a material is measured by

the quantity magnetization or magnetic susceptibility. The magnetization M is defined as the

magnetic moment per unit volume and it is related to the magnetic susceptibility per unit volume F as

MH

F [CGS] or 0MH

PF [SI]

The susceptibility F is dimensionless. Substances with a negative magnetic susceptibility are called diamagnetic and substances with positive magnetic susceptibility are called paramagnetic.


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Volume susceptibility (Fv) vF SI: dimensionless CGS: emu (electromagnetic unit)

Mass susceptibility (Fm) m vF F U SI: 3 1.m kg

CGS: 3 1.cm g

Mole susceptibility (Fmol) mol mMF F U SI: 3 1.m mol

CGS: 3 1.cm mol

where M is the molar mass and U is the density in 3.kg m (SI) or 3.g cm (CGS)


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Quantum theory of Diamagnetism: Theory of Paramagnetism: (a) Langevin paramagnetism: This paramagnetism is due to bound electrons in the orbitals,

having total angular momentum J= . Due to external magnetic field, the degenerate energy levels

split into degenerate states. These degenerate states are separated by U HP , where Bg JP P

is the magnetic moment of an atom or ion, BP is Bohr magnetron and g is the Landes splitting

factor given by, ( 1) ( 1) ( 1)1 2 ( 1)J J S S L Lg

J J

If the magnetic system is in thermal equilibrium with the crystal lattice, the distribution of the atoms

amongst the various energy levels will be governed by the Maxwell-Boltzmann statistics. The

probability that an atom or ion will be in a state n will be proportional to exp nB

Ek T

, where nE is the

energy of nth state. Hence for any temperature T, such that B Bk T gJ HP!! , the occupation of the

levels will obey the Maxwell-Boltzmann statistics, as shown in the fig. The net magnetization in this

case will be


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( )B JM NgJ B xP BB

gJ Hxk TP

where ( )JB x is the Brillouin function defined as

2 1 2 1 1 1( ) coth coth2 2 2 2JJ JB x x x

J J J J

In the limit, 1x i.e., B Bk T gJ HP!!

0Jm

1/ 2Jm

3 / 2Jm

1/ 2Jm

3/ 2Jm '

J = 3/2

U HP

32 B

gP P

32 B

g HP'


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31coth ...3 45x xx

x

The magnetization equation is 2 2( 1)

3B

B

Ng J J HMk T

P

Hence the susceptibility reduces to

2 2 2( 1)3 3

B

B B

Ng J JM N CH k T k T T

P PF 2 2 ( 1)Bg J JP P & 2 2( 1)

3B

B

Ng J JCk

P

Where P is the effective Bohr magnetron and C is the curie constant.

(b) Pauli paramagnetism: This type of paramagnetism arises because of free electrons in metals. Previously it was believed that, the paramagnetism shown by conduction electrons is

similar to that due to bound electrons, 2

B

NMk TP . However the experimental results on most of the


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non-ferromagnetic metals contradicted this result. Later it was shown by Pauli that with the

application of Fermi-Dirac distribution would correct the theory as required.

According to the Fermi-Dirac theory, the states of the electrons are filled up to a level known as

Fermi level. Till Fermi level, both the electrons with up spin and electrons with down spin are

equally populated. The application of the external magnetic field will try to line up the spin direction

parallel to the field. Most conduction electrons in a metal, however, have no possibility of turning

over when a field is applied, because most of orbitals in the Fermi sea with parallel spin are

already occupied. Only the electrons within a range Bk T of the top of the Fermi distribution have a

chance to turn over in the field, thus only the fraction / FT T of the total number of electrons

contribute to the susceptibility. Hence 2 2

B F B F

N H T N HMk T T k TP P|

Which is independent of temperature and of the observed order of magnitude.

A detailed mathematical derivation gives the accurate expression for conduction electron

magnetization as,


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> @22 22

2

ln ( )1

6B

Fd g E

M Hg E k TdE

SP

In the limit, B Fk T E or FT T , one may neglect the second order terms and the pauli

paramagnetic susceptibility is

2

2 33p F B F

M Ng EH k T

PF P

where we have used, 3 3( )2 2F F B F

N Ng EE k T

.


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(c) Van Vleck paramagnetism: We consider an atomic or molecular system which has no magnetic moment in the ground state, by which we mean that the diagonal matrix element of the

magnetic moment operator zP is zero.

Suppose that there is a non-diagonal matrix element 0zs P of the magnetic moment

operator, connecting the ground state 0 with the excited state s of energy 0sE E' above the

ground state. Then by standard perturbation theory the perturbed ground state in a weak field

( z HP ' ) has a moment 2

2 00 ' 0 ' zz

H s PP #'

and the upper state has a moment 2

2 0' ' zz

H ss s

PP # '

There are two interesting cases to consider:


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Case (a). Bk T' . The surplus population in the ground state over the excited state is

approximately equal to / 2 BN k T' , so that the resultant magnetization is 2

2 02

z

B

H s NMk T

P ' '

which gives for the susceptibility 2

0zB

N sMH k T

PF

Here N is the number of molecules per unit volume. This contribution is of the usual Curie form,

although the mechanism of magnetization here is by polarization of the states of the system,

whereas with free spins the mechanism of magnetization is the redistribution of ions among the

spin states. We note that the splitting does not enter in (37).

Case (b). Bk T' !! . Here the population is nearly all in the ground state, so that 2

2 0zNH sMP

'

The susceptibility is


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22 0zN sM

HPF '

independent of temperature. This type of contribution is known as Van Vleck paramagnetism. In

general, ' is crystal field energy, leading to inhomogeneous splitting of d- and f- energy levels. S.No Types examples Susceptibility F 1 Dia-magnetic 1Water(H2O)

1Helium (He) -9.035 x 10-6 vF (SI) -9.85 x 10-10 vF (SI)

2 Para-magnetic (1) Langevin 1Oxygen (O2) @ 300K 3.73 x 10-7 vF (SI)

(2) Pauli

2Vanadium(V)

2Platinum (Pt)

1Aluminum (Al)

5 x 10-6 3 1.cm g

1 x 10-6 3 1.cm g

2.2 x 10-5 vF (SI) (3) Van Vleck 3PrNi5

4Tm3Al5O12 5FeS2

a-axis: 0.038 emu.mol-1

c-axis: 0.082 emu.mol-1

0.6 emu (mol-f.u)-1


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0.2 x 10-6 1.emu g

3 Ferro-magnetic (1) Ferro Iron 3.9 x 106

(2) Antiferro Terbium 9.51x 10-2

(3) Ferri Magnetite, Fe3O4 2500 1 Wikepedia; 2 Kittel; 3 Selected topics in magnetism-Gupta & Multani; 4 Physica B, 329333, 669 (2003);

5 solid state commun. 22, 153, (1977)


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Fig: Temperature dependence of various Paramagnetism & diamagnetism

Fig: Temperature dependence of Tm3Al5O12 (Van Vleck paramagnet shown by dashed line)

Fig: Perfect diamagnetism exhibited by Ba(Fe0.9Co0.1)2As2 superconductor (Tc~22K).


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Fig: Paramagnetism in Bulk & nano Platinum. The bulk Pt is a Pauli paramagnet whereas nano particles of Pt shows Langevin paramagnetism

Fig: Inverse of susceptibility of a ferromagnet (MnCl2), paramagnet (FeSO4)& diamagnet (NaCl)


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Ferromagnetism Vs Superparamagnetism

H=0 H=Ha H=0


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Ferromagnetism: Origin of exchange Consider a simple model with just two electrons which have spatial coordinates r1 and r2 respectively. The wave

function for the joint state can be written as a product of single electron states, so that if the first electron is in

state 1( )a r\ and the second electron is in state 2( )b r\ . Then the joint wave function is 1 2( ) ( )a br r\ \ . For electrons the overall wave function must be antisymmetric so the spin part of the wave function must either

be an antisymmetric singlet state SF (S = 0) in the case of a symmetric spatial state or a symmetric triplet state

TF (S = 1) in the case of an antisymmetric spatial state. Therefore we can write the wave function for the singlet case ES and the triplet case ET as

> @1 2 1 2 1 21( , ) ( ) ( ) ( ) ( )2S a b b a Sr r r r r r\ \ \ \ \ F

> @1 2 1 2 1 21( , ) ( ) ( ) ( ) ( )2T a b b a Tr r r r r r\ \ \ \ \ F

where both the spatial \ and spin parts F of the wave function are included. The energies of the two possible states are

1 2*S S SE H dr dr\ \


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1 2*T T TE H dr dr\ \ > @ > @^ `1 2 1 2 1 2 1 2 1 2 1 21* *( ) *( ) *( ) *( ) * ( ) ( ) ( ) ( )2S S S a b b a S a b b a SE H dr dr r r r r H r r r r dr dr\ \ \ \ \ \ F \ \ \ \ F

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2

*( ) *( ) ( ) ( ) * ( ) * ( ) ( ) ( )12 *( ) * ( ) ( ) ( ) *( ) *( ) ( ) ( ) *

a b a b a b b aS

b a a b b a b a S S

r r H r r r r H r rE

r r H r r r r H r r dr dr

\ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ F F

> @ > @^ `1 2 1 2 1 2 1 2 1 2 1 21* *( ) *( ) *( ) *( ) * ( ) ( ) ( ) ( )2T T T a b b a S a b b a SE H dr dr r r r r H r r r r dr dr\ \ \ \ \ \ F \ \ \ \ F

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2

*( ) *( ) ( ) ( ) *( ) *( ) ( ) ( )12 * ( ) *( ) ( ) ( ) *( ) *( ) ( ) ( ) *

a b a b a b b aT

b a a b b a b a T T

r r H r r r r H r rE

r r H r r r r H r r dr dr

\ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ F F

with the assumption that the spin parts of the wave function SF and TF are normalized. The difference between the two energies is

1 2 2 1 1 22 *( ) *( ) ( ) ( )S T a b a bE E r r H r r dr dr\ \ \ \


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Had there been no exchange, 0S TE E . However due to exchange of particles, 0S TE E z and

the total energy of the system is lowered by an amount 1 2S SJJJG JJG

< , where J is the exchange constant

given as

1 2 2 1 1 22 *( ) *( ) ( ) ( )S T a b a bJ E E r r H r r dr dr\ \ \ \ According to Heisenberg model, the exchange Hamiltonian for a system may be written as

i jS Sex iji j

H Jz

JG JJG< Depending on the system, this exchange may be of various types, such as

(a) Direct exchange

(b) Indirect Exchange

(i) Super exchange

(ii) Double exchange

(iii) Indirect exchange in metals (RKKY)

[Ref: Magnetism in condensed matter: Stephen Blundell]


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Simple Ferromagnet

Simple Antiferromagnet

Ferrimagnet

Canted Antiferromagnet

Paramagnet


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Magnetic domains: Domain structure is a natural consequence of the various contributions to the energyexchange, anisotropy, and demagnetizationof a ferromagnetic body.

.T D K HE E E E : TE -total energy, HE -

magnetization energy (due to external

magnetic field H + internal exchange

energy), DE -self energy of magnetization

in its own field, KE - anisotropy energy.

Domain structures are often more

complicated than our simple examples, but

domain structure always has its origin

in the possibility of lowering the energy

of a system by going from a saturated

configuration with high magnetic energy to

a domain configuration with a lower energy.


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Demagnetization factor: We may understand the origin of domains by considering the structures shown in Fig. 30,

each representing a cross section through a ferromagnetic single crystal. In (a) we have a single

domain; as a consequence of the magnetic poles formed on the surfaces of the crystal this

configuration will have a high value of the magnetic energy. The DE for iron is of the order of 2 610M | erg/cm2. This large magnetic energy may be decreased by introducing a domain

structure, although it may be accomplished only at the expense of increasing the other energy

terms. In (b) the magnetic energy is reduced by roughly one-half by dividing the crystal into two

domains magnetized in opposite directions. There is now, however a boundary or transition region

between the two domains in which the atomic moments are not parallel to each other and

moreover do not lie along easy directions. Hence subdivision into additional linear domains

magnetized alternately parallel and antiparallel to the long edge is likely to occur as long as it will

reduce total energy TE . In (c) with N domains the magnetic energy is reduced to

approximately 1/N of the magnetic energy of (a), because of the reduced spatial extension of the

field.


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Certain domain arrangements make the demagnetization energy zero: an example is fig (d) & (e).

The flux circuit is completed within the crystalthus terms called closure domains, are most likely

to occur if their magnetization lies along an easy direction. Such may be the case for cubic crystals

like iron.

It is the demagnetization energy term that is primarily responsible for the occurrence of domains.

Now, it is to be recalled that the origin of the demagnetization energy is the classical dipole-dipole

interaction, which is very much smaller than the exchange interaction between adjacent atoms.

Hence it may at first appear anomalous that a domain structure rather than a state of uniform

magnetization is usually favoured. The reason is that the dipole-dpole forces are long range,

dropping off slowly with distance, whereas the exchange forces are short range being limited

almost to the nearest neighbours. Thus, overall, a domain structure is to be expected, whereas

over short distances, that is, within one domain, the magnetization is expected to be uniform or

almost uniform.

Therefore, as a general rule, a domain configuration is expected that will reduce, if not remove, the

uncompensated poles on the surface of the specimen. Moreover, it is reasonable to expect that


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uncompensated poles will not occur along domain walls, since they would create a magnetic field,

and hence magnetic energy. The condition for the avoidance of poles on domain walls is that the

component of the magnetization normal to the wall be continuous across the wall. This condition is

fulfilled for fig (d) and (e). Here the domain walls that separate the vertical and closure domains

bisects the 90o angle between the magnetization vectors of these two domains.


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Anisotropy energy: There is energy in a ferromagnetic crystal which directs the magnetization along certain crystallographic axes called directions of easy magnetization. This energy is called

the magnetocrystalline or anisotropy energy. This term arises because of highly directional and

anisotropic nature of d- and f-orbitals.

Cobalt is a hexagonal crystal. The hexagonal c-axis is the direction of easy magnetization at room

temperature, as shown in Fig. 27. The magnetization of the crystal sees the crystal lattice through

orbital overlap of the electrons. In cobalt the anisotropy energy density is given by

2 41 2sin sinKU K KT Tc c

where T is the angle the magnetization makes with the hexagonal axis. At room temperature

1K c 4.1 x 106 erg/cm3 ; 2K c 1.0 x 106 erg/cm3.

Iron is a cubic (BCC) crystal, and the cube edges are the directions of easy magnetization. The

anisotropy energy of iron magnetized in an arbitrary direction with direction cosines 1D , 2D , 3D referred to the cube edges is


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2 2 2 2 2 2 2 2 21 1 2 2 3 3 2 2 1 2 3KU K KD D D D D D D D D At room temperature in iron 1K 4.2 x 105 erg/cm3 and 2K 1.5 x 105 erg/cm3.

Fe [BCC]

Ni [FCC]


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Domain walls: A Bloch wall in a crystal is the transition layer that separates adjacent regions (domains) magnetized in different directions. The entire change in spin direction

between domains does not occur in one discontinuous jump across a single atomic plane, but


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takes place in a gradual way over many atomic planes (Fig. 29). The exchange energy is lower

when the change is distributed over many spins.

This behaviour may be understood by interpreting the Heisenberg equation 11

2N

p pp

U J S S

classically. Here J is the exchange integral and S is the spin quantum number.

Consider N spins each of magnitude S on a line or a ring, with nearest-neighbour spins coupled by

the Heisenberg interaction:


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Here J is the exchange integral and pS= is the angular momentum of the spin at site p. If we

treat the spins pS as classical vectors, then in the ground state (fig 8a) and the exchange energy

of the system is 20 10U NJS .

What is the energy of the first excited state? Consider an excited state with one particular spin

reversed, as in Fig. 8b.

0 1 2 2 3 3 4 4 5 5 62 2 2 2 2

2

2210

U NJ S S S S S S S S S SNJ S S S S SNJS

G G G G G G G G G G

1 1 2 2 3 3 4 4 5 5 6

2 2 2 2 2

2

222

U NJ S S S S S S S S S SNJ S S S S SNJS

G G G G G G G G G G

This increases the energy by 28JS , so that 21 0 8U U JS . But we can form an excitation of much

lower energy if we let all the spins share the reversal, as in Fig. 8c. That is, the two end spins are

oppositely oriented and intermediate spins gradually turning. In the above fog, if we consider first

spin to be at 0 degree and last spin as 180 degree, the intermediate spins will be at, 36, 72, 108,

144 degrees respectively. So the energy of the excited state will be,


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2 1 2 2 3 3 4 4 5 5 62 2 2 2 2

2 2

2

2 (0.81 0.81 0.81 0.81 0.81 )2 4.05 8.1

U NJ S S S S S S S S S S

NJ S S S S SNJS NJS

u

G G G G G G G G G G

Thus, longer the length of spin flip, lower will be the energy.

If a total change of S (180o spin flip) occurs in N equal steps, the angle between neighbouring spins is / NS , and the exchange energy per pair of neighbouring atoms is

2 2( / )exw JS NS . The total exchange energy of a line of N + 1 atoms is 2 2 /exNw JS NS

The wall would thicken without limit were it not for the anisotropy energy, which acts to limit

the width of the transition layer. The spins contained within the wall are largely directed away from

the axes of easy magnetization, so there is an anisotropy energy associated with the wall,

roughly proportional to the wall thickness.


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Consider a wall parallel to the cube face of a simple cubic lattice and separating domains

magnetized in opposite directions. We wish to determine the number N of atomic planes

contained within the wall. The energy per unit area of wall wE is the sum of contributions from

exchange and anisotropy energies:

w ex anisE E E The exchange energy is given approximately by (55) for each line of atoms normal to the plane of

the wall. There are 21/ a such lines per unit area, where a is the lattice constant. Thus 2 2 2/ex JS NaV S

The anisotropy energy is of the order of the anisotropy constant times the thickness Na, or

anis KNaV ; therefore 2 2

2wJS KNa

NaSV (56)

This is a minimum with respect to N when


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2 2

2 20w JS Ka

N N aV Sw w (57)

or 2 2

3JSN

KaS (58)

For order of magnitude, N | 300 in iron. The total wall energy per unit area on our model is

2 2 2 2 3 2 2

2 2 2 2 3

1/2 1/2 1/2 1/2

1/2 1/2

2 2( )

2

wJS JS Ka JSKNa Ka

Na a JS KaJ SK K J S

a aJS Ka

S S SVS

S S

S

(59)

in iron wV | 1 erg/cm2 . Accurate calculation for a 180 wall in a (100) plane gives 2

122wJS K

aV .


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3.magnetism

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