Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications

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BookID 160928 ChapID 09 Proof# 1 - 29/07/09

9.9 Ferromagnetism 271

atoms per unit volume is written as M = Ng

(βg

BJ), where the 445

function B

(x) is called the Brillouin function. If the magnetic ﬁeld B is 446

small compared to 500 T at room temperature, M becomes 447

M 

J(J +1)

and we obtain the Curie’s law for the paramagnetic susceptibility:

448

Para





at high temperature, (g

BJ  k

T ). 449

In the presence of the magnetic ﬁeld B, the number of electrons of spin 450

up (or down) per unit volume is 451



∞

dEf

(E)g (E ∓ μ

B) .

For ζ  μ

B and k

T  ζ, the magnetization M(= μ

−

− n

)) reduces 452

to 453

M  μ



g(ζ)+

T )



(ζ)



with ζ = ζ

−

T )



(ζ

)

g(ζ

)

. Since g(ζ)=





1/2

, we obtain the 454

(quantum mechanical) expression 455

2ζ



1 −





+ ···



for the Pauli spin (paramagnetic) susceptibility of a metal.

456

In quantum mechanics, a dc magnetic ﬁeld can alter the distribution of 457

the electronic energy levels and the orbital states of an electron are described 458

by the eigenfunctions and eigenvalues given by 459

|nk

 = L

−1

y+ik



x +

¯hk

mω



; E

¯h

+¯hω



n +



The quantum mechanical (Landau) diamagnetic susceptibility of a metal

460

becomes 461

= −

2ζ



e¯h

∗



= −

2ζ



∗



Appearance of m

∗

(not m) indicates that the diamagnetism is associated with 462

the orbital motion of the electrons. 463

In a metal, as we increase B, the Landau level at k

= 0 passes through the 464

Fermi energy ζ and the internal energy abruptly decreases. Many physically 465

observable properties of the system are periodic functions of the magnetic 466

ﬁeld. The periodic oscillation of the diamagnetic susceptibility of a metal at 467

low temperatures is known as the de Haas–van Alphen eﬀect. Oscillations 468

in electrical conductivity are called the Shubnikov–de Haas oscillations. 469

Uncorrected Proof

BookID 160928 ChapID 10 Proof# 1 - 29/07/09

10 3

Magnetic Ordering and Spin Waves 4

10.1 Ferromagnetism 5

10.1.1 Heisenberg Exchange Interaction 6

The origin of the Weiss eﬀective ﬁeld is found in the exchange ﬁeld between 7

the interacting electrons on diﬀerent atoms. For simplicity, assume that atoms 8

A and B are neighbors and that each atom has one electron. Let Ψ

and Ψ

be the wave functions of the electron on atoms A and B, respectively. The 10

Pauli principle requires that the wave function for the pair of electrons be 11

antisymmetric. If we label the two indistinguishable electrons 1 and 2, this 12

means 13

Ψ(1, 2) = −Ψ(2, 1). (10.1)

The wave function for an electron has a spatial part and a spin part. Let

i↑

and η

i↓

be the spin eigenfunctions for electron i in spin up and spin down 15

states, respectively. There are two possible ways of obtaining an antisymmetric 16

wave function for the pair (1,2). 17

=Φ

)χ

(1, 2) (10.2)

=Φ

)χ

(1, 2). (10.3)

The wave function Ψ

has a symmetric space part and an antisymmetric 18

spin part, and the wave function Ψ

has an antisymmetric space part and 19

a symmetric spin part. In (10.2) and (10.3), the space parts are 20

√

[Ψ

(1)Ψ

(2) ± Ψ

(2)Ψ

(1)] , (10.4)

and χ

are the spin wave functions for the singlet (s = 0) spin state 21

(which is antisymmetric) and for the triplet (s = 1) spin state (which is 22

symmetric). 23

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BookID 160928 ChapID 10 Proof# 1 - 29/07/09

276 10 Magnetic Ordering and Spin Waves

(1, 2) =

√

[η

1↑

2↓

− η

1↓

2↑

]; S

=0 (10.5)

(1, 2) =

⎧

⎨

⎩

1↑

2↑

√

[η

1↑

2↓

+ η

1↓

2↑

] S

1↓

2↓

= −1

(10.6)

If we consider the electron–electron interaction

V =

, (10.7)

we can evaluate the expectation value of V in state Ψ

or in state Ψ

.Since 25

V is independent of spin it is simple enough to see that 26

Ψ

|V |Ψ

 = Φ

|V |Φ

 (10.8)

= Ψ

(1)Ψ

(2) |V |Ψ

(1)Ψ

(2) + Ψ

(1)Ψ

(2) |V |Ψ

(2)Ψ

(1).

When we do the same for Ψ

,weobtain 27

Ψ

|V |Ψ

 = Φ

|V |Φ

 (10.9)

= Ψ

(1)Ψ

(2) |V |Ψ

(1)Ψ

(2)−Ψ

(1)Ψ

(2) |V |Ψ

(2)Ψ

(1).

The two terms are called the direct and exchange terms and labeled V

and J, 28

respectively. Thus, the expectation value of the Coulomb interaction between 29

electrons is given by 30

V  =

+ J for the singlet state (S =0)

−J for the triplet state (S =1)

(10.10)

Now, S =

and S

)

. Therefore, 32

(

)

−

S(S +1)−

. Here, we have used 33

the fact that the operator S

has eigenvalues S(S +1) and

and

have 34

eigenvalues

(

+1)=

. Thus, one can write 35

(

−

if S =0

if S =1.

Then, we write

V  = V

+ J



1 − S



= V

−

J−2J

. (10.11)

Here, −2J

denotes the contribution to the energy from a pair of atoms 37

(or ions) located at sites 1 and 2. For a large number of atoms we need only 38

sum over all pairs to get 39

E = constant −



i=j

(10.12)

Uncorrected Proof

BookID 160928 ChapID 10 Proof# 1 - 29/07/09

10.1 Ferromagnetism 277

Normally, one assumes that J

is nonzero only for nearest neighbors and 40

perhaps next nearest neighbors. The factor

is introduced in order to avoid 41

double counting of the interaction. The introduction of the interaction term 42

−2J

is the source of the Weiss internal ﬁeld which produces ferromag- 43

netism. If z is the number of nearest neighbors of each atom i, then for atom 44

i we have 45

= −2JzS

= −g

(10.13)

10.1.2 Spontaneous Magnetization 47

From our study of paramagnetism we know that 48

M = Ng

(x), (10.14)

where x =

Local

. Here, B

Local

is B +λM , i.e. it includes the Weiss ﬁeld. 49

If we plot M vs. x, we get the behavior shown in Fig. 10.1. But for B =0, 50

Local

= λM. Therefore, x =

SλM

. If we plot this straight line x vs. M 51

on the panel of Fig. 10.1 for diﬀerent temperatures T we ﬁnd the behavior 52

shown in Fig. 10.2. Solutions (intersections) occur only at (M =0,x =0)for 53

T>T

.ForT<T

there is a solution at some nonzero value of M, i.e. 54

Fig. 10.1. Schematic plot of the magnetization M of a paramagnet as a function

of the dimensionless parameter x deﬁned by x =

Local

Fig. 10.2. Schematic plot of the magnetization M of a paramagnet for various

diﬀerent temperatures, in the absence of an external magnetic ﬁeld, as a function of

the dimensionless parameter x deﬁned by x =

Local

Uncorrected Proof

BookID 160928 ChapID 10 Proof# 1 - 29/07/09

278 10 Magnetic Ordering and Spin Waves

(T )=Ng

),x

SλM

The Curie temperature T

is the temperature, at which the gradient of the 55

line M =

Sλ

and the curve M = Ng

(x) are equal at the origin. 56

Recall that, for small x, B

(x)=

(S+1)x

+ O(x

). Then the T

is given by 57

λN





S(S +1)



. (10.15)

It is not diﬃcult to see that M

(T )vs.T looks like Fig. 10.3. 59

If a ﬁnite external magnetic ﬁeld B

is applied, then we have 60

M =

Sλ

−

. (10.16)

Plotting this straight line on the M–x plane gives the behavior shown in

Fig. 10.4. 62

Fig. 10.3. Schematic plot of the spontaneous magnetization M

as a function of

temperature T

Fig. 10.4. Schematic plot of the magnetization M of a paramagnet, in the presence

of an external magnetic ﬁeld B

, as a function of the dimensionless parameter x

deﬁned by x =

Local

Uncorrected Proof

BookID 160928 ChapID 10 Proof# 1 - 29/07/09

10.1 Ferromagnetism 279

10.1.3 Domain Structure 63

If all the magnetic moments in a ﬁnite sample are lined up, then there will 64

be ﬂux emerging from the sample as shown in Fig. 10.5. There is an energy 65

density

8π

H(r)·B(r) associated with this ﬂux emerging from the sample, and 66

the total emergence energy is given by 67

U =

8π



r H(r) · B(r) (10.17)

The emergence energy can be lowered by introducing a domain structures

as shown in Fig. 10.6. To have more than a single domain, one must have a 69

domain wall, and the domain wall has a positive energy per unit area. 70

Fig. 10.5. Schematic plot of the magnetic ﬂux around a sample with a single domain

of ﬁnite spontaneous magnetization

Fig. 10.6. Domain structures in a sample with ﬁnite spontaneous magnetization.

(a) Pair of domains, (b) domains of closure

Uncorrected Proof

BookID 160928 ChapID 10 Proof# 1 - 29/07/09

280 10 Magnetic Ordering and Spin Waves

Fig. 10.7. A chain of magnetic spins interacting via Heisenberg exchange interac-

tion. (a) Single domain, (b) a domain wall, (c) gradual spin ﬂip

10.1.4 Domain Wall 71

Consider a chain of magnetic spins (Fig. 10.7a) interacting via Heisenberg 72

exchange interaction 73

= −2J



i,j

·s

where the sum is over all pairs of nearest neighbors. Compare the energy

of this conﬁguration with that having an abrupt domain wall as shown in 75

Fig. 10.7b. Only spins (i)and(j) have a misalignment so that 76

ΔE = H

(i ↑,j ↓) − H

(i ↑,j ↑)

= −2J(

)(−

) −



−2J(

)(

)



= J.

(10.18)

Energetically it is more favorable to have the spin ﬂip gradually as shown

in Fig. 10.7c. If we assume the angle between each neighboring pair in the 78

domain wall is Φ, we can write 79

)

= −2Js

·s

= −2Js

cos Φ. (10.19)

Now, if the spin turns through an angle Φ

(Φ

= π in the case shown in 80

Fig. 10.7b) in N steps, where N is large, then Φ



within the domain 81

wall, and we can approximate cos Φ

by cos Φ

≈ 1 −

. Therefore, the 82

exchange energy for a neighboring spin pair will be 83

)

= −2JS



1 −



(10.20)

The increase in exchange energy due to the domain wall will be

= N





= JS

. (10.21)

Clearly, the exchange energy is lower if the domain wall is very wide. In fact, 86

if no other energies were involved, the domain wall width Na (where a is the 87

atomic spacing) would be inﬁnite. However, there is another energy involved, 88

the anisotropy energy. Let us consider it next. 89

Uncorrected Proof

BookID 160928 ChapID 10 Proof# 1 - 29/07/09

10.1 Ferromagnetism 281

10.1.5 Anisotropy Energy 90

We realize that crystals are not spherically symmetric, but have ﬁnite point 91

group symmetry. In real crystals, certain directions are easy to magnetize 92

and others are hard. For example, Co is a hexagonal crystal. It is easy to 93

magnetize Co along the hexagonal axis, but hard to magnetize it along any axis 94

perpendicular to the hexagonal axis. The excess energy needed to magnetize 95

the crystal in a direction that makes an angle θ with the hexagonal axis can 96

be written 97

= K

sin

θ + K

sin

θ>0. (10.22)

For Fe, a cubic crystal, the 100 directions are easy axes and the 111 direc-

tions are hard. The anisotropy energy must reﬂect the cubic symmetry of the 99

lattice. If we deﬁne α

=cosθ

as shown in Fig. 10.8, then an approximation 100

to the anisotropy energy can be written 101

≈ K



+ α



+ K

. (10.23)

The constants K

and K

in (10.22) and (10.23) are called anisotropy 102

constants. They are very roughly of the order of 10

erg/cm

. 103

Fig. 10.8. Orientation of the magnetization with respect to the crystal axes in a

cubic lattice

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BookID 160928 ChapID 10 Proof# 1 - 29/07/09

282 10 Magnetic Ordering and Spin Waves

Fig. 10.9. Rotation of the magnetization in a domain wall

Clearly, if we make a domain wall, we must rotate the magnetization away 104

from one easy direction and into another easy direction (see, for example, Fig. 105

10.9). To get an order of magnitude estimate of the domain wall thickness 106

we can write the energy per unit surface area as the sum of the exchange 107

contribution σ

and the anisotropy contribution σ

108

, (10.24)

where a is the atomic spacing. The anisotropy energy will be proportional to

109

the anisotropy constant (energy per unit volume) times the number of spins 110

times a. 111

≈ KNa  10

− 10

J/m

. (10.25)

Thus the total energy per unit area will be

112

σ =

+ KNa. (10.26)

113

The σ has a minimum as a function of N, since the exchange part wants N 114

to be very large and the anisotropy part wants it very small. At the minimum 115

we have 116

N 





1/2

≈ 300. (10.27)

The width of the domain wall is δ = Na  πS(

)

1/2

, and the energy per 117

unit area of the domain wall is σ  2πS(

)

1/2

. 118

10.2 Antiferromagnetism 119

For a Heisenberg ferromagnet we had an interaction Hamiltonian given by 120

H = −2J



i,j

· s

, (10.28)

and the exchange constant J was positive. This made s

and s

align parallel 121

to one another so that the energy was minimized. It is not uncommon to have 122

Uncorrected Proof

BookID 160928 ChapID 10 Proof# 1 - 29/07/09

10.4 Zero-Temperature Heisenberg Ferromagnet 283

Fig. 10.10. Sublattice structure of spins in a ferrimagnet

spin systems in which J is negative. Then the Hamiltonian 123

H =2|J|



i,j

·s

(10.29)

will attempt to align the neighboring spins antiparallel. Materials with J < 0

124

are called antiferromagnets. 125

For a antiferromagnet, the magnetic susceptibility increases as the tem- 126

perature increases up to the transition temperature T

|J|

,knownasthe 127

N´eel temperature.AboveT

, the antiferromagnetic crystal is in the standard 128

paramagnetic state. 129

10.3 Ferrimagnetism 130

In an antiferromagnet we can think of two diﬀerent sublattices as shown in 131

Fig. 10.10. If the two sublattices happened to have a diﬀerent spin on each 132

(e.g. up sublattice has s =

, down sublattice has s = 1), then instead of an 133

antiferromagnet for J<0, we have a ferrimagnet. 134

10.4 Zero-Temperature Heisenberg Ferromagnet 135

In the presence of an applied magnetic ﬁeld B

oriented in the z-direction, 136

the Hamiltonian of a Heisenberg ferromagnet can be written 137

H = −



i,j

J(R

− R

· S

− gμ



. (10.30)

Here, we take the usual practice that the symbol S

represents the total angu- 138

lar momentum of the ith ion and is parallel to the magnetic moment of the 139