Marder M.P. Condensed Matter Physics

Подождите немного. Документ загружается.

24.

Classical Theories of Magnetism and

Ordering

24.1 Introduction

The ancient Greeks knew of the attraction between lodestone and iron; magnets

receive brief mention from Plato and Aristotle. Curious statements echo back from

antiquity, that "a loadstone rubbed with garlic does not attract iron...." [Gilbert

(1600),

p. 2], but the practical use of lodestone as a magnetic compass is also

quite old, dating at least before 1000 A.D. Unlike electrified amber, a lodestone's

attractive force is permanent, but strangely selective. It attracts iron, but not other

metals such as silver or gold. A magnet split in half produces two weaker magnets.

And "[a]pply a red-hot iron rod to a magnetized needle and the needle stands still,

not turning to the iron; but as soon as the temperature has fallen somewhat it at

once turns to it" [Gilbert (1600), p. 107].

If magnetism is the oldest great mystery of solids, it has also remained one of

the most difficult to explain. One reason is that the origin of magnetism is relent-

lessly quantum mechanical. No solid built of charged particles could have magnetic

properties in a world completely ruled by classical physics. Most magnetic effects

result from quantum-mechanical interactions of electrons with one another, inter-

actions that cannot even be accommodated within the single-electron theory that

has dominated study so far. To make matters even more interesting, magnetic ions

interact with each other over large distances, and their response to external stimuli

tend to be hysteretic and spatially complex. Some of the very simplest questions

one can possibly pose in the subject of magnetism are still unsolved.

24.2 Three Views of Magnetism

24.2.1 From Magnetic Moments

The macroscopic theory of magnetic media can begin with the claim that within

the solid there lives a population of magnetic moments M, whirling loops of charge

producing currents according to

7mag = cVxM. (24.1)

When one defines

H = B-4TTM (24.2)

723

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

724

Chapter 24. Classical Theories of Magnetism and Ordering

and separates y

mag

from other currents, the second of

Eqs.

(20.10b) becomes

Vxß=

mag

-f

/ext

-| Currents are separated into (24.3)

C C C Öt "external" and "magnetic."

= 4^xÄ+^ + I^ (24.4)

c c ot

^VxH =

^ + if. (24.5)

c c ot

The rest of Maxwell's equations remain unchanged, featuring B as before. In a

linear magnetic medium, the magnetic induction B and magnetic field H are related

B = fiH, (24.6)

where \i is the magnetic permeability, and the material is called paramagnetic if

\i >

and diamagnetic is ß < 1. The magnetic susceptibility is defined to be

y = . In general, \ is a tensor. (24.7)

When magnetic response is linear, 47r% =

/U —

1, so paramagnets have x > 0, and

diamagnets have x < 0. Because x is dimensionless in both SI and cgs systems of

units,

one must be warned that the susceptibility in SI is defined to be 4ir greater

than the susceptibility in cgs. Unit conversions provide no warning of this conven-

tion.

24.2.2 From Conductivity

The magnetic dipole density M seems to be a new physical addition to the problem

of electromagnetics, but so long as it is linearly related to B, it can be derived from

a special case of the phenomenology introduced by Eq. (20.11) relating current j

to electric field E. The argument has a weakness, but it is nevertheless interesting

to watch magnetism emerge from this point of view. Working in the space of

Fourier transform variables q and u, divide the electric field into longitudinal and

transverse components according to

= ^-^-, È

= Ê-È

. (24.8)

Then a medium with magnetic permeability ß is one in which the relation between

current and field is

j=^(l--)È

(24.9)

4TTILÜ V [l'

ß j fp' / \ \ — — -> Reexpressed '

real space, using

=>^f = 1— VxVx£ theidentity (24.10)

Ot 4lT V /// Vx Vx£ = VV-£-V

£.

Three Views of Magnetism

725

= — (l--)Vx— From Eq. (20.10b). (24.11)

4ir

n ) dt

-f c ( 1 \ -* — ->

=> j = (1 Vxfß- So) Bo is a time-independent field. (24.12)

47T

[I'

, \\- - 1 3D FromEq. (20.10b), and setting

=>Vxß= 1 JVxßH — B

to zero; the need to set B

to (24.13)

^ /i ' C Ot zero is the technical flaw in this

derivation.

- B -* -* 1 ÖD

=»Vx- = Vx# = - —. (24.14)

// c o?

24.2.3 From a Free Energy

The weakness in deriving Eq. (24.14) from Eq. (24.8) appears in Eq. (24.12); a

time-independent magnetic field Bo arises, and it must arbitrarily be set to zero

to obtain the conventional form of magnetic response. In addition, the derivation

assumes a linear relation between B and H. The goals of a proper thermodynamic

account of magnetism are first to remove this ambiguity, and second to deal with

the fact that many important magnetic phenomena are not linear. Without placing

magnetism on such a foundation, it is not possible to describe some of the more in-

teresting forms of magnetic response such as formation of ferromagnetic domains,

or superconductivity.

Begin by supposing the existence of an energy functional

£{ß(r)} (24.15)

that gives the energy of a magnetic system when the magnetic induction B(r) is

specified. In addition to isolated systems, it is often useful to consider a system

such as the one sketched in Figure 24.1. A sample is placed within a large box,

surrounded by current-carrying wires that constitute its only contact with the ex-

ternal world. The contents of the box do work on the outside world only if an

electric field E arises. If so, the field does work at a rate J dr E

■

ext

;

equivalently,

the external world does work on the box at the rate

p The integral is taken over the box containing

/ d7 F(7)

■

7 (~r) k"

sam

ar,

d wires. The box needs to be (1A

ft)

I \ J jextV )■ i

enough that at its outer edges the inter- ^ ' '

action between wires and sample has become

very small.

Figure 24.1. A sample is placed within a

box and influenced by the outside world only

through currents y

ext

that are created by wires

passing into the box and that are physically

separated from the sample.

726

Chapter 24. Classical Theories of Magnetism and Ordering

Supposing, however, that energy

£ as a

functional

induction

B in

the box

known, the change in energy can be computed in another way. Define

H(r)

4-7T—zr—-.

See

Appendix

B for the

definition

func-

(24.17)

ÖB(r) tional derivatives.

Then for any small change in B(r), the change in energy is

- -.

= — / dr

H(r)-SB(r). There is no need to assume that H is a linear (24.18)

4-7T

function of B.

In particular,

fdrHVxÈ (24.20)

4-7T

A-K

4TT

- - - - - - 1

Use V

■

x ß)

[£-Vx//-V-(//x£)j

=(VXÂ)-B-(VXB)-A.

(24.21)

The second term of

Eq.

(24.21) can be converted

■

to a

sur

ace

integral

energy radiated

out of

the

/94 22^

box.

For

large enough

box, and

slow enough

^ ' '

Comparing Eqs. (24.22) and (24.16) and requiring them to be equal for arbitrary

fields E imposes on

{B} the constraint

4-7T

H{7)

— ;

ext

. (24.23)

Although H is produced by the external currents, it cannot be computed by pretend-

ing the sample to be altogether absent and finding

the

field

that would be produced

without it. The sample imposes boundary conditions on this equation, and the

solution depends upon them.

From this starting point the rest of magnetostatics can be recovered. The mag-

netization is defined by

M{r)

= ~{B{r)-H{r))-

(24.24)

for linear materials the permeability and susceptibility can be defined as before.

Equilibrium States. An isolated system in equilibrium at temperature

and with

specified induction B(r) will choose a state that minimizes

5(T,B)

E(B)-TS.

(24.25)

Using Eq. (24.24), changes in the free energy can be written

ÖJ = -SOT +

drH{?)- 5M{r)

+ ^- j

dr SH

(r). (24.26)

Magnetic Dipole Moments

727

If the system is connected to the world by external currents, as in Figure 24.1,

is no longer the appropriate thermodynamic potential. By altering its magnetiza-

tion, the system can extract energy from the sources that maintain the current

e]it

fixed. Under these circumstances the thermodynamic potential to minimize is not

but instead S, where

3 = 3"-— /drB{r)-H{7). (24.27)

47T J

A brief calculation to verify this claim is the subject of Problem 3. Small changes

in external currents produce changes in S

63 = -— I drB(r)-SH(r) (24.28)

47T J

= - f drM{r)-6H(r)-^- j dr H{r)

■

SH(r). (24.29)

When the field H is employed as an independent variable, then thermodynamic

equilibrium is given also by minima of

S = S + ^- / drH

{r) (24.30)

Φ7T J

for which

ÖS

=-SOT-

dfM-SH.

(24.31 )

The thermodynamic potential 9, which describes an ensemble at fixed temperature

T and field H, is a common starting point for studies of magnetic systems.

B versus H. A source of confusion is the question of when to employ B and when

to employ H. There are two competing considerations. As indicated in Eq. (24.23),

the source for H is external currents. Experiments that control external currents

therefore control H more directly than B, and they are frequently reported in terms

of H. In particular, for a long thin cylindrical sample with its axis pointing along

Eq. (24.23) implies that no matter how the sample responds to B, the field H is

constant everywhere in space, because the component of H tangential to the sur-

face is continuous, and the normal component vanishes everywhere except at the

ends.

It is natural in this case to speak of H as the experimentally applied field. On

the other hand, the microscopic field perceived by nuclei or localized electrons is

clearly B. Thus experimental results should feature H, while microscopic Hamil-

tonians should employ B. The fact that this convention is not standard is probably

due to the fact that the magnetic permeability of many metals differs from

only

by parts in 10

(Table 25.4), so that in these materials B and H are equal for all

practical purposes.

24.3 Magnetic Dipole Moments

The magnetic dipole moment density M is really just a convenient way of viewing

closed loops of current, but it is worth exploring the properties of these dipoles a

728

Chapter 24. Classical Theories of Magnetism and Ordering

bit further to find their energy in external magnetic fields. The magnetic dipole

moment of a current distribution can be defined as

f 1

ffl = I d~p—f x j(7). In order for the magnetic moment to be inde- (24.32)

J 2c pendent of the choice of origin, the integral

of the current over space must vanish.

The Lorentz force on any current distribution is

p = - j dr J(r) x B{r). (24.33)

If the current distribution j forms a closed loop and then vanishes except in the

vicinity of some point, say the origin, then it makes sense to expand the magnetic

induction B as a Taylor series about that point, obtaining

F=-

[ dr J(r) x [5(0) + (r• V)fl(0) +. . .] (24.34)

0 + (ffl X V) X B

See

Problem 1 to verify this claim. (24.35)

= V(m

•

See

Problem 1 to verify this claim. (24.36)

=4> [J = —ffi. B. U is the potential energy of the dipole in an (24.37)

external field.

According to Eq. (24.37), magnetic dipoles should tend to align along external

fields to minimize their energy, and solids should all be paramagnetic. However,

Table 24.1. Susceptibilities of insulating elements near 290 K

Element

(10~

mole~

-19.18

-5.24

-6.70

-5.88

-20.18

-7.99

-4.00

-1.88

-45.68

-28.49

Element

(10

-6

mole"

)

-12.04

-7.02

-26.63

-15.39

-23.69

-3.09

-37.00

-43.42

-43.33

AU entries are diamagnetic. The data are in cgs; susceptibilities

in SI are defined to be 4n times greater. The recorded values are

obtained by dividing the dimensionless susceptibility x by moles

per cm

. Source: Landolt and Börnstein (1959), vol. 2, part 9 and

Grigoriev and Meilkhov (1997).

Magnetic Dipole Moments 729

the calculation assumes that somehow the current j flowing through the dipole

remains constant as the dipole orients itself in an external field, and it neglects the

work involved in maintaining the current.

In fact, a free charge circling in a magnetic field is diamagnetic. A positively

charged particle under the influence of a magnetic field in the positive z direction

circles in a clockwise direction, and according to Eq. (24.32), or the right-hand

rule,

it produces a magnetic moment in the negative z direction.

Both diamagnetism and paramagnetism are possible because some magnetic

moments really are permanent, while others arise only in response to applied fields.

The internal current loops of electrons are fixed by quantum mechanics, and they

cannot be altered by external fields. Therefore electrons orient their intrinsic mag-

netic moments in the direction of applied inductions and enhance them, producing

paramagnetic behavior with fi > 1. However, placing electrons in a box and con-

sidering the magnetic contributions due to their circular orbits in the external fields

tends to lead to diamagnetic behavior,

< 1. Examples of diamagnetic materials

appear in Table 24.1.

Figure 24.2. Schematic view of

Faraday

balance.

cylindrical sam-

ple is placed within a coil whose

windings increase in density to pro-

duce a magnetic induction with

both constant and linearly increas-

ing components. The resulting force

on the sample is measured as a

change in its weight.

Experimental Phenomena. Measuring the magnetic properties of materials usu-

ally comes down to the ability to make an accurate measurement of magnetic mo-

ments, either those that arise in response to an external applied field or those that

arise spontaneously. One classic technique is the

Faraday

balance,

shown in Fig-

ure 24.2. A long cylindrical sample is placed inside a coil where the density of

the windings increases linearly in the vertical direction, so that when current flows

through the coil it produces a magnetic induction of the form

{z) = Bn

~\~

zB\. The component of the magnetic field in the (24.38)

vertical direction increases linearly in the ver-

tical direction.

The constant part of

the

induction,

Bo,

induces a magnetic moment m in the sample,

while the gradient in the induction, dB

/dz = B\, produces a force mB\, according

to Eq. (24.36).

730

Chapter 24. Classical Theories of Magnetism and Ordering

24.3.1 Spontaneous Magnetization of Ferromagnets

So long as the response of a solid to magnetic fields remains linear, magnetism

is captured by Eq. (24.14). Most condensed matter specimens are either diamag-

netic or paramagnetic, particularly at elevated temperature. However, the study of

magnetism gains new interest when it begins to focus upon a set of spectacular

phenomena in which magnetic dipoles take on a life of their own and cannot be

described by any function of the instantaneous magnetic field, let alone a linear

one.

Below a critical temperature T

of 1042 K, iron develops a spontaneous mag-

netization, which rises continuously to a maximum value at very low temperatures.

This magnetization can be measured by conventional means, such as the Faraday

balance, but it is also possible to measure directly the intense magnetic fields seen

at each iron nucleus that are a signature of the magnetic order. Two separate ex-

perimental techniques are employed. The first is the Mössbauer effect, and data of

this type from which the internal magnetic fields were extracted appeared in Figure

13.20.

The second is nuclear magnetic resonance (NMR), the analog of electron

spin resonance discussed in Section

22.4.3.

The magnetic field at the nucleus is de-

duced by using the known magnetic moment m/ of the nucleus and then searching

for resonant absorption of oscillatory radiation when the nucleus is driven between

two spin states in a static magnetic field. Because the mass of nuclei is much larger

than the mass of electrons, the characteristic frequency for resonant absorption is

in the megahertz for nuclei, as opposed to gigahertz for electrons, but the principle

of the two resonant absorption experiments is the same. Resulting measurements

of the internal field in iron versus temperature appear in Figure 24.3.

The natural unit for discussing magnetic moments is the Bohr magneton

= ^, (24.39)

2mc

whose value is

9.27-10-

ergG

. (24.40)

Figure 24.3. Internal magnetic

fields in iron, with temperature mea-

sured in terms of the Curie tem-

perature T

— 1042 K, and internal

fields in terms of their room temper-

ature value

HQ.

The magnetization

falls off in a similar but not identical

fashion, disappearing at the same

temperature. [Source: Preston et al.

(1962).]

Temperature T/T

21,

= 9.27-10

_zl

cmesu

Magnetic Dipole Moments 731

Magnetic moments for selected elements and compounds appear in Table 24.2.

Ferromagnets exhibit a specific heat peak in the vicinity of the critical tempera-

ture T

where ferromagnetism first appears, called the Curie temperature and shown

in Figure 24.4(A). This peak is characteristic of second-order phase transitions, and

will be discussed further in Section 24.6. Above the transition temperature, they

are paramagnets with magnetic susceptibility, as shown in Figure 24.4(B) obeying

X^Y~Q

]

(24.41)

0 is sometimes called the Curie-Weiss temperature, and is intended to provide an

accurate description of susceptibility at high temperatures. It does not necessarily

coincide with the Curie temperature, although the two are always of the same order

of magnitude.

.i.i.i. i .

50 100 150 200 250 300

Temperature T (K)

Figure 24.4. (A) Specific heat of iron attributed to magnetic effects in the vicinity of the

ferromagnetic transition temperature. [Source: Hofmann et al. (1956) p. 53.] (B) Magnetic

susceptibility \ °f

EuO.

l/x is plotted and is linear, showing that \

l/(^

—

), and the

Curie-Weiss and Curie temperatures coincide. Source: Matthias et al. (1961), p. 160.]

Iron's neighbors in the periodic table, nickel and cobalt, are also ferromagnetic.

The outermost electrons of these elements are two As electrons forming a conduc-

tion band. Right below them in energy is an incomplete d shell, the electrons

of which group together to produce a large magnetic moment. The interatomic

interactions are large enough that the d electrons form a narrow band of mobile

electrons, but simultaneously they display features of localized moments. Sections

26.4 and 26.7 will introduce some of the theoretical models used in such cases.

Several of the rare earths are also ferromagnetic, but the localization of the

associated moments is more complete, so the difficulties in understanding these

elements are less severe than it is for the transition metals.

24.3.2 Ferrimagnets

While the magnetization of iron drops smoothly and continuously to zero as tem-

perature T approaches T

from below, some rare earth magnets have a more compli-

1 10

C/5

(A)

500 1000 1500 2000 (

Temperature T (K) (B

732 Chapter 24. Classical Theories of Magnetism

and

Ordering

Table 24.2. Properties

magnetically ordered materials

Compound

9 mi Compound T

(K) (K) (fjL

) (K) M

CoO

CuO

MnO

NiO

312

291

230

100

122

523

23.9

1394

289

1043

302

628

-330

-745

-610

-2470

1415

157

108

1100

289

650

0.59

3.8

0.5

1.72

10.65

7.12

2.2

7.97

10.9

0.6

10.9

FeFe2Û4

(magnetite)

FeNiFe0

FeLiFeC>4

FeCuFe0

FeCoFe0

858

943

728

793

4.1

2.3

2.6

1.3

3.7

The symbols

a and f

refer

antiferromagnets

and

ferromagnets,

and the

ferrites

la-

beled

fi are

ferrimagnetic.

refers

magnetic moment

per

atom

and is

measured

in units

of the

Bohr magneton,

eh/2mc

[Eq.

(24.39)]

at 0 K. The

rare earth

ferromagnets

do not

become paramagnets

their ferromagnetic Curie temperature,

but instead make

transition

antiferromagnetism, and become paramagnets above

a second critical temperature. Source: Landolt

and

Bernstein (1959), vol.

part

Slick (1980), and Grigoriev and Meilkhov (1997).

cated behavior, shown

Figure 24.5.

At a

compensation temperature,

the

magne-

tization reaches zero,

but it

passes through this value

and out the

other side before

eventually reaching zero

for

good

at T

. The

interpretation

this behavior

that

unit cells contain magnetic moments pointing

competing directions,

and

tending

to diminish

temperature increases

different rates. This view

confirmed

ex-

plicitly

neutron scattering. Materials

this type

are

called ferrimagnets;

at the

compensation temperature

the two

moments exactly balance.

The magnets known through

the

centuries such

magnetite (lodestone)

are

of this type.

The

thermodynamics

ferrimagnets

different from that

ferro-

magnets.

In the

high-temperature paramagnetic phase,

the

magnetic susceptibility

is observed

behave

■y

= . That is, ferrimagnets are distinguished by a (24.42)

T + 101 negative Curie-Weiss temperature.

The idea

compensating magnetic moments within each unit cell

is due to

Néel (1948),

and the

magnetic transition temperature

ferrimagnets

called

the Néel temperature.