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Fundamentals of the Physics of Solids / 14-Magnetically Ordered Systems

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14

Magnetically Ordered Systems

A particularly interesting class of crystalline solids is constituted by those materials that exhibit regularity not only in the spatial arrangement of atoms but also in the alignment of their magnetic moments. It was shown in Chapter 3 that in a stand-alone atom or ion the orbital angular momentum L and the spin S of electrons on incomplete shells give rise to a magnetic moment −gJμBJ , where J is the dimensionless total angular momentum. Since the d- and f -electrons of ionic cores are usually localized in solids, too, these atomic moments are observed in the crystalline phase as well – although the crystalline field due to neighboring atoms may split the multiply degenerate ground state of the free ion, and therefore the moment may be modified. When the interaction between magnetic moments – namely, the quantum mechanical exchange interaction – is su ciently strong on the scale of thermal energies, the magnetic moments of neighboring atoms may mutually align each other in some direction. It may occur that, in contrast to paramagnets, the orientation of each spin is rigidly fixed, however no correlation is observed in the orientation of magnetic moments, and the correlation function of spins drops o rapidly with distance. Since this static disorder is similar to the situation encountered in glasses, where atomic positions show a similar disorder, such systems are referred to as spin glasses. We shall analyze the properties of such materials in Chapter 36 of Volume 3. In the present chapter we shall be concerned with materials in which magnetic moments are aligned in some – usually crystallographically determined, high-symmetry – direction in such a way that their orientation shows long-range correlations. Such structures are said to be magnetically ordered.

First we shall get acquainted with the magnetic structure of ferromagnets, antiferromagnets, and ferrimagnets, which are the most common families of magnetically ordered materials. Then we shall examine the interactions that are responsible for magnetic ordering. At su ciently high temperatures thermal fluctuations disrupt this order; the mechanism is similar to the melting of crystal lattices. First the simplest description provided by the mean-field theory will be presented, and then the behavior around the critical point will be

450 14 Magnetically Ordered Systems

examined, using a phenomenological approach. After a brief discussion of the role of anisotropy in magnetic ordering we shall survey some properties of the domain structure, which plays an important role in technical applications of magnets. The quantum mechanical treatment of the behavior of magnetically ordered systems will be the subject of the next chapter.

14.1 Magnetic Materials

Atomic magnetic moments in paramagnetic materials become partially ordered upon the application of an external magnetic field. In their crystalline state the majority of the elements show this behavior. However, in certain 3d metals, rare-earth metals, and their compounds atomic magnetic moments become spontaneously ordered, aligned in a particular direction below a characteristic temperature. Based on the relative orientation of moments, ferromagnetic, antiferromagnetic, and ferrimagnetic materials are distinguished. Below we shall give a brief overview of these three types of magnetic materials, together with their most important properties.

14.1.1 Ferromagnetic Materials

A material is said to be ferromagnetic when the atomic magnetic moments of equal magnitude are aligned in a common direction by the interactions among them.1 In the ground state this is perfectly true, however at higher temperatures alignment in a common direction is valid only in an average sense. Above a critical temperature TC called the Curie temperature the order of the magnetic moments is completely destroyed.

Among transition metals with an incompletely filled 3d shell bcc iron (α- Fe) and fcc nickel become ferromagnetically ordered at relatively high temperatures, so they behave as magnetic materials at room temperature. Cobalt has two ferromagnetically ordered forms: ε-Co, which has an hcp structure and is stable at room temperature, and α-Co, which has an fcc structure and is stable at higher temperatures. Their magnetic transition temperatures (of the cubic structure for cobalt) and their saturation magnetization values are listed in Table 14.1. It is still common practice to express saturation magnetization in CGS units. To convert it to SI values, 1 gauss (CGS) is equivalent (=@ ) to 103 A/m (SI). Comparison of di erent sources is sometimes di cult because some authors use the magnetic polarization J = μ0M , while others prefer the saturation value of the magnetic moment per unit mass (σs).

In addition to the saturation magnetization, we have also given the moment μ per atom, which is calculated from it using Avogadro’s number and the

1In a broader sense a system may be called ferromagnetic even when the ordered moments are not aligned completely parallel. The requirement is that one component should point in the same direction, while the perpendicular components must cancel out.

14.1 Magnetic Materials

451

Table 14.1. The critical temperature TC, saturation magnetization Ms, ordered moment μ, and two parameters that appear in the high-temperature susceptibility: the e ective magneton number pe , and the paramagnetic Curie temperature Θ for the ferromagnetic 3d elements

Element

TC

Ms

μ (T = 0)

pe

Θ

(K)

(103 A/m)

(μB)

(T > Tc)

(K)

Fe

1043

1752

2.226

3.13

1101

Co

1388

1446

1.715

3.15

1415

Ni

627

5210

0.619

1.61

650

 

 

 

 

 

 

molar volume. The reader may realize that the values are di erent from those obtained for free ions. For comparison, we also give the magnetic moment – or more precisely, the e ective magneton number pe obtained from the susceptibility measured in the high-temperature paramagnetic phase. Considering that the saturation moment is |ge|SμB for localized spins of magnitude S, while the e ective moment in the paramagnetic phase is |ge|(S(S + 1))1/2 μB, the closeness of the lowand high-temperature values indicates that in metallic ferromagnets, too, the momentum that is responsible for magnetism comes predominantly from core electrons, rather than conduction electrons – the mobile electrons which are not bound to the ion cores, and which are responsible for metallic conduction. If this were not the case, the atomic magnetic moment would vanish at the same time as magnetization as the system is heated above the Curie temperature. Nevertheless the deviations from the free-ion values indicate that the ferromagnetic character of the elements of the iron group cannot be explained solely in terms of the localized moments of the ion cores: conduction electrons contribute as well. In this chapter we shall deal only with the model of localized spins. The magnetic properties due to mobile electrons will be discussed in Chapters 16 and 33.

Provided the moments are localized, their orientation in a magnetic structure can be represented by arrows drawn at the magnetic atoms. The ferromagnetic structures in simple, face-centered, and body-centered cubic crystals are shown in Fig. 14.1.

Since the moment of each magnetic atom points in the same direction in the ordered state, the translational symmetry of the lattice is not altered by the appearance of magnetic order. The symmetry of the magnetic structure is nonetheless lower than that of the nonmagnetic crystal because magnetic moments single out a preferred direction, and hence some of the rotational or reflection symmetries of the nonmagnetic state may be broken. For example, in cubic crystals magnetic moments may be aligned in the direction of a crystallographic axis. This axis is then no longer equivalent with the two others, and so the initial cubic symmetry is reduced to tetragonal.

452 14 Magnetically Ordered Systems

Fig. 14.1. Orientation of ordered magnetic moments in simple, face-centered, and body-centered cubic crystals

Ferromagnetic materials are found among the metallic and nonmetallic compounds of several elements with an incomplete 3d shell, not only the three listed in Table 14.1. Certain rare-earth metals and some of their compounds also show ferromagnetic properties at low temperatures. A handful of them are shown in Table 14.2, along with the critical temperature of the ferromagnetic phase and the saturation magnetization extrapolated to T = 0. As we shall see, the listed rare-earth metals are not purely ferromagnetic, as magnetic moments are not perfectly aligned in the ground state, but they possess a nonvanishing magnetization.

Table 14.2. Curie temperature and saturation magnetization (in units of 103 A/m) for some ferromagnetic rare-earth metals and their compounds, as well as ferromagnetic compounds containing 3d elements

Element

TC (K)

Ms

Compound

TC (K)

Ms

Compound

TC (K)

Ms

Gd

293

2000

EuO

77

1910

Au2MnAl

200

323

Tb

219

1440

EuS

16.5

1184

Cu2MnAl

630

726

Ho

20

2550

GdCl3

2.2

550

MnBi

620

675

Er

18

 

FeB

598

 

MnAs

670

870

Dy

85

 

Fe2B

1043

 

CrBr3

37

270

As it was mentioned on page 178, in 3d metals the orbital angular momentum is quenched, and so the localized moment comes entirely from spins. Consequently the electron g-factor, ge appears in the moment. In rare-earth ions, where no such quenching occurs, the magnetic moment is calculated from the total angular momentum J and the Landé factor gJ . Measured data for the saturation magnetic moment are usually in excellent agreement with the expected value of the ordered moment for trivalent ions, calculated as gJ μBJ from the data given in Table 3.6.

In each of the above compounds only one component has a nonvanishing magnetic moment. There exists a special group of ferromagnetic materials in

14.1 Magnetic Materials

453

which two di erent types of atom possess a nonzero ordered moment. A few examples are given in Table 14.3.

Table 14.3. Curie temperature and ordered moment (in Bohr magnetons) for some ferromagnetic 3d compounds

Compound

TC (K)

μ

μ

Fe3Pt

400

μFe = 2.7 μB

μPt = 0.5 μB

CoPt

750

μCo = 1.6 μB

μPt = 0.25 μB

CoPt3

290

μCo = 1.64 μB

μPt = 0.26 μB

14.1.2 Antiferromagnetic Materials

It was first suggested by L. Néel2 in 1932 that in addition to ferromagnets with a macroscopic magnetization there may exist another class of materials, in which localized magnetic moments are aligned but the net spontaneous magnetization is nonetheless zero.

This happens because the magnetic moments of half of the atoms point in one direction, while those of the other half in the opposite direction. However, the cancellation of the magnetic moments occurs already on atomic scales, since the atoms with the two spin orientations are arranged in a checkerboardlike pattern; the magnetic moment of an atom is compensated by that of a nearest neighbor. Such materials are called antiferromagnetic. The transition temperatures TN – called Néel temperature – of some antiferromagnetic materials are listed in Table 14.4. Comparison with Table 14.2 reveals that some rare-earth metals have both ferromagnetic and antiferromagnetic phases. In such cases the high-temperature paramagnetic phase is first transformed into an antiferromagnetic phase (the net spontaneous magnetization remains zero), and then, at a lower temperature, into the ferromagnetic phase (a nonzero net spontaneous magnetization appears).

The rare-earth metals listed in the first column crystallize in hexagonal close-packed structure; the oxides of composition AB in sodium chloride structure; the oxides and fluorides of composition ABC3 in perovskite structure; the oxides of composition AB2O4 in spinel structure; the fluorides of composition XF2 in tetragonal rutile structure – and antiferromagnetic materials of even more complex structure abound.

In contrast to ferromagnetic systems, many di erent antiferromagnetic structures are possible even in the simplest crystal lattices, as the requirement of zero net moment can be satisfied in various ways. We saw in Chapter 5 that

2Louis Eugène Félix Néel (1904–2000) was awarded the Nobel prize in 1970 “for [his] fundamental work and discoveries concerning antiferromagnetism and ferrimagnetism which have led to important applications in solid state physics”.

454 14 Magnetically Ordered Systems

Table 14.4. Néel temperature TN for some antiferromagnetic materials

Element

TN (K)

Compound

TN (K)

Compound

TN (K)

Cr

311

MnO

118

FeS

593

Mn

100

FeO

185

MnF2

72

Ce

12.5

CoO

291

FeF2

79

Nd

19.2

NiO

515

CoF2

38

Sm

106

CuO

230

FeF3

394

Eu

90.5

NdFeO3

760

CoF3

460

Dy

178

LaFeO3

750

K2NiF4

97

Ho

132

KMnF3

88

α-Fe2O3

948

Er

84

KNiF3

275

Cr2O3

318

Tb

230

NiCr2O4

65

MnPt

975

Tm

56

GeFe2O4

10

Mn3Pt

485

even when the restriction of collinearity is imposed – i.e., magnetic moments can only point in two opposite directions, up and down, for example –, a whole wealth of magnetic structures can still be found. Without going into details, we shall just present the antiferromagnetic structures that may occur in cubic crystals.

In one of the simplest antiferromagnetic structures occuring in simple cubic crystals with a monatomic basis the magnetic moments on adjacent lattice sites are directed oppositely. This structure, called type-G antiferromagnet, is shown in Fig. 14.2(c). The atoms with upward (downward) pointing spin form a face-centered cubic sublattice. The magnetic structure is therefore made up of two interpenetrating ferromagnetic fcc sublattices of opposite magnetization. The six nearest neighbors of each atom are on the other sublattice. Consequently the magnetic primitive cell is just the primitive cell of the fcc sublattice.

(a) Type A

(b)

Type C

(c)

Type G

Fig. 14.2. Antiferromagnetic structures in simple cubic lattices

14.1 Magnetic Materials

455

Other antiferromagnetic structures are also possible in a simple cubic crystal. Two of them are shown in parts (a) and (b) of Fig. 14.2. In the first case, called type-A antiferromagnet, magnetic moments are all parallel in the base plane but oppositely magnetized planes alternate along the perpendicular direction. The primitive translation vector of this magnetically ordered structure is twice as long in this direction as in the magnetically disordered state. In the second case, called type-C antiferromagnet, atoms with up and down spin are arranged in a checkerboard-like pattern in the base plane, and identical planes are stacked in the perpendicular direction. At first sight the primitive cell would seem to have been doubled in two directions. However, this is not so: the new primitive vectors are along the old face diagonals. The volume of the magnetic primitive cell is thus just twice the volume of the chemical primitive cell.

While in the previous case (part a) moments were aligned ferromagnetically in the planes (100), now (part b) they are aligned in the planes (110). In the perpendicular direction [110] oppositely magnetized planes alternate. Note that in the first case (part c) atomic magnetic moments are aligned ferromagnetically in the planes (111), and the magnetization direction alternates in subsequent planes. Therefore the magnetic moment density shows periodic variations along the direction [111].

Assuming that magnetic moments are strictly localized on atoms, its spatial distribution can be written as

 

 

μ(r) = μ(Rm)δ(r − Rm) ,

(14.1.1)

m

where the sum is over the position of each atom with a magnetic moment. Function μ(Rm) is defined in discrete lattice points, therefore it can be expanded into a Fourier series according to (C.1.49):

1

 

 

μ(Rm) =

 

μq eiq·Rm ,

(14.1.2)

N

 

 

 

q

 

where the sum is over the wave vectors q inside the Brillouin zone. Note that the spatial distribution of the magnetic moment can be characterized by a single Fourier component μq for the antiferromagnetic structures presented above. In the three cases the vectors q are

qA = (π/a)(0, 0, 1) , qC = (π/a)(1, 1, 0) , qG = (π/a)(1, 1, 1) . (14.1.3)

These vectors of the reciprocal space correspond to high-symmetry points of the Brillouin zone: a face center, an edge center, and a vertex – that is, points X, M , and R in Fig. 7.2. Similar conclusions are drawn for the antiferromagnetic structures in bodyand face-centered cubic lattices. This is not accidental – however its explanation requires the Landau theory of second-order phase transitions.

456 14 Magnetically Ordered Systems

Up to now we have always spoken of up and down spins. The actual orientation of the moments with respect to the crystallographic axes – and thereby the possible directions of the vector μq – is determined by crystalline anisotropy. Energetically favorable directions usually correspond to some highsymmetry directions. This is why moments were drawn parallel to one of the crystallographic axes, although they can equally point in the direction of a face or body diagonal. We shall discuss this in more detail later.

(a)

(b)

(c)

Fig. 14.3. Simple antiferromagnetic structures in body-centered cubic lattices

Three simple structures are possible in a body-centered cubic crystal; they are shown in Fig. 14.3. In structure (a) the atomic magnetic moments at the vertices point in the opposite direction as at the body centers. Atoms with upward and downward pointing moments form two interpenetrating simple cubic lattices. The nearest neighbors of each atom are on the opposite sublattice. Ferromagnetically coupled planes are perpendicular to the crystallographic axes. In structure (b) moments are coupled ferromagnetically along one edge of the cube, and antiferromagnetically along the two others. Ferromagnetic planes are now perpendicular to one of the face diagonals. In both magnetic structures the relative orientation of magnetic moments can be specified by a single vector q, namely

q

1

= (2π/a)(0, 0, 1),

and q

2

= (2π/a)( 1

, 1

, 0) .

(14.1.4)

 

 

 

2

2

 

 

This is not the case for structure (c). Here the moment of the atom at a vertex of the cube is antiparallel to the moments of the six second-nearest neighbors, located at adjacent vertices. The relative orientation is characterized by the vector

q

3

= (2π/a)( 1

, 1

, 1 ) .

(14.1.5)

 

2

2

2

 

This means that atomic moments are coupled ferromagnetically within planes (111), however, owing to the complex phase factor, the relative orientation of the moments at body centers with respect to those at vertices is not fixed. To understand the underlying physical reason consider the nearest neighbors of an atom: the moment points upward at four of them and downward at the four others. As the e ects of the nearest neighbors cancel out, the direction of the

14.1 Magnetic Materials

457

moment at the cube center is not specified by the moments at the vertices; instead, it is determined by the interactions with more distant neighbors. When the orientation of the moment at a particular cube center is fixed, then the moments at other cube centers cannot be chosen at will any more: an alternating pattern has to emerge. From a magnetic viewpoint the lattice can be decomposed into four face-centered cubic sublattices. The freedom in the choice of the orientation of the spin at the center manifests itself in the mathematical description as the requirement that both q3 and −q3 must appear in the Fourier representation of the moment density. The phases of the corresponding Fourier components determine the orientation of the atomic moment at the cell center. When the upper sign is chosen in

1

 

ei(q3·Rm ±π/4) + ei(q3 ·Rm ±π/4)

 

μ(Rm) = 2

μ

(14.1.6)

= μ [cos(q3 · Rm) sin(q3 · Rm)] ,

the structure shown in part (c) is obtained; when the lower sign is chosen, the spin at the center points in the opposite direction.

In terms of the primitive vectors (7.2.7) of the reciprocal of the bcc lattice the q are expressed as

q1 = 12 (b1 + b2 − b3) , q2 = 12 b3 , q3 = 14 (b1 + b2 + b3) . (14.1.7)

It is readily seen in Fig. 7.7 that these vectors q are high-symmetry points – H, P , and N – at the boundary of the corresponding Brillouin zone.

The simple magnetic structures that occur in face-centered cubic crystals and their customary notation are shown in Fig. 14.4. For the overwhelming majority of antiferromagnetic materials with this type of Bravais lattice the alignment of the moments corresponds to the structures of the first or second type, although structures of the third and fourth types also exist in nature.

The spatial distribution of magnetic moments can be characterized by a single vector q in type-I and type-II antiferromagnets, while in type-III and type-IV structures both qi and −qi appear, in the form given by (14.1.6). Expressed in terms of the primitive vectors (7.2.12) of the reciprocal lattice, the vectors q characterizing the magnetic structure are

qI = (2π/a)(0, 0, 1) = 12 (b1 + b2) ,

qII = (2π/a)( 12 , 12 , 12 ) = 12 (b1 + b2 + b3) ,

(14.1.8)

qIII = (2π/a)(1, 12 , 0) = 14 (b1 + 2b2 + 3b3) , qIV = (2π/a)( 12 , 12 , 0) = 14 (b1 + b2 + 2b3) .

By way of example, an even more complicated magnetic structure is also shown in Fig. 14.4. This could be called a four-q version of type-II antiferromagnets. It is derived from the simple type-II structure by flipping the atomic

458 14 Magnetically Ordered Systems

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TypeI

 

 

 

 

 

 

Type III

TypeII

 

 

 

 

4q Type II

Type IV

Fig. 14.4. Collinear magnetic structures in face-centered cubic crystals

moments at the face centers. This is possible because nearest-neighbor interactions do not determine the relative orientation of these moments with respect to the moments at the cube vertices. Specification of the structure requires four vectors

q = (2π/a)( 1

,

1

, 1 ) ,

q = (2π/a)( 1

,

1

,

1 ) ,

(14.1.9)

2

 

2

2

2

 

2

 

2

q = (2π/a)( 1

,

 

1

, 1 ) ,

q = (2π/a)(

 

1

,

1 , 1 ) ,

 

2

 

2

2

2

 

2

2

 

and the magnetic moments can be given as

 

 

 

 

 

 

μ(Rm) = 21 μ −eiq·Rm + eiq ·Rm + eiq ·Rm + eiq ·Rm .

(14.1.10)

Apart from the type-IV structure, which occurs very rarely anyway, the antiferromagnetic structures are again represented by wave vectors q that correspond to high-symmetry points of the Brillouin zone, namely X, L, and W .

As it was mentioned in Chapter 8, neutron di raction is ideally suited to the determination of magnetic structures. If neutrons interacted only with nuclei, the location of the di raction peaks would permit only the determination of atomic positions, i.e., the crystal structure. However, through their spin and magnetic moment, neutrons also interact with electrons on incomplete shells that carry a nonzero moment. If these moments are arranged in an ordered magnetic structure, the Bragg condition of elastic magnetic scattering can only be met by the vectors of the reciprocal of the magnetic Bravais lattice, i.e., of the lattice spanned by the primitive translation vectors of the magnetic structure. Since the magnetic primitive cell may be larger than the chemical one, the primitive vectors of the magnetic reciprocal lattice may be shorter