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Fundamentals of the Physics of Solids / 09-The Structure of Real Crystals

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9

The Structure of Real Crystals

In the previous chapters, where the symmetries of the crystalline state were listed and di raction was studied, crystals were assumed to be ideal, with a strict periodicity. But real crystals are never ideal. No matter how carefully they are grown, there are always some departures from perfect order. The specific types of defects that appear in the sample and their numbers are determined, besides preparation conditions, by the history of the sample. Every single property of the material is a ected by these defects – but not to the same extent. Mechanical and elastic properties of solids are most sensitive to structure. These properties cannot even be interpreted without a proper understanding of structural defects. This is why it is of the utmost importance in materials science to know the real structure of crystals, whether they occur naturally or are grown for a specific purpose.

Other properties, for example electric and magnetic properties in general, are relatively insensitive to structure although they also depend to a small degree on the presence of defects. Any deviation from the regular structure modifies the state of electrons, too, so when these are studied defects in imperfect crystals cannot be altogether ignored. In fact e ects of impurities and the ensuing disorder on electronic states has become a hot research topic in the past decades leading to a lot of new discoveries. In this chapter we shall present the most characteristic types of deviation from the ideal lattice structure. The e ects of defects on electronic states will be discussed in Chapters 17 and 36.

Departure from the perfectly periodic arrangement appears in crystals for various reasons and in various forms. In materials of stoichiometric composition – even when each atom is at its proper place in the primitive cell – natural isotopes are randomly distributed at the available positions, unless special preparation processes are employed. This hardly influences the properties of solids. However, when it comes to potential scattering of neutrons by nuclei, di erent isotopes may have highly disparate scattering lengths, and so besides the Bragg peaks an additional, smeared-out incoherent background

274 9 The Structure of Real Crystals

is observed because of isotopic disorder. This has to be taken invariably into account in the evaluation of measured data.

It is impossible to prepare a sample that is chemically 100% pure. Even the most e ective purification techniques fail to eliminate a small amount of impurities – perhaps as little as a few parts per million (ppm). When growing crystals, impurities inevitably find their way into the sample – but they can also be introduced on purpose, to modify some property of the material that is sensitive to the presence of defects. In this case they are called dopants rather than impurities.

An impurity that occupies the place of an atom in the crystal is known as a substitutional impurity. If the size of the impurity atom is appreciably di erent from the original one, the lattice is locally distorted, as illustrated in Fig. 9.1. This kind of lattice distortion occurs for all other defect types, too, although this will not always be shown in the figures.

Fig. 9.1. Deformation of the lattice around a substitutional impurity atom that is larger or smaller than the original one

From a structural point of view, lattice defects that arise from the irregular arrangement of atoms are more interesting than substitutional impurities. These defects are customarily classified by the region of the sample over which they extend.

1.Point defects are not point-like deviations from perfect crystalline structure in the mathematical sense, however such defects do not extend over more than a few lattice constants in each direction.

2.For line defects deviation from the ideal structure extends over small distances along two directions, however in the third direction – along a straight or curved line – it may extend over the whole crystal.

3.For planar defects deviation from the ideal structure extends over a plane or a curved surface while it is limited to a few lattice constants in the third dimension. They are also called interfacial defects because the defect region forms an interface between two regular regions.

4.For volume or bulk defects the structure in a macroscopic three-dimensional region is di erent from the rest of the sample.

9.1 Point Defects

275

Below we shall discuss these four types of defect in separate sections. We shall deal with structural questions only.

9.1 Point Defects

Starting with the structure of an ideal crystal, three types of point defect are possible. One of them, an impurity atom occupying the position of an atom in the crystal has already been mentioned. Below we shall discuss the two other cases, the internal defects of pure samples.

Compared to the regular structure atoms may be missing. When a site in the lattice is vacant, one may say that a vacancy is present in the lattice. This situation is illustrated in Fig. 9.2(a); deformation of the lattice around the vacancy is not shown. Several references call this a Schottky defect, however we shall adhere to commoner usage and reserve this term for the case when two oppositely charged ions leave their sites in an ionic crystal, creating a pair of vacancies. Another type of defect is shown in Fig. 9.2(b): here an atom appears at an interstitial site, that is in the empty region among the atoms located at the sites of a regular lattice.

(a)

(b)

Fig. 9.2. Point defects in ideal crystal structures: (a) vacancy; (b) interstitial

If the ideal crystal is considered as the ground state of the solid, crystals with vacancies and/or interstitials can be considered as excited states of crystalline matter, since – as we shall see – in thermal equilibrium at finite temperatures these defects are present in finite concentrations.

9.1.1 Vacancies

First we shall demonstrate that vacancies are indeed generated thermally in any crystal. Consider an ideal crystal with N atoms, and assume that n vacancies are formed via the di usion of n atoms to the surface. We shall denote the energy required to remove one atom – the formation energy of a vacancy – by ε0; when vacancies are su ciently far apart and thus interactions among them can be neglected, the formation energy of n vacancies is E = 0. The total internal energy of the system is

276

9 The Structure of Real Crystals

 

 

E = E0 + 0 ,

(9.1.1)

where E0 is the internal energy of the ideal crystal.

 

 

Here we are not interested in the determination of the energy ε0. Note

however that since the formation of vacancies gives rise to a change in volume, the quantity directly accessible to measurements is formation enthalpy. Vacancy formation energies for some simple metals are listed in Table 9.1. It should be noted that these are much smaller than binding energies – precisely because of the deformation of the lattice.

Table 9.1. Vacancy formation energies (in eV) and their thermal equilibrium concentrations at the melting point in some simple metals

Element

Cu

Ag

Au

Al

ε0 (eV)

1.07

1.09

0.98

0.78

cv

4.5 × 105

3.5 × 105

17 × 105

6 × 105

In spite of the increase in energy due to the vacant sites, the concentration of vacancies remains finite in thermal equilibrium at finite temperatures as the increase in energy is compensated for by the increase in entropy due to disorder, and thus free energy will be lower than in the ordered state. The configurational entropy of the disordered state can be determined by assuming that the atoms missing from the vacancies have di used to the surface and occupy sites that can be considered as the continuation of the crystal. A sample containing N atoms and n vacancies can therefore be considered to have N + n lattice sites, of which n are vacant. Assuming that the n vacancies are distributed randomly over the N + n sites, the entropy associated with disorder is

Sconfig = kB ln

N + n

= kB ln

(N + n)!

.

(9.1.2)

n

 

 

 

N ! n!

 

When besides the number of lattice sites the number of vacancies is also large, the Stirling formula (C.3.29)

ln n! = n + 21 ln n − n + 21 ln 2π + O (1)

(9.1.3)

can be used. Keeping only leading-order terms,

 

 

 

Sconfig ≈ kBN ln

N + n

+ kBn ln

N + n

(9.1.4)

 

 

.

N

n

In addition to this, the finite temperature entropy S0 of the perfectly regular crystal needs to be taken into account.

The formation of vacancies is accompanied by a change in the sample volume as atoms move to its surface. Thermal equilibrium therefore occurs at

9.1 Point Defects

277

the minimum of the Gibbs free energy (sometimes also known as free enthalpy or Gibbs potential)

G = E − T S + pV .

(9.1.5)

Provided that the displacement of atomic positions about the vacancies is a small perturbation that can be ignored, the total volume of the sample increases from N v0 to (N + n)v0, where v0 is the specific volume per atom. Substituting the expressions for internal energy, entropy, and volume into the Gibbs free energy formula,

G ≈ E0 + 0 − T S0 − kBT N ln

N + n

− kBT n ln

N + n

 

N

n

(9.1.6)

+ p(N + n)v0 .

To obtain the equilibrium number of vacancies this expression has to be minimized:

∂G

≈ ε0 − kBT ln

N + n

+ pv0 = 0 .

(9.1.7)

∂n

 

n

If the number of vacancies is much smaller than the number of lattice sites

(n N ), we have

 

n = N e(ε0+pv0 )/kBT .

(9.1.8)

Vacancy formation energies are on the order of an eV, so pv0 can be ne-

glected compared to them:

 

n = N eε0 /kBT .

(9.1.9)

The equilibrium number of thermally generated vacancies shows strong temperature dependence. As indicated by the data in the last line of Table 9.1, close to the melting point the concentration of such defects is on the order of 104–105. If the sample contained only thermally generated vacancies, their number would be entirely negligible at room temperatures because of the exponential temperature dependence. When a sample is cooled quickly from high temperatures (quenching), high nonequilibrium vacancy concentrations may freeze into it. A part of them can be removed by annealing.

When thermally excited atoms di use from the interior of the crystal to the surface leaving vacancies behind, the dimensions of the crystal increase more rapidly than the lattice constant (the microscopic distance between atoms at regular lattice sites). This is why di erent thermal expansion coe cients are obtained from di raction measurements that are sensitive to atomic distances and variations of the macroscopic dimensions of the sample.

If the crystal is built up of more than one kind of atom, and vacancies appear on each sublattice independently, then the thermal vacancy concentration can be determined separately for each kind because formation energies depend on the atom and its environment. If removing a jth type atom requires an energy εj – and, as before, the contribution due to the change in volume can

278 9 The Structure of Real Crystals

be neglected – then the Gibbs free energy of a sample containing nj vacancies of the jth type is

G = E0 + nj εj − T S0

kBT Nj ln

 

Nj + nj

 

 

Nj

 

j

j

 

 

 

(9.1.10)

 

 

Nj + nj

 

 

 

kBT nj ln

 

 

 

,

 

nj

 

j

 

 

 

 

where Nj is the number of jth type atoms in the crystal. (The same expression gives the Helmholtz free energy as the volume term is absent.) This expression is minimized by

nj = Nj eεj /kB T .

(9.1.11)

Obviously, defects with the lowest formation energy are the most abundant. In most cases only these need to be taken into account.

9.1.2 Interstitials

So far it has been assumed that when a vacancy is generated the atom moves to the surface of the crystal. This is often not the case: it may also occupy a position that is not occupied in a perfect crystal. We have seen that by considering atoms in a crystalline structure as spheres there may be large enough regions among them for a new atom to fit in (especially when neighboring atoms are slightly displaced). The narrow space among the original atoms is an interstice, so an atom occupying it is called an interstitial atom or simply an interstitial.

The energy of an interstitial atom depends on its exact position. Local minima of the potential due to neighboring atoms are to occur at positions that are surrounded symmetrically by atoms of the regular lattice. For a simple cubic crystal such are the body center and the face centers of the cube. In face-centered cubic crystals two such characteristic sites exist, as shown in Fig. 7.13. One of them is the center of the cube at 12 12 12 ; the second is

the center of an octant, at 1 1 1 (which quadrisects the space diagonal) or any

4 4 4

equivalent position. In relation to Fig. 7.13 we also saw that the local environments around the interstitial are di erent at the two sites. In the first case the interstitial atom is surrounded by six first neighbors in an octahedral geometry, thus the site has an octahedral (cubic) symmetry. The local point group of reflections and rotations that leave this point invariant is the group Oh. In the second case the defect site has four nearest neighbors that are arranged tetrahedrally. Local symmetries are elements of the tetrahedral point group Td. As mentioned in Chapters 6 and 7, the pattern of energy level splitting may be di erent for atoms in the two environments. This can be used for their identification, through optical spectroscopy or nuclear magnetic resonance.

Two interstitial sites are distinguished in a body-centered cubic lattice, too. As shown in Fig. 7.8, two empty sites of high symmetry are found among the

9.1 Point Defects

279

atoms at the vertices. Edge centers (at 12 00 and equivalent positions, including face centers) are called octahedral sites – although the six neighbors do not form a regular octahedron and the symmetry is only tetragonal. They are also called “small” sites because they o er relatively little space. The lattice needs to be deformed to a lesser extent to make room for an interstitial atom when the atom occupies one of the tetrahedral sites, 14 12 0 or an equivalent position. These sites are called “large”.

The equilibrium concentration of interstitials can be evaluated along the same lines as that of vacancies, and the result (9.1.9) applies to this case, too. However, the theoretical determination of the formation energy is now much more di cult than for vacancies: an interstitial deforms the lattice more than a vacancy does, as it has to make some room for itself among other atoms, and so the contribution due to the deformation of the crystal is more important. Because of the larger deformation the formation energy is also somewhat higher, on the order of a few eV, consequently interstitials are more di cult to generate thermally than vacancies.

Further types of atomic configurations are possible when the additional atom is allowed to displace one or several neighbors substantially. An important case is when an atomic position is occupied by two atoms symmetrically on either side of the site in a regular crystal. This is in fact an interstitial pair, and the configuration is called a split interstitial. Such situations are shown in Fig. 9.3. For example, in the primitive cell of a body-centered cubic crystal the atom originally at the body center and the additional atom may be arranged symmetrically on both sides of the center, in the 110 direction. In a possible split interstitial configuration of a face-centered cubic lattice an atom at a face center is displaced in the 100 direction to make enough space for another atom on the other side of the face.

(a)

(b)

Fig. 9.3. Split interstitials: (a) in the [110] direction in a body-centered cubic lattice; (b) in the [100] direction in a face-centered cubic lattice

In copper, the prototype of face-centered cubic crystals, this defect – a split interstitial along the 100 direction – has a lower energy than the configuration in which a copper atom is simply placed at the interstitial site.

While in these dumbbell defects two atoms share the space of one in a perfect crystal – making, of course, some extra space for themselves among the neighbors –, it may also happen that three atoms share two lattice positions, or

280 9 The Structure of Real Crystals

[001]

[110]

(a)

(b)

Fig. 9.4. Crowdion defects in a body-centered cubic crystal along the [111] direction: (a) three atoms sharing two lattice sites; (b) four atoms sharing three lattice sites

four atoms share three lattice positions, etc. along a line. Such configurations extending over a few atomic distances in one direction are called crowdions. Figure 9.4 shows the atomic arrangement in a plane section of a body-centered cubic crystal cubic crystal containing C3/2 and C4/3 types of crowdion along the [111] direction.

Crowdions tend to have higher energies than split interstitials, and thus occur much more rarely naturally.

9.1.3 Pairs of Point Defects

As we have seen, in thermal equilibrium the concentrations of vacancies and interstitials are both finite and small enough for treating them as if each defect had been generated independently of the others. Strictly speaking this is not true. When vacancies are thermally generated, some of the atoms that leave their lattice sites move to the surface, while others become trapped at interstitial sites. Which of these possibilities occurs is of particular interest in ionic crystals. A missing ion breaks local charge neutrality, which has to be restored by another nearby defect, since charge neutrality must be valid in any relatively small region of space otherwise the Coulomb energy would become excessively large and the configuration would be energetically highly unfavorable. Restoration of charge neutrality can occur via the formation of

Schottky defects or Frenkel defects.

Schottky Defects

It was first pointed out by W. Schottky (1930) that in ionic crystals built up of positively and negatively charged ions a vacancy appearing on the sublattice of cations is accompanied by a nearby vacancy on the sublattice of anions. Although both ions di use to the surface, charge neutrality is maintained locally inside the sample. This defect – a pair of oppositely charged nearby vacancies – is called a Schottky defect (although several references use this term for a single vacancy). The alternative terms vacancy pair and divacancy are also widely used. This type of defect is illustrated in Fig. 9.5.

Equation (9.1.9), or its generalization, (9.1.11) cannot be used to determine the concentration of Schottky defects as the derivation of these formulas was

 

 

 

 

 

 

 

 

 

9.1 Point Defects 281

 

 

 

 

Schottky defect

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

vacancy pair

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Frenkel defect

Fig. 9.5. Schottky and Frenkel defects in an ionic crystal

based on the independence of individual defects. When the charge of the jth type ion is qj , charge neutrality requires

qj nj = 0 .

(9.1.12)

j

When minimizing the Gibbs free energy (9.1.10), this can be taken into account by a Lagrange multiplier. The quantity to be minimized is then

G + λ

 

(9.1.13)

qj nj .

 

j

 

Along the same lines as above, the calculation yields

 

nj = Nj e(εj +λqj )/kBT .

(9.1.14)

The so-far undetermined multiplier λ has to be determined from the equation for charge neutrality,

 

 

qj Nj e(εj +λqj )/kBT = 0 .

(9.1.15)

j

For simplicity, consider a crystal built up of an equal number N of oppositely charged anions (q) and cations (−q). When vacancy formation energies are denoted by ε+ and ε, (9.1.14) tells us that the equilibrium numbers n+ and nare given by

n+ = N e(ε++λq)/kB T ,

n= N e(ελq)/kB T .

(9.1.16)

The condition for neutrality is now n+ = n, which implies λq = (ε−ε+)/2, and so

282 9 The Structure of Real Crystals

n+ = n= N e(ε++ε)/2kBT .

(9.1.17)

The number of Schottky defects is thus determined by the average of the two formation energies. As listed in Table 9.2, formation energies are typically around 2 eV for alkali halides, i.e., much smaller than the binding energy of the ion pair, on the order of 10 eV.

Table 9.2. Formation energies of Schottky and Frenkel defects (in eV) in some simple ionic crystals

Schottky defects

 

Frenkel defects

Compound ε0 (eV)

Compound ε0 (eV)

 

 

 

 

 

LiF

2.51

 

AgCl

1.4

NaCl

2.28

 

AgBr

1.1

NaBr

1.72

 

CaF2

2.6

KCl

2.28

 

SrF2

2.3

KI

1.60

 

BaF2

1.9

CsCl

1.86

 

SrCl2

1.9

Since we shall not discuss in detail the properties of ionic crystals elsewhere, it should be mentioned here that in those ionic crystals (e.g., alkali halides), where Schottky defects are the most important defects, electric conductivity can be interpreted in terms of the motion of vacancies. As massive ions move slowly, resistivity is several orders of magnitude larger than in ordinary metals. Compared to metals, where resistivity is typically on the order of μΩ cm, values between 102 and 108 Ω cm are observed in ionic crystals. Because of the thermal generation of vacancies electrical conductivity increases with temperature, showing thermally activated behavior.

The fact that vacancies and interstitials carry charge plays an important role in determining the properties of ionic crystals. If a negatively charged ion is missing, the hole has an e ective positive charge, and can therefore bind an electron, restoring charge neutrality. The defect consisting of an electron bound to a vacancy is called a color center or F -center.1 Defects in which two or three electrons are bound by two or three neighboring vacancies are called

M - and R-centers.2

1The name refers to the property that electrons in the attractive potential of the vacancy are on hydrogen-like orbitals, so they can absorb light of specific frequencies, giving a particular color to the crystal. The name F -center comes from the German word for color, Farbe.

2M -centers are also called di-F -centers or F2-centers.