Point defects in non-metallic crystals

4.2.2 Point defects in non-metallic crystals

Point defects in non-metallic, particularly ionic, struc- tures are associated with additional features (e.g. the requirement to maintain electrical neutrality and the possibility of both anion-defects and cation-defects existing). An anion vacancy in NaCl, for example, will

be a positively-charged defect and may trap an elec- tron to become a neutral F-centre. Alternatively, an anion vacancy may be associated with either an anion interstitial or a cation vacancy. The vacancy-interstitial pair is called a Frenkel defect and the vacancy pair a Schottky defect, as shown in Figure 4.5. Interstitials are much more common in ionic structures than metal- lic structures because of the large ‘holes’ or interstices that are available.

In general, the formation energy of each of these two types of defect is different and this leads to different defect concentrations. With regard to vacancies, when

E f C >E f , i.e. the formation will initially produce more cation than anion vacancies from dislocations and boundaries as the temperature is raised. However, the electrical field produced will eventually oppose the production of further cations and promote the

Figure 4.4 Climb of a dislocation, (a) and (b) to annihilate, formation of anions such that of equilibrium there (c) and (d) to create a vacancy .

will be almost equal numbers of both types and the

Defects in solids 87

Figure 4.5 Representation of point defects in two-dimensional ionic structure: (a) perfect structure and monovalent ions, (b) two Schottky defects, (c) Frenkel defect, and (d) substitutional divalent cation impurity and cation vacancy .

combined or total concentration c of Schottky defects can only be transferred by the diffusion of the charge at high temperatures is ¾10 .

carrying defects through the oxide. Both p- and n-type Foreign ions with a valency different from the host

semiconductors are formed when oxides deviate from cation may also give rise to point defects to maintain

stoichiometry: the former arises from a deficiency of charge neutrality. Monovalent sodium ions substituting

cations and the latter from an excess of cations. for divalent magnesium ions in MgO, for example,

Examples of p-type semiconducting oxides are NiO, must be associated with an appropriate number of

PbO and Cu 2 O while the oxides of Zn, Cd and Be are either cation interstitials or anion vacancies in order

n -type semiconductors.

to maintain charge neutrality. Deviations from the sto- ichiometric composition of the non-metallic material

4.2.3 Irradiation of solids

as a result of excess (or deficiency) in one (or other) There are many different kinds of high-energy radi- atomic species also results in the formation of point

ation (e.g. neutrons, electrons, ˛-particles, protons, defects.

deuterons, uranium fission fragments, -rays, X-rays) An example of excess-metal due to anion vacancies

and all of them are capable of producing some form is found in the oxidation of silicon which takes place at

of ‘radiation damage’ in the materials they irradiate. the metal/oxide interface. Interstitials are more likely

While all are of importance to some aspects of the to occur in oxides with open crystal structures and

solid state, of particular interest is the behaviour of when one atom is much smaller than the other as, for

materials under irradiation in a nuclear reactor. This is example, ZnO (Figure 4.6a). The oxidation of copper

because the neutrons produced in a reactor by a fis- to Cu 2 O, shown in Figure 4.6b, is an example of non-

sion reaction have extremely high energies of about stoichiometry involving cation vacancies. Thus copper

2 million electron volts (i.e. 2 MeV), and being elec- vacancies are created at the oxide surface and diffuse

trically uncharged, and consequently unaffected by the through the oxide layer and are eliminated at the

electrical fields surrounding an atomic nucleus, can oxide/metal interface.

travel large distances through a structure. The resul- Oxides which contain point defects behave as semi-

tant damage is therefore not localized, but is distributed conductors when the electrons associated with the

throughout the solid in the form of ‘damage spikes.’ point defects either form positive holes or enter the

The fast neutrons (they are given this name because conduction band of the oxide. If the electrons remain

2 MeV corresponds to a velocity of 2 ð 10 7 ms ⊳ locally associated with the point defects, then charge

are slowed down, in order to produce further fission,

Figure 4.6 Schematic arrangement of ions in two typical oxides. (a) Zn >1 O, with excess metal due to cation interstitials and (b) Cu <2 O, with excess non-metal due to cation vacancies .

88 Modern Physical Metallurgy and Materials Engineering by the moderator in the pile until they are in thermal

equilibrium with their surroundings. The neutrons in

a pile will, therefore, have a spectrum of energies which ranges from about 1/40 eV at room temperature (thermal neutrons) to 2 MeV (fast neutrons). However, when non-fissile material is placed in a reactor and irradiated most of the damage is caused by the fast neutrons colliding with the atomic nuclei of the material.

The nucleus of an atom has a small diameter (e.g.

10 10 m), and consequently the largest area, or cross- section, which it presents to the neutron for collision

Figure 4.7 Formation of vacancies and interstitials due to is also small. The unit of cross-section is a barn, i.e.

particle bombardment (after Cottrell, 1959; courtesy of the

10 28 m 2 so that in a material with a cross-section Institute of Mechanical Engineers) .

of 1 barn, an average of 10 9 neutrons would have to pass through an atom (cross-sectional area 10 19 m 2 )

for one to hit the nucleus. Conversely, the mean free annihilating each other by recombination. However, it

is expected that some of the interstitials will escape about 0.3 m. If a metal such as copper (cross-section,

path between collisions is about 10 9 atom spacings or

from the surface of the cascade leaving a correspond-

4 barns) were irradiated for 1 day ⊲10 5 s⊳ in a neutron ing number of vacancies in the centre. If this number

2 flux of 10 is assumed to be 100, the local concentration will be 17 m s 1 the number of neutrons passing through unit area, i.e. the integrated flux, would be

100/60 000 or ³2 ð 10 .

10 22 nm 2 and the chance of a given atom being hit Another manifestation of radiation damage concerns the dispersal of the energy of the stopped atom into ⊲D integrated flux ð cross-section⊳ would be 4 ð 10 ,

the vibrational energy of the lattice. The energy is

i.e. about 1 atom in 250 000 would have its nucleus deposited in a small region, and for a very short struck.

time the metal may be regarded as locally heated. To For most metals the collision between an atomic

distinguish this damage from the ‘displacement spike’, nucleus and a neutron (or other fast particle of mass

m where the energy is sufficient to displace atoms, this

) is usually purely elastic, and the struck atom heat-affected zone has been called a ‘thermal spike’. To mass M will have equal probability of receiving any

raise the temperature by 1000 ° kinetic energy between zero and the maximum E max

C requires about 3R ð 4E n Mm/⊲M C m⊳ 2

4.2 kJ/mol or about 0.25 eV per atom. Consequently,

neutron. Thus, the most energetic neutrons can impart

a 25 eV thermal spike could heat about 100 atoms an energy of as much as 200 000 eV, to a copper

, where E

n is the energy of the fast

of copper to the melting point, which corresponds atom initially at rest. Such an atom, called a primary

to a spherical region of radius about 0.75 nm. It is ‘knock-on’, will do much further damage on its sub-

very doubtful if melting actually takes place, because sequent passage through the structure often producing

the duration of the heat pulse is only about 10 to secondary and tertiary knock-on atoms, so that severe

10 s. However, it is not clear to what extent the local damage results. The neutron, of course, also con-

heat produced gives rise to an annealing of the primary tinues its passage through the structure producing fur-

damage, or causes additional quenching damage (e.g. ther primary displacements until the energy transferred

retention of high-temperature phases).

Slow neutrons give rise to transmutation products. copper) necessary to displace an atom from its lat-

in collisions is less than the energy E d (³25 eV for

Of particular importance is the production of the noble tice site.

gas elements, e.g. krypton and xenon produced by fis- The damage produced in irradiation consists largely

sion in U and Pu, and helium in the light elements of interstitials, i.e. atoms knocked into interstitial posi-

B, Li, Be and Mg. These transmuted atoms can cause tions in the lattice, and vacancies, i.e. the holes they

severe radiation damage in two ways. First, the inert leave behind. The damaged region, estimated to con-

gas atoms are almost insoluble and hence in association tain about 60 000 atoms, is expected to be originally

with vacancies collect into gas bubbles which swell pear-shaped in form, having the vacancies at the cen-

and crack the material. Second, these atoms are often tre and the interstitials towards the outside. Such a

created with very high energies (e.g. as ˛-particles displacement spike or cascade of displaced atoms is

or fission fragments) and act as primary sources of shown schematically in Figure 4.7. The number of

knock-on damage. The fission of uranium into two vacancy-interstitial pairs produced by one primary

new elements is the extreme example when the fis-

sion fragments are thrown apart with kinetic energy is about 1000. Owing to the thermal motion of the

knock-on is given by n ' E max / 4E d , and for copper

³100 MeV. However, because the fragments carry atoms in the lattice, appreciable self-annealing of the

a large charge their range is short and the damage damage will take place at all except the lowest tem-

restricted to the fissile material itself, or in materi- peratures, with most of the vacancies and interstitials

als which are in close proximity. Heavy ions can be

Defects in solids 89 accelerated to kilovolt energies in accelerators to pro-

an easier process and the activation energy for migra- duce heavy ion bombardment of materials being tested

tion is somewhat lower than E m for single vacancies. for reactor application. These moving particles have a

Excess point defects are removed from a mate- short range and the damage is localized.

rial when the vacancies and/or interstitials migrate to regions of discontinuity in the structure (e.g. free sur-

4.2.4 Point defect concentration and

faces, grain boundaries or dislocations) and are annihi-

annealing

lated. These sites are termed defect sinks. The average number of atomic jumps made before annihilation is given by

most sensitive properties to investigate the point defect concentration. Point defects are potent scatterers of

(4.5) electrons and the increase in resistivity following

n D Az v t

m /kT a ]

where A is a constant ⊲³1⊳ involving the entropy of migration, z the coordination around a vacancy, v the

f /kT Q ] Debye frequency ⊲³10 (4.4) / s⊳, t the annealing time at the ageing temperature T a and E m the migration energy of

where A is a constant involving the entropy of the defect. For a metal such as aluminium, quenched formation, E f is the formation energy of a vacancy

to give a high concentration of retained vacancies, the and T Q the quenching temperature. Measuring the

annealing process takes place in two stages, as shown resistivity after quenching from different temperatures

in Figure 4.8; stage I near room temperature with an enables E 4

f 0 activation energy ³0.58 eV and n ³ 10 , and stage II versus 1/T Q . The activation energy, E m , for the

C with an activation energy of movement of vacancies can be obtained by measuring

in the range 140–200 °

¾1.3 eV.

the rate of annealing of the vacancies at different Assuming a random walk process, single vacancies p annealing temperatures. The rate of annealing is

would migrate an average distance ( n ð atomic spac- inversely proportional to the time to reach a certain

ing b) ³30 nm. This distance is very much less than value of ‘annealed-out’ resistivity. Thus, 1/t 1 D A either the distance to the grain boundary or the spac-

m /kT 1 ] and 1/t 2 D m /kT 2 ] and by

ing of the dislocations in the annealed metal. In this eliminating A we obtain ln ⊲t 2 /t 1 ⊳DE m [⊲1/T 2 case, the very high supersaturation of vacancies pro-

⊲ 1/T 1 ⊳ ]/k where E m is the only unknown in the duces a chemical stress, somewhat analogous to an expression. Values of E f and E m for different materials

osmotic pressure, which is sufficiently large to create are given in Table 4.1.

new dislocations in the structure which provide many At elevated temperatures the very high equilibrium

new ‘sinks’ to reduce this stress rapidly. concentration of vacancies which exists in the structure

The magnitude of this chemical stress may be gives rise to the possible formation of divacancy and

estimated from the chemical potential, if we let dF even tri-vacancy complexes, depending on the value

represent the change of free energy when dn vacancies of the appropriate binding energy. For equilibrium

are added to the system. Then, between single and di-vacancies, the total vacancy

dF/dn D E f 0 C kT ln c concentration is given by

D kT ln ⊲c/c 0 ⊳

c v D c 1v C 2c 2v where c is the actual concentration and c 0 the equilib- and the di-vacancy concentration by

rium concentration of vacancies. This may be rewrit- ten as

c 2v D Azc 1v 2 exp [B 2 /kT ] where A is a constant involving the entropy of forma-

tion of di-vacancies, B 2 the binding energy for vacancy pairs estimated to be in the range 0.1–0.3 eV and z a configurational factor. The migration of di-vacancies is

Table 4.1 Values of vacancy formation . E f / and migration

. E m / energies for some metallic materials together with the self-diffusion energy . E SD /

Energy Cu Al Ni

(eV) E f 1.0–1.1 0.76 1.4 0.9 2.13 3.3 1.05 Figure 4.8 Variation of quenched-in resistivity with

E D 2.0–2.2 1.38 2.9 1.4 2.89 5.2 3.45 temperature of annealing for aluminium (after Panseri and Federighi, 1958, 1223) .

90 Modern Physical Metallurgy and Materials Engineering

D ⊲kT/b 3 ⊳ [ln ⊲c/c 0 ⊳ ]

where dV is the volume associated with dn vacancies and b 3 is the volume of one vacancy. Inserting typical values, KT ' 1/40 eV at room temperature, b D

0.25 nm, shows KT/b 3 ' 150 MN/m 2 . Thus, even a moderate 1% supersaturation of vacancies i.e. when ⊲c/c 0 ⊳D

1.01 and ln ⊲c/c 0 ⊳D

0.01, introduces a

c equivalent to 1.5 MN/m 2 .

The equilibrium concentration of vacancies at a Figure 4.9 Variation of resistivity with temperature

temperature T 2 will be given by c 2 D f /kT 2 ]

produced by neutron irradiation for copper (after Diehl) .

and at T 1 by c 1 D f /kT 1 ]. Then, since

1 1 around room temperature and is probably caused by ln ⊲c 2 /c 1 ⊳ D ⊲E f /k⊳ T 1 T 2 the annihilation of free interstitials with individual vacancies not associated with a Frenkel pair, and also

the chemical stress produced by quenching a metal the migration of di-vacancies. Stage IV corresponds to from a high temperature T 2 to a low temperature T 1 the stage I annealing of quenched metals arising from

is vacancy migration and annihilation to form dislocation

loops, voids and other defects. Stage V corresponds

D ⊲kT/b ⊳ ln ⊲c 2 /c

c 1 ⊳ D ⊲E f /b 3 ⊳

to the removal of this secondary defect population by T 2 self-diffusion.

For aluminium, E f is about 0.7 eV so that quench- ing from 900 K to 300 K produces a chemical stress

2 of about 3 GN/m 4.3 Line defects . This stress is extremely high, sev- eral times the theoretical yield stress, and must be

4.3.1 Concept of a dislocation

relieved in some way. Migration of vacancies to grain boundaries and dislocations will occur, of course, but

All crystalline materials usually contain lines of struc- it is not surprising that the point defects form addi-

tural discontinuities running throughout each crystal tional vacancy sinks by the spontaneous nucleation of

or grain. These line discontinuities are termed dislo- dislocations and other stable lattice defects, such as

cations and there is usually about 10 10 to 10 12 m of voids and stacking fault tetrahedra (see Sections 4.5.3 1 dislocation line in a metre cube of material. Disloca-

and 4.6). tions enable materials to deform without destroying the When the material contains both vacancies and

basic crystal structure at stresses below that at which interstitials the removal of the excess point defect

the material would break or fracture if they were not concentration is more complex. Figure 4.9 shows

present.

the ‘annealing’ curve for irradiated copper. The

A crystal changes its shape during deformation by resistivity decreases sharply around 20 K when the

the slipping of atomic layers over one another. The interstitials start to migrate, with an activation

theoretical shear strength of perfect crystals was first energy E m ¾ 0.1 eV. In Stage I, therefore, most

calculated by Frenkel for the simple rectangular-type of the Frenkel (interstitial –vacancy) pairs anneal

lattice shown in Figure 4.10 with spacing a between out. Stage II has been attributed to the release of

interstitials from impurity traps as thermal energy 1 This is usually expressed as the density of dislocations supplies the necessary activation energy. Stage III is

10 to 10 12 m .

Figure 4.10 Slip of crystal planes (a); shear stress versus displacement curve (b) .

Defects in solids 91 the planes. The shearing force required to move a plane

of atoms over the plane below will be periodic, since for displacements x < b/2, where b is the spacing of atoms in the shear direction, the lattice resists the applied stress but for x > b/2 the lattice forces assist the applied stress. The simplest function with these properties is a sinusoidal relation of the form

mmm

is the maximum shear stress at a displacement D b/4. For small displacements the

m D (4.7)

and since b ' a, the theoretical strength of a perfect Figure 4.11 Schematic representation of (a) a dislocation loop, (b) edge dislocation and (c) screw dislocation This calculation shows that crystals should be rather .

strong and difficult to deform, but a striking exper- imental property of single crystals is their softness,

the dislocation line is rarely pure edge or pure screw, which indicates that the critical shear stress to pro-

but it is convenient to think of these ideal dislocations

since any dislocation can be resolved into edge and This discrepancy between the theoretical and experi-

duce slip is very small (about 10 5 µ or ³50gf mm 2 ).

screw components. The atomic structure of a simple mental strength of crystals is accounted for if atomic

edge and screw dislocation is shown in Figure 4.13 planes do not slip over each other as rigid bodies but

and 4.14.

instead slip starts at a localized region in the structure and then spreads gradually over the remainder of the

4.3.3 The Burgers vector

plane, somewhat like the disturbance when a pebble is It is evident from the previous sections that the Burgers dropped into a pond.

vector b is an important dislocation parameter. In any In general, therefore, the slip plane may be divided

deformation situation the Burgers vector is defined by into two regions, one where slip has occurred and the

constructing a Burgers circuit in the dislocated crystal other which remains unslipped. Between the slipped

as shown in Figure 4.12. A sequence of lattice vectors and unslipped regions the structure will be dislocated

is taken to form a closed clockwise circuit around (Figure 4.11); this boundary is referred to as a dislo-

the dislocation. The same sequence of vectors is then cation line, or dislocation. Three simple properties of

taken in the perfect lattice when it is found that the

a dislocation are immediately apparent, namely: (1) it circuit fails to close. The closure vector FS (finish- is a line discontinuity, (2) it forms a closed loop in the

start) defines b for the dislocation. With this FS/RH interior of the crystal or emerges at the surface and

(right-hand) convention it is necessary to choose one (3) the difference in the amount of slip across the dis-

direction along the dislocation line as positive. If this location line is constant. The last property is probably

direction is reversed the vector b is also reversed. the most important, since a dislocation is characterized

The Burgers vector defines the atomic displacement by the magnitude and direction of the slip movement

produced as the dislocation moves across the slip associated with it. This is called the Burgers vector,

b plane. Its value is governed by the crystal structure

, which for any given dislocation line is the same all because during slip it is necessary to retain an identical along its length.

lattice structure both before and after the passage of the dislocation. This is assured if the dislocation has