1. Anandh Subramaniam & Kantesh Balani
POINT DEFECTS IN CRYSTALS
Overview
Vacancies & their Clusters
Interstitials
Defects in Ionic Crytals  Frenkel defect  Shottky defect
MATERIALS SCIENCE
&
ENGINEERING
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur
URL: home.iitk.ac.in/~anandh
AN INTRODUCTORY E-BOOK
Part of
A Learner’s Guide
Advanced Reading
Point Defects in Materials
F. Agullo-Lopez, C.R.A. Catlow, P.D. Townsend
Academic Press, London (1988)

2. Point defects can be considered as 0D (zero dimensional) defects.
The more appropriate term would be ‘point like’ as the influence of 0D defects spreads into a small region around the defect.
Point defects could be associated with stress fields and charge.
Point defects could associate to form larger groups/complexes → the behaviour of these groups could be very different from an isolated point defect.  In the case of vacancy clusters in a crystal plane the defect could be visualized as an edge dislocation loop.
Point defects could be associated with other defects (like dislocations, grain boundaries etc.)  Segregation of Carbon to the dislocation core region gives rise to yield point phenomenon.  ‘Impurity’/solute atoms may segregate to the grain boundaries.
Based on Origin Point defects could be Random (statistically stored) or Structural. More in the next slide
Based on Position Point defects could be Random (based on position) or Ordered. More in the next slide

3. Based on origin Point Defects Based on position Point Defects
Point defects can be classified as below from two points of view.
The behaviour of a point defect depends on the class (as below) to which it belongs.
Based on
origin
Point Defects
Statistical
Structural
Arise in the crystal for thermodynamic reasons
Arise due to off-stoichiometry in an compound (e.g. in NiAl with B2 structure Al rich compositions result from vacant Ni sites)
Based on
position
Point Defects
Random
Ordered
Occupy random positions in a crystal
Occupy a specific sublattice
Vacancy ordered phases in Al-Cu-Ni alloys (V6C5, V8C7)

4. Based on source Point Defects
Intrinsic
Extrinsic
No additional foreign atom involved
Atoms of another species involved
Vacancies
Self Interstitials
Anti-site defects
In ordered alloys/compounds
Note: Presence of a different isotope may also be considered as a defect

5. Vacancy Interstitial Non-ionic crystals Impurity Substitutional
Alloying element/Dopant
Substitutional
0D (Point defects)
Frenkel defect
Ionic crystals
Other ~
Schottky defect
Imperfect point-like regions in the crystal about the size of 1-2 atomic diameters.
The extent of the distortion field may however extend to a larger distance.
Point defects can be created by ‘removal’, ‘addition’ or displacement of an atomic species (atom, ion).
Defect structures in ionic crystals can be more complex and are not discussed in detail in the elementary introduction.

6. Vacancy Missing atom from an atomic site is called a vacancy.
Atoms around the vacancy displaced from their equilibrium positions.
This gives rise to a stress field in the vicinity of the vacancy.
Based on their origin vacancies can be:  Random/Statistical (thermal vacancies, which are required by thermodynamic equilibrium) or  Structural (due to off-stoichiometry in a compound).
Based on their position vacancies can be random or ordered. (Ordered defects become part of the crystal structure and are ‘no longer defects’ in the usual sense).
Vacancies play an important role in diffusion of substitutional atoms and in many other processes/effects in materials science, including climb of edge dislocations, some forms of creep and increased resistivity.
Non-equilibrium concentration of vacancies can be generated by:  quenching from a higher temperature  bombardment with high energy particles  plastic deformation.  off-stoichiometry in ordered compounds. Etc.
Neighbouring atoms are displaced from their equilibrium position in a perfect crystal

7. Interstitial Impurity Substitutional Impurity/Alloying Element/Dopant
Revise Voids in Crystals
A ‘foreign’ element added (called as impurity/alloying element/dopant based on the context) can go to an interstitial site (between atoms) or may substitute for an atom of the host.
Overlaid to illustrate the relative size of atom and void (usually the insterstitial atom is bigger than the void)
Interstitial
Compressive & Shear Stress Fields
Impurity
Or alloying element
Compressive stress fields
Substitutional
Substitutional Impurity/Element  Foreign atom replacing the parent atom in the crystal  E.g. Cu sitting in the lattice site of FCC-Ni
Interstitial Impurity/Element  Foreign atom sitting in the void of a crystal  E.g. C sitting in the octahedral void in HT FCC-Fe
Tensile Stress Fields

8. In some (rare) situations the same element can occupy both a lattice position and an interstitial position ► e.g. B in steel.
By using ion irradiation or some other ‘strong forces’ an substitutional atoms may be forced to occupy an interstitial position.
The diffusion mechanism of these two types of point defects (interstitial vs substitutional) is different. This is because for the diffusion of substitutional atom the neighbouring site has to be vacant; while in the case of interstitial diffusion the neighbouring site is usually vacant (as the solubility of interstitial atoms is small).

9. Interstitial C sitting in the octahedral void in HT FCC-Fe
rOctahedral void / rFCC atom = 0.414
rFe-FCC = 1.29 Å  rOctahedral void = x 1.29 = 0.53 Å
rC = 0.71 Å
 Compressive strains around the C atom
Solubility limited to 2 wt% (9.3 at%)
Interstitial C sitting in the octahedral void in LT BCC-Fe
rTetrahedral void / rBCC atom = 0.29  rC = 0.71 Å
rFe-BCC = Å  rTetrahedral void = 0.29 x = Å
► But C sits in smaller octahedral void- displaces fewer atoms
 Severe compressive strains around the C atom
Solubility limited to wt% (0.037 at%)

10. Why are vacancies referred to as equilibrium thermodynamic defects?
In these discussions we will keep in view metallic crystals like Fe, Cu, Zn, etc.
Formation of a vacancy leads to ‘missing bonds’ and distortion of the lattice.
The potential energy (Internal energy & Enthalpy) of the system increases.
Work required for the formation of a point defect → Enthalpy of formation (Hf) [kJ/mol or eV/defect].
Though it costs energy to form a vacancy, its formation leads to increase in configurational entropy (the crystal without vacancies represents just one state, while the crystal with vacancies can exist in many energetically equivalent states, corresponding to various positions of the vacancies in the crystal → ‘the system becomes configurationally rich’).
 at some temperature above zero Kelvin there is an equilibrium concentration/number of vacancies (at ‘low’ temperatures no vacancies may be stable). Refer to a calculation later for the calculation of the T at which the first vacancy becomes stable.
These type of vacancies are called Thermal Vacancies (and will not leave the crystal on annealing at a temperature at which these are stable→ Thermodynamically stable).
Note: up and above the equilibrium concentration of vacancies there might be a additional non-equilibrium concentration of vacancies which are present. This can arise by quenching from a high temperature, irradiation with ions, cold work etc.
When we quench a sample from high temperature part of the higher concentration of vacancies present (at higher temperature there is a higher equilibrium concentration of vacancies present) may be quenched-in at low temperature

11. Enthalpy of formation of vacancies (Hf)
1 eV= × J
Crystal Kr Cd Pb Zn Mg Al Ag Cu Ni
kJ / mol
7.7 38 48 49 56 68
106
120
168
eV / vacancy
0.08
0.39
0.5
0.51
0.58
0.70
1.1
1.24
1.74
Note that the second row is in kJ per mole of vacancies while the 3rd row is eV per vacancy.

12. Revise chapter on equilibrium before this computation
Calculation of equilibrium concentration of vacancies
Revise chapter on equilibrium before this computation
Let nv be the number of vacancies, N the number of sites in the lattice
Assume that concentration of vacancies is small i.e. nv/N << 1
 the interaction between vacancies can be ignored ( )
 Hformation (nv vacancies) = nv . Hformation (1 vacancy)
Let Hf be the enthalpy of formation of one vacancy (assumed constant for now).
()
G = H  T S
S = Sconfigurational = Sstate with vacancies – Sstate without vacancies=perfect crystal
(1)
G (putting n vacancies) = nv.Hf  T Sconfig
Configurational entropy
Calculating Sconfig:  S = k.ln(). For the state without vacancies (perfect crystal), the number of configurations is ‘1’  Sperfect crystal = k.ln(1) = 0. Hence, Sconfigurational = Sstate with vacancies.  In a lattice with N atoms (which could be NAvagadro = N0) there are nv vacancies and (N–nv) filled sites. The possible number of configurations () is given by: (i.e. the possible number of ways I can chose nv vacant sites from a perfect lattice containing N sites).
(2)
Continued…

13. User R instead of k if Hf is in J/mole (instead of J/atom)
From equations (1), (2)
Using Sterling’s approximation:
ln(n!)= [nln(n) – n]
zero
from ()
For energy minimum 
Assuming nv << N
k = kB = Boltzmann constant
= 1.38  10–23 J/K
= 8.62  10–5 eV/K
User R instead of k if Hf is in J/mole (instead of J/atom)

14. Variation of G with vacancy concentration at a fixed temperature
T (ºC)
n/N
500
1 x 1010
1000
1 x 105
1500
5 x 104
2000
3 x 103
Hf = 1 eV/vacancy
= 0.16 x 1018 J/vacancy
Close to the melting point in FCC metals Au, Ag, Cu the fraction of vacancies is about 104 (i.e. one in 10,000 lattice sites are vacant).
Metal
n/N at Tm Cu
2 x 104 Kr
3 x 103 Cd
6.2 x 104 Al
9 x 104

15. Even though it costs energy to put vacancies into a crystal (due to ‘broken bonds’), the Gibbs free energy can be lowered by accommodating some vacancies into the crystal due to the configurational entropy benefit that this provides
Hence, certain equilibrium concentration/number of vacancies are preferred at T > 0K
Q & A
At what temperature does the first vacancy become stable in a Cu crystal?
This we can determine by substituting nv = 1 in the equation below we can determine the temperature.
Data:
No. of atoms in the crystal = NAvagadro
Hf (Cu) = 1.24 eV/vacancy
kB = 8.62  10–5 eV/K
nv = 1
We assume that the missing atom goes to the surface. With the assumption that the number of surface sites is small– we need not worry about this one atom!

16. Ionic Crystals
In ionic crystal, during the formation of the defect the overall electrical neutrality has to be maintained (or to be more precise the cost of not maintaining electrical neutrality is high)

17. Frenkel defect
Cation being smaller can get displaced to interstitial voids.
This kind of self interstitial costs high energy in simple metals and is not usually found [Hf(vacancy) ~ 1eV; Hf(interstitial) ~ 3eV].
E.g. in AgI & CaF2 the cation can form a self interstitial.  Ag interstitial concentration near melting point:  in AgCl of 103,  in AgBr of 102.
nF → no. of Frenkel defects in a MX crystal
HF → enthalpy of formation of a Frenkel defects
Ni → no. of interstitial sites available

18. Schottky defect
A Schottky defect consists of a pair of anion and cation vacancies→ this maintains charge neutrality.
E.g. Alkali halides
Missing Anion
Missing Cation
The total number of configurations is now the (number of ways the cation vacancy can be arranged)  (the number of ways the anion vacancy can be arranged). total = anion . cation
Factor ‘2’  the increase is steeper as compared to vacancies
ns → no. of Schottky defects
Hs → enthalpy of formation of a Schottky defect

19. Typical enthalpies of formation of Schottky and Frenkel defects.
Schottky Defects
Compound
Hs (J)
 10–19
Hs (eV)
LiI
2.08
1.30
LiBr
2.88
1.80
NaCl
3.69
2.30
CaO
9.77
6.10
MgO
10.57
6.60
Frenkel Defects
Compound
HF (J)
 10–19
Hs (eV)
-AgI
1.12
0.70
AgBr
1.92
1.20
AgCl
2.56
1.60
CaF2
4.49
2.80
ZrO2
6.57
4.10
Even in solids like LiI with low Hs (enthalpy of formation of Schottky defects) (1.30 eV) the fraction of defects at RT (300K) is small (1.210–11). At 1000K the fraction is 5.310–4.
This implies there are very few Schottky defects at RT.
Depending on the values of Hs & HF both these defects may be present in a crytal (though one of them dominates in most systems).

20. Other defects due to charge balance (/neutrality condition)
If Cd2+ replaces Na+ → one cation vacancy is created
Schematic

21. Defects due to off stiochiometry
ZnO heated in Zn vapour → ZnyO (y >1)
The excess cations occupy interstitial voids
The electrons (2e) released stay associated to the interstitial cation
Schematic

22. Other defect configurations: association of ions with electrons and holes
M2+ cation associated with an electron
X2 anion associated with a hole

23. F centre absorption energy (eV)
How do colours in some crystals arise due to colour centres?
Actually the distribution of the excess electron (density) is more on the +ve metal ions adjacent to the vacant site
Colour centres
(F Centre)
Violet colour of CaF2 → missing F with an electron in lattice
Ionic Crystal
F centre absorption energy (eV)
LiCl
3.1
NaCl
2.7
KCl
2.2
CsCl
2.0
KBr
LiF
5.0
Red
Visible spectrum: nm

24. Some more complications: an example of defect association
Two adjacent F centres giving rise to a M centre

25. The choice of antisite or vacancy is system specific
Structural Point defects
In ordered NiAl (with ordered B2 structure)  Al rich compositions result from vacancies in Ni sublattice.
In Ferrous Oxide (Fe2O) with NaCl structure there is a large concentration of cation vacancies.  Some of the Fe is present in the Fe3+ state  correspondingly some of the positions in the Fe sublattice is vacant  leads to off stoichiometry (FexO where x can be as low as 0.9 leading to considerable concentration of ‘non-equilibrium’ vacancies).
In NaCl with small amount of Ca2+ impurity:  for each impurity ion there is a vacancy in the Na+ sublattice.
Antisite on Al sublattice ← Ni rich side
Al rich side → vacancies in Ni sublattice
NiAl
Antisite on Al sublattice ← Fe rich side
Al rich side → antisite in Fe sublattice
FeAl
The choice of antisite or vacancy is system specific

26. FeO heated in oxygen atmosphere → FexO (x <1)
Vacant cation sites are present
Charge is compensated by conversion of ferrous to ferric ion:
Fe2+ → Fe3+ + e
For every vacancy (of Fe cation) two ferrous ions are converted to ferric ions → provides the 2 electrons required by excess oxygen

27. Point Defect ordering
Using the example of vacancies we illustrate the concept of defect ordering
As shown before, based on position vacancies can be random or ordered
Ordered vacancies (like other ordered defects) play a different role in the behaviour of the material as compared to random vacancies

28. Crystal with vacancies
Schematic
Origin of A sublattice
Origin of B sublattice
Crystal with vacancies
As the vacancies are in the B sublattice these vacancies lead to off stoichiometry and hence are structural vacancies
Vacancy ordering
Examples of Vacancy Ordered Phases: V6C5, V8C7

29. Vacancy Ordered Phases (VOP)
Me6C5 trigonal ordered structures (e.g. V6C5 → ordered trigonal structure exists between ~ K) (The disordered structure is of NaCl type (FCC lattice) with C in non-metallic sites)  Space group: P31  The disordered FCC basis vectors are related to the ordered structure by:
Atom
Wyckoff Position x y z
Vacancy
3(a)
1/9
8/9
1/6 C1
4/9
5/9 C2
7/9
2/9 C3
1/3 C4 C5 V1
1/12 V2 V3 V4 V5 V6
3(A)

30. Complex and Associated Point Defects

31. Association of Point defects (especially vacancies)
Point defects can occur in isolation or could get associated with each other (we have already seen some examples of these).
If the system is in equilibrium then the enthalpic and entropic effects (i.e. on G) have to be considered in understanding the association of vacancies.
 If two vacancies get associated with each other (forming a di-vacancy) then this can be visualized as a reduction in the number of bonds broken, leading to an energy benefit (in Au this binding energy is ~ 0.3 eV).  but this reduces the number of configurations possible with only dissociated vacancies.  The ratio of mono-vacancies to divacancies increases with increasing temperature.
Similarly an interstitial atom and a vacancy can come together to reduce the energy of the crystal  would preferred to be associated.
Non-equilibrium concentration of interstitials and vacancies can condense into larger clusters.  In some cases these can be visualized as prismatic dislocation loop or stacking fault tetrahedron).
Point defects can also be associated with other defects like dislocations, grain boundaries etc.
We had considered a divacancy. Similar considerations come into play for tri-vacancy formation etc.
Click here to know more about Association of Defects
Concept of Defect in a Defect & Hierarchy of Defects
Click here to know more about Defect in a Defect

32. Note: these are structural vacancies
Complex Point Defect Structures: an example
The defect structures especially ionic solids can be much more complicated than the simple picture presented before. Using an example such a possibility is shown.
In transition metal oxides the composition is variable
In NiO and CoO fractional deviations from stoichiometry (103 - 102) → accommodated by introduction of cation vacancies
In FeO larger deviations from stoichiometry is observed
At T > 570C the stable composition is Fe(1x)O [x (0.05, 0.16)]
Such a deviation can ‘in principle’ be accommodated by Fe2+ vacancies or O2 interstitials
In reality the situation is more complicated and the iron deficient structure is the 4:1 cluster → 4 Fe2+ vacancies as a tetrahedron + Fe3+ interstitial at centre of the tetrahedron + additional neighbouring Fe3+ interstitials
These 4:1 clusters can further associate to form 6:2 and 13:4 aggregates
Note: these are structural vacancies
Continued…

33. Schematic
4:1 cluster → 4 Fe2+ vacancies as a tetrahedron + Fe3+ interstitial at centre of the tetrahedron + additional neighbouring Fe3+ interstitials
The figure shows an ideal starting configuration- the actual structure will be distorted with respect to this depiction

34. Methods of producing point defects
Growth and synthesis Impurities may be added to the material during synthesis
Thermal & thermochemical treatments and other stimuli  Heating to high temperature and quench  Heating in reactive atmosphere  Heating in vacuum  e.g. in oxides it may lead to loss of oxygen  Etc.
Plastic Deformation
Ion implantation and irradiation  Electron irradiation (typically >1MeV) → Direct momentum transfer or during relaxation of electronic excitations)  Ion beam implantation (As, B etc.)  Neutron irradiation

35. What is the equilibrium concentration of vacancies at 800K in Cu
Solved
Example
What is the equilibrium concentration of vacancies at 800K in Cu
Data for Cu:
Melting point = 1083 C = 1356K
Hf (Cu vacancy) = 120  103 J/mole
k (Boltzmann constant) = 1.38  1023 J/K
R (Gas constant) = J/mole/K
First point we note is that we are below the melting point of Cu
800K ~ 0.59 Tm(Cu)
If we increase the temperature to 1350K (near MP of copper)
Experimental value: 1.0  104

36. Solved
Example
If a copper rod is heated from 0K to 1250K increases in length by ~2%. What fraction of this increase in length is due to the formation of vacancies?
Data for Cu:
Hf (Cu vacancy) = 120  103 J/mole
R (Gas constant) = J/mole/K
Cu is FCC (n = 4)
Continued…

38. Funda Check
What is the difference between a Vacancy, a Void and a Hole?
These 3 words are technical terms in materials science and are often used in more than one context.
Vacancy is typically a missing atom from its site, but is sometimes used in the context of a missing electron from its shell (“vacancy in the L shell”).
Void can come in two forms: (a) inter-atomic voids in crystals and (b) ‘macroscopic’* void (which is missing matter in a material).
A hole is a missing electron in the valence band. Instead of treating the (n1) negatively electrons in the valence band, we consider a positively charged hole (in the valence band).
* Macroscopic as compared to the inter-atomic void.