1. Defect chemistry – a general introduction
Truls Norby

2. Brief history of structure, stoichiometry, and defects
Early chemistry had no concept of stoichiometry or structure.
The finding that compounds generally contained elements in ratios of small integer numbers was a great breakthrough!
Understanding that external geometry often reflected atomic structure.
Perfectness ruled. Non-stoichiometry was out.
Intermetallic compounds forced re-acceptance of non-stoichiometry.
But real understanding of defect chemistry of compounds is less than 100 years old.

3. Perfect structure
Our course in defects takes the perfect structure as starting point.
This can be seen as the ideally defect-free interior of a single crystal or large crystallite grain at 0 K.

4. Close-packing
Metallic or ionic compounds can often be regarded as a close-packing of spheres
In ionic compounds, this is most often a close-packing of anions (and sometimes large cations) with the smaller cations in interstices

5. Some simple classes of oxide structures with close-packed oxide ion sublattices
Formula
Cation:anion coordination
Type and number of occupied interstices
fcc of anions
hcp of anions MO
6:6
1/1 of octahedral sites
NaCl, MgO, CaO, CoO, NiO, FeO a.o.
FeS, NiS
4:4
1/2 of tetrahedral sites
Zinc blende: ZnS
Wurtzite: ZnS, BeO, ZnO
M2O
8:4
1/1 of tetrahedral sites occupied
Anti-fluorite: Li2O, Na2O a.o.
M2O3, ABO3
6:4
2/3 of octahedral sites
Corundum:
Al2O3, Fe2O3,
Cr2O3 a.o.
Ilmenite: FeTiO3
MO2
6:3
½ of octahedral sites
Rutile: TiO2, SnO2
AB2O4
1/8 of tetrahedral and 1/2 of octahedral sites
Spinel: MgAl2O4
Inverse spinel: Fe3O4

6. The perovskite structure ABX3
Close-packing of large A and X
Small B in octahedral interstices
Alternative (and misleading?) representation

7. We shall use 2-dimensional structures for our schematic representations of defects
Elemental solid
Ionic compound

8. Defects in an elemental solid
From A. Almar-Næss: Metalliske materialer.

9. Defects in an ionic compound

10. Defect classes
Electrons (conduction band) and electron holes (valence band)
0-dimensional defects
point defects
defect clusters
valence defects (localised electronic defects)
1-dimensional defects
Dislocations
2-dimensional defects
Defect planes
Grain boundaries (often row of dislocations)
3-dimensional defects
Secondary phase

11. Perfect vs defective structure
Perfect structure (ideally exists only at 0 K)
No mass transport or ionic conductivity
No electronic conductivity in ionic materials and semiconductors;
Defects introduce mass transport and electronic transport; diffusion, conductivity…
New electrical, optical, magnetic, mechanical properties
Defect-dependent properties

13. Point defects – intrinsic disorder
Point defects (instrinsic disorder) form spontaneously at T > 0 K
Caused by Gibbs energy gain as a result of increased entropy
Equilibrium is a result of the balance between entropy gain and enthalpy cost
1- and 2-dimensional defects do not form spontaneously
Entropy not high enough.
Single crystal is the ultimate equilibrium state of all crystalline materials
Polycrystalline, deformed, impure/doped materials is a result of extrinsic action

14. Defect formation and equilibrium
Free energy vs number n of defects
Hn = nH
Sn = nSvib + Sconf
G = nH - TnSvib - TSconf
For n vacancies in an elemental solid:
EE = EE + vE K = [vE] = n/(N+n)
Sconf = k lnP = k ln[(N+n)!/(N!n!)]
For large x: Stirling: lnx!  xlnx - x
Equilibrium at dG/dn = 0
= H - TSvib - kT ln[(N+n)/n] = 0
n/(N+n) = K = exp(Svib/k - H/kT)

15. Kröger-Vink notation for 0-dimensional defects
A = chemical species
or v (vacancy)
s = site; lattice position
or i (interstitial)
c = charge
Effective charge = Real charge on site minus charge site would have in perfect lattice
Notation for effective charge:
• positive
/ negative
x neutral (optional)
Point defects
Vacancies
Interstitials
Substitutional defects
Electronic defects
Delocalised
electrons
electron holes
Valence defects
Trapped electrons
Trapped holes
Cluster/associated defects

16. Perfect lattice of MX, e.g. ZnO

17. Vacancies and interstitials

18. Electronic defects

19. Foreign species

20. Protons and other hydrogen defects
H+ H H-

21. How can we apply integer charges when the material is not fully ionic?

22. The extension of the effective charge may be larger than the defect itself

23. ……much larger….

24. …but when it moves, an integer number of electrons also move, thus making the use of the simple defect and integer charges reasonable

25. Defects are donors and acceptorsEv

26. Defect chemical reactions
Example: Formation of cation Frenkel defect pair:
Defect chemical reactions must obey three rules:
Mass balance: Conservation of mass
Charge balance: Conservation of charge
Site ratio balance: Conservation of host structure

27. Defect chemical reactions obey the mass action law
Example: Formation of cation Frenkel defect pair:

28. Notes on mass action law
The standard state is that the site fraction of the defect is 1
Standard entropy and enthalpy changes refer to full site occupancies. This is an unrealisable situation.
Ideally diluted solutions often assumed
Note: The standard entropy change is a change in the vibrational entropy – not the configurational.

29. Electroneutrality
The numbers or concentrations of positive and negative charges cancel, e.g.
Often employ simplified, limiting electroneutrality condition:
Note: The electroneutrality is a mathematical expression, not a chemical reaction. The coefficients thus don’t say how many you get, but how much each “weighs” in terms of charge….

30. Site balances
Expresses that more than one species fight over the same site:
Also this is a mathematical expression, not a chemical reaction.

31. Defect structure; Defect concentrations
The defect concentrations can now be found by combining
Electroneutrality
Mass and site balances
Equilibrium mass action coefficients
Two defects (limiting case) and subsequently for minority defects
Brouwer diagrams
or three or more defects simultaneously
More exact solutions
…these are the themes for the subsequent lectures and exercises…