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Module 2: Defect Chemistry and Defect Equilibria
Introduction
Introduction
Materials in general consist of defects which can be divided into a variety of categories such as point
defects or 0-D defects, line defects or 1-D defects and 2-D or surface defects. These defects play an
important role in determining the properties of ceramic materials and in this context, the role of point
defects is extremely important. In this module, we will learn about various point defects, the role of
stoichiometry i.e. cation and anion excess and deficit, the role of foreign atoms on the defect
chemistry. Subsequently, we will adopt a simple thermodynamic basis for calculating their
concentration in equilibrium and then will extend the Gibbs-Duhem relation for chemical systems to
the defects in ceramics considering them to be equivalent to the dilute solutions, an approximation
which is fairly valid. This will lead us to the determination of defect concentrations as a function of
partial pressure of oxygen which is an important exercise to establish the defect concentration vs pO 2
diagrams, called Brower’s diagrams.
The Module contains:
Point Defects
Kroger-Vink Notation in a Metal Oxide, MO
Defect Reactions
Defect Structures in Stoichiometric Oxides
Defect Structures in Non-Stoichiometric Oxides
Oxygen Deficient Oxides
Dissolution of Foreign Cations in an Oxide
Concentration of Intrinsic Defects
Intrinsic and Extrinsic Defects
Units for Defect Concentration
Defect Equilibria
Defect Equilibria in Stoichiometric Oxides
Defect Equilibria in Non-Stoichiometric Oxides
Defect Structures involving Oxygern Vacancies and Interstitials
Defect Equilibrium Diagram
A Simple Procedure for Constructing at Brower's Diagram
Extent of Non-Stoichiometry
Comparative Behaviour of TiO 2 and MgO vis-à-vis Oxygen Pressure
Electronic Disorder
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Examples of Intrinsic Electronic and Ionic Defect Concentrations
Summary
Suggested Reading:
Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal
Oxides (Science & Technology of Materials), P.K. Kofstad, John Wiley and
Sons Inc.
Physical Ceramics: Principles for Ceramic Science and Engineering, Y.-M.
Chiang, D. P. Birnie, and W. D. Kingery, Wiley-VCH
Introduction to the Thermodynamics of Materials, David R. Gaskell, Taylor and
Francis.
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Module 2: Defect Chemistry and Defect Equilibria
Point Defects
2.1 Point Defects
Point defects are caused due to deviations from the perfect atomic arrangement or
stoichiometry. These could be missing lattice ions from their positions, interstitial ions or
substitutional ions (or impurities) and valence electrons and/or holes.
Usually, point defects in metals are electrically neutral whereas in ionic oxides, these are
electrically charged.
Ionic defects
Occupy lattice positions
Can be either of vacancies, interstitial ions, impurities and substitutional ions
Electronic defects
Deviations from a ground state electron orbital configuration give rise to such defects
when valence electrons are excited into higher energy orbitals/ levels and lead to
formation of electron or holes.
Defects are present in most oxides and are easily understood. Hence most examples in the
following section use examples of oxides.
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Module 2: Defect Chemistry and Defect Equilibria
Kroger-Vink Notation in a Metal Oxide (MO)
2.2 Kroger–Vink notation in a metal oxide, (MO)
Kroger-vink notations are typically used to depict the atomic defects with charges. Following tables
provide the most common notations.
Regular Sites
M m: normal or regular occupied metal or
cation site
Oo : normal or regular occupied oxygen or
anion site
Point Defects (a • (dot) means a positive charge and a ' (prime) means a
negative charge)
Oxygen (anion) vacancy
VO
Metal (cation) vacancy
VM
Oxygen (anion) interstitial
Oi
Metal (cation) interstitial
Mi
Vacant interstitial site
Vi
Foreign cation
Mf
Foreign cation on regular
Mfm
metal site
Foreign
cation
interstitial site
on
Mfi
A normal cation or anion in
an oxide with zero effective MM x or O O x
charge
Charged oxygen vacancy:
VO • or VO ••
Charged metal vacancy
VM ' or VM ''
Charged metal or oxygen
Mi •• and O i ''
interstitial
Neutral cation
vacancies
and anion
electrons and holes
VM x or VO x
e' or h •
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Module 2: Defect Chemistry and Defect Equilibria
Defect Reactions
2.3 Defect Reactions
Rules for writing defect reactions
Ratio of regular cation and anion sites is always constant.
Mass balance to be preserved.
Electrical neutrality is to be always preserved.
Both ionic and electronic defect compensations are possible determined by the energetics.
We will assume complete ionization of defects.
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Module 2: Defect Chemistry and Defect Equilibria
Defect Structures in Stoichiometric Oxides
2.4 Defect Structures in Stoichiometric Oxides
Charged point defect is a defect which is ready to be ionized and provides a complimentary
electronic charged defect. Various such combinations are possible such as
Cation and anion vacancies (VM and VO )
Vacancies and interstitial ion of same kind i.e. VO and O i or VM and Mi
Misplaced atoms interchanged (M O and O M ) - interchanged
Vacancies and misplaced atoms for the same kind of atom (VM + MO )
Interstitial and misplaced atoms i.e.,O i and MO
Interstitial atoms i.e. Mi and O i
Among all of these, the first two are most important as these are regularly seen in many important
oxides. The first is called Schottky disorder while the second is called as Frenkel disorder.
2.4.1Schottky Disorder
This defect normally forms at the outer or inner surfaces or dislocations. It eventually diffuses
into the crystal unit as equilibrium is reached.
Figure 2.1 Schottky
Disorder
The defect reaction is written as
0 (or Null)
VM ''+ V0 ••
This defect is preferred when cations and anions are of comparable sizes.
Examples are rocksalt structured compounds such as NaCl, MgO, Corundum, Rutite etc..
2.4.2 Frenkel Disorder
:
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Figure 2.2 Frenkel Defect
This defect can form inside the crystal.
It forms where cations are appreciably smaller then anions.
Defect reaction is written as
0
VM '' +
Mi ••
In cases where anions form the disorder, then it is called as Anti-Frenkel. The corresponding
defect reaction in that case would be
0
V0 ••+ O i ''
Examples of compounds showing this defect are AgBr type compounds such as AgBr, AgI
etc.
2.4.3 Intrinsic Ionization
Thermal creation of electron hole pair and is depicted by
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Module 2: Defect Chemistry and Defect Equilibria
Defect Structures in Non-Stoichiometric Oxides
2.5 Defect Structures in Non - Stoichiometric Oxides
Mainly of two types
i. Oxygen deficient (or excess metal)
ii. Metal deficient (or excess oxygen)
Nonstoichiometry necessitates presence of point defects and extent of non-stoichiometry
determines the concentration of Defects.
In such oxides, electrical neutrality is preserved via the formation of point defects and
electronic changes.
Intrinsic ionization is always a possibility.
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Module 2: Defect Chemistry
Defect Structures in Non-Stoichiometric Oxides
2.5.1 Oxygen Deficient Oxides
Formation of oxygen vacancies or metal interstitials or both are possible.
Formation occurs only at the surface.
2.5.1.1 If oxygen vacancies are the dominating defects
Depicted by MO2-x (x is the extent of non-stoichiometry) and overall reaction as
MO2 MO2-x + x /2 O 2 ↑
Due to loss of oxygen, possible defect reactions would be
Electronic compensation leading to creation oxygen vacancies and of electrons
O0
VO ••+ ½ O 2 + 2e'
Ionic compensation leads to formation of oxygen vacancies and reduction of metal ions
on their sites.
O0
VO ••+ ½ O 2 + 2M' M
2.5.1.2 If metal interstitials are the dominating defects then,
Depicted as (M 1+y O 2 is the extent of non-stoichiometry)
Possible defect reactions are
Ionic compensation leading to the formation of metal interstitials and reduction of metal
ions on their sites
MM
Mi •••• + 4 M' M
OR
Electronic compensation leading to the formation of metal interstitials and free electrons
M
Mi •••• + 4e'
Creation of quasi-free electrons (extra charge is represented as M’)
Conduction occurs due to transport of electrons
Typically n-type conductors.
Example: TiO2 , ZrO 2 , CeO2 , Nb2 O 5
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Module 2: Defect Chemistry and Defect Equilibria
Defect Structures in Non-Stoichiometric Oxides
2.5.2 Metal Deficient Oxides
Formation of either metal vacancies or oxygen interstitials (excess oxygen)
Formation occurs typically at the surface.
The following cases are possible:
2.5.2.1 If metal deficiency is dominating defect then
Depicted as metal deficient oxide M1-y O (y is the extent of non-stoichiometry)
Possible defect reaction is that of electronic compensation.
Creation of holes
Conduction due to holes i.e. a p- type conductor
Examples of oxides showing this characteristics are MnO, NiO, CoO, FeO etc.
2.5.2.2 If metal deficiency is dominating defect then
Oxides depicted as MO2+x
Oxygen interstitials can form due to following reaction
P-type conductor
Example can be an oxide like UO 2.
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Module 2: Defect Chemistry and Defect Equilibria
Dissolution of Foreign Cations in an Oxide
2.6 Dissolution of Foreign Cations in an Oxide
2.6.1 Case-1: Parent oxide is MO and foreign oxide is Mf2 O3 .
The following scenarios are likely:
i. Mf 3+ occupies M2+ sites in MO giving rise to an extra positive charge on the metal site and a
free electron according to the following defect reaction
(i)
ii. Alternatively for a metal deficient oxide MO, creates metal vacancies as
(ii)
iii. For an oxygen deficient oxide, oxygen vacancies are compensated as
(iii)
Reaction (iii) results in the reduction in vacancy concentration, while reactions (i) and (ii) result
in increase in the electron concentration or metal vacancy concentration.
iv. Reaction (i), for a p-type conductor, can be alternatively expressed as following
(iv)
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Module 2: Defect Chemistry
Concentration of Intrinsic Defects
2.7 Concentration of Intrinsic Defects
Let us consider the formation of Frenkel defects in a halide, MX, i.e.
MM + X X
V M ' + Mi •+ X X
Change in the free energy (ΔG) upon formation of
ΔG f energy per pair
'n' Frenkel defect pairs at an expense of
(2.1)
where ΔSC is the change in configurational entropy and is positive. Equilibrium concentration of
defects is found by minimizing ΔG w. r. t. n i.e. the concentration at which free energy is minimum.
Change in entropy is given by
(2.2)
where W is the number of ways in which defects can be arranged.
Now, as per the defect reaction shown above, number of Frenkel pairs (n) would lead to the
formation equal number of interstitials (n i ) as well as vacancies (n v ) i.e.
(2.3)
Assume that total number of lattice sites = N
Number of ways to arrange the vacancies, W v is
(2.4)
Ways to arrange the interstitials (assuming that N lattice sites are equivalent to N interstitial sites), W i
are
(2.5)
Total number of possible configurations
(2.6)
So, entropy change will now be
OR
or
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(2.7)
For large values of N, Sterling’s approximant
can be applied which leads to
(2.8)
and total free energy change is
(2.9)
(2.10)
Figure 2.3 Equilibrium Vacancy
Concentration
Now, if vacancies were stable defects, then at certain concentration, the free energy change has to
be minimum, as shown in the figure. Hence, at equilibrium, we can safely write that
Now at equilibrium,
(2.11)
We can also assume
since number of vacancies is much smaller than number of lattice
sites in absolute terms.
This results in
(2.12)
Now we know that ΔG f = ΔH f - TΔSv where ΔH enthalpy of Frenkel defect formation and ΔSv =
vibrational entropy change.
Hence Equation (2.12) further simplifies to
(2.13)
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Assuming that exp (ΔSv /2kT ) ~1 as vibrational entropy change is very small, and hence
(2.14)
Similarly, for Schottky defects, you can work out that
(2.15)
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Module 2: Defect Chemistry and Defect Equilibria
Intrinsic and Extrinsic Defects
2.8 Intrinsic and Extrinsic Defects
2.8.1 Intrinsic behavior
Defect which can be determined from the intrinsic defect equation and is temperature
dependent, increasing with increasing temperature.
2.8.2 Extrinsic behavior
Extrinsic defects are defects caused by impurities consisting of aliovalent cations.
Defect concentration depends upon impurity concentration which is constant and independent
of temperature. Only at very high temperatures, intrinsic behavior again dominates, and the
cross-over temperature depends upon the defect formation energy.
2.8.3 Example
Defect formation energies for some ceramic materials are
Here, one can see the relation with the melting point that melting point of MgO is
~2825°C while it is ~801°C for NaCl. So, at any given temperature NaCl will have
much larger defect concentration than MgO. However, at the same homologous
temperature, defect concentrations can be quite similar.
Interestingly, while the highest achievable purity level in MgO is 1 ppm, in NaCl, it is 50
ppm. Typically, these impurities consist of aliovalent cations which give rise to defects,
called extrinsic defects. Thus the concentration of extrinsic defects is much greater than
intrinsic defect concentration in MgO. As a result, defects in NaCl are likely to be
intrinsic but MgO is most likely to contain extrinsic defects.
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Module 2: Defect Chemistry and Defect Equilibria
Units for Defect Concentration
2.9 Units for Defect Concentration
Defect concentration fraction, n/N , is nothing but the ratio of number of defects, n, relative to
number of occupied lattice sites N i.e. defect concentration fraction. The denominator should actually
be n+N but since, N>>n, it can be approximated as n+N ~ N. Commonly used units for
concentration is #/cm 3 or cm -3
Typical defect concentration in ceramics ~ 1 ppm.
So, if the density of atoms in a solid ~10 23 cm -3 , 1 ppm concentration would be equivalent to 10 17
cm -3 .
Conversion of mole fraction to number per unit volume can be the following:
No. of formula units per unit volume =
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria
2.10 Defect Equilibria
2.10.1 Thermodynamics of Defect Reactions
A defect reaction can be treated like a chemical reaction allowing us to relate the thermodynamic
variables like pO 2 temperature to the free energy change or enthalpy change which can be
determined using experimental techniques. This allows us to establish, for example, an equilibrium
diagram between defect concentration and pO 2 , helping us to identify various regions which may be
useful under practical conditions. For detailed chemical thermodynamics, you should refer to the
appropriate subject or books.
So, if a chemical system consists of n 1 + n 2 + ---- +ni moles of constituents 1, 2, 3, ………..,i, the
partial molar free energy of Ith constituents is given as
(2.16)
Then, according to the Gibbs Duhem equation, at equilibrium
(2.17)
In a chemical reaction
Free energy change can be written as
(2.18)
where ΔG o
. Free energy change is standard state i. e. at unit activities.
At equilibrium, ΔG o = 0, , hence
(2.19)
where, K is equilibrium or reaction constant and
.
In addition, free energy can be expressed as
(2.20)
where
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which leads to K = K0 exp (-ΔH 0 /RT), where K0 = ΔS0 and R is the gas constant.
Alternatively,
(2.21)
This is an important outcome as it shows that we can treat the defects in a solid as solutes in a
solvent.
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria in Stoichiometric Oxides
2.11 Defect Equilibria in Stoichiometric Oxides
The defects which we usually consider in stoichiometric oxides are Schottky and Frenkel defects and
following paragraphs so analysis for both these kinds of defects for an oxide MO.
2.11.1 Schottky Defects
Defect reaction in an oxide MO is written as
Equilibrium constant for this reaction is
KS = [
] [V M '']
Here square brackets i.e. [ ] are used for concentration.
Equilibrium constant can be also be expressed as
(2.22)
where ΔG S is the molar free energy of defect formation and is ΔH S - TΔSS , where ΔH S is the
enthalpy for defect formation and ΔSS is the entropy change which is mainly vibrational in nature
and can be assumed to be constant. This leads to
(2.23)
If Schottky defects dominate, then
[
]
(2.24)
Here, as one can see, defect concentrations are independent of pO 2 .
2.11.2 Frenkel defects
For an oxide MO
which leads to
(2.25)
OR
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At reasonably low defect concentrations when
=
[M i ••] and [V M ''] << MM and MM ˜1
Thus
[M i ••][VM '']=KF
(2.26)
If Frenkel defects dominate, then
[M i ••]= [V M '']
i.e.
[M i ••]= [V M ''] = KF1/2
(2.27)
I n a similar manner what we did above for Schottky defects, one can now write
[M i ••]= [V M '']
(2.28)
Again we can see that the defect is independent of pO 2 .
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria in Non-Stoichiometric Oxides
2.12 Defect Equilibria in Non-Stoichiometric Oxides
2.12.1 Oxygen Deficient Oxides
In the following sections, we take example of MO type oxygen deficient oxides with cases when
either oxygen vacancies may dominate or metal excess may dominate in the form of metal
interstitials or when both kinds of defects are simultaneously present.
Case I: when oxygen vacancies are dominant defects
Case II: when metal interstitials (metal excess) dominate
Case III: both kinds of defects are present.
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria in Non-Stoichiometric Oxides
2.12.1.1 Case I: When oxygen vacancies are dominant defects
Here, the defect reaction can be expressed as
The reaction constant will be
K=[
]
(2.29)
Here,we assume that [OO ] = 1. From the above reaction, to maintain the electrical neutrality,
n e = 2 [V O ••].
Thus
(2.30)
or
(2.31)
or
(2.32)
and
[
]=
(2.33)
The relationship between the defect concentration and pO 2 can be seen in the schematic figure
where defect concentration varies as pO 2 -1/6 . This makes sense because concentration of vacancies
will go down as we supply more oxygen to the material.
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Figure 2. 4
Defect concentration vs pO2 in an oxygen
deficient oxide with oxygen vacancy as dominating defect
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria in Non-Stoichiometric Oxides
2.12.1.2 Case II: If interstitial metal or metal excess is present
Here, the defect reaction will be
The equilibrium constant can be written as
(2.34)
Both [M M ] and [OO ] can be assumed to be ~1 if [M i ••] << [M M ] and [OO ].
According to the electrical neutrality condition
ne = 2 [ Mi ••]
(2.35)
Thus
OR
ne
(2.36)
The plot will be similar to that of the above case.
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria in Non-Stoichiometric Oxides
2.12.1.3 Case III: Simultaneous presence of oxygen vacancies and metal
interstitials
Such a scenario is often found in ceramic oxides like TiO2 , and Nb2 O 5 .
Consider a metal oxide (MO 2 ) with doubly charged oxygen vacancies and metal ion interstitials. The
corresponding defect reaction is
OR
Assuming
, the defect equilibrium can be written as
[VO••] n e2 pO2 1/2 =K1
(2.37)
[
(2.38)
]
According to the electrical neutrality condition
ne
= 2[ V O••] + 2[ Mi ••]
(2.39)
Two limiting cases can be considered:
When [V0 •• ] >>[Mi•• ]
= [ V O••] =
(2.40)
And
(2.41)
i.e.
[ Mi ••]
(2.42)
As you can see, under such conditions, [M i ••] decreases more rapidly with increasing pO 2 . This is
commonly observed in TiO2 and Nb2 O 5 where [V 0 ••] can be 10 10 times higher than [M i ••].
••
••
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When [M i ] >>[V 0 ]
Following similar exercise as above, we can calculate
[ Mi ••] =
=
(2.43)
and
[
]=
(2.44)
Here, [V 0 ••] increases with increasing pO 2 while keep decreasing with increasing pO 2 but at a
different rate.
Figure 2. 5
Defect concentration vs pO2 in
an oxygen deficient oxide with oxygen
vacancy as dominating defect
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria in Non-Stoichiometric Oxides
2.12.2 Metal Deficient Oxides
Now we turn towards the case of MO type oxides with deficient of metal which can be reflected
either by metal vacancies or oxygen interstitials or presence of both. Here we do analysis only for
metal vacancies while other two cases can be done in a similar fashion as shown in previous
paragraph.
For MO oxide, assuming complete ionization of vacancies, we can write
whose equilibrium constant will be
(2.45)
If
then
(2.46)
According to the electrical neutrality condition
(2.47)
Again, the concentration of defects is proportional to pO 2 1/6 .
One can do similar exercise for the cases when oxygen interstitial is the main defect and also when
there is mixed presence of metal vacancies and oxygen interstitials. This is left to the readers to
perform themselves.
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibria in Non-Stoichiometric Oxides
2.12.3 Intrinsic Ionization
Intrinsic ionization leads to the formation of electrons and holes via
Equilibrium constant is
(2.48)
If n e = n h ,
(2.49)
Again, one sees that concentration of electron and holes are independent of oxygen pressure.
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Module 2: Defect Chemistry and Defect Equilibria
Defect Structures involving Oxygern Vacancies and Interstitials
2.13 Defect Structures involving Oxygen Vacancies and
Interstitials
Depending upon the partial pressure of oxygen, an oxide may be oxygen deficient (or metal excess)
or metal deficient (or oxygen excess).
Let us consider the following conditions in an oxide MO:
Low pO 2 i.e. oxygen vacancies dominate.
High pO 2 i.e. oxygen interstitials dominate.
At intermediate pO 2 i.e. oxide is stoichiometric.
Assuming that both oxygen vacancies and oxygen interstitials are doubly charged (fully ionized), the
defect reactions can be written as follows:
At low pO 2
The defect reaction can be written as
+[
] + 2e'
The corresponding reaction constant, assuming [M M ] and [OO ] =1, would be
[
]
(2.50)
At high pO 2
The defect reaction is
and hence the reaction constant is
(2.51)
At intermediate pO 2
Stoichiometric defects are likely to prevail i.e. either via intrinsic ionization or Anti-Frenkel
defects.
Intrinsic ionization of electrons and holes
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and corresponding reaction constant is
(2.52)
Similarly formation of oxygen Frenkel defects (Anti-) leads to
with reaction constant as
.[
]
(2.53)
From the above four relations, we can write
(2.54)
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Module 2: Defect Chemistry and Defect Equilibria
Defect Structures involving Oxygern Vacancies and Interstitials
2.13.1 Limiting Conditions
Now we need to determine the limiting condition for determining the boundaries of pO 2 across which
various defect concentrations can be plotted as a function of oxygen partial pressure. These three
regions are regions of
Low pO 2 ,
Intermediate pO 2 , and
High pO 2
These regions depict oxygen deficit (or metal excess), stoichiometric composition and oxygen excess
(or metal deficiency) respectively. Following sections eluciate the process for determining these
boundaries for a metal oxide with either of oxygen deficit, stoichiometric composition and oxygen
excess for an oxide considering anti-Frenkel defects.
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Module 2: Defect Chemistry and Defect Equilibria
Defect Structures involving Oxygern Vacancies and Interstitials
2.13.1.1 Low pO 2 i.e. oxygen deficit
At large oxygen deficit, we can assume that
[
]
(2.55)
Thus from (2.50)
(2.56)
Substituting in (2.53)
(2.57)
And from (2.52)
(2.58)
Combining (2.56)-(2.58) and using (2.55), we get the following condition
(2.59)
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Module 2: Defect Chemistry and Defect Equilibria
Defect Structures involving Oxygern Vacancies and Interstitials
2.13.1.2 Excess oxygen i.e. high pO 2
At large oxygen excess, we can assume that
[
]
(2.60)
In such a situation, from (2.51), we get
(2.61)
and from (2.53), we get
[
]=
(2.62)
Now, from (2.52), we get
(2.63)
Now combining (2.61)-(2.63) and using (2.60), we get an important condition i.e.
(2.64)
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Module 2: Defect Chemistry and Defect Equilibria
Defect Structures involving Oxygern Vacancies and Interstitials
2.13.1.3 Stoichiometric Condition, i.e., Intermediate pO 2
Case - I: Intrinsic ionization dominates i.e.
The defect reaction is
The corresponding reaction constant is
OR
[
] and
(2.65)
Here both n e and n h are independent of pO 2 while the point defect concentrations are given as from
(2.50)
(2.66)
and from (2.51)
(2.67)
Case – II: Internal disorder and anti-Frenkel defects dominate i.e.
The reactions are
(2.68)
Now, since
and [
] are independent of pO 2 , using (2.50) and (2.51) respectively, n e and
n h are given as
(2.69)
(2.70)
The above equations provide the limiting conditions of oxygen partial pressure separating three
regimes of oxygen pressures with variations of defect concentration vs pO 2 obtained. From this we
can plot a defect concentration vs pO 2 plot, also called as Brouwer’s Diagram. Such diagrams are
extremely important in defect chemistry to understand the dominating defects which govern the
physical processes.
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Module 2: Defect Chemistry and Defect Equilibria
Defect Equilibrium Diagram
2.14 Defect Equilibrium Diagram
2.14.1 Frenkel defects dominating at stoichiometric composition
The following diagram is obtained when Frenkel defect dominates i.e. the internal disorder of the
material dominates in the intermediate pressure range.
The best way to draw the diagram is to first draw the central region i.e. making Vo = O i and then
extend the lines of Vo and O i into low and high pressure region with appropriate slopes depending
upon the oxide stoichiometry. Then, draw the electron and hole concentrations, n and p, in the low
and high pressure regions respectively since their relationship to Vo and O i is straightforward. Then
extend these in the intermediate region and low/high pO 2 region depending according to the slopes
obtained from the analysis. This process yields the diagram as shown in the figure below.
Figure 2. 6
Concentration of ionic defects vs pO2 with
Oxygen Frenkel defects dominating at stoichiometric
composition
2.14.2 Intrinsic ionization dominating at stoichiometric composition
Using the proceedure similar to that explained in the previous slide except that in the central region
now n=p as intrinsic ionization dominates at the stoichiometric composition, we obtained the following
figure. There are subtle differences as we can observe by comparing the two figures.
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Figure 2. 7
Concentration of ionic defects vs pO2 with
intrinsic ionization dominating at stoichiometric composition
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Module 2: Defect Chemistry and Defect Equilibria
A Simple Procedure for Constructing at Brouwer's Diagram
u
2.15 A Simple General Procedure for Constructing a Brouwer's
Diagram
1. First one needs to determine how many defects are relevant. This can be, to a large extent,
determined by crystal structure, solute concentration and electrical conductivity or diffusion
rates. For example, one can neglect Frenkel defects i.e. interstitial for closed packed
structures where Schottky defects can be dominant i.e. KF << KS .
2. Write independent defect reactions for each defect and K values e.g. oxidation and reduction
are not independent, related through intrinsic electronic reaction.
Intrinsic defect formation mechanism
Oxidation or reduction (not both)
3. N-1 equations are obtained for N defect concentrations and another electronically apparition.
4. Define regions of pO 2 .
5. Observe which defect concentration decreases or increases with the change of pO 2 .
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Module 2: Defect Chemistry and Defect Equilibria
Extent of Non-Stoichiometry
2.16 Extent of Non-Stoichiometry
In highly stoichiomteric pure oxides such as MgO, Al 2 O 3 , ZrO 2 , the extent of oxidation or
reduction is very small. These are often characterized by large energy for oxidation or
reduction. Changes in oxygen pressure have very little effect on the defect concentration.
When cations are of fixed valence, the tendency for retaining the stoichiometry is even larger.
Oxides containing multivalent cations, such as transition elements, are much more prone to be
non-stoichiometric. Examples are TiO2+x , BaTiO3-x and SrTiO3-x where Ti4+ ions can be
easily reduced to Ti3+ creating oxygen deficiency of order 1% within the limits of the stability
of oxide i.e. before decomposition and phase change.
Transition metal mono-oxide series Ni 1-x O, Co1-x O, Mn 1-x O and Fe 1-x O are the oxides in
which a fraction of the divalent cations is easily oxidized to the divalent state resulting in
cation deficiency, x . The deficiency is ~5x10-4 % for Ni 1-x O, ~1% for Co1-x O, ~0.1% for
Mn 1-x O and ~0.15% for Fe 1-x O. FeO is seldom stoichiometric and it has a minimum nonstoichiometry of 0.05%.
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Module 2: Defect Chemistry and Defect Equilibria
Example: Comparative Behaviour of TiO 2 and MgO vis-à-vis Oxygen Pressure
2.17 Example: Comparative Behavior of TiO 2 and MgO vis-à-vis
Oxygen Pressure
Magnesium oxide in intrinsic form, primarily contains Schottky defects. However, under oxidizing
conditions, defect reaction would be
with the equilibrium constant as
which is experimentally determined to be
It can be seen from the above equations that Kox increases as the temperature increases; i.e. MgO
can be oxidized at high temperatures producing extrinsic Mg vacancies and holes if no other defects
are present.
Further from electrical neutrality condition,
It should be noted that though actual concentrations can be very small but they still do vary with
temperature and oxygen pressure.
For example, at 80% of Tm (2480 K) in air i.e. pO 2 = 0.21 atm
which is equivalent to a deficiency of x=0.6 ppm in Mg 1-x O,
While in pO 2 of 10 -9 MPa,
which is equivalent to a deficiency level of x=0.04 ppm in Mg 1-x O.
In comparison, TiO2-x is more non-stoichiometric and prone to having oxygen deficiency. The
reduction reaction is
From the electrical neutrality condition
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[
]
and
The reaction constant is
=[
]
MPa.cm-1 .
and is experimentally determined to be
Now, at 0.8 Tm , i.e. 1690 K
In air
[
]=
which is equivalent to a deficiency (x) of 93 ppm.
While at a pO 2 of 10 -9 MPa,
[
]=
which is equivalent to a deficiency of x~0.27% in TiO2-x and makes TiO2-x an n-type semiconductor
due to the resulting high electron concentration.
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Module 2: Defect Chemistry and Defect Equilibria
Electronic Disorder
2.18 Electronic Disorder
Unlike intrinsic point defeats, intrinsic electronic defects are optically or thermally created.
This occurs in materials having a forbidden energy gap between conduction and the valence band
and are categorized as semiconductors and insulators (see footnote**).
Defect density is in number per unit volume of the crystal.
Disorder implies elevation of electrons into higher energy levels creating vacant states in lover
energy bands which are called as holes.
Excitation of electrons across the bandgap into conduction band
Bandgap (Eg) for semiconductors is typically below 2.5 eV e.g. Si has bandgap of ~1.1 eV
whereas for insulators it is typically above 2.5-3 eV.
The band diagram for a semiconductor or insulator can be seen below.
Figure 2.8 Band diagram for an insulator
Band gap energy values for a few selected materials are shown in the
table below:
Si
1.1 eV
NaCl
7.3 eV
Ge
0.7 eV
MgO
7.8 eV
Diamond
5.4 eV
NiO
4.2 eV
GaAs
1.43 eV
FeO
2.0 eV
GaP
2.25 eV
BaTiO3
2.8 eV
BN
4.8 eV
TiO2
3.0 eV
CdTe
1.44 eV
UO 2
5.2 eV
ZnO
3.2 eV
SiO2
8.5 eV
3.1 eV
MgAl2 O 4
7.8 eV
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** Basically, materials having a well defined band gap show conduction band, band of higher energy
and valence band, bands of lower energies with maximum of valence band and minimum of conduction
band separated by the forbidden energy gap i.e. Eg . The position of Fermi energy, EF, lies in this
forbidden gap. At 0 K, all the states in the valence band are filled while the states in the conduction
band are empty. Another way to express this is that all the energy states below EF are filled while
those above EF, are empty at 0 K. It is just that for basic physics reasons, carriers cannot reside in the
forbidden energy gap.
Elementary physics of bands in materials can read from any book related to solid state physics or
electronic properties of materials as listed in the bibliography.
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Module 2: Defect Chemistry and Defect Equilibria
Electronic Disorder
2.18.1 For a Pure or Intrinsic Compound (semiconductor or insulator)
Concentrations of electrons (n e ) and holes ((nh ) are equal at any temperature and are given as
(2.71)
(2.72)
where N c and N v are defined as effective conduction and valence band density of states and are
expressed as
and
(2.73)
Here me * and mh * are the effective mass of electron and holes respectively, k is the Boltzmann’s
constant, h is the Planck’s constant and T is the temperature. The value of N c and N v is ~10 19 cm -3
at 300 K.
Electron and hole concentrations in the conduction and valence bands are equal i.e.
(2.74)
Fermi energy is defined as
(2.75)
where
.
In oxides, me * and mh * are generally 2-10 times larger than the mass of free electron, me . Since,
the atomic density of solids is about 10 23 cm -3 , the density of states is about four order of
magnitude lower than that in semiconductors.
Moreover, if the second term in equation (2.75) is small and is actually zero if me * = mh * , then
Fermi level is quite close to the center of the band gap.
Hence for an intrinsic compound where n e = n h .
(2.76)
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Fermi level usually remains in the middle of the band gap but can shift up or down when materials is
doped. Typically EF moves up, from the center of the bandgap towards EC , for n-type doping and
moves down for p-type doping.
The above expressions have striking similarity to the concentration of lattice defects where
(2.77)
The density of electronic states may be thought of as equivalent to the density of vacancies in the
lattice sites.
The excitation of electrons across the band gap can be depicted by a chemical defect reaction as
follows
The equilibrium constant is
(2.78)
At 300K
(2.79)
Here Ki is not unit-less unlike the reaction constant in the defect reactions because n e and n h have
the units of cm -3 .
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Module 2: Defect Chemistry and Defect Equilibria
Examples of Intrinsic Electronic and Ionic Defect Concentrations
2.19 Examples
2.19.1 Intrinsic electronic and ionic defect concentrations in MgO
Consider that a material like MgO usually has Schottky defects with enthalpy of formation (ΔH F) ) of
about 7.7 eV. Its band gap is about 7.65 eV which decreases at a rate of 1 meV per K as MgO is
heated.
The question is that in case of an absolutely pure and stoichiometric MgO, which defects are likely to
be created and present in higher concentrations at a temperature of say 1400°C or 1673 K?
We can calculate the Schottky defect concentration as
Electron and hole concentrations are calculated as
In MgO,
and
where mo is the mass of free electron and is 9.1×10 31 kg.
At 1673K, Eg = 7.85 eV – (1570*1*10 3 ) eV = 6.28 eV .
Hence, n e = n h = 4.6*10 10 cm 3
Now magnesium vacancy concentration can be calculated as
[
]
Hence, at 1400°C, despite high energy of Schottky defect formation, the vacancy concentration will be
slightly larger than the electronic carrier concentration due to thermal excitation.
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Module 2: Defect Chemistry and Defect Equilibria
Examples of Intrinsic Electronic and Ionic Defect Concentrations
2.19.2 Role of Donor and Acceptors
In semiconductors such as Si, donors such as As and P are used used for n-type behavior and
acceptor atoms such as B and Al are used for p-type behavior. These donor and acceptor atoms
basically create donor and acceptor energy levels very close to conduction and valence band
respectively, such that the difference with the band edges is approximately equal to kT at room
temperature.
In Ionic solids, all the ionic defects with non-zero effective charge can be viewed as either a donor or
acceptor. Obviously, defects with positive charge act as donors while those with negative charge act
as acceptors.
Figure 2.9 Figure showing the positions of
impurity energy levels in the band diagram of
MgO
For example, oxygen vacancy can be viewed as donor according to the following reaction
The ionization energies are
These energy levels are situated with respect to the conduction and valence band edges in the band
gap.
For example in MgO, Al acts as a donor while Na acts as an acceptor according to the following
reactions:
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Similarly, Cl acts as a donor while N acts as an acceptor.
In case of BaTiO3 , substitution of Ba by La leads to an electron i.e. La acts as a donor whereas Al
and Fe substitution on Ti sites leads to creation of holes and hence these are termed as acceptor
impurities. Y atom can replace either Ba or Ti due to its intermediate size .
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Module 2: Defect Chemistry and Defect Equilibria
Examples of Intrinsic Electronic and Ionic Defect Concentrations
2.19.3 Electronic vs Ionic Compensation of Solutes
Here we will discuss which of the electronic or ionic compensation of solute incorporation in oxides is
favoured and what are the conditions determining this.
In oxide semiconductors, the effectiveness of a donor or an acceptor is not only governed by their
ionization energies, it is also governed by the extent of oxidation and reduction, even in case of
shallow dopants with smaller ionization energies. This is due to the fact that an aliovalent impurity in
an ionic compound can be charge compensated by ionic defects (ionically compensated ) or by
electrons or holes (electronically compensated ) or by a combination of the two. Variables governing
the extent of these are pO 2 , dopant concentration and temperature.
We will take the example of Nb 2 O 5 doping in TiO2 .
The defect reactions are written as
Ionic compensation
(1)
Electronic
compensation
(2)
Combination of the two reactions i.e. ((1) – (2)) leads to
(3)
Equation (3) shows that as pO 2 increases, oxidation is favored and hence formation of titanium
vacancies is more likely. Similarly, as the temperature reduces, oxidation is again favored.
Thus Nb doping of TiO2 tends to be compensated by VTi '''' if Nb2 O 5 concentration is large, pO 2 is
high and the temperature is low ,whereas the inverse conditions favour the electronic compensation.
In any case, the electrical neutrality condition requires that
(4)
Similar effects are observed in care of titanates such as BaTiO3 .
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Module 2: Defect Chemistry and Defect Equilibria
Examples of Intrinsic Electronic and Ionic Defect Concentrations
2.19.4 Point Defects and Crystal Density in ZrO2 when doped with CaO
CaO doping in ZrO 2 allows stabilization of high temperature tetragonal or cubic polymorphs of ZrO 2
at room temperature as a metastable phase.
CaO doping also introduces the defects according to
(1)
OR
(2)
From (1), oxygen vacancies formed allow ZrO 2 to possess high ionic conductivity as a result cubic
ZrO 2 is used as electrolyte. When CaO in ZrO 2 dissolves substitutionally, it has to be compensated
by creation of a positive charge.
These substitutions also lead to a change in crystal density of ZrO 2 .
Figure 2.10 Density of ZrO2 as a function of CaO
doping
For more case studies, readers are referred to Chapter 2 of Physical Ceramics by Chiang,
Birnie and Kingery (see bibliography).
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Module 2: Defect Chemistry and Defect Equilibria
Summary
Summary
In this module, we discussed the defect formation in ceramic, with references to oxides. While
defects such as Schottky defects maintain the stoichiometry of the materials, most oxide ceramics
are prone to the non-stoichiometry. This non-stoichiometry results in defects such as ion vacancies
or interstitials and compensating charged defects either via ionic compensation or electronic
compensation. While defect concentration in the intermediate oxygen partial pressure (around
atmospheric conditions) are independent of the partial pressure of oxygen, the defects in nonstoichiotemetric oxides either at high or low pressures are strongly dependent on the partial pressure
of oxygen. This can be effectively understood through the construction of Brower diagrams. Finally,
we looked at the electronic disorder and evaluated the conditions to compare the ionic and electronic
defect concentrations.
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