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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 ''] [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
