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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 pointdefects or 0-D defects, line defects or 1-D defects and 2-D or surface defects. These defects play animportant role in determining the properties of ceramic materials and in this context, the role of pointdefects is extremely important. In this module, we will learn about various point defects, the role ofstoichiometry i.e. cation and anion excess and deficit, the role of foreign atoms on the defectchemistry. Subsequently, we will adopt a simple thermodynamic basis for calculating theirconcentration in equilibrium and then will extend the Gibbs-Duhem relation for chemical systems tothe defects in ceramics considering them to be equivalent to the dilute solutions, an approximationwhich is fairly valid. This will lead us to the determination of defect concentrations as a function ofpartial pressure of oxygen which is an important exercise to establish the defect concentration vs pO2

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 TiO2 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 MetalOxides (Science & Technology of Materials), P.K. Kofstad, John Wiley andSons 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 andFrancis.

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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 orstoichiometry. These could be missing lattice ions from their positions, interstitial ions orsubstitutional ions (or impurities) and valence electrons and/or holes.

Usually, point defects in metals are electrically neutral whereas in ionic oxides, these areelectrically 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 defectswhen valence electrons are excited into higher energy orbitals/ levels and lead toformation of electron or holes.

Defects are present in most oxides and are easily understood. Hence most examples in thefollowing 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 tablesprovide the most common notations.

Regular Sites

Mm: normal or regular occupied metal orcation site

Oo: normal or regular occupied oxygen oranion site

Point Defects (a • (dot) means a positive charge and a ' (prime) means anegative 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 regularmetal site

Mfm

Foreign cation oninterstitial site

Mfi

A normal cation or anion inan oxide with zero effectivecharge

MMx or OO

x

Charged oxygen vacancy: VO• or VO

••

Charged metal vacancy VM' or VM

''

Charged metal or oxygeninterstitial Mi

•• and Oi''

Neutral cation and anionvacancies VM

x or VOx

electrons and holes 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 complimentaryelectronic 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. VOand Oi or VMand Mi

Misplaced atoms interchanged (MOand OM) - interchanged

Vacancies and misplaced atoms for the same kind of atom (VM + MO)

Interstitial and misplaced atoms i.e.,Oi and MO

Interstitial atoms i.e. Mi and Oi

Among all of these, the first two are most important as these are regularly seen in many importantoxides. 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 diffusesinto the crystal unit as equilibrium is reached.

Figure 2.1 SchottkyDisorder

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 correspondingdefect reaction in that case would be

0 V0••+ Oi

''

Examples of compounds showing this defect are AgBr type compounds such as AgBr, AgIetc.

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-stoichiometrydetermines the concentration of Defects.

In such oxides, electrical neutrality is preserved via the formation of point defects andelectronic 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 O2↑

Due to loss of oxygen, possible defect reactions would be

Electronic compensation leading to creation oxygen vacancies and of electrons

O0 VO••+ ½ O2+ 2e'

Ionic compensation leads to formation of oxygen vacancies and reduction of metal ionson their sites.

O0 VO••+ ½ O2+ 2M'M

2.5.1.2 If metal interstitials are the dominating defects then,

Depicted as (M1+yO2 is the extent of non-stoichiometry)

Possible defect reactions are

Ionic compensation leading to the formation of metal interstitials and reduction of metalions 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, ZrO2, CeO2, Nb2O5

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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-yO (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 UO2.

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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 Mf2O3.

The following scenarios are likely:

i. Mf3+ 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) resultin 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+ XX VM' + Mi•+ XX

Change in the free energy (ΔG) upon formation of 'n' Frenkel defect pairs at an expense ofΔGf energy per pair

(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 theformation equal number of interstitials (ni) as well as vacancies (nv) i.e.

(2.3)

Assume that total number of lattice sites = NNumber of ways to arrange the vacancies, Wv is

(2.4)

Ways to arrange the interstitials (assuming that N lattice sites are equivalent to N interstitial sites), Wiare

(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 VacancyConcentration

Now, if vacancies were stable defects, then at certain concentration, the free energy change has tobe 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 latticesites in absolute terms.

This results in

(2.12)

Now we know that ΔGf = ΔHf - 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 temperaturedependent, 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 independentof temperature. Only at very high temperatures, intrinsic behavior again dominates, and thecross-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 havemuch larger defect concentration than MgO. However, at the same homologoustemperature, defect concentrations can be quite similar.

Interestingly, while the highest achievable purity level in MgO is 1 ppm, in NaCl, it is 50ppm. Typically, these impurities consist of aliovalent cations which give rise to defects,called extrinsic defects. Thus the concentration of extrinsic defects is much greater thanintrinsic defect concentration in MgO. As a result, defects in NaCl are likely to beintrinsic 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 tonumber of occupied lattice sites N i.e. defect concentration fraction. The denominator should actuallybe n+N but since, N>>n, it can be approximated as n+N ~ N. Commonly used units for

concentration is #/cm3 or cm-3

Typical defect concentration in ceramics ~ 1 ppm.

So, if the density of atoms in a solid ~1023 cm-3, 1 ppm concentration would be equivalent to 1017

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 thermodynamicvariables like pO2 temperature to the free energy change or enthalpy change which can be

determined using experimental techniques. This allows us to establish, for example, an equilibriumdiagram between defect concentration and pO2, helping us to identify various regions which may be

useful under practical conditions. For detailed chemical thermodynamics, you should refer to theappropriate subject or books.

So, if a chemical system consists of n1 + n2 + ---- +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 ΔGo . Free energy change is standard state i. e. at unit activities.

At equilibrium, ΔGo = 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 = K0exp (-ΔH0/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 asolvent.

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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 andfollowing 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 = [ ] [VM'']

Here square brackets i.e. [ ] are used for concentration.Equilibrium constant can be also be expressed as

(2.22)

where ΔGS is the molar free energy of defect formation and is ΔHS - TΔSS, where ΔHS 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 pO2.

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

=

[Mi••] and [VM''] << MM and MM˜1

Thus

[Mi••][VM'']=KF (2.26)

If Frenkel defects dominate, then

[Mi••]=[VM'']

i.e.

[Mi••]=[VM'']=KF

1/2 (2.27)

In a similar manner what we did above for Schottky defects, one can now write

[Mi••]=[VM''] (2.28)

Again we can see that the defect is independent of pO2.

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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 wheneither oxygen vacancies may dominate or metal excess may dominate in the form of metalinterstitials 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,

ne = 2 [VO••].

Thus

(2.30)

or

(2.31)

or

(2.32)

and

[ ] = (2.33)

The relationship between the defect concentration and pO2 can be seen in the schematic figure

where defect concentration varies as pO2-1/6. 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 oxygendeficient oxide with oxygen vacancy as dominating defect

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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 [MM] and [OO] can be assumed to be ~1 if [Mi••] << [MM] 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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2.12.1.3 Case III: Simultaneous presence of oxygen vacancies and metalinterstitials

Such a scenario is often found in ceramic oxides like TiO2, and Nb2O5.

Consider a metal oxide (MO2) with doubly charged oxygen vacancies and metal ion interstitials. The

corresponding defect reaction is

OR

Assuming , the defect equilibrium can be written as

[VO••] ne

2 pO21/2=K1 (2.37)

[ ] (2.38)


ne = 2[VO••] + 2[Mi

••] (2.39)

Two limiting cases can be considered:

When [V0••] >>[Mi

••]

= [VO••] = (2.40)

And

(2.41)

i.e.

[Mi••] (2.42)

As you can see, under such conditions, [Mi••] decreases more rapidly with increasing pO2. This is

commonly observed in TiO2 and Nb2O5 where [V0••] can be 1010 times higher than [Mi

••].

•• ••

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When [Mi ] >>[V0 ]

Following similar exercise as above, we can calculate

[Mi••] = = (2.43)

and

[ ] = (2.44)

Here, [V0••] increases with increasing pO2 while keep decreasing with increasing pO2 but at a

different rate.

Figure 2. 5 Defect concentration vs pO2 inan oxygen deficient oxide with oxygenvacancy as dominating defect

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2.12.2 Metal Deficient Oxides

Now we turn towards the case of MO type oxides with deficient of metal which can be reflectedeither by metal vacancies or oxygen interstitials or presence of both. Here we do analysis only formetal vacancies while other two cases can be done in a similar fashion as shown in previousparagraph.

For MO oxide, assuming complete ionization of vacancies, we can write

whose equilibrium constant will be

(2.45)

If then

(2.46)


(2.47)

Again, the concentration of defects is proportional to pO21/6.

One can do similar exercise for the cases when oxygen interstitial is the main defect and also whenthere is mixed presence of metal vacancies and oxygen interstitials. This is left to the readers toperform themselves.

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2.12.3 Intrinsic Ionization

Intrinsic ionization leads to the formation of electrons and holes via

Equilibrium constant is

(2.48)

If ne = nh,

(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 andInterstitials

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 pO2 i.e. oxygen vacancies dominate.

High pO2 i.e. oxygen interstitials dominate.

At intermediate pO2 i.e. oxide is stoichiometric.

Assuming that both oxygen vacancies and oxygen interstitials are doubly charged (fully ionized), thedefect reactions can be written as follows:

At low pO2

The defect reaction can be written as

+ [ ] + 2e'

The corresponding reaction constant, assuming [MM] and [OO] =1, would be

[ ] (2.50)

At high pO2

The defect reaction is

and hence the reaction constant is

(2.51)

At intermediate pO2

Stoichiometric defects are likely to prevail i.e. either via intrinsic ionization or Anti-Frenkeldefects.

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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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 pO2 across which

various defect concentrations can be plotted as a function of oxygen partial pressure. These threeregions are regions of

Low pO2,

Intermediate pO2, and

High pO2

These regions depict oxygen deficit (or metal excess), stoichiometric composition and oxygen excess(or metal deficiency) respectively. Following sections eluciate the process for determining theseboundaries for a metal oxide with either of oxygen deficit, stoichiometric composition and oxygenexcess for an oxide considering anti-Frenkel defects.

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2.13.1.1 Low pO2 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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2.13.1.2 Excess oxygen i.e. high pO2

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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2.13.1.3 Stoichiometric Condition, i.e., Intermediate pO2

Case - I: Intrinsic ionization dominates i.e.

The defect reaction is

The corresponding reaction constant is

OR

[ ] and (2.65)

Here both ne and nh are independent of pO2 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 pO2 , using (2.50) and (2.51) respectively, ne and

nh are given as

(2.69)

(2.70)

The above equations provide the limiting conditions of oxygen partial pressure separating threeregimes of oxygen pressures with variations of defect concentration vs pO2 obtained. From this we

can plot a defect concentration vs pO2 plot, also called as Brouwer’s Diagram. Such diagrams are

extremely important in defect chemistry to understand the dominating defects which govern thephysical 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 thematerial dominates in the intermediate pressure range.

The best way to draw the diagram is to first draw the central region i.e. making Vo = Oi and then

extend the lines of Vo and Oi 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 lowand high pressure regions respectively since their relationship to Vo and Oi is straightforward. Then

extend these in the intermediate region and low/high pO2 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 withOxygen Frenkel defects dominating at stoichiometriccomposition

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 regionnow n=p as intrinsic ionization dominates at the stoichiometric composition, we obtained the followingfigure. 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 withintrinsic 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

2.15 A Simple General Procedure for Constructing a Brouwer'sDiagram

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 diffusionrates. For example, one can neglect Frenkel defects i.e. interstitial for closed packedstructures 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 reductionare 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 pO2.

5. Observe which defect concentration decreases or increases with the change of pO2.

u

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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, Al2O3, ZrO2, the extent of oxidation or

reduction is very small. These are often characterized by large energy for oxidation orreduction. 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 stabilityof oxide i.e. before decomposition and phase change.

Transition metal mono-oxide series Ni1-xO, Co1-xO, Mn1-xO and Fe1-xO 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 Ni1-xO, ~1% for Co1-xO, ~0.1% for

Mn1-xO and ~0.15% for Fe1-xO. FeO is seldom stoichiometric and it has a minimum non-

stoichiometry of 0.05%.

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Module 2: Defect Chemistry and Defect Equilibria Example: Comparative Behaviour of TiO2 and MgO vis-à-vis Oxygen Pressure

2.17 Example: Comparative Behavior of TiO2 and MgO vis-à-visOxygen Pressure

Magnesium oxide in intrinsic form, primarily contains Schottky defects. However, under oxidizingconditions, 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 defectsare present.

Further from electrical neutrality condition,

It should be noted that though actual concentrations can be very small but they still do vary withtemperature and oxygen pressure.

For example, at 80% of Tm (2480 K) in air i.e. pO2 = 0.21 atm

which is equivalent to a deficiency of x=0.6 ppm in Mg1-xO,

While in pO2 of 10-9 MPa,

which is equivalent to a deficiency level of x=0.04 ppm in Mg1-xO.

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

= [ ]

and is experimentally determined to be MPa.cm-1 .

Now, at 0.8 Tm, i.e. 1690 K

In air

[ ] =

which is equivalent to a deficiency (x) of 93 ppm.

While at a pO2 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 bandand 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 loverenergy 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 eVwhereas 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 thetable 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 UO2 5.2 eV

ZnO 3.2 eV SiO2 8.5 eV

3.1 eV MgAl2O4 7.8 eV

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** Basically, materials having a well defined band gap show conduction band, band of higher energyand valence band, bands of lower energies with maximum of valence band and minimum of conductionband 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 conductionband 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 orelectronic 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 (ne) and holes ((nh) are equal at any temperature and are given as

(2.71)

(2.72)

where Nc and Nv 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 Nc and Nv is ~1019 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 1023 cm-3, the density of states is about four order ofmagnitude 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 ne = nh.

(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 isdoped. 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 thelattice sites.

The excitation of electrons across the band gap can be depicted by a chemical defect reaction asfollows

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 ne and nh 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 Examples2.19.1 Intrinsic electronic and ionic defect concentrations in MgO

Consider that a material like MgO usually has Schottky defects with enthalpy of formation (ΔHF)) 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 isheated.

The question is that in case of an absolutely pure and stoichiometric MgO, which defects are likely tobe 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 9.1×1031 kg.

At 1673K, Eg = 7.85 eV – (1570*1*103 ) eV = 6.28 eV .

Hence, ne = nh = 4.6*1010 cm3

Now magnesium vacancy concentration can be calculated as

[ ]

Hence, at 1400°C, despite high energy of Schottky defect formation, the vacancy concentration will beslightly larger than the electronic carrier concentration due to thermal excitation.

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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 andacceptor atoms such as B and Al are used for p-type behavior. These donor and acceptor atomsbasically create donor and acceptor energy levels very close to conduction and valence bandrespectively, such that the difference with the band edges is approximately equal to kT at roomtemperature.

In Ionic solids, all the ionic defects with non-zero effective charge can be viewed as either a donor oracceptor. Obviously, defects with positive charge act as donors while those with negative charge actas acceptors.

Figure 2.9 Figure showing the positions ofimpurity energy levels in the band diagram ofMgO

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 bandgap.

For example in MgO, Al acts as a donor while Na acts as an acceptor according to the followingreactions:

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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 acceptorimpurities. Y atom can replace either Ba or Ti due to its intermediate size.

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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 isfavoured and what are the conditions determining this.

In oxide semiconductors, the effectiveness of a donor or an acceptor is not only governed by theirionization energies, it is also governed by the extent of oxidation and reduction, even in case ofshallow dopants with smaller ionization energies. This is due to the fact that an aliovalent impurity inan ionic compound can be charge compensated by ionic defects (ionically compensated ) or byelectrons or holes (electronically compensated ) or by a combination of the two. Variables governingthe extent of these are pO2, dopant concentration and temperature.

We will take the example of Nb2O5 doping in TiO2.

The defect reactions are written as

Ionic compensation (1)Electroniccompensation (2)

Combination of the two reactions i.e. ((1) – (2)) leads to

(3)

Equation (3) shows that as pO2 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 Nb2O5 concentration is large, pO2 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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2.19.4 Point Defects and Crystal Density in ZrO2 when doped with CaO

CaO doping in ZrO2 allows stabilization of high temperature tetragonal or cubic polymorphs of ZrO2

at room temperature as a metastable phase.

CaO doping also introduces the defects according to

(1)

OR

(2)

From (1), oxygen vacancies formed allow ZrO2 to possess high ionic conductivity as a result cubic

ZrO2 is used as electrolyte. When CaO in ZrO2 dissolves substitutionally, it has to be compensated

by creation of a positive charge.

These substitutions also lead to a change in crystal density of ZrO2.

Figure 2.10 Density of ZrO2 as a function of CaOdoping

For more case studies, readers are referred to Chapter 2 of Physical Ceramics by Chiang,Birnie and Kingery (see bibliography).

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Summary

Summary

In this module, we discussed the defect formation in ceramic, with references to oxides. Whiledefects such as Schottky defects maintain the stoichiometry of the materials, most oxide ceramicsare prone to the non-stoichiometry. This non-stoichiometry results in defects such as ion vacanciesor interstitials and compensating charged defects either via ionic compensation or electroniccompensation. While defect concentration in the intermediate oxygen partial pressure (aroundatmospheric conditions) are independent of the partial pressure of oxygen, the defects in non-stoichiotemetric oxides either at high or low pressures are strongly dependent on the partial pressureof 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 electronicdefect concentrations.