Simple and Complex Defects
Nalini Vajeeston
Department of Chemistry, University of Oslo
FERMiO, Gaustadalleen 21
NO-0349 Oslo, Norway
Outline
 Introduction
 Kröger -Vink notations
 Proton defects in oxides
 Simple defect materials
● Acceptor doped-LaNbO4
● Phosphates and pyrophosphates
 Complex defect materials ● Ba2In2O5
● Mayenite (Ca12Al14O33)
 Summary
● Ba3La(PO4)3
Introduction
• Defects
Deviations from the ideal structures are present at any temperature and
occur naturally in all crystalline compounds. These deviations or
imperfections are called defects.
Defects in stoichiometric compounds
(crystal composition is unchanged)
Defects in non-stoichiometric compounds
(formed by introducing dopants or impurities)
(composition is changed)
Schottky and Frenkel defects.
Cation vacancies, interstitial anions,
oxygen vacancies and electronic holes.
• Defect Structure
A complete description of the point and electronic defects in a compound and
their concentrations as a function of the partial pressures of the constituents
and the temperature is termed the defect structure of the compound.
Kroger-Vink notations for simple defects
Defect Type
Notation
Defect Type
Notation
Non-metal vacancy at nonmetal site
vX
Impurity non-metal (Y) at
non-metal site
YX
Metal vacancies at metal
site
vM
Impurity metal (A) at metal
site
AM
Neutral vacancies
vXMvXX
Non-metal vacancies with
positive effective charge*
v•X
Metal vacancies with
negative effective charge*
v/M
Interstitial metal
Mi
Interstitial non-metal
Xi
Intertitial metal with positive
effective charge*
M•i
Interstitial non-metal with
negative effective charge*
X/i
Free positive hole
h•
Free electron
e/
Substitutional hydroxide
OH•O
* The effective charge is the charge that the defect has with respect to the normal crystal lattice.
Defect reactions
Mass balance
• The defect reaction must balance with respect to the mass.
• Vacancies, which only represent empty sites, have zero mass and do not count.
• Electronic defects are not considered to count in the mass balance.
Ratios of regular lattice sites
• The ratio(s) of the number of regular cation and anion lattice sites in a
crystalline compound is constant.
• No sites are created in the formation of electronic defects.
Electroneutrality
• The total effective charge is the same before and after the formation of the
defects.
(The net charge on the left and right hand sides of a reaction equation must
be the same).
Hydrogen defects in metal oxides
When a metal oxide is equilibrated in gas mixtures with hydrogen containing
gases, e.g. H2O, hydrogen will dissolve in the metal oxide.
The extent of the dissolution of hydrogen will depend on the defect structure
of the oxide and the ambient oxygen and hydrogen activities.
The dissolution of protons from water vapour may in these terms be written
H 2 O g   2O Ox  2OH O  2e / 
Defect equilibrium:
K
1
O 2 g 
2
 2
O
2
[OH ] n p o2
1
2
[OOx ] 2 p H 2O
The concentration of protons in metal oxides is dependent on the partial
pressures of both the ambient oxygen and water vapour as well as the
concentration of electronic defects.
Effect of water vapour on oxygen-deficient M2O3
Undoped, oxygen-deficient oxide:
The predominant defects  electrons and oxygen vacancies
Electroneutrality condition in dry environments
n  2[VO ]
n  2[VO ]  21 / 3 K 1/3[OOX ]1 / 3 pO12/ 6
VO
PO2 = constant; PH2O = varied
1/ 2
Proton concentration is propotional to p H 2O
In wet environments:
The predominant defects: protons
Electroneutrality condition:
n  [OH O ]
n  [OH O ]  K 1/ 4 pH12/O4 pO12/ 8
Brouwer plot of effects of water vapour
on defect concentrations in oxygen
deficient M2O3-d
Effect of water vapour on acceptor-doped M2O3
Equilibrium constant:
Hydration reaction:

O
H 2O  v  O  2OH
x
O
[OH O ] 2
ΔS 0
 ΔH 0
K  exp
exp
 
R
RT
[vO ][OOx ] p H 2O

O
Electroneutrality:
2[vO ]  [OH O ]  [A/ ]  constant
Oxygen vacancies and protons
compensate the acceptor doping.
[A / ] - [OH O ]
[v ] 
2

O
Concentration of protons:

8[A / ]

[O]Kp H 2O  1  1  x

[O O ]Kp H 2O


[OH O ] 
4




Brouwer plot of the effect of water vapour
(at constant oxygen pressure)
on defect concentrations in acceptor-doped,
oxygen deficient M2O3.
Proton defects in Phosphates (Sr-doped LaPO4)

½ M 2 P2O7  M Ln  ½( P2O7 )2 PO4
/


½( P2O7 )2 PO4  ½ H 2O( g )  ( HPO4 )2 PO4
Substitution of divalent metals for rare earth metals
leads to condensation of orthophosphate ions, i.e.
formation of pyrophosphate ions as oxygen deficits.
Protons dissolve into phosphates forming hydrogen
phosphate groups through the equilibrium between
the condensed phosphate ions and water vapor in
ambient atmosphere.
Monoclinic monazite type structure
The electroneutrality condition of an acceptor-doped phosphates
with oxygen vacancies and protons
( HPO )  2( P O )   M   constant

4 PO4
2

/
7 2 PO4
Ln
Proton defects in Pyrophosphates:
Hydration reaction:
3( P2O7 )Px2O7  H2O( g )  ( P2O8 )//P2O7  2( HP2O7 )P2O7
Equilibrium constant:
( P O )  ( HP O ) 
K
( P O )  pH O
2
//
8 P2O7
2
x
3
7 P2O7
2
2

7 P2O7
 exp
Shydr
R
2
exp
 H hydr
RT
Electroneutrality:
   

2 ( P2O8 ) //P2O7  A  ( HP2O7 )P2O7
/

( HP O )   A ( HP O ) 
2
( HP O )
2
3

7 P2O7

7 P2 O7
/
2
  A  
2

7 P2O7
TiP2O7

 2KpH 2O ( P2O7 ) Px2O7
3
 
2 KpH O( P O ) 
2 A/

 
3  2 A/ 3  27  2 KpH 2O ( P2O7 ) Px2O7

 

 2 A/ 3 27  2 KpH 2O ( P O ) x

2 7 P2 O7


3

3
3 3
3 3
Cubic superstructure
0
1
3
/
3

2

2
2 KpH 2O ( P2O7 ) Px2O7
32
1
3
x
7 P2 O7
2

3
3
 
   27  2KpH O( P O ) 
4 A
/ 3
2

x
7 P2 O7
2
4 A/ 3  27  2 KpH 2O ( P2O7 ) Px2O7

3



1
3
3



1
3

Acceptor doped LaNbO4
LaNbO4 exists in two different polymorphs
Low temperature phase  Monoclinic-Fergusonite-type structure
High temperature phase Tetragonal-Scheelite structure
Monoclinic
Tetragonal
LaNbO4 may have a tendency to dissolve ptotons by interacting
with ambient water vapor:
H 2 O(g)  VO  OO  2OHO
Electroneutrality condition: 2[vO ]  [OH O ]  [A/ ]  constant
The condensed coordination
polyhedra of Nb3O11 found
by theoretical calculation.
Condensation occurs when oxygen vacancy is
formed in phosphate.
Based on this, oxygen vacancy in phosphate

can be expressed as P2 O 7 2PO
4
Same condensation can occur in LaNbO4.
Oxygen vacancy in LaNbO4 can be written as .

Nb3 O11 3NbO
4
more complicated than phosphate.
Ba2In2O5 (or BaInO2.5)
Ba2In2O5 is an oxygen deficient perovskite
ordered into the brownmillerite-structure
at low temperatures.
Around 930 °C it disorders into the perovskite.
The oxygen vacancy conductivity jumps
two orders of magnitude.
Brownmillerite-structure
Defects:
The disordered phase has 5 oxide ions and
1 oxide ion vacancy vO sharing the same
perovskite site.
what is the compensating negative effective charge?
●The disordering can be seen as a anion-Frenkel-disorder
(the formation of oxide ion vacancies and interstitials)
● Acceptor
New nomenclature
Perovskite
Ba2In2O5 (or BaInO2.5)- new nomenclature
Disordered Ba2In2O5 oxide ions and vacancies on the oxide ion sublattice.
The perfect oxide ion site is statistically occupied 5/6 with an oxide ion and
1/6 with a vacancy 56 O
Each oxide ion occupying the site to a degree of 5/6 has a formal charge -2
the site statistically has a charge of -2 · 5/6 = -5/3
Real charge of oxide ion = -2
Its effective charge = -2 - (-5/3) = -1/3.
Real charge of vacancy = 0
Effective charge = 0-(-5/3) = +5/3.
The oxide ion is denoted in the expanded Kröger-Vink nomenclature as O
Oxide ion vacancy 
v
5
3
5O
6
Site occupancy sum
interms of mole fraction
Electroneutrality condition
1
3
1/
5
6
6
[O53O ]  53 [v 53O ]
1
/
3
5
O
6
5

3
5
O
6
[O ]  [v ]  6
1
/
3
5
O
6
[O ]  5
and
5

3
5
O
6
[v ]  1
1/
3
5
O
6
Defect chemical reactions with Ba2In2O5
Reduction and oxidation
5
1
/
3
5
O
6
O
v
5

3
5
O
6
 2e /  12 O2 ( g )
KR 
Equilibrium coefficient:

1
[v 53O ][e / ]2 pO2 2
6
1
/
[O53O ]
6
Electroneutrality:
1
3
1
/
3
5
O
6
5

3
5
O
6
[O ]  [e / ]  53 [v ]
Electrons minority defects
5
Equilibrium coefficient:
1
3
KR 

1
[v 53O ][e / ]2 pO2 2
6
5

3
5
O
6
The corresponding oxidation reaction is
5

3
5
O
6
[O ]  [v ]
5
3
1

[e / ]2 pO2 2
5
5[v ]
Solve this with respect to the
concentration of electrons and obtain
1
/
3
5
O
6
1
1
[e / ]  (5 K R ) 2 pO24
1
2
5

1
O2 ( g )  v53O  O53O  2h
6
/
6
sum of the reduction and oxidation reactions yields the intrinsic ionisation of electrons
0  2e /  2h
or
0  e /  h
Hydration reaction:
H 2O( g )  vO  OOx  2OH O
Hydration reaction for disordered Ba2In2O5:
5

1
2

H 2O( g )  v53O  O53O  2OH 53O
6
/
6
6
Equilibrium coefficient:
2
KH 
Hydroxide defects < two native defects are dominating
and constant,
1
The concentration of hydroxide defects takes on a p H2 2 O
dependency.
[OH 53 O ]2
5

3
5
O
6
6
1/
3
5
O
6
[v ][O ] pH 2 O
To increase the water vapour partial pressure,the hydroxide defects
become dominating,
Electroneutrality
2
3
2

3
5
O
6
1
/
3
5
O
6
[OH ]  [O ]
1
3
The 6 oxygen sites are disorderly filled with 4 oxide ions and 2 hydroxide ions
overall formula Ba2In2O4(OH)2, or BaInO2(OH).
Zr Doping
BaO  ZrO2  Ba
x
Ba

In
1
/
3
5
O
6
 Zr  3O
The electroneutrality becomes
1
3
1
/
3
5
O
6

In
5

3
5
O
6
[O ]  Zr  [v ]
5
3
At 50 % substitution one oxygen vacancy and eleven oxide ions
out of twelve positions; Ba4In2Zr2O11.
At high doping levels,
it gets the same whether one considers it to be
Zr-doped Ba2In2O5 or In-doped BaZrO3.
Mayenite, Ca12Al14O33
Unit cell 
(Ca24Al28O64)4+ · 2O2-
lattice framework with 12 nano-cages
extra-framework oxide ions are randomly
distributed in the nano-cages and can be
replaced by F-,Cl-,OH- and H- ions.
Each nano-cage contains two crystallographic
positions for the oxide ion.
It is reasonable to assume that only one oxide
ion can be fitted in a cage at any time, and that
the energy barrier between the two positions is
small enough that the oxide ion is effectively
delocalized over the two positions at elevated
temperatures.
In this way, each oxide ion occupies
one out of 6 available cages.
J. Medvedeva
The defect situation in mayenite can be described as one
with an inherently deficient sublattice (the oxide ions in
the extra-framework nanocages).
The site is denoted as 1/6 occupancy of oxide ions as the perfect state,
consequently with a charge of -2/6 = -1/3.
Effective charge of O2-: (-2) – (-1/3) = -5/3
Effective charge of cage: (0) – (-1/3) = +1/3
Effective charge of OH-: -1 – (-1/3) = -2/3
OH
2
/
3
1
O
6
O
5
/
3
1
O
6
v
1

3
1
O
6
The real charge of the species
minus the real charge of the
perfect reference lattice
The electroneutrality in the pure, dry material then reads
 53 /   13  
5O1   v1 
 6O   6O 
Mayenite has a strong tendency to become hydrated by replacing
the oxide ions with hydroxide ions.
Hydration reaction:
H 2O( g )  O
5
/
3
1
O
6
v
1

3
1
O
6
 2OH
Equilibrium constant:
2
/
3
1
O
6
2
/

3
K  OH 1 
O
6 

2
1
1
 53 /   13   1
O1 O  v1 O  pH 2 O
 6   6 
The new electroneutrality:
2
5
1
/
/ 
 



3
3
3
2OH 1   5O1   v1 
O
6 

 6O   6O 
Site limitation:
2
5
1
/
/ 
 



3
3
3
OH 1 O   O1 O   v1 O   6
6 

 6   6 
The site sum of 6 enforces concentrations to refer to
fractions of one formula unit mayenite Ca12Al14O33,
or molar fraction of the same
The concentration of hydroxide ions as a function of water
vapour partial pressure and temperature:
2
 6 KpH 2 O  4 ( KpH 2 O ) 2  5 KpH 2 O
/

3
OH 1 O  
4  KpH 2 O
6 

Systems with disordered occupancy by several cccupants
Ba3La(PO4)3
Eulytite structure
The four cations (three divalent Ba2+ and one
trivalent La3+) disorderly occupies the same site.
The site is statistically occupied by
(3·2 + 1·3)/4 = 9/4 = 2¼ positive charges.
New nomenclature:
3Ba  1La
Site:
4
Ba2+
or
La3+
9

4
or
Ba3 La
4
9

4
ion would be denoted 
Ba3 / 4 La1 / 4
or
Ba
1
4
/
3 Ba 1La
4
or
9

4
3

4
La3 Ba 1La
4
An abbreviation for the complex site expression can be useful in such cases.
9
Thus, defining
M
9

4
3Ba  1La 4

4
Ba2+
La3+



allows to denote
1
4
/
BaM
3

4
LaM
The electroneutrality reads
1
4
Ba  La 
1
4
/
M
3
4
3

4
M
Acceptor doping with an excess of Ba2+ to dissolve protons, in phosphates
represented as hydrogen phosphate defects on phosphate sites,
The new electroneutrality reading
1
4
Ba   La  (HPO ) 
1
4
/
M
3
4
3

4
M

4 PO4
Assume the compound is perfectly stoichiometric,
we consider a minor concentration of additional defects formed by oxidation.
(oxygen interstitials and electron holes)
 
p  2 Oi//  pO1 /26
Since the two defects of disordered Ba2+ and La3+ are dominating in numbers,
the holes end up as minor defects with the familiar
p p
1/ 4
O2
dependency they attain when ionic defects rule.
Summary
● Proton defects in oxides, phosphates and pyrophosphates are explained.
● Defect chemistry of acceptor doped-LaNbO4 was discussed.
● Defect structures of Mayenite (Ca12Al14O33) and Ba3La(PO4)3 are derived.
● The new extension of the Kröger-Vink nomenclature, and defects present
in the disordered Ba2In2O5 have been discussed.
References
● Defects and transport in crystalline solids
Per Kofstad and Truls Norby
● A Kröger-Vink-compatible notation for defects in inherently defective sublattices
Truls Norby - to be submitted.
● High-temperature protonic conduction in acceptor doped-LaPO4
K. Amezawa, S. Kjelstrup, T. Norby, and Y. Ito, Electrochem. Soc. 145 (1999) 3313.
● High-temperature protonic conduction in TiP2O7 and Al-doped TiP2O7
Nalini Vajeeston*, Reidar Haugsrud, Helmer Fjellvåg, Truls Norby - to be submitted.
● High temperature hydration and conductivity of mayenite, Ca12Al14O33
Ragnar Strandbakke, Camilla Kongshaug, Reidar Haugsrud, Truls Norby - to be submitted.
● Local condensation of oxygen vacancies in t-LaNbO4 from first principle calculations
Akihide Kuwabara,*, Reidar Haugsrud, Svein Stølen,Truls Norby –submitted.