Lattice defects in oxides.
Correlations between defects, properties and
crystal structures
Defects in oxides
Lattice or point defects:
Extended defects:
• Vacancies (oxygen, cation)
• Crystallographic shear
• Interstitial ions
• Dislocations
• Foreign atoms at regular sites (doping/solid solutions) • Grain boundaries
• Defect pairs and clusters
• Electronic defects (free electrons and holes)
Defect chemistry. Lattice defect can be treated as chemical entities (energy of
formation) using defect reactions (mass-action law, equilibrium constant).
Kroger-Vink notation for defects. Defect charge is referred to the perfect crystal.
Binary oxides
(ZnO)
Ternary oxides
(ZnAl2O4)
Cation vacancy
VZn''
Oxygen vacancy
VO
Interstitial cation
Zni
Interstitial oxygen
Oi''
Foreign ion (donor)

AlZn
Foreign ion (acceptor)
Antisite defects

'
AlZn
ZnAl
'
NaZn
Defects in oxides
Rules for defect reactions:
• Site relation. The number of sites must be in the correct proportion (MaXb: M/X = a/b).
• Sites can be created or destroyed taking into account the site relation.
• Mass balance.
• Electroneutrality condition.
1eV = 96.5 kJ/mol
Frenkel disorder
AgBr
'
AgAg  Vi 
 Agi  VAg
Schottky disorder
''
MgMg  OO MgO

VMg
 VO  Mgsurf  Osurf
''
zero MgO

VMg
 VO
Defects and entropy
N atoms arranged at (N+n) sites with n vacancies
G  G0  nHv  T Sconf  nSvibr 
 N  n !
Sconf  k B ln W  k B ln 
 N!n! 
Hv  0;
Sconf  0;
Thermodynamic probability for distinguishable particles
No!
W
no !n1!n2!...nr !
-
Svibr  0
N0: total number of
atoms
ni: number of atoms on
the i-th energy state
Defects and nonstoichiometry in binary oxides
Oxygen nonstoichiometry (TiO2, CeO2, Nb2O5, V2O5)
'
1
OO CeO

2  VO  2CeCe  O2
2
'
CeCe
 Ce3
1
Oo CeO

2 VO  2e '  O2
2
Metal nonstoichiometry (FeO, NiO, MnO)

1
O2 FeO

 OO  VFe''  2 Fe Fe
2

FeFe
 Fe3
Fe1-yO
1
O2 FeO

 OO  VFe''  2h 
2
Defects and nonstoichiometry in binary oxides
1
Oo CeO

2 VO  2e '  O2
2
K1  [VO ]n2 pO1/22
CeO2-x
x: fraction of vacant sites in CeO2-x
K1'  [VO ]3 pO1/22  x3 pO1/22
x  pO12 / 6
1/ n
O2
x p
n = 6: doubly ionized vacancies
n = 4: singly ionized vacancies
n = 2: neutral vacancies
Isolated
defects
In general:
n ≤6 for isolated defects or defect complexes
G O2  O2  O02  RT ln pO2  nRT ln x
- log x in CeO2-x
Defect ordering
Formation of subphases

O
Electroneutrality
Defect complexes
V   12 x
n  e'
-GO2 (kcal/mol)
 
n  2 VO
Equilibrium constant
Defects and nonstoichiometry in binary oxides. Formation of shear planes
Ordering of defects and formation of superstructures is observed for large
deviations from stoichiometry (TiO2-δ, Nb2O5-δ, WO3-δ, ReO3-δ, etc.). Elimination of
oxygen vacancies by formation of metal-rich shear planes is a common
mechanisms (crystallographic shear).
Formation of shear planes in ReO3
(left) and WO3 (right) by elimination
of oxygen vacancies
Lattice defects and nonstoichiometry in perovskites
BaTiO3
Partial Schottky disorder (TiO2–rich side)
Partial Schottky disorder (BaO-rich side)
Full Schottky disorder
Oxygen nonstoichiometry
1eV = 96.5 kJ/mol
Lattice defects and nonstoichiometry in perovskites
Ti-rich
Ba-rich
BaTiO3
Lattice defects, nonstoichiometry and phase transitions in perovskites
Cubic (paraelectric) – tetragonal (ferroelectric) phase transition in BaTiO3
Enthalpy of transition
1200°C
Ti-rich
Transition temperature
Ba-rich
Ba-rich
1320°C
Ti-rich
Lattice defects and electrical conductivity in perovskites
Electrical conductivity
 
Z: numero di cariche; e: carica
dell’elettrone; : mobilità
c: concentrazione
   zi e i ci
n  e'; p  h
(1)m = -1/6
BaTiO3
(2)m = -1/4
(1) At low p(O2) << p0(O2)
1
OO  O2  VO  2e '
2
K1  VO n2 pO1/22
n  2 VO
 
 
n  pO12 / 6
(2a) At intermediate p(O2) and R = Ba/Ti < 1
V   V   aR
''
Ba

O
n  pO12 / 4
(2b) At intermediate p(O2) with R = 1
and acceptor impurities
n-type
p0(O2)
n=p
Reduction (1): H2-3 eV
Oxidation (3): H1 eV
p-type
A   2V 


O
n  pO12 / 4
(3) At high p(O2) p(O2) > p0(O2)
1
O2  VO  OO  2h 
2
 
K 2  p 2 VO
1
pO12 / 2
 
p  VO pO12 / 4
Doping of perovskites: controlling defect nature and concentration
Acceptor doping: the substitutional impurity has a lower charge than the regular
and bring less oxygen into the lattice
Fe2O3  2BaO BaTiO
3  2BaBa  2FeTi'  VO  5OO
1
O2  Fe2O3  2 BaO BaTiO
3  2 Ba Ba  2 FeTi'  2h   6OO
2
'
Na2O  2TiO2 BaTiO
3  2NaBa
 2TiTi  VO  5OO
Donor doping: the substitutional impurity has a higher charge than the regular and
bring more oxygen into the lattice
1

La2O3  2TiO2  O2  2 LaBa
 2TiTi  2e '  6OO
2
BaTiO3

''
La2O3  3TiO2  2LaBa
 VBa
 2TiTi  9OO
1
Nb2O5  2 BaO BaTiO
3  O2  2 NbTi  2 Ba Ba  2e '  6OO
2
''
Nb2O5  BaO BaTiO
3  2NbTi  VBa
 BaBa  6OO
BaTiO3
Doping of perovskites: influence of doping on electron conductivity
Donor doped compounds:
• Black colour;
• Good conductivity (>10-2 S/cm) even at RT;
• Some show metallic conduction (103 S/cm, La:SrTiO3);
1

La2O3  2TiO2 BaTiO
3  O2  2 LaBa
 2TiTi  2e '  6OO
2

La2O3  3TiO2 BaTiO
3  2LaBa
 VBa''  2TiTi  9OO
1
OO  O2  VO  2e '
2
La:BaTiO3
1200°C
p0(O2)
Doping of perovskites: influence of doping on electron conductivity
Acceptor doped compounds
• Light colour;
• Good conductivity at high temperature
• Many are insulators at RT;
• Can be fired in reduced atmosphere retaining their dielectric properties.
p0(O2)
Mn2O3  2BaO BaTiO
3  2BaBa  2MnTi'  VO  5OO
1
O2  Mn 2O3  2 BaO BaTiO
3  2 Ba Ba  2MnTi'  2h   6OO
2
Doping of perovskites: from isolated defect to oxygen vacancy ordering and
formation of layered structures
Due to the high dielectric constant (20-1000) and structural stability, perovskites can
accomodate a large concentration of foreign aliovalent impurities (good solvent) and related
charge compensating defects (cation or oxygen vacancies). The simple model of randomly
distributed isolated defects (no association) holds up to high dopant concentration (few at.% for
acceptors, 10 at.% for donors). At higher dopant concentration, ordering of defects, formation
of shear planes and layered structures is observed.
Fe2O3  2SrO SrTiO
3  2SrSr  2FeTi'  VO  5OO
   
SrFexTi1 xO3 x / 2 ; FeTi'  2 VO
x = 0: SrTiO3; x = 1: Sr2Fe2O5
At T < 700°C, oxygen vacancy ordering occurs in
Sr2Fe2O5 (brownmillerite structure).
For intermediate compositions, intergrowth of
perovskite blocks and brownmillerite layer with
general formula AnBnO3n-1.
Perovskites doped with high concentration of
acceptor impurities (10-20 at.%) shows high ionic
(oxygen) and electronic (holes related to
transition metals Ti, Fe, Co, Nb) conductivity.
Application
as
mixed
conductors
in
electrochemical devices.
ABO3
A2B2O
5
Sr2Fe2O5
A4B4O11
Doping of perovskites: from isolated defect to oxygen vacancy ordering and
formation of layered structures
La1-xSrxFeO3 with 0 < x <0.25 can be accurately described as an acceptor-doped
perovskite with randomly distributed defects
At low p(O2):
2SrO  2Fe2O3 LaFeO
3  2SrLa'  2FeFe  VO  5OO
1
OO  O2  VO  2e '
2
Fe2+
p-type
n-type
At high p(O2)
1
O2  VO  OO  2h 
2
Fe4+
p-type

n-type
Doping of perovskites: excess oxygen, shear planes and layered structures
Reducing atmosphere, black conducting
ceramics (up to 103 Scm-1) , random
distribution of defects, x up to 0.3
x
La2O3  (1  x) SrO  TiO2 
2
x
La x Sr1 xTiO3  O2
2
Oxidizing atmosphere, less conducting,
O excess accomodated by formation of
shear planes when x > 0.17, by small
isolated defects when x<0.17
x
La2O3  (1  x) SrO  TiO2 
2
Lax Sr1 xTiO3
δ = x/2
x = 0, δ = 0 : SrTiO3
x = 1, δ = 0.5 : La2Ti2O7
cubic
distorted
orthorhombic
The structure can be described as the
intergrowth of perovskite layers (SrTiO3) and
La2Ti2O7 layers. A layered perovskite with
general formula:
La4Srn-4TinO3n+2
SrTiO3
La2Ti2O7
Layered perovskites
A typical example: Ruddlesden-Popper phases SrO(SrTiO3)n or Srn+1TinO3n+1
Sr3Ti2O7
P
SrO
P
SrO
SrTiO3+Sr3Ti2O7
P
Sr3Ti2O7
(n = 2)
Aurivillius compounds: ((Bi2O2)2+(Bim-1TimO3m+1)2-)
Ferroelectric & piezoelectric materials with high TC