Defects and nonstoichiometry
Simple intrinsic point defects
The thermodynamics of defect formation
Extrinsic defects
Defects in nonstoichiometric materials
Defect clustering
Solid solutions
Extended defects
– CS planes and shear structures
Defects in crystals
It is not possible to make crystals that are prefect
in every respect
– some are more perfect than others
It takes energy to create defects in crystals
The presence of defects increases the entropy of
the crystal
– above absolute zero always expect some intrinsic
defects
Stoichiometry
Many solid materials are non-stoichiometric
– all that really matters is charge balance
Non-stoichiometry is common amongst
transition metal compounds
– FexO where 0.957 >x > 0.833
– YBa2Cu3O7-x, 1 > x > 0
Non-stoichiometry can control properties
Non-stoichiometric compounds
TiO x
“TiO ”
0.65 < x < 1.25
“TiO 2 ”
1.998 < x < 2.000
VO x
“VO ”
0.79 < x < 1.29
Mn xO
“MnO ”
0.848 < x< 1.000
NixO
“NiO ”
0.999 < x < 1.000
LixV2 O 5
0.2 < x < 0.33
The thermodynamics of defect formation
All
macroscopic samples of materials contain some
defects as defect formation is entropically favored
– when defect formation is enthalpically very unfavorable there
may be very small numbers of defects
Types of defect
Defects may occur in isolation due to the increase
in entropy of the crystal
– intrinsic point defects
May occur in isolation to balance the presence of
an impurity
– extrinsic point defect
Defect may occur throughout the crystal
– extended defect
Intrinsic point defects
Two common types of intrinsic point defect
– Schottky and Frenkel
A Schottky defect consists of charge balancing
cation and anion vacancies
– Found in NaCl
A Frenkel defect is a charge balancing interstitial
and vacancy
– can have cation or anion Frenkel defects
Schottky and Frenkel defects
Shottky defect in NaCl
- both cation and anion are missing from
their regular lattice sites
-at room temp on 1 in 1015 sites are vacant in NaCl
-200 kJmol-1 creation energy
Cation Frenkel defect in AgCl
-cation is displaced from regular lattice
site onto interstitial site
- 130 kJ mol-1 creation energy
Frenkel defects
Frenkel defects may occur on either the anion or
cation sublattice
Cation Frenkel defects are more common than
anion defects
– cations are smaller than anions and hence easier to
accommodate in interstitial positions
Fluorite structures (CaF2, SrF2, ZrO2, UO2) are
good at accommodating anion Frenkel defects
Kroger-Vink notation for defects
Defect is denoted by symbol of atom involved or by V if it is a vacancy
Superscript • indicates a net charge of +1, superscript ‘ indicates a net
charge of –1. Superscript x indicates no net charge
Subscript indicates nature of site in crystal lattice, s for surface, I for
interstitial, element symbol for normal lattice site
Examples:
–
–
–
–
–
–
V’Na
sodium ion vacancy net charge –1
V•Cl
chloride ion vacancy net charge +1
x
Na Na, ClxCl Na and Cl on their normal lattice sites
Cd•Na cadmium on Na site net charge +1
Ag•i silver on interstitial site net charge +1
F’i fluoride on interstitial site net charge -1
Estimation of defect concentration
It is possible to calculate the equilibrium
concentration of defects in a solid using statistical
mechanics
ns ~ N exp(-∆HS / 2RT)
nF ~ (NNi)1/2 exp(-∆HF/2RT)
Defect concentration depends upon the energy
needed to form a defect and the temperature
Typical values of the defect concentration
Most simple ionic solids have low defect
concentrations
However, small changes in the energetics for defect
formation can lead to high defect concentrations
Values of ns/N
T/K
∆HS = 5 x 10-19 J ∆HS = 1 x 10-19 J
300
6.12 x 10-27
5.72 x 10-6
1000
1.37 x 10-8
2.67 x 10-2
Defects in AgCl
Ag+ + Vi Agi+ + VAg
K = [Agi+][VAg] / [Ag+][Vi]
Let N be the number of lattice sites and Ni the
number of interstitial sites
– Ni = [VAg] = [Agi+]
– [Ag+] = N - Ni
[Vi] = αN
– number of interstitials is simply related to number of
lattice sites for most materials
Defects in AgCl continued
K ~ Ni2 / α N2
– Substitute into equilibrium constant
∆G = -RT lnK, so
[VAg] = Ni = N α1/2 exp(-∆G/2RT)
∆Hf for defects
Color centers
Electrons trapped in vacant sites give rise to
colored materials
– color centers
– color arises due to transitions between electron in a box
levels
Trapped electrons can be produced by
– irradiation of the sample
– treatment with an electron donor like sodium or
potassium vapor
F, H and V centers
Irradiation can lead to defects where an electron has bee lost or added
Treatment with alkali metal vapor can lead to excess electrons in material
F Center – electron trapped in anion vacancy
Example of color center as trapped electron leads to
absorption in visible
H Center – interstitial Cl atom
bonds to lattice ClV Center – electron removed
from lattice anion site, resulting
Cl atom pairs with neighboring Cl-
Imaging plates
Color centers are useful in medical X-rays
using BaFBr:Eu2+ phosphors
BaFBr:Eu2+ phosphors
Extrinsic point defects
If cationic impurities are introduced into a solid and the
dopant does not have the same valence as the cation it is
replacing extrinsic defects will be introduced
– Fe1-xO has cation vacancies
– Ca2+ in ZrO2 - anion vacancies
– Y3+ in ZrO2
- anion vacancies
– Ca2+ or Cd2+ in NaCl
- cation vacancies
Real crystals contain both intrinsic and extrinsic defects
– the dominant defect type depends upon temperature and
doping/nonstoichiometry level
Defect clusters and aggregates
Point
them
defects interact and effect the structure around
– This may lead to clustering
Even
in something as simple as NaCl, cation and
anion vacancies tend to pair up as they are
electrostatically attracted to one another
Nonstoichiometric 3d oxides
FeO
Wustite is a very well studied example of a
nonstoichiometric compound
The compound “FeO” is not stable
The stoichiometry is always Fe1-xO
The iron oxygen phase diagram
The nature of the defects in “FeO”
Density measurements confirm that the
nonstoichiometry is incorporated by having vacant
iron sites
There is Fe(III) present to charge compensate the
system
The defect structure of “FeO”
The defect structure is more complicated than
random iron vacancies and Fe(III)
Koch clusters in Fe1-xO
The Fluorite structure
Defect clusters in UO2+x
Excess
oxygen is
incorporated in interstitial
sites
– This leads to displacement
of oxygens from normal
sites
– Arrangement of defects is
similar to structure of U4O9
» Can view defects as forming
clusters of U4O9 in UO2
matrix
The defect structure of TiO
“TiO” spans the composition range TiO0.65 TiO1.25
The stoichiometric phase TiO has many vacancies
– At high temperatures the vacancies are disordered
– At low temperatures the Ti and O vacancies exist in an
ordered array
Defects in TiO
Based
on NaCl
structure
– 1 in every six atoms
is missing
– vacancies order at low
temp
The structure of TiO1.25
Based on NaCl with all anions present, but has
ordered Ti vacancies
Order disorder
Many materials show temperature dependent
ordering phenomena
Spinels frequently show temperature dependent
ordering
– Mgtet[Al2]octO4 (normal) and Mgtet[MgTi]octO4
(inverted) but other compositions may be partially
inverted and the degree of inversion may depend on
synthesis temperature
Substitutional solid solutions
In
many compounds it is possible to replace a metal
atom or ion with another element that has similar size
and bonding requirements
– In metal alloy can replace metal atom with another element
that is within 15% size
– Can get complete solid solution formation between Al2O3
and Cr2O3 – Al2-xCrxO3
– Exstensive solid solution formation is favored by high
temperatures due to the disorder associated with the solid
solution
Criteria for solid solution formation
Typically,
for an ionic solid the ion size difference
should be less than 15-20% to get complete solid
solution formation
– > 30% size difference usually precludes solid solution
formation
End
member of solid solution should hve same
structure if complete solid solutions is to form
– Zn2SiO4 and Mg2SiO4 have different metal coordination
» So Zn2-xMgxSiO4 and Mg2-xZnxSiO4 have different structures
Interstitial solid solutions
Some
solid solutions involve inserting
atoms into interstitial sites in a parent
structure
– PdHx 0 < x < 0.7 - hydrogen occupies
interstitial sites in fcc Pd
– Carbon in interstitial sites of fcc Fe
Aliovalent substitution
If
you replace an ion by one with a different
oxidation state (aliovalent substitution) there has
to be a charge compensation mechanism
Cation vacancies
– Dope calcium into NaCl – Na1-2xCaxVxCl
– Replacement of Mg2+ by Al3+ in spinel
» [Mg1-3xVxAl2x]tet[Al2]octO4
– Oxidation of NiO
» Ni2+1-3xVxNi3+2xO
Aliovalent substitution
Interstitial
anions
– Not common due to limited size of interstitial sites but
occurs for fluorite structure
» Ca1-xYxF2+x
» U4+1-xU6+xO2+x
Anion
vacancies
– Important in ionic conductors
» Zr1-xCaxO2-x
Interstitial
0.1 < x < 0.2
cations
– Lix(Si1-xAlx)O2 stuffed quartz structure (0 < x < 0.5)
Characterizing solid solutions
Can
determine if solid solution forms by
measuring lattice constants of material using
x-ray diffraction
– Lattice constants typically vary linearly with solid
solution composition
» Vegard’s law
Can
work our mechanism of solid solution
formation with the aid of density
determination
Magneli phases
The oxides of metals such as W, Mo and Ti
display a wide range of compositions
– WO3-x, MO3-x, TiO2-x
Magneli realized that these compounds were best
represented as homologous series of phases rather
than solid solutions
– TinO2n-1, MonO3n-1 etc
Crystallographic shear structures
The homologous series can be formed by
incorporating crystallographic shear planes into the
structures
– these are extended defects
The shear planes change the stoichiometry of the
material
– at the CS plane may have face sharing rather than edge
sharing or edge sharing rather than corner sharing
Can get shear planes in 2 or 3-D leading to
– slab structures and block structures
Molybdenum and tungsten oxides
A wide variety of tungsten and molybdenum
oxides are known
Many of them belong to homologous series
– MnO3n-1 or MnO3n-2
– for example, Mo4O11, Mo5O14 and Mo6O17
Defects in molybdenum and
tungsten oxides
Why are there so many different oxides?
The parent oxide WO3 has a ReO3 structure at high
temperature
All of these different stoichiometry oxides can be
derived from WO3 by incorporating an ordered array
of planar defects
– crystallographic shear planes
The ReO3 structure
The incorporation of a shear plane
Homologous series
Each member of the homologous series has a
different repeat distance between shear planes
Consider W11O32
Block structures
Crystallographic
shear planes running
in two directions can
lead to double shear
or block structures
– W4N26O77 consists
of 4 x 4 and 3 x4
blocks
Tungsten bronzes
MxWO3
– M is an alkali metal or alkaline earth or H
They can be prepared by
– electrocrystallization of melts
– treatment of WO3 with alkali metal sources
– hydrogen spillover
There are a variety of possible crystal structures
Are used as bronze pigments
The structures of tungsten bronzes
NaxWO3 - often ReO3 based
KxWO3 - 0.19 < x < 0.33 from reaction of K with
WO3 is hexagonal
– potassium is bigger than sodium and needs a larger site
KxWO3 - x < 0.19 regular intergrowth structure
Tetragonal bronzes are known for Na and K
Tungsten bronze structure types
Tetragonal tungsten bronze
Hexagonal tungsten bronze
Intergrowths
It is possible to combine slabs of simple
structures together to build up a solid
If the slabs grow together in an order array
you have an ordered intergrowth
– new structure type if order is long range
If slabs are randomly stacked together you
have a random intergrowth
Intergrowth tungsten bronzes
Double rows hexagonal
structure intergrown
with ReO3 structure
Single rows hexagonal
structure intergrown
with ReO3 structure
Hexagonal
KxWO3 stable for 0.19 < x 0.33, lower
values of x can be accommodated by intergrowing
with ReO3 type WO3
Intergrowth bronze BaxWO3
Single
rows of hexagons can be seen. Some
of sites are not filled by Ba2+
Stacking faults
Many structures can be thought of as
consisting of an ordered stack of layers
Sometimes this ordering of the layers
breaks down
– ABCABCABC..... normal
– ABCBCABCABC...... with stacking fault
Antiphase domain boundaries
At
an antiphase boundary the ordering pattern within a
crystal structure abruptly changes
– This could be a change in metal atom or cation ordering, for
example in CuAu where there is ordering of Cu and Au