Ionic Conductors: Characterisation of Defect
Structure
Lectures 5-6
Defects in Crystalline Solids
Dr. I. Abrahams
Queen Mary University of London
Lectures co-financed by the European Union in scope of the European Social Fund
Thermal Defects
Crystalline solids are those that show a regular repeating structure, e.g.
NaCl. However all crystals are imperfect above 0 K, i.e, atoms are
missing or displaced from their normal lattice positions.
e.g. NaCl
ccp Cl– ions with Na+ in every octahedral site
At room temperature a small number of Cl– and Na+ ions are missing
(ca.1 in 1015 atoms missing).
Defects have important effects on the properties of materials. In ionic
conductors it is the presence of defects that facilitates ionic conduction.
Where the influence of the defect (apart from elastic strains) extends for
just a few Angstroms this is called a point defect. Where the defect
invoves a number of point defects extending over longer distances then it
may be described as an extended defect.
Lectures co-financed by the European Union in scope of the European Social Fund
Classical Point Defects
Point defects are the simplest cases, where a single atom/ion or pair of
ions (in ionic AB structures) is missing or displaced.
Sites where an atom is missing are known as vacancies.
Atoms or ions displaced into a part of the structure that is normally vacant
are called interstitials.
Three main types of point defect will be considered:
Schottky Defects
Frenkel Defects
Colour Centres
Lectures co-financed by the European Union in scope of the European Social Fund
Schottky Defects
A Schottky defect in a metal consist of a single vacancy. In ionic crystals
of the type AB the Schottky defect consists of a pair of missing ions.
e.g. NaCl
One Na+ and one Cl– absent
from their normal lattice sites.
These absent positions are
described as vacancies.
Electroneutrality is maintained
Lectures co-financed by the European Union in scope of the European Social Fund
HF of a Schottky defect in NaCl is ca. 220 kJ mol-1
Therefore the heat of formation is high and explains why only 1 in 1015
Na+ and Cl– ions are missing at room temperature. However despite the
low concentration this can have an important influence on properties.
For example the presence of defects can lead to ionic conduction (small
in NaCl).
Ionic conduction
occurs as a result of
Na+ ions moving into a
vacancy.
Many ionic solids
exhibit Schottky
defects e.g. NaCl,
LiCl, MgO etc.
Lectures co-financed by the European Union in scope of the European Social Fund
Frenkel Defects
Frenkel defects consist of anions or cations displaced from their normal
lattice sites into interstitial sites. Interstitial sites are ones which are not
normally occupied in the perfect crystal.
e.g. AgCl (rocksalt)
Ag+ ion displaced from it normal octahedral position to an interstitial
tetrahedral site.
Lectures co-financed by the European Union in scope of the European Social Fund
HF for a Frenkel defect in AgCl ca. 130 kJ mol-1. Therefore again only a
very small number of defects are present at room temperature.
NaCl and AgCl both have the rocksalt structure, but have different point
defects.
In any compound the defect with the lowest HF will form.
Frenkel defects have lower HF in AgCl.
Schottky defects have lower HF in NaCl.
NaCl is more ionic than AgCl.
Interstitial Ag+ is more stable than interstitial Na+.
Lectures co-financed by the European Union in scope of the European Social Fund
Colour Centres
The F– centre is the best known colour centre (F for farbenzentre). F –
centres occur when an electron becomes trapped on an anion vacancy.
If NaCl is heated in Na
vapour, Na ionises at the
surface. Na+ remains at the
surface, but e- migrates into
the bulk, where it becomes
trapped on an anion vacancy.
The resulting solid is
greenish-yellow in colour and
is non-stoichiometric.
Na1+Cl (<<1)
Colour centres can also be created by irradiation (e.g. X-rays).
Because the e – is unpaired, ESR (electron spin resonance) can be used to
study colour centres.
Lectures co-financed by the European Union in scope of the European Social Fund
Energetics of defect formation
Unlike extended defects such as dislocations and grain boundaries, point
defects are present in relatively high concentrations at thermodynamic
equilibrium.
Consider a perfect crystal. Introducing a single defect results in a greater
degree of disorder i.e. a gain in entropy due to the number of possible sites
at which the vacancy may be located. This configurational entropy S is
given by:
S  k ln W
where k is the Boltzman constant and W is the number of possible
configurations of locating n defects over N sites and is given by:
W
N!
N  n !n!
Lectures co-financed by the European Union in scope of the European Social Fund
The driving force behind defect formation is a lowering of the Gibbs’ free
energy (G), where:
G = H – TS
Starting with the perfect crystal, the more
defects created, the more energy required
(H increases).
The more defects created the greater the
disorder (S increases).
Initially
TS >> H  – ve G
As more defects are created H increases, but disorder decreases and
therefore the TS term becomes small.
Later
H >> TS  + ve G
Therefore no more defects are spontaneously created.
Lectures co-financed by the European Union in scope of the European Social Fund
The number of defects increases until G becomes +ve, corresponding to
a minimum in the G curve. This is therefore the equilibrium defect
concentration.
At higher temperatures the TS term becomes larger and therefore we
see a larger equilibrium defect concentration.
Lectures co-financed by the European Union in scope of the European Social Fund
Thus if the enthalpy of formation of a single defect is HF and the
additional entropy associated with the addition of this defect is S. Then
for n defects the change in free energy will be:
G  nH F  T S  nS '
The equilibrium defect concentration of defects ne corresponds to the
minimum in the G curve and can be found by differentiating G with
respect to n, i.e. dG/dn = 0
 S' 
ne
 H 
 exp   exp   F 
N  ne
 kT 
k
where N is the total number of atoms.
Since ne << N:
 S' 
ne
 H 
 exp   exp   F 
N
 kT 
k
Lectures co-financed by the European Union in scope of the European Social Fund
Non-Stoichiometric Defects
Schottky defects in NaCl and Frenkel defects in AgCl are termed
stoichiometric defects as they leave the stoichiometry of the crystal
unchanged.
The F-centre in NaCl is an example of a non-stoichiometric defect
since the stoichiometry is changed with respect to the pure crystal.
Another example of a non-stoichiometric defect occurs in wustite (FeO).
This compound exhibits the rocksalt structure and like NaCl shows
Schottky defects. However in this case:
No of cation vacanies > No of anion vacancies
i.e. a stoichiometry of Fe1-xO
Electroneutrality is maintained by oxidation of Fe(II) to Fe(III) and so the
formula may be more correctly written as:
Fe(II)1- yFe(III)2y/3Vy/3O
(y = x/3)
Lectures co-financed by the European Union in scope of the European Social Fund
Defect Notation
The IUPAC accepted notation for defect formulae is the
Kröger-Vink notation. Symbols are of the form:
A
c
s
A is the species and can be:
an element symbol, e.g. Li
e for electrons
V for vacancy
h for hole.
C is the effective charge and can be:
 for an effective negative charge
 for an effective positive charge
x for an effective neutral charge.
S is the site of the species and can be:
the element symbol for the atom that normally resides in
that site, e.g. Li
i for interstitial
F.A. Kröger and H.J. Vink, in" Solid State Physics, vol. 3, editors: F. Seitz, D. Turnbull, p.273-301, (1956)
Lectures co-financed by the European Union in scope of the European Social Fund
Defect Equations
An atom in its normal lattice site is assumed to be neutral. Vacancies and
interstitials are assumed to have an effective charge. Similarly aliovalent
substitution introduces an effective charge. As in normal chemical
equations the charges should balance.
e.g.
Schottky defect in NaCl
NaNax + ClClx  VNa + VCl
Frenkel defect in AgCl
AgAgx  Agi + VAg
Solid solution formation in YSZ
2ZrZrx + OOx  2YZr + VO
Lectures co-financed by the European Union in scope of the European Social Fund
Defect Clusters
Simple point defects i.e. interstitial/vacancies in some cases can cluster
together in a solid to minimise energy.
Consider a vacancy in a ccp metal. The energy of defect formation will
involve contributions arising from breaking bonds. To place a second
defect next to the first will involve lower energy since less bonds have to
be broken. The difference in energy between two isolated defects and
two paired defects is termed the binding enthalpy, HB:
H F (pair )  2 H F (isolated )  H B 
where HF is the enthalpy of formation. The number of vacancy pairs np
is given by:
 2 H F (isolated)  H B 

n p  exp  
kT


Lectures co-financed by the European Union in scope of the European Social Fund
Since:
 S' 
ne
 HF 


 exp   exp  

N
 kT 
k
Then if ni is the number of isolated vacancies:
ni2
 HB 
 q exp  

np N
 kT 
where q is factor dependent on the entropy of defect formation and the
number of possible positions for the vacancy pair over N atom sites.
The ratio of isolated vacancies and vacancy pairs at equilibrium is
given by:
 S' 
 2 H F (isolated)  H B  
n1

 q exp    exp  
n2
kT


 k
This means that the number of vacancy pairs at equilibrium increases
with increasing temperature.
Lectures co-financed by the European Union in scope of the European Social Fund
Defect Clusters in Ionic Conductors
Defect clusters in ionic conductors can be important when considering
possible conduction mechanisms:
e.g. LISICON
Lithium SuperIonic CONductor
LISICON is a solid electrolyte, which shows high Li+ ion conductivity. It is
a solid solution between Li4GeO4 and Li2ZnGeO4 and has a general
formula:
Li2+xZn1-xGeO4
Li2ZnGeO4 has hcp O2– ions with Li+, Zn2+ and Ge4+ in half of the
available tetrahedral sites.
The octahedral sites are vacant. As Li4GeO4 is introduced in the solid
solution, some of the Li+ directly replaces Zn2+ on the tetrahedral sites.
However, in order to maintain electroneutrality, extra Li+ is required,
which enters the vacant octahedral sites.
Lectures co-financed by the European Union in scope of the European Social Fund
The octahedral sites share faces with filled tetrahedral sites. This is
energetically unfavourable and so Li+ ions in the tetrahedral sites are
displaced into vacant tetrahedra.
This results in the formation of defect clusters
Lectures co-financed by the European Union in scope of the European Social Fund
Extended Defects
We have seen how individual point defects can come together in
crystalline solids to minimise free energy.
Extended defects occur over much larger scales. These include:
Crystallographic shear structures
Stacking faults
Antiphase grain boundaries
Dislocations
Low angle grain boundaries
Lectures co-financed by the European Union in scope of the European Social Fund
Crystallographic Shear Structures
Crystallographic shear is the term given, when one or more parts of a
structure are translated with respect to the remainder of the structure.
This results in complex structures where the ideal structure may be
limited to columns, blocks or layers separated by planes of different
structure/composition.
This type of structure is common in certain transition metal oxides that
show variable oxidation state of the metal.
e.g. MoO3
MoO3 exhibits the ReO3 type
structure with corner sharing
MoO6 octahedra.
O:Mo ratio = 3
Lectures co-financed by the European Union in scope of the European Social Fund
MoVI is readily reduced to lower oxidation states such as MoIV
In partially reduced molybdenum oxides such as M9O26 the average
oxidation state is 5.78.
In this structure there is less oxygen per Mo: O:Mo = 2.889. This means
that in order to maintain an octahedral geometry, some of the Mo
octahedra must share edges.
These are arranged in groups of four edge sharing octahedra which are
distributed at regular intervals in the structure to give a shear plane.
Ref: W.H.
McCarroll, K.V.
Ramanujachary,
Encyclopaedia of
Inorganic
Chemistry, J.
Wiley and Sons.
Lectures co-financed by the European Union in scope of the European Social Fund
This leaves the ReO3 type blocks separated from each other by the
shear planes.
In fact Mo9O26 is part of a homologous series of MonO3n-1 oxides (n = 8,
12-14).
In all the ReO3 block extended infinitely in 2 dimensions, but their length
in the third dimension varies with n.
Similar series are seen in Ti oxides TinO2n-1 (n = 4 to 10) and Nb oxides.
In H-Nb2O5 shear planes occur without change in oxidation state. A
double shear occurs resulting in a very complex structure containing
different size blocks of ReO3 units.
Lectures co-financed by the European Union in scope of the European Social Fund
Stacking Faults
Stacking faults tend to occur in layered structures. Essentially these
involve an interruption of the normal layer sequence.
e.g. Co
Co can crystallise in hcp or ccp forms. These forms differ only in their
stacking sequence i.e.
hcp ABABAB
ccp ABCABC
So for example stacking faults can occur in the hcp arrangement as:
ABABACABAABABACABABAB
Similarly in ccp metals a stacking fault can occur in the (111) plane.
ABCABCBABCABCABCBABCABC
Lectures co-financed by the European Union in scope of the European Social Fund
Antiphase Boundaries
Closely related to stacking faults are anitiphase boundaries. They involve
a translation of one part of the structure to another such that like parts of
the structure face each other across the antiphase boundary. Thus the
periodicity changes phase from  to -.
Lectures co-financed by the European Union in scope of the European Social Fund
Dislocations
Dislocations occur where many individual point defects come together.
They are created by thermal energy and rapid crystal growth.
They can occur in all types of solid, but are very important in metals where
they greatly affect their properties.
There are two important classes:
(1) Edge dislocations
This is where a line of
vacancies split a plane of atoms
into two half-planes.
Atoms are perturbed around the
ends of the half-planes resulting
in a stressed region.
Lectures co-financed by the European Union in scope of the European Social Fund
(2) Screw dislocations
This is where an extra half plane of atoms becomes attached to the
surface. The step eventually propagates on the surface resulting in a full
step.
It is called a screw
dislocation because when
viewed perpendicular to the
step, the atoms appear to
spiral through the crystal.
Under applied stress or
heat dislocations will
migrate. This allows for
the hardening and
softening of metals.
Lectures co-financed by the European Union in scope of the European Social Fund
Berger’s Vector
Dislocations are characterised by the Burger’s vector.
e.g. Consider the atoms around a perfect part of a crystal
Travelling from atom to atom a
circuit of the perfect part of the
crystal arrives at the starting
atom in the upper plane
In contrast a circuit around an edge
dislocation arrives at a different
atom in the upper plane.
The Burger’s vector is defined as the vector between the starting
atom and the last atom.
Lectures co-financed by the European Union in scope of the European Social Fund
Grain Boundaries
Most materials are not single crystals, but polycrystalline, i.e. a mosaic of
small crystals (crystallites) held together in a ‘jig-saw’ like pattern. The
regions where the individual crystallites are held together are called grain
boundaries.
In some cases, there is a very small angle (1<<) between domains. The
region separating the domains is called a sub-grain boundary.
Lectures co-financed by the European Union in scope of the European Social Fund
Low Angle Grain Boundaries
Low angle grain boundaries can be formed by
collections of dislocations.
When several edge dislocations come
together with the same orientation, the
accumulated distortions caused in the lower
planes results in a re-alignment of these
lower planes at an angle to the upper planes.
This is known as a low angle grain
boundary.
Grain boundaries act as barriers to movement of defects and
dislocations.
They are very important in ionic conductors where they can lower the
total conductivity.
Lectures co-financed by the European Union in scope of the European Social Fund
Locking of dislocations
Adding a second component can often lock or trap a defect or dislocation.
Forming a solid solution can achieve this.
For example this is used in steel manufacture where carbon or boron are
added to iron.
Lectures co-financed by the European Union in scope of the European Social Fund