MME 467
Ceramics for Advanced Applications
Lecture 17
Defects in Ceramics 1
Ref: Barsoum, Fundamentals of Ceramics, Ch6, McGraw-Hill, 2000
Prof. A. K. M. B. Rashid
Department of MME, BUET, Dhaka
Topics to discuss....
q  Defects and their classifications
q  Point defects and their notations
q  Thermodynamics of point defects
1 Introduction
q  A perfect crystal has
v  A completely ordered structure
v  Its atoms at their equilibrium positions
v  Their electrons distributed in the lowest-energy states
q  Real crystals are not perfect
v  they contains imperfections
q  Many properties are markedly dependent on
small deviation from the ideal structure.
Defects and their classes
Electronic defects
1. Electron hole: electrons excited into higher energy levels, leaving vacant
positions in the normally-filled electronic energy-level bands
2. Excitons: excited electron remains closely associated with the electron hole,
forming a electron-electron hole pair
Atomic defects
Occur as isolated points, lines and surfaces in the structure
during the arrangement of atoms in the space lattice.
1. Point defects: Vacancy, interstitials
2. Line defects: Grain and twin boundaries
3. Planer defects: Edge and screw dislocations
4. Volume defects: Pores, cracks, and inclusions
2 Point defects – why bother?
q  Point defects dictate several properties:
v  Diffusion, which determines sintering and sometimes creep
v  Electrical conductivity
v  Dielectric constants
v  Most dramatic example: YBa2Cu3O7
v  Metal oxidation
v  Solid electrolytes
q  In ceramic system, more is known about point
defects than about the structure of dislocations,
grain boundaries, or free surfaces.
q  In metals, point defects are straightforward.
v  only one type of atom, charge neutrality is not an issue
q  In ceramics, the situation is more complex.
v  defects must form by preserving charge neutrality all the time
v  defects always occur in neutral “bunches”
q  Three categories of point defects in ionic ceramics:
1. Stoichiometric defects
2. Non-stoichiometric defects
Intrinsic defects
3. Extrinsic defects (a.k.a. impurities)
3 Stoichiometric defects
q  Crystal chemistry (cations/anions ratio)
does not change.
q  Important defects in this category include:
1. Vacancies and interstitials
2. Schottky and Frenkel defects
Schottky Defect
Vacancy – vacancy pair
Frenkel Defect
Vacancy – interstitial pair
Non-stoichiometric defects
q  Result in a change in crystal composition, and
a transfer of mass across the boundaries of
the crystal
Example 1: removal of an oxygen atom from a ceramic oxide
this results in a vacancy with a charge of two and two electrons in the conduction band.
Example 2: addition of extra oxygen in interstitial site
the defect site in this case acts as an acceptor, and two holes are created in the valence band.
MaOb
Stoichiometric oxide compound
Ma+δOb
Non-stoichiometric oxide (metal excess)
MaOb+δ
Non-stoichiometric oxide (oxygen excess)
4 Extrinsic defects
q  Created as a result of the
presence of impurities in the
host crystals.
Substitutional
q  Impurities of such type may
form interstitial and
substitutional defects.
Interstitial
Example:
CaO doped ZrO2
Ca
Zr
Only cations shown
Cation
vacancy
Substitutional
impurity
M
O
M
O
M
O
M
O
O
M
O
M
O
M
O
M
M
L
M
O
M
O
M
O
O
M
O
M
O
M
O
M
I
Anion
vacancy
Interstitial
impurity
M
Cation
interstitial
Misplaced
atoms
M
O
M
O
M
O
M
O
O
O
O
M
M
M
O
M
M
O
M
O
M
O
M
O
Various defects found in binary ceramics.
Misplaced atoms can only occur in covalent ceramics due to charge considerations.
5 Kroger-Vink Defects Notations
Main Symbol, A
C
AB
The entity occupying the defect site.
It is either the species involved (i.e., the chemical symbol
of the element), or the letter V for vacancy.
Examples are M, X, V, or substitution elements.
Superscript, C
For the excess (or, effective electric) charge
associated with the site.
It is the difference between the real charge of the
defect species and that of the species that would
have occupied the site in a perfect crystal.
C
AB
The superscript is a prime (’) for each negative
charge, a dot (.) for every positive charge, or an x
for zero effective charge.
Subscript, B
For the type of site occupied.
It is either the crystallographic position occupies
by species involved, or the letter i for interstitial.
Examples include M, X, or i
6 Example: Lattice defects in MX binary compound ( M+2, X–2)
Defect
Excess charge
Notation
Vacancy on M sub-lattice
0 – (+2) = –2
VM”
Vacancy on X sub-lattice
0 – (–2) = +2
VXŸŸ
M atom in interstitial site
+2 – (0) = +2
MiŸŸ
X atom in interstitial site
–2 – (0) = –2
Xi ”
M atom misplaced in X site
+2 – (–2) = +4
MXŸŸŸŸ
X atom misplaced in M site
–2 – (+2) = –4
XM””
0
VMVX
+3 – (+2) = +1
LMŸŸ
–1 – (–2) = +1
YX’
Free electron
–1
e’
Electron hole
+1
hŸ
Di-vacancy on M and X sites
Impurity cation
L+3 on
M site
Impurity anion Y–1 on X site
Excess charge = Present charge – Previous charge; Ÿ = positive charge, ’ = negative charge
Examples:
1. What are the probable defects in pure Al2O3 ?
(1) Vacancy on Al+3 sub-lattice:
’’’
VAl
(2) Vacancy on O-2 sub-lattice:
VO
˜˜
˜˜˜
(3)
Al+3
into interstitial site:
(4) O-2 into interstitial site:
Al i!
’’
O i!
7 2. Consider the possible defects that can occur in
pure NaCl.
(1) Vacancy on Na+ sub-lattice:
VNa’
(2) Vacancy on Cl- sub-lattice:
V Cl
(3) Na+ into interstitial site:
Na
(4) Cl- into interstitial site:
’
Cl i!
˜
˜
3. Consider the possible defects formed due to
the addition of CaCl2 to NaCl.
˜
(1)
Ca on Na site:
Ca Na
(2)
Ca into interstitial site:
Ca i!
˜˜
4. Consider the possible defects formed due to
the addition of KCl to NaCl.
X
(1)
K on Na site:
K Na
˜
(2)
K into interstitial site:
K i!
8 Thermodynamics of point defects
q  Why vacancies are an unavoidable
thermodynamic necessity?
q  How many vacancies are there at a
given temperature?
v  Each defect and its concentrations in a material can be
described in terms of associated energy of formation and
other thermodynamic properties.
Approach:
Find an expression for G of perfect crystal and one for defective crystal;
the one with the lower (more negative G) wins…
Free Energy of a Perfect Crystal
Gperf = Hperf – T Sperf
S = Sconfig + ST + ....
Sconfig = – k N ln
ST
Sconfig = Configuration entropy, associated
with atomic disorder and rearrangement of atoms
and/or defects on a given number of lattice sites
ST = Thermal entropy, associated with
vibration with the atoms and rearrangement of
atoms/ions over existing energy levels.
N
n
+ n ln
N+n
N+n
kT
= 3 N k ln
+1
hν
n = vacant site
N = atom site
k = Boltzmann’s constant
h = Plank’s constant
ν = vibration frequency of bond
9 For perfect crystal, Sconfig = 0
since there is only one way of arranging N atoms on N lattice sites.
Sperf = ST = 3 N k
ln
kT
+ 1
hν
Gperf = Hperf – T Sperf
Gperf =
Hperf – 3 N k T
ln
kT
+ 1
hν
Free Energy of a Defective Crystal
Gdef = Hdef – T Sdef
q  If it cost hd to form one defect,
total enthalpy change for n defects is:
Hdef = Hperf + n hd
Sdef = Sconfig + ST
Sconfig = – k N ln
N
n
+ n ln
N+n
N+n
10 ST = 3 N k ln
kT
+1
hν
(For perfect crystal)
q  To find ST for defective crystal, let us assume that
atoms near a vacancy vibrate with a different frequency ν’ than the rest, and the
remainder of atoms will be unaffected and continue to vibrate with the frequency ν.
q  If the coordination number of vacancy is ζ, the
total number of atoms affected is ζn. Then, for
the defective crystal,
ST = 3k (N - ζn) ln
kT
+1
hν
vibration term unaffected by vacancies
+ 3k ζn ln
kT
+1
hν’
vibration affected by vacancies
Gdef = Hperf + nV hd
– kT
3 (N - ζn) ln
kT
+1
hν
+ 3ζn ln
kT
+1
hν’
–
Gperf =
N ln
Hperf – 3 N k T
N
n
+ n ln
N+n
N+n
ln
kT
+ 1
hν
11 ΔG = Gdef - Gperf
= n hd + 3kTζn ln
+ kT
N ln
ν’
ν
N
n
+ n ln
N+n
N+n
q  Free energy change upon introduction of n defects
in a otherwise perfect crystal is a function of both
n and T.
Let us plot this function ….. Ignoring the 2nd term.
defects increases with increasing temperature.
12 q  The number of vacancies at which ΔG is minimum
is the equilibrium number of vacancies at that
temperature.
neq
N + neq
neq
N
neq
N
neq
≈
N
= exp
= exp
= exp
hd – T ΔSvib
kT
Δgd
Δg = hd – T ΔSvib
kT
Δsvib = 3ζk ln (ν/ν’)
Δgd
Δg = hd – T ΔSvib
kT
Δsvib = 3ζk ln (ν/ν’)
q  The expression does not contain any Sconfig term
but depends solely on the free energy associated with the formation of single
defects, hd.
q  The equilibrium number of vacancies increases
exponentially with temperature
at high temperature, configuration entropy term becomes more important
relative to the enthalpy term, which in tern shifts n to higher values (Fig. 6.2b).
13 Next Class
Lecture 18
Defects in Ceramics 2
14