Transcript Document 7319176

Defect chemistry – a general introduction
Truls Norby
Department of Chemistry
University of Oslo
Centre for Materials Science
and Nanotechnology (SMN)
FERMIO
Oslo Research Park
(Forskningsparken)
[email protected]
http://folk.uio.no/trulsn
Brief history of structure, stoichiometry,
and defects
•
Early chemistry had no concept of stoichiometry or structure.
•
The finding that compounds generally contained elements in ratios of
small integer numbers was a great breakthrough!
•
Understanding that external geometry often reflected atomic structure.
•
Perfectness ruled. Non-stoichiometry was out.
•
Intermetallic compounds forced re-acceptance of non-stoichiometry.
•
But real understanding of defect chemistry of compounds is less than
100 years old.
Perfect structure
•
Our course in defects
takes the perfect structure
as starting point.
•
This can be seen as the
ideally defect-free interior
of a single crystal or large
crystallite grain at 0 K.
Close-packing
•
Metallic or ionic
compounds can often be
regarded as a closepacking of spheres
•
In ionic compounds, this
is most often a closepacking of anions (and
sometimes large cations)
with the smaller cations in
interstices
Some simple classes of oxide structures
with close-packed oxide ion sublattices
Formula
Cation:anion
coordination
Type and number
of occupied
interstices
fcc of anions
hcp of anions
MO
6:6
1/1 of octahedral
sites
NaCl, MgO, CaO, CoO,
NiO, FeO a.o.
FeS, NiS
MO
4:4
1/2 of tetrahedral
sites
Zinc blende: ZnS
Wurtzite: ZnS, BeO, ZnO
M2O
8:4
1/1 of tetrahedral
sites
occupied
Anti-fluorite: Li2O, Na2O
a.o.
M2O3, ABO3
6:4
2/3 of octahedral
sites
Corundum:
Al2O3, Fe2O3,
Cr2O3 a.o.
Ilmenite: FeTiO3
MO2
6:3
½ of octahedral
sites
Rutile: TiO2, SnO2
AB2O4
1/8 of tetrahedral
and 1/2 of
octahedral
sites
Spinel: MgAl2O4
Inverse spinel: Fe3O4
The perovskite structure ABX3
•
•
•
Close-packing of large A
and X
Small B in octahedral
interstices
Alternative (and
misleading?)
representation
We shall use 2-dimensional structures for
our schematic representations of defects
•
Elemental solid
•
Ionic compound
Defects in an elemental solid
From A. Almar-Næss: Metalliske materialer.
Defects in an
ionic compound
Defect classes
•
Electrons (conduction band) and electron
holes (valence band)
•
0-dimensional defects
–
–
–
•
1-dimensional defects
–
•
Dislocations
2-dimensional defects
–
–
•
point defects
defect clusters
valence defects (localised electronic defects)
Defect planes
Grain boundaries (often row of dislocations)
3-dimensional defects
–
Secondary phase
Perfect vs defective structure
• Perfect structure (ideally exists only at 0 K)
• No mass transport or ionic conductivity
• No electronic conductivity in ionic materials
and semiconductors;
• Defects introduce mass transport and
electronic transport; diffusion, conductivity…
• New electrical, optical, magnetic,
mechanical properties
• Defect-dependent properties
Point defects – intrinsic disorder
•
Point defects (instrinsic disorder)
form spontaneously at T > 0 K
–
–
•
1- and 2-dimensional defects do
not form spontaneously
–
–
•
Caused by Gibbs energy gain as a
result of increased entropy
Equilibrium is a result of the balance
between entropy gain and enthalpy
cost
Entropy not high enough.
Single crystal is the ultimate
equilibrium state of all crystalline
materials
Polycrystalline, deformed,
impure/doped materials is a result
of extrinsic action
Defect formation and equilibrium
Free energy vs number n of defects
Hn = nH
Sn = nSvib + Sconf
G = nH - TnSvib - TSconf
For n vacancies in an elemental solid:
EE = EE + vE
K = [vE] = n/(N+n)
Sconf = k lnP = k ln[(N+n)!/(N!n!)]
For large x: Stirling: lnx!  xlnx - x
Equilibrium at dG/dn = 0
= H - TSvib - kT ln[(N+n)/n] = 0
n/(N+n) = K = exp(Svib/k - H/kT)
Kröger-Vink notation for 0-dimensional
defects
•
•
Point defects
–
–
–
•
Electronic defects
–
–
•
Vacancies
Interstitials
Substitutional defects
Delocalised
• electrons
• electron holes
Valence defects
• Trapped electrons
• Trapped holes
Cluster/associated defects
Kröger-Vink-notation
A
c
s
A = chemical species
or v (vacancy)
s = site; lattice position
or i (interstitial)
c = charge
Effective charge = Real charge on site
minus charge site would have in
perfect lattice
Notation for effective charge:
•
positive
/
negative
x
neutral
(optional)
Perfect lattice of MX, e.g. ZnO
Zn
Zn
O
2
Zn
x
Zn
2O
O
x
O
vi
v
x
i
Vacancies and interstitials
v
//
Zn
Zn
v

i

O
O
//
i
Electronic defects
e
/
/
Zn
Zn
h

Zn
O

Zn

O
Foreign species
Ga
/
Zn

Zn
N
/
O
Ag

O
F
Li

i
Protons and other hydrogen defects
H+
H
H-
H i
OH

O
(OH)

O
OHi/
(2(OH))
H ix
H

O
x
MO2
How can we apply integer charges when
the material is not fully ionic?
v

O
The extension of the effective charge may
be larger than the defect itself
(4M M vO )

……much larger….
(4M M 4OO vO )

…but when it moves, an integer number of
electrons also move, thus making the use of the
simple defect and integer charges reasonable
(4M M 4OO vO )
 v

O

Defects are donors and acceptors
E
H i
Ec
Ga Zn
v Ox
Ag
/
Zn
v O
v
x
Zn
v O
v /Zn
v //Zn
Ev
Defect chemical reactions
Example: Formation of cation Frenkel defect pair:
Zn xZn  vix  v //Zn  Zn i 
Defect chemical reactions must obey three rules:
•
Mass balance: Conservation of mass
•
Charge balance: Conservation of charge
•
Site ratio balance: Conservation of host structure
Defect chemical reactions obey the mass
action law
Example: Formation of cation Frenkel defect pair:
Zn xZn  vix  v //Zn  Zn i 
KF 
av // aZn 
Zn
i
aZn x av x
Zn
KF 
i
av // aZn 
Zn
i
aZn x av x
Zn
i
[v //Zn ] [Zn i  ]
[v //Zn ][Zn i  ]
[Zn] [i]
//




[v
][Zn
Zn
i ]
x
x
x
x
[Zn Zn ] [v i ] [Zn Zn ][v i ]
[Zn] [i]
0
ΔS vib
 ΔG 0
 ΔH 0
 [v ][Zn ]  exp
 exp
exp
RT
R
RT
//
Zn

i
Notes on mass action law
• The standard state is that the site fraction of the defect
is 1
• Standard entropy and enthalpy changes refer to full site
occupancies. This is an unrealisable situation.
• Ideally diluted solutions often assumed
• Note: The standard entropy change is a change in the
vibrational entropy – not the configurational.
KF 
av // aZn 
Zn
i
aZn x av x
Zn
i
0
0
0
ΔS

ΔG

Δ
H
 [v //Zn ][Zn i  ]  exp
 exp vib exp
RT
R
RT
Electroneutrality
•
The numbers or concentrations of positive and
negative charges cancel, e.g.
2[v //Zn ]  2[Oi// ]  [Ag /Zn ]  [NO/ ]  [e / ]  2[Zn i  ]  2[vO ]  [Ga Zn ]  [OHO ]  [h  ]
•
Often employ simplified, limiting electroneutrality
condition:
2[v //Zn ]  2[Zn i  ]
or
[v //Zn ]  [Zn i  ]
Note: The electroneutrality is a mathematical expression, not a
chemical reaction. The coefficients thus don’t say how many
you get, but how much each “weighs” in terms of charge….
Site balances
•
Expresses that more than one species fight over the same
site:
[O Ox ]  [v O ]  [OHO ]  [O]
•
(  1 in ZnO )
Also this is a mathematical expression, not a chemical
reaction.
Defect structure; Defect concentrations
•
•
The defect concentrations can now be found by combining
–
Electroneutrality
–
Mass and site balances
–
Equilibrium mass action coefficients
Two defects (limiting case) and subsequently for minority defects
–
•
or three or more defects simultaneously
–
•
Brouwer diagrams
More exact solutions
…these are the themes for the subsequent lectures and exercises…