defect chemistry – a general introduction
DESCRIPTION
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.TRANSCRIPT
[email protected] http://folk.uio.no/trulsn
Department of ChemistryUniversity of Oslo
Centre for Materials Science and Nanotechnology (SMN)
FERMIOOslo Research Park (Forskningsparken)
Defect chemistry – a general introduction
Truls Norby
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 close-packing of spheres
• In ionic compounds, this is most often a close-packing 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: MgAl2O4Inverse 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– point defects– defect clusters– valence defects (localised electronic defects)
• 1-dimensional defects– Dislocations
• 2-dimensional 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
– Caused by Gibbs energy gain as a result of increased entropy
– Equilibrium is a result of the balance between entropy gain and enthalpy cost
• 1- and 2-dimensional defects do not form spontaneously
– 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 = nHSn = nSvib + Sconf
G = nH - TnSvib - TSconf
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 dG/dn = 0= H - TSvib - kT ln[(N+n)/n] = 0
n/(N+n) = K = exp(Svib/k - H/kT)
Kröger-Vink notation for 0-dimensional defects
• Point defects– Vacancies– Interstitials– Substitutional defects
• Electronic defects– Delocalised
• electrons• electron holes
– Valence defects• Trapped electrons • Trapped holes
• Cluster/associated defects
csA
• Kröger-Vink-notation
A = chemical species or v (vacancy)
s = site; lattice position or i (interstitial)
c = chargeEffective charge = Real charge on site
minus charge site would have in perfect lattice
Notation for effective charge:• positive/ negativex neutral (optional)
Perfect lattice of MX, e.g. ZnO
xZnZn
2ZnZn
-2OOxOO
ivxiv
Vacancies and interstitials
iZn
//Znv
Ov
//iO
Electronic defects
/ZnZn
/e
hZnZnOO
Foreign species
ZnGa
/ZnAg
/ONOF
iLi
Protons and other hydrogen defects
H+ H H-
OOH
iH
O(OH)
/iOH
OH
xiH
xMO2(2(OH))
How can we apply integer charges when the material is not fully ionic?
Ov
The extension of the effective charge may be larger than the defect itself
)v(4M OM
……much larger….
)v4O(4M OOM
…but when it moves, an integer number of electrons also move, thus making the use of the simple defect and integer charges reasonable
)v4O(4M OOM
O v
Defects are donors and acceptors
E
xOv
Ov Ov
Ec
Ev
ZnGa
/ZnAg
//Znv/
ZnvxZnv
iH
Defect chemical reactions
Example: Formation of cation Frenkel defect pair:
Defect chemical reactions must obey three rules:
• Mass balance: Conservation of mass
• Charge balance: Conservation of charge
• Site ratio balance: Conservation of host structure
i//Zn
xi
xZn ZnvvZn
Defect chemical reactions obey the mass action law
Example: Formation of cation Frenkel defect pair:
i//Zn
xi
xZn ZnvvZn
]][Zn[v]][v[Zn]][Zn[v
[i]][v
[Zn]][Zn
[i]][Zn
[Zn]][v
i//Znx
ixZn
i//Zn
xi
xZn
i//Zn
//
xi
xZn
iZn
vZn
ZnvF aa
aaK
RTH
RΔS
RTΔG
aa
aaK vib
vZn
ZnvF
xi
xZn
iZn
000
i//Zn
Δexpexpexp]][Zn[v//
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.
RTH
RΔS
RTΔG
aa
aaK vib
vZn
ZnvF
xi
xZn
iZn
000
i//Zn
Δexpexpexp]][Zn[v//
Electroneutrality
• The numbers or concentrations of positive and negative charges cancel, e.g.
• Often employ simplified, limiting electroneutrality condition:
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….
][Zn][vor ]2[Zn]2[v i//Zni
//Zn
][h][OH][Ga][v2]2[Zn][e][N][Ag][O2]2[v OZnOi//
O/Zn
//i
//Zn
Site balances
• Expresses that more than one species fight over the same site:
• Also this is a mathematical expression, not a chemical reaction.
) in ZnO 1 ( [O]][OH][v][O OOxO
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
– Brouwer diagrams
• or three or more defects simultaneously
– More exact solutions
• …these are the themes for the subsequent lectures and exercises…