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Defect chemistry – a general introduction
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Transcript of Defect chemistry – a general introduction
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[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
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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.
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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.
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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
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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
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The perovskite structure ABX3
• Close-packing of large A and X
• Small B in octahedral interstices
• Alternative (and misleading?) representation
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We shall use 2-dimensional structures for our schematic representations of defects
• Elemental solid
• Ionic compound
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Defects in an elemental solid
From A. Almar-Næss: Metalliske materialer.
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Defects in an ionic compound
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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
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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
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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
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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)
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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)
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Perfect lattice of MX, e.g. ZnO
xZnZn
2ZnZn
-2OOxOO
ivxiv
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Vacancies and interstitials
iZn
//Znv
Ov
//iO
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Electronic defects
/ZnZn
/e
hZnZnOO
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Foreign species
ZnGa
/ZnAg
/ONOF
iLi
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Protons and other hydrogen defects
H+ H H-
OOH
iH
O(OH)
/iOH
OH
xiH
xMO2(2(OH))
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How can we apply integer charges when the material is not fully ionic?
Ov
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The extension of the effective charge may be larger than the defect itself
)v(4M OM
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……much larger….
)v4O(4M OOM
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…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
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Defects are donors and acceptors
E
xOv
Ov Ov
Ec
Ev
ZnGa
/ZnAg
//Znv/
ZnvxZnv
iH
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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
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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//
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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//
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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
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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
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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…