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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!

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brief history of structure stoichiometry and defects
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
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
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
the perovskite structure abx 3
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
We shall use 2-dimensional structures for our schematic representations of defects
  • Elemental solid
  • Ionic compound
defects in an elemental solid
Defects in an elemental solid

From A. Almar-Næss: Metalliske materialer.

defect classes
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 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 – 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
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)

kr ger vink notation for 0 dimensional defects
Kröger-Vink notation for 0-dimensional defects
  • Kröger-Vink-notation
  • 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)
  • Point defects
    • Vacancies
    • Interstitials
    • Substitutional defects
  • Electronic defects
    • Delocalised
      • electrons
      • electron holes
    • Valence defects
      • Trapped electrons
      • Trapped holes
  • Cluster/associated defects
slide24

…but when it moves, an integer number of electrons also move, thus making the use of the simple defect and integer charges reasonable

defect chemical reactions
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
defect chemical reactions obey the mass action law
Defect chemical reactions obey the mass action law

Example: Formation of cation Frenkel defect pair:

notes on mass action law
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.
electroneutrality
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….

site balances
Site balances
  • Expresses that more than one species fight over the same site:
  • Also this is a mathematical expression, not a chemical reaction.
defect structure defect concentrations
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…