Defect chemistry – a general introduction

1. Defect chemistry – a general introduction Truls Norby

2. 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.

3. 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.

4. 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

5. Some simple classes of oxide structures with close-packed oxide ion sublattices

6. The perovskite structure ABX3 • Close-packing of large A and X • Small B in octahedral interstices • Alternative (and misleading?) representation

7. We shall use 2-dimensional structures for our schematic representations of defects • Elemental solid • Ionic compound

8. Defects in an elemental solid From A. Almar-Næss: Metalliske materialer.

9. Defects in an ionic compound

10. 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

11. 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

13. 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

14. 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)

15. 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

16. Perfect lattice of MX, e.g. ZnO

17. Vacancies and interstitials

18. Electronic defects

19. Foreign species

20. Protons and other hydrogen defects H+ H H-

21. How can we apply integer charges when the material is not fully ionic?

22. The extension of the effective charge may be larger than the defect itself

23. ……much larger….

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

25. Defects are donors and acceptors E Ec Ev

26. 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

27. Defect chemical reactions obey the mass action law Example: Formation of cation Frenkel defect pair:

28. 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.

29. 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….

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

31. 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…