MENA 3200 Energy Materials Materials for Electrochemical Energy Conversion Part 2

1. MENA 3200 Energy Materials Materials for Electrochemical Energy Conversion Part 2 Electrical conductivity – defects and transport Diffusion and stability Truls Norby

2. ConductivityFundamentals of electrical conductivityConductivity requirements

3. Resistivity and resistance • Charged particles in an electric field E feel a force F • The force sets up a net flux density and current density i • The ratio ρ(rho) = E/i is termed resistivity and is an intensive materials property • Resistivity has units (V/m)/(A/m2) = (V/A)m = ohm*m = Ωm • For an object we may instead express a current I and voltage U • The ratio R = U/I (Ohm’s law) is termed resistance and is an extensive property for the object • Resistance has units V/A = ohm = Ω • The resistanceof a current-carryingobject is obtained from theresistivityρ, lengthl, and cross-sectional area a: R = ρ*l/a

4. Conductivity and conductance • Conductivity σ (sigma) is the inverse ofresistivity: σ = 1/ρ • Conductance G is the inverse ofresistance: G = 1/R • The units for G and σare S (siemens) and S/m, respectively. • (Other/olderunits for conductancecompriseΩ-1, ohm-1, and mho) • G = σ*a/l

5. Exercises • A rectangular solid sample has a length of 2 cm and a cross-section with sides 5 x 5 mm2. Electrodes for merasurements are painted on its far faces. • If its conductivity is 1000 S/cm, what is its conductance? • And its resistance? • A circular disk has thickness 2 mm and diameter 2 cm. We paint electrodes on its two faces and measure the resistance. • If the resistance is 10 Ω, what is theresistivity? • If the conductance is 10 S, what is the conductivity?

6. Total conductivity, transport numbers • The conductivityof a substance has contributions from all species, mechanisms, and pathwaysofcharge carriers: • Electronic and ionic • Electronic: electrons and holes • Ionic: cations and anions • Or more detailed, for instance, protons, oxide ions, and metal cations • Mechanisms: vacancies and interstitials • Microstructuralpathways: bulk, grainboundaries, surfaces… • The total conductivity is a sum ofpartialconductivities over all species, mechanisms, and pathways: • The fractionofthe total conductivity (and ideallythefractionofanycurrentgoingthroughthesubstance) is termedthe transport number or transferencenumber for s:

7. Exercise • Normally, only one or two charge carriers, defects, mechanisms, or pathways dominate to the extent that we need to take them into account. The others can be neglected. • What dominates the conductance in • Si? As-doped Si? • Pt? • NaCl(s)? NaCl(aq)? • H2O(l)? HCl(aq)? • Y-doped ZrO2? • La2NiO4+δ? • Alumina single crystal? Dense alumina ceramic? Porous alumina ceramic?

8. Exercise • One can enhance or depress selected contributions for measurements or use • Discuss how you might affect the contributions below in the case of solid samples: • Electronic conductivity vs oxide ion conductivity • Proton conductivity • Bulk conductivity • Grain boundary conductivity • Outer surface conductivity • Inner surface (pore wall) conductivity

9. Series resistance contributions • Till now, we have looked at parallel possibilities that add to conductance and give more current • There are also many sources to series problems that add to resistance and give less current (more voltage): • Bulk resistance • Traps • Grain boundary resistance • Electrode (contact) resistance • Note the difference between grain boundary conduction and grain boundary resistance • What is the source of each one? • How can they be affected?

10. Conductivity; charge, concentration, and mobility • The conductivity of a species s is given by its charge zs, volume concentration cs, and charge mobility us. • The charge is an integer multiple zs of e or F, depending of whether the concentration is given in number of particles or moles of particles per unit volume: • The concentration cs may arise from different models comprising doping and thermodynamics for electrons and/or point defects. • Charge mobility us is the product of mechanical mobility Bs and charge zse:

11. Charge mobility; itinerant carriers (metallic mobility) • In materials with metallic mobility (itinerant electrons or holes, broad bands) the mobility is determined by scattering, and the mobility is proportional to the mean free length between scattering events and inversely proportional to the electron or hole effective mass and the mean velocity at the mobile electrons’ energy level (Fermi level): • Scatterersaredefects (e.g. impurities) or phonons (latticevibrations) • Both contribute to resistance in series: • Typical temperature dependencies: • Typically, impurities dominate at low T and lattice vibrations at high T.

12. Charge mobility; diffusing carriers • For ions that move by defects in materials and for non-itinerant (trapped) electrons in semiconductors, the mobility of the ionic defect or electronic species is determined by diffusion; thermally activated jumps from site to site: • Note that usT (and thus σsT) is an exponential function of 1/T, and therefore the activation enthalpy may be extracted from the slope of a plot of ln(usT) orlog(usT) versus 1/T (similar to an Arrhenius plot). • Such electronic charge carriers are called small polarons – the electron deeply trapped in the relaxation of the lattice around itself. Small polaronmobilities are orders of magnitude smaller than itinerant (metallic) mobilities. • Electronic charge carriers trapped in more shallow relaxations are called large polarons and have intermediate mobilities.

13. Concentrations cS of charge carriers - overview • Metals: Concentration of electrons approx. equal to the concentration of valence electrons • Electronic semiconductors: Concentration of electrons n or holes p fixed by donor or acceptor dopants • Solid ionic conductors: Concentration of defects (e.g. oxygen vacancies or protons) fixed by acceptors or structural disorder • Liquid ionic conductors: Concentration of ions…

14. Conductivity of components and defects • For foreign species, like protons in an oxide, the conductivity of the defect is simply e.g. • But for a component, like oxide ions in an oxide, conductivity can be expressed in terms of the component or the defect • Components need defects to move, and defects need components to move

15. Exercise • Which is bigger? Cd or Cc? • Which is bigger: ud or uc? • Which is faster? The component atoms or the defects?

16. In order to understand, analyse, and affect the conductivity in crystalline solids, we need to understand defect concentrations Introductory on defect chemistry

17. Briefhistoryofdefects • Earlychemistryhadnoconceptof stoichiometry or structure. • The findingthat compounds generallycontained elements in ratiosofsmallintegernumberswas a greatbreakthrough! H2O CO2 NaCl CaCl2 NiO • Understandingthatexternalgeometryoftenreflectedatomicstructure. • Perfectnessruled. Variable composition (non-stoichiometry) wasout. • However, variable composition in some intermetallic compounds became indisputable and in the end forcedre-acceptanceofnon-stoichiometry. • But real understandingofdefectchemistryof compounds mainlycameabout from the 1930s and onwards, attributable to Frenkel, Schottky, Wagner, Kröger…, manyofthemphysicists, and almost all German! Frenkel Schottky Wagner

18. Notice the distortions of the lattice around defects The size of the defect may be taken to be bigger than the point defect itself Defects in an elemental solid (e.g. Si or Ni metal) Adapted from A. Almar-Næss: Metalliske materialer, Tapir, Oslo, 1991.

19. Defects in an ionic solid compound

20. Bonding • Bonding: Decrease in energy when redistributing atoms’ valence electrons in new molecular orbitals. • Three extreme and simplified models: • Covalent bonds: Share electrons equally with neighbours! • Strong, directional pairwise bonds. Forms molecules. Bonding orbitals filled. • Soft solids if van derWaals forces bond molecules. • Hard solids if bonds extend in 3 dimensions into macromolecules. • Examples: C (diamond), SiO2 (quartz), SiC, Si3N4 • Metallic bonds: Electron deficiency: Share with everyone! • Atoms packed as spheres in sea of electrons. Soft. • Only partially filled valence orbital bands. Conductors. • Ionic bonds: Anions take electrons from the cations! • Small positive cations and large negative anions both happy with full outer shells. • Solid formed with electrostatic forces by packing + and – charges. Lattice energy.

21. Formal oxidation number • Bonds in compounds are not ionic in the sense that all valence electrons are not entirely shifted to the anion. • But if the bonding is broken – as when something, like a defect, moves – the electrons have to stay or go. Electrons can’t split in half. • And mostly they go with the anion - the most electronegative atom. • That is why the ionic model is useful in defect chemistry and transport • And it is why it is very useful to know and apply the rules of formal oxidation number, the number of charges an ion gets when the valence electrons have to make the choice

22. Bonding – some important things to note • Metallic bonding (share of electrons) and ionic bonding (packing of charged spheres) only have meaning in condensed phases. • In most solids, any one model is only an approximation: • Many covalent bonds are polar, and give some ionic character or hydrogen bonding. • Both metallic and especially ionic compounds have covalent contributions • In defect chemistry, we will still use the ionic model extensively, even for compounds with little degree of ionicity. • It works! • …and we may understand why.

23. Formal oxidation number rules • Fluorine (F) has formal oxidation number -1 (fluoride) in all compounds. • Oxygen (O) has formal oxidation number -2 (oxide) , -1 (peroxide) or -1/2 (superoxide), except in a bond with F. • Hydrogen (H) has oxidation number +1 (proton) or -1 (hydride). • All other oxidation numbers follow based on magnitude of electronegativity (see chart) and preference for filling or emptying outer shell (given mostly by group of the periodic table).

24. Point defects

25. We will now start to consider defects as chemical entities We need a notation for defects. Many notations have been in use. In modern defect chemistry, we use Kröger-Vink notation (after Kröger and Vink). It describes any entity in a structure; defects and “perfects”. The notation tells us Whatthe entity is, as the main symbol (A) Chemical symbol or v (for vacancy) Wherethe entity is, as subscript (S) Chemical symbol of the normal occupant of the site or i for interstitial (normally empty) position Its charge, real or effective, as superscript (C) +, -, or 0 for real charges or ., /, or x for effective positive, negative, or no charge Note: The use of effective charge is preferred and one of the key points in defect chemistry. We will learn what it is in the following slides Kröger-Vink notation

26. The effective charge is defined as the charge an entity in a site has relative to (i.e. minus) the charge the same site would have had in the ideal structure. Example: An oxide ion O2- in an interstitial site (i) Real charge of defect: -2 Real charge of interstitial (empty) site in ideal structure: 0 Effective charge: -2 – 0 = -2 Effective charge

27. Example: An oxide ion vacancy Real charge of defect (vacancy = nothing): 0 Real charge of oxide ion O2- in ideal structure: -2 Effective charge: 0 – (-2) = +2 Example: A zirconium ion vacancy, e.g. in ZrO2 Real charge of defect: 0 Real charge of zirconium ion Zr4+ in ideal structure: +4 Effective charge: 0 – 4 = -4 Effective charge – more examples

28. Dopants and impurities Y3+ substituting Zr4+ in ZrO2 Li+ substituting Ni2+ in NiO Li+ interstitials in e.g. NiO Electronic defects Defect electrons in conduction band Electron holes in valence band Kröger-Vink notation – more examples

29. Cations, e.g. Mg2+ on normal Mg2+ sites in MgO Anions, e.g. O2- on normal site in any oxide Empty interstitial site Kröger-Vink notation – also for elements of the ideal structure (constituents)

30. Silicon atom in silicon Boron atom (acceptor) in Si Boron in Si ionised to B- Phosphorous atom (donor) in Si Phosphorous in Si ionised to P+ Kröger-Vink notation of dopants in elemental semiconductors, e.g. Si

31. Protonic defects • Hydrogen ions, protons H+ , are naked nuclei, so small that they can not escape entrapment inside the electron cloud of other atoms or ions • In oxidic environments, they will thus always be bonded to oxide ions –O-H • They can not substitute other cations • In oxides, they will be defects that are interstitial, but the interstitial position is not a normal one; it is inside an oxide ion. • With this understanding, the notation of interstitial proton and substitutional hydroxide ion are equivalent.

32. A few tips: • Defects and charges are done seemingly a little different in elemental semiconductors and ionic solids • The donor and acceptor dopants are by tradition entered in doping reactions neutral in the former and effectively charged (ionised to their preferred valency) in the latter. Don’t let it confuse or disencourage you. • Physicists use + and – for effective and real charges alike, and actually don’t differentiate them much. Don’t let physicists confuse or disencourage you , and be kind with them . • Don’t mix real and effective charges in one reaction equation or electroneutrality consideration. • Use effective charges only in defect chemistry, which can only refer to one single phase. • Use real charges in all cases of exchange of charge between phases, like in electrochemistry. • I use v and i for vacancy and interstitial, while Kröger and Vink (and most of the rest still) use V and I.

33. Electroneutrality

34. Electroneutrality • One of the key points in defect chemistry is the ability to express electroneutrality in terms of the few defects and their effective charges and to skip the real charges of all the normal structural elements •  positive charges =  negative charges can be replaced by •  positive effective charges =  negative effective charges •  positive effective charges -  negative effective charges = 0

35. The number of charges is counted over a volume element, and so we use the concentration of the defect species s multiplied with the number of charges z per defect Example, oxide MO with oxygen vacancies, acceptor dopants, and defect electrons: If electrons dominate over acceptors, we can simplify: Note: These are not chemical reactions, they are mathematical relations and must be read as that. For instance, in the above: Are there two vacancies for each electron or vice versa? Electroneutrality

36. Examples of some important defect chemical reactions

37. Stoichiometric compounds – intrinsic disorders Disorders that do not exchange mass with the surroundings, and thus do not affect the stoichiometry of the compound.

38. Schottky disorder in MO M2+ new structural unit O2- or, equivalently:

39. Frenkel disorder in MO M2+ O2-

40. Anion (anti-)Frenkel disorder in MO M2+ O2-

41. Intrinsic electronic ionisation Three equivalent reaction equations: Consider charges, electrons and sites: Simpler; skip sites: Simplest; skip valence band electrons:

42. Valence defects – localised electrons and holes Example: Fe2O3 Example: Ilmenite FeTiO3

43. Nonstoichiometric compounds – exchange of components with the surroundings Disorders that exchange mass of one of the components with the surroundings, and thus change the stoichiometry of the compound. We will take the first one – oxygen deficiency – in small steps, then the other ones more briefly.

44. Oxygen deficiency “Normal” chemistry: The two electrons of the O2- ion are shown left behind Defect chemistry: More realistic picture, where the two electrons are delocalised on neighbouring cations

45. Oxygen deficiency The two electrons of the O2- ion are shown left behind The two electrons are loosely bonded since the nuclear charge of the former O2- ion is gone. They get a high energy close to the state of the reduced cations…the conduction band. The vacancy is a donor.

46. Ionisation of the oxygen vacancy donor Electrons excited to conduction band delocalised over entire crystal, mainly in orbitals of reduced cation

47. Oxygen deficiency – overall reaction

48. Defect reactions involving foreign elementsSubstituentsDopants

49. Foreign elements; some terminology • Foreign elements are often classified as • impurities – non-intentionally present • dopants – intentionally added in small amounts • substituents – intentionally substituted for a host component (we tend to call it all doping and dopants) • They may dissolve interstitially or substitutionally • Substitutionally dissolved foreign elements may be • homovalent – with the same valency as the host it replaces • heterovalent – with a different valency than the host it replaces. • Also called aliovalent • Heterovalent metals • Higher valent metals will sometimes be denoted Mh (h for higher valent). • Lower valent metals will sometimes be denoted Ml (l for lower valent).

50. Doping of semiconductors • In covalently bonded semiconductors, the valence electrons will strive to satisfy the octet rule for each atom. • As example, we add P or B to Si. • Si has 4 valence electrons and forms 4 covalent bonds. • Phosphorous P has 5 valence electrons. When dissolved in the Si structure it thus easily donates its extra electron to the conduction band in order to become isoeletronic with Si. • Boron B has 3 valence electrons. When dissolved in the Si structure it thus easily accepts the lacking electron from the valence band in order to become isoeletronic with Si.