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

2. Conductivity Fundamentals of electrical conductivity Conductivity 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/m 2 ) = (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 resistance of a current-carrying object is obtained from the resistivity ρ, length l, and cross-sectional area a: R = ρ*l/a

4. Conductivity and conductance Conductivity σ (sigma) is the inverse of resistivity: σ = 1/ρ Conductance G is the inverse of resistance: G = 1/R The units for G and σ are S (siemens) and S/m, respectively. (Other/older units for conductance comprise Ω -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 mm 2. 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 the resistivity? ◦ If the conductance is 10 S, what is the conductivity?

6. Total conductivity, transport numbers The conductivity of a substance has contributions from all species, mechanisms, and pathways of charge 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 ◦ Microstructural pathways: bulk, grain boundaries, surfaces… The total conductivity is a sum of partial conductivities over all species, mechanisms, and pathways: The fraction of the total conductivity (and ideally the fraction of any current going through the substance) is termed the transport number or transference number 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)? ◦ H 2 O(l)? HCl(aq)? ◦ Y-doped ZrO 2 ? ◦ La 2 NiO 4+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 z s, volume concentration c s, and charge mobility u s. The charge is an integer multiple z s of e or F, depending of whether the concentration is given in number of particles or moles of particles per unit volume: The concentration c s may arise from different models comprising doping and thermodynamics for electrons and/or point defects. Charge mobility u s is the product of mechanical mobility B s and charge z s e:

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): Scatterers are defects (e.g. impurities) or phonons (lattice vibrations) 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 u s T (and thus σ s T) is an exponential function of 1/T, and therefore the activation enthalpy may be extracted from the slope of a plot of ln(u s T) or log(u s T) 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 polaron mobilities 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 c S 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? C d or C c ? Which is bigger: u d or u c ? 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. Brief history of 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! H 2 O CO 2 NaClCaCl 2 NiO Understanding that external geometry often reflected atomic structure. Perfectness ruled. Variable composition (non- stoichiometry) was out. However, variable composition in some intermetallic compounds became indisputable and in the end forced re-acceptance of non-stoichiometry. But real understanding of defect chemistry of compounds mainly came about from the 1930s and onwards, attributable to Frenkel, Schottky, Wagner, Kröger…, many of them physicists, and almost all German! Frenkel Schottky Wagner

18. Defects in an elemental solid (e.g. Si or Ni metal) Notice the distortions of the lattice around defects ◦ The size of the defect may be taken to be bigger than the point defect itself 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 der Waals forces bond molecules. ◦ Hard solids if bonds extend in 3 dimensions into macromolecules.  Examples: C (diamond), SiO 2 (quartz), SiC, Si 3 N 4 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. Kröger-Vink notation 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 What the entity is, as the main symbol (A) ◦ Chemical symbol ◦ or v (for vacancy) Where the 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

26. Effective charge 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 O 2- 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

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

28. Kröger-Vink notation – more examples Dopants and impurities Y 3+ substituting Zr 4+ in ZrO 2 Li + substituting Ni 2+ in NiO Li + interstitials in e.g. NiO Electronic defects Defect electrons in conduction band Electron holes in valence band

29. Kröger-Vink notation – also for elements of the ideal structure (constituents) Cations, e.g. Mg 2+ on normal Mg 2+ sites in MgO Anions, e.g. O 2- on normal site in any oxide Empty interstitial site

30. Kröger-Vink notation of dopants in elemental semiconductors, e.g. Si 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 +

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. Electroneutrality 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?

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 or, equivalently: new structural unit M 2+ O 2-

39. Frenkel disorder in MO M 2+ O 2-

40. Anion (anti-)Frenkel disorder in MO M 2+ O 2-

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: Ilmenite FeTiO 3 Example: Fe 2 O 3

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 The two electrons of the O 2- ion are shown left behind More realistic picture, where the two electrons are delocalised on neighbouring cations “Normal” chemistry: Defect chemistry:

45. Oxygen deficiency The two electrons of the O 2- ion are shown left behind The two electrons are loosely bonded since the nuclear charge of the former O 2- 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 elements Substituents Dopants

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.

51. Doping of semiconductors

52. Doping of Si with P (donor) or B (acceptor)

53. Doping of ionic compounds: M 1-x O doped substitutionally with Mh 2 O 3 Mh 3+ substituting M 2+ will constitute donor-doping, giving effectively positive dopants. M 1-x O contains M vacancies and electron holes. The doping may thus be compensated by producing M vacancies: or – less relevant - by consuming electron holes. This is a reduction reaction and releases oxygen:

54. Ni 1-x O doped substitutionally with Li 2 O Li + and Ni 2+ are similar in size, so Li + may substitute Ni 2+. This will constitute acceptor-doping with effectively negative dopants. (This is utilised in Li-doped NiO for p-type conducting electrodes for fuel cells, batteries etc.) Ni 1-x O contains nickel vacancies and electron holes. The doping may thus be compensated by consuming Ni vacancies or – better - by producing electron holes. This is an oxidation reaction and requires uptake of oxygen

55. ZrO 2-y doped substitutionally with Y 2 O 3 Y 3+ will form effectively negative defects when substituting Zr 4+ and thus acts as an acceptor. It must be compensated by a positive defect. ZrO 2-y contains oxygen vacancies and electrons The doping is thus most relevantly written in terms of forming oxygen vacancies:

56. ZrO 2-y doped substitutionally with Y 2 O 3 Note: Electrons donated from oxygen vacancy are accepted by Y dopants; no electronic defects in the bands.

57. Hydration – dissolution of protons from H 2 O Water as source of protons. Equivalent to other oxides as source of foreign elements. Example: Hydration of acceptor-doped MO 2, whereby oxygen vacancies are annihilated, and protons dissolved as hydroxide ions. The acceptor dopants are already in, and are not visible in the hydration reaction in this case

58. Ternary and higher compounds With ternary and higher compounds the site ratio conservation becomes a little more troublesome to handle, that’s all. For instance, consider the perovskite CaTiO 3. To form Schottky defects in this we need to form vacancies on both cation sites, in the proper ratio: And to form e.g. metal deficiency we need to do something similar: (But oxygen deficiency or excess would be just as simple as for binary oxides, since the two cations sites are not affected in this case …)

59. Doping of ternary compounds The same rule applies: Write the doping as you imagine the synthesis is done: If you are doping by substituting one component, you have to remove some of the component it is replacing, and thus having some left of the other component to react with the dopant. For instance, to make undoped LaScO 3, you would probably react La 2 O 3 and Sc 2 O 3 and you could write this as: Now, to dope it with Ca 2+ substituting La 3+ you would replace some La 2 O 3 with CaO and let that CaO react with the available Sc 2 O 3 : The latter is thus a proper doping reaction for doping CaO into LaScO 3, replacing La 2 O 3.

60. Back to where we started: Conductivity Product of charge, charge mobility, and concentration s can be a constituent or a defect

61. Which ions can we use? Let us look at what are the fuels for fuel cells H 2 CH 4 Diesel CH 3 OH C 2 H 5 OH CH 3 OCH 3 NH 3 They will all be reformed or cracked into H 2 externally or at the anode catalyst. H 2 is effectively the fuel at all fuel cell anodes – to be oxidised to H 2 O by O 2 in air, via an ionic conduction transport

62. Fuel cells – types according to electrolyte Aqueous polymer AqueousMoltenSolid SOFCO 2- PCFCH+H+ MCFCCO 3 2- AFCOH - PAFCH3O+H3O+ PEMFCH3O+H3O+ PC-SOFC (PCFC) BaCe 0.9 Y 0.1 O 3-d SAFC CsHSO 4 HT-PEMFC Phosphonated PBI No solid OH - conductors No good H + conductors among hydroxides No solid OH - conductors No good H + conductors among hydroxides

63. Conductivity of electrolyte Must be fully ionic t ion > 0.99 Requires large band gaps, typically > 3 eV Preferably > 0.01 S/cm Requires ionic disorder (high concentration, high mobility) ◦ Liquid state (aqueous solution, molten salt) ◦ Solid state crystals with intrinsic disorder ◦ Solid state doped crystals

64. Conductivity requirements for the electrodes The job is to transport electrons But transport of ions is also welcome What kind of materials can we use?

65. Conductivity requirements for the interconnects The job is to transport electrons only What kind of materials can we use?

66. Stability What are the stability issues for our three materials classes? Electrolyte Electrodes ◦ Anode ◦ Cathode Interconnect

67. Microstructure What are the microstructural requirements for our three materials classes? Electrolyte Electrodes ◦ Anode ◦ Cathode Interconnect

68. How can we now refine the selection criteria for Electrolyte? Electrode? Interconnects? Any suggestions for actual materials? At 80°C for the PEMFC At 800°C for the SOFC?