Fundamentals of Solid State Ionics I Defect Chemistry Tutorial - PowerPoint PPT Presentation

The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) for the Fuel Cells and Hydrogen Joint Technology Initiative under grant agreement n° [621244]. Fundamentals of Solid State Ionics – I – Defect Chemistry Tutorial lecture at the Summer School Valencia, September 23-25, 2015 Main purposes Truls Norby Introduce defect chemistry to newbies Focus on some important principles and good practices for oldies Outline What are defects and why are they important? Department of Chemistry Random diffusion and ionic conductivity University of Oslo Defect reactions and equilibrium thermodynamics Centre for Materials Science Examples include MO, ZrO 2 , BaZrO 3 and Nanotechnology (SMN) Li ion battery materials FERMIO Oslo Research Park Computational defect chemistry (Forskningsparken) Summarising advice truls.norby@kjemi.uio.no http://folk.uio.no/trulsn

Stoichiometric compounds; Point defects form in pairs: Intrinsic point defect disorders • Schottky defects – Cation and anion vacancies • Frenkel defects – Cation vacancies and interstitials • Anti- or anion-Frenkel defects – Anion vacancies and interstitials

Stoichiometric compounds: Electronic defects: Intrinsic electronic disorder • Dominates in undoped semiconductors with moderate bandgaps Defect electrons in the conduction band and electron holes in the valence band

Random diffusion and self diffusion • Mass transport in crystalline solids is driven by thermal energy kT • Leads to random diffusion • If the diffusing species is a constituent it is also called self-diffusion • Two most important mechanisms: Vacancy mechanism Interstitial mechanism • Defects are needed in both

Diffusivity is a difficult entity to understand. First warm-up: Diffusivity: a matter of geometry and jump rates Number of neighbouring sites Orthogonal directions Jump rate • Constituent by vacancy     2 2 1 1 Likelyhood of target D s s ZX mechanism r , c c v site to be vacant 6 6 Jump distance Rate of sufficiently energetic attempts • Vacancy     2 2 1 1 D s s Z r , v v 6 6 • Constituent by interstitial Likelyhood     2 2 1 1 D s s ZX mechanism to be r , c c i 6 6 interstitial • Interstitial        2 2 2 1 1 1 D s s Z ( 1 X ) s Z r , i i i 6 6 6   ΔG ΔS ΔH     ω m m m exp exp exp RT R RT

Diffusivity – next exercise to look at what it is: The Nernst-Einstein relation – linking mobility and diffusivity • Application of a force F i gives the randomly diffusing particles i a net drift velocity v i : v  B F i i i • The proportionality B i is called mechanical mobility («beweglichkeit») • Mechanical mobility B i (beweglichkeit) is the diffusivity D i over the thermal energy kT : D i  i  D B kT i B i kT • This is the Nernst-Einstein relation

Electrical field; force, flux density, and current density • An electrical field is the downhill gradient in electrical potential:  d   E dx • It gives rise to a force on a charged particle i given as  d    F z eE z e i i i dx • The flux density j i is the volume concentration c i multiplied with the drift velocity v i :    j c v c B F c B z eE i i i i i i i i i • Current density by multiplication with charge:    2 i z ej z ec v c B ( z e ) E i i i i i i i i i

Mobilities and conductivity • We now define a charge mobility u i Charge mobility u is in physics often u  z eB denoted μ . We here use u to avoid i i i confusion with chemical potential. • We then obtain for the current density:   2 i c B ( z e ) E z ec u E i i i i i i i • We now define electrical conductivity σ i   z ec u Very important! Know it! i i i i Charge x concentration x charge mobility • and obtain    i z ec u E E This is one form of Ohm’s law. i i i i i Conductivity has units S/cm or S/m.

Ionic conductivity for vacancy mechanism Volume concentration • Constituent by vacancy mechanism of vacancies  2 2 2 1 ( z e ) c D ( z e ) c s ZX      i i , c i , c i i , c i , v 2 6 z ec u ( z e ) c B i i i , c i , c i i , c i , c kT kT • Vacancy  2 2 2 1 ( z e ) c D ( z e ) c s Z      2 i i , v i , v i i , v 6 z ec u ( z e ) c B i i i , v i , v i i , v i , v kT kT Volume concentration Charge mobility of vacancies of vacancies (~ concentration independent) Regardless of whether you consider the constituent or the defect, you need the concentration of the defect – indirectly or directly.

Before we move on... Formal oxidation number – integer charges • We know that bonds in ionic compounds are not fully 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 cannot split in half. • And mostly they go with the anion - the most electronegative atom. • That is why the ionic model applies in defect chemistry and transport • And it is why it is very useful to know and apply the rules of formal oxidation numbers, the number of charges an ion gets when the valence electrons have to make the choice • z are integer numbers

Defect chemistry • Allows us to describe processes involving defects • Allows application of statistical thermodynamics – Equilibrium coefficients; Enthalpies and entropies • Yields defect structure (concentrations of all defects) under given conditions • The defect concentrations for transport coefficients (e.g. conductivity) • Requires nomenclature • Requires rules for writing proper reactions • Additional requirements: Electroneutrality, site balances…

Kröger-Vink notation • In modern defect chemistry, we use Kröger-Vink notation. It can describe any entity in a crystalline structure; defects and “perfects”. C • A Main symbol A, a subscript S, and a superscript C: • What the entity is, as the main symbol ( A ) S – Chemical symbol – or v (for vacancy) Kröger and Vink used uppercase V • Where the entity is – the site - as subscript ( S ) for vacancies and I for interstitial sites, perhaps because that is natural – Chemical symbol of the normal occupant of the site for nouns n German. – or i for interstitial (normally empty) position I say: How would you then do defect • Its charge , real or effective, as superscript ( C ) chemistry for vanadium iodide VI 3 ? – +, -, or 0 for real charges I claim that lowercase v and i are much better in all respects, and – or . , / , or x for effective positive, negative, or no charge hereby use v and i. Basta. • The use of effective charge of a few defects over the real charge of all the “perfects” is preferred and one of the key points in defect chemistry. – We will learn what it is in the following slides

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) 2 - O Real charge of defect: -2 i Real charge of interstitial (empty) site in ideal structure: 0 // O Effective charge: -2 - 0 = -2 i

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   v Effective charge: 0 - (-2) = +2 O • 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 //// v Effective charge: 0 - 4 = -4 Zr

Kröger-Vink notation – more examples • Dopants and impurities / Y Y 3+ substituting Zr 4+ in ZrO 2 Zr  Li Li + interstitials i • Electronic defects / e Defect electrons in conduction band  h Electron holes in valence band

We will now make use of the thermodynamics of chemical reactions comprising defects In order to do that correctly, we need to obey 3 rules for writing and balancing defect chemical reaction equations: • Conservation of mass - mass balance • Conservation of charge - charge balance • Conservation of site ratio (host structure)

We will now use Schottky defect pair as our simple example to learn many things: Schottky defects in MO • We start by writing the relevant defect formation reaction:        x x // x x M O v v M O M O M O M O • which we can simplify to     // 0 v v O M • We then write its equilibrium coefficient:   // [ v ] [ v ]    O M K a a X X     // // S v v v v [O] [M] O M O M Activities a For point defects, activities The site fraction is the are expressed in terms of concentration of defects over site fractions X the concentration of sites