Fundamentals of Solid State Ionics I Defect Chemistry

Transcription

1 The research leadng to these results has receed fundng from the European Unon's Seenth Framework Programme (FP7/ ) for the Fuel Cells and Hydrogen Jont Technology Intate under grant agreement n [6144. Fundamentals of Sold State Ioncs I Defect Chemstry Tutoral lecture at the Summer School Valenca, September 3-5, 015 Truls Norby Man purposes Introduce defect chemstry to newbes Focus on some mportant prncples and good practces for oldes Department of Chemstry Unersty of slo Centre for Materals Scence and Nanotechnology (SMN) FERMI slo Research Park (Forsknngsparken) truls.norby@kjem.uo.no utlne What are defects and why are they mportant? Random dffuson and onc conductty Defect reactons and equlbrum thermodynamcs Examples nclude M, Zr, BaZr 3 L on battery materals Computatonal defect chemstry Summarsng adce

2 Stochometrc compounds; Pont defects form n pars: Intrnsc pont defect dsorders Schottky defects Caton and anon acances Frenkel defects Caton acances and ntersttals Ant- or anon-frenkel defects Anon acances and ntersttals

3 Stochometrc compounds: Electronc defects: Intrnsc electronc dsorder Domnates n undoped semconductors wth moderate bandgaps Defect electrons n the conducton band and electron holes n the alence band

4 Random dffuson and self dffuson Mass transport n crystallne solds s dren by thermal energy kt Leads to random dffuson If the dffusng speces s a consttuent t s also called self-dffuson Two most mportant mechansms: Vacancy mechansm Intersttal mechansm Defects are needed n both

5 Dffusty s a dffcult entty to understand. Frst warm-up: Dffusty: a matter of geometry and jump rates Consttuent by acancy mechansm rthogonal drectons D Jump rate 1 r, c 6 s c 1 6 s ZX Number of neghbourng stes Lkelyhood of target ste to be acant Jump dstance Rate of suffcently energetc attempts Vacancy D r, s s Z Consttuent by ntersttal mechansm D 1 r, c 6 s c 1 6 s ZX Lkelyhood to be ntersttal Intersttal D r 1, s s Z( 1 X ) s Z ΔGm ω exp exp RT ΔS R m exp ΔH RT m

6 Dffusty next exercse to look at what t s: The Nernst-Ensten relaton lnkng moblty and dffusty Applcaton of a force F ges the randomly dffusng partcles a net drft elocty : B F The proportonalty B s called mechancal moblty («beweglchket») Mechancal moblty B (beweglchket) s the dffusty D oer the thermal energy kt: B D kt D B kt Ths s the Nernst-Ensten relaton

7 Electrcal feld; force, flux densty, and current densty An electrcal feld s the downhll gradent n electrcal potental: E d dx It ges rse to a force on a charged partcle gen as F z ee z e d dx The flux densty j s the olume concentraton c multpled wth the drft elocty : j c c B F c B z ee Current densty by multplcaton wth charge: z ej z ec c B ( z e) E

8 Mobltes and conductty We now defne a charge moblty u u z eb We then obtan for the current densty: Charge moblty u s n physcs often denoted μ. We here use u to aod confuson wth chemcal potental. c B ( z e) E z ec u E We now defne electrcal conductty σ z and obtan ec u Very mportant! Know t! Charge x concentraton x charge moblty z ec u E E Ths s one form of hm s law. Conductty has unts S/cm or S/m.

9 Ionc conductty for acancy mechansm Consttuent by acancy mechansm Vacancy kt ZX s c e z kt D c e z B c e z u ec z c c c c c c c, 6 1,,,,,,, ) ( ) ( ) ( kt Z s c e z kt D c e z B c e z u ec z 6 1,,,,,,, ) ( ) ( ) ( Volume concentraton of acances Charge moblty of acances (~ concentraton ndependent) Volume concentraton of acances Regardless of whether you consder the consttuent or the defect, you need the concentraton of the defect ndrectly or drectly.

10 Before we moe on... Formal oxdaton number nteger charges We know that bonds n onc compounds are not fully onc, n the sense that all alence electrons are not entrely shfted to the anon. But f the bondng s broken - as when somethng, lke a defect, moes the electrons hae to stay or go. Electrons cannot splt n half. And mostly they go wth the anon - the most electronegate atom. That s why the onc model apples n defect chemstry and transport And t s why t s ery useful to know and apply the rules of formal oxdaton numbers, the number of charges an on gets when the alence electrons hae to make the choce z are nteger numbers

11 Defect chemstry Allows us to descrbe processes nolng defects Allows applcaton of statstcal thermodynamcs Equlbrum coeffcents; Enthalpes and entropes Yelds defect structure (concentratons of all defects) under gen condtons The defect concentratons for transport coeffcents (e.g. conductty) Requres nomenclature Requres rules for wrtng proper reactons Addtonal requrements: Electroneutralty, ste balances

12 Kröger-Vnk notaton In modern defect chemstry, we use Kröger-Vnk notaton. It can descrbe any entty n a crystallne structure; defects and perfects. Man symbol A, a subscrpt S, and a superscrpt C: What the entty s, as the man symbol (A) Chemcal symbol or (for acancy) Where the entty s the ste - as subscrpt (S) Chemcal symbol of the normal occupant of the ste or for ntersttal (normally empty) poston A C S Kröger and Vnk used uppercase V for acances and I for ntersttal stes, perhaps because that s natural for nouns n German. Its charge, real or effecte, as superscrpt (C) +, -, or 0 for real charges or., /, or x for effecte poste, negate, or no charge I say: How would you then do defect chemstry for anadum odde VI 3? I clam that lowercase and are much better n all respects, and hereby use and. Basta. The use of effecte charge of a few defects oer the real charge of all the perfects s preferred and one of the key ponts n defect chemstry. We wll learn what t s n the followng sldes

13 Effecte charge The effecte charge s defned as the charge an entty n a ste has relate to (.e. mnus) the charge the same ste would hae had n the deal structure. Example: An oxde on - n an ntersttal ste () Real charge of defect: - - Real charge of ntersttal (empty) ste n deal structure: 0 Effecte charge: = - //

14 Effecte charge more examples Example: An oxde on acancy Real charge of defect (acancy = nothng): 0 Real charge of oxde on - n deal structure: - Effecte charge: 0 - (-) = + Example: A zrconum on acancy, e.g. n Zr Real charge of defect: 0 Real charge of zrconum on Zr 4+ n deal structure: +4 Effecte charge: 0-4 = -4 //// Zr

15 Kröger-Vnk notaton more examples Dopants and mpurtes Y 3+ substtutng Zr 4+ n Zr L + ntersttals / Y Zr L Electronc defects Defect electrons n conducton band Electron holes n alence band / e h

16 We wll now make use of the thermodynamcs of chemcal reactons comprsng defects In order to do that correctly, we need to obey 3 rules for wrtng and balancng defect chemcal reacton equatons: Conseraton of mass - mass balance Conseraton of charge - charge balance Conseraton of ste rato (host structure)

17 We wll now use Schottky defect par as our smple example to learn many thngs: Schottky defects n M We start by wrtng the releant defect formaton reacton: M x M x // M M x M x whch we can smplfy to 0 // M We then wrte ts equlbrum coeffcent: K a a // X X // S M M [ [ // [ M [M Acttes a For pont defects, acttes are expressed n terms of ste fractons X The ste fracton s the concentraton of defects oer the concentraton of stes

18 Schottky defects n M 0 // M K s are often smplfed. There are arous reasons why: Because you sometmes can do t properly; Because the smplfcaton often s a reasonable approxmaton; Because you are perhaps not nterested n the dfference between the exact and smplfed K (ths often means that you dsregard the possblty to assess the entropy change); Because nether the full nor smplfed forms make much sense n terms of entropy, so they are equally useful or accurate (or naccurate), and then we may well choose the smplest. If we express concentratons n molar fractons (mol/mol M), then [M = [ = 1, and we may smplfy to K S [ // [ [ M // [ M [ [M

19 Schottky defects n M 0 // M NTE: At equlbrum, an equlbrum coeffcent expresson s always ald and must be satsfed at all tmes! K S [ [ // M Thus the product of the concentratons of oxygen and metal acances s always constant (at constant T). We may well stress ths by nstead wrtng: [ [ // M K S Whle K S represents nformaton about the system, we hae two unknowns, namely the two defect concentratons, so ths s not enough. We need one more pece of ndependent nput.

20 Schottky defects n M 0 // M The second pece of nput s the electroneutralty expresson. If the two defects of the Schottky par are the domnatng defects, we may wrte [ [ // M or [ [ // M It s now mportant to understand that ths s NT an eternal truth the electroneutralty statement s a choce: We choose to belee or assume that these are the domnatng defects. The next step s to combne the two sets of nformaton; we nsert the electroneutralty nto the equlbrum coeffcent: [ [ [ [ // M // M [ // M K K [ K 1/ S // M S S K 1/ S [ [ // M Vola! We hae now found the expresson for the concentraton of the defects. In ths case, they are only a functon of K S.

21 Schottky defects n M 0 From the general temperature dependency of K, K we obtan [ S // M ΔG exp RT [ 0 S ΔS exp R 0 SS R ΔH exp RT H RT // 1/ M KS exp exp 0 S 0 S 0 S Note: Ths not the Gbbs energy change (whch becomes zero at equlbrum) It s the standard Gbbs energy change. What does standard refer to? The square root and number arse from the reacton contanng defects. ln or log defect concentratons s 1/T (an t Hoff plots): ln[ log[ ln[ log[ 0 SS R H R // M 0 SS Rln10 0 S 1 T 0 H S Rln10 // M 1 T ln10=.303

22 Schottky defects n M 0 // M an t Hoff plot Standard entropy and enthalpy changes can be found from ntercept wth y axs and slope, respectely, after multplcaton wth R and -R. log[ plots can be more ntellgble, but requre the addtonal multplcatons wth ln10 =.303. ΔS S0 /R ln [ -ΔH S0 /R [.. =[ M // The standard enthalpy change can hae any alue: Fndng t s a result! The standard entropy change can be estmated: Fndng t s therefore a control! Dare to try? Get nterested n pre-exponentals and entropes! 1/T Man contrbuton to entropy changes s gas s condensed phases: ~10 J/molK!

23 Recap before we moe on 0 // M The soluton we found assumes that the two Schottky defects are domnatng. The standard entropy and enthalpy changes of the Schottky reacton refer to the reacton when the reactants and products are n the standard state. For defects, the standard state s a ste fracton of unty! Ths s a hypothetcal state, but neertheless the state we hae agreed on as standard. Therefore, the entropy as dered and used here s only ald f the pont defect concentratons are entered (plotted) n unts of ste fracton (whch n M happens to be the same as mole fracton). ther speces gases, electrons, condensed phases should be expressed as acttes, referrng to ther defned standard states, f possble. The model also assumes dealty,.e. that the acttes of defects are proportonal to ther concentratons. It s a dlute soluton case.

24 Now a detour to a more dffcult and perhaps controersal case; electronc defects Intrnsc onsaton of electronc defects For conducton band electrons and alence band holes, the releant reacton s / 0 e h N C The equlbrum coeffcent may be wrtten / [e [h K a / a e h N N C V n N C Here, the acttes of electrons and holes are expressed n terms of the fracton of ther concentraton oer the densty of states of the conducton and alence bands, respectely. The reason s that electrons behae quantum-mechancally and therefore populate dfferent energy states rather than dfferent stes. p N V N V * 8m ekt h 3/ * 8m hkt h 3/ The standard state s accordng to ths: n 0 = N C and p 0 = N V

25 Intrnsc onsaton of electronc defects If we choose to apply the concepts of standard Gbbs energy, entropy, and enthalpy changes as before, we obtan K n N C p N V G exp RT Ths s possble and useful, but not commonly adopted. 0 S exp R In semconductor physcs t s nstead more common to use smply: 0 H exp RT 0 K / [ / e [ h n p N C N V exp E RT g Ths states that the product of n and p s constant at a gen temperature, as expected for the equlbrum coeffcent for the reacton. Howeer, the concept of actty s not appled, as standard states for electronc defects are not commonly defned. For ths reason, we here use a prme on the K / to sgnfy the dfference to a normal K from whch the entropy could hae been dered.

26 Intrnsc onsaton of electronc defects From K n N C p N V exp G RT 0 exp S R 0 exp H RT 0 and K / [ / e [ h n p N C N V exp E RT g we see that the band gap E g s to a frst approxmaton the Gbbs energy change of the ntrnsc onsaton, whch n turn conssts manly of the enthalpy change. We shall not enter nto the fner detals or of the dfferences here, just stress that np = constant at a gen temperature. Always! Physcsts mostly use E g /kt wth E g n ev per electron, whle chemsts often use E g /RT (or ΔG 0 /RT) wth E g n J or kj per mole electrons. Ths s a tral conerson (factor 1 ev = J/mol = kj/mol).

27 Intrnsc onsaton of electronc defects If we choose that electrons and holes domnate the defect structure; n p We nsert nto the equlbrum coeffcent expresson and get n p n n p K K / N C N V E exp RT / 1/ 1/ ( NC NV ) exp g E g RT A logarthmc plot of n or p s 1/T wll thus hae a slope that seems to reflect E g / as the apparent enthalpy. Because of the temperature dependences of the densty of states t should howeer be more approprate to plot nt -3/ or pt -3/ s 1/T to obtan a slope that reflects E g / more correctly.

28 xygen defcent oxdes xygen acances are formed accordng to It s common for most purposes to neglect the dson by N C, to assume [ x = 1 and to remoe p 0 = 1 bar, so that we get ) ( 1 / g e x 1/ 0 C 1/ 0 C 1/ ) ( / N n [ [ [ [ N n [ [ x x g e p p p p a a a a K x Ths bg expresson may seem unnecessary, but s meant to help you understand 1/ / [ C p n K N K Then fnally, a case of nonstochometry, nolng onc and electronc defects: I use agan the prme n K / to sgnfy ths neglectance

29 xygen defcent oxdes We now choose to assume that the oxygen acances and electrons are the two domnatng defects. The electroneutralty then reads [ n We now nsert ths nto the equlbrum coeffcent and get K / 3 1/ 4[ p We fnally sole wth respect to the concentraton of defects: [ ( 1 4 K / ) 1/3 p 1/6 n [ (K / ) 1/3 p 1/6

30 xygen defcent oxdes We splt K / nto a pre-exponental and the enthalpy term: n [ (K / ) 1/3 p 1/6 (K /,0 1/3 exp 3 ) H RT 0 p 1/6 From ths, to a frst approxmaton, a plot of the logarthm of the defect concentratons s 1/T wll ge lnes wth slope of ΔH 0 /3R The number 3 relates to the formaton of 3 defects n the defect reacton x / 1 e ( g)

31 xygen defcent oxdes n [ (K / ) 1/3 p 1/ 6 By takng the logarthm: / 1 log n log log[ 1 3 log( K) 6 log p we see that a plot of logn s logp ges a straght lne wth a slope of -1/6. Ths knd of plot s a Brouwer dagram Note that log[.. s a parallel lne log = 0.30 unts lower.

32 Electroneutralty ne of the key ponts n defect chemstry s the ablty to express electroneutralty n terms of the few defects and ther effecte charges and to skp the real charges of all the normal structural elements poste charges = negate charges can be replaced by poste effecte charges = negate effecte charges poste effecte charges - negate effecte charges = 0

33 Electroneutralty The number of charges s counted oer a olume element, and so we use the concentraton of the defect speces s multpled wth the number of charges z S s z Example: M wth oxygen acances, metal ntersttals, and electrons: [ s zs [ s 0 [M -[e / 0 or [ [M [e / If oxygen acances domnate oer metal ntersttals we can smplfy: [ [e / Note: These are not chemcal reactons, they are mathematcal relatons and must be read as that. For nstance, n the aboe: Are there two acances for each electron or ce ersa?

34 Equlbra and electroneutraltes In defect chemstry, we combne nformaton from equlbrum coeffcents and electroneutralty expressons There s a potental ptfall For defect equlbra, you should use ste fractons n order to get the entropes rght Dfferent defects hae dfferent reference frames (ther host sublattces) For electroneutraltes, you must use olume concentratons, molar fractons, or formula unt fractons All defects must hae the same frame when countng ther charges They can be the same, but are n general not

35 Impurtes Dopng Substtuton

36 We wll only stop at a few mportant ponts for a sngle mportant case - YSZ: Zr -y doped substtutonally wth Y 3 Note: Dopng reactons are almost neer at equlbrum! They are most often fxed or frozen! Y / x 3 YZr 3 What would t take to hae them n equlbrum? Note: Electrons donated from oxygen acancy are accepted by Y dopants; no electronc defects n the bands. Dopant (secondary) phase must be present as source and snk Temperature must be ery hgh

37 Phase dagrams and defect chemstry All sold solutons and ther phase boundares are determned by defect thermodynamcs But suprsngly few studes attempt at takng adantage of ths, e.g. to ratonalse solublty and phase dagram studes

38 xde on conductors xde on conducton of YSZ Zr 0.9 Y The conductty has to a frst approxmaton a smple temperature dependency gen only by the moblty and hence random dffusty of the constant concentraton of oxygen acances. I hae chose to neglect two thngs: * nly a plot of log(σt) would ge a truly straght lne (remember why?) * Defects nteract: xygen acances and acceptor dopants assocate, lowerng the concentraton of free moble acances - or ther moblty f you prefer at lower temperatures.

39 can be hydrated to become proton conductors Y: BaZr 3 : A proton conductng oxde Zr 0.9 Y Ba BaZr 0.9 Y H (g) x H E a,h + 3 E a, From Kreuer,.K-D.

40 Three sldes for the noce on ternary and hgher compounds Ternary and hgher compounds Wth ternary and hgher compounds the ste rato conseraton becomes a lttle more troublesome to handle, that s all. For nstance, consder the peroskte CaT 3. To form Schottky defects n ths we need to form acances on both caton stes, n the proper rato: 0 // Ca //// T 3 And to form e.g. metal defcency we need to do somethng smlar: 3 (g) // Ca //// T 3 x 6h but oxygen defcency or excess would be just as smple as for bnary oxdes, snce the two catons stes are not affected n ths case

41 What f a ternary oxde has a strong preference for one of the caton defects? It can choose to make a selecton of the defects by throwng out one of the components, n order to not brake the ste rato conseraton rule. Example: Schottky defects n AB 3 wth only A and acances: A x A x // A A(g) Example: xdaton of AB 3 by formng metal defcency only on the A ste: x 1 // AA (g) A h A(s) Note: Choce of A(s) (secondary phase) or A(g) (eaporaton) are arbtrarly hosen to llustrate the possbltes

42 Dopng of ternary compounds The same rule apples: Wrte the dopng as you magne the synthess s done: If you are dopng by substtutng one component, you hae to remoe some of the component t s replacng, and thus hang some left of the other component to react wth the dopant. For nstance, to make undoped LaSc 3, you would probably react La 3 and Sc 3 and you could wrte ths as: 1 1 x x x La 3 Sc 3 La La Sc Sc 3 Now, to dope t wth Ca + substtutng La 3+ you would replace some La 3 wth Ca and let that Ca react wth the aalable Sc 3 : Ca 1 / x 5 x 1 Sc 3 Ca La Sc Sc The latter s thus a proper dopng reacton for dopng Ca nto LaSc 3, replacng La 3.

43 Defect chemstry of battery materals?

44 Sold-state L on conductor: L : La /3 T 3 The peroskte has two structurally dfferent A stes; /3 La, and 1/3 empty: La /3 1/3 T 3 Substtute 1 L for 1 La on the La ste, and add L on the empty ste: La /3-x L 3x T 3 or (La /3-x L x )(L x 1/3-x )T 3 [L // La [L Dopng reacton: L // 4 x x T 3 LLa 3 L T T 3

45 LFeP 4 cathode materal Man defect dsorder s L defcency Can be wrtten n seeral ways: Wrtten as an extracton of L : x 1 / 1 L L (g) L h L (s) 4 More releant: Extracton of L(s) to the anode: L x L / L h L(s) Een more releant: Extracton of L + ons to the electrolyte: L x L / L h L e ften donor doped. Total electroneutralty: [D [h [ / L - Normally, neer mx real and effecte charges For battery electrode materals, t may stll be useful: Both types of charges must then be consered separately

46 Computatonal defect chemstry Generate a computatonal cell wth many atoms (ons) and few defects Try to make t charge neutral Establsh boundary condtons by surroundng the cell wth copes of tself Calculate energy mnmum by densty functonal theory (DFT) Defect formaton Gbbs energy; dfference between defecte and perfect lattce; Chemcal potental of gas speces: Defect concentratons: c defect f ΔE N exp- kbt defect Numercally ft to electroneutralty. You enter p s (e.g. p ) and you obtan the Ferm leel μ e You can obtan all defect concentratons s T, p, dopng leel, etc. x 1 (g) e The standard entropy of gases s a frst approxmaton of entropes, that enables you to calculate equlbrum defect concentratons at fnte T, p, etc. We can also calculate lattce and hence defect entropes a further refnement. /

47 Summarsng adce Be honest! Admt and admre your defects! Ramble! That s what your defects do and keep you don! Learn! The nomenclature, the three rules, and wrtng electroneutraltes! Combne! Defect equlbra and the lmtng electroneutralty! Practce! Be brae! Do the statstcal thermodynamcs rght (standard states and ste fractons) and get the pre-exponentals and entropes. Check! Combne DFT and defect chemstry! Become an Almghty Computatonal Defect Chemst! (ACDC) Not a UCDP