Defect Chemistry 1
What is a defect? Fundamental definition Any deviation from the perfect crystal lattice is a defect! Macroscopic defects like porosities and cracks have an overall negative influence on mechanical as well as electrical properties of a material. In the fabrication of electroceramics these macroscopic defects are diminished as much a possible to have high density and crack free materials. Most important defects affecting electrical properties: Atomic defects Electronic defects 2
Atomic (point) defects in Oxides Atomic defects: properties depend on position, localized Missing ions: - oxygen ions, oxygen vacancies - cations, cation vacancies Substituted ions Interstitial ions FeO 3
Electronic defects in oxides Electronic defects: properties depend on energy levels available for the electrons, delocalized over the entire lattice electrons positive holes e are placed in energy levels(energy band model, valence and conduction bands separated by a gap). A positive hole is formed when an e jumps from the valence to the conduction band across the gap. A positive hole is a missing electron in the valence band with a relative charge of +1. 4
Defect notations Kröger Vink notation (Symbol for type) position subscript V O = vacancy on oxygen position V M = vacancy on metal position, V Fe O i = interstitial oxygen ion M i = interstitial cation Y Zr = Y-ion on Zr-ion position 5
Defects charges Relative charge Charge relative to the charge normally present in the position of the defect Examples: Zr Zr relative charge = zero, but Y Zr relative charge = -1 Zr Y relative charge = +1 FeO: V Fe vacancy relative charge = -2 6
Charges of oxygen vacancies Formation of oxygen vacancies: Oxygen atoms are removed from the crystal Oxygen ions how many electron in outer orbital? 8 Oxygen atoms how many electrons in outer orbital? 6 Oxygen vacancy how many electrons left? 2 Rel. Charge? Zero! But these electrons can easily migrate to neighbouring ions forming vacancies with one or zero electrons present. Thus V O with rel. charges of zero, +1 and +2 can be formed! 7
Relative charges of interstitial ions O 2- ions, O i -2 O 2- moves to an interstitial position which original charge is clearly zero so that the generated interstitial ion will have a charge of -2 Cl -1 ions, Cl i, rel. charge = -1 Na +1 ions, Na i,? +1 Zr 4+ - ions, Zr i, rel. charge = +4 8
Notation: relative charges Relative charges are indicated by a superscript: neutral - x positive charges black dots negative charges - apostrophes Neutral: V O x Positive charges: V O, V O Negative charges: V Fe 9
Three typical reactions: Formation of defects - high temperatures: intrinsic defects Due to intrinsic reactions, reactions characteristic of the material itself without the influence of foreign substances, at high temperatures where the lattice becomes partially unstable due to thermal vibrations. Also termed as thermal defects - substitution substitutions with aliovalent cations, i.e. cations with a different valence than that of the host lattice. - reaction with surrounding atmosphere reductions at low oxygen pressure or oxydations at high oxygen pressure 10
Intrinsic defects Defects are normally formed in pairs: Stoichiometry must be maintained! -Frenkel defect: cation vacancy and interstitial cation in a MO oxide: M Mx = V M + M i -Anti-Frenkel defect: oxygen vacancy and interstitial oxygen ion in a MO oxide: O Ox = V O + O i 11
Intrinsic defects -Schottky defect: oxygen vacancy and cation vacancy Stoichiometry must be maintained! Formed when both an oygen ion and a cation leave the lattice, for instance by evaporation from the surface of the crystal. In this case oxygen vacancy cations vancancy pairs are formed. In all reactions it is also clear that electrical neutrality is stricly obeyed. It is of course not possible create an electrically charged material by these reactions. MO: O Ox + M Mx = V O + V M M 2 O 3 : 3O Ox + 2M Mx = 3V O + 2V M 12
Defects formed by reaction with surronding atmosphere: reduction MO = MO 1-x + x/2 O 2 O Ox + M Mx = V O + 2M M + 1/2O 2 Oxides with cations easily reduced! No Stoichiometric defects! 13
Defects formed by reaction with surronding atmosphere: oxidation MO + y/2o 2 = M 1-y O Note clusters! Cations easily oxidized! No Stoichiometric defects! 14
Formation of interstitial oxygen ions- O i High oxygen pressures 1/2 O 2 + 2M Mx = O i + 2M M the electrons needed to form oxygen ions are provided from adjacent cations which becomes oxidized as for the formation of cation vacances The formation of interstitial oxygen ions thus preferentially takes place in oxides which are easily oxidized. Preferentially occurring in oxides where cations are easily oxidized FeO 15
Defects formed by substitution Substitution of cations The defects formed depend on the valency of the subsituting cations relative to that of the host cations Lower valency: ZrO 2 doped with CaO: Ca 2+ replaces Zr 4+ CaO(ZrO 2 ) = Ca Zr + V O + O O x Oxygen vacancies (positive defects) are formed to maintain electrical neutrality Doping zirconia with an oxide with a cation with a lower valence is thus an efficient way to form good oxygen ion conductors. A dopant very much used in zirconia is yttria, Y 2 O 3. 16
Defects formed by substitution Higher valency Y 2 O 3 doped with ZrO 2 : the defect formed by the substitution will be Zr Y and the compensating defect with an opposite charge will be interstitial oxygen ions, O i. 2 ZrO 2 (Y 2 O 3 ) = 2 Zr Y + O i + 3O O Same valency MgO doped with CaO No defects are formed, but if the two are ions have different sizes doping will create tensions in the lattice which can have an influence on for instance the mechanical properties of the material. 17
Defect Reactions Now we must consider the "reactions" associated with defect formation. In particular, formation of oxygen vacancies and reduced cations at reduced oxygen partial pressures. Can defects in a solid be considered as ions in a solution? Yes if these conditions are met: - random distribution of defects - no interactions - high mobility Law of mass action can be used Find the dependece of defects concentrations on the oxygen pressure 18
Rules which must be obeyed For a reaction occurring in a solid neutrality must be maintained,, i.e., chargeconservation, electroneutrality, ratio between cation and anion positions must be constant, mass balance, i.e., conservation of matter the total number of positions can be changed, but not the ratio, site balance, i.e, lattice site conservation Note that if a site of a constituent of an multicomponent crystal is created or destroyed, an appropriate number of complementary sites must be created or destroyed to maintain site stoichiometry. Thus we can say for hypothetical compound M a X b : a(x X +V X ) = b(m M +V M ) Vacancies count as lattice sites! 19
Formation of oxygen vacancies O x x o +2 Ce Ce V o +2 Ce Ce + 1 2 O 2 The solid phase where the reaction occurs can be considered a non-ideal solution = = γ can be considered constant within a relatively small composition range Substituting and including γ values in the equilibrium constant 20
& log p(o 2 ) = =! each time one oxygen vancancy is formed two reduced cations are formed to maintain the charge balance Neutrality condition: " At constant T " $! 21
Brouwer plots - V O Slope depends on the type of oxygen vacancy 22
[V M ] & log po 2 : Fe 1-y O A double negatively charged iron vacancy is formed together with two oxdized cations and an oxygen ion in an oxygen position in the lattice. The oxygen atoms in the atmosphere need two e for the formation of oxygen ions and these e are delivered by adjacent Fe 2+ ions, which become oxidized. Equilibrium Constant: Neutrality condition: Substituting (3) in (2): % = &! + " $ The exponent for the formation of cation vacancies has the same value but with opposite sign as compared to the exponent for the formation of oxygen vacancies. N.B. the concentration of the Fe vacancies correspond to the deviation, y, from the stoichiometric composition.
Brouwer plots - V M Brouwer plots for the three types of iron vacancies will thus also be straight lines but with slopes of 1/6, 1/4 and 1/2. Single charged and neutral vacancies consists of clusters with the compositions shown. + " $ % = &! % % % % + " * = &! + % % % % + + " = &!
Brouwer plot for O i Oxidation: Neutrality condition:! + " $,! The oxygen pressure dependence is the same as that obtained for the concentration of V Fe. The formation of the intrinsic anti-frekel defects consisting of an oxygen vacancy oxygen interstitial is independent of the oxygen pressure. 25
Brouwer plot: many defects Construction: -log [defect] vs log(po 2 ) - 3 p(o 2 ) regions; - one type of defect dominates in each region - sharp transition between regions is an approximation How can we establish the dependence on P(O 2 ) for each defect? 26
Oxygen ion conductors: defect reations Defects formed by doping with lower valency cation (A 2 O 3 in MO 2 ): A 3O ' 2 O3( MO2) 2AM + VO + Vacancies needed for O 2- conduction. Vacancies concentration only depends on dopant concentration and not on oxygen pressure. O 1 At low oxygen P, a reduction can take place (better if high T). O is removed from the lattice forming oxygen vacancy and electrons which may reduce the cation. O 1 2 O V O + 2e + O2 2 27
Oxygen ion conductors: defect reations At high p(o 2 ), oxidation can occur uptaking oxygen ions in oxygen vacancies forming normal oxygen ion sites in the lattice. The 2e needed are supplied by neighbouring cations where two holes are formed. 1 2 O 2 + VO OO + 2h 3 Interstitial oxygen may also be formed as Anti-Frenkel at high temperature: O O O A + V '' 4 i O 28
Oxygen ion conductors: defect reations The last process that can occur is intrisic ionization, where an e is jumping from the valence band into the conduction band. In this reaction both a free electron and a free positive hole is formed. 0 e ' + h 5 A The overall neutrality conditions is obtained including all the reactions, all the concentrations of positive defects on one side and all the negative defect on the orther side in the equation. h [holes] p, e [electrons] n. 6 29
Defect concentrations and p(o 2 ) Neutrality condition Each region of the Brouwer plot for many defects then obeys a simplified neutrality condition established by one member from each side of the overall neutrality equation This generates the following regions: Low PO 2 range High PO 2 range High PO 2 range n = 2[ V ] p = 2 O " O i [ A ' ] M Intermediate [ ' ] p = A M = 2[ V O ] NB: the condition p = n is only a point, as the intrinsic ionization is independent of the oxygen pressure. 30
Calculation for region [ ].. 1 2e O 2 OO V O + + V ( ) 1 6 o O 2 2 K (..) [..] 2 V V n p( O ) 1 2 o = p ( ) 1 6 n p O o 2 2 Eq. 2 O e ' + h K = np i p ( ) + 1 6 p O 2 Eq. 5 O O O '' i + V O K = AF [ ][ ] O.. V.. i O [ ].. O ( ) + 1 6 p O i 2 Eq. 4 31
Oxygen ion conductors: Brouwer plot YSZ, SDC, GDC For the n = 2[V O ] region n (electrons) are dominating,and the oxygen conductor is a n-conductor in low oxgen pressure range. For the regions to the right it is p (positive holes) which are dominating, and the material becomes a p-conductor in these high pressure ranges. In the intermediate pressure rangev O is dominating and the material is therefore an oxygen ion conductor in intermediate pressure range. 32
Conductivity plot σ = σ + σ + σ total σ = 2e[ V ] µ ( V ) + enµ ( n) + ep ( p) ion total O O µ n p where: e is the electronic charge n and p are the concentration of electrons and positive holes µ is the mobility of the three types of defect (µ) mobilities of e and h 1000 mobility of the oxygen vacancies n-conduction will dominate at low oxygen pressure p-conduction will dominate at high oxygen pressure ionic conduction will only dominate in the medium pressure range This range is called the electrolytic domain. In this range the oxygen ion conduction is of course constant as the vacancy concentration is determined by the dopant concentration. 33
From Brouwer to Conductivity plot The slopes for the oxygen pressure dependence of the different defects in the Brouwer plot are maintained in the conductivity plot σ σ + σ + σ total = t ion +t n +t p =1 ion n p 34
Conductivity plot There are regions with mixed conduction, (ionic and electronic or positive hole conduction). In these regions t i decreases from 1 to zero. These regions are quite small because of the high mobilities of e and hcompared to the mobility of the oxygen vacancies. 35
Conductivity plot The electrolytic (or ionic) domain: the region between the oxygen pressures where the transport number of vacancies, or that of electrons and positive holes, becomes 0.5. In this region the oxygen ion conduction is constant (ti=1). The domain boundary P n indicates the p(o 2 ) where t i = t n = 0,5, or σ i = σ n. The domain boundary P p indicates the p(o 2 ) where t i = t p = 0,5, or σ i = σ p P n and P p are important as they show the oxygen pressure at which the p and n conduction take over from the ionic conduction. This is important in many appications of oxygen ion conductors, where the conductor can short circuit at these pressures! 36
Total conductivity total σ = σ + σ + σ total ion n [ ] ( ) V.. V.. en ep o o n p σ = 2 + + e µ µ µ p Note: mobility of electronic defects much larger than that of ions even small concentration of these carriers will give an overall electronic conductivity. Trasport number i.e. the ratio of the current carried by a carrier : t ion +t n +t p =1 37
Dependence on temperature Both carrier concentration and mobility are thermally activated. Arrhenius equation describe the temperature dependence of both ionic and electronic conduction: ( Q ) σ = exp kt σ 0 Where: σ 0 factor depending on temperature Q activation energy k Boltzman constant T absolute temperature * correct formula is: σt = exp( Q kt ) σ 0 38
Arrhenius Plot log (σ i ) vs 1/T plot for typical oxygen conductors formed by doping 39
Domain boundaries of stabilized zirconia Ionic domain The ionic domain between P p and P n becomes smaller with increasing temperature and it dissapears completely at a characteristic temperature. Ionic conduction is therefore not possible above this temperature. 40
What determines ionic conductivity? Host oxide Type and concentration of dopant Temperature structure and composition of the host oxide determine the overall electrical properties no pure oxides show ionic conduction except at very high T where oxygen vacancies are formed by an intrinsic Anti-Frenkel reaction Focus on oxides where oxygen vacancies are formed by doping with oxides with aliovalent cations and where high concentrations of vacancies can be obtained The conductivity which can be obtained with a given host oxide clearly depends on the type and the concentration of the dopants. T: very important since the migration of the ions is a thermally activated process. In many cases, the defects are, especially at low T, bound in defect clusters which inhibit the free movements of the defects and thus decreases the conductivity. The dissociation of these clusters is T dependent. 41
Host oxides/dopants Flurite oxide structure FCC (face centered cubic) Examples: ZrO 2, ThO 2, CeO 2 doped with Y 2 O 3, CaO 42
σ dependence on defect concentration A maximum for all dopants. σ increases with oxygen vacancy concentration (due to increasing dopant concentration) until a concentration is reached where the defects start to interact reducing the mobility of oxygen vacancies (thus σ). Conductivity of zirconiavsdifferent dopant amount Two composition fields. The limiting compostion between the two ranges is generally considered to be x = 0.08 (in Zr 1-x M x O 2-x ). 1. the dilute solution range below the composition of the maximum in σ (0<x<0,08) 2. the concentrate solution range above this composition (x>0.08) The concentration giving the maximum σ lies in the concentrated range. 43