The three rules for writing defect reactions

Transcription

1 Introducton As dscussed n the prevous chapter several dfferent types of pont defects may be formed n metal odes and other norganc compounds. In prncple, all types of defects wll be present, but n general, only a small number of dfferent defects wll predomnate. When defect-dependent propertes are to be nterpreted, t s mportant that the defect concentratons as a functon of temperature and the actvtes of the crystal components are known. In order to understand these functons, t s necessary to formulate chemcal reacton equatons for the formaton (or annhlaton) of the defects. The defect formaton may ether occur nternally n the sold or through reactons wth the envronment. In the followng, the rules for formulatng defect reactons wll be descrbed and appled to dfferent defect structure systems, whle n the net chapter, condtons for equlbrum and equatons relatng equlbrum defect concentratons to temperature, actvtes (partal pressures) of the components n a compound, mpurty concentratons, etc., wll be dscussed. The three rules for wrtng defect reactons From a thermodynamc pont of vew a sold contanng pont defects consttutes a sold soluton where the pont defects are dssolved n the sold. In analogy wth lqud solutons, the sold may be consdered to be the solvent and the pont defects the solute. Smlarly, the defect equlbra may be treated n terms of the thermodynamcs of chemcal reactons and solutons. In analogy wth reactons n aqueous solutons, the rules for wrtng defect reactons nclude requrements of electroneutralty and mass balance. But for defect reactons n a crystallne compound t s also necessary to requre that the rato of regular structure stes n the metal and oygen sublattces must be mantaned, even f the total number of stes may ncrease or decrease durng the reacton. The rules may be summarsed as follows: Conservaton of mass mass balance The defect reacton must balance wth respect to the mass,.e. the number and types of atoms nvolved n the defect reacton must be the same before and 1

2 after the defect formaton or annhlaton. Vacances, whch only represent empty stes, have zero mass and do not count. Also electronc defects are consdered not to count n the mass balance. 1 Conservaton of charge The compounds are and should reman electrcally neutral. Wth the perfect crystal as reference, ths means that the total effectve charge s the same before and after the formaton or annhlaton of the defects. Ths means that the net charge on the left and rght hand sdes of a reacton equaton must be the same. Ths charge may be counted n terms of effectve charges or of real charges. However, effectve and real charges may n general not be balanced aganst each other, and one must therefore avod usng both effectve and real charges n the same equaton. Conservaton of the rato of structure stes The rato(s) of the number of caton and anon structure stes n a crystallne compound s constant. For nstance, n an ode the rato of regular and stes s 1:1 regardless of whether the actual composton s stochometrc or nonstochometrc. Correspondngly, n a compound 3 the rato of regular caton to anon stes s :3, and f 3 oygen stes are created n 3 through a defect reacton, two regular -stes - vacant or flled - n the metal sublattce must also smultaneously be created. Ths rule s specal to defect chemstry n crystallne solds, and requres specal attenton and some practce to avod ts ptfalls. As an ecepton, ths rule does not apply to odes wth nfntely adaptve structures. Despte the requrement of the constancy n the rato of regular stes, the total number of regular structure stes may change n a defect reacton, and therefore, the defect equaton may nclude the formaton or annhlaton of structure stes as long as the proper ratos are mantaned. It should be noted that no stes are created n the formaton of electronc defects. In the many defect equatons and equlbra whch wll be dscussed, surface atoms wll not be consdered separately. All atoms wll be consdered to be bulk atoms, and ths means that the treatment only apples to suffcently large 1 ne may choose to count and balance also the mass of electrons, but then one must be sure to nclude also non-defect electrons (electrons n the valence band). ne may choose to operate wth stes or energy levels - for electronc defects, and one must then take nto account the empty levels of the conducton band and the occuped levels of the valence band.

3 crystallte szes for whch the number of surface atoms s nsgnfcant compared to that of the bulk atoms. Defect reactons generally do not result n sgnfcant changes n the number of surface atoms. In the formaton of a vacancy, for nstance, a bulk atom s transferred to the surface, but n the same process a prevous surface atom, n turn, becomes a bulk atom. (It s worth mentonng, however, that the creaton of the new bulk atom has consequences for the energy nvolved, as we shall dscuss n the net chapter.) Eamples of reacton equatons for defects We wll now go drectly to formaton of defects n onc compounds such as metal odes, as ths wll llustrate the three rules. As we move on, the reader s encouraged to check the equatons and the three rules aganst each other to promote understandng of how they apply from the smpler to the somewhat more challengng cases. In vew of the many types of pont defects that may be formed n norganc compounds and that each type of defect may have varyng effectve charge, numerous defect reactons may n prncple be formulated. In the followng, a few smple cases wll be treated as eamples. Frst, we wll consder defect structure stuatons n stochometrc compounds (Schottky, Frenkel and ntrnsc electronc dsorders) and then defect structure stuatons n nonstochometrc odes wll be llustrated. Fnally, eamples of defect reactons nvolvng foregn elements wll be consdered. The treatment and eamples of defect reactons wll not only provde tranng n applyng the rules of defect reactons, but also broaden the descrpton of defect structures and of the ndvdual pont defects. Stochometrc compounds ntrnsc dsorders As descrbed n the prevous chapter, the defect structures n stochometrc compounds contan equvalent concentratons of negatvely and postvely charged pont defects. These are formed as a result of nternal equlbra n the crystal and do not nvolve reactons wth the surroundngs. For ths reason the defect structures n stochometrc compounds are also termed nternal dsorder. Schottky dsorder As descrbed n the prevous chapter, the Schottky dsorder nvolves the presence of equvalent amounts of caton and anon vacances. In an ode ths means that the crystal contans equal concentratons of metal and oygen vacances. The overall formaton of such a defect par wthn the crystal nvolves 3

4 the transfer of a par of catons and anons on regular structure stes from the bulk to the surface. In realty the defects are formed at eternal and nternal surfaces or dslocatons and subsequently dffuse nto the crystal untl they are randomly dstrbuted. In wrtng the equaton, one s only nterested n the ntal and fnal states, and one dsregards the knetcs of the defect reacton. If one starts wth a par of catons and anons on regular structure stes wthn the crystal, one must also take nto account that the formaton of the Schottky par results n the formaton of two new structure stes, and the overall equaton may thus be wrtten = v v // (.1) However, n ths equaton the normally occuped stes on both sdes may be cancelled, and the net equaton therefore becomes 0 (.) = v // v where 0 (nl) desgnates a perfect crystal. Fgure -1. Formaton of Schottky defect par n. Left: Perfect schematc structure. ddle: ne and one have left vacances n the structure and formed a new structural unt on the surface. Rght: Equvalent representaton. In terms of the three rules of defect chemcal reactons, both (.1) and (.) conserve mass, both conserve charge (beng neutral on ether sde), and both conserve the structure ste raton of 1:1 by formng one new and one new ste. Frenkel dsorder For the Frenkel dsorder the predomnant defects are ether lmted to the catons and anons, and the dsorder nvolves the presence of equal numbers of vacances and ntersttal ons n a sublattce n a crystal. In the formaton of a Frenkel defect par, a caton on a normal ste s transferred to an ntersttal ste, and no new structure stes are created n the process. If the ntersttal on and the resultng vacancy are assumed to be doubly charged, the formaton of a Frenkel defect par may be wrtten 4

5 // v = v (.3) Note that we choose to have an empty ntersttal ste as a reactant. Ths has not been usual n defect chemstry, but t helps vsualse that such a ste s needed, and t keeps track of t through the reacton. A correspondng equaton may be wrtten for the formaton of an anon or ant Frenkel defect par (anon vacancy and anon ntersttal). Fgure -. Formaton of Frenkel-type defect pars n. Left: Perfect schematc structure. ddle: Frenkel defect par; ne leaves a caton vacancy and takes an ntersttal poston. Rght: Ant (anon) Frenkel defect par; vacancy and ntersttal. Intrnsc onsaton of electrons The ectaton of an electron from the valence band to the conducton band, thereby leavng an electron hole n the valence band, s wrtten 0 = e / h (.4) Fgure -3. Intrnsc onsaton of an electron-hole par over the band gap between the valence band edge E v and conducton band edge E c. As mentoned above we can choose to apply the conservaton of mass of electrons n whch case we mght rewrte the equaton as follows: e = e / h (.5) 5

6 We can also choose to keep track of the stes or energy levels for electrons; e h = h e v c v / c (.6) However, by conventon, the three are equvalent, and the frst and smplest suffces. As mentoned earler, the electronc defects may be localzed and then denoted valence defects. In these cases the reactons are connected wth ndvdual atomc stes. For nstance the ntrnsc onzaton (dsproportonaton) of Fe 3 ons nto Fe and Fe 4 ons would be wrtten Fe (.7) Fe / = Fe Fe Fe Fe Smlarly, ntrnsc onzaton may also take place by charge transfer. In lmente, FeT 3, we may for nstance observe the reacton Fe T 4 = Fe 3 T 3, whch n terms of defects s wrtten: Fe T = Fe T Fe T Fe / T (.8) Nonstochometry The rato of caton to anon structure stes s the same whether a compound s stochometrc or nonstochometrc. But as nonstochometry means that there s an ecess or defct of ether catons or anons, nonstochometry also means that there s an ecess of a certan type or types of defects relatve to that n the stochometrc condton. If the predomnatng type of defects s charged, electronc defects of the opposte effectve charge are created n order to conserve electrcal neutralty. The etent of nonstochometry and the defect concentratons n norganc compounds are functons of temperature and actvtes (e.g. epressed as partal pressures) of ther components. Frst we consder, as an eample the formaton of oygen defcency n an ode. The overall reacton may be wrtten ( y g y s) = ( s) ( ) (.9) From ths equaton t s qualtatvely seen usng le Chateler's prncple that the oygen defct ncreases wth decreasng oygen pressure. Conversely, for odes 6

7 wth ecess oygen the nonstochometry ncreases wth ncreasng oygen pressure. In the followng varous defect reactons whch are encountered n odes wth oygen or metal ecess or defct wll be consdered. As the actvty of the metal component s usually neglgbly small compared to that of the oygen actvty under most epermental condtons, the nonstochometry n metal odes s correspondngly a result of the nteracton and echange of oygen between the metal ode and the surroundng gas atmosphere. In the followng eamples of formaton of nonstochometrc defects, only cases where the metal odes nteract wth gaseous oygen are llustrated. However, t should be borne n mnd that the correspondng nonstochometry and defects may be formed by nteracton wth metal f t s epermentally feasble to control the actvty of the metal component n the surroundngs of the crystal. ygen-defcent odes An oygen vacancy s formed by the transfer of an oygen atom on a normal ste to the gaseous state. There s no change n the number of structure stes. Ths defect reacton may be wrtten 1 = v g) (.10) ( In ths equaton t s assumed that the oygen vacancy s neutral,.e. the two electrons of the - on that were there have stayed behnd. They are assocated wth the vacancy or ts mmedate neghbourhood and the vacancy as a result has zero effectve charge. Fgure -4. Formaton of a neutral oygen vacancy n. Left: Schematc perfect structure. ddle: An oygen atom escapes as gas and leaves a vacancy wth two assocated electrons. Rght: The two electrons shown delocalsed on neghbourng catons. As descrbed n Chapter 1, the two electrons trapped at or near the vacancy may, dependng on the temperature and vacancy concentraton, be ected and transferred away from the vacancy. Thus, the oygen vacancy acts as a donor and may become sngly and doubly charged: v = v e / (.11) 7

8 v = v e / (.1) Fgure -5. Ionsaton of an oygen vacancy. Left: v. ddle: / v e. Rght: / v e. Fgure -6. Electron energy band pcture of the same onsaton steps. The formaton of the doubly onzed oygen vacancy can be wrtten as the total reacton: = v / 1 e ( g) (.13) At hgh temperatures, the onsaton s typcally complete, so that ths equaton s one of the most commonly encountered n hgh temperature defect chemstry. We wll thus use t etensvely. In the equatons above the free electrons are consdered delocalsed n the conducton band. If they are localsed at a metal on on a normal structure ste the last equaton could be reformulated as follows: / 1 = v ( g) (.14) If ths takes place n the ode the valence of the atoms then partly reduces from 4 to 3 when the ode becomes non-stochometrc ( -y ). 8

9 des wth ecess metal The oygen defcency n an ode may alternatvely be equvalent to the presence of ecess metal relatve to stochometrc composton, and n ths case the predomnant defects consttute ntersttal atoms. Correspondngly the composton of the ode should then be wrtten 1. The formaton of an ntersttal atom n ths ode nvolves the transfer of a regular atom to an ntersttal ste. As a metal structure ste s annhlated n ths process, two oygen structure stes must smultaneously be annhlated n the oygen sublattce and ths s acheved by transferrng two oygen atoms to the gas phase. The defect equaton then becomes v = ( g ) (.15) The neutral ntersttal atoms may be successvely onsed to sngly, doubly, trply or quadruply charged ntersttal ons, e.g. / = e, etc. (.16) From ths we can state that the metal ntersttal, lke the oygen vacancy, s an electron donor. It may be noted that n both cases t s the effectvely neutral or ncompletely onzed defects that s a donor, not the fully onzed defects. Fgure -7. Formaton of caton ntersttals. Left: Schematc perfect structure. ddle: Formaton of metal ntersttal by loss of. Rght: Electron energy band dagram of the process. We have untl now wrtten the formaton of metal defects by echange wth oygen n the surroundng atmosphere. We may n prncple choose to wrte the same n terms of echange wth the metal component n the surroundngs. For nstance, metal ntersttals n can be formed n equlbrum wth the metal vapour: ( g) v = e / (.17) 9

10 etal-defcent odes In metal-defcent compounds metal vacances are the predomnatng pont defects. In an ode a metal vacancy s formed by reactng oygen gas wth the ode. Ths creates new oygen structure stes and therefore an equvalent number of new metal structure stes - whch are vacant - are also formed: 1 ( g)= (.18) v In ths reacton t s assumed that the vacancy that s formed s effectvely neutral, that s, two electrons are taken from the surroundngs of the vacancy n order to form the - on. The lackng electrons may be taken up from the valence band, formng electron holes here; v // = v h (.19) The overall reacton s thus // = v h 1 ( g) (.0) Fgure -8. Formaton of caton defcency n. Left: ygen gas and perfect schematcal structure. Rght: Caton vacancy formed by makng new formula unt on surface. Fgure -9. Electron energy band structure changes accompanyng the formaton and onsaton of caton defcency. 10

11 By eperence, ths s the most dffcult to comprehend and reproduce of the smple defect reactons. New stes are formed n the rato requred to mantan the rato of the structure. ne type of stes s empty as formed. The formaton of electron holes may alternatvely be epressed n terms of valence defects, so that overall reacton becomes: // = v 1 ( g) (.1) Ths correspondngly means that the valence for some atoms changes from for to 3; creaton of charged metal vacances nvolves odaton. The metal vacancy, whether t s effectvely neutral (electrons mssng from ts neghbourhood) or onzed (electrons suppled from the rest of the crystal) llustrates that the effectve sze of a pont defect may etend beyond the ste tself; the charge may be dstrbuted over at least the closest neghbours of the defect. Accordngly, defect chemstry works fne under the onc model and wth the assumpton of nteger charges, even f the compound s far from deally onc the pont defect s just a lttle bgger than one ste. It works because n defect chemcal as n other chemcal reactons, the electrons have to choose to go or stay no half electrons are nvolved. It then does not matter how close that electron was of the pont defect as long as t was assocated wth t. For most defect chemcal consderatons, t then also suffces to consder the defect as a true pont defect, and that the electrons are assocated wth the pont defect tself. des wth ecess oygen In metal odes wth ecess oygen the predomnatng pont defects are ntersttal oygen atoms or ons. The formaton of a neutral ntersttal oygen atom through reacton of oygen wth the ode s wrtten 1 g) = (.) ( v No new structure stes are formed n ths reacton. The neutral ntersttal oygen atoms may n prncple be onsed to yeld electron holes and ntersttal ode ons wth negatve effectve charges, so that the total reacton becomes, for nstance, // v = h 1 ( g) (.3) Summary of fundamental reactons for ode a b For reference, the table below contans all the fundamental reactons we have been through, wth fully onsed defects, and oygen as component to echange for non-stochometry. The reactons are made for a bnary ode of 11

12 general formula a b,.e., contanng catons wth formal odaton state a b. It may be noted that the reactons nvolvng no metal defects are unaffected by the coeffcents a and b. Table -1. Summary of fundamental reactons for formaton of fully onzed defects n bnary ode a b. ygen gas chosen for reactons creatng non-stochometry. Schottky dsorder Frenkel dsorder 0 b = v ab / a v a a = b b v v / Ant-Frenkel dsorder // v = v Intrnsc electronc onsaton 0 = e / h ygen defcency / 1 = v e ( g) etal ecess b a b / b v b = e g) etal defcency ygen ecess a b a g = v b / b b a ( ) a a h 1 ( g) // v = h a a ( Defect reactons for elemental solds Before we contnue, we brefly stop by some reactons for elemental solds, e.g. metals such as N and halfmetals such as S or Ge. In comparson wth compounds especally onc compounds the equatons are smple and the three rules of defect reactons manly reduce to one: mass balance. Wth the element as eample we can descrbe the formaton of a par of vacancy and ntersttal a Frenkel par as: v = v (.4) What we may call Schottky defects nvolves only vacances of. The atoms removed can be placed n a new structure ste on the surface of the crystal, = v or or removed to the gas phase, 0 = v (.5) = v (g) (.6) but the latter are equvalent as the sold and gaseous phases are n nvarable equlbrum wth each other. In semconductng solds the vacances and ntersttal atoms may have a tendency to accept or donate electrons, respectvely, from the valence band or to the conducton band and become effectvely charged. Wth S as eample we thus have: 1

13 v S / = vs h (.7) S / = S e (.8) Dssoluton of foregn elements The presence of mpurtes or dopants may sgnfcantly affect or even control the concentratons of the natve defects n a compound. The effects are to a large etent dependent upon the relatve valences of the ons n the parent compound and those of the mpurtes or dopants and what stes the mpurtes and dopants occupy. When the prncpal valence s the same as that of the host ste, we say that the foregn element s homovalent. When t s unequal to that of the host ste, we say that the foregn element s heterovalent or alovalent. Foregn alovalent elements may dssolve n the host structure wth zero effectve charge. However, ther dfference from the host atom makes them easly donate or accept one or more electrons they are donors or acceptors. We wll brefly analyse the reason for ths and descrbe the processes n terms of defect reactons. When boron, B, or phosphorous, P, dssolve as dopants n elemental slcon, S, they have one too few or one too many valence electrons, respectvely, compared to S. However, they are n ths state effectvely neutral snce the electron number n each case s compensated by the nuclear charge. Nevertheless, the mssng or etra electrons form local states n the band structure that easly accept or donate electrons: B S / = BS h (.9) PS = P S e / (.30) The neutral defects, the acceptor and donor, were here formed by dssolvng the foregn elements n the same odaton state as the host, namely 0. It s the desre of the crystal to fulfl the octet rule around each atom the covalent nature of the bond that causes the onzaton n ths case. We may do somethng smlar wth onc compounds such as odes. For nstance, hypothetcal L, contanng dvalent lthum, L, may be dssolved n N to form effectvely neutral L speces; L. Snce L s unstable n N odaton state, t easly accepts an electron from the valence band of the crystal: N L N / = LN h (.31) 13

14 Thus, n ths case, t s the formal odaton state accordng to an onc model that s the argument for why the acceptor s onsed. Smlar arguments hold for donors n onc compounds. Fgure -10. Electron energy band representaton of pure S, S wth un-onsed and onsed P donors (left) and smlarly B acceptors (rght). When dealng wth alovalent foregn elements t s common to assume drectly the onsaton of the donor or acceptor nto an effectvely charged state. We often refer to the cases as donor- or acceptor-doped systems, but keep n mnd that t s the unonsed speces that consttutes the donor or the acceptor. Effects of dssoluton of alovalent odes n the metal-defcent ode 1-. Let us assume that the parent ode s metal-defcent 1- and that the // majorty pont defects n 1- are charged metal vacances, e.g., v, compensated by electron holes, h. The ode s thus a p-type electronc conductor. Assume now that a hgher valent ode h 3 s added. (h s used as an arbtrary chemcal symbol for a hgher valent metal.) Some h 3 dssolves n the 1-. The h-ons have n ths case a valence of 3, and f t s assumed that the h-ons occupy normal -stes n 1-, the dssolved h-ons wll have one postve effectve charge, h. These addtonal postve effectve charges must for electroneutralty reasons be balanced by creaton of an equvalent concentraton of negatve effectve charges or annhlaton of an equvalent concentraton of postve effectve charges. Let us consder these two alternatves. The dssoluton of h 3 may n the frst case be compensated by the formaton of addtonal -vacances wth two negatve effectve charges. In ths case the equaton for the dssoluton of h 3 s wrtten // 3 = h v h 3 (.3) and thus the dssoluton of h 3 n 1- ncreases the concentraton of metal vacances n the parent ode. But as wll be dscussed n detal n the net chapter an ncreased concentraton of charged metal vacances smultaneously results n a decrease n 14

15 the concentraton of electron holes. The dssoluton of h 3 may thus alternatvely be wrtten wth a defect reacton nvolvng annhlaton of electronc holes: h 1 h = h ( 3 g ) (.33) When the dssoluton of h 3 s wrtten n ths way, the equaton emphasses that the concentraton of electron holes s reduced. However, as the concentraton of metal vacances and electron holes are nterrelated, t should be emphassed that both processes take place. Actually, the dssoluton of the hgher valent caton decreases the concentraton of all effectvely postve defects and ncreases the concentraton of all negatvely charged defects. Qualtatvely, ths may smply be seen as a consequence of the electroneutralty equaton or, f one wshes, of Le Chateler's prncple. Detals of these aspects wll be further dscussed usng defect dagrams n Chapter 4. If one has the choce of wrtng a dopng reacton by creaton or annhlaton of defects, t s usually more meanngful to choose the creaton of defects, snce ths can descrbe the dopng reacton to levels beyond the defect concentratons that ested n the undoped ode, and snce ths descrbes the defects that wll domnate when the dopng level gets hgh. So far we have added a hgher valent dopant ode to 1-. Let us alternatvely consder what takes place when a lower valent dopant ode, l, s added to 1-. If the l ons dssolve substtutonally n 1-, the dssolved l / ons have one negatve effectve charge, l. Ths wll be compensated by the formaton of postve effectve charge or the annhlaton of negatve effectve charge. In our eample ode ths wll manly affect electron holes and metal vacances, and the two defect reactons may thus be wrtten 1 / ( g) = l h l (.34) l = // / v l (.35) Thus, the addton of l to 1- has the opposte effect of addton of h 3 ; t ncreases the concentraton of electron holes and decreases the concentraton of metal vacances. ne may note that the frst of the two reactons s an odaton and thus nvolves uptake of oygen, whle the latter s not. It s typcal of such dssolutons of foregn alovalent elements n odes that they can be charge compensated by means of electronc defects n whch case we are dealng wth reducton or odaton and uptake or release of oygen or by means of pont defects n whch case we have no change n odaton states and no echange of oygen gas. 15

16 -y Effects of dssoluton of alovalent odes on the oygen-defcent ode The eamples n the prevous secton show the effects of addtons of hgher and lower valent ode to the p-conductng, metal-defcent 1-. Let us also brefly consder the effects of dopng an oygen-defcent ode -y wth hgher and lower valent odes, respectvely. The predomnatng defects n -y are oygen vacances compensated by defect electrons. The ode t thus an n-type electronc conductor. When the dopant ode s hgher valent, e.g. h 5, and the dopant caton, h 5, dssolves substtutonally, the dopant ons get one effectve postve charge, h. Ths charge must be compensated ether by formaton of negatve effectve charges n the form of electrons or by annhlaton of postve effectve charges, oygen vacances. These defect reactons may be wrtten h / 1 = h e 4 ( 5 g ) (.36) 5 v = h h 5 (.37) Thus when -y s doped wth a hgher valent odes h 5 the concentraton of electrons s ncreased and the concentraton of oygen vacances s decreased. When the same ode, -y, s doped wth a lower valent ode, l 3, / the negatve effectve charge of the dssolved atoms l s compensated by annhlaton of electrons or formaton of oygen vacances: / 1 / 3 e ( g) = l (.38) l 4 / 3 = l v l 3 (.39) The latter s eemplfed by yttra-stablsed zrcona where the acceptor dopng s compensated by moble oygen vacances that lay ground for ths materal as a sold electrolyte: Y / 3 = Zr v Y 3 (.40) Ths mportant eample s llustrated n Fgure -11. Numerous other eamples may be formulated, dependng on the defect structure of the parent ode, the valence of the foregn on and the ste t 16

17 occupes. It may be noted that a foregn caton dssolvng ntersttally always wll have a postve effectve charge and thus affect the defect structure n a smlar manner as a hgher valent caton dssolved substtutonally. We shall return to ths and other eamples n subsequent chapters and n connecton wth the revew and dscusson of a few ndvdual odes systems. Fgure -11. Y-dopng of Zr. Left: Y 3 s added to Zr. ddle: Zr gets new Zr stes and stes n rato 1:. Y dssolves substtutonally and one oygen vacancy compensates the effectve charge of two dopant ons. Rght: Electron energy band representaton of the process. Foregn anons n odes In odes, homovalent foregn anons comprse S -, whle alovalent foregn anons comprse F - and N 3-. They can enter as mpurtes durng synthess, or dssolve from gaseous speces under reducng atmospheres, e.g., H S g) = S H ( ) (.41) ( g / 3 NH ( g) = N v 3H ( 3 g ) (.4) Dssoluton of hydrogen and water vapour n metal odes When metal odes are eposed to gas atmospheres contanng water vapour or other hydrogen contanng gases, hydrogen wll dssolve n the odes. Under odzng or mldly reducng condtons, the hydrogen atoms onse to protons and assocate wth oygen atoms on normal structure stes and thereby form hydrode ons on normal oygen stes, hydrogenaton as H. We may thus for nstance wrte the H = H = H e / (.43) (see Fgure -1) n whch case the protons dssolved are charge compensated by the formaton of defect electrons. In terms of defect chemstry the dssolved proton, located on a normal ode on as hydrode, may also be consdered to consttute an ntersttal hydrogen on, and as such t s also n the lterature 17

18 alternatvely wrtten H. ne just has to bear n mnd that the protons do not occupy regular ntersttal postons (vods). Fgure -1. Schematc hydrogenaton of an ode and onsaton of the hydrogen ntersttal atoms nto protons n H groups and electrons. The electrons may nteract wth other defects n the ode so that the protons n effect are compensated by formaton of other negatve defects or by the annhlaton of postve defects. From the dssoluton reacton and through the nteracton wth natve defects n the ode t s clear that the dssoluton of hydrogen n metal odes s dependent both on the partal pressure of the hydrogen source (e.g. water vapour or hydrogen) and of oygen. These aspects wll be descrbed n more detal n a later chapter. Under reducng condtons, where hydrogen s stable n odaton state 0 (as H n the gas phase) we may foresee neutral hydrogen atoms dssolved n odes, probably ntersttally, as H, as mentoned above. Under even much more reducng condtons could also hydrde ons be epected to become stable, e.g. as dssolved substtutonally for ode ons, as the defect H. Protons may also dssolve from water vapour as a source. The dssoluton of hydrogen from ts ode, H, s n prncple smlar to dssoluton of other foregn catons. However, the possblty of a controlled water vapour pressure and the fast dffuson of protons makes t much easer to attan and vary (and more dffcult to completely avod) an equlbrum content of protons n the ode. f partcular nterest s the reacton between water vapour and oygen vacances, by whch an acceptor-doped ode compensated by oygen vacances n the absence of water (dry state) becomes domnated by protons when hydrated: v = H H (g) (.44) 18

19 Fgure -13. Hydraton of oygen vacances n acceptor-doped. Ternary and hgher compounds We have so far concentrated on bnary compounds, mostly odes, and only touched upon elemental solds. Ternary and hgher compounds fall, however, under eactly the same rules of wrtng defect reactons. A typcal ternary compound s a ternary ode such as perovskte, CaT 3. As an eample of defect reactons for ths case, we consder frst the formaton of Schottky defects. When we form new structure stes n ths reacton, we need to form vacances on both Ca and T stes to mantan the rato between them, n addton to the approprate number of oygen vacances: 0 // //// = vca vt 3v (.45) If we further consder the uptake of oygen by formaton of caton vacances and electron holes, we agan have to balance the caton stes: // //// = vca vt 3 6h 3 ( g) (.46) Smlar prncples should be appled also n cases where one and the same element s dstrbuted on dfferent crystallographc stes. For nstance, Y 3 has a structure where all ode ons are not strctly equal. Smlarly, dstorted perovsktes may have unequal oygen stes. In the pyrochlore structure, A B 7, there are 6 oygen stes of one type and 1 of slghtly dfferent coordnaton and energy (and one whch s structurally empty and thus to be regarded as an ntersttal ste). In prncple the formaton or annhlaton of crystal unts has to mantan the rato between those dfferent stes n all such cases. However, ths s so far hardly ever practced n defect chemstry. Contrary to bnary odes, ternary and hgher odes can have nonstochometry not only n terms of the oygen-to-metal rato, but also nternally between the varous catons. Ths s n practce often a result of synthess. For nstance, t may be dffcult to wegh n eactly equal numbers of moles of Ca and 19

20 T precursors when syntheszng CaT 3, so that the syntheszed materal has a permanent number of vacances on one of the caton stes. Such non-stochometry may also be a result of equlbra. For nstance, f A-ste defcency s energetcally favourable over B-ste defcency n the compound AB 3, we may at very hgh temperatures (e.g., durng snterng) see a preferental evaporaton of the A component. For a perovskte A B 4 3 we can for ths case wrte: A // A = v A v A( g) (.47) Durng odaton we mght smlarly see a preferental ncorporaton of A-ste vacances, resultng n a precptaton of an A-rch phase: A 1 // ( g) = v h A( s ) A A (.48) It may be noted that these reacton equatons do not volate the ste rato conservaton requrement of the ternary ode. When we earler doped elementary or bnary compounds the reacton was farly straghtforward. When we dope a ternary or hgher compound, however, the reacton may be less obvous we have some choces. It s qute common, however, to do the synthess and wrte the equaton n such a way that one takes out a correspondng amount of the host element that s substtuted. If we, for nstance, want to dope LaSc 3 wth Ca substtutng for La, we go for a composton La 1- Ca Sc 3. In order to see how we wrte the dopng reacton n ths case we frst just look at the trval normal synthess: Sc = La La Sc Sc La 3 (.49) Accordngly, we then wrte the defect reacton for the dopng n the way that we let there be Sc 3 reserved for the Ca: / Sc = CaLa Sc Sc v 1 Ca (.50) Summary Defect reactons can be wrtten n a smlar manner as ordnary chemcal reactons. They must confer wth the requrements for conservaton of mass, 0

21 charge, and ratos of structure stes, but are allowed to ncrease or decrease the total number of structure unts. We have treated defect reactons mostly n cases for bnary onc compounds, notably odes. Reactons for elemental crystallne solds follow the same basc rules and prncples, but wthout the ste rato conservaton requrement. Also hgher compounds, e.g. ternary odes, follow the same rules, only often wth slghtly more comple ste conservaton consderatons and more defects. Normally, one should seek to use reactons that create rather than annhlate defects, and one should avod usng reactons that form more defects than necessary (as such reactons are then a sum of smpler reactons). We have formulated defect reactons whch descrbe ntrnsc onc and electronc dsorder, nonstochometry, varable onsaton of pont defects, and substtutonal dssoluton of alovalent catons and anons. Alovalent elements may be compensated by electronc defects or by pont defects, of whch the former nvolve red-o-reactons. We have treated hydrogen defects specally, as they arse from a specal source and n ts most stable form, the proton, take on a specal sze and type of defect. Problems The Problems n ths chapter manly provde tranng n 1) formulatng a reacton n terms of the approprate reactants and products, and ) balancng and checkng them wth respect to a) mass conservaton, b) charge conservaton, and c) ste rato conservaton. 1. Wrte a reacton for the formaton of Schottky defects n NaCl.. Wrte a reacton for the formaton of Schottky defects n. 3. Wrte a reacton for the formaton of Schottky defects n Cu. 4. Wrte a reacton for the formaton of anon Frenkel defects n. 5. Wrte a reacton for the formaton of anon Frenkel defects n CaF. 6. Wrte a reacton for a charge transfer between the caton and anon n Ce,.e. for reducton of the cerum on and odaton of the ode on. Wrte the same process as an ntrnsc onsaton assumng delocalsed electronc defects. (Last part s trval just to check your understandng.) 7. Wrte a reacton for the formaton of fully onsed oygen vacances and electrons when oygen s lost n the reacton 3 = 3-y y/ (g). 8. Wrte reactons for the drect formaton of fully onsed metal ntersttals n, 3, and. 1

22 9. Wrte reactons for the drect formaton of fully onsed metal vacances n, 3, and. 10. Wrte reactons for formaton of sodum vacances n NaCl by echange wth a) chlorne gas and b) sodum gas. 11. We have n the tet clamed that charges can be assocated drectly wth deal pont defects, even f the compound s not deally onc, and that ths normally does not affect the results of the calculatons. Under what condtons wll t start to matter that the charge of the defect actually etends outsde the defect tself? 1. Wrte a reacton for dssoluton of Ca nto the ode Zr -y 13. Wrte a reacton for dssoluton of f nto - 3 when we assume that the foregn metal f s small and therefore dssolves ntersttally. 14. Wrte a reacton for dssoluton of Ca substtutonally nto the anon- Frenkel domnated Y Wrte a reacton for dssoluton of Zr substtutonally nto the anon- Frenkel domnated Y Wrte a reacton for dssoluton of CaF nto Ca, assumng that F - dssolves substtutonally and that Ca s domnated by Schottky defects n the pure, undoped state. 17. Wrte a reacton for dssoluton of protons n an ode wth water vapour as source and wth oygen ntersttals as compensatng defects. 18. Wrte a reacton for dssoluton of protons n an ode 1- wth water vapour as source. 19. Consder an ode 3 whch s acceptor-doped by dssolvng the ode l. The dopants are compensated by oygen vacances. Wrte a defect reacton for the dssoluton of l. Wrte a reacton for the dssoluton of protons from water vapour nto ths acceptor doped ode, by annhlaton of the oygen vacances. Fnally, wrte a reacton for the dssoluton of l n the presence of water vapour compensated by protons drectly. Can the latter reacton be constructed as a sum of the two prevous reactons? 0. Hydrde ons H - have been suggested to dssolve n odes accompaned by protons by dsproportonaton of hydrogen gas. Wrte defect reactons for ths n the case that the hydrde dssolves a) ntersttally, and b) substtutonally. (Assume n both cases that the ode s perfectly stochometrc and has no defects as a startng pont). 1. Wrte a defect reacton for the formaton of Schottky dsorder n spnel gal 4.. Wrte defect reactons for formaton of A-ste Frenkel, B-ste Frenkel, and anon Frenkel dsorders n a perovskte AB 3 (assumng both catons are trvalent). 3. Suggest a defect reacton for formaton of Schottky-type dsorder n Lan 3 when t s assumed that caton vacances are formed n the La sublattce only.

23 4. Wrte a defect reacton for dssoluton of Sc 3 n CaT 3 durng synthess, assumng that Sc substtutes T formng acceptors compensated by oygen vacances. 5. Wth reference to Problem n Chapter 1, the materal of your choce, and ts full and lmtng electroneutralty: Straghtforward: Wrte a reacton for the formaton of the defects you have selected to be domnatng. ore dffcult: Wrte a reacton that replaces the domnant postve defect wth another postve defect. Do smlar wth a domnant negatve defect. 3

24 Answers and hnts to selected Problems, Ch F = v / Na vcl = v / Cu v / F v = F vf 6. Ce / Ce = Ce Ce or 7. hnt: easer than you mght thnk 0 = e / h / 3 8. : - 3 : 3 v = 6e ( g) : - 9. : - 3 : /// = v 3 6h 3 ( g) : / Cl g) = vna Cl Cl h 1. Ca( s) 13. / ( Na = v h Na( g) // = CaZr v f = /// 3 ( s) 3f 3 v 14. Hnt: forms oygen vacances 15. Hnt: Forms oygen ntersttals // 16. CaF ( s) = Ca Ca vca F Na Na 17. // v = H H ( g) // = v H H ( g) 0 // /// = vg val 4v. Hnt: easy solutons thnk smple 3. - / 4. Ca Sc 3( s) = CaCa ScT 5 v