or the formation of a Schottky pair in MO, that we here choose to write; x O or the excitation of electronic defects, that we here choose to write;

Transcription

1 3. Defect equlbra 3. Defect equlbra Introducton After hang learned to formulate defect reactons n a compound, we now turn our nterest towards the equlbrum relatons for those reactons. These correlate the equlbrum concentratons of the dfferent defects wth temperature, acttes (e.g. partal pressures) of the components n the compound, and other parameters whch affect the defect structure. From a thermodynamc pont of ew a sold contanng pont defects consttutes a sold soluton where the pont defects are dssoled n the sold. In analogy wth lqud solutons the sold may be consdered to be the solent and the pont defects the solute. Thus, the defect equlbra may be treated n terms of the thermodynamcs of chemcal reactons and solutons. As eamples of defect equlbra, we may use the formaton of a acancy n an elemental sold; E + E E E E or the formaton of a Schottky par n, that we here choose to wrte; or the ectaton of electronc defects, that we here choose to wrte; e h + e or the formaton of oygen defcency; e ( g) From any startng pont (gen concentratons of reactants and products), a reacton may proceed to the rght (forward) or left (backward). The proceedng of a reacton n some drecton wll be accompaned by changes n the nternal energy U of the system. The change s called U. At constant pressure (whch we most normally consder) the change n energy s gen by the change n enthalpy, H; H. There wll also be a change n the dsorder (chaos) of the system, epressed by the entropy S. The entropy change s denoted S. The frst and second law of thermodynamcs state that n an solated system 3 the energy s constant and the entropy can only ncrease or reman constant. These laws are emprcal we cannot proe them, just eperence them. Equlbrum thermodynamcs n closed systems that we shall now consder s one such eperence. The solated system of the Unerse or a closed thermobottle are not 3 An solated system s a system that echanges nether mass nor heat wth ts surroundngs. The Unerse s beleed to be such a system. A closed nsulated thermobottle s a good eample of a practcally solated system. 3.1

2 3. Defect equlbra ery practcal laboratores or process contaners for our purpose. We want to consder a closed system 4,.e. one that keeps the chemcals nsde but echanges heat. From a tetbook n thermodynamcs you may learn how the 1 st and nd laws of thermodynamcs predct that any process or reacton n a closed system wll proceed spontaneously (oluntarly) only as long as T S > H, that s, as long as H -T S <. You may say that the effect of dsorder must oercome the energy cost n the closed system for the process to run by tself. For ths purpose we see that the enthalpy H of a system can be splt nto Gbbs (free) energy G and the energy gen by the entropy and temperature: H G + TS. The change n the Gbbs (free) energy by a chemcal reacton s thus gen by dg dh TdS (3.1) and, as stated aboe, a reacton proceeds as long as dg <. If dg >, t wll proceed backwards to make dg <. A reacton reaches equlbrum at dg. Ths etremely mportant concluson we can use to understand and calculate chemcal equlbra. 5 A reacton at equlbrum does n fact not mean that t has stopped, but that the forward and backward reactons run equally fast. We wll later on consder equlbrum thermodynamcs also from the ewpont of the mass-acton. Ths wll allow us to use the equlbrum constant, the man new tool of ths chapter, plus prode more nsght to help understand chemcal equlbrum. The creaton of sngle, unassocated pont defects n an elemental, crystallne sold ncreases the nternal energy of the system and the enthalpy of the defect formaton s poste. But the confguratonal entropy of the system also ncreases, and the equlbrum concentraton of the defects wll be reached when the Gbbs energy of the system s at mnmum. Thermodynamcally, pont defects wll thus always be present n a crystal aboe. All types of defects wll n prncple be formed. Howeer, the free energes of formaton of the dfferent types and systems of defects can hae wdely dfferent alues, and correspondngly t s often found that certan defects predomnate n a partcular sold. The relate concentratons of the dfferent types of defects wll be a functon a temperature and other arables. Thus, defect equlbra wth large poste enthalpy of formaton, for nstance, whch are not faoured at low temperatures, may become mportant at hgh temperatures. 4 A closed system s one where the mass (chemcal enttes) of the system s confned, but where energy (heat) can be echanged wth the surroundngs. The epresson for the Gbbs energy arses as a result of the defnton of entropy and of the frst law of thermodynamcs whch preseres the total energy of the system and the surroundngs. 5 It has been sad that chemsts hae the ablty to accept ts applcablty and to use t for a lfetme often wthout eer really understandng ts deraton or bascs. y eperence s that physcsts hae the tendency to reject t and to be unable to use t for the same reasons. 3.

3 3. Defect equlbra Pont defects and equlbrum n an elemental sold - a smple statstcal thermodynamc approach Before consderng the thermodynamcs of defect formaton and of defect reactons n compounds such as metal odes and the correspondng use of the law of mass acton, the smpler case of formaton of one type of defect n an elemental sold wll be treated by means of statstcal thermodynamcs. Consder a perfect crystal of an elemental sold E wth N atoms as startng materal. Let n acances be formed accordng to the defect-chemcal reacton E E E + E E (3.) where E E s an E-atom on an E-ste and E s a acant E-ste. The acances are formed by mong an E-atom from an E-ste n the bulk to the surface of the crystal. The total number of stes then becomes N+n. The change n Gbbs energy can be epressed n terms of the enthalpy of formaton of each acancy, H, and the entropy assocated wth the formaton of the n defects, S. Fgure 3-1. Left: Elemental sold consstng of 39 atoms on 3 stes (n the -dmensonal pcture of t). Rght: 39 atoms and one acancy, spread oer 4 stes. The enthalpy change on formng n acances s thus n H. It epresses the energy cost assocated wth creatng a new, empty ste. The entropy change can be dded nto two parts: A bratonal entropy change, S b, reflects the entropy created by bratons assocated wth each new acancy. A confguratonal entropy, S conf, arses from the dstrbuton of n acances among N+n stes. The bratonal entropy change s assocated wth the bratons n the neghbourhood of each acancy, and the total bratonal entropy change s thus proportonal to the number of acances, n S b. The change n Gbbs free energy assocated wth the formaton of n acances may accordngly be wrtten G n H T S n H T ( n S + S ) n ( H T S ) T S (3.3) b conf b conf It may be noted that we hae assumed that H and S b are constant,.e. ndependent of defect concentratons. Ths may only be epected to hold true for small concentratons of defects, where defect-defect-nteractons are neglgble. Boltzmann was the frst to propose that the confguratonal entropy (of mng) can be epressed n terms of the thermodynamc probablty, W, by the relaton 3.3

4 3. Defect equlbra S conf k lnw (3.4) where k s Boltzmann's constant. The thermodynamc probablty W represents n ths case the number of dstngushable ways whereby n acances may be dstrbuted on N+n lattce stes and s gen as W ( N + n )! (3.5) N! n! We recognse that for the perfect crystal n so that W1 and S conf. As the number of acances ncreases, S conf ncreases and we hae S conf S conf. For large numbers of N and n, whch s typcal for real crystals, Strlng's appromaton ( ln! ln for >>1) may be appled, whereby Eq. (3.5) takes the form N + n N + n Sconf k( N ln + n ln ) (3.6) N n By nserton nto the epresson for the change n free energy Eq. (3.3) we get N + n N + n G n ( H T Sb ) kt ( N ln + n ln ) (3.7) N n The changes n enthalpy, entropy, and Gbbs energy are llustrated n Fgure 3-. Fgure 3-. Eample of the araton n enthalpy, bratonal and confguratonal entropy, and Gbbs energy wth the concentraton of acances n an elemental sold. 3.4

5 3. Defect equlbra At equlbrum G and thus G wll be at a mnmum wth respect to n,.e. d Gdn ; d G H T S dn V b n + kt ln N + n (3.8) n The term represents the fracton of the total number of stes whch are N + n acant and s through rearrangement gen by n N + n Sb H ep ( ) ep( ) (3.9) k kt Snce the formaton of the acances s wrtten E E E + E E, the equlbrum constant may, followng the law of mass acton, be wrtten n Sb H a X ep ( ) ep( ) (3.1) E E N + n k kt a E s the actty and X n represents the ste fracton of acances n N n E + the crystal. S b n the epressons aboe represents the bratonal entropy change and H the enthalpy change per acancy. If one wants to epress the same propertes per mole of acances, k must be substtuted by the gas constant R N A k where N A s Aogadro's number. Relatons correspondng to Eq. (3.1) may be dered for other types of pont defects. At any temperature T> all solds wll followng Eq. (3.1) contan pont defects. In the eample aboe we hae shown that the equlbrum constant, contanng defect acttes as ste fractons, s gen by an enthalpy change and an entropy term whch consttutes the bratonal entropy change as one goes from the reactants to the products. It does not contan the (confguratonal) entropy change. Ths apples to chemcal reactons and ther equlbrum constants n general. Furthermore, t llustrates that the bratonal entropy change enters wthout other contrbutons f the equlbrum constant s epressed properly, as acttes. Each actty s untless, that s, t s some concentraton dded by the concentraton n the standard, reference state. For defects, acttes are gen as ste fractons. When acttes properly use standard states as reference, the enthalpy and bratonal entropy changes are called standard enthalpy and entropy changes, H and S, respectely. We may combne the standard enthalpy and entropy changes nto a standard Gbbs energy change: Sb H G ep( ) ep( ) ep( ) R RT RT (3.11) 3.5

6 3. Defect equlbra We wll see later that ths standard Gbbs energy change s the Gbbs energy change we hae when the reacton proceeds wth all reactants and products n ther standard reference state (actty of unty). It s not the same as the arable Gbbs energy change calculated for the progresson of the defect formaton reacton earler. What happened to the confguratonal entropy? Should t not be part of the entropy that balances the reacton at equlbrum? And what s? Look at Eq. (3.11). The rght hand sde tells us a the standard enthalpy, entropy, and Gbbs energy changes how much the reacton wants to proceed when all reactants and products are n ther standard states. If G < t s a faourable reacton. s the quotent of all product acttes oer all reactant acttes when equlbrum s attaned. In other words, tells us how much the acttes hae to deate from the standard state n order to balance the force from G to dre the reacton forward ( G <, >1) or backward ( G >, <1). ne may thus say that the confguratonal entropy s n the. In our eample of the elemental sold, the equlbrum quotent contaned only the actty of the product, the ste fracton of acances. What were the standard states n our eample? For atoms, t s a ste fracton of 1,.e. the perfect crystal s the standard state. For the acances t s also a ste fracton of 1,.e. an empty crystal! Ths eemplfes that standard states of defects are rtual. Thermodynamcs of chemcal reactons As a bass for the subsequent consderatons of thermodynamcs of defect equlbra the followng secton prodes a bref reew of some general aspects of thermodynamcs of chemcal reactons. In treatng chemcal reactons and systems consstng of two or more consttuents, t s conenent to ntroduce the concept of partal molar thermodynamc quanttes. As regards equlbra, the partal molar Gbbs energy s partcularly mportant. The partal molar Gbbs energy s commonly termed the chemcal potental and wrtten µ. The physcal sgnfcance of a partal molar property s the ncrease n the alue of that property resultng from the addton of 1 mole of that consttuent to such a large quantty of the system that ts composton remans essentally unchanged. If a system conssts of n 1 + n n moles of consttuents 1,, and, the partal molar free energy for the ""th consttuent s gen by G µ ( ) T,p,n1,,... (3.1) n n The Gbbs energy of a system s n terms of chemcal potentals of the consttuents gen by G n1 µ 1 + nµ +... nµ (3.13) 3.6

7 3. Defect equlbra When a chemcal reacton takes place, the numbers of arous consttuents change, and dg changes correspondngly. For an open system (.e. where we may add or remoe heat and mass) at constant temperature and pressure, dg s gen by dg T, p µ 1dn1 + µ dn +... µ dn (3.14) At equlbrum we hae dg µ dn (3.15) T, p It may also be shown that under equlbrum condtons n1 dµ 1 + ndµ +... n dµ (3.16) Ths equaton s one form of the Gbbs-Duhem equaton. In a chemcal reacton, e.g. aa + bb cc + dd (3.17) the change n Gbbs energy s gen by the dfference n the total free energy n the fnal and ntal state G cµ + dµ ( aµ + bµ ) B (3.18) C D A The chemcal potental of the consttuent "" n the mture can be wrtten µ µ + RT ln a (3.19) where a s the actty of consttuent "" n the mture and µ s the chemcal potental of consttuent "" at a chosen standard state of unt actty. The actty of pure solds or lquds s usually taken as unty at atmospherc pressure (1 bar or 1 5 Pa), whle for gases the actty s unty when the partal pressure of the gas consttuent s 1 bar. In solutons the actty may be equated wth the mole fracton, X, n deal solutons: a X (3.) In non-deal solutons the actty s related to the mole fracton through the actty coeffcent, γ, a γ X (3.1) 3.7

8 3. Defect equlbra γ s thus equal to unty n deal solutons. It may be noted that n a system consstng of seeral phases, the condton of equlbrum mples that the chemcal potental of consttuent "" s the same n all phases. If one ntroduces the epresson for the chemcal potental n Eq. (3.19) nto Eq. (3.18) t follows that G for the reacton n Eq. (3.17) s gen by A c B d ac ad G G + RT ln (3.) a b a a where G cµ + dµ ( aµ + bµ ) represents the change n Gbbs energy n C D A B the standard state,.e. at unt acttes. G s a constant under specfed standard states. At equlbrum G such that G a RT ln a C D a b A c a a B d equlbrum (3.3) where the quotent at equlbrum s called the equlbrum coeffcent, : a a c d C ad a b A a B equlbrum (3.4) s termed the equlbrum coeffcent as t relates the rato of acttes of products and reactants when equlbrum has been attaned at a gen temperature. It takes a constant alue at any gen temperature and s thus also often called the equlbrum constant. G may as n Eq. (3.1) be epressed n terms of the standard enthalpy change, H, and entropy change, S, such that G H T S RT ln (3.5) Ths may be rewrtten n the form H S H ep( ) ep( ) ep( ) (3.6) RT R RT Thus, we hae arred at the same type of epresson for the equlbrum constant as we dd n our nttal eample from statstcal thermodynamcs. From that, we know that the entropy term contaned n ep( S R) contans the change n bratonal entropy terms between the reactants and products n ther standard states, and that t does not contan the confguratonal entropy change noled by the system when the acttes are dfferent from the standard states. From the equaton aboe we see that the temperature dependence of s gen by 3.8

9 3. Defect equlbra d ln H (3.7) d(1 T) R and that a plot of ln s 1T (an t Hoff plot) wll ge S R as the ntercept and - H R as the slope. Ths general treatment s applcable to any deal solutons whether these are gaseous, lqud, or sold, and t apples also to defect chemstry as we hae eemplfed ntally. Thermodynamcs and pont defects Equlbrum thermodynamcs may as stated aboe be appled to defect reactons. Howeer, before so dong t s of nterest to consder the thermodynamc propertes of the nddual pont defects. Vrtual chemcal potentals of pont defects The regular atoms on ther normal stes and the pont defects occupy partcular stes n the crystal structure and these hae been termed structural elements by röger, Steltjes, and Vnk (1959) (see also röger (1964)). As dscussed n Chapter, the rules for wrtng defect reactons requre that a defnte rato of stes s mantaned due to the restrant of the crystal structure of the compounds. Thus f a normal ste of one of the consttuents n a bnary compound s created or annhlated, a normal ste of the other consttuent must smultaneously be created or annhlated. As dscussed aboe the chemcal potental of a consttuent represents the dfferental of the Gbbs energy wth respect to the amount of the consttuent when the pressure, temperature and the amounts of all other consttuents are kept constant. When a structure element s created, the number of complementary structure elements can not be kept constant due to the requrement of a defnte ste rato, and t s therefore not possble to assgn a true chemcal potental to a structure element. Howeer, röger et al. showed that one may get around ths dffculty by assgnng a rtual chemcal potental ζ (zeta) to each separate structure element and that the rtual chemcal potental behaes lke a true chemcal potental and may be related n a smlar way to actty: ζ ζ + RT ln a (3.8) The rtual potental dffers from the true chemcal potental by an undefned constant whch s ncorporated n ζ. For ths reason the absolute alue of the rtual chemcal potental can not be determned epermentally. In real changes n crystals where the ste rato s mantaned and complementary structure elements (or buldng unts) are created or annhlated, the undefned constants n the rtual chemcal potental of each separate structure element cancel out, and the oerall change s charactersed by real changes n 3.9

10 3. Defect equlbra Gbbs free energes; dg dµ dζ drtlna. At constant temperature and wth deal condtons (small concentratons of defects), a change n actty corresponds to the change n concentraton, and we hae dg dµ RTdlna RTdln(c c ) RTdlnc. röger et al. ponted out that a smlar treatment apples to aqueous electrolytes. As we hae mentoned before, we hae not only the problem of measurng true chemcal potentals µ of defects and other structure elements, we also hae the problem of establshng or een magnng the standard state of ste fracton of unty: For oygen acances n a smple ode ths would for nstance mean a crystal where all ode ons are mssng,.e. a crystal wth only catons. Ideal and non-deal models As descrbed n the precedng secton the defect reactons and equlbra can be descrbed as chemcal reactons and treated n terms of the law of mass acton. It s emphassed that the equlbrum constants relate acttes of the structure elements noled n the defect reacton, but under deal condtons and when the structure elements can be assumed to be randomly dstrbuted oer the aalable stes, the acttes equal the concentratons of the structure elements. Ths s a ald assumpton for ery dlute solutons. Howeer, for larger defect concentratons nteractons take place between the defects, etc. and n prncple acttes should be used nstead of concentratons. The nteractons between charged defects may be accounted for by usng the Debye-Hückel theory n analogy wth the nteractons of ons n aqueous solutons. Ths requres knowledge of the relate delectrc constant, the smallest dstance between charged defects, and other parameters for the sold. Debye- Hückel correctons hae, for nstance, been worked out and tested for caton acancy defects n metal-defcent Co 1- and N 1-. At nfnte dluton the metal acances can be consdered to hae two negate charges, Co, and the equalent number of electron holes s assumed to be randomly dstrbuted n the ode. Wth an ncreasng concentraton of pont defects and electron holes, there wll be an ncreasng assocaton of the metal acances (wth two effecte negate charges) and the postely charged electron holes. In these terms, the metal acances can be consdered surrounded by a cloud of electron holes. Ths results n a deaton from a random dstrbuton of electron holes and a correspondng deaton from the deal soluton model. For the case of Co 1- the Debye-Hückel correcton can eplan the ncreasng deaton from dealty wth ncreasng defect concentraton. Howeer, ths model n ts present form does not satsfactorly eplan arous electrcal conductty results for Co 1- and so far apparently fals to ge a full descrpton of the defect structure and defectdependent propertes of the ode. An alternate approach s to consder that the assocaton of defects leads to the formaton of new "assocated" defect speces. These "new" defects are, n turn, treated n terms of the deal soluton models. Specfcally, n the case of metal acances n, for nstance, Co 1- and N 1- t s assumed that the metal 3.1

11 3. Defect equlbra acances wth two effecte charges become assocated wth one electron hole to ge a metal acancy wth one negate effecte charge,, as has already been descrbed n Chapter. At ths stage ths smple approach appears to ge an equally consstent descrpton of the defect structure and defect-dependent propertes of Co 1- as the much more complcated Debye-Hückel model. Howeer, t s to be epected that future deelopments wll prode a more detaled understandng of defect nteractons and how they nfluence defectdependent propertes of norganc compounds. Despte ts shortcomngs the deal soluton approach wll be used n ths book n treatng defect equlbra. Defect nteractons wll generally be treated by assumng the formaton of assocated defects as s, for nstance, eemplfed aboe for the formaton of sngly charged metal acances through the assocaton of an electron hole wth a doubly charged metal acancy. Furthermore, the defect equlbra wll be treated usng the law of mass acton. Concentratons of the structure elements may be epressed n many unts. In semconductor physcs t s common to epress the concentratons n number per cm 3. Ths has the dsadantage, howeer, that when two compounds wth dfferent molecular (packng) denstes contan the same number of defects per cm 3, the number of defects per molecular unt s dfferent n the two compounds, and ce ersa. Furthermore, the molecular unt sze (packng densty, or unt cell parameters) of a compound changes as a functon of ts nonstochometry and temperature, and n ths case the unt of defects per cm 3 does not unequocally reflect the relate defect concentraton per molecular unt or atom ste under dfferent condtons. In the followng, concentraton wll n most cases be epressed as the number of defects or atoms per molecular unt or ste,.e. n molar or ste fractons, whle we for some purposes need to use the number of defects per cm 3. As mentoned before, the alue of the equlbrum constant depends on the unts of concentraton that are employed, but t s a smple matter to conert the alues of the equlbrum constant from one system to another. Co Eamples of defect equlbra n pure, stochometrc metal odes In ths secton we wll see how we wrte defect reactons for selected smple cases of combnatons of defects, epress the equlbrum constants, and combne ths wth an electroneutralty condton to epress the concentraton of domnatng defects as a functon of temperature and acttes. We wll when possble stay wth proper statstcal thermodynamcs, such that entropy changes n the defect reacton can be nterpreted as a change n the bratonal entropy from reactants to products n ther standard states. Appromatons wll be mplemented n order to sualse the smplcty of the method. 3.11

12 3. Defect equlbra Schottky defects As a frst llustraton let us consder the defect equlbrum for Schottky defects n the ode and where the metal and oygen acances are doubly charged. The defect equaton for ther formaton s gen n Chapter as + (3.9) The correspondng defect equlbrum may at low defect concentratons be wrtten S [ [ X X (3.3) [ [ where S s the equlbrum constant for a Schottky defect par, X represent ste fractons, and square brackets epress concentratons of speces. [ and [ represent the concentraton of oygen and metal stes, respectely. If we let square brackets epress concentraton n mole fracton (.e. the number of moles of the speces per mole of ), then [ [ 1, and then S [ [ (3.31) When the concentratons of the defects are epressed n ste fractons, or properly conerted to mole fractons as here, S s related to the standard Gbbs energy cange G S, enthalpy change H S, and entropy change S S, of formaton of pars of doubly charged Schottky defects: S GS [ [ ep(- ) ep( RT S R S H ) ep(- RT S ) (3.3) It may be useful to repeat that the entropy change S S does not nclude the confguratonal entropy changes assocated wth the formaton of the defects, but only the change n bratonal entropy assocated wth the acances formed. When the Schottky par domnates the defect structure, we may see from the reacton equaton or from the electroneutralty condton that n the concentratons of the metal and oygen acances are equal: [ [ (3.33) Inserton nto the precedng equaton yelds [ [ SS ep( R 1 S ) ep( H RT S ) (3.34) Under these condtons the defect concentratons are only dependent on the temperature; they are ndependent of the acttes of the components and. The slope n a an t Hoff plot of ln[ or ln[ s 1T would be - H S R, see 3.1

13 3. Defect equlbra Fgure 3-3. The factor results from the square root of the equlbrum constant, n turn resultng from the fact that there are two defects created, n comparson wth the one we got n the element used as eample earler. Fgure 3-3. an t Hoff plot of the logarthm of defect concentratons s nerse absolute temperature yelds ntercept and slope related to the standard entropy and enthalpy, respectely, of the defect formaton reacton, dded by the number of defects formed and R. It may be useful to note that the procedure we hae appled s to wrte down the defect reacton and the mass acton law epresson for ts equlbrum constant, and the prealng electroneutralty condton. Thus, we hae two ndependent equatons for the two unknowns (proded that ether S or all of S S, H S, and T are known). Frenkel defect pars In our net eample we consder Frenkel defect pars n. If, for the sake of llustraton, the Frenkel defects n are assumed to be doubly charged, the defect equaton for ther formaton s gen as + + (3.35) It may be noted that a acant ntersttal ste has been ncluded to keep track of the stes where the ntersttals are gong. The correspondng defect equlbrum s wrtten as F X X X X [ [ [ [ [ [ [ [ [ [ [ [ (3.36) If there s one ntersttal ste per, and f the defect concentratons are small such that [ [ 1, we obtan smply F [ [ (3.37) 3.13

14 3. Defect equlbra If Frenkel defects predomnate n the pure stochometrc compound, the concentratons of ntersttal caton and acances are then equal, and we obtan: [ [ SF ep( R - H ) ep( RT 1 F F ) (3.38) where S F and H F are the standard entropy and enthalpy changes of formaton of Frenkel defect pars. Under these condtons the concentraton of the Frenkel defect pars s ndependent of the acttes of the metal and oygen components. Intrnsc onsaton of electrons For many odes, and partcularly when consderng defect structure stuatons close to stochometry, t s essental to take nto account the ntrnsc onsaton of electrons. Ths can nclude localsed defects (alence defects) or delocalsed defects (alence band and conducton band). In the case of alence defects, the onsaton of a pure bnary metal ode a b may typcally be assgned to med alency of the metal and thus be wrtten as (3.39) + The equlbrum constant may then be epressed as ste fractons, whch n our case mmedately smplfes nto an epresson contanng only olume or molar concentratons: [ [ (3.4) [ If ths ntrnsc onsaton domnates the defect structure, we hae [ [ (3.41) Inserton nto the equlbrum constant yelds [ [ S H ep( )ep( R RT 1 [ [ ) (3.4) In the case of delocalsed electrons we wrte the onsaton reacton e + h (3.43) If we descrbe ths as a chemcal reacton, the product of the acttes of the electrons and holes should reman constant at a gen temperature. Howeer, 3.14

15 3. Defect equlbra electrons and holes n the band model do not occupy stes, but energy states, and thus the actty mght be represented as the fracton of concentraton oer densty of states: [e [h n p a a (3.44) e h N N N N C V C V where s the equlbrum coeffcent and n and p denote the concentratons of electrons and holes, respectely. N C and N V denote the densty of states n the conducton band and alence band edges, respectely. If we apply statstcal thermodynamcs to ths we mght assgn to standard Gbbs energy, entropy and enthalpy changes: n p G H S H ep, ep ep ep (3.45) N N RT RT R RT C V Howeer, the use of entropy s not commonly adopted for reactons nolng band electrons and holes, as the concept of the standard states, that would mply nn C and pn V, are not commonly accepted. In physcs, one nstead uses smply 6 E g [ e [ h n p NC NV ep (3.46) RT where E g s the band gap. We see from ths that NC NV and that the band gap would correspond to the standard Gbbs energy change of the ntrnsc onsaton reacton. The prme on ths and others to follow s used here to denote an equlbrum coeffcent that has been multpled wth denstes of state such that t gets unts (e.g. m -6 or cm -6 as here) and cannot be drectly related to an entropy. ur bref dscusson of electronc onsaton as seen from both defect chemcal and physcs perspectes may seem confusng, and s n fact not well resoled n tetbooks or n the lterature. Ths should not preent us from comng out wth a smple concluson: The product np s constant at any gen temperature, and ts araton wth temperature s roughly gen by the band gap, E g. If the concentratons of electrons and electron holes predomnate, then the electroneutralty condton s n p and nserton nto the equlbrum coeffcent yelds 6 Physcsts mostly use E g kt wth E g n ev (per electron), whle chemsts often use E g RT (or G RT) wth E g n J per mole electrons. Ths s a tral conerson (factor 1 ev Jmol kjmol). 3.15

16 3. Defect equlbra E 1 1 g n p ( ) ( NC NV ) ep( ) (3.47) RT The densty of states s often under certan assumptons appromated as N C * 8πm ekt h 3 and N V * 8πm hkt h 3 (3.48) * * where the constants hae ther usual meanngs, and m e and m h are the effecte masses of electrons and holes, respectely. Effecte masses are sometmes known, or may be to a frst appromaton be set equal to the rest mass of electrons, whereby N C and N V may be estmated. It may be noted that they hae unts of m -3 and that they contan temperature dependences of T 3. Intrnsc onzaton of electrons and holes s, for nstance, concluded to domnate the defect structure of Fe 3 and Cr 3 at hgh temperatures. Defect equlbra n nonstochometrc odes Here, we wll employ manly the same methodology as n the precedng secton, but when delocalsed defect electrons and holes are ntroduced we cannot stay strctly wth statstcal thermodynamcs and the equlbrum thermodynamcs cannot be related n the same smple manner to the change n bratonal entropy any longer. ygen defcent odes In odes wth oygen defct, the predomnant defects are oygen acances. If these are fully onsed (doubly charged), and f the electronc defects are localsed as alency defects, ther formaton reacton can be wrtten: ( g) + (3.49) The correspondng equlbrum coeffcent shall be epressed n terms of acttes. Wrtng these as ste fractons for the defects, and as partal pressure for the gas speces (the dson by the standard state of 1 bar s omtted for smplcty) we get X X X X p 1 [ [ [ [ p 1 (3.5) 3.16

17 3. Defect equlbra If these oygen acances and the compensatng electronc alence defects are the predomnatng defects n the oygen defcent ode, the prncple of electroneutralty requres that [ [ (3.51) By nserton, and assumng the ode s and that [ [ 1 (small defect concentratons) we then obtan: S H 1 6 [ [ ( ) p ep( ) ep( )p (3.5) 3R 3RT Ths shows that the concentratons of the domnatng defects (defect electrons and oygen acances) ncrease wth decreasng oygen partal pressure. It furthermore shows the relaton between defect concentratons and entropy and enthalpy changes of the defect reacton. The factor 3 that enters results from the formaton of 3 defects n the defect reacton. Now, let us do the same treatment, but for an ode where defect electrons are delocalsed n the conducton band. The formaton of oygen acances s then wrtten: e ( g) (3.53) The equlbrum coeffcent should be a a e a 1 ( g ) [ [ n N p p 1 C [ [ N (3.54) a C p [ [ n p 1 It s common for most purposes to neglect the dson by N C, to assume [ 1 and to remoe p 1 bar, so that we get (3.55) 1 [ n p Ths means that N C and that the epresson s ald for small concentratuons of defects. If these oygen acances and the compensatng electrons are the predomnatng defects n the oygen defcent ode, the prncple of electroneutralty requres that n [ (3.56) By nserton we then obtan: 3.17

18 3. Defect equlbra H 1 6 n [ ( ) p (, ) ep(- )p (3.57) 3RT and delberately use a pre-eponental nstead of an entropy change. therwse, the soluton s the same as n the case wth localsed alence defects for the electrons. A plot of log n or log[ s log p (at constant temperature) wll ge straght lnes wth a slope of 16. Such plots are called Brouwer dagrams, and they are commonly used to llustrate schematcally the behaour of defect concentratons under smplfed lmtng cases of domnatng defects. Fgure 3-4. Brouwer dagram for n [ as the electroneutralty condton. If we want to nestgate the more comple stuaton of nolng neutral and partally onsed oygen acances and the ectaton of the trapped electrons (descrbed n Chapter ) we start by wrtng down the stepwse reactons and ther correspondng equlbra: 1 ( g) (3.58) + [ 1 p (3.59) [ + e (3.6) [ n (3.61) [ 1 + e (3.6) [ n (3.63) [ 3.18

19 3. Defect equlbra where, 1, and are the respecte equlbrum constants. Now the prncple of electroneutralty requres that n [ + [ (3.64) The concentratons of the electrons and the neutral, sngly, and doubly charged oygen acances are related through the equatons aboe, and by combnaton epressons for each of the defects may be obtaned. The electron concentraton s gen by n (3.65) 3 1 1( + n) p Ths equaton has two lmtng condtons. If n <<, then n (3.66) [ ( ) p 1 Under these condtons the concentratons of electrons and oygen acances are relately small, the doubly charged acances are the predomnatng oygen acances, and the concentratons of electrons and these oygen acances are 1 6 proportonal to p. f course, ths stuaton corresponds to the smple case presented ntally for the doubly charged acances, and obously. If, on the other hand, n >>, then n (3.67) [ ( ) p 1 Under these condtons the concentraton of electrons s relately large, and the predomnant oygen acances are then sngly charged. Furthermore, the 1 4 concentratons of electrons and oygen acances are then proportonal to p. A general tendency smlar to that of oygen defcent odes apples to metal defcent odes; n the ode 1- the metal acances are doubly charged at ery small deatons from stochometry and tend to become sngly charged wth ncreasng nonstochometry. It may be noted that the neutral oygen acances are not affected by charged defects, nor do they affect the electroneutralty. de wth ecess metal We wll as an eample consder the formaton of fully onsed ntersttal metal ons and complementary electrons n an ode wth ecess metal + 3. The equaton for the formaton of ntersttal metal ons wth three effecte poste charges and three complementary electrons s gen by 3.19

20 3. Defect equlbra e ( g ) + 4 (3.68) The correspondng defect equlbrum s gen by n p [ [ (3.69) [ where s the equlbrum constant. If these defects are the predomnatng ones, and we as before assume small defect concentratons, the electroneutralty condton and the oygen pressure dependence of the ntersttal metal ons and the electrons s 3[ (3.7) n (3 ) p Thus, also n ths case the concentraton of the defects ncreases wth decreasng oygen actty and s now proportonal to p. Ths oygen pressure dependence s dfferent from that for formaton of sngly as well as doubly charged oygen acances. In such a case t would thus n prncple be possble to decde from measurements of electron concentraton (e.g. a electrcal conductty) or of nonstochometry as a functon of oygen actty whether the predomnatng defects are trply charged ntersttal metal ons or oygen acances wth one or two charges. Had the metal ntersttals predomnantly been doubly charged n the ode + 3, a p dependence would hae been the result, and addtonal studes of defect-dependent propertes (e.g. self-dffuson of metal or oygen) would be needed to dstngush ths stuaton from that of sngly charged oygen acances. It may be noted that t s not only the absolute number of charges on a defect that determnes the p dependence, but also the dfference between the actual charge and the charge gen by the nomnal alence of the atoms noled. Thus, n the ode (or 1+) doubly charged metal ntersttals wll ge a -16 p dependence of the defect concentratons (as opposed to the case aboe) and n ths case be ndstngushable from doubly charged oygen acances. Smultaneous presence of oygen acances and ntersttal metal ons In an ode a b where the rato of metal to oygen s larger than the stochometrc rato a:b t s a pror dffcult to predct whether ntersttal metal ons or oygen acances predomnate. In prncple both types of defects may be mportant, at least n certan regons of nonstochometry. In the followng, let us consder a case where t may be necessary to take nto account the smultaneous presence of ntersttal metal ons and oygen acances. Consder an ode wth a stochometrc composton. Let us further assume for the sake of llustraton that when the ode s nonstochometrc the mportant pont defects are doubly charged oygen acances and doubly charged ntersttal metal ons. (It may be noted that the metal ntersttals are not fully 3.

21 3. Defect equlbra onsed n ths case.) The composton of the nonstochometrc ode may accordngly be wrtten 1+y -y. The defect equatons for the formaton of these two types of defects are wrtten e ( g) + + e + ( g) (3.71) (3.7) The correspondng defect equlbra (assumng [ and [ to be equal to unty) then become 1-1 n p [ (3.73) [ - -1 n p [ [ (3.74) [ The electroneutralty condton s gen by n [ + [ (3.75) Two lmtng condtons may be consdered: When [ >> [ then n (3.76) [ ( ) p as obtaned also earler for the same condtons. By nsertng ths relatonshp for n nto the equlbrum constant for formaton of the mnorty defects, metal ntersttals, we obtan [ (3.77) 3 3 ( ) p Under these condtons the mnorty concentraton of metal ntersttals, [, ncreases rapdly wth decreasng oygen partal pressure, more rapdly than the two domnatng defects, and may eentually catch up wth them and become domnatng at the epense of oygen acances, as llustrated n the fgure below. 3.1

22 3. Defect equlbra Fgure 3-5. Schematc presentaton defect concentratons as a functon of oygen pressure n an ode contanng both doubly charged oygen acances and doubly charged ntersttal metal ons, compensated by electrons. When, as the other lmtng case, [ >> [ then n (3.78) [ ( ) p By nsertng ths relatonshp for n nto the equlbrum constant for formaton of the mnorty defect, oygen acances, we obtan [ ( ) p (3.79) It may be noted that under these condtons the concentraton of oygen acances ncreases wth the oygen pressure, as shown n the fgure. The oerall stuaton may be llustrated schematcally as shown n Fg.3.3, n whch the concentratons of the three defects are plotted as a functon of the partal pressure of oygen. In ths dagram t s assumed that [ s 1 5 tmes larger than [ at 1 atm, and wth ths assumpton t s seen that the ntersttal catons wll predomnate at a partal pressure below about 1-17 atm. any odes wth hgh meltng ponts are stable oer large oygen pressure ranges and t s not mprobable that ths hypothetcal stuaton, wth oygen acances predomnatng at hgh partal pressures of oygen (and small deatons from stochometry) and ntersttal catons predomnatng at low partal pressures (relately large deatons from stochometry) may generally apply to oygen defcent odes at eleated temperatures. In actual cases more than one onsaton state of the defects may of course also hae to be consdered. In ths secton we hae employed and llustrated a ery mportant sequence of actons often used to sole deal stuatons n defect chemstry, but also mplctly used n tradtonal aqueous and other chemstry. We wrte down the sequence of operatons n a numbered lst for clarty: 3.

23 3. Defect equlbra 1. Wrte down the full electroneutralty condton contanng all charged defects of nterest.. Wrte down a number of ndependent chemcal reactons and correspondng equlbrum constant epressons f you hae n defects you need normally n-1 such equlbra. The n th epresson s the electroneutralty tself. 3. Decde one set (normally a par) of domnatng charged defects, and smplfy the electroneutralty condton to ths lmtng stuaton. 4. Insert ths lmtng condton nto an approprately chosen defect equlbrum to obtan the epresson for the concentraton of the two domnatng defects. (Normally, smplfcatons such as assumng small defect concentratons are made here.) 5. Insert the obtaned epresson nto another equlbrum epresson and sole to obtan an epresson of the concentraton of a mnorty defect. 6. Etrapolate to see whch mnorty defect comes up to become domnatng as condtons change. Then repeat from step 3 untl all defects hae been dealt wth under all smplfed defect stuatons. n the bass of such an eercse one may construct schematc dagrams of how defect pars domnate defect structures and how mnorty defects behae and eentually take oer. Transton zones are not soled eplctly n ths way and are thus often drawn sharp and schematcally. Logarthmc depctons are common, and such plots are then called Brouwer dagrams. etal-defcent ode The reactons and defect equlbra for the formaton of sngle, unassocated neutral metal acances and the subsequent ectaton of electron holes n may for small defect concentratons be wrtten 1 ( g) + (3.8) (3.81) 1 [ [ p + h (3.8) [ 1 p (3.83) [ + h (3.84) 3.3

24 3. Defect equlbra [ p (3.85) [ where, 1, and are the equlbrum constants for the respecte defect equlbra. When the metal acances and ther complementary electron holes are the predomnatng defects, the electroneutralty condton reads p [ + [ (3.86) Through combnatons of the equlbrum constant epressons and electroneutralty aboe, the concentraton of the separate pont defects and electronc defects can be ealuated. In an ode 1- ths leads to relatonshps smlar to that for oygen acances wth the ecepton that the concentratons of electron holes and metal acances always ncrease wth ncreasng oygen pressure. The concentraton of electron holes s, for nstance, determned by the epreesson 3 1 p 1( + p) p (3.87) As regards the electron holes two lmtng condtons may be consdered: At low concentratons of electron holes (small deatons from stochometry; p << ) we hae p (3.88) ( ) p 1 At hgh concentratons of electron holes ( p >> ) we obtan p (3.89) ( ) p 1 The oygen pressure dependence of the concentraton of electron holes and the total concentraton of the charged metal acances (deaton from stochometry) correspondngly changes from p to p wth ncreasng deaton from stochometry etal odes wth ecess oygen. The defect equlbra nolng the formaton of ntersttal oygen atoms or ons n odes wth ecess oygen may be set up followng the same treatment as appled to the other defects whch hae been dealt wth aboe. The concentraton of ntersttal oygen speces ncreases wth p, and the formaton of charged oygen ntersttals s accompaned by the formaton of electron holes. 3.4

25 3. Defect equlbra Eamples of defect structures nolng both oygen defcency and ecess In the precedng eamples of defect structures n non-stochometrc odes t has been assumed that the odes ether hae an oygen defct (or ecess metal) or ecess oygen (or metal defct). In many odes the predomnatng defect may change from one type to another dependng on the oygen actty. As an llustraton of such a defect structure stuaton a hypothetcal case wll be consdered where an ode predomnantly contans oygen acances at reduced oygen acttes and ntersttal ode ons at hgh oygen acttes. In an ntermedate regon the ode wll be stochometrc or close to stochometrc. Some odes wth the fluorte structure ehbt such a defect structure, e.g. U ±. For the sake of smplcty n llustratng such a defect structure stuaton t wll be assumed that both the ntersttal ode ons and the oygen acances are doubly charged. In ths case t wll then be necessary to consder the followng equlbra, for the formaton of oygen acances, oygen ntersttals, electrons and holes, and for anon Frenkel pars: (3.9) 1 [ n p (3.91) 1 [ p p np (3.9) [ [ (3.93) AF In these equlbra we hae assumed small defect concentratons such that the concentratons of normal lattce stes and empty ntersttal stes hae been assumed constant and equal to unty and thus omtted from the epressons. It should be noted that the defect equlbra are nterrelated, and through a combnaton of the equatons t may be shown that AF. Thus, only three out of the four equlbra are suffcent to descrbe the defect structure of the ode. The full electroneutralty condton s gen by [ + p [ + n (3.94) We wll eplore the defect structure of the ode by consderng lmtng condtons and follow the procedure lsted before to construct Brouwer dagrams. ygen defct. At large oygen defct the followng appromaton may be made 3.5

26 3. Defect equlbra n [ >> [, p (3.95) We frst nsert ths nto the approprate equlbrum to fnd the concentratons of the domnatng defects n the usual manner: n (3.96) [ ( ) p and then nsert ths nto other equlbra to fnd the concentratons of the mnorty defects. By nsertng the epresson for the concentraton of acances nto the anon Frenkel equlbrum, we obtan for the concentraton of ntersttals: [ AF ( ) p (3.97) By nsertng the epresson for the concentraton of electrons nto the ntrnsc electronc equlbrum, we obtan for the concentraton of holes: p (3.98) ( ) p ygen ecess For relately large ecess oygen, that s, when p [ >> [, n (3.99) we may n a manner analogous to the precedng case, dere the followng relatons: p (3.1) [ ( ) p [ (3.11) AF ( ) p n (3.1) ( ) p Stochometrc condton At or close to stochometry two alternate lmtng condtons must be consdered, namely domnance by ntrnsc electronc onsaton or by anon Frenkel dsorder. If ntrnsc onsaton of electrons predomnates, and thus ( ) 1 p n >> [,[ (3.13) 3.6

27 3. Defect equlbra The concentratons of electrons and electron holes (n and p) are then ndependent of oygen pressure. The pont defect concentratons are obtaned by nserton of the epresson for the concentratons of the electronc defects nto the approprate equlbra, and we obtan: [ (3.14) 1 p 1 [ p (3.15) A Brouwer dagram llustratng ths case of stochometrc condton, along wth oygen defct and ecess, s shown below. Fgure 3-6. Schematc presentaton of oygen pont defects and electronc defects as a functon of oygen pressure n an ode whch dependng on the partal pressure of oygen may hae an ecess or defct of oygen. Intrnsc electronc ectaton s assumed to predomnate at stochometrc composton. If, on the other hand, anon Frenkel dsorder predomnates under stochometrc condtons, and thus 1 1 p n ( ) [ [ AF >>, (3.16) 3.7

28 3. Defect equlbra then [ and [ are ndependent of the partal pressure of oygen, whle the concentratons of electronc defects are gen by n ( ) AF p (3.17) p ( ) AF p (3.18) A Brouwer dagram llustratng ths case of stochometrc condton, along wth oygen defct and ecess, s shown below. Fgure 3-7. Schematc presentaton of the concentraton of oygen pont defects and electronc defects as a functon of oygen pressure. ygen defects are assumed to predomnate at stochometrc composton. From the dagrams we can conclude that at low oygen acttes the ode has oygen defcency and wll be an electronc n-type conductor, because the moblty of the electrons s always much hgher than that of oygen acances. At hgh oygen partal pressures the ode wll correspondngly hae oygen ecess and be a p-type electronc conductor. At ntermedate oygen acttes t may be be a med n- and p-type electronc conductor n the case of ntrnsc electronc dsorder, whle t may ehbt onc or med oncelectronc conducton n the case of anon Frenkel dsorder. Smlar dagrams and analyses may be made for many other combnatons of non-stochometrc and stochometrc defect stuatons of pure (undoped) odes and other onc compounds. 3.8

29 3. Defect equlbra Summary Chemcal reactons for defects can be formulated and treated usng the mass-acton law. As for other chemcal reactons, equlbrum constants can be defned n terms of the acttes of the defects and other speces. Under the normal constrctons we can appromate acttes wth concentratons of defects and partal pressures of gases. The equlbrum constants can also be epressed n terms of the standard Gbbs energy change of the reacton, whch n turn s a functon of the standard entropy and enthalpy changes. We hae seen that the standard entropy change of the standard Gbbs energy change contans only bratonal terms of the standard states proded that the confguratonal terms are properly handled n epressng the equlbrum condtons. Ths s often possble n defect dealsed defect chemstry snce we can apply classcal statstcal thermodynamcs. Howeer, for non-classcal defects such as delocalsed electrons and holes ths approach s less meanngful, and the entropy change may then be nterpreted n a less straghtforward manner. We hae ntroduced the use of a prme (as n ) to denote equlbrum constants that are not epressed accordng to statstcal thermodynamcs and where the entropy change n the Gbbs energy change may hae other than bratonal contrbutons. The mass acton epressons can be combned wth the full or lmted cases of the electroneutralty condton to obtan eact or appromate (lmtng case) epressons for the concentraton of defects. Such concentratons are typcally a functon of the oygen partal pressure and temperature. It s common to llustrate defect structures for odes by plottng log defect concentratons s log p (Brouwer dagrams) or ln or log defect concentratons s 1T (an t Hoff plots). We hae shown these prncples and technques through eamples of ntrnsc onc and electronc dsorder, arous types of nonstochometry and arable onsaton of pont defects. We hae restrcted our treatment to smple cases and made smplfcatons where possble. Assumpton of small defect concentratons hae allowed the assumptons that the concentratons of normal lattce atoms and empty ntersttals stes are constant, wth acttes equal to unty. (Larger defect concentratons can to a frst appromaton be taken nto account by ncludng mass and ste balances nto the epressons used to sole the defect structure.) Lterature röger, F.A. (1964) The Chemstry of Imperfect Crystals, North-Holland, Amsterdam, and Wley, New York röger, F.A., Steltjes, F.H. and Vnk, H.J. (1959), Phlps Res. Rept. 14, 3.9