Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 2 Part 9 potx

26.2 Point Defects in Ionic Crystals 453 1. In undoped crystals with Schottky disorder, charge neutrality requires equal site fractions of cation and anion vacancies. Then C V C (0) = C V A (0) = exp  S SP 2k B  exp  − H SP 2k B T  =  K SP (T ) . (26.3) 2. In undoped crystals with Frenkel disorder, charge neutrality requires equal numbers of vacancies and interstitials. For Frenkel disorder in the cation sublattice we have C V C (0) = C I C (0) = exp  S FP 2k B  exp  − H FP 2k B T  =  K FP (T ) . (26.4) In Eqs. (26.3) and (26.4) the zero in the arguments of the site fractions refers to intrinsic conditions. Frenkel disorder in the anion sublattice can be treated in an analogous way. 3. If the formation enthalpies of Schottky and Frenkel pairs are similar, cation vacancies, anion vacancies, and cation interstitials will be present. The relative concentrations of the various defects must then be deter- mined by satisfying Eq. (26.1) and Eq. (26.2) simultaneously, and in addition the condition of charge neutrality which requires that C V C = C V A + C I C , (26.5) provided all ions have the same charge. 4. Heterovalent doping of ionic crystals means that some of the matrix ions are replaced by foreign ions of different valence. For example, if Ca 2+ ions replace a few Na + ions in NaCl, an equal number of cation vacancies must be added to preserve charge neutrality. Thus, the equilibrium con- centration of defects is determined by the arrangement which maintains charge neutrality and at the same time satisfies Eqs. (26.1) or (26.2) (see below). Vacancy-pairs: Relative to the perfect lattice cation and anion vacancies bear opposite charges and attract each other by Coulomb forces. When two Table 26.1. Dominant type of disorder in ionic crystals Material Structure Dominant disorder Alkali halides (NaCl, KCl, ) NaCl Schottky Alkaline earth oxides NaCl Schottky AgCl, AgBr NaCl Cation Frenkel CsCl, TlCl, TlBr CsCl Schottky BeO, MgO, . . . Wurtzite Schottky Alkaline earth fluorites, CeO 2 ,ThO 2 CaF 2 Anion Frenkel 454 26 Ionic Crystals vacancies form a nearest-neighbour pair, V P , in addition to Coulomb energy, binding energy is also gained through lattice relaxation. It is then convenient to consider the pair as a distinct defect, which is formed by the reaction of a negatively charged cation vacancy, V − C , and a positively charged anion vacancy, V + A , according to 1 V − C + V + A  V P . (26.6) The vacancy-pair (Fig. 26.2) is a neutral defect and the concentration of pairs is given by C P = C V C C V A Z P exp  ∆G P k B T  = Z P exp  − G SP − ∆G P k B T  , (26.7) where Z P is the number of distinct orientations of the pair (Z P =6inthe NaCl structure). ∆G P is the Gibbs free energy released when two isolated vacancies form a pair. Since vacancy-pairs are neutral entities, their motion does not contribute to ionic conduction. However, pairs contribute to diffu- sion. 26.2.2 Extrinsic Defects Doping Effects: In ionic crystals, additional defects are created by intro- ducing aliovalent impurities (dopants), which differ in charge from the host ions. For example, if a divalent cation impurity i is substituted in the cation sublattice of a NaCl crystal, a cation vacancy must be present to compensate for the excess positive charge of the impurity. Depending on the temperature, the additional cation vacancy will be either isolated or, at low temperatures, associated with the impurity (Fig. 26.2). The effect of aliovalent impurities on conductivity and diffusion of ionic solids has been the subject of many studies. It provides a powerful tool for investigating the types and relative mobilities of defects. The power of the technique comes from the fact that there are definite relations between the doping concentration and the defect concentrations. In practice, doping often means doping with divalent cations because it is easier from an experimental point of view than doping with aliovalent anions. This is a consequence of the higher solubility of divalent cations as compared to divalent anions in halide crystals. Let us consider explicitly an alkali halide (e.g., NaCl) doped with a di- valent alkaline earth halide (e.g., CaCl 2 or SrCl 2 ). In this case, the divalent cation is accompanied by one extra cation vacancy V C (Fig. 26.2). The re- lation between the atomic fractions of impurities, C i , and those of defects is then 1 In the physico-chemical literature the so-called Kr¨oger-Vink notation is used. In the present case this would correspond to: V − C ≡ V  C and V + A ≡ V  A . 26.2 Point Defects in Ionic Crystals 455 C i + C V A = C V C . (26.8) As a consequence, the defect concentration is varied by the impurity concen- tration. In an undoped NaCl-type crystal, according to Eq. (26.3) the con- centrations of cation vacancies, C V C (0), and anion vacancies, C V A (0), must be equal. If divalent impurities are added, the concentrations of cation and anion vacancies are different, but the Schottky product, Eq. (26.1), is still fulfilled. Using Eq. (26.8) to substitute for C V A yields C V C (C V C − C i )=exp  − G SP k B T  = C 2 V C (0) = C 2 V A (0) . (26.9) Equation (26.9) assumes as a useful approximation that the impurities and the defects are all distributed at random. It can be rewritten to give a quadratic equation in C V C . The physically acceptable root is C V C = C i 2  1+  1+ 4C 2 V C (0) C 2 i  1/2  . (26.10) This equation simplifies in two limiting cases: 1. Intrinsic region: In this region we have C V C (0)  C i .Then, C V C ≈ C V C (0) . (26.11) This case applies for undoped material or, since C V C (0) increases ex- ponentially with temperature, it may also apply to doped materials at elevated temperatures. 2. Extrinsic region: In this region we have C V C (0)  C i . Then, C V C ≈ C i . (26.12) Hence the total concentration of cation vacancies is fixed by the dopant concentration. Since C V C (0) decreases exponentially with decreasing tem- perature, and since no material is absolutely free from multivalent impu- rities, there is always a temperature below which C V C (0)  C i is fulfilled. Impurity-vacancy Complexes: At very low temperatures impurity atoms and cation vacancies form impurity-vacancy complexes. A divalent impurity attracts vacancies bearing the opposite effective charge. When these two de- fects are in their most stable configuration, they are considered as a separate defect, an associated impurity-vacancy pair (iV C ). Unless the impurity is small compared to the size of the ion for which it substitutes, the most stable configuration is a nearest-neighbour configuration. If it is small, the next- nearest-neighbour configuration is more stable [18, 19]. The association and dissociation reaction can be written as i + + V − C  iV C . (26.13) 456 26 Ionic Crystals The divalent impurity ion (i + ) provides an excess of one unit of positive charge relative to the perfect crystal, while the cation vacancy (V − C )–amiss- ing cation – introduces one unit of negative charge. Neutral impurity-vacancy complexes form by association. In the extrinsic region, thermal vacancies can be neglected and the total vacancy concentration equals the analytical doping concentration C i . The concentration of free vacancies is C V − C = C i − C iV C = C i (1 − p) , (26.14) where C iV C is the concentration of complexes and p ≡ C iV C C i (26.15) is an abbreviation for the fraction of impurity ions associated with a vacancy. p is called the degree of association. The site fraction of unpaired impurities, i + ,isC i − C iV C . Hence the law of mass action for Eq. (26.13) yields C iV C (C i − C iV C )(C i − C iV C ) = Z iV C exp  ∆G iV C k B T  (26.16) or p (1 − p) 2 = C i Z iV C exp  ∆G iV C k B T  . (26.17) Z iV C denotes the number of distinct orientations of the iV C complexes (Z iV C = 12 for nearest-neighbour complexes; Z iV C = 6 for next-nearest neighbour complexes in the NaCl structure). ∆G iV C is the Gibbs energy of association. 26.3 Methods for the Study of Defect and Transport Properties Defect properties in ionic crystals are usually infered from studies of the ionic conductivity, tracer diffusion of cations and anions, isotope effect, hetero- diffusion, electromigration, dielectric and anelastic relaxation. In Table 26.2 we have listed some of the experiments that have been or can be made on ionic crystals to provide information about defect properties. The experimental procedures for the measurement of tracer diffusion, isotope effect, and dc conductivity are discussed in Chaps. 9, 13, and 16, respectively, and need not be repeated here. Tracer techniques can also be applied to study diffusion in an electric field (electromigration). In an experiment suggested by Chemla [20] one measures, at the same time, the diffusion coefficient from the spreading-out of the tracer distribution (Gaussian solution for a thin tracer layer) as well as the mobility from the drift of the Gaussian due in an electric field. The 26.3 Methods for the Study of Defect and Transport Properties 457 Table 26.2. Experimental techniques for the determination of transport properties in ionic crystals. For definition of the symbols see Sect. 26.4 Type of Measurement Properties obtained Conductivity of pure crystal σ = σ C + σ A ; D σ Conductivity of crystal doped with heterovalent cations Mobility of cation vacancies; H M V C Conductivity of crystal doped with heterovalent anions Mobility of anion vacancies; H M V A Cation tracer diffusion in pure crystal D ∗ C (total)=D ∗ C + D ∗ CP Anion tracer diffusion in pure crystal D ∗ A (total)=D ∗ A + D ∗ AP Cation tracer isotope effect Information on D ∗ CP Anion tracer isotope effect Information on D ∗ AP Cation tracer diffusion in electric field Information on D ∗ C Anion tracer diffusion in electric field Information on D ∗ A Impurity diffusion in cation sublattice Information on defect complexes Impurity diffusion in anion sublattice Information on defect complexes Dielectric relaxation Reorientation of defect complexes Anelastic relaxation Reorientation of defect complexes free vacancies drift along the field because of their electric charge, thereby giving an independent measure of their mobility, whereas vacancy pairs are unaffected by the field since they are neutral. Unfortunately, the method is technically very difficult. In order to maintain an electric field in an ionic crystal, a steady dc current must pass through the electrodes. This often causes deteriorations at the crystal-electrode contact interface. Nevertheless, electromigration measurements have been made for a few ionic crystals and have furthered our knowledge about these materials. Complexes of aliovalent impurities and vacancies possess electric and me- chanical dipole moments. Such entities profoundly affect many of the prop- erties of an ionic crystal, including electrical as well as mechanical ones. Methods of dielectric and anelastic relaxation (see Chap. 14) may be used to observe the reorientation of the complex in an externally applied electric or stress field, respectively. These measurements give direct information about the kinetics of the reorientation process. Combining this information with in- formation on the kinetics of migration from diffusion measurements provides a rather complete picture of the various ionic motions processes, which take place in the presence of aliovalent impurities. A review of this topic has been given by Nowick [13]. 458 26 Ionic Crystals 26.4 Alkali Halides 26.4.1 Defect Motion, Tracer Self-diffusion, and Ionic Conduction The physical interpretation of numerous experimental studies about ionic conductivity, tracer self-diffusion of cations and anions, isotope effect, elec- tromigration, impurity diffusion, and relaxation performed on alkali halides requires a model of the perfect crystal perturbed mainly by the four kinds of point defects shown in Fig. 26.2: Schottky defects, heterovalent impurity ions, impurity-vacancy complexes, and vacancy-pairs. Most frequently, defects are observed not as a consequence of their static properties, but because of their mobility. Tracer Self-diffusion: Diffusion mainly occurs by exchange jumps of ions with vacancies: alkali ions exchange with cation vacancies, halide ions ex- change with anion vacancies. Neglecting for the moment contributions of vacancy-pairs, the tracer self-diffusion coefficients of cations and of anions, D ∗ C and D ∗ A ,aregivenby D ∗ C = f V a 2 C V C ω V C , (26.18) D ∗ A = f V a 2 C V A ω V A . (26.19) f V is the correlation factor for the vacacy mechanism (see Chap. 7) and a the lattice parameter 2 . The jump rate of point defects has been discussed in Chap. 5. Derivations based either on absolute rate theory or on many-body theory of equilibrium statistics have resulted in an expression which can be written for the jump rate of cation vacancies, ω V C ,as ω V C = ν 0 C exp  − G M V C k B T  = ν 0 C exp  S M V C k B  exp  − H M V C k B T  (26.20) and for the jump rate of anion vacancies, ω V A ,as ω V A = ν 0 A exp  − G M V A k B T  = ν 0 A exp  S M V A k B  exp  − H M V A k B T  . (26.21) ν 0 C and ν 0 A are the attempt frequencies for cation and anion jumps, G M V C and G M V A Gibbs free energies, S M V C ,S M V A and H M V C ,H M V A the corresponding entropies and enthalpies of vacancy migration (superscript M), respectively. The vacancy fractions in Eqs. (26.18) and (26.19) depend on the purity of the material and on its temperature: In the intrinsic region,whereC V C = C V C (0) = C V A (0) = C V A , Eqs. (26.18) and (26.19) can be rewritten as 2 Sometimes in the literature, the cation-anion distance is denoted by a.Then a factor of 4 must be included in Eqs. (26.18) and (26.19). 26.4 Alkali Halides 459 D ∗ C = f V a 2 ν 0 C exp  S SP /2+S M V C k B     D 0 C exp  − H SP /2+H M V C k B T  , (26.22) D ∗ A = f V a 2 ν 0 A exp  S SP /2+S M V A k B     D 0 A exp  − H SP /2+H M V A k B T  . (26.23) In this region, the variation of D ∗ C and D ∗ A with temperature stems from the fact that both the vacancy site fractions and the jump rates are Arrhenius activated. D 0 C and D 0 A are pre-exponential factors. The tracer diffusivities and the conductivity are independent of the purity of the specimen so that they are ‘intrinsic’ properties of the material. Apart from vacancy-pair corrections, the relevant activation enthalpies of self-diffusion of cations and anions, ∆H C and ∆H A ,are: ∆H C = H SP 2 + H M V C , (26.24) ∆H A = H SP 2 + H M V A . (26.25) In alkali halide crystals, cation vacancies are more mobile than anion va- cancies, i.e. ω V C >ω V A . This is mainly a consequence of different sizes of the ions. Therefore, although both types of vacancies are present in equal numbers, the cation diffusivity in the intrinsic region is faster than the anion diffusivity: D ∗ C >D ∗ A . (26.26) The difference is, however, not very large since it is a consequence of different defect mobilities only. In the extrinsic region with divalent cation dopants, the concentration of cation vacancies is fixed, i.e. C V C ≈ C i . The equation for D ∗ C then becomes D ∗ C = f V a 2 C i ω V C = f V a 2 C i ν 0 C exp  S M V C k B     D 0  C exp  − H M V C k B T  . (26.27) In Eq. (26.27) we have assumed that the cation vacancies are not associ- ated with impurities. This is justified for not too low temperatures. The pre- exponential factor D 0  C in Eq. (26.27) is much smaller than D 0 C in Eq. (26.22) and it is proportional to the dopant concentration C i . The anion vacancy concentration, C V A , is suppressed, since the Schottky product Eq. (26.1) de- mands that C V A is inversely proportional to the concentration of heterova- lent cation impurities. For doping with divalent anion impurities, an equation analogous to Eq. (26.27) holds for anion diffusion. 460 26 Ionic Crystals The equations developed above cannot completely account for the experi- mental findings. The discrepancy is due to the existence of neutral vacancy- pairs. These contribute to self-diffusion but not to the ionic conductivity. The total tracer diffusivities, D ∗ C (total)andD ∗ A (total), are sums of the diffu- sivities mediated by isolated vacancies and by vacancy-pairs, D ∗ CP and D ∗ AP : D ∗ C (total)=D ∗ C + D ∗ CP , (26.28) D ∗ A (total)=D ∗ A + D ∗ AP . (26.29) Interactions between cation and anion vacancies become appreciable at high vacancy concentrations. In thermal equilibrium the population of cation and anion vacancies increases with temperature according to Eq. (26.7). Thus, vacancy-pair formation in equilibrium is relevant in particular at high tem- peratures. Concomitantly, the pair contributions to tracer self-diffusion, D ∗ CP and D ∗ AP , increase with increasing temperature. They can be written as [11]: D ∗ CP =2a 2 f CP ω CP C P , (26.30) D ∗ AP =2a 2 f AP ω AP C P . (26.31) These expressions contain rates for exchange jumps with cation or anion va- cancies of the pair, ω CP and ω AP , and the relevant correlation factors for cation or anion tracer diffusion, f CP and f AP . Note that the correlation factors are different for cations and anions and different from those of iso- lated vacancies. As the lifetime of the vacancy pair is long compared to the time between atomic jumps, the pair motion involves both cation and anion jumps. The correlation factors are thus functions of the ratio ω CP /ω AP .More explicitly, the pair contributions can be written as D ∗ CP =2a 2 f CP ν 0 CP Z P exp  − G SP − ∆G P + G M CP k B T  , (26.32) D ∗ AP =2a 2 f AP ν 0 AP Z P exp  − G SP − ∆G P + G M AP k B T  . (26.33) ν 0 CP and ν 0 AP denote attempt frequencies, whilst G M CP and G M AP are Gibbs energies of migration for cation and anion jumps of the pair, respectively. Ionic Conductivity: When a solid is placed in an electrical circuit which maintains a voltage across it, a force is exerted on charged particles. In metals and semiconductors essentially all of the current is carried by the electrons. In ionic solids at elevated temperatures, electricity is conducted through the solid by the motion of ions. Cation and anion defects move so as to let current flow in the external circuit. To derive an equation relating the conductivity and the diffusion coefficient, it is necessary to account for the force exerted on the ions by the electric field. This reasoning led us to the Nernst-Einstein relation in Chap. 11. The total ionic dc conductivity, σ dc , of an alkali halide 26.4 Alkali Halides 461 crystal is the sum of the contributions from cation and anion vacancies and can be written as: σ dc = σ C + σ A . (26.34) σ C and σ A denote the partial conductivities of the cation and the anion sublattice, respectively. For ionic conductors, conductivity and diffusivity are related since the same ionic species are involved in charge and mass transport. As discussed in Chap. 11, the so-called charge (or conductivity) diffusion coefficient of ions, D σ , is introduced. We then ascribe partial conductivities, σ C and σ A , to partial charge diffusivities, D σC and D σA ,via D σC ≡ k B Tσ C N C q 2 and D σA ≡ k B Tσ A N A q 2 , (26.35) where N C and N A are the number densities of cations and anions, and q the charge of the ions. For alkali halides, we have N C = N A = N ion and we obtain a total charge diffusion coefficient via D σ ≡ D σC + D σA = k B Tσ dc N ion q 2 . (26.36) D σ has the dimensions of a diffusion coefficient. However, it does not strictly correspond to any diffusion coefficient that can be measured by way of Fick’s laws (see the remarks in Chap. 11). A schematic Arrhenius plot of transport phenomena in an alkali halide crystal is shown in Fig. 26.3. In the extrinsic region, the number of vacancies is dominated by the impurity level and approaches the doping concentration. Then, the temperature dependence of σ dc arises mainly from the migration enthalpy of the corresponding vacancies. These are cation (anion) vacancies in material doped with heterovalent cations (anions). In the intrinsic region, the tracer diffusivities of cations and anions are described by Eqs. (26.28) and (26.29), and the total charge diffusion by Eq. (26.36). Experimental data indi- cate that the behaviour in Fig. 26.3 is still somewhat idealised. For example, at low temperatures the conductivity deviates downward from the extrinsic line (see Fig.26.4). This is a consequence of the formation of impurity-vacancy complexes. Haven Ratio: The Haven ratios for cations and anions, H R C and H R A , relate tracer and charge diffusivities of both sublattices via: H R C ≡ D ∗ C (total) D σC and H R A ≡ D ∗ A (total) D σA . (26.37) Let us for the moment suppose that tracer diffusion and ionic conduction occur by isolated cation and anion vacancies. Then both Haven ratios are equal to the correlation factor of vacancy diffusion. For an fcc lattice we have f V =0.781. We then would arrive at 462 26 Ionic Crystals Fig. 26.3. Schematic diagram of charge diffusivity, D σ , and tracer diffusivities of anions and cations, D ∗ A and D ∗ C , in alkali halides. Parallel lines in the extrinsic region correspond to different doping contents C i D ∗ C + D ∗ A = f V (D σC + D σA ) ≡ f V D σ = f V k B Tσ dc C ion q 2 . (26.38) This equation is based on the assumption that only isolated cation and anion vacancies mediate diffusion. If vacancy-pairs act as additional vehicles of dif- fusion, deviations from Eq. (26.38) occur. Vacancy-pairs are neutral and do not contribute to the conductivity, but make contributions to tracer diffusion of both ionic species. 26.4.2 Example NaCl The amount of experimental data available on transport properties of alkali halides and especially of NaCl is very large. Several reviews supply catalogues of experimental values for transport properties of alkali halide crystals [12– 16, 21]. Rather than attempt to update these reviews, we illustrate some of the main features by presenting typical results for NaCl. Conductivity and Self-diffusion: Figure 26.4 shows the conductivity of a NaCl single crystal doped with divalent Sr 2+ ions according to Beni ` ere et al. [22]. An intrinsic and an extrinsic region of ion conduction can clearly be distinguished. The dashed lines represent extrapolations of the intrinsic and extrinsic parts and show where deviations occur. The slope of the intrin- sic part (above about 650 ◦ C) corresponds to an activation enthalpy, which equals half of the formation enthalpy of Schottky pairs plus the migration enthalpies of cation vacancies (see Eq. 26.24). The slope of the linear ex- trinsic part reflects the migration enthalpy of cation vacancies. 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Burton (Eds.), Academic Press, 197 5, p.381 A.L Laskar, Proc of the NATO ASI Series on Diffusion in Materials, A.S Laskar, J.L Bocquet, G Brebec, C Monty (Eds.), Kluwer Academic Publishers, 199 0, p 4 59 A.L Laskar, Diffusion in Ionic Solids: Unsolved Problems, in: Diffusion in Solids – Unsolved Problems, G.E Murch (Ed.), Trans Tech Publications, Z¨rich, Switzerland, 19 92 , p 20 7 u F Beni`re, Diffusion in Alkali... Vol X, 197 0 R.G Fuller, Ionic Conductivity (Including Self-Diffusion), in: Point Defects in Solids, J M Crawford Jr., L.M Slifkin (Eds.), Plenum Press, 19 72, p 103 A.S Nowick, Defect Mobilities in Ionic Crystals Containing Aliovalent Ions, in: Point Defects in Solids, J.M Crawford Jr., L.M Slifkin (Eds.), Plenum Press, 19 72, p 151 W.J Fredericks, Diffusion in Alkali Halides, in: Diffusion in Solids –... by the brilliant and perceptive English scientist Michael Faraday (1 791 –1867) He reported in 1833 that Ag2 S conducted electricity in the solid state and observed that hot PbF2 also conducted electricity [1] Around 190 0, the German Nobel laureate in chemistry of 1 92 0 Walther Nernst (1864– 194 1) observed that mixed oxides of ZrO2 and Y2 O3 glowed white-hot, when a current was passed through them at high . finds D 2 = 8 3 a 2 ω 2 f 2 p 1+p , (26 .43) where p is the degree of association given by Eq. (26 .15). When p is unity, the diffusivity reaches a saturation value D 2 (sat)= 4 3 a 2 ω 2 f 2 (26 .44) and. f V a 2 ν 0 C exp  S SP /2+ S M V C k B     D 0 C exp  − H SP /2+ H M V C k B T  , (26 .22 ) D ∗ A = f V a 2 ν 0 A exp  S SP /2+ S M V A k B     D 0 A exp  − H SP /2+ H M V A k B T  . (26 .23 ) In this. Electromigration: The isotope effect (see Chap. 9) of the radioisotope pair 22 Na/ 24 Na has been measured on single crystals of NaCl between 5 89 and 796 ◦ CbyRothman et al. [ 29 ] and by Barr and Le Claire [30].