TRANSPORT ACROSS BOUNDARY LAYERS IN IONIC CRYSTALS PART II: STATIONARY CHEMICAL DIFFUSION

PII: S0022-3697(98)00065-1

Pergamon

J. Phys. Chem Solids Vol 59, No. 9, pp. 1555–1569, 1998 0022-3697/98/$ - see front matter 䉷 1998 Elsevier Science Ltd. All rights reserved

TRANSPORT ACROSS BOUNDARY LAYERS IN IONIC CRYSTALS PART II: STATIONARY CHEMICAL DIFFUSION J. JAMNIK†* and J. MAIER Max-Planck Institute fu¨r Festko¨rperforschung, Heisenbergstrabe 1, 70569 Stuttgart, Germany (Received 23 September 1997; accepted 17 April 1998) Abstract—Defect concentration and charge density profiles are derived for a mixed conductor MO 1þd with two carriers under a small chemical potential gradient. The description starts from the formulation of the flux equations for the diffusing species of relevance (e.g. O 2¹ and 2e ¹ in an oxide), takes account of the Poisson equation and is restricted to dilute systems and linear regime. Several examples are discussed in detail: (i) it is shown analytically that chemical diffusion involving carriers with different diffusion coefficients leads to a kinetic build-up of subsurface space charges; (ii) the influence of sluggish surface reactions on the steady state flux and the concentrations profiles is studied on both a ‘‘black box’’ and a mechanistic level; (iii) the impact of different prototype grain boundaries (amorphous blocking core, depletion of carriers in adjacent space charge layers) on chemical diffusion is treated in detail numerically. 䉷 1998 Elsevier Science Ltd. All rights reserved Keywords: interfaces, D. diffusion, D. defects

1. INTRODUCTION

Chemical diffusion is the fundamental process which describes the diffusion part of compositional changes. It is a major step in chemical reactions, sintering, transport of a component through electrodes or chemical filters. It is also involved in electrochemical polarisation or impedance experiments using electrodes which are reversible for electrons but not reversible for ions or vice versa. Another application of general importance is the kinetics of coulometric titration. The concept of chemical diffusion has been well worked out for the electroneutral bulk. Wagner [1] has opened the field in this regard. Further, contributions are due to Yokota [2], Heyne [3], Dudley and Steele [4], Schmalzried et al. [5] and others. Maier [6] generalised the concept to materials in which internal reversible reactions take place, e.g. to internal valence changes and reversible trapping reactions being in local equilibrium. Such source and sink effects will be neglected in the present context. Equilibrium space charge effects in solid ionic and mixed conductors have been considered by different authors. The influence on the ionic conductivity has been studied in detail in Ref. [7]. Jamnik et al. [8] discussed the boundary defect chemistry by taking explicit account of spatial and structural differences of core and space charge regions. This core–space charge model is used as a starting point for the transport considerations set out in the following. *Corresponding author. Tel: 0711 689-1730; Fax: 0711 6891722 †On leave from the National Institute of Chemistry, Hajdrihova 19, SI-1000 Ljubljana, Slovenia.

Adarnczyk and Nowotny [9] tackled the problem of chemical diffusion through subsurface space charge layers, although still assuming that the deviations from equilibrium do not lead to additional charging. On the other hand Lyubornirsky et al. [10] took Poisson’s equation into account in their modelling of chemical impurity diffusion through p–n junctions in semiconductors, but neglected the influence of electrons. This approximation is very good for materials exhibiting a high density of electronic carriers, but it does not apply to mixed conductors in general. Nachev and Balkanski [11] used a lattice–gas model to take account of structural changes at boundaries in single carrier systems. In this paper, we will calculate the steady state of a chemical diffusion process through a crystal with one ionic and one electronic carrier. The crystal can be inhomogeneous regarding the underlying structure (cores of grain boundaries and surfaces) and/or with respect to point defect concentrations (space charge regions). Two deficiencies of the traditional approach will be remedied: (i) the treatment takes full account of Poisson’s equation and does not presuppose local electroneutrality; and (ii) the transport of both carriers is considered on a discrete level, which allows for structural inhomogeneities on the atomic scale. The use of the Boltzmann ansatz restricts the consideration to dilute defects, but makes the treatment feasible. The assumption of linear transport equations is another restriction. Hence, only small changes of the component partial pressure from their equilibrium (reference) values are addressed. In the first part of the present paper (Section 2) a discrete model, based on Part 1 (Ref. [12]), considering

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J. JAMNIK AND J. MAIER

2.1. Basic equations We study the transport in an oxide bicrystal of composition MO 1þd, extending from x ¼ ¹ L to x ¼ L. We assume a laterally homogeneous grain boundary and surface properties such that the system is effectively one-dimensional (Fig. 1). Following the procedure given in Ref. [12] we divide the boundary regions (MO 1þd –MO 1þd and MO 1þd –O 2) into the boundary cores and the adjacent space charge layers. The core structure is allowed to be different from the otherwise homogeneous MO 1þd crystal. The equilibrium space charge regions exhibit the ideal MO 1þd crystal structure

Jj ¼ k~j, i cj, i ¹kj, i þ 1 cj, i þ 1

(1)

k~j;i and kj;i þ 1 being the electric-potential-dependent kinetic rate parameters for the transition of carriers from the ith into the (i þ 1)th layer and backward, respectively [Fig. 2(c)]. Note that in the steady state, the fluxes J j are position-independent. Since we consider the transport in a linear regime, it holds for the deviation, dq, of the relevant quantity, q, from the equilibrium value q e: dq p q e. Therefore, the influence of the electric potential difference D if ⬅ f iþ1 ¹ f i on the rate constants can be approximated by k~j, i ⬇ k~j,e i (1 ¹ zj eDi df=2kB T) and by an analogous expression for k j,iþ1 [12]. The symbol z j denotes the charge number of the carrier j (z j can be positive or negative), the symbol, e, the absolute value of the electronic charge, and the symbols k B and T the Boltzmann constant and the temperature, respectively. We also write the concentrations as sums of equilibrium quantities and perturbations. Neglecting higher order terms and taking into account, that the equilibrium flux vanishes, viz. k~j,e i cej, i ⬅ kj,e i þ 1 cej, i þ 1 , we obtain: ! " # dcj, i þ 1 dcj, i zj e e e ~ (df þ ¹ e ¹ dfi ) : dJj ¼ ¹ kj, i cj, i cej, i þ 1 cj, i kB T i þ 1 ~

2. DESCRIPTION OF THE MODEL

apart from differences in the point defect concentrations (i.e. same standard chemical potentials m o). We divide the crystal into layers stacked parallel to the surfaces [Fig. 2(a)]. Their thickness, s, is taken to be the average jump length of charge carriers. For the sake of simplicity we assume that the core thicknesses are equal to s as well and that the crystal thickness 2L is a multiple of s: 2L ¼ ns (n being an integer). Thus, the core of the grain boundary is entirely comprised by a single layer [Fig. 2(a)]. Each layer, i, is characterised by a mean density of ionic and electronic carriers, c j,i (defined per unit area rather than volume) [Fig. 2(b)]. The subscript, j, denotes here and hereafter the type of the carriers (j ¼ 1 for ionic defects and j ¼ 2 for electronic defects). Whenever the chemistry is specified, we identify j ¼ 1 with oxygen interstitials, O⬙i , and j ¼ 2 with holes, h˙, respectively. Each layer is also described by a mean electric potential, f i, and by mean carrier chemical potentials, m j,i. The discrete electric field, E i, and the carrier flux densities, J j,i, are defined at inter-layer sites as shown in Fig. 2(b). We describe the steady state net flux between two neighbouring layers phenomenologically according to: ~

the transport of ionic and electronic defects which are coupled by the Poisson equation will be outlined. The results are grouped into Sections 3–5. In Section 3, two general transport coefficients are derived analytically. They relate the steady state component flux and the terminal voltage to the partial pressure ratio at two different sides of the crystal. In Section 4, the impact of the surface properties on the stationary chemical diffusion is discussed. We show, analytically, that due to different carrier (ions and electrons) mobilities, subsurface space charge layers appear as a result of chemical polarisation, even if we start from a space charge free, initial situation. Numerical results address the charging of the surface core and show the influence of slow surface reaction steps on the defect distributions as well as on chemical and electrical potential profiles. In Section 5, numerical results regarding the chemical diffusion through a grain boundary are discussed. Two simple cases are treated separately: (i) a core with reduced ionic mobility (e.g. a very thin glassy phase separating the neighbouring grains, Section 5.1); and (ii) a core with enhanced ionic and electronic defect concentrations (Section 5.2). Finally, in Section 5.3, the transport through a grain boundary with adjacent equilibrium space charge layers (e.g. in SrTiO 3) is considered.

~

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~

Fig. 1. Master example as used in the text. Equilibrium concentrations of defects (O⬙i , h˙) in the MO 1þd bicrystal are determined by the equilibrium (initial) partial pressure of oxygen. The changing intensity of the grey colour indicates a positiondependent equilibrium carrier density in space charge and core layers.

(2) The first term in the brackets is obviously the difference of the chemical potential changes which together with the second term, viz. the difference of the electrical potential changes, constitutes the driving force for the flux. The two terms are Xcoupled by Gauss’ equation, ee0 (Ei ¹ Ei ¹ 1 ) ¼ e j zj cj, i , in which e 0 denotes the permittivity of free space, and e the macroscopic dielectric constant of the crystal. This coupling can

Transport across boundary layers

1557

elegantly be taken into account by introducing new variables dE j (‘‘virtual electric fields’’ [12]): ezj dEj, i ¹ dEj, i ¹ 1 ¼ dc (3) ee0 j, i

þ dE 2,i. Since eqn (3) defines dE j’s up to an additive constant, we complete the definition by the boundary condition: (4) (dE1 ¹ dE2 )x ¼ L ¼ CE

the sum of which is of course the electric field dE i ⬅ dE 1,i

with an arbitrary constant C E (see Ref. [12] for details).

Fig. 2. The bicrystal under consideration is divided into layers (a), each of which is characterised by mean carrier concentrations, c j,i, and electrical potential f i (b). Transition of carriers between the layers is described by simple kinetics sketched in Fig. 2(c). The ⫽ thresholds DG⫽ → and DG← depend on the electric field, E i, which can be imposed externally or created internally due to a chemical diffusion process.

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J. JAMNIK AND J. MAIER

After substituting dc j,i in eqn (2) by dE j,i and expressing the potential difference, D idf, by ¹ sdE i, the transport eqn (2) reads: " L2D cej, ⬁ e dEj, i þ 1 ¹ dEj, i 1 dJO ¼ ¹ j jj, i zj evj s cej, i þ 1 #  dEj, i ¹ dEj, i ¹ 1 e ¹ (dE þ dE ) ð5Þ þ j 1, i 2, i : j, i cej, i q The conductivities, jej, i ¼ lzj leuej, i cej, i cej, i þ 1 =s2 , as ⬀ defined in Ref. [12] using the mobility uej, i q  k~e ke , characterise the transition of carriers ~

j, i j, i þ 1

between the ith and (i þ 1)th layers [Fig. 2(b)]. The density of carriers in the region remote from boundaries is the individual Debye lengths are denoted by cej, ⬁ and q defined by LDj ⬅ ee0 kB T=e2 z2j cej, ⬁ . Since, in the case of chemical diffusion, the net electric current has to vanish in the steady state, the individual carrier fluxes, dJ j appearing in eqn (2), can be replaced by the component flux dJ O ⬅ dJ j/n j. The symbols n j denote the stoichiometry coefficients: if dJ O is the flux of oxygen, with dJ 1 and dJ 2 being the fluxes of oxygen interstitials and holes, respectively, then n 1 ¼ 1 and n 2 ¼ 2. Note that S jn jz j ¼ 0. The transport eqn (5) is a difference (differential in continuum picture) equation in which virtual fields are unknowns and the flux, dJ O, appears as a parameter. The overall driving forces, i.e. the perturbations of the component partial pressures at x ¼ ¹ L and at x ¼ L, come into play via boundary conditions as shown in the next section.

2.2. Boundary conditions In equilibrium, the chemical potentials of ionic and electronic defects at the boundaries are related to the

Fig. 3. Sketch of the boundary conditions for the left hand side boundary. Black box level: the terminal quantities, E ext and dm O from one side and E 1 and c j,2 from the other side are connected by boundary conditions eqns (7) and (8). Mechanistic level: spatial positions of individual reaction steps eqns (19a) and (19b), (transfer steps are denoted by jej, 1 ) are indicated.

component partial pressures in the gas phase according to mO(gas) ¼ mO2 ¹ ¹ 2me ¹ . The deviation from this equality acts as a driving force for a component flux, J O, across the boundary. In the linear regime, J O ( ¼ dJ O) is given by: dJO ¼ Ls {dmO(gas) ¹ (dmO2 ¹ ¹ 2dme⬘ )},

(6)

in which L s denotes the overall reactivity of surface reactions. Mechanistically, the surface process may be very complicated. It consists of several reaction pathways, e.g. adsorption of O 2 from the gas phase, dissociation, ionisation and then incorporation into the lattice. However, we do not need to specify the different elementary steps of the surface reaction for the analytical considerations in Sections 3 and 4. Thus, the part of the crystal in which this surface reaction occurs we preliminarily consider as a ‘‘black box’’, the size of which we identify with the size of surface layers just for the purpose of clarity (Fig. 3). In order that eqn (6) acts as an appropriate boundary condition for the transport eqn (5), both perturbations, dmO(gas) and (dmO2 ¹ ¹ 2dme ¹ ), should refer to the terminals of the black box. Hence, the term (dmO2 ¹ ¹ 2dme ¹ ) in eqn (6) describes the component chemical potential in the first layer next to the core (i ¼ 2, see Fig. 3). According to the exemplary defect chemistry which we have chosen (an oxide with interstitials and holes), (dmO2 ¹ ¹ 2dme ¹ ) can be expressed by (dmO⬙i þ 2dmh˙) , i.e. by the chemical potentials of charge carriers relevant for the transport in the rest of the crystal.‡ Since perturbations of defect chemical potentials are proportional to the virtual field differences (see eqn (3), dmj;i ⬀ dcj;i ⬀ dEj;i ¹ dEj;i ¹ 1 Þ , eqn (6) can be immediately recast into the boundary condition for the left hand side boundary: ! ee0 kB T X nj dEj, 2 ¹ dEj, 1 : (7) dJO ¼ Ls dmO(gas) ¹ e cej, 2 j zj (The expression for the right hand side boundary is analogous [12].) Eqn (6), and consequently also the boundary condition (eqn (7)), operate only with the sum dmO⬙ þ 2dmh˙ without i any restriction with respect to the individual quantities, dmO⬙ and 2dmh˙. Nevertheless, these latter quantities deteri mine the overall driving forces for O⬙i and h˙. In conventional treatments, the ‘‘missing link’’ was provided by imposing the local electroneutrality condition, viz. and dcO⬙ þ 2dch˙¼ 0 , which however, is violated within the i boundary regions. In the cases in which the core remains uncharged (for generalisation see Section 4.2) we require, instead, the continuity of the dielectric displacement through the black box, yielding the electrostatic boundary ‡Eqn (5) can in principle be applied even if the type of ionic or electronic defect changes with position. See Ref. [12] for this generalisation.

Transport across boundary layers

1559

Fig. 4. Equivalent circuit for steady state chemical diffusion. Flux through resistors is proportional to the differences of electrochemical potentials. ‘‘Ground’’ refers to the initial (equilibrium) value of the oxygen chemical potential in the surrounding gas.

condition:

obtained, viz.: Eext ¼ e(dE1, i ¼ 1 þ dE2, i ¼ 1 )

for the left hand side of the crystal. The symbol E ext denotes the external electric field which may be applied in a contact-less mode. Owing to overall electroneutrality, the electric field strengths on both sides of the crystal have to be identical. In addition, the virtual fields have to satisfy eqn (4). This yields the boundary conditions for the right hand side of the crystal: 1 eE1, i ¼ n ¼ (Eext þ CE ) 2 1 edE2, i ¼ n ¼ (Eext ¹ CE Þ: 2

(9a) (9b)

3. RESULTS: ANALYTICAL DERIVATION OF TERMINAL QUANTITIES (COMPONENT FLUX, ELECTROCHEMICAL POTENTIAL DIFFERENCES)

Two general coefficients which determine the terminal quantities will be calculated in this section. One is the chemical diffusion resistance, L ¹1, which relates the steady state component flux to the outer difference of , p(l) , , the superthe component partial pressures (p(r) O2 O2 scripts (r) and (l) denote the right and the left hand side of the crystal, respectively):§ JO ¼ L

(r) (l) (dpO ¹ dpO )kB T 2 2 e pO2

(10)

and the parameter U j, which determines the fraction of the overall force driving individually ionic and electronic defects: dm˜ j(r) ¹ dm˜ j(l) ¼ Uj

(r) (l) (dpO ¹ dpO )kB T 2 2 : peO2

nX ¹1

(8)

(11)

Both quantities L and U j are easily measured, the former one, e.g. by a permeation experiment, and the latter one by an electrical measurement.k To calculate L, we multiply both sides of eqn (5) by s=jei, j ⬅ Rj, i , which is the inter-layer resistance for the carrier j, and sum up from i ¼ 2 to i ¼ n ¹ 1. The result

ezj nj JO i¼2

dEj, n ¹ dEj, n ¹ 1 cej, n ! nX ¹1 dEj, 2 ¹ dEj, 1 dEi þ s ¹ cej, 2 i¼2

Rj, i ¼ ¹ L2Dj cej, ⬁

ð12Þ

we multiply with ez jn j, sum over and express the virtual field differences by J O using boundary conditions (eqn (7)). Since S jz jn j ¼ 0, the electric field terms cancel and the solution reads: L ¹ 1 ¼ (ezj nj )2

nX ¹1 i¼2

(R1, i þ R2, i ) þ

1 Ls(l)

þ

1 Ls(r)

(13)

The result formally agrees with the conventional expression for the ambipolar resistance. Note, however, that the R j,i’s and L s’s also refer to space charge regions and core regions. The chemical diffusion resistance, L ¹1, is easily represented by the equivalent circuit shown in Fig. 4. The common feature is that all inter-layer resistances, R j,i, and also the reciprocals of surface reactivities appear in series. This is intuitively expected and can also be derived without use of Poisson’s equation, because the conditions 2JO⬙ ¼ Jh˙ and divJ j ¼ 0 are also valid in boundary regions i (in the steady state!). The summands Ls¹ 1 ’s appear in series since the surface reactions are running in series with the transport. Those R j,i’s which refer to space charge layers and cores, appear in the same way as in the bulk simply because there is no possibility for ionic and electronic defects to be recombined or created within the crystal (or in other words divJ j ¼ 0). On the continuum level, the sums in eqn (13) can be replaced by „ integrals 0L jj¹ 1 dx which have already been solved for specific cases, e.g. Debye or Schottky space charge layers as outlined in Ref. [13]. The resulting bulk and space charge resistances were used to interprete impedance measurements in e.g. AgCl [14] and SrTiO 3 [15]. As §Eqns (10) and (11) are defining for L and U j. These two quantities can, with respect to multiplicative constants, also be defined in a different way. kThe measurement of the terminal voltage using an ionically or electronically reversible probe yields U 1 or U 2, respectively.

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J. JAMNIK AND J. MAIER

shown here, the same resistances, but combined in series, also apply to steady state chemical diffusion in the linear regime.* The same series resistances are of significance for the rate of diffusion-controlled reaction kinetics. The parameter U j, which determines the terminal voltages is derived from eqn (2) as follows: we divide both sides of eqn (2) by k~j,e i cej, i , sum over i and substitute J j by eqn (13). After rearrangement the result reads: 1 Xn R nj i ¼ 2 j, i   (14) Uj ¼ X 1 1 n ¹2 (R þ R ) þ (z en ) þ 1, i 2, i j j i¼2 Ls(l) Ls(r) ⱕ 1, which implies Obviously 兺 jn jU j Dm˜ O⬙ þ 2Dm˜ h˙ ⱕ 1=2DmO2(gas) . As seen from eqn (14), i the equality only holds if the surface reactions are very (r) fast (L(l) s , Ls , → ⬁). Generally, however, this is not the case. Let us assume that the adsorption of oxygen from the gas phase on the crystal surface is very slow. Then a finite part of the overall driving force is spent to drive the slow adsorption process. Yet, for the remaining part of 1=2DmO2(gas) the above ‘‘equality’’ applies. This is also evident from Fig. 4, if Kirchhoff-like laws, referring to electrochemical potential drops over resistances, R j,i and Ls¹ 1 ’s, are applied. The use of L s in eqn (14) presupposes that there is no electric potential drop across the ‘‘black boxes’’ which comprise the overall surface reactions. This assumption can be relaxed if surface reactions are considered mechanistically (see Section 4.2). Then L s is interpreted as the reactivity of a single reaction step (e.g. oxygen ionisation), while the reactivities of the incorporation reactions are covered by extending the sums in eqn (14) from i ¼ 1 to i ¼ n þ 1. The naive picture discussed here is in quantitative agreement with rigorous computations as far as the steady state terminal quantities are considered, but fails in describing the situation inside, e.g. concentration profiles close to the boundaries. This will be shown in the following part of the paper.

component flux has been neglected (the situation in which the core may be charged, will be considered numerically in the next section). In this simple case the crystal is also free of equilibrium space charge layers (‘‘flat level’’). Then, cej, i and jej, i are independent of position (independent of subscript qi). If, in addition,   the total Debye length, X 2 2 e , is greater than s, we may LD ¼ ee0 kB T=e j zj cj, ⬁ rewrite eqn (5) in a continuum form:   1 ⳵2 ¹ L2Dj jej 2 dEj þ jej (dE1 þ dE2 ) (15) JO ¼ zj enj ⳵x in which dE j are continuous functions of the position coordinate x. For the sake of simplicity, we study the chemical diffusion driven by a small increase of the partial pressure at one side and an equally small decrease of the partial (r) ¹ dp(l) pressure at the other side: dpO O2 . Then, the boundary 2 conditions (eqns 9a and 9b) apply to both the left and the right hand side of the system [12]. Introducing the ratios, lj ⬅ LDj =LD (by definition 1=l21 þ 1=l22 ¼ 1), the solution of eqn (15) for dE j(x) is readily obtained [12]. The sum, dE 1 þ dE 2, viz. the electric field perturbation inside of the crystal, reads [Fig. 5(a)]: 0 1 x x cosh cosh B C L L DC D : (16) dE ¼ dEint (JO )B þ Eext @1 ¹ LA L cosh cosh LD LD The first term stems from the chemical diffusion and the

4. RESULTS: CONCENTRATION AND POTENTIAL PROFILES IN THE SINGLE CRYSTAL

4.1. Non-chargeable core: analytical solution Analytical solutions for the concentration profiles in the presence of non-zero component flux are only possible for very simple situations, viz. for a single crystal with atomically smooth surface planes and a sufficiently low defect density in the surface cores.† The latter restriction is needed, since in the boundary conditions considered so far, any charging of the surface core due to a non-vanishing *This is an exact result. In Ref. [9] an approximation was derived neglecting majority carriers. †In this case any charging of boundary regions is very unfavourable (energetical point of view).

Fig. 5. Electric field and concentration profiles for a pure single crystal with ideal (defect free) surface cores. Parameters used in calculations are: z 1 ¼ ¹ 2, z 2 ¼ 1 and ce2 ¼ 2ce1 . The internal electric field which is built-up kinetically stems from different defect diffusion coefficients (we set D1 ¼ 1=40D2 ).

Transport across boundary layers

second one from the external field. The chemical diffusion part starts with a zero value at the surfaces and reaches, after a few L D’s, a constant value of dEint (JO ) ⬅ z1 en1 JO {(l21 je1 ) ¹ 1 ¹ (l22 je2 ) ¹ 1 }, in the bulk of the crystal [Fig. 5(a)]. dE int vanishes if l21 je1 ¼ l22 je2 , or more concisely if D 1 ¼ D 2 (D j’s are defect diffusion coefficients related to mobility by Einstein relations Dj ¼ kB T=Celzj luj ). Differentiation of dE j with respect to x yields, by definition (see eqn (3)), the concentration profiles [Fig. 5(b)]: x e sinh n ez c L j j j D cj (x) ¹ cej ¼ ¹ JO x ¹ (dEint (JO Þ ¹ Eext )LD ˜ L kB T D cosh LD (17) The left hand side of eqn (17) denotes the deviation of the carrier concentration from its equilibrium (reference) value; the right hand side includes two terms: the conventional one which assumes local electroneutrality and a term taking account of space charges. The conventional term is linear in x, just as predicted by Fick’s law. Indeed, ˜: the proportionality constant D 1 ˜ ⬅ kB T D e2

z21 ce1

þ

1 z22 ce2

1 1 þ je1 je2

(18)

is precisely the conventionally derived chemical diffusion coefficient. However, the second term is new, and describes the subsurface space charge layers. These charges are necessary, according to Gauss’ law, for the electric field to change from E ext outside of the crystal to dE int inside. As we see from eqn (17), E ext influences the carrier concentration profiles, but it does not affect the flux (the chemical diffusion resistance L ¹1 is independent of E ext, see eqns (10) and (13); obviously the diffusional and the drift parts of J O change in a compensating manner). This is in agreement with the energy conservation principle: since in the steady state of the chemical diffusion no net electric charge is transported, E ext cannot do any work on the system. Though E ext does not affect the flux in the linear approximation, in the non-linear regime, however, the surface carrier density can be significantly modified by high values of E ext (of the order of 10 6 V cm ¹1), and consequently also the surface reaction rate. Such an influence of E ext on the mass flux would then be a catalytic effect.

4.2. Real core: numerical solutions 4.2.1. Surface reactions on a mechanistic level. In the previous section, we assumed that the surfaces of a single crystal are atomically smooth and nearly free of any point defects. Such an assumption is of course difficult to

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realise experimentally [as atomic force microscopy studies show (see for example [16]), it may appear on a very small scale, e.g. on a surface area of 10 ⫻ 10 nm]. On a larger scale, the surface exhibits a variety of charged point defects: kinks, jogs, surface adatoms, surface vacancies, adsorbed and possibly ionised species from the surrounding gas atmosphere (O 2¹), charged impurities segregated from the bulk of the crystal, etc. In addition there are many frozen-in defects depending on preparation conditions. The individual steps of the surface reaction, 1=2O2(gas) N O2 ¹ ¹ 2e ¹ , depend on the surface defect chemistry. To make the treatment feasible, we assume that adsorption sites are located within the surface core layer in which also the ionisation reaction occurs.* Further, we make no difference between the ionic core defects (O⬙core ) and adsorbed oxygen ions (we distinguish, however, between the standard chemical potentials of point defects in the core layer and in the layers next to it). Then the surface reaction can be split into the adsorption of the oxygen from the gas atmosphere into the core layer and the subsequent ionisation reaction forming two core defects: 1 O N Ocore 2 2(gas)

(19a)

Ocore N O⬙core þ 2h˙core

(19b)

Location at which reaction steps (eqns (19a and b)) occur is indicated in Fig. 3. Without substantial restriction of generality we will always assume that the first step is very fast. Since the second step cannot be affected by electric field (all species are at the same coordinate, see Fig. 3), the reaction rate is simply determined by the ‘‘surface reactivity’’, k s. This constant enters the boundary conditions similarly to L s in eqn (6) (for details see Ref. [12]). On this level, the boundary conditions refer to the defect densities in the surface core directly and not to the first layer next to it (as was the case using the black box approach). Thus, we can now easily verify the influence of the core disorder on the defect concentration profiles caused by the steady state component flux. 4.2.2. Parameters used in numerical calculations Since the transport in the linear regime depends on the equilibrium defect densities, we outline here our corespace charge model [8] of the boundary. We emphasise the main materials parameters† which we will vary in the next sections in order to highlight different cases of interest. Defining the density of available lattice sites for oxygen interstitials, NO⬙ , and the effective density of i hole states, Nh˙, the law of mass action for the oxygen *This may be close to reality if significant surface relaxation occurs. †The choice of these parameters is, of course, not unique.

J. JAMNIK AND J. MAIER

incorporation reaction reads:  ! !2 s e e: ceO⬙ p c O 2(gas) h i ¼ NO⬙ ¹ ceO⬙ Nh˙ ¹ ceh˙ po i

" ⫻ exp

(20)

i

! # 1 o o o ¹ mO⬙ ¹ 2mh˙ =kB T : m i 2 O2(gas)

N j’s are generally different for the bulk and core layers (in the core they may even depend on the sample preparation). Exclusively for simplicity’s sake we set in the calculations NO⬙ , c ¼ Nh˙, c ¼ Nj, ⬁ . Since moO⬙ and m0h˙ are the i i only two quantities in eqn (20) which reflect the structural inhomogeneities within the material we introduce the sum moO⬙ þ 2moh˙ ⬅ go as a parameter.* This materials i

parameter is different for the core (goc ) and the bulk region (go⬁ ). According to eqn (20), goc ⬍ go⬁ implies that in a pure and space charge free material, defect density in the core of the grain boundary is higher than within the grains, and in a material characterised by goc ⬎ go⬁ the opposite is true. The different structure of the core with respect to the bulk usually leads to a defect redistribution, viz. to a build-up of equilibrium space charge layers adjacent to the core. The width of the distribution is characterised by L D and the magnitude of the distribution by the equilibrium space charge potential Dfe ⬅ fec ¹ fe⬁ . If we neglect the configurational entropy of core defects and effects due to impurity segregation, Df e can be expressed as Dfe ⬇ f (moj, ⬁ ¹ moj, c , cej, ⬁ , goc , go⬁ , zj , nj ). The explicit expression for a univalent case, viz. z 1 ¼ 1 ¼ ¹ z 2, is simpler and will be given in Section 5.3. Equilibrium concentration contours in space charge layers and core defect densities are then easily calculated as e.g. shown in Ref. [8]. The decisive material parameters used are cej, ⬁ , goc , go⬁ and Df e. 4.2.3. Numerical results: high defect density on the surface We consider a single crystal with a high equilibrium defect density in surface cores [we set goc ¼ 1=2go⬁ , see Fig. 6(a)] and, for the sake of simplicity, without any equilibrium space charge layers (Df e ¼ 0) and a very fast ionisation reaction (k s → ⬁). We assume that the bulk defect concentrations and diffusion coefficients are the same as in the case considered previously (Fig. 5), viz. ceO⬙ ¼ 1=2ceh˙ and DO⬙ ¼ 1=40Dh˙, and that the thresholds i i ⫽ DG⫽ → and DG← for transition from the bulk into the core are the same as in the bulk, viz. kj,e ic þ 1 ¼kj,e ⬁ . (index ic denotes here and hereafter specifically the core layer and ⬁ refers to bulk). The thresholds for transitions from the core into the bulk are of course higher [see Fig. 6(a), note that cej, core q cej, ⬁ ] and are already determined by the equilibrium densities according to k~j,e ic cej, ic ¼kj,e ic þ 1cej, ic þ 1 . Consequently, the jej, i ’s ( ¼ k~j,e i cej, i ) are, as in the previous case (Fig. 5), position-independent. The comparison of the concentration changes (with respect to the equilibrium level) in Fig. 5(b), Fig. 6(b) shows that in the case of defect-rich cores, the kinetically built-up surface charge is located mainly in the cores and only a small part of it is carried by the subsurface space charge layers; if goc p go⬁ , then all the surface charge is accommodated in the core. In the opposite limit, goc q go⬁ (not shown here) numerical results are very similar to the analytically derived profiles [eqn (17), Fig. 5(b)]. Although the direct experimental data concerning goc =go⬁ are not available, we believe, on the basis of interfacial capacitance measurements in many ionic solids, that ~

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~

~

Fig. 6. Pure single crystal with high defect density in surface cores. (a) Initial situation on microscopic level: goc ¼ 1=2go⬁ . Different carrier diffusion coefficients, D1 ¼ 1=40D2 , are indicated by different barrier heights for O⬙i and h˙. Temperature and oxygen partial pressure are chosen such that L D ¼ 5s (five barriers within one Debye length). Other parameters used in calculations are: z 1 ¼ ¹ 2, z 2 ¼ 1 and ce2 ¼ 2ce1 . Deviations of the carrier concentrations and electric field from initial values are displayed in (b) and (c). In this figure and also in all following figures in which numerical results are present, staircase type plots reflect the discretisation.

*Of course we could also use DGoincorporation , comprising the total argument of the exponential function in eqn (20).

Transport across boundary layers

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Fig. 7. Pure single crystal with slow surface reactions. (a) Initial situation: goc ¼ go⬁ , D 1 ¼ D 2, z 1 ¼ ¹ 2, z 2 ¼ 1 and ce2 ¼ 2ce1 . Ionisation reaction eqn (19b) is parametrised by the ‘‘surface reactivity’’ ks ¼ 10 ¹ 2 je⬁ =ðe2 LD Þ and transfer of oxygen ions between the surface e e ¹2 e k1, ⬁ . Deviations of the electric field, defect concentrations and core and the lattice by the rate constants k~1, ic þ 1 ¼ k1, ic þ 1 ¼ 10 component chemical potential from their equilibrium values are shown in (b)–(d). Staircase type of plot [especially evident in (b)] reflects the discrete nature of numerical computations. The step width equals an average jump length of defects. ~

oxygen ions between the core and the first lattice layer by e e e k~1, ic ¼ k1, ic þ 1 ¼ 1=100k1, ⬁ [Fig. 7(a), upper part]. For electronic defects we assume that the threshold for this exchange is the same as in the bulk [Fig. 7(a), bottom part]. The difficulty of the ion incorporation is reflected by an abrupt change of the ionic concentration between the core and the layer next to it and by forming adjacent space charge layers [Fig. 7(c)]. The corresponding electric field is given in Fig. 7(b). Please note, that in this case, in contrast to the situation displayed in Figs 5 and 6, both surface regions are overall neutral (E ext ¹ dE int ¼ 0). Since an electric field perturbation cannot affect the rate of the surface reaction eqn (19b), this step itself does not distort the linearity of the profiles, but just decreases their slopes.* This is easy to understand considering the component chemical potential profile displayed in Fig. 7(d). The very first and the last steps are due to ionisation reactions. Since we fixed the overall difference (r) (l) ¹ dmO , a slow ionisation reaction reduces the dmO remaining difference dmO li ¼ n þ 1 ¹ dmO li ¼ 1 which drives the flux through the crystal. ~

~

unless very careful surface etching and annealing procedures have been undertaken, the defect density in surface cores is much higher than in the bulk. Then, the conventionally calculated defect concentration profiles for chemical diffusion in single crystals are valid even in a spatially broader range than calculated in the previous section [Fig. 5(b)]; the local electroneutrality condition is, to a good approximation, only broken in the surface cores [Fig. 6(b)]. 4.2.4. Numerical results: slow surface reactions. So far, we have assumed that all surface reactions were fast. In reality, very often the inverse situation is encountered. In the framework of our model three transport steps are involved in the ‘‘overall incorporation’’ of oxygen: (i) the ionisation reaction (eqn (19b)), characterised by the surface reactivity k s; (ii) the exchange of O⬙i and h˙ between the surface core and the layer next to it (here crystal structure changes are involved); and (iii) the transport of both defects through the adjacent space charge layer (the equilibrium defect density changes). To show the influence of (i) and (ii), we calculated the concentration profiles for an impurity-free single crystal (ceO⬙ ¼ 1=2ceh˙) characterised by i DO⬙ ¼ Dh˙, goc ¼ go⬁ [Fig. 7(a)], and vanishing equilibrium i e space charge layers (Df ¼ 0). The ionisation reaction is parametrised by ks ¼ 10 ¹ 2 je⬁ =ðe2 LD Þ and the exchange of

*We expect that a slow oxygen adsorption, viz. the reaction step eqn (19a) which we neglected, would lead to similar effects as step eqn (19b).

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J. JAMNIK AND J. MAIER 5. GRAIN BOUNDARY IN A BICRYSTAL

Although in equilibrium, the grain boundary exhibits similarities to two crystal surfaces pressed back to back, there are two important differences with respect to transport. First, the sum of the chemical potentials of ionic and electronic defects in the core is not fixed by the external component partial pressure. Second, in the core of our grain boundary (as well as in the bulk) there is usually no neutral or simply ionised oxygen. Therefore, the individual fluxes of ionic and electronic defects are not coupled by a chemical reaction such as 1=2O2 N O2 ¹ ¹ 2e ¹ , which would lead to internal sources and sinks. The similarities to the surfaces are reflected by the fact that: (i) the rate constants for transition of defects through the core; and (ii) the concentrations of defects in core and space charge layers may differ from the bulk values. By increasing complexity, we first concentrate (Section 5.1, Section 5.2) on core effects (Df e ¼ 0) and then in Section 5.3, we study the influence and the modification of space charge layers already present in equilibrium. Since, in this section, we want to emphasise phenomena which are more specific to a grain boundary, we omit the previously considered surface effects purposely (we assume that surface reactions are very fast and that DO⬙ ¼ Dh˙).

expect, the behaviour of minority carriers to be determined by Fick’s law: JO ¼ Dj

⳵ dc : ⳵x j

(21)

Indeed Fig. 8(a), Fig. 9(a) show that this is obeyed: in the situation displayed by Fig. 8, the minority carrier diffusion coefficient, Dh˙, is position-independent [Fig. 8(c), bottom part], hence dch˙ is a linear function of x. In Fig. 9, the minority carriers (ions) are impeded by the core. Nonetheless, Fick’s law is still valid to a very good approximation: the slope of the concentration profile dcO⬙ [Fig. 9(a)] is proportional to DO¹⬙ 1 (x); since at x ¼ i i 0 (core location) the flux of O⬙i is impeded [see Fig. 8(c), upper part], dcO⬙ , changes very steeply across the core. i

i

5.1. Blocking amorphous core

*This implicitly means that we consider a doped crystal. However, we omit impurities in the core.

Fig. 8. Doped bicrystal. The flux of majority carriers is impeded by the grain boundary core. Temperature, initial oxygen partial pressure and amount of impurities are such that ce2 ¼ 1=10ce1 and that L D ¼ 5s. Other parameters are: z 1 ¼ ¹ 2, z 2 ¼ 1 and D 1 ¼ D 2. (c) The grain boundary is considered to be an amorphous core with a low permeability for O⬙i . We set e e ~e k~1, ic ¼ k1, ic þ 1 ¼ 1=30k1, ⬁ . Deviations of defect concentrations and component chemical potential from initial values are shown in (a) and (b). ~

Often a second ‘‘impurity’’ phase is present which wets the grain boundary core (interphase). If this is a thin glassy phase, it may well be possible that the flux of ionic defects is impeded by the core, while the electrons or holes can penetrate (e.g. tunnel) through. To highlight this situation we numerically treat the chemical diffusion through a doped bicrystal for two cases. The conditions (dopant level, oxygen partial pressure and temperature) are chosen such that, in the first case, in the bulk, the initial ionic defect concentration prevails over the electronic carrier concentration* (ceO⬙ ¼ 10ceh˙, Fig. 8) and that i in the second case the opposite situation is realised e e (cO⬙ ¼ 1=10ch˙, Fig. 9). In both scenarios, we omit initial i space charges and set in the bulk DO⬙ ¼ Dh˙. We paramei trise the transition through the core by ionic rate constants e e e (mobility) k1, ic ¼ k1, ic þ 1 ¼ 1=30k1, ⬁ the hole mobility being invariant [Fig. 8(c); the core is identified with the region between the icth and (ic þ 1)th layer; here the migration threshold for ions is much higher than in the bulk]. Fig. 8(a), Fig. 9(a) show the numerically calculated deviations of defect concentrations from their equilibrium values as functions of position. Let us adopt for a moment a naive standpoint. The chemical diffusion flux is determined by the carrier with the smallest jej , thus in our two cases by the minority carriers. Then one may

Transport across boundary layers

In contrast to minority carriers the concept of Fick’s law for majority carriers fails completely in boundary regions. The dashed lines in Fig. 8(a), Fig. 9(a) show the hypothetical profiles for majority carriers as calculated from eqn (21). We see that such a solution would lead to an electrically charged bulk (the electric charge is proportional to the difference between ionic and electronic profiles), which is an unacceptable result (the total electrostatic energy of the system would be very high and would increase with the size of the system). Hence, within the bulk the correctly calculated profiles [Fig. 8(a), Fig. 9(a)] follow the behaviour of the minority carriers (to maintain electroneutrality). The formation of boundary space charge layers can be interpreted as follows: in the first case [Fig. 8(a)] in which majority carriers are impeded by the core, space charge layers build up to provide an additional driving force for the majority carriers to overcome the core barrier, and in the second case [Fig. 9(a)] space charge layers allow for a continuous adjustment of the majority carrier density, and dch˙, at one side of the grain boundary to the values at the other side. Fig. 8(b), Fig. 9(b) display the changes of the component chemical potential with position. As easily proven by eqn (2), the well-known expression for the ambipolar

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flux: JO ¼ ¹ (zj enj ) ¹ 2

(22)

is valid also in non-homogenous stationary systems. The ambipolar conductivity (prefactor in eqn (22)) can always be approximated by the smallest jej . In the case considered in Fig. 8, the relevant jej is position-independent, hence the dm O profile is linear [Fig. 8(b)]. In the other case (Fig. 9) the minority carriers are impeded by the core, thus an abrupt step in the dm O profile in the core region [Fig. 9(b)] occurs.

5.2. Grain boundary core with high point defect density In contrast to the previous situation in which the core was characterised by an interlayer region with a high migration threshold and no sites for oxygen ions and electrons, the core in this case denotes a grain boundary containing a high density of these charge carriers. This is especially relevant for the transport along the boundary. Of course such a grain boundary does not represent any barrier to the perpendicular flux. It does, however, strongly influence the carrier density profiles in the boundary region. In order to demonstrate this, we again examine a bicrystal under such conditions that in the bulk ceO⬙ ¼ 1=10ceh˙. We parametrise the core defect chemistry i by goc ¼ 3=4go⬁ , and Dfe ¼ 0 [Fig. 10(a)], i.e. we assume a higher defect density in the core than in the bulk, but an absence of equilibrium space charge layers; the latter assumption will be relaxed in the next section. The bulk diffusion coefficients are set equal, DO⬙ ¼ Dh˙, and the i thresholds for transitions from both bulk sides into the core to be the same as in the bulk, viz. k~j,e ic ¹ 1 ¼ kj,e ic þ 1 ¼ k~j,e ⬁ . The thresholds for transitions out of the core are automatically higher [since goc ⬍ go⬁ , see Fig. 10(a)] and already determined by kj,e i cej, i ¼ kj,e i þ 1 cej, i þ 1 . Hence, jej, i ’s ( ⬀ kj,e i cej, i ) are position-independent. Thus the only deviation from bulk properties is an enhanced defect density in the core. Fig. 10(b) shows the changes of the individual carrier chemical potentials, dmj, i ⬀ dcj, i =cej, i , upon an increased partial pressure of oxygen at the right hand side. At the left hand side the partial pressure was fixed to the (initial) equilibrium value by the boundary conditions, hence, (dmO⬙ þ 2dmh˙)lx ¼ ¹ L ¼ 0. The component chemical poteni tial, dm O, exhibits a linear profile as expected, due to the invariance of jej, i ’s (eqn (22)). On the other hand, the individual chemical potentials, dmO⬙ and dmh˙, are i depressed and enhanced in the grain boundary region [Fig. 10(b)], respectively. As shown in Fig. 10(c) there is also a build-up of kinetic space charges, although the initial grain boundary was free of space charge. To understand the origin of kinetic space charges in this case (note that DO⬙ ¼ Dh˙) we assume for a moment, that, ~

Fig. 9. Doped bicrystal. Here the flux of minority carriers is impeded by the grain boundary core: ce2 ¼ 10ce1 . All other parameters are the same as in Fig. 8. Carrier concentrations (a) and component chemical potential (b) are displayed as a function of position from which initial position-independent values were subtracted.

je1 je2 ⳵ dm þ je2 ⳵x O

je1

i

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J. JAMNIK AND J. MAIER

by analogy to the bulk, also in the grain boundary region the only driving forces for the carriers are the gradients of the chemical potentials. Then, owing to the position invariance of jej, i ’s, dm j profiles would be linear [Fig. 10(b), dashed lines] implying ‘‘harmless’’ relative changes of the concentration in the core and in the region close to it. Since cej, core q cej, i⫽core , the absolute changes of concentrations in the core would then be much higher than in the neighbouring layers [even higher than the numerically calculated cej, core displayed in Fig. 10(c)] leading to an extremely charged core without any counter charge. On the other hand, linear dc j profiles would not result in any space charges, but would lead to extremely deformed dm j profiles and thus to a chemically very

unfavourable situation. The actual dm j and dc j contours [Fig. 10(b), Fig. 10(c)] are a compromise between both extremes (linear dm j or dc j), or in other words correspond to a favourable partition of the energy to the electrostatic and chemical degrees of freedom.

5.3. Grain, boundary with equilibrium space charges As a final example, we consider almost the same bicrystal as in Section 5.2, viz.ceion ¼ 1=10ceeon , goc ¼ 1=2go⬁ , but Df e ⫽ 0, i.e. we no longer neglect equilibrium space charges. For the purpose of computational simplicity we restrict ourselves to univalent defects (z 1 ¼ 1 ¼ ¹ z 2, and n 1 ¼ 1 ¼ n 2) and assume equal densities of the available defect sites for the positive and negative carriers independent of position. We assume that the impurities (dopants) in the bicrystal, even though not mobile enough to redistribute during the chemical diffusion experiment, attained thermodynamic equilibrium profiles. Still we neglect any impurity in the core. Then, setting fe⬁ ¼ 0, the position-dependence of the space charge potential reads (see for example [17]):      lx ¹ x0 l efe0 efe (x) ¼ 4kB Tarctanh exp ¹ tanh LD 4kB T (23) The locus x 0 denotes the edge of the space charge layer (in the discrete picture the layer next to the core). The potential fe0 , which refers to x 0, is approximately given by [8]: " !# ce1, ⬁ 1 o e o o o ef0 ⬇ (m1, ⬁ ¹ m1, c ) ¹ (m2, ⬁ ¹ m2, c ) þ kB Tln e 2 c2, ⬁ (24) fe0

Fig. 10. Doped bicrystal: ce2 ¼ 10ce1 , z 1 ¼ ¹ 2, z 2 ¼ 1 and D 1 ¼ D 2. (a) The grain boundary core is non-blocking but with much higher defect density than in the bulk: goc ¼ 3=4go⬁ . Deviations of chemical potentials of individual defects and of the component from initial values are displayed in (b). Dashed lines indicate hypothetical linear profiles to be expected in a single crystal. Deviations of concentrations from initial values are shown in (c). Core concentrations were previously divided by a factor of four for diagrammatic reasons.

can only be obtained numerically. The exact value of The procedure outlined, e.g. in Ref. [8], also automatically yields the value of the potential in the core and the core defect densities. For the example treated in the following, we set the standard chemical potentials such that 1 goc ¼ go⬁ 2 [Fig. 11(a), note that go ¼ mo1 þ mo2 ] and that the right hand side of eqn (24) equals 3k BT. Numerically obtained values for efe0 and eDfe were 2.07 and 2.31 k BT, respectively. Eqn (23) determines the equilibrium defect density profiles in the space charge regions, cej (x) ¼ cej, ⬁ exp( ¹ zj fe (x)=kB T), on the continuum level. Discrete values, which we provide by averaging: ÿ
Z i ¹ 1 þ 12 s (25) cej, i ⬅ ÿi ¹ 1 ¹ 1
s cej (x)dx 2

are displayed in Fig. 11(b). According to the assumption,

Transport across boundary layers

~

goc ¼ 1=2go⬁ , native defects are easier to form in the core than in the bulk. The adjacent space charge layers reflect an accumulation of electronic defects and, on a one order of magnitude smaller level, a depletion of ionic defects [Fig. 11(b)]. In addition to the concentrations we also have to choose the mobility values. As in the previous section we set in the bulk D ion ¼ D eon. In space charge layers, the rate constants k~j,e i and kj,e i þ 1 are modified due to a nonvanishing equilibrium electric field [ ⬀ ⳵=⳵xfe (x)] [Fig. 11(a)]. According to the definition

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q uj, i ⬀ k~j,e i kj,e i þ 1 , the position-dependence of jj, i cancels within the space charge layers. Thus, the jej, i ’s simply follow the concentration profiles. In the core region, densities and mobilities both have to change since the standard chemical potentials change [see Fig. 11(a)]. Following the procedure described in the previous section we assume that the rate constants for transition into the core are the same as in the bulk. The complete equilibrium picture which we used in the calculations of the chemical diffusion, is shown in Figs. 11(a) and (b).

~

Fig. 11. Doped bicrystal with initial space charges at the grain boundary. (a) Parameters used in calculations: z 1 ¼ 1, z 2 ¼ ¹ 1, D 1 ¼ D 2 and goc ¼ 1=2go⬁ . (b) Equilibrium (initial) concentrations of ionic (ion) and electronic (eon) carriers (ce2, ⬁ ¼ 10ce1, ⬁ . (c) Deviations of carrier concentrations from equilibrium values due to steady state chemical diffusion. Values in the core dcion =ce1, ⬁ ¼ 7:9 and dceon =ce1:⬁ ¼ 0:27, are not shown due to diagrammatic reasons. Deviations of chemical potentials of ionic and electronic carriers from their equilibrium values are shown in (d). Deviations of the component chemical potential and of the electric potential from equilibrium values are displayed in (e) and (f).

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J. JAMNIK AND J. MAIER

Fig. 11(e) displays the deviations of the component chemical potential from the position-independent equilibrium profile. First we recognise that the component chemical potential changes with position much more steeply in the grain boundary region than in the bulk. This reflects the ‘‘diffusion resistance’’ of the grain boundary region stemming from the depletion of the minority charge carriers. On the black box level (Section 3) the excess grain boundary diffusion resistance reads: ¹1 ¼ (ezj nj )2 (Reon, gb þ Rion, gb ): Lgb

(26)

If we approximate the sum by the minority carrier resistance, and use the known expression for the excess space charge resistance [18]:     Rj, ⬁ eDfe 2LD exp ¹1 (27) Rj, gb ¼ L 2kB T we obtain with the parameters L ¼ 8L D and eDf e ¼ ¹1 2.06 k BT, the result Lgb =L⬁¹ 1 ¼ 1=3. The subscript ⬁ here and in the above equation refers to the crystal of the same thickness, but without grain boundary (the total ¹1 ). The bicrystal diffusion resistance equals L⬁¹ 1 þ Lgb ratio of 1/3 implies that one third of the overall component chemical potential drop occurs over the grain boundary region. By extrapolating the bulk profile [Fig. 11(e)] to x ¼ L, this is readily verified. While the local behaviour of the component chemical potential can be expressed in closed form (eqn (22)), the individual dm j can only be obtained from the numerical solution of the detailed transport eqn (5). These profiles are to a certain extent similar to the grain boundaries with a high core defect density irrespective of the absence [Fig. 10(b)] or presence [Fig. 11(d)] of adjacent equilibrium space charge layers. The basic difference is, that in the case of initially existing space charge layers, the minority carrier chemical potential exhibits an overall step across the boundary region. In contrast to the component chemical potential, the concept of the grain boundary terminal quantities cannot be applied to dmj ’s. As shown in Fig. 11(f), a kinetically built-up finite electric potential difference exists across the grain boundary region. Hence, the change of the defect chemical potential across the boundary region is not the proper driving force. The electric potential difference cancels for the component chemical potential, which enables the use of eqn (26). In the case displayed in Fig. 11, the grain boundary space charge layers make the chemical diffusion through the boundary more difficult than in the bulk. The reason is of course the grain boundary depletion of the carriers with the smallest je⬁ in the bulk. If the equilibrium space charge potential were of opposite sign, then the carriers with the greatest je⬁ would be depleted and the carriers with the smallest je⬁ accumulated. This would enhance the chemical diffusion through the boundary.

An experimental example which the above model may address, is a grain boundary in SrTiO 3. It is known — mostly from impedance measurements — that the equilibrium space charge layers adjacent to grain boundaries in acceptor-doped SrTiO 3 are composed of depletion layers of oxygen vacancies and holes while the dopant Fe⬘Ti -concentration is very roughly position-independent within the space charge layers. Such a situation can be better described by the parabolic potential contour [19], widely used in the semiconductor literature, and by a doped core, rather than using eqns (23) and (24). The general features of chemical diffusion through such Schottky-type space charge layers would nevertheless be the same as discussed here. The decrease of the chemical diffusion flux by such a grain boundary, as reflected by Fig. 11, was recently observed experimentally by in situ measurements of concentration profiles in SrTi0 3 bicrystals after a sudden change of the oxygen partial pressure [20]. Such time-dependent problems will be discussed more precisely and in detail elsewhere (Part III) [21]. 6. CONCLUSIONS

The theoretical procedure described in Part I was applied to stationary chemical diffusion. It is restricted to materials dominated by two charge carriers without internal sources and sinks, and applies to dilute situations and small driving forces (linear regime). The conventional Wagner’s treatment (using electroneutrality a priori) fails if applied to boundary regions. Concentration profiles are explicitly derived for a single crystal upon exposure to a component partial pressure difference. If the defect diffusion coefficients are different, chemical diffusion leads to a charging of the crystal surfaces (core) and to the formation of adjacent space charge layers, even if we start with an initially space charge-free situation. More generally, any spatial variation (due to interfaces) of equilibrium carrier concentrations and/or interfacial rate constants which are different for ionic and electronic defects, leads to a kinetic charging of the inhomogeneous region. This is shown quantitatively and specifically for several cases: (i) sluggish surface reactions; (ii) amorphous grain boundary core (blocking for ions); (iii) highly disordered grain boundary core; and (iv) a grain boundary with adjacent equilibrium space charge layers. If in the core region only the rate constants (mobilities) differ with respect to bulk values [(i), (ii)], then — surprisingly at first glance — Fick’s law is a good approximation for minority carriers, but it fails for majority carriers. In situations in which the equilibrium defect density differs from bulk values [(iii), (iv)], the conventional description of concentration profiles fails completely within the boundary regions. Numerically

Transport across boundary layers

calculated carrier concentration and chemical potential profiles can, in all cases, be understood as a compromise between deviations from stoichiometry and local electroneutrality. Acknowledgements—We thank Miran Gabersˇcˇek for helpful discussions and reading the manuscript.

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10. Lyubomirsky, L., Lyahovitskaya, V. and Cahen, D., Appl. Phys. Lett., 1997, 70, 613. 11. Nachev, I. and Balkanski, M., Physica Scripta, 1994, 49, 371. 12. Jamnik, J. and Maier, J., Ber. Bunsenges. Phys. Chem., 1997, 101, 23. 13. Maier J., in Proceedings of the 17th Riso International Symposium on Materials Science: High Temperature Electrochemistry: Ceramics and Metals, ed. F. W. Poulsen, N. Bonanos, S. Linderoth, M. Mogensen and B. ZachauChristiansen. Riso National Laboratory, Roskilde, Denmark, 1996. p. 61–76. 14. Jamnik, J., Habermeier, H.-U. and Maier, J., Physica, 1995, B204, 57. 15. Denk, I., Claus, J. and Maier, J., J. Electrochem. Soc., 1997, 144, 3526. 16. Hegenbart, G. and Mu¨ssig, Th., Surf. Sci. Lett., 1992, 275, L655. 17. Kliewer, K. L. and Koehler, J. S., Phys. Rev., 1965, 140, A1226. 18. Jamnik, J., Appl. Phys., 1992, A55, 518. 19. Hagenbeck, R., Schneider-Sto¨rmann, L., Vollmann, M. and Waser, R., Mater. Sci. Engng, 1996, B39, 179. 20. Denk, I., Noll, F. and Maier, J., J. Am. Ceram. Soc., 1997, 80, 279. 21. Jamnik J. and Maier J., J. Electrochem. Soc., 1998, 145, 1762.