Fundamentals of electrical conduction in ceramics

Fundamentals of electrical conduction in ceramics

7

Steffen Grieshammera,b, Roger A. De Souzab a Helmholtz-Institut M€unster (IEK-12), Forschungszentrum J€ulich GmbH, M€unster, Germany, b Institute of Physical Chemistry, RWTH Aachen University, Aachen, Germany

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Introduction to electrical conductivity

1.1 Conductivity of ceramic materials The electrical conductivity is, arguably, the most important property of ceramics that are to be used as electrolytes and electrodes for electrochemical energy conversion devices such as fuel cells and batteries. While ceramics are widely considered as insulators, there are numerous examples where the electrical conductivity can attain high values; it can vary by orders of magnitude depending on composition and temperature. As an example, we have given in Fig. 1 the total conductivities of selected highly conductive ceramics. The conductivities range over orders of magnitude and exhibit substantial differences in their dependence on temperature. Indeed, some materials show high conductivity (>1 mS cm
1) even at room temperature, while for others such a conductivity is reached only at temperatures above 500 °C. The differences in the conductivity behavior are due to the type of the charge carrier, the underlying migration mechanism, and the material’s electronic and crystallographic structure. In principle, all charged species in a material contribute to its electrical conductivity: cations, anions, and electrons. The total conductivity is given, therefore, by the sum of the partial conductivities of all carriers: σ total ¼

X

σi

(1)

At this point, it is worth making a special distinction between ionic and electronic charge carriers that lead to ionic and electronic conductivity, respectively. In principle, all materials are mixed ionic and electronic conductors, but the values of the partial conductivities differ in general by orders of magnitude and therefore such materials are described as either purely electronic or purely ionic conductors. Indeed, in most cases, the total conductivity is dominated by one or perhaps two species, with all others having only negligible contributions to the total conductivity. The nature of the conductivity is important for various applications: For electrolytes in fuel cells or solid-state batteries, high ionic conductivity is required to limit the cell resistance for feasible layer thicknesses, whereas the electronic conductivity Advanced Ceramics for Energy Conversion and Storage. https://doi.org/10.1016/B978-0-08-102726-4.00007-7 © 2020 Elsevier Ltd. All rights reserved.

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Fig. 1 Total conductivity of selected ceramic materials as a function of the inverse temperature. (a) Zr0.85Y0.15O1.925 (Han et al., 2007), (b) Ce0.8Gd0.2O1.9 (Balazs and Glass, 1995), (c) La0.9Sr0.1Ga0.8Mg0.2O2.85 (Huang et al., 1996), (d) Sr0.9Ti0.6Fe0.4O3-δ (Fagg et al., 2001), (e) Li7La3Zr2O12 (Murugan et al., 2007), (f ) Na3.1Sc2Si0.4P2.6O11.8 (Guin et al., 2016), (g) Li1.5Al0.5Ti1.5(PO4)3 (Ma et al., 2016), (h) Na-β00 -alumina, single crystal (Engstrom et al., 1981).

should be negligible to prevent leakage currents. In contrast, for electrodes and gas separation membranes both high ionic conductivity and electronic conductivity are required to enable high electrochemical performance. For each charge carrier we can express its partial conductivity in terms of its charge q, its concentration c, and its electrical mobility μ: σ ¼ qcμ

(2)

As the charge of a particular charge carrier is generally fixed, the conductivity depends on the two other quantities, concentration and mobility, both of which can vary by orders of magnitude. It should be noted that in a perfect ionic lattice with neither electronic nor ionic defects no conductivity is possible. It is the defects, either in the ion lattice or in the electronic structure, which enable conductivity and, in most cases, it is more reasonable to examine the concentration and mobility of defects rather than those of the regular lattice elements. In this chapter, we will discuss the concentration and mobility of defects, including their dependence on temperature and composition. During the writing of this chapter, we referred to several textbooks and review articles (Chiang et al., 1997; Gross and Marx, 2018; Grosso and Pastori Parravicini, 2014; Howard and Lidiard, 1964; Kittel, 1976; Maier, 2004; Waser and Ielmini, 2016).

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Our aim is to introduce the reader to the basics of defect chemistry, diffusion processes, and electrical conductivity, and we will illustrate these topics by suitable examples where appropriate.

1.2 Types of conductors Electrical conduction can occur by electrons and/or ions. In the case of electrons and holes, the conduction behavior depends mainly on the band structure of the material, as illustrated in Fig. 2. When atoms approach each other closely enough for the atomic orbitals to overlap in a solid, crystal bands are formed from the orbitals. These bands are filled, to some degree, with the electrons that were present in the atomic orbitals. Metals are characterized by partially filled bands with the Fermi level EF lying within a band. When an electric field is applied, only those electrons close to the Fermi level can move to unoccupied states in the same band resulting in an electronic current. In semiconductors, the valence band and conduction band are separated by a bandgap and the Fermi level lies within the gap. However, if the bandgap is small enough and the temperature is high enough for considerable thermal excitation of electrons from the valence band to the conduction band, charge carriers in both bands are generated: electron holes in the valence band and electrons in the conduction band. The distinction between insulators and semiconductors is not unambiguous. Strictly speaking, every semiconductor is a insulator, if the bandgap is large enough at a sufficiently low temperature such that excitation of electrons is negligible. For example, the bandgap in the elemental semiconductor silicon is 1.1 eV but ZnO with a bandgap of about 3.2 eV is still considered as a semiconductor. In contrast, the typical insulator α-Al2O3 (corundum) has a bandgap of 8.8 eV. While metallic behavior is usually not associated with ceramic materials, several ceramics show metallic conductivity such as ReO3 (with its half-filled conduction band). In addition, there are some ceramic materials, such as YBa2Cu3O7, that show superconductivity (i.e., the conductivity becomes infinite and the resistance becomes zero) if the temperature is low enough (liquid nitrogen temperatures for YBa2Cu3O7). We will not go into any detail on this topic but we will focus on ceramic materials with finite conductivity above room temperature. Besides electronic conduction, ionic conduction is possible in ceramics with metallic behavior, semiconductors, and insulators. The motion of ions is enabled by the thermally activated hopping of ions within the lattice. Fig. 2 Band diagram of a metallic conductor, semiconductor, and insulator. White: empty states; Blue(gray in print version): occupied states.

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Concentration of charge carriers

2.1 Types of defects Electronic defects exist in the band structure of the material and include electrons in the conduction band and electron holes in the valence band. For reason of brevity, we will refer to these in the following only as electrons and holes, respectively. Defects in the ionic lattice of the structure can be classified according to their dimensionality. We focus here on zero-dimensional (point) defects, which are present at finite temperature even in single crystals. Indeed these point defects are the source of ionic conductivity and they can be described by common chemical equations. Typical point defects are illustrated in Fig. 3 including a vacancy, where a single ion is missing; an interstitial, where an ion occupies a position that is normally vacant; and a dopant or impurity, where a regular ion is replaced by a different ion. Further point defects can include valence changes of an ion or antisite defects, for example, a cation sitting on an anion site. Point defects in the lattice can be formed by intrinsic defect disorder, by doping or by reduction and oxidation. Intrinsic defect disorder implies the formation of defects within the lattice by thermal excitation. The disorder is intrinsic since there is no exchange with other phases and defect concentrations solely depend on temperature. In ionic compounds, charged defects can only occur as defect pairs to satisfy the condition of global charge neutrality. The two most common intrinsic disorders are Frenkel and Schottky disorder as depicted in Fig. 4.

Fig. 3 Illustration of (A) vacancy, (B) interstitial, and (C) dopant.

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Fig. 4 Illustration of (A) Frenkel and (B) Schottky disorder.

In the case of Frenkel disorder, one ion is displaced from its regular position to an interstitial position resulting in the formation of a vacancy and an interstitial. For Frenkel disorder of the anion lattice, the expressions anti-Frenkel or anion-Frenkel are also used. In the case of Schottky disorder, anions and cations are displaced to the surface leading to the formation of corresponding vacancies. To satisfy the charge neutrality condition, vacancies have to be formed according to the stoichiometric ratio of the crystal. While Frenkel disorder always leads to the formation of a defect pair (vacancy and interstitial), Schottky disorder can result in multiple defects, such as the Schottky quintet (two aluminum vacancies and three oxygen vacancies) in Al2O3. The predominant disorder type depends on the structure and composition, and predicting which disorder is dominant is not possible in general. Nevertheless, some assumptions can be made based on structure type and ionic radii. For example, in close-packed structures like the rock salt structure of NaCl, the formation of interstitials is unlikely, especially for the generally larger anions, and hence, Schottky disorder can be expected to be the dominating disorder type. The defect chemistry of a ceramic material can be deliberately modified by doping. It is important to note that the term doping is commonly used for ceramics, even for dopant fractions of several percents, for which one should, strictly speaking, use the term substitution. For the elemental semiconductors, doping is typically up to the ppm range. Of particular interest is aliovalent doping, since charge neutrality demands the simultaneous formation of oppositely charged defects for charge compensation. For example, the substitution of Al3+ in Al2O3 by Mg2+ reduces the positive charge in the lattice, which could be compensated ionically by either oxygen vacancies or cation interstitials. Again, the compensation mechanism depends on structure and composition, and similar considerations as for intrinsic disorder are applicable. Since the doping ions originate from an extrinsic source these defects are considered as extrinsic. Oxidation and reduction can lead to the formation of point defects as well. For oxides, the most common process is the exchange of oxygen with the surrounding

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gas atmosphere. At low oxygen activities, oxygen is released from the oxide, and at high activities, it is incorporated. Different from intrinsic disorder and doping, these processes lead to nonstoichiometry that cannot be explained by ionic defects alone but requires electronic defects to be involved. A consistent designation whether these defects are considered as intrinsic or extrinsic does not exist (and it is not really that important).

€ ger-Vink-notation 2.2 Kro In a real lattice, a crystallographic defect is a deviation from the ideal lattice at a given lattice position and thus defects can be defined as “defect” ¼ “real-ideal” with respect to charge and element. This definition is used in the Kr€oger-Vink notation to describe defects in a solid. In Kr€ oger-Vink notation, each lattice element is described in the form SεL where S is the symbol of the actual element, L is the symbol of the particular lattice site in the ideal crystal and ε is the charge relative to the ideal charge at this lattice site. The symbol S can be any elemental symbol and additionally V or v to denote a vacancy. The symbol L can be any elemental symbol and additionally i to indicate an interstitial site. To distinguish relative charges from absolute charges, the symbol ε is denoted with / for one negative charge, for one positive charge, and  in the neutral case. An Mg2+-ion on an Al3+-site in Al2O3 would thus be denoted as Mg/Al and an oxygen vacancy as VO . Further examples are given in Table 1. In the same manner, electrons in the conduction band and holes in the valance band are denoted as e/ and h , respectively. In addition, associates of different defects are combined in parentheses, for example, a defect associate of an oxygen vacancy and an Mg dopant in Al2O3 is given   / by VO
MgAl . We can regard defects (including vacancies) as quasichemical species and therefore we can formulate quasichemical reactions for defects using Kr€oger-Vink notation. The reactions for Schottky, Frenkel, and anti-Frenkel disorder in Al2O3 are given in Eqs. (3)–(5), respectively. Schottky l



l



///  2Al Al + 3OO Ð 2VAl + 3VO + Al2 O3 

(3)

Table 1 Examples for Kr€oger-Vink notation in Al2O3.

S L ε SεL

Al3+vacancy

Al3+interstitial

O22vacancy

O22 -interstitial

Mg2+ on Al3+-site

V Al
3 V/// Al

Al i +3 Ali

V O +2 VO

O i
2 O//i

Mg Al
1 Mg/Al





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Frenkel ///  Al Al + Vi Ð VAl + Ali



(4)

Anti-Frenkel //  O O + Vi Ð VO + Oi 

(5)

Likewise, we can formulate the incorporation of MgO into Al2O3 as Al2 O3

/ + VO + 2O 2MgO ! 2MgAl O 

(6)

As mentioned above, this is not the only possible incorporation mechanism and as long as we have no other evidence, all possibilities are correct if they conserve mass, charge, and the site balance of the host crystal. This constrains that no charge or mass can be created or destroyed in the reaction and likewise the ratio between anion and cation sites is maintained.

2.3 Electronic defects For electronic conduction, free electrons in the conduction band or holes in the valence band have to be present. In metals, partially filled bands exist and conduction is already possible at zero temperature upon application of an electric field. At finite temperatures, electrons are excited across the Fermi energy to previously unoccupied states. The formation of each electron above the Fermi energy is accompanied by the formation of an electron hole below the Fermi energy. For the concentrations of electrons and holes, we thus have ce ¼ ch. To estimate the concentration of electrons and holes, we have to know the number of available states N(E) and the probability that these states are occupied f(E). ce ðEÞ ¼ N ðEÞf ðEÞ

(7)

The energy-dependent number of available states per unit volume is called the density of states N(E) and can be approximated from the theory of a free electron gas according to N ð EÞ ¼

 3=2 E1=2 2meff e 2π 2 h2

(8)

Here the energy is referred to the band minimum and meff e is the effective mass of the electrons. In solids, the effective mass determines how the electrons or electron holes react to an applied electric field. The effective mass can be much smaller or much larger than the mass of a free electron (Kittel, 1976). The density of states near the

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Fig. 5 Illustration of (A) density of states, (B) Fermi-Dirac function (T > 0), and (C) density of occupied states in a metal near the Fermi level. Based on Kasap, S.O., 2006. Principles of Electronic Materials and Devices, third ed. McGrawHill, Boston.

conduction band edge shows a parabolic behavior for increasing energy as depicted in Fig. 5A. The probability of occupation of the states is given by the Fermi-Dirac function: f ð EÞ ¼

1   E
EF 1 + exp kB T

(9)

At absolute zero, the function has a step-like shape with all states occupied below the Fermi energy and all states unoccupied above the Fermi energy. At finite temperatures, electrons from occupied states are excited to unoccupied ones and the probability function is smeared out close to the Fermi energy (cf. Fig. 5B). The density of occupied states can be obtained from the product of N(E) and f(E) (cf. Fig. 5C). In semiconductors, the electrons have to be excited from the valence band to the conduction band, which are separated by the bandgap Eg. For N(E) and f(E) we can apply the same considerations as for metals. For energies with E
EF ≫ kBT, which is true for the conduction band in a semiconductor, the Fermi-Dirac function can be expressed by the simpler Boltzmann equation: f ðEÞ ¼ exp

  EF
E kB T

(10)

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Combining Eqs. (8), (10) gives the energy-dependent density of occupied states. To obtain the total density of occupied states in the conduction band we have to integrate from the minimum of the conduction band to infinity. Assuming identical effective masses of electrons and electron holes, the Fermi level lies in the center of the bandgap and the concentration of electrons and holes is  ce ¼ ch ¼ 2

2πmeff e kB T h2

3=2

  Eg exp
2kB T

(11)

From a different (and simpler) starting point, we can write the excitation of the electrons as a quasichemical reaction of the form nil Ð e= + h

(12)

Applying the mass action law and charge neutrality condition we obtain   Eg ce ¼ ch ¼ Ke1/2 ¼ K01/2 exp
2kB T

(13)

where K1/2 0 corresponds to the prefactor derived in Eq. (11).

2.4 Defect formation 2.4.1 Intrinsic The formation of intrinsic defects in a given ceramic material is only controlled by the temperature. To obtain the equilibrium concentration of defects we have to minimize the Gibbs energy of the system at a given temperature. The formation requires the expenditure of an enthalpy Δh per defect. In addition, the introduction of a defect will lead to a perturbation of the phonon modes and consequently change the vibrational entropy Δs per defect compared to the ideal lattice. The Gibbs energy of formation for one defect is therefore given by Δg ¼ Δh
TΔs

(14)

It is easy to understand that Δg is positive, as the lattice would otherwise be instable (on account of the unlimited formation of this defect). However, the formation of defects is favored by the configurational entropy. The introduction of defects in the lattice increases the number of possible configurations and the defects are thus entropically stabilized. As an example, let us consider Frenkel disorder in an oxide AO. For simplicity, we assume that the number of lattice sites NL of one sublattice equals the number of interstitial sites NI such that NL ¼ NI ¼ N. As charge neutrality demands the formation of one interstitial for each vacancy, we obtain the relation NV ¼ Ni ¼ NF, where NV and

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Ni are the numbers of vacancies and interstitials, respectively, and NF is the number of Frenkel pairs. For the total Gibbs energy of the lattice we obtain G ¼ G0 + ΔG ¼ G0 + NF ΔgF
TΔSC

(15)

where ΔgF is the Gibbs energy per Frenkel pair and ΔSC is the configurational entropy given by ΔSC ¼ kB ln ðΩÞ

(16)

Ω is the number of possible arrangements of defects in the lattice. For each sublattice with N sublattice sites and NF defects the total number of configurations for this sublattice is given by Ω¼

N! ðN
NF Þ!NF !

(17)

Combining Eqs. (15)–(17) gives us an expression for the change of the Gibbs energy depending on the number of defects. Here we have to consider that we have a disorder in both the regular lattice and the interstitial lattice. As we have NL ¼ NI and NV ¼ Ni there is no difference between the number of configurations of interstitial ions on interstitial sites and of vacancies on regular sites and we come to the expression ΔG ¼ NF ΔgF
2kB T ln

N! ðN
NF Þ!NF !

(18)

with Stirling’s approximation, ln N ! ¼ N ln N
N, we can eliminate the factorials from the equation to obtain  ΔG ¼ NF ΔgF
2kB T ln N ln

N N
NF + NF ln NF N
NF

 (19)

Our task is now to find the value of NF that minimizes the Gibbs energy by differentiating ΔG with respect to NF. ∂ΔG NF ¼ ΔgF + 2kB T ln 0 N
NF ∂NF

(20)

Rearranging Eq. (20) yields the site fraction nF, that is, the number of defects per number of lattice sites,   NF NF ΔgF nF ¼  ¼ exp
N N
NF 2kB T

(21)

where we assume that NF ≪ N. For practical reasons, it can be more convenient to define the molar fraction [def], that is, the fraction of defect def per formula unit.

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For an oxide AaOb, the two quantities are related by [VA] ¼ a  nVA. With the volume per formula unit (f.u.), we can directly obtain the concentration of defects cdef ¼ [def ]/Vf. u.. One should be cautious as the definitions are not clear throughout the literature and [def ] might as well represent the site fraction or the concentration of defects. Alternatively, we can approach the defect fraction from a chemical viewpoint and write the reaction for Frenkel disorder in AO as  // A A + Vi >VA + Ai



(22)

We get the corresponding equilibrium constant from the mass action law: //

  V A Ai ΔgF KF ¼   ¼ exp
kB T AA V i 

(23)

 Since we assume small defect fractions, we have [A A ] ¼ [Vi ] ¼ 1 and by inserting the charge neutrality condition, we come to the same result as in Eq. (21):



 



ΔgF VA// ¼ Ai ¼ KF1/2 ¼ exp
2kB T 

(24)

We can use the same formalism for Schottky disorder in the oxide AaOb, where the activity of the pure crystal is unity:  aA A + bOb Ð aVA + bVO + Aa Ob

(25)

b   ½VA a VO ΔgS KS ¼ a b ¼ exp
kB T A O A O

(26)





Here, we drop the charge indicator for cation vacancies, which depends on the com

position parameters a and b. With the charge neutrality condition b½VA  ¼ a VO we obtain 

 ½VA  ¼ a  exp


ΔgS ða + bÞkB T

 (27)

So far, we only looked at individual intrinsic disorders. However, in reality, all defect equilibria are coupled. Consider an oxide AO with both Frenkel and Schottky-disorder

KF ¼ VA// Ai

(28)



KS ¼ VA// VO

(29)





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and the charge neutrality condition



2 VA// ¼ 2 Ai + 2 VO 



(30)

Consequently, we have three equations with three unknowns. Analytical solutions of this set of equations do not have simple forms. However, the intrinsic disorder in a crystal is generally dominated by one type of mechanism. Let us assume here that

KF ≫ KS and accordingly VA// , Ar ≫ VO We can now use the so-called Brouwer-approximation to simplify the charge neutrality condition to one positive and one negative defect: 







VA// ¼ Ai 

(31)



For the majority defects VA// , Ai , we get the same expression as before whereas the

fraction of minority defects VO is given by 





KS VO ¼ 1/2 KF 

(32)

2.4.2 Doping While intrinsic disorder depends only on temperature, defect concentrations can be directly modified by doping. A special role in this context is played by aliovalent dopants. In analogy to semiconductors, dopants with a lower oxidation state such as Y3+ in ZrO2 are often referred to as acceptor dopants, while dopants with higher oxidation state, such as Nb5+ in ZrO2 are referred to as donor dopants. For the dopant incorporation, several mechanisms are possible and the predominant one depends on the corresponding solution energy. For M3+ ions on Zr4+ sites in ZrO2 compensation by oxygen vacancies, M3+ interstitials or Zr4+ interstitials are imaginable, resulting in the following reactions: ZrO2

M2 O3 ! 2MZr/ + VO + 3O O ZrO2



(33)

2M2 O3 ! 3MZr/ + Mi + 6O O 

ZrO2

(34)

  Zr Zr + 2OO + 2M2 O3 ! 4MZr/ + Zri + 8OO 

(35)

Calculated solution energies for M2O3 in ZrO2 are shown in Fig. 6 for the three mechanisms. It is apparent that for all trivalent dopants the mechanism from Eq. (33) is favorable. For further analysis, we consider AgI with intrinsic defects dominated by Frenkel disorder. Doping with CdI2 leads to the formation of Cd2+ ions on Ag+ sites, which are

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Fig. 6 Calculated solution energy of different M2O3 oxides in ZrO2 according to Eqs. (33)–(35). Based on Zacate, M.O., Minervini, L., Bradfield, D.J., Grimes, R.W., Sickafus, K.E., 2000. Defect cluster formation in M2O3-doped cubic ZrO2. Solid State Ionics 128(1–4), 243–254.

compensated by additional silver vacancies. We can now define three regions depending on the dopant fraction x. At sufficiently high levels of doping the defect,h chemistry i h willi be dominated by / extrinsic defects and charge neutrality reduces to CdAg ¼ VAg ¼ x. The fraction

of the interstitials, in this case, is given by Agi ¼ KF =x. At very low doping levels and high temperatures, intrinsic defects will dominate

h / i and charge neutrality simplifies to Agi ¼ VAg ¼ KF⁄ . 



1



2

In the transition region between the intrinsic and extrinsic regime, have i hall defects i

h / to be considered in the charge neutrality equation Agi + CdAg ¼ VAg . With the 



mass action law for Frenkel disorder we obtain h

i

KF

¼ VAg/ CdAg + VAg/ 

(36)

and after rearranging

VAg/

2




x VAg/
KF ¼ 0

We can solve the quadratic equation to obtain the molar fraction of vacancies:

(37)

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Fig. 7 Illustration of the transition from intrinsic to extrinsic regime in CdI2-doped AgI depending on the dopant fraction.



VAg/



x ¼ + 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 + KF 4

(38)

Fig. 7 illustrates the dependence of the defect fractions on the doping level. At low dopant level, we have the intrinsic regime and the defects are dominated by Frenkel disorder. In the intermediate region, intrinsic disorder and doping contribute, while at high doping level [V/Ag] is proportional to the dopant fraction x.

2.4.3 Reduction and oxidation If an oxide is in equilibrium with the external atmosphere, the lattice exchanges oxygen with the atmosphere. The reduction can be written as = O O Ð VO + 2e + 1=2O2ðgÞ 

(39)

The mass action law for this reaction is given by

2   VO e= aO2 1=2 ΔgR 

KR ¼ ¼ exp
kB T OO 

(40)

where the activity of oxygen can be approximated by the oxygen partial pressure relative to the standard pressure pO2/p°. Under oxidizing conditions, we can write the incorporation of oxygen as  VO + 1=2O2ðgÞ Ð O O + 2h 

(41)

Combining Eqs. (39), (41) gives twice the generation equation of electrons and holes Eq. (12), showing that the three reactions are not independent. Only two of these three equations are required to fully describe the system.

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Reduction and oxidation lead to a nonstoichiometry of the lattice AO1
δ. For δ < 0 the material is cation deficient and for δ > 0 it is anion deficient. Typical examples are cation-deficient transition metal oxides such as FeO1
δ and CoO1
δ and oxygen-deficient CeO2
δ and SrTiO3
δ. It should be noted that the value of δ gives no information about the nature of the defects, that is, if these are interstitials or vacancies.

2.4.4 Localization Electronic defects formed during oxidation or reduction are not necessarily delocalized in the conduction or valence band but can localize at specific atoms. One example is FeO1
δ where Fe2+ ions are partly oxidized to Fe3+ and compensated by Fe vacancies. Formally, the oxidation can be written in analogy to Eq. (41): //  + O 1=2O2ðgÞ Ð VFe O + 2h

(42)

The localization is written as  Fe Fe + h Ð FeFe 

(43)

And the equilibrium is described by ½FeFe 

 Fe Fe ½h  

Kloc ¼

(44)

€ ger-Vink-diagram 2.5 Kro As shown in the previous section, defect concentrations are influenced by various defect reactions including intrinsic disorder, doping, and exchange with the atmosphere. In reality, all defect reactions will occur simultaneously, and defects are coupled by the corresponding equilibrium constants and charge neutrality. We will now discuss simultaneous defect equilibria, where we are particularly interested in the dependence on the oxygen partial pressure. Consider a simple oxide of the composition AO with intrinsic disorder dominated by anti-Frenkel defects. In addition, we are interested in the exchange of oxygen with the atmosphere and electronic excitation. Consequently, we include four types of defects, namely, oxygen vacancies and interstitials as well as electrons and holes, in the analysis. The concentrations of the defects are determined by the following equilibria: h i K e ¼ ½ h  e =

(45)



KaF ¼ Oi// VO 

(46)

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2 VO e= aO2 1=2 

KR ¼ OO 

(47)

In addition, the charge neutrality condition has to be satisfied: h i

2 Oi// + e= ¼ 2 VO + ½h : 

(48)

As a result, we have four equations with four variables. As an analytical solution is not possible in the general case, a numerical solution can be obtained. However, the problem can be simplified by using the Brouwer approximation considering only one positive and one negative majority defect in the charge neutrality condition depending on oxygen partial pressure. At low oxygen partial pressure, the lattice is reduced, with oxygen vacancies and electrons as dominating defects. Eq. (48) can then be simplified, by considering only



the majority defects, to e= ¼ 2 VO ¼ x. Inserting this relation in Eq. (47) results in 3 (49) K ¼ 4 VO aO2 1=2 



and



VO



h i K 1=3 aO2
1=6 ¼ 1=2 e= ¼ 4

(50)

Using this result in Eqs. (46), (45) we obtain the proportionalities for the minority defects: ½h  pO2 1=6

Oi// pO2 1=6

(51) (52)

In the intermediate pO2 range, the defect chemistry is dominated either by intrinsic defects or electronic defects. Considering a large bandgap insulator, one would find that the concentration of intrinsic defects is much larger than that of electronic defects and the charge neutrality condition is, hence, given by

pffiffiffiffiffiffiffi Oi// ¼ VO ¼ KaF pO2 0 

(53)

By inserting Eq. (53) into Eqs. (47), (45) we get the partial pressure dependencies of

electronic defects with e= pO2
1=4 and ½h  pO2 1=4 . Going onward to high oxygen partial pressures, we find that the charge neutrality is

given by 2 Oi// ¼ ½h . By combining Eqs. (45)–(47) we obtain the relation with the oxygen partial pressure: KF Ke 2 aO 1=2 KR ¼ // 2 2 O i ½h 

(54)

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Inserting the charge neutrality condition leads to Oi// pO2 1=6 and ½h  pO2 1=6 and



correspondingly VO pO2
1=6 and e= pO2
1=6 . Even though we do not know the exact values for the equilibrium constants, we can draw a qualitatively correct Kr€ oger-Vink or Brouwer diagram, representing the relation between oxygen partial pressure and defect fractions in a double logarithmic plot. For the construction, we have to consider the following aspects: 

1. For each defect and partial pressure range, we know the dependence on the partial pressure, that is, the slope in the Kr€oger-Vink diagram given by the exponent in the partial pressure. 2. For each partial pressure range, we know the majority and minority defects, which are reflected in the relative position of the defects in the diagram. 3. The individual partial pressure ranges have to be connected such that we get a continual transition between the regions.

For our oxide AO dominated by anti-Frenkel disorder, we get the plot of Fig. 8. We can summarize the construction of a Kr€ oger-Vink diagram as follows: 1. Define all the defects you want to consider, such as intrinsic, electronic, and dopant defects. 2. Note all involved reaction equations and corresponding mass action laws such as intrinsic disorder, redox reaction, and electronic excitation. 3. Consider small defect concentrations such that the regular lattice elements equal approximately the ideal occupation. 4. Derive the complete charge neutrality condition.

Fig. 8 Kr€oger-Vink diagram for a pure, large bandgap oxide AO with dominating anti-Frenkel disorder.

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5. Apply the Brouwer approximation for each partial pressure range, that is, reduce the charge neutrality condition to one positive and one negative defect. 6. Apply the charge neutrality condition to determine the fraction of majority defects. 7. Use the fractions of majority defects to solve for the minority defects. 8. Draw the Kr€oger-Vink diagram for each partial pressure range based on the results. The slope of each curve corresponds to the exponent in the pO2-dependence. Consider the relative position of the curves regarding majority and minority defects.

2.6 Examples 2.6.1 Doped cerium oxide We now continue with doping effects and consider cerium oxide doped with gadolinium oxide. In this case, Gd3+ dopants reside on Ce4+ sites and are compensated by oxygen vacancies. CeO2

Gd2 O3 ! 2GdCe/ + VO + 3O O 

(55)

At very high or very low oxygen partial pressures the behavior will not change com



pared to our previous example and the charge neutrality will be given by e= ¼ 2 VO in the low-pressure range and 2[O//i ] ¼ [h ] in the high-pressure range. In the intermediate region, we assume that doping dominates over intrinsic disorder. Increasing the partial pressure from the low partial pressure region depletes the fraction of electrons whereas the fraction of vacancies is still high due to the dopants. Consequently, we /



¼ 2 VO ¼ x ¼ const. and can rewrite obtain the charge neutrality condition GdCe Eq. (47) as 

l



2 ½GdCe/  e= aO2 1=2

KR ¼ 2 O O

(56)



For the minority defects we get the proportionalities e= pO2
1=4 , ½h  pO2 1=4 and //

Oi pO2 0 . A further increase of the oxygen partial pressure fills up the oxygen vacancies and compensation of the dopant ions takes place by electron holes instead, that is, /

GdCe ¼ ½h  ¼ x. With Eq. (45) we obtain for the reduction reaction:

VO Ke2 aO2 1=2 KR ¼ 2x2 

(57)





From this expression, we obtain VO pO2
1=2 and with Eq. (46), Oi// pO2 1=2 . The fraction of the electronic defects is independent of the oxygen partial pressure. Combining the four partial pressure regions, we get the complete Kr€oger-Vink diagram as shown in Fig. 9. 

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Fig. 9 Kr€oger-Vink diagram for acceptor-doped cerium oxide. Based on Tuller, H.L., Bishop, S.R., 2010. Tailoring material properties through defect engineering. Chem. Lett. 39(12), 1226–1231.

2.6.2 Cobalt oxide Co1
δO is a typical example of a cation-deficient p-type semiconductor. The charge //

neutrality is, therefore, expected to be given by VCo ¼ ½h  and the incorporation of oxygen can be written as //  + O 1=2O2ðgÞ Ð VCo O + 2h

(58)

Applying the mass action law and inserting the charge neutrality condition yields a partial pressure dependency of pO1/6 2 . In contrast, thermogravimetric measurements revealed a slope of 1/4, as shown in Fig. 10. This observation can be explained by an additional reaction between Co vacancies and electron holes leading to the formation of singly charged vacancies. // + h Ð VCo/ VCo

(59)

Combining Eqs. (58), (59) and applying the mass action law results in 

O ½VCo/ ½h  K ¼ O 1=2 aO2 K1 For the vacancies, we thus obtain ½VCo/  pO2 1=4 .

(60)

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Fig. 10 Nonstoichiometry of Co1
δO depending on partial pressure. Lines are fits with a slope of 1/4. Data from Bransky, I., Wimmer, J.M., 1972. The high temperature defect structure of CoO. J. Phys. Chem. Solids, 33(4), 801–812.

2.6.3 Doped strontium titanate Strontium titanate is an ABO3 perovskite with Sr2+ on the A site and Ti4+ on the B site. Due to the close-packed structure, vacancies are favored over interstitials, and the dominant intrinsic disorder is SrO partial Schottky. //  Sr Sr + OO Ð VO + VSr + SrO 

(61)

//

V V aSrO KS ¼ O  Sr 

OO SrSr 

(62)

Acceptor doping is usually achieved by doping of B-sites with trivalent ions such as Al3+ or Fe3+. Donor doping is feasible either by substitution of the A-site by trivalent dopants such as Y3+ or La3+ or by pentavalent doping on the B-site, for example, by Nb5+ or Ta5+. In the following, we will focus on doping with donors D irrespective of the doping site. At low partial pressures, oxygen vacancies, and electrons dominate as for doped ceria. Increasing partial pressure leads to compensation of donor dopants by electrons =

e ¼ ½D  ¼ x. For oxygen vacancies we obtain

K O O
1/2 aO 2 (63) VO ¼ x2 //

pO2 1=2 . and ½h  pO2 0 and VSr l



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For further increase of the partial pressure, we have the charge neutrality condition 2[V//Sr] ¼ [D ] ¼ x:

With Eq. (62) we get for the oxygen vacancies VO pO2 0 and for electronic

defects we get ½h  pO2 1=4 and e= pO2
1=4 . In experiments, the highly oxidizing regime with the charge neutrality //

2 VSr ¼ ½h  is hardly accessible. Nevertheless, we can write the oxidation as l



//  1=2O2 + Sr Sr Ð VSr + 2h + SrO

(64)

where we assume the formation of SrO on the surface. From the equilibrium constant // 
1/2 K ¼ VSr ½h aO2 (65) //





¼ ½h  pO2 1=6 , e= pO2
1=6 , and VO pO2
1=6 . we obtain 2 VSr The resulting Kr€ oger-Vink diagram is presented in Fig. 11. 

2.7 Grain boundaries So far we have only addressed point defects but in a real polycrystalline sample extended defects, such as dislocations (one-dimensional) and grain boundaries (two-dimensional) are present depending on the thermal history of the sample. Such defects are not present at equilibrium because their configuration entropy is insufficient to compensate for the formation of energy (see Section 2.4). Surfaces or heterointerfaces, as extended defects, are of course present because of the finite amount of material present. Here we will focus on grain boundaries and surfaces. Although these higher dimensional defects are not in equilibrium, the point defects in the grain boundary region are in equilibrium, which deviates from the situation in the bulk. Since the structure in the region of the grain boundary is altered from the bulk structure the energetics for point defects are modified as well. In ionic materials, the formation of space charge layers has been observed especially in oxides. For materials such as CeO2 or SrTiO3, the energy of formation for oxygen vacancies is decreased in the core of the grain boundary (see Fig. 12A). As a result, the vacancies accumulate in the core and the resulting electrical field leads to a depletion of the vacancies near the grain boundary (see Fig. 12B). The concentration of charge carriers for transport across the boundary is therefore greatly diminished.

3

Mobility of charge carriers

3.1 Atomistic viewpoint Besides the concentration of defects, the conductivity is determined by the mobility of the defects, which can vary by orders of magnitude. In general, electronic defects are much more mobile than ionic defects and, consequently the electronic conductivity can exceed the ionic conductivity even for ce, ch ≪ cV, ci. In order to describe electronic mobility, one needs to take into account scattering processes due to defects and

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Fig. 11 Kr€oger-Vink diagram for donor-doped strontium titanate.

lattice vibrations (phonons). The interested reader is referred to the relevant literature (Gross and Marx, 2018; Grosso and Pastori Parravicini, 2014); here we focus on the mobility of ions as charge carriers. From the atomistic viewpoint, one can distinguish between three different migration mechanisms (see Fig. 13). In the vacancy mechanism, an ion jumps to a neighboring vacant site; in effect, the ion and the vacancy change places. In the interstitial Fig. 12 (A) Variation of the Gibbs formation energy close to the grain boundary. (B) The resulting vacancy profile close to the grain boundary. Based on De Souza, R.A., 2009. The formation of equilibrium space-charge zones at grain boundaries in the perovskite oxide SrTiO3. Phys. Chem. Chem. Phys. 11(43), 9939–9969.

Fundamentals of electrical conduction in ceramics

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mechanism, an interstitial ion moves to a neighboring free interstitial site. In the interstitialcy mechanism, an interstitial ion forces a neighboring ion from its regular lattice position onto an empty interstitial site, taking in the process the regular lattice position. For each of these migration mechanisms an energy barrier has to be overcome; ion motion is thus a thermally activated process.

3.2 Diffusion 3.2.1 Random walk Without any external driving force, a single particle in an otherwise empty, isotropic lattice performs a random walk. As an example, we monitor a single vacancy in a one-dimensional lattice. The vacancy performs a series of consecutive steps along the z-direction and in every step, the vacancy moves a distance Δzi of length l. The total displacement of the vacancy is given by the sum of all individual steps: Δz ¼

X

Δzi

(66)

i

To obtain the mean value hΔzi we can either perform n-independent runs for one vacancy or monitor an ensemble of n noninteracting vacancies; in any case, we obtain the result hΔzi ¼

1X Δz n

(67)

Fig. 13 Transport mechanisms on the atomic level: (A) vacancy mechanism, (B) interstitial mechanism, and (C) interstitialcy mechanism.

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As the movement of the vacancies is uncorrelated hΔzi equals zero. However, the mean squared displacement h(Δz)2i is not zero and we obtain D

ðΔzÞ

2

E

*

XX

+

Δzi Δzj ¼ hΔz  Δzi ¼ i j * + X XX

2 ðΔzi Þ + Δzi Δzj ¼ i

i

(68)

j6¼i

Δzi and Δzj are uncorrelated and the product can be positive or negative with equal probability. Thus, the second term on the right-hand side becomes zero. The first term is identical to nl2 as for any individual jump the quantity (Δzi)2 equals l2 and we obtain D

ðΔzÞ

2

E

* ¼

X

+ ðΔzi Þ

2

¼ nl2

(69)

i

If we introduce a constant frequency Γ for the vacancy jumps, the time for n jumps is n/Γ. We remember that in each step the jump can be in forward or backward direction and define the diffusion coefficient D as a characteristic quantity for the diffusion for the vacancies in one dimension: D¼

nl2 l2 ¼ Γ 2t 2

(70)

In other words, the characteristic length z that one vacancy travels is proportional to the square root of the time and the diffusion coefficient: z¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E

pffiffiffiffiffiffiffiffi ðΔzÞ2 ¼ 2Dt

(71)

Assuming isotropic diffusion in each direction, we can define the diffusion coefficient for the three-dimensional case where the vacancy can jump to Z neighboring positions with frequency Γ: l2 D ¼ ZΓ 6

(72)

3.2.2 Macroscopic In a lattice with a homogeneous distribution of vacancies or other mobile particles, the diffusion does not lead to any change in the distribution. In contrast, in the case of a concentration gradient, the diffusion will lead to a net flux of particles. Phenomenologically, the flux can be related to the concentration gradient through Fick’s first law, the constant of proportionality being the diffusion coefficient:

Fundamentals of electrical conduction in ceramics

j ¼
D

dc dx

301

(73)

The negative sign indicates that the flux is down the gradient and the flux leads to a decrease of the gradient. In most cases, the overall concentration of particles within a system does not change, that is, particles are neither generated nor destroyed. If we look at a small volume element or length element in the one-dimensional case, the difference between the inward flux and the outward flux equals the change of the concentration within the element. This is expressed by the continuity equation


dj dc ¼ dx dt

(74)

Combining Eqs. (73), (74) leads to Fick’s second law in one dimension   dc d dc ¼ D dt dx dx

(75)

Assuming that the diffusion coefficient is independent of the position we can write dc d2 c ¼D 2 dt dx

(76)

The one-dimensional expressions for the diffusion flux in Eq. (73) and the diffusion equation in Eq. (75) can be generalized to the three-dimensional case: j ¼
Drc

(77)

dc ¼ r  ðDrcÞ dt

(78)

Here, j is the vector flux, the Nabla operator r acts on the concentration field c and D is a symmetric (second-order) tensor. We will now take a closer look at the properties of the diffusion coefficient D. So far we followed the random walk of noninteracting vacancies in an otherwise occupied lattice and from Eq. (70) we can obtain the diffusion coefficient of the vacancies DV. From a practical point of view, we are more interested in the diffusion of the ions and the corresponding diffusion coefficient Dion. Since we know that the jump of one vacancy involves the jump of an ion in the opposite direction, the total jump frequency of ions and vacancies must be identical: cV ΓV ¼ cion Γion for the diffusion coefficient of the ions we thus obtain

(79)

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Advanced Ceramics for Energy Conversion and Storage

Dion ¼

½V  DV ½ion

(80)

For diluted defects we have [V] ≪ [ion] and can write the diffusion coefficient in terms of the site fraction nV: Dion  nV DV

(81)

The same expression holds true for interstitial ions. In an intrinsically Frenkel disordered material, both vacancies and interstitials can contribute to the diffusion of the ions and thus Dion nion ¼ DV nV + Di ni

(82)

For diluted defects, the diffusion coefficients of the defects are independent of the defect concentration. We will see in Section 3.5 that this is not generally the case in concentrated defect solutions. The diffusion coefficient of the ion in Eq. (82) is also called the self-diffusion coefficient. Experimentally this is impossible to measure as individual ions are indistinguishable. In practice, tracer diffusion experiments using stable or radioactive isotopes, which are chemically identical but distinguishable by mass or radioactive signature, allow the motion of chemically indistinguishable particles to be followed. The measured tracer diffusion coefficient is generally not identical to the selfdiffusion coefficient due to the correlated motion of the tracer ions. D∗ ¼ f ∗ Dion

(83)

The exact value of the correlation factor depends on the type of mechanism (vacancy, interstitial, and interstitialcy) as well as on the crystal structure. In general, smaller coordination numbers lead to smaller values of f ∗. Some examples are given in Table 2.

3.2.3 Temperature dependence Phenomenologically, the temperature dependence of the diffusion coefficient can be expressed by an Arrhenius-type equation,   EA (84) D ¼ D0 exp
kB T where D0 is a constant and EA is the activation energy. That is, when one plots the natural logarithm of experimentally determined values of the diffusion coefficient against inverse temperature one finds in a large number of cases a straight line defining an activation energy. What this activation energy actually refers to is a complex subject and we discuss this in detail below.

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Table 2 Tracer correlation factors for selected lattices and diffusion mechanisms (Waser and Ielmini 2016). Lattice

Mechanism

f∗

Diamond Simple cubic Bcc cubic Fcc cubic Any lattice Diamond

Vacancy Vacancy Vacancy Vacancy Interstitial Interstitialcy

0.5 0.6531 0.7272 0.7815 1 0.727

On a microscopic level, the activation energy is related to the difference of the Gibbs energy of the system between the ground state and the transition state, as illustrated in Fig. 14. While the ions vibrate around their equilibrium position, the Gibbs activation energy of migration determines the probability of a transition to a neighboring free site and the jump frequency is given by 

ΔGmig Γ ¼ υ0 exp
kB T

 (85)

where υ0 is the attempt frequency of an ion jump. The Gibbs energy can be separated into the corresponding activation enthalpy and activation entropy and for the selfdiffusion coefficient of the ions we obtain     ΔSmig ΔHmig Z 2 exp
Dion ¼ nV l υ0 exp kB kB T 6

(86)

It should be noted that the definition of the attempt frequency is not consistent in the literature and might include the entropic contribution or not. If we consider ΔHmig and ΔSmig to be independent of the temperature and compare Eqs. (84), (86) we see that EA equals ΔHmig, assuming nV to be constant. Fig. 14 Illustration of the energy barrier for the migration of an ion to a neighboring site.

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3.2.4 Intrinsic and extrinsic regime We now consider an oxide AO with intrinsic disorder dominated by anti-Frenkel defects and lightly doped with a lower valent oxide B2O that leads to the formation of oxygen vacancies. The temperature dependence of the diffusion in this system is illustrated in Fig. 15. In contrast to Eq. (84), we have no straight line but can identify two different regimes with an activation energy at low temperature and an activation energy at high temperature with EA, HT > EA, LT. We write the diffusion coefficient of oxygen ions as   ΔHmig DO ¼ nV κ exp
kB T

(87)

where we include all constants in κ. In the low-temperature regime, thermal excitation of defects is unlikely and the fraction of vacancies is set by the fraction of dopant ions

2 VO ¼ BA/ ¼ x. For the diffusion coefficient, we obtain 

  ΔHmig 1 DO ¼ xκ exp
kB T 2

(88)

with Eq. (84) we see that EA ¼ ΔHmig. In the high-temperature regime, the defect fraction is dominated by intrinsic defect disorder and the fraction of vacancies is given by



VO



  pffiffiffiffiffiffiffiffi ΔgAF ¼ KAF ¼ exp
2kB T

Combining Eq. (87) leads to Fig. 15 Illustration of the diffusion coefficient in the extrinsic and intrinsic regime. Dotted and dashed lines indicate the intrinsic and extrinsic contribution, respectively.

(89)

Fundamentals of electrical conduction in ceramics

  Δhmig ΔhAF 1 0
DO ¼ κ exp
kB T 2kB T 2

305

(90)

with Eq. (84), we see that EA ¼ ΔHmig + ΔhAF/2. Based on these considerations we can identify the two temperature regimes as an extrinsic regime, where the vacancy fraction is dominated by doping and the activation energy equals the migration enthalpy; and an intrinsic regime, where the vacancy fraction is dominated by antiFrenkel disorder and the corresponding enthalpy of formation adds to the activation energy. We see that the phenomenological activation energy could include various contributions and we will discuss further effects later on.

3.3 Conductivity While diffusion takes place without an external driving force, electrical conduction is the response to an external electric field. As ions are charged particles, an external electric field modifies the energy barriers as illustrated in Fig. 16. The energies of ground states and transition states are altered depending on particle’s charge and direction in the electric field. Consequently, the migration barriers are shifted in a zeroth-order approximation as well according to (Mott and Gurney, 1950; Verwey, 1935) ΔG

mig ¼ ΔGmig 

qlE 2

(91)

It should be noted that at higher electric fields, deviations from this approximation occur (Genreith-Schriever and De Souza, 2016). We can split the movement of an ion into a contribution in the + z-direction and
z-direction:   ΔG

mig z  ¼ t l υ0 exp
kB T 2

!

Fig. 16 Effect of an electrical field on the energy landscape.

(92)

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Without an external field, both parts will be identical and the net displacement equals zero. With a field we can apply Eq. (91) and obtain the displacement of the ions in the field direction:      ΔGmig
qlE=2 ΔGmig + qlE=2 l
exp
hzi ¼ t υ0 exp
kB T kB T 2

(93)

For weak electric fields with qlE/2 ≪ kBT we can simplify Eq. (93) to   ΔGmig ql2 Eυ0 exp
hzi ¼ t 2kB T kB T

(94)

The drift velocity of the ions in the electric field is given by v¼

  ΔGmig ql2 hzi Eυ0 exp
¼ 2kB T kB T t

(95)

Using the definition of the mobility in accordance to Ohm’s law, we obtain   ΔGmig v ql2 ql2 μ¼ ¼ υ0 exp
¼ Γ kB T 2kB T E 2kB T

(96)

The comparison with Eq. (70) shows that the electrical mobility of the ions is related to the diffusion coefficient by the Nernst-Einstein relation: μ¼

qD kB T

(97)

It should be noted that this relation is strictly true only for diluted, noninteracting defects. In contrast to diffusion, which leads to a broadening of a delta-distributed tracer, the electrical field leads to an additional shift along the field direction as shown in Fig. 17. Combining Eqs. (2), (97) we can obtain the diffusion coefficient from conductivity measurements: Dσ ¼ σ

kB T cq2

(98)

The relation between the conductivity diffusion coefficient and the tracer diffusion coefficient is called the Haven ratio HR ¼

D∗ Dσ

(99)

For noninteracting defects HR equals the tracer correlation coefficient but can deviate significantly in the case of interacting defects.

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Fig. 17 Illustration of the diffusional broadening of a delta-distributed tracer concentration and the effect of the drift in the electrical field. Based on Murch, G., 1982. The haven ratio in fast ionic conductors. Solid State Ionics 7 (3), 177–198.

For the temperature dependence of the ionic conductivity, we obtain an Arrheniuslike behavior: lnðσT Þ ¼ ln σ 0


EA kB T

(100)

3.4 Examples 3.4.1 Doped ceria We now come back to our example of Gd2O3-doped CeO2, as introduced in Section 2.6. The ionic conductivity is dominated by the migration of oxygen vacancies. In addition, we have electronic charge carriers and the total conductivity is given by σ total ¼ σ O + σ e + σ h

(101)

Ceria can be easily reduced according to = O O Ð VO + 2e + 1=2O2ðgÞ 

(102)

However, for typical doping levels of 10%–20%, the concentration of electrons formed by reduction is much lower than the number of oxygen vacancies even at oxygen partial pressure as low as 10
20 bar. Consequently, the charge neutrality is dom /



inated by GdCe ¼ 2 VO . Nevertheless, the mobility of electrons is orders of magnitude larger than the mobility of oxygen vacancies. With the individual mobilities and the Kr€oger-Vink diagram, we can plot the partial conductivities depending on oxygen partial pressure as presented in Fig. 18. As the concentration of vacancies is fixed by the dopant fraction, the ionic conductivity does not change with partial pressure. Hole conduction 

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plays a role only at exceedingly high oxygen partial pressures. At partial pressures around 1 bar, the total conductivity is dominated by the ionic contribution, while at low partial pressures electron conductivity is in the same range or even exceeds ionic conductivity. The proportion of the partial conductivities is characterized by the transference number ti ti ¼

σi σ total

(103)

For our example in Fig. 18, the ionic transference number decreases while the electronic transference number increases with decreasing oxygen partial pressure. The region where the ionic conductivity is much larger than the electronic conductivity, that is, tion  1, is called the ionic domain. It is noted that no consistent definition of the exact meaning of “much larger” exists but usually, a factor around 102 is assumed (Tuller and Nowick, 1975). In Fig. 18, the ionic domain for two temperatures is indicated by the shaded areas. It is apparent that the extent of the ionic domain decreases with increasing temperature since the creation of electronic charge carriers is favored at higher temperatures. The extent of the ionic domain is of interest, for example, for the application as solid electrolyte, which demands high ionic but low electronic conductivity. Operation outside of the ionic domain leads to leakage currents and thus lowered efficiency of an electrochemical cell.

3.4.2 LSCF Perovskite structured lanthanum strontium cobalt ferrite (LSCF) is a suitable material for oxygen permeation membranes and solid oxide fuel cell cathodes exhibiting both high ionic and high electronic conductivity. Doping the parent structure LaCoO3 with strontium on the La site increases the ionic conductivity through the formation of Fig. 18 Conductivity diagram for Gd-doped ceria for two temperatures adapted from Steele (1994) with hole conductivities extrapolated from Lubke and Wiemhofer (1999). Shaded areas indicate the ionic domains with tion > 0.99.

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oxygen vacancies while doping iron on the Co-sites increases the chemical stability (Bouwmeester et al., 2004). The incorporation of Sr on the La site is compensated by both oxygen vacancies and localized holes:  2SrO + 2La La + 0:5O2 + 2FeCo ! La2 O3 + 2SrLa/ + 2FeCo

(104)

 2SrO + 2La La + OO ! La2 O3 + 2SrLa/ + VO

(105)





In addition, the material is in equilibrium with atmospheric oxygen and reducing the partial pressure removes oxygen from the lattice.   2FeCo =CoCo + O O ! 0:5O2 + VO + 2FeCo =CoCo

(106)





2 VO + CoCo + FeCo ¼ ½SrLa/ 

(107)













With decreasing oxygen partial pressure, the concentration of oxygen vacancies increases at the expense of electron holes, which results in a decrease of the total electrical conductivity as observed in Fig. 19. The same is true for the temperature dependence. For increasing temperature, the endothermic reaction in Eq. (106) is shifted to the right side leading to increasing nonstoichiometry. Simultaneously the concentration of electronic charge carriers is diminished leading to decreasing conductivity with increasing temperature.

3.4.3 Sodium chloride The transition from the extrinsic to the intrinsic regime has been measured for nominally pure NaCl, as shown in Fig. 20 (Etzel and Maurer, 1950). The conductivity is due to sodium vacancies, which are formed in the intrinsic regime by Schottky disorder:  Na Na + ClCl Ð VNa/ + VCl + NaCl 

(108)

At low temperatures, the defect chemistry is dominated by impurities such as Cd2+ ions and the concentration of sodium vacancies is fixed. The activation energy of the conductivity thus equals the enthalpy of the ionic mobility:   σ0 0:85 eV σ ¼ exp
T kB T

(109)

At high temperatures, additional vacancies are formed due to Schottky disorder and the concentration depends on the temperature. In accordance with Eq. (90) we can calculate the energies of migration and formation from the experimental data:

310

Fig. 19 (A) The nonstoichiometry and (B) the total conductivity in La0.6Sr0.4Co0.2Fe0.8O3
δ depending on oxygen partial pressure. Data from Bouwmeester, H.J.M., Den Otter, M.W., Boukamp, B.A., 2004. Oxygen transport in La0.6Sr0.4Co1yFeyO3-d. J. Solid State Electrochem. 8(9), 599–605.

Fig. 20 Conductivity of nominally pure NaCl depending on temperature. Data from Etzel, H.W., Maurer, R.J., 1950. The concentration and mobility of vacancies in sodium chloride. J. Chem. Phys. 18(8), 1003–1007.

Advanced Ceramics for Energy Conversion and Storage

Fundamentals of electrical conduction in ceramics

    σ0 0:85 eV 2:02 eV exp
σ ¼ exp
T kB T 2kB T

311

(110)

3.5 Interactions So far we have relied on the critical assumption of noninteracting defects forming an ideal solution. At high defect concentrations, this assumption is not valid anymore, although a critical concentration is not easy to predict and generally depends on the structure, dielectric properties and charge of the defects. When moving from an ideal solid solution to the concentrated system, we generally have to change the site fractions in the previous equations to the corresponding activities, that is, introducing an activity coefficient γ. We can try to approximately calculate the activity coefficient or circumvent the need for activities by explicitly defining defect associates and we will discuss both approaches shortly. In an ionic crystal, the interaction between defects can be expected to be mainly of electrostatic nature. The interaction energy of two defects with charges q1 and q2 for a given distance r can then be calculated from Coulomb’s law: EC ¼

q1 q2 4πεr ε0 r

(111)

€ 3.5.1 Debye-Huckel For defects still in the dilute regime, we can apply the Debye-H€uckel theory known from liquid electrolyte solutions to obtain an approximation of γ. The theory is based on the picture of a charged defect being placed in a cloud of opposite charge. For simplicity we consider only defects with z+ ¼ j z
j here and define the radius of the spherical cloud by the Debye length:  λ¼

εr ε0 k B T 2q2 NA c

1=2 (112)

Assuming that λ is much larger than the radius of the centered point defect we can write the average interaction energy for each defect as ΔHD ¼


q2 8πεr ε0 λ

(113)

This quantity is always negative due to the attractive interaction between the charged defect and the cloud that leads to stabilization of the defect. In fact, this is just the difference between the chemical potential of the defects in the ideal system and in the real system. We can, therefore, relate the quantity ΔHD to the activity coefficient by

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Advanced Ceramics for Energy Conversion and Storage

kB T lnðγ Þ ¼ μreal
μideal ¼


q2 8πεr ε0 λ

(114)

and see that ln(γ) is proportional to c1/2. However, this approximation is only valid for fairly dilute defects and for higher concentration, phenomenological corrections were suggested, for example, with a proportionality of ln(γ) to c1/3. A particular drawback of this approach is that it neglects any structural aspects of the ionic lattice. For small distances between pairs of defects, the structural peculiarities, as well as interactions beyond electrostatics will play a more important role and an explicit treatment of defect associates might be more useful.

3.5.2 Associates We can define associates such as defect pairs or larger defect cluster as new quasichemical species. One type of associate is the already introduced localization of an electronic defect on an ion or on an ionic defect. In the following, we will focus on the association of ionic defects. As a start, we consider intrinsically Frenkel disordered AgI. Due to the opposite charge, the defects are likely to be forming neutral defect pairs:   (115) VAg/ + Agi Ð VAg/
Agi 



For the reaction, we obtain the corresponding mass action law:  

VAg/
Agi



Kass ¼ VAg/ Agi 



(116)

with the charge neutrality condition, we obtain the total number of silver vacancies and interstitials:

VAg/





 

  ¼ VAg/ + VAg/
Agi ¼ KF1/2 1 + Kass KF1/2 

total

(117)

Apparently, the total number of vacancies is increased by the factor (1 + KassK1/2 F ), due to the stabilizing effect of the interactions. However, as the aggregates are neutral they will not contribute to the ionic conductivity and the association should have no measurable effect on the conductivity. Now we turn to the extrinsic regime of AgI doped with CdI2, where Ag-vacancies compensate the dopants. The vacancies now form neutral clusters with the dopant ions:   VAg/ + CdAg Ð VAg/
CdAg 



h  i VAg/
CdAg i Kass ¼

h VAg/ CdAg

(118)





(119)

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313

In this case, the total fraction of defects is defined by the dopant fraction x: x¼

h  i h i + CdAg VAg/
CdAg 



For strongly associated defects we can assume

(120) h  i h i / VAg
CdAg ≫ CdAg and, con



sequently x  [(V/Ag
CdAg)]. In combination with Eq. (119) and the charge neutrality condition we get for the fraction of nonassociated Ag-vacancies: l



VAg/





x ¼ Kass

1=2 (121)

Consequently, the concentration of vacancies that contribute to the conductivity is not linear in the dopant fraction but depends on its square root as shown in Fig. 21. It is noted that at high defect fractions the definition of distinct pairs or clusters becomes somewhat arbitrary. Consider, for example, cerium oxide with a Gd content of 20%. In the first and second cation shell around an anion position, there are 4 and 12 cation positions, respectively. Statistically, the chance to find a vacancy without a gadolinium ion in the first two shells is thus very small and we have no longer isolated defect pairs but a network of interacting defects. In addition, for highly ordered defect structures the change of the configurational entropy has to be considered. In the derivation of the Boltzmann-approach in Section 2.4, we assumed a statistical distribution of the defects that defines the number of microstates Ω. For strongly associated defects this statistical treatment is no longer correct as the configurational space shrinks and the original Boltzmann approach is not valid anymore (Grieshammer and Martin, 2017). In this sense, the application of simple models might be limited and simulations such as Monte Carlo or molecular dynamics are necessary to describe the effect of defect interactions correctly. Fig. 21 Illustration of the transition from nonassociated regime to the associated regime in CdI2doped AgI.

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3.5.3 Example: Doped ceria A classic example of defect interactions and the effect on ionic conductivity is doped ceria. The doping with trivalent oxides leads to the formation of oxygen vacancies. Consequently, one could expect a steady increase in the conductivity with increasing dopant fractions. However, defect interactions lead to a shift of the maximum to lower dopant fractions typically around 10%–20% as shown in Fig. 22. Two effects can be separated here: Large dopant ions situated along the migration path of the oxygen ion lead to an increase of the migration barrier and thus a blocking of the migration. In addition, the dopant ions attract the oppositely charged vacancies leading to a trapping effect (Grieshammer et al., 2014; Koettgen et al., 2018). The formation of a cluster  /  GdCe -VO has been used to explain the kink in the experimental curve in Fig. 22 (Zhan et al., 2001). The simple picture is that at high temperatures, the defects are not associated and the conductivity is given by 

Fig. 22 (A) The ionic conductivity of Gd-doped ceria in the bulk interior depending on the dopant fraction (Tianshu et al., 2002). (B) Arrhenius-type diagram for Ce0.9Sm0.1O1.95 (Zhan et al., 2001).

Fundamentals of electrical conduction in ceramics



ΔHmig σT ¼ σ 0 exp
kB T

315

 (122)

In contrast, at low temperatures, the vacancies are trapped at dopant ions and in each migration step the barrier is increased by the effective association enthalpy, leading to the conductivity expression   eff ΔHmig + ΔHass σT ¼ σ 0 exp
kB T

(123)

Based on these considerations the effective association enthalpy can be obtained as the difference of the macroscopic activation energy at low and high temperature. We use the word “effective” here because it is not necessarily the microscopic association energy related to the interaction of one dopant ion and one oxygen vacancy. The conduction in heavily doped materials is a complex process including interactions in several distances and various jump environments for the moving ions with different migration barriers. The presented formalism is, therefore, a drastic simplification and a deeper understanding is only possible by performing simulations (Grieshammer et al., 2014; Koettgen et al., 2018).

3.5.4 Example: Strontium titanate A further example of the interaction between dopant ions and oxygen vacancies is perovskite structured SrTiO3. Ni2+ on the Ti4+ sites leads to the creation of oxygen //

vacancies and in the oxidizing atmosphere, the charge neutrality NiTi ¼ VO can be assumed. Experimental measurements revealed two distinct activation energies for Ni-doped SrTiO3 in the low-temperature regime (1.0 eV) (Waser, 1991) and nominally pure SrTiO3 in the high-temperature regime (0.62 eV) (De Souza et al., 2012) as depicted in Fig. 23. In a simple approximation, it can be assumed that part of the oxygen vacancies is immobilized by association with Ni-dopants: 

  // //  Ð VO
NiTi VO + NiTi 



(124)

As a result, the conductivity depends on the concentration of free vacancies. Based on the equilibrium constant of association Ka and the total amount of nickel dopants [Ni//Ti]tot, Schie et al. (2014) evaluated the concentration of free vacancies which is given by

i 1=2

//

1 h 4Ka NiTi +1
1 VO ¼ tot 2Ka 

(125)

Again, such a treatment ignores that activation barriers may also be affected by the presence of the dopant. A superior approach, therefore, is to consider the trapping

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Fig. 23 Conductivity of acceptor-doped SrTiO3 in the high and low-temperature regime according to the experiment and model. Based on Schie, M., Waser, R., De Souza, R.A., 2014. A simulation study of oxygenvacancy behavior in strontium titanate: beyond nearestneighbor interactions. J. Phys. Chem. C 118(28), 15185–15192.

effect, as the time the vacancies are trapped by the dopants compared to the time the vacancies are free. The conductivity is then modified by the factor: α¼

tfree tfree + ttrapped

(126)

where tfree and ttrapped depend on the rates of trapping, detrapping, and free migration.

3.6 Grain boundary conductivity So far we have focused on conductivity within the bulk material but in polycrystalline samples extended defects exist, in particular, grain boundaries. We can thus differentiate between bulk conductivity within a single grain and grain-boundary conductivity across or along the grain boundary separating two grains. In principle, the grainboundary conductivity can be higher or lower than the bulk value for both cases (across or along the grain boundary). Individual grain boundaries are characterized by their tilt and twist axes, their tilt and twist angles, and the interface plane. In a typical polycrystalline sample, grain boundaries with different parameters exist, and measurements such as electrochemical impedance spectroscopy give an (weighted) average over all types. In metals, the diffusional transport along dislocations has been demonstrated to be accelerated due to a more open structure compared with that of the bulk. In contrast, for ceramics the picture is not as clear. There are cases, O in Al2O3 ((Nakagawa et al., 2006), Mg in MgO (Sakaguchi et al., 1992), and Ni in NiO (Atkinson and Taylor, 1979)) where fast diffusion along dislocations has been unambiguously observed. There are also other cases (e.g., O in SrTiO3 (Schraknepper et al., 2018)) where arrays of dislocations block diffusion rather than accelerating it. The key factor here appears

Fundamentals of electrical conduction in ceramics

317

to be the structure. In those structures in which the migration of an ion has a highenergy barrier, the structural perturbation of a dislocation yields a decrease in barrier height. In contrast, in those cases in which the barrier is low and ion transport is facile, the structural perturbation increases the barrier, making ion transport more difficult (Metlenko et al., 2014). As an example, we show the ionic conductivity of Gd2O3-doped ceria in Fig. 24. Depending on the dopant fraction and temperature, the bulk conductivity can be an order of magnitude higher than the grain boundary conductivity. Likewise, the activation energy for bulk conductivity is considerably smaller than for grain boundary conductivity. Several reasons could lead to a hindrance of transport across the grain boundary. The mismatch between the adjacent grains might result in an increased migration barrier. In addition, the formation of secondary phases at the grain boundary could create an insulating layer between the grains. Formation of SiO2 at the interface is a known

Fig. 24 Bulk and grain boundary conductivity for Gd-doped ceria. (A) Dependence on the Gd-fraction at 400 °C. (B) Arrhenius-type plot for doping fraction of 0.15 (Tianshu et al., 2002).

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source for increased resistance. Furthermore, in ionic materials the formation of a space charge layers as described in Section 2.7 can lead to a depletion of mobile defects in the region of the grain boundary and consequently a decrease of the conductivity. In general, one should carefully check whether reported conductivity values refer to bulk, grain boundaries, or overall conductivities.

References Atkinson, A., Taylor, R.I., 1979. The diffusion of Ni in the bulk and along dislocations in NiO single crystals. Philos. Mag. A 39 (5), 581–595. Balazs, G., Glass, S., 1995. AC impedance studies of rare earth oxide doped ceria. Solid State Ionics 76 (1–2), 155–162. Bouwmeester, H.J.M., Den Otter, M.W., Boukamp, B.A., 2004. Oxygen transport in La0.6Sr0.4Co1-yFeyO3-d. J. Solid State Electrochem. 8 (9), 599–605. Chiang, Y.-M., Birnie, D.P., Kingery, W.D., 1997. Physical Ceramics. J. Wiley, New York. De Souza, R.A., Metlenko, V., Park, D., Weirich, T.E., 2012. Behavior of oxygen vacancies in single-crystal SrTiO3: equilibrium distribution and diffusion kinetics. Phys. Rev. B. 85(17). Engstrom, H., Bates, J.B., Brundage, W.E., Wang, J.C., 1981. Ionic conductivity of sodium beta00 -alumina. Solid State Ionics 2 (4), 265–276. Etzel, H.W., Maurer, R.J., 1950. The concentration and mobility of vacancies in sodium chloride. J. Chem. Phys. 18 (8), 1003–1007. Fagg, D.P., Kharton, V.V., Kovalevsky, A.V., Viskup, A.P., Naumovich, E.N., Frade, J.R., 2001. The stability and mixed conductivity in La and Fe doped SrTiO3 in the search for potential SOFC anode materials. J. Eur. Ceram. Soc. 21 (10), 1831–1835. Genreith-Schriever, A.R., De Souza, R.A., 2016. Field-enhanced ion transport in solids: Reexamination with molecular dynamics simulations. Phys. Rev. B. 94 (22). Grieshammer, S., Martin, M., 2017. Influence of defect interactions on the free energy of reduction in pure and doped ceria. J. Mater. Chem. A 5 (19), 9241–9249. Grieshammer, S., Grope, B.O.H., Koettgen, J., Martin, M., 2014. A combined DFT + U and Monte Carlo study on rare earth doped ceria. Phys. Chem. Chem. Phys. 16 (21), 9974. Gross, R., Marx, A., 2018. Festk€orperphysik, third ed. De Gruyter, Berlin; Boston. Grosso, G., Pastori Parravicini, G., 2014. Solid State Physics, second ed. Academic Press, an imprint of Elsevier, Amsterdam. Guin, M., Tietz, F., Guillon, O., 2016. New promising NASICON material as solid electrolyte for sodium-ion batteries: correlation between composition, crystal structure and ionic conductivity of Na3+xSc2SixP3
xO12. Solid State Ionics 293, 18–26. Han, M., Tang, X., Yin, H., Peng, S., 2007. Fabrication, microstructure and properties of a YSZ electrolyte for SOFCs. J. Power Sources 165 (2), 757–763. Howard, R.E., Lidiard, A.B., 1964. Matter transport in solids. Rep. Prog. Phys. 27, 161–240. Huang, K., Feng, M., Goodenough, J.B., 1996. Sol-gel synthesis of a new oxide-ion conductor Sr- and mg-doped LaGaO3 perovskite. J. Am. Ceram. Soc. 79 (4), 1100–1104. Kittel, C., 1976. Introduction to Solid State Physics, fifth ed. Wiley, New York. Koettgen, J., Grieshammer, S., Hein, P., Grope, B.O.H., Nakayama, M., Martin, M., 2018. Understanding the ionic conductivity maximum in doped ceria: trapping and blocking. Phys. Chem. Chem. Phys. 20, 14291–14321.

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Lubke, S., Wiemhofer, H.D., 1999. Electronic conductivity of Gd-doped ceria with additional Pr-doping. Solid State Ionics 117 (3–4), 229–243. Ma, Q., Xu, Q., Tsai, C.-L., Tietz, F., Guillon, O., Dunn, B., 2016. A novel sol-gel method for large-scale production of nanopowders: preparation of Li1.5Al0.5Ti1.5(PO4)3 as an example. J. Am. Ceram. Soc. 99 (2), 410–414. Maier, J., 2004. Physical Chemistry of Ionic Materials: Ions and Electrons in Solids. Wiley, Chichester; Hoboken, NJ. Metlenko, V., Ramadan, A.H.H., Gunkel, F., Du, H., Schraknepper, H., Hoffmann-Eifert, S., Dittmann, R., Waser, R., De Souza, R.A., 2014. Do dislocations act as atomic autobahns for oxygen in the perovskite oxide SrTiO3? Nanoscale 6 (21), 12864–12876. Mott, N.F., Gurney, R.W., 1950. Electronic Processes in Ionic Crystal, second ed. Oxford University Press, Oxford. Murugan, R., Thangadurai, V., Weppner, W., 2007. Fast lithium ion conduction in garnet-type Li7La3Zr2O12. Angew. Chem. Int. Ed. 46 (41), 7778–7781. Nakagawa, T., Nakamura, A., Sakaguchi, I., Shibata, N., Lagerl€ of, K.P.D., Yamamoto, T., Haneda, H., Ikuhara, Y., 2006. Oxygen pipe diffusion in sapphire basal dislocation. J. Ceram. Soc. Jpn. 114 (1335), 1013–1017. Sakaguchi, I., Yurimoto, H., Sueno, S., 1992. Self-diffusion along dislocations in single-crystals MgO. Solid State Commun. 84 (9), 889–893. Schie, M., Waser, R., De Souza, R.A., 2014. A simulation study of oxygen-vacancy behavior in strontium titanate: beyond nearest-neighbor interactions. J. Phys. Chem. C 118 (28), 15185–15192. Schraknepper, H., Weirich, T.E., De Souza, R.A., 2018. The blocking effect of surface dislocations on oxygen tracer diffusion in SrTiO3. Phys. Chem. Chem. Phys. 20 (22), 15455–15463. Steele, B.C.H., 1994. Oxygen-transport and exchange in oxide ceramics. J. Power Sources 49 (1–3), 1–14. Tianshu, Z., Hing, P., Huang, H., Kilner, J., 2002. Ionic conductivity in the CeO2–Gd2O3 system (0.05 Gd/Ce 0.4) prepared by oxalate coprecipitation. Solid State Ionics 148 (3), 567–573. Tuller, H.L., Nowick, A.S., 1975. Doped ceria as a solid oxide electrolyte. J. Electrochem. Soc. 122 (2), 255–259. Verwey, E.J.W., 1935. Electrolytic conduction of a solid insulator at high fields the formation of the anodic oxide film on aluminium. Physica 2 (1), 1059–1063. Waser, R., 1991. Bulk conductivity and defect chemistry of acceptor-doped strontium-titanate in the quenched state. J. Am. Ceram. Soc. 74 (8), 1934–1940. Waser, R., Ielmini, D., 2016. Resistive Switching From Fundamentals of Nanoionic Redox Processes to Memristive Device Applications. Wiley-VCH, Weinheim. Zhan, Z.L., Wen, T.L., Tu, H.Y., Lu, Z.Y., 2001. AC impedance investigation of samariumdoped ceria. J. Electrochem. Soc. 148 (5), A427–A432.

Further reading Bransky, I., Wimmer, J.M., 1972. The high temperature defect structure of CoO. J. Phys. Chem. Solids 33 (4), 801–812. De Souza, R.A., 2009. The formation of equilibrium space-charge zones at grain boundaries in the perovskite oxide SrTiO3. Phys. Chem. Chem. Phys. 11 (43), 9939–9969.

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Kasap, S.O., 2006. Principles of Electronic Materials and Devices, third ed. McGraw-Hill, Boston. Murch, G., 1982. The haven ratio in fast ionic conductors. Solid State Ionics 7 (3), 177–198. Tuller, H.L., Bishop, S.R., 2010. Tailoring material properties through defect engineering. Chem. Lett. 39 (12), 1226–1231. Zacate, M.O., Minervini, L., Bradfield, D.J., Grimes, R.W., Sickafus, K.E., 2000. Defect cluster formation in M2O3-doped cubic ZrO2. Solid State Ionics 128 (1–4), 243–254.