Ion conduction and redistribution at grain boundaries in oxide systems

Progress in Materials Science 89 (2017) 252–305

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Ion conduction and redistribution at grain boundaries in oxide systems q Giuliano Gregori, Rotraut Merkle, Joachim Maier ⇑ Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany

a r t i c l e

i n f o

Article history: Received 25 December 2016 Received in revised form 19 April 2017 Accepted 22 April 2017 Available online 29 April 2017 Keywords: Grain boundaries Interfaces Point defects Space charge effects Ionic conductivity

a b s t r a c t The review provides a comprehensive overview on the major findings regarding ion redistribution at interfaces in oxide systems and its effects on the electrical transport properties. As far as interfaces are concerned, grain boundaries and hence polycrystalline materials are to the fore. Heterocontacts and hence composites are only considered if relevant for the general understanding. Selected examples refer to oxide ceramics but also to composites. As far as the fundamental properties are concerned, major emphasis is laid on the impact on ion conductivity. Purely electronic effects (e.g. interfaces in semiconductors, boundaries in superconductors, or the formation of two-dimensional electron gases at the interface between two insulators) as well as phenomena related with solid/fluid interfaces are not addressed. Ó 2017 Elsevier Ltd. All rights reserved.

Contents 1. 2.

3.

q

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Charge carrier concentration and conductivity at interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Conductivity of polycrystalline materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Chemical effects in polycrystalline materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Nanocrystalline ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Stoichiometry polarization caused by grain boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Ion mobility effects at interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Identifying and quantifying space-charge effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detailed discussion of selected examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Perovskites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. SrTiO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. BaTiO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. (Pb,La)(Zr,Ti)O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4. BaZrO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5. (La, Sr)GaO3
d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6. Li3xLa2/3
xTiO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Fluorites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

⇑ Corresponding author.

E-mail addresses: [email protected] (R. Merkle), [email protected] (J. Maier). http://dx.doi.org/10.1016/j.pmatsci.2017.04.009 0079-6425/Ó 2017 Elsevier Ltd. All rights reserved.

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List of symbols and abbreviations Variables Description cdop dopant concentration concentration of defect j cj cj;bulk bulk concentration of defect j cmaj;bulk bulk concentration of the enriched majority charge carrier cd ambipolar concentration chemical capacitance Cd C bulk bulk capacitance C gb grain boundary capacitance grain size dg Dd chemical diffusion coefficient D tracer diffusion coefficient e elementary charge (1.602  10
19 C) F Faraday constant I current  k tracer effective surface rate constant kB Boltzmann constant (1.381  10
23 J/K) mass action constant of a defect formation reaction Kr Mj exponent of the dependence of the defect concentration on dopant concentration n ideality parameter of a constant phase element Q gb Nj exponent of the dependence of the defect concentration on pO2 pO2 oxygen partial pressure Qcore charge density at grain boundary core Q gb effective capacity of constant phase element R gas constant or, electrical resistance, cf. context Rbulk bulk resistance Rgb grain boundary resistance Rd chemical resistance mobility of charge carrier j uj ti ionic transference number T temperature V voltage Y jj conductance along the direction parallel to the direction of the current zdop charge number of the dopant zj charge number of mobile charge carrier j charge number of the enriched majority charge carrier zmaj cr;j characteristic exponent of the dependence of the defect concentration in K r dgb thickness of the depletion zone D/0 space charge potential D/0 ¼ /0
/bulk k Debye length k screening length (Mott-Schottky case) e0 vacuum permittivity (8.854  10
12 F/m) eeff effective dielectric constant er relative dielectric constant ebulk bulk dielectric constant l chemical potential l standard chemical potential l~ electrochemical potential qbulk bulk resistivity qgb grain boundary resistivity rk conductivity along the grain boundaries parallel to the direction of current r? conductivity through the grain boundaries perpendicular to the direction of current reon electronic conductivity rsgb specific grain boundary conductivity rion ionic conductivity rj;bulk bulk conductivity of charge carrier j rm effective measured conductivity (including parallel and perpendicular boundary contributions) rm;gb macroscopic or measured grain boundary conductivity rd ambipolar conductivity

253

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sd /ðxÞ /0 /bulk

mgb x xbulk xgb

4.

5.

chemical diffusion relaxation time space charge potential as a function of the coordinate x space charge potential at x = 0 bulk space charge potential (usually set equal to zero) grain boundaries volume fraction angular frequency bulk angular frequency grain boundary angular frequency

3.2.1. Ceria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Acceptor doped zirconia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Grain boundary transport properties of other structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Corundum (a-Al2O3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Rock salt structured oxides (MgO, NiO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Rutile TiO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4. Wurtzite (ZnO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5. Apatite-related structures (La9.33(SiO4)6O2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6. Spinel (Li4Ti5O12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.7. Garnet (Li7La3Zr2O12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comprehensive discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Materials overview: similarities and differences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Atomistic understanding of defect segregation to grain boundary cores. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Effects on ionic transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Blocking of ionic transport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Fast O diffusion along grain boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3. Accelerated cation diffusion along grain boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4. Mobility and strain effects (thin films) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Approaches for grain boundary engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277 279 280 280 281 282 284 285 285 285 286 286 287 291 291 291 292 292 294 295 296 296

1. Introduction Ion conduction in solids is a highly relevant phenomenon, as it is significant for enabling compositional changes in ionic materials and hence for preparation and degradation of ionic materials; it is most important for the electric function of electroceramics. Ion conduction is the key characteristic of solid electrolytes and hence for the use in fuel cells, chemical sensors or chemical reactors. Here the electronic contribution is desired to be as small as possible. Archetypal materials are zirconia and ceria. If both ionic and electronic conductivity are significant, one speaks of mixed conductors; such materials are in principle suited for electrodes, permeation membranes and chemical sensors. The electrode function of mixed conductors is particularly relevant for Li- or Na-based batteries where beyond transport also storage of ions and electrons is required. Of particular significance is the mixed conductor as model material for discussing charge carrier chemistry on a general level (here SrTiO3 is most significant). In various materials ionic conduction is unwanted, and if present may lead to degradation (e.g. BaTiO3 capacitors). Ion conduction is enabled by ionic point defects such as vacancies or interstitials. Point defects are – in spite of the positive formation enthalpy – relevant centers even in thermodynamic equilibrium owing to the gain of configurational entropy on forming them. Yet, not all point defects are present in their equilibrium concentration, as frozen-in situations or the presence of dopants show. Higher dimensional defects are – apart from surfaces or the heterophase contacts which are necessary for constituting the Wulff-shape – non-equilibrium structure elements. The configurational entropy gain is minute, while the formation energy is large. (We ignore here domain boundaries that might be even energetically favored.) Nonetheless, in terms of practical significance such non-equilibrium interfaces are of high significance for ion conduction. They are typically frozen-in structure elements and set the boundary conditions for point-defect redistribution. Hence, charge carrier concentrations at interfaces are very different from the bulk values. This may lead to accumulation or depletion effects. Furthermore the core of the interface itself represents a locus of varied structure, and hence of transport properties. Particularly difficult to treat are cases in which the interfaces are structurally smeared out towards the bulk. We mostly refer here to the abrupt structural model where structure and hence the local chemical potential of a given point defect behaves as a step function. This is not only a convenient assumption but in many cases of interest quite accurate.

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In this contribution we concentrate on grain boundaries, ignore surfaces and consider solid/solid heterointerfaces only if they are relevant for understanding the charge carrier redistribution phenomena. Dislocations as one-dimensional defects, which play an intermediate thermodynamic and kinetic role between point defects and interfaces, are addressed only marginally. Also stacking faults (which are less abundant in ionic solids compared to metals) are not in the focus. Speaking about grain boundaries, the manifold of these structure elements is immense. They range from low-angle (tilt or twist) boundaries to large-angle boundaries or even veritable, sometimes amorphous, interphases frequently containing different structural elements than the bulk. The grains they connect are not only differently oriented – they may differ in size, shape and composition. In view of the scope of our paper, this contribution uses interfacial thermodynamic (see Refs. [1–3]) and atomistic approaches sporadically and strategically whenever appropriate, while the charge carrier chemistry (point defect situation) is treated and reviewed more systematically. 2. Theoretical considerations 2.1. Charge carrier concentration and conductivity at interfaces Transport properties of ionic crystals are caused by ionic and/or electronic defects such as excess or deficient particles, as described by bulk defect chemistry. For more details the reader is referred to Refs. [4–7]. Defects are described by the KrögerVink notation, which specifies the type of the defect as well as its site (subscript) and charge relative to the perfect lattice (superscript 0 = negative,  = positive). As characteristic centers we consider ionized oxygen interstitials ðO00i Þ and vacancies  ðVO Þ as well as excess electrons ðe
Þ and holes ðh Þ. For simplicity we only consider the oxygen sublattice to reversibly interact with the outer atmosphere [8]. The extension to more carriers in equilibrium or to associates is straightforward. Note the following relations for the defects’ chemical potentials l

lO2
¼ li ¼
lv le
¼ ln ¼
lP

ð2:1Þ

where i; v ; n; p refer to the building units interstitial ðO00i
Vi Þ, vacancy ðVO
OO Þ, excess electron (e
¼ excess electron in  conduction band minus free electronic site in conduction band, h ¼ vacant e
-site in valence band minus regular e
in valence band). At elevated temperature where bulk transport and surface exchange kinetics is sufficiently fast, the oxide equilibrates with the gas phase according to

OOx

1 O2 þ VO þ 2e0 2

ð2:2Þ

In general for a binary oxide under dilute conditions and for a simple defect chemistry the equilibrium concentration cj;bulk of any charge carrier is given by M

cj;bulk ðT; cdop ; pO2 Þ ¼ aj pO2 Nj cdopj

Y

K r ðTÞcrj :

ð2:3Þ

r

as a function of the control parameters T, oxygen partial pressure pO2 , and dopant concentration cdop (comprising immobile foreign but also native defects). The dopant concentration influences the defect concentrations via the electroneutrality conP dition j zj cj;bulk ¼ 0 with defect charge zj. The Kr’s are the mass action constants of the internal and external defect reactions (e.g. K O ¼ pO2 cVO c2e0 for reaction (2.2) where cOxO  const: is implicitly included in KO). They are responsible for the typically sensitive dependence of cj;bulk on T. Fig. 2.1 gives some examples, along with defect-chemical diagrams showing the variation of cj;bulk with the control parameters pO2 ; T, cdop. Note that the K’s are composed of the l0j ’s. The exponents N j ; M j ; crj depend on the actual defect chemical model. In multinary systems in which other sublattices are affected by compositional changes, other partial pressures need to be considered. In various cases of interest Eq. (2.3) applies, namely when the other sublattices are frozen. Then the concentration of these defects is determined by preparational conditions, and simply enter the term cdop as dopants. If it is the cationic sublattice that is mobile, Eq. (2.3) is still correct for a binary, while for a multinary the definition of the chemical environment is less trivial. The respective bulk conductivities cj;bulk (cf. Fig. 2.1) follow by multiplication with the molar charge zjF and the mobility uj that for dilute conditions is only a function of temperature:

rj;bulk ¼ cj  uj zj F

ð2:4Þ

Typically, the mobility of electronic defects is several orders of magnitude larger than that of ionic defects. For oxides with fluorite and perovskite type structure, the mobility of oxygen defects (VO ) is again much higher than that of cation defects (cation vacancies), making dopant cations essentially immobile. On the other hand, in many binary oxides cation defects are the majority ionic carriers (see examples in Section 3). Regarding the temperature dependence of the conductivity, it is important to note that it contains contributions from the carrier’s migration barrier as well as the T-dependence of their concentrations (unless that is fixed by doping). Before studying realistic polycrystalline materials, understanding of the electric behavior of a bicrystal is necessary. It obviously matters if one considers transport along or parallel to the interface. At this moment we do not specify the nature

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Fig. 2.1. Point defect concentrations (bottom row) and conductivities (top row) as function of dopant content and pO2 at a constant temperature, reon, rion = electronic, ionic conductivity. (a) for a material in which the Schottky reaction determines the intrinsic defect concentrations in undoped samples and donor dopants are compensated by cation vacancies (e.g. SrTiO3), (b) in which the Frenkel reaction determines the intrinsic defect concentrations in undoped samples and donor dopants are compensated by oxygen interstitials (e.g. CeO2). Since the acceptor doped case for (b) is similar to that in (a) it is not displayed. The predominant electroneutrality conditions for the respective regimes are indicated. All defect concentrations that are not fixed by the dopant concentration will change with temperature according to the T-dependence of the respective mass action constants.

of the boundary, nor do we discuss their energies and degree of freedom (orientation, compositional deviation, etc.). For such questions the reader is referred to [3,9]. We simply parametrize the boundary core by local standard chemical potentials that are different from the bulk values. The electric interface effects have (at least) two contributions, one from the core region that exhibits a structure different from the bulk, the second from the space charge zones adjacent. Even though grain boundaries are a consequence of grain to grain orientation we will as first approximation ignore effects of orientation on transport. Irrespective of this, the space charge regions are naturally inhomogeneous, which itself gives rise to an electric anisotropy. If one measures along the interface, the highly conducting parts of the profile are dominant, whereas the more resistive ones are dominant in measurements across the boundary. In the following we ignore structural inhomogeneities (lateral or across) in the core regions and calculate the charge carrier concentration profiles under idealized conditions. Then we calculate the conductivity effects in both directions assuming spatially invariant mobilities (for mobility effects see Section 2.7). This is the abrupt core-space charge model used in Ref. [10]. Owing to the different structure that the grain boundary core exhibits, characterized by a standard chemical potential l0j;core of the charged constituents differing from that of the grain interior l0j;bulk , there will be a charging. This charging is a consequence of the individual carrier redistribution in order to satisfy constancy of the electrochemical potential

l~ j;core ¼ l~ j;bulk [11–13]. If all the majority carriers are mobile, such charging leads eventually to steep Gouy-Chapman accumulation and depletion profiles. In the case that the mobile majority carrier is compensated by an immobile one (typically a dopant) and is depleted as a consequence of the structural situation, the lack of sufficient screening leads to rather extended Mott-Schottky zones. These conditions hold for ionic as well as electronic carriers. For simplicity we only consider oxygen ions and electrons to be mobile. The electronic and ionic picture is coupled (cf. Figs. 2.2c and 2.3c) as all the carriers feel the same electric field and are in equilibrium to satisfy the relation

1 2

l~ O2

2l~ e
¼ lO ¼ lO2 ;

ð2:5Þ

where lO , the chemical potential of the component oxygen, is given by the oxygen partial pressure lO2 ¼ l0O2 þ RT ln pO2 . In order to determine the grain boundary core charge, we need to know the l0j;core ’s for all species in the gb core (or more precisely the relevant differences), and we assume complete equilibrium with the gas phase. Fig. 2.2a shows the defect thermodynamics for a situation in which the majority compensating defect is mobile (GouyChapman case). Owing to the structurally different situation, l0j;core in general differs from the respective bulk value, which results in the gb core having a nonzero charge. In Fig. 2.2 we exemplarily assume that oxygen vacancies have a lower l0core in the gb core. From the requirement of equal electrochemical potential of the positive and negative majority carriers in the gb core and in the grain (violet line in Fig. 2.2a) the sign and magnitude of the core charge follow. For the situation depicted in Fig. 2.2, this leads to a positive charging of the core and a corresponding space charge potential. The core charge may be primarily electronic as assumed typically in semiconductor physics. If however the ionic charge carriers are in majority in the oxide systems to be discussed here, it is very likely that the excess core charge is ionically dominated.

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257

Fig. 2.2. (a) (Electro)chemical potential of VO in gb core and space charge zone and space charge potential / for Gouy-Chapman case with positive core charge and accumulation of acceptor dopants A00 in the space charge zone. The fact that the difference between l0V and lV is smaller in the core than in the O O grain interior corresponds to a higher VO concentration in the core. For simplicity, we assumed here that only VO segregate to the gb core, not A00 . (b) Corresponding concentration profiles on a logarithmic scale, note the stronger relative accumulation/depletion of doubly charged defects owing to their higher charge. (c) Coupling of the chemical potentials of electronic and ionic carriers.

Fig. 2.3. (a) (Electro)chemical potential of VO in gb core and space charge zone and space charge potential / for Mott-Schottky case with positive core charge. Note also the different shape of the potential profile compared to the Gouy-Chapman case (cf. Eq. (2.9) vs. (2.10)). (b) Corresponding concentration profiles (the acceptor dopant A0 is regarded as immobile) on a logarithmic scale; the actual example shows an inversion case for the electronic defects. (c) Coupling of the chemical potentials of electronic and ionic carriers.

To fulfill global electroneutrality, this excess charge has to be compensated by an adjacent space charge zone with carrier accumulation/depletion profiles cj ðxÞ. The electrochemical potential of the mobile defect is constant throughout the whole system. Together with the Poisson-Boltzmann relation (1-dimensional; setting /bulk = 0) 2

d /ðxÞ 2

dx

¼


F X cj;bulk zj e
zj F/ðxÞ=RT

e

ð2:6Þ

j

this determines the defect concentration profiles extending from x = 0 towards the bulk x = 1. For simplicity we characterize the dielectric behavior of the core layer by a dielectric constant ecore while the bulk value is ebulk (as first approximation, we use ecore ¼ ebulk [14]). As the real situation is discrete, it is necessary to assume that between core (x = s) and space charge zone (x = 0), a carrier-free transition zone with thickness  bond length and an effective dielectric constant eT is present [15]. If that zone is free of charge carrier, the electrical potential increases linearly in this transition zone. The necessity of taking account of such a transition zone is well established for liquid/solid contacts (neglecting it leads to grossly wrong interfacial capacitances) but as well for solid/solid contacts [12,16,17]. The effect of this transition zone becomes relevant mainly for Gouy-Chapman situations with steep potential gradients (i.e. for pronounced boundary charge), allowing to at least semiquantitatively describing such situations (see e.g. [17,18]) before a completely discretized atomistic model has to be used. It is less important for Mott-Schottky cases (cf. the different contribution of the red dotted line relative to /0 in Figs. 2.2a and 2.3a). For such cases – in particular if the boundary charge is not extremely high – good agreement between experimental data and modelling results may be found despite neglecting the charge carrier free transition zone (see e.g. Refs. [13,19,168]). The extension of the Gouy-Chapman space charge zone is on the order of the Debye length

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k¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e0 er RT

ð2:7Þ

2z2maj F 2 cmaj;bulk

which is the smaller the higher the concentration cmaj;bulk of the compensating majority carrier is (note that for highly doped samples with cmaj;bulk in the range of several percent, one obtains k  0.2 nm, for which the continuum ansatz approaches the limit of its validity). Since by accumulation a higher charge density can be built up compared to depletion, for a given core charge the width of the space charge zone is smaller in the Gouy-Chapman case than for the Mott-Schottky case. The expressions for the space charge potential profile /(x) and the relation of /0 to the core charge density Qcore are given in Table 2.1. Fig. 2.2b shows exemplary defect concentration profiles. It is important to note that for given space charge potential /0, defects with higher charge are accumulated/depleted much more strongly (cf. also Section 2.5). A collection of typical space charge situations (including mixed cases) exemplified for ceria is given in Table 2.2; the mathematical (approximate) expressions can be found in Refs. [20–22]. Integrations of the cj and 1=cj profiles allow one to then calculate the conductance contributions of the space charge zones parallel and perpendicular to the interface (examples can be found in Ref. [20]). The mean conductivities hrk i, hr? i then follow by multiplication with the spatially invariant mobility (assuming dilute conditions). For perpendicular transport through depletion zones, the maximum depletion in the most resistive depletion determines the overall conductivity. Analogously, the highest accumulation determines the transTable 2.1 Comparison of Gouy-Chapman and Mott-Schottky situations. The concentration of accumulated or depleted carrier j is related by cj ðxÞ ¼ cj;bulk e
zj F/ðxÞ=RT to the space charge potential profile /(x) (note the convention /bulk = 0). z = carrier charge, F = Faraday constant.

Extension of space charge zone

Gouy-Chapman case

Mott-Schottky case

Debye length rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ 2z2 e0Fe2rcRT (2.7)

Depletion width qffiffiffiffiffiffiffiffiffiffiffi 4zj F/0 (2.8) k ¼ k RT

2RT x zmaj F/0 =2RT Þ (2.9) /ðxÞ ¼ /0 þ zmaj;bulk F lnð1 þ 2k e

for x  2k and strong accumulation (effect of depleted carrier negligible)

/ðxÞ ¼ /0 ðkx
1Þ2 (2.10) for x  k*, neglecting the minority carrier accumulation

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jz jF/0 Q core ¼ 8RT e0 er cmaj;bulk sinhð maj 2RT Þ (2.11) for symmetrical cases z1 =
z2

Q core ¼ 2k zmaj Fcmaj;bulk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2.12) ¼ 8e0 er zmaj F/0 cmaj;bulk

maj

Space charge potential profile

gb core charge density

maj;bulk

Table 2.2 Defect concentration profiles and space charge potentials calculated for ceria assuming a constant gb core charge density of +0.3 C/m2 (A) or
0.3 C/m2 (B). Red = [VO ], blue = [e0 ], yellow = [dopant]. For the Mott-Schottky case (A4) as well as for the mixed cases (A3), (B1) a change of the predominant conductivity (electronic or ionic) between bulk and grain boundaries is expected (depending also on the carrier mobilities). Adapted from Ref. [22] by permission of the PCCP Owner Societies.

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port parallel to accumulation layers. The effect of depletion zones on parallel transport and accumulation zones on perpendicular transport is negligible unless nanograined materials are concerned (cf. Section 2.4). The presence of space charge effects also modifies the temperature dependence of the conductivity (for details see e.g. [23]). In the so-called Mott-Schottky case where the compensating majority defect (e.g. acceptor dopant A0 ) is immobile and hence not in interfacial equilibrium, the compensation of the gb core charge occurs by depletion of the mobile carrier (VO ) as illustrated in Fig. 2.3. Neglecting the accumulation of minority defects, the space charge potential and majority carrier 2

2

profiles can be analytically calculated from the Poisson equation d /ðxÞ=dx ¼
F q=e where the charge density q is constant and given by the immobile defect (concentration corresponding to the far-reaching depletion of the mobile carrier in the space charge zone). The width of the depletion zone k⁄ exceeds the Debye length k

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4zj F/0 k ¼k RT 

ð2:8Þ

and is directly related to the core charge (cf. Table 2.1). Correspondingly, for constant Qcore the space charge potential /0 is temperature independent in the Mott-Schottky case (while it increases with T for the Gouy-Chapman case, cf. Eq. (2.11)). Exemplary Mott-Schottky concentration profiles are shown in Fig. 2.3b. Defects with sign opposite sign to the core charge are accumulated. This may even lead to ‘‘inversion” situations where the minimum of electronic defect concentrations occurs not at the beginning of the space charge zone but rather at x > 0 (more details in Section 2.7). For given values of /0, lV ;bulk
lV ;core can directly be calculated while the core charge density obtained from Eq. (2.12) O

O

and zdop cdop;bulk further yields lV ;core
l0V ;core as long as no site saturation in the core occurs (for such situations see numerO

O

ically calculated examples in Ref. [13]). Within this regime, one finds that the space charge potential is essentially determined by the free energy of segregation l0j;core
l0j;bulk of defect j that determines the core charge (Fig. 2.4a). For given values of

l0j;core
l0j;bulk (and sufficiently below site saturation) the core charge density depends strongly on the respective

bulk concentration (Fig. 2.4b). On the other hand, if the core charge Qcore is approximately constant (e.g. by approaching site saturation), a higher dopant concentration strongly decreases /0 according to Eq. (2.12), see examples in Fig. 4.5. The comparison in Table 2.1 shows that for a given core charge the Gouy-Chapman case does not only lead to a smaller width of the space charge zone but also to a smaller space charge potential than a Mott-Schottky situation (cf. Eq. (2.9) vs. Eq. (2.10); see also Ref. [22] for numerical examples). It is important to note that for strong accumulation the assumptions of ideally dilute conditions and spatially invariant mobility may break down. Defect interactions modifying defect concentrations as well as mobilities (well known for bulk situations, e.g. in highly doped ceria and zirconia [24,25]) are to be expected. Also, for high defect concentration with k approaching atomic bond lengths the limits of a continuum model become perceptible. Not only has site saturation to be considered, but the restriction of ions to discrete sites/layers also affects the carrier concentration profiles as well as interfacial capacitance and storage behavior at heterocontacts (see e.g. [26,18]). Gradient corrections are usually the tool to be introduced to correct for very steep profiles [27], and have been recently applied to space charge problems [28]. However, in [28] the respective prefactors have not been derived from a fundamental theoretical treatment, but adjusted as fit parameter to match the experimental dopant dependence of the bulk conductivity. Furthermore, the validity of the Cahn Hilliard

Fig. 2.4. Exemplary model calculations for the Mott-Schottky case (500 K, er = 150 cf. SrTiO3, the core charge density is expressed as number of VO per unit cell in the gb core which is assumed to be one unit cell thick). (a) Space charge potential as function of l0j;core
l0j;bulk and dopant concentration. (b) Core charge density as function of l0j;core
l0j;bulk and dopant concentration.

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approach for charged situations is unclear [4]. The experience in semiconductor physics suggests such corrections not to be first order effects. 2.2. Conductivity of polycrystalline materials As outlined in the introduction, the electric transport properties of a polycrystal consisting of crystallites glued together via grain boundaries can significantly differ from the respective properties of a single crystal. As additional complications the contacts may be laterally inhomogeneous, e.g. by including pores or second solid phase particles. Moreover, the microstructure itself may not be even approximately periodic. Given the knowledge of the individual contributions, the superposition of them to the overall conductivity rm is in realistic cases only numerically accessible, e.g. by finite element calculations. In spite of all this complexity, a lot is to be learned from idealized microstructures. Here we restrict ourselves to the most simple case, a uniform brick-layer type of microstructure as shown in Fig. 2.5a. As we allow for a finite thickness of the core region one also has to consider conductivity/resistivity contributions from this core layer. In spite of the great degree of idealization this model has proven to be astonishingly well applicable to real situations. In general cases where boundaries may even be laterally inhomogeneous and/or grain sizes and shapes are different (Fig. 2.5b) and where grain boundary densities may be rather large, numerical approaches do best service. Fig. 2.5c shows that for an inhomogeneous current distribution stemming from geometrical deviations from the brick-layer situation, the more precise modelling does not lead to substantial deviations from the brick-layer approach (mean field approach) [29]. A brick layer model is not only a useful approximation for the conductivity of a polycrystalline material, but also for its capacitive behavior. Within this model, the capacitance of the grain boundary semicircle is given by (see e.g. [30–33])

C gb ¼ e0 er

dg dgb

ð2:13Þ

where dg is the grain size and dgb the thickness of the gb depletion layer. In a bricklayer model the superposition of the different contributions is rather trivial as long as the volume fraction mgb of grain boundaries is small, the boundary zones have a homogeneous conductivity and there are no special effects in the zones where boundary layers overlap (edge zones, contributing on the order of m2gb ) or even where the edge zones overlap (corner zones, contributing on the order of

rm ¼

m3gb ). The superposition is given by [34]

rbulk r?gb þ ð2=3Þmgb rkgb r?gb r?gb þ ð1=3Þmgb rbulk

ð2:14Þ

where rkgb ; r? gb represent the conductivity parallel and perpendicular to the grain boundaries. There are different expressions for such a superposition in the literature [35], that differ on the order of higher powers of mgb . The obvious reason is evident from Fig. 2.6 which considers the simplified situation in 2D. If the corner zone (in 3D this already occurs for edges) is of significance, it matters how the different resistors are connected. In reality, the current lines near the corner are inhomogeneous. Such effects can be significant if the corner zones offer distinctly different conductivities and/or if nanocrystals are considered such that

mgb is not small compared to 1. In cases where one does not have to distinguish between rkgb and

r?gb , an effective medium approach is very useful. The result is the classic Maxwell-formula

rm ¼

rgb ð3rbulk
2ðrbulk
rgb Þmgb Þ : 3rgb þ ðrbulk
rgb Þmgb

ð2:15Þ

Fig. 2.5. (a) Idealized brick layer model, (b) a more realistic microstructure of ceramic materials, (c) calculated impedance for the brick layer model (line) and realistic microstructure (solid circles) of a ceramic with blocking grain boundaries. Reproduced with permission from Ref. [29]. Copyright 2014, The Electrochemical Society.

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Fig. 2.6. Ratio of effective to bulk conductivity for different volume fractions of blocking grain boundaries. Curves (a) and (b) calculated from the two variants of the brick layer model as indicated in the inset, (c) calculated from Eq. (2.15) (note that /1 in [36] corresponds to 1
mgb here). Reproduced from Ref. [36] by permission of the PCCP Owner Societies.

Even though derived for situations in which concentration effects are absent, it is – as shown in Ref. [36] – also valid in cases where electric potential differences must be replaced by differences in the electrochemical potentials. This is particularly relevant for space charge effects. Moreover, Eq. (2.15) applies for the ambipolar conductivities for a steady state permeation experiment. In real composites the morphology may drastically change as function of volume fractions (and hence formally the proportion of parallel and serial effects). A formal possibility to take account of that is a mixing rule in which the volume fraction appears in the exponent of the conductivities involved [37]. Let us now concentrate on the range in which Eq. (2.16) is valid. In Ref. [34] it is shown how space charge effects can be implemented. As mentioned, they make the boundary effects necessarily anisotropic. i.e. it matters whether one considers transport along and across the boundaries. In Eq. (2.16) r? and rk have to be replaced by the mean conductivities addressed above that differ for the two directions. An example is the anisotropy of AgCl grain boundaries (Fig. 2.7, [38]). As far as the parallel transport is concerned, the conductive space charge zones dominate, while for transport across boundaries, core effects dominate. This leads (i) to a lowered high frequency impedance semi-circle, as parallel boundaries short-circuit the bulk and (ii) to an appearance of a low frequency semicircle owing to the transport across boundaries. If the samples

Fig. 2.7. Microcontact impedance spectra of a polycrystalline AgCl sample. Diamonds: measured at the grain interior, squares: contacted at a highly conductive grain boundary. Reproduced from Ref. [38] by permission of the PCCP Owner Societies.

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are annealed and the grains have grown, the low frequency semi-circle has become smaller while the high frequency semicircle has become larger. For details, the reader is referred to Ref. [34]. 2.3. Chemical effects in polycrystalline materials In general the total conductivity has various additive contributions from ions and electrons. Each of them is composed of charge carriers of opposite signs (interstitials, vacancies; excess electrons, holes). In the space charge zones their relative proportions vary substantially owing to the field effect that is opposite for carriers of opposite signs (cf. Figs. 2.2 and 2.3). This holds especially true if ionic carriers of higher valence are referred to. Hence at grain boundaries not only the dominating transport mechanism may change but also the ionic transference number tion = rion/(rion + reon). The examples treated in Sections 2.5 and 3.1.1 show the significance of the effect, and also show how different contributions can be deconvoluted [39]. While the individual conductivity contributions sum to the total value, the ambipolar conductivity
1


1 rd ¼ ðr
1 ¼ ðrion reon Þ=ðrion þ reon Þ ion þ reon Þ

ð2:16Þ

is obtained by summing up the inverse conductivities of the carriers. The ambipolar conductivity is decisive in chemical experiments such as permeation or stoichiometry variations, see e.g. [40]. In the same way as the electrical conductivity has to be complemented by capacitive elements in order to describe the transients, the ambipolar conductivity alone describes chemical steady state experiments but when complemented with a chemical capacitance then also time dependencies. The latter measures the change of composition if the chemical potential is varied [41]. Most importantly the chemical diffusion coefficient Dd is proportional to the product of the inverse chemical resistance and the inverse chemical capacitance. The first is proportional to the ambipolar conductivity rd defined above and the latter to the ambipolar concentration cd , both composed of contributions from ionic and electronic carriers (in the simplest case of dilute monovalent defects 1=cd ¼ 1=cdion þ 1=cdeon ). Concentration changes by fast transport along grain boundaries are extensively treated in the literature [42–45]. If one is only interested in the relaxation time of chemical diffusion in a polycrystalline ceramic, the following consideration is helpful and instructive: The diffusional relaxation time for the pure bulk problem (chemical diffusion in a single crystal of thickness L) is

sd ¼ Rdbulk C dbulk /

L2 Ddbulk

ð2:17Þ

Now let us consider the diffusion into a brick-layered ceramic with grain size l. If the diffusion along and across the grain boundaries is very fast (compared to bulk), sd is determined by the diffusion within the grains, l being the decisive diffusion length (cf. Fig. 2.8, [46]) and L the macroscopic sample dimansion:

Ddm ¼

L2 l

2

Ddbulk /

L2

rdbulk

2

d l cbulk

ð2:18Þ

The opposite case is realized in a ceramic with hardly permeable grain boundaries (of width dgb). Then Rd is determined by Rdgb

while C d is determined by the bulk value C dbulk leading to

l rgb : dgb cdbulk d

Ddm /

ð2:19Þ

As the ambipolar conductivity value is determined by the smaller contribution the space charge splitting in a GouyChapman zone always leads to a slowing down when compared with the bulk [47]. Also a Mott-Schottky situation leads

Fig. 2.8. Diffusion into a ceramic (grain size = l) with grain boundaries with high diffusivity (left) or blocking character (right); the darker the gray the higher the local diffusivity. Reproduced from Ref. [46] with permission of Slovenian Chemical Society.

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to decreased Dd since at least one of the defects involved is depleted. For an in-depth treatment of time and space dependent concentration changes across grain boundaries the reader is referred to Refs. [48,49]. 2.4. Nanocrystalline ceramics While the situation is comparatively straightforward if the boundary effects are well separated by the grain interior with bulk behavior, the situation becomes delicate if the boundaries interfere (Fig. 2.9, [50]). Such mesoscopic situations have been treated in a 1-dimensional case. Recently, nanocrystalline SrTiO3 has been investigated where the grains are so small that again a quasi-homogeneous situation is met, yet with parameters determined by the boundary layer thermodynamics ([51,52], see also Figs. 3.6 and 3.7). In such extreme cases the degree of anisotropy has decreased but not completely disappeared. The disappearance of unperturbed bulk is indicated in Fig. 2.9. At the very right hand side crystallinity has completely disappeared and the amorphous state is realized. Nanocrystallinity also has an important effect on the classic picture of grain boundary diffusion [42–44]. Essentially two effects are found worthwile in Refs. [45,53,54] and investigated numerically: (i) Perpendicular and parallel pathways are simultaneously of importance. (ii) Neighboring boundaries interfere. 2.5. Stoichiometry polarization caused by grain boundaries The previously discussed phenomena allowed for investigating or even changing grain boundary properties. Blocking grain boundaries can even become the cause for stoichiometry polarization of the adjacent bulk material [39]. It is well known that electrodes which block selectively ions or electrons lead to a stoichiometry polarization in the bulk under applied dc current (e.g. increased oxygen deficiency at an ion-blocking electrode, caused by accumulation of VO and corresponding e0 accumulation). Such measurements with selectively blocking electrodes allow for determination of the partial conductivities and the chemical diffusion coefficient. Grain boundaries can have a similar effect, even if reversible electrodes are used. This is the consequence of the above described strong variation in the transference number in the space charge zones when ionic and electronic carriers have different charge, and the resulting filtering effect. This phenomenon is particularly obvious and relevant in the non-linear regime where serious local property changes in the bulk occur. An example is given in Section 3.1.1. 2.6. Ion mobility effects at interfaces Depending on the actual misorientation angle, low-angle grain boundaries can be treated as resulting from arrays of edge dislocations. In the framework of resistive switching phenomena, it has been proposed that dislocations offer preferential pathways for oxygen transport [55,56]. However, recent experimental and theoretical studies provided evidences of the contrary, namely that oxygen diffusion is hindered and the energy barrier associated to the oxygen migration is larger in the dislocations cores than in the bulk [57–59]. While the calculations find a strongly decreased oxygen vacancy formation energy in the dislocation cores for SrTiO3 as well as for ceria, the migration barrier along the dislocation cores are higher than the bulk value, see Fig. 2.10. Note that owing to the negative VO segregation energy also the barrier for the VO jump away from the dislocation core is much higher than the bulk value. This low VO mobility along dislocations in perovskite and fluorite structured oxides is different from the behavior of dislocations in metals, which typically exhibit fast transport that was related to the larger free volume in such extended defects (note that in metals the whole situation at the grain boundaries strongly differs from that in oxides since defect are not charged and no compensating defects are required). Regarding the resistive switching, rather electron accumulation layers around dislocation cores than enhanced VO diffusivity at dislocations plays an important role (cf. [57,60] and references therein). As far as transport along dislocations and also along interfaces is concerned, it seems pertinent to formulate the following ‘‘rule of thumb”: In materials of very low bulk mobilities such higher dimensional defects may offer pathways of enhanced mobility (e.g. Al2O3 cf. Section 3.3.1, also many metals). In materials with high bulk mobilities (silver halides, perovskites and fluorites with highly mobile VO ) such higher dimensional

Fig. 2.9. Left to right: transition of polycrystalline material to nanocrystalline and finally amorphous state. Reprinted with permission from Ref. [50]. Copyright 2014 American Chemical Society.

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Fig. 2.10. Migration barrier of VO in the bulk and along an edge dislocation core in SrTiO3 (pair potential simulation). Reprinted with permission from Ref. [58]. Copyright 2015 American Chemical Society.

defects are very likely to show depressed mobilities (related to the locally decreased symmetry). Also for epitaxial YSZ-CeO2 heterostructures with perceptible dislocation density at the interfaces no effect on lateral oxygen ion transport could be found [446]. 2.7. Identifying and quantifying space-charge effects Grain boundaries with enhanced conductivity cannot be recognized from a single impedance spectrum; the response will qualitatively not differ from that of a single crystal (as the effects are in parallel and the interfacial capacitance is negligible the dominating capacitance is the dielectric capacitance of the sample, and it is not possible to recognize if the resistance represents the whole bulk or is carrier through some highly conductive paths). Fast gb transport can then only be identified by a systematic variation of grain size (but ensuring that other materials parameters remain unchanged). In the case of samples with sufficiently large grains, space-resolved tracer diffusion measurements can deliver direct evidence for fast gb transport as later discussed in Fig. 4.8. Blocking grain boundaries lead to the appearance of a second semicircle (in addition to the bulk response at the highest frequencies) in the impedance spectrum at lower frequencies, see e.g. [35]. Here the effects are in series with the interfacial capacitance being very large. The peak angular frequency of a semicircle is given by x ¼ R
1 C
1 , and C gb  C bulk leads to well-separated gb semicircles. The response of not completely reversible electrodes typically appears at even lower frequencies. Impedance spectra are usually fitted by equivalent circuits (for alternative methods see e.g. [61,62]). As long as stoichiometry polarization at grain boundaries is not relevant, the spectra can be represented with a serial arrangement of two parallel RC combinations (if present, gb stoichiometry polarization leads to an additional Warburg-type arc [39]). While the brick layer model predicts a perfect semicircle for blocking grain boundaries, real samples often exhibit deformed/depressed gb semicircles. This behavior can phenomenologically be modelled by replacing the gb capacitor Cgb in the equivalent circuit by a constant phase element Qgb which contains an ideality parameter n (n = 1 for perfect semicircle). This nonideality can be viewed to represent a distribution of the characteristic times for the grain boundaries, corresponding to a certain distribution in the resistive and capacitive properties of the individual grain boundaries in a polycrystalline sample. From Qgb the capacitance which has the most important contribution can be calculated according to 1
n C gb ¼ ðRgb Q gb Þ

1=n

see e.g. [63]. Within the brick layer model, the local (‘‘specific”) gb conductivity conductivity

ð2:20Þ

rsgb is obtained from the ‘‘macroscopic” gb

rm;gb ¼ L=ðA  Rgb Þ by

rsgb ¼ rm;gb

dgb C bulk ¼ rm;gb : dg C gb

ð2:21Þ

I.e. when it is possible to extract both bulk and gb capacitance from the impedance spectrum, rsgb can be calculated without explicit knowledge of grain size dg and thickness of the depletion zone dgb . However, if one wants to calculate dgb from Eq. (2.13), the grain size needs to be (approximately) known. Several reasons can lead to a blocking behavior of grain boundaries: (i) One very obvious cause is the formation of a continuous secondary phase with lower conductivity. This was observed e.g. in early investigations of ceria or zirconia with a high concentration of Si impurities, which lead to an insulating SiO2 phase along the gb (see e.g. [64]). Such a phase is expected to be visible in TEM investigations, and to exhibit a different activation energy (and possibly pO2 dependence) compared to the grain interior. dgb of the low conducting layer calculated from C gb should match the thickness of the secondary phase.

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(ii) Closely related is the current constriction phenomenon, where the insulating phase does not cover the gb completely but leaves only small ‘‘clean” contact areas with correspondingly higher current densities (a similar effect is observed for samples with low density where grains touch only at narrow sinter necks). In this case, the grain boundary resistance is expected to show almost identical T- and pO2 dependence as the bulk. The low-frequency semicircle results from the fact that at low frequencies the current is concentrated to the clean contact, while at high frequencies the ac signal is transmitted capacitively at the whole gb area (see e.g. [65,66], Fig. 2.11). The value of C gb depends on the thickness of the blocking layer as well as on the area fraction of the clean contact. The behavior is more complicated and no longer bulk-like if the current lines are non-linear throughout the grain. (iii) The third possibility is a blocking effect caused by space charge depletion zones, which exhibits several characteristic phenomena. The first is the fact that – at least for small dopant concentrations – the width of the low conducting layer is much larger than the thickness of a structurally distorted gb core region or secondary phase (0.5–1 nm), cf. Eq. (2.8). The second point is that dgb decreases with increasing dopant concentration, most pronounced for Mott-Schottky situations (Eqs. (2.7), (2.8) and Fig. 2.12). Third, the activation energy of rgb is larger than the bulk value (zjF/0  Ea,gb
Ea,bulk if the T-dependence of /0 is negligible, cf. [23]). Finally, a very strong evidence for space charge zones is a decrease of Rgb with applied dc bias (measured by I-V curves or impedance spectra with additional dc bias). However, in order to observe this effect it is necessary that the voltage drop at each individual gb is in a comparable range as the space charge potential. This requires measurements on individual grain boundaries or at least very large-grained samples and large applied voltages. Fig. 2.13 shows the changes of the defect concentration profiles at a grain boundary under different applied bias values [68]. For small bias values (ohmic regime), the maximum depletion (? Rgb ) and width of the depletion zone (? C gb ) hardly change, but for larger values (varistor regime) the increased minimum VO concentration leads to a lower Rgb , while the larger width of the depletion zone decreases C gb . The combination of these two features is a very strong proof for space charge depletion zones. The identification of space charge effects as the origin of blocking grain boundary behavior by these characteristics will further be exemplified for SrTiO3, BaZrO3 and CeO2 in Section 3. When space charge depletion zones are identified as the dominating origin of the blocking behavior, the space charge potential /0 can be extracted by several methods. For Mott-Schottky situations and charge zj of the mobile carrier, /0 is obtained from the ratio of bulk and gb resistivities

qgb ezj F/0 =RT ¼ qbulk 2zj F/0 =RT

ð2:22Þ

If the impedance spectra show the peak angular frequencies xbulk, xgb of the bulk and gb semicircles, /0 is easily calculated from

xbulk ezj F/0 =RT ¼ xgb 2zj F/0 =RT

ð2:23Þ

without the need for microstructural information of the sample (grain size) [69]. However, this equation holds only as long as no inversion occurs. In the case of inversion between p- and n-type electronic carriers, the most resistive layer is not directly adjacent to the gb core any more but shifted outward, and the maximum resistivity is fixed by the crossover point with the accumulated carrier (not directly varying with /0 any more). Inversion can be recognized from the fact that the C gb value expected for /0 extracted by Eq. (2.23) and bulk permittivity, dopant concentration is significantly larger than the mea-

Fig. 2.11. Simulation of a grain boundary which is largely covered by an insulating phase (a) potential distribution: for low frequencies the current is concentrated at the clean contact; at high frequencies the ac signal is homogeneously transmitted across the gb (b) corresponding impedance spectrum with high frequency arc = bulk conductivity and low frequency arc = current constriction effect. Reprinted from Ref. [65], copyright 1996, with permission from Elsevier.

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Fig. 2.12. Thickness dgb of the depletion layers at the grain boundaries in polycrystalline Y-doped CeO2, extracted from the grain boundary capacitance. Data taken from Ref. [67].

Fig. 2.13. Simulation of a blocking grain boundary (Mott-Schottky situation) under increasing values of dc bias UDC (a) VO concentration profile, (b) profile of the space charge density. Reprinted from Ref. [68] with the permission of AIP Publishing.

sured value (dgb increases further with increasing /0 also in the inversion regime). Equations describing the inversion case can be found in [70]. For SrTiO3, a space charge potential of about 0.6 V already suffices to move into the inversion regime (which is easily reached for bicrystals with a tilting angle larger than about 5°). The space charge potential leads also to a higher activation energy of rgb compared to rbulk, however the difference is not exactly equal to zj e0 /0 but contains an additional term originating from the T-dependence of /0 (see e.g. [23], valid for MottSchottky regime without inversion). Finally, the space charge potential can also be determined from the measured dc current-voltage behavior. A recent theoretical treatment showed that for Mott-Schottky situations of predominant ionic conductors with a constant excess charge of the gb core the I-V relation should follow a power law [71,72]. The value of /0 is then obtained from the slope of log(I) vs log(V). Again, the applied bias per individual grain boundary has to become roughly comparable to /0 in order to operate in the varistor regime. Furthermore, the electrodes have to be reversible, and stoichiometry polarization at the grain boundaries (cf. Fig. 3.1c) should be absent. The effect of Gouy-Chapman cases is sensitively perceived for accumulation layers leading to enhanced conductivity (however, as discussed at the beginning of this section the impedance spectra will appear ‘‘bulk-like”). Such effects have been investigated in polycrystalline silver halides [73]. Potentials of several hundred millivolts are equivalent to substantial conductivity enhancement effects that for not too large grain sizes may dominate the overall conductivity. The parallel conductance is then given by

Y k ¼ 4krbulk ezF/0 =2RT

ð2:24Þ

If the carriers dominating the conductivity are not in majority, enhancement effects can even stem from Mott-Schottky situations. Such a case was discussed in CaF2/BaF2 heterolayers [74]. At higher temperatures the situation switched to GouyChapman layers.

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(a)

(d) (b)

(c)

(e)

Fig. 3.1. Impedance spectra acquired for Fe-doped SrTiO3 bicrystals at T = 400 °C and pO2 = 0.1 bar. The misorientation angles are (a) 2.3°, (b) 5.4°, and (c) 7.8°. In the top-right hand corner of each graph is a schematic diagram of the dislocation spacing. Reprinted from Ref. [70] with the permission of AIP Publishing. (d) Concentration profiles obtained from in situ optical measurements of oxygen chemical diffusion acquired at different diffusion times. The vertical dotted line at the normalized distance of 0.7 corresponds to the boundary: Here an abrupt change of the concentration is evident. Modified with permission from Ref. [47]. Copyright 1999, The Electrochemical Society. (e) Optical image corresponding to sharp concentration profile changes at the boundaries of a tricrystal.

Combinations of Mott-Schottky and Gouy-Chapman situations, e.g. acceptor dopants being mobile under sintering conditions but frozen-in at measurement temperatures (cf. Fig. 3.13) typically require the use of approximations (see e.g. [75]) or numerical calculations (e.g. [21]). 3. Detailed discussion of selected examples After having illustrated the theoretical background and the analytical tools that are employed to carry out a quantitative analysis of the space-charge effects, a selection of relevant materials is presented, which on one hand illustrates for representative examples the relevance of boundary effects, and on the other hand refers to materials that are significant for applications. 3.1. Perovskites Perovskites encompass a variety of compounds exhibiting a large spectrum of functional properties, which are extremely relevant for several technological applications. If we focus our attention on the electrical properties only (e.g. dielectric as well as electrical transport), BaTiO3 was the first perovskite, for which both bulk defect chemistry and grain boundary effects had been systematically investigated [76]. Nonetheless because of the simplicity of the phase relations, SrTiO3 has been undoubtedly the material that has been most extensively studied in the context of the purpose of this contribution, so that it can be certainly considered here as the very model system. For this reason, we start this section with SrTiO3 and only later we refer to other relevant perovskites. 3.1.1. SrTiO3 Strontium titanate has been extensively investigated in the context of a variety of relevant fields including solid state ionics, thermoelectricity, memristors, superconductivity [55,77–81]. The systematic study of its bulk defect chemistry has

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revealed that depending on temperature and environment (oxidizing vs reducing) as well as on the doping situation, the electrical transport properties are dominated by three different mobile charge carriers: excess electrons (corresponding to Ti3+), electron holes (corresponding to O
), and oxygen vacancies [82–86]. Interstitial ions have a very high energy owing to the dense structure; cation vacancies are present but immobile under most conditions (their concentration can be subsumed under the dopant content). Such a versatility has made this material a prototypical system for the investigation of fundamental aspects of the electrical transport properties of perovskites. Donor doped SrTiO3 is a good n-type electronic conductor that owing to the lack of ionic carriers hardly allows for stoichiometric bulk equilibration. Here grain boundary effects play a dominant role. In acceptor doped SrTiO3 (equally in heavily Schottky disordered SrTiO3) one can easily observe p-type, n-type and ionically dominated transport regimes (cf. Fig. 2.1a). Numerous contributions have demonstrated over the years that the grain boundaries of acceptor-doped SrTiO3 hinder the transport of the positively charged mobile carriers (holes and oxygen vacancies), while they favor instead the transport of electrons. In impedance spectroscopy in oxidizing conditions the blocking behavior of the gb for holes and VO leads to the appearance of a low-frequency semicircle as shown in Fig. 3.1a–c for SrTiO3 tilt bicrystals with different misorientation angles. Note that the effective thickness of the boundary layer is the width of the depletion zones (typically k⁄ = 5–50 nm for SrTiO3, depending on the dopant concentration and gb core charge) rather than the width of the gb core region. The blocking behavior manifests itself also in optical in situ measurements of oxygen chemical diffusion carried out on bicrystals Fig. 3.1d), proving the crucial role of boundaries on the overall transport properties of this material [47,70]. Dedicated experiments relying on Wagner-Hebb polarization and using selectively blocking electrodes (pO2,YBCO|SrTiO3| Au,pO2) allowed for distinguishing between electronic and ionic transport and, more importantly, for establishing that while the grain interior exhibits a perceptible ionic transport (in particular at lower T), the conductivity at the grain boundaries is exclusively electronic (Fig. 3.2) [87]. This feature corresponds to the space charge splitting shown in Fig. 2.3a according to which for a positively charged core p-type and oxygen vacancy carriers are depleted, the latter much more strongly according to the higher charge. The increase of the n-type carriers can lead to an inversion from p-type bulk to n-type boundary conductivity [88].

Fig. 3.2. Arrhenius diagram of the total (solid symbols), ionic (open triangles) and electronic (open circles) conductivity for bulk and grain boundary, respectively, of a polycrystalline SrTiO3 ceramic. Remarkably, the electric conduction in the bulk (a) has a perceptible ionic contribution whereas at the grain boundaries (b) is exclusively electronic. Reproduced with permission from Ref. [87]. Copyright 2001, The Electrochemical Society.

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This enormous variation of the ratio electronic conductivity/ionic conductivity can lead to a filtering effect causing stoichiometry polarization of the bulk as described above. Arc 2 in Fig. 3.3 shows the impedance of the grain boundaries (blocking of holes) while arc 3 with its characteristic Warburg part reflects the bulk polarization [39]. Under dc load, such effects can lead to degradation phenomena (‘‘electrocoloration”, similarly as for not completely reversible electrodes). For a tricrystal, this is demonstrated in Fig. 3.1e (the darker the color, the higher the local concentration of Fe4+, i.e. of trapped holes). Such peculiar properties are due to the presence of a positive space charge potential [89–93]. Further evidences were provided for example by d.c.-biased impedance spectroscopy studies performed on SrTiO3 bicrystals as well as on microcontacted SrTiO3 ceramics as shown in Fig. 3.4 [92,93]. In the last case, the ability of placing microcontacts of single adjacent grains allowed for a statistical study of the resistive and capacitive behavior of a rather large number of different grain boundaries.

Fig. 3.3. Impedance spectra acquired using YBCO electrodes from (a) a single crystal and (b) a bicrystal. While in the first case only one contribution is detected (as expected), several contributions appear in the second case: Arc 2 is almost a perfect semicircle and arc 3 corresponding to a Warburg behavior. Reprinted from Ref. [39] with the permission of AIP Publishing.

Fig. 3.4. Microcontacts experiment on a SrTiO3 ceramic. (a) Optical image of the contacts and (b) variation of the impedance spectrum as a function of the applied d.c. bias: Note that only the low frequency part pertaining to the grain boundary contribution changes with bias. Reprinted from Ref. [93] with permission of John Wiley & Sons.

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(b) 0.20

2.3

5.4

7.8

565

{12}

Δφ / mV

(a)

560

0.15

Qcore / Cm

-2

555

580

600

620

640

660

680

T/ K

0.10 570

{ }

(c) Δφ / mV

0.05

565

560

0.00 0

2

4

6

8

θ / deg

10

12

14 555 1 10

10

2

3

10

10

4

10

5

pO2 / Pa Fig. 3.5. (a) Core charge as a function of the misorientation angle of four SrTiO3 bicrystals (T = 400 °C and pO2 = 0.1 bar). (b) and (c) Space-charge potential as a function of the temperature and the oxygen partial pressure, respectively. Reprinted from Ref. [94], copyright 2005, with permission from Elsevier.

Fig. 3.6. (a) Impedance spectra acquired from samples with grain size of 80 nm, 200 nm and 2500 nm. The microcrystalline sample exhibits two contributions (see inset) whereas the spectra of nanocrystalline samples are characterized only by one semicircle. (b) Calculated concentration profiles: for small grains the carrier depletion extends through the whole grain. Reprinted from Ref. [51] with the permission of AIP Publishing.

The origin of the space charge potential lies in the positive excess charge of the grain boundary core. In low angle grain boundaries, such an excess of positive charge at the grain boundary core (and the corresponding space charge potential D/0 ) can be traced back to the chemistry of the individual dislocations cores constituting the grain boundary (see Section 2). Fig. 3.5 displays the variation of the core charge for a series of bicrystals having different misorientation angles as well as the (rather weak) temperature and pO2 dependence of the space charge potential. As evidenced by independent transmission electron microscopy studies, the origin of such excess positive charge in the gb core can be explained by oxygen vacancy segregation in the core, which results from the necessity of structurally accommodating grains with different crystallographic orientations, see e.g. Ref. [95] and more detailed discussion in Section 4.2. Notably, thanks to the rather large dielectric constant of this oxide even at high temperatures, the extent of the space charge zone is broad particularly in those cases in which the impurity or dopant concentration is low. The investigation of nanocrystalline nominally undoped SrTiO3 (with a residual acceptor concentration on the order of 100 ppm) revealed the intriguing situation of samples in which the average grain size was smaller than the extent of the space charge layer (mesoscopic situation) [96]. Fig. 3.6a compares the spectrum acquired from a nanocrystalline sample (average grain size 80 nm) to the one collected from a sample with an average grain size of 2500 nm. For the nanosized ceramics, the separation of bulk and gb semicircle is absent in the impedance spectra because the carrier depletion extends almost homogeneously throughout the whole grains (Fig. 3.6b). Fig. 3.7 shows the electrical conductivity of mesoscopic SrTiO3 with an average grain size of 30 and 50 nm (in which 2k is on the order of 80 nm) [51,52]. Remarkably, compared with the bulk transport properties of a sample with the same nominal composition the p-type conductivity (characterized by the positive slope in oxidizing conditions) drops by almost 4 orders of magnitude while under reducing atmosphere the n-type conductivity (negative slope) is increased by almost 3 orders of

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Fig. 3.7. Oxygen partial pressure dependence of the electrical conductivity obtained from a set of different samples containing 100 ppm acceptor impurities. The solid squares refer to a nanocrystalline sample with an average grain size of 30 nm, the diamonds to a sample with an average grain size of 80 nm, the triangles to the grain boundary contribution of a microcrystalline sample and the open squares to its bulk properties. The green lines were calculated from Ref. [85] for an acceptor concentration of 100 ppm. Reprinted from Ref. [52] with permission of John Wiley & Sons.

magnitude (the difference being due to non-flat profiles). Moreover, in the nanocrystalline samples the p-n transition is shifted up to 12 orders of magnitudes in terms of oxygen partial pressure [52]. It is noteworthy that all these effects were quantitatively explained by a change of the concentration profiles due to space-charge effects occurring at the grain boundaries. All the considerations made above refer to the case of SrTiO3 containing acceptors (either in the form of undesired impurities or intentionally added dopants). The grain boundary properties are significantly different if one considers the situation of donor-doped SrTiO3 (e.g. with La replacing Sr or Nb replacing Ti). This is because, for such compositions, depending on temperature and oxygen partial pressure, strontium vacancies instead of conduction electrons compensate the positive charge resulting from the donors [86]. Owing to the almost complete lack of oxygen vacancies, equilibration with pO2 is extremely sluggish and dominated by comparatively faster grain boundary diffusion [97]. If grain boundary diffusion of oxygen is comparatively fast the oxidation front develops perpendicularly to the boundaries into the grains. This oxidized zone is rather insulating. Its extent can be far larger than the characteristic extent of a space charge layer [98,99]. A striking effect consists of using such frozen-in situations characterized by highly conducting interior and insulating shells as a capacitor material with high effective dielectric constant (eeff  e  dgrain/dlayer).

Fig. 3.8. Conductive AFM measurements at a R5 grain boundary in n-conducting Nb-doped SrTiO3. Top: current measured from tip to grounded bicrystal, the gb appears a low-conductive region. Bottom: current from tip flowing to contact at the left side of the bicrystal, high as long the tip is located left of the gb and suddenly decreasing when the tip is right of the gb. Reprinted with permission from Ref. [103]. Copyright 2004 by the American Physical Society.

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Irrespective of the kinetics the equilibrium space charge behavior is interesting in itself. A blocking behavior of boundaries in Nb-doped SrTiO3 bicrystal was observed by using 4 point a.c. and d.c. measurements, and attributed to a negative space charge potential [100]. Measurements of a Nb-doped SrTiO3 R5 bicrystal by atomic force microscopy also indicated a negative charge of the gb core (Fig. 3.8) [103]. Holographic TEM measurements also indicated a negative space charge potential (negative gb core charge) in Nb-doped SrTiO3 [101]. Here it is important to note that Chiang and Takagi [102] set the potential at the gb to zero (opposite to the typically used convention of / = 0 in the grain interior, see also Section 2) which leads to negative / in the space charge region for positive core charge, and Ref. [101] follows this sign convention. The reasons (cation vacancy segregation) that lead to a negative gb core charge for donor-doped SrTiO3 are discussed in more detail in Section 4.2. Note that such a negative space charge potential leads to enriched oxygen vacancies offering a straightforward explanation of fast oxygen transport (in addition to possible mobility effects; see also Section 4.3.2.).

3.1.2. BaTiO3 BaTiO3 has been extensively used in capacitors thanks to its ferroelectric properties (large bulk dielectric permittivity) and, when donor-doped (e.g. with La replacing Ba or Nb replacing Ti), as a n-type semiconductor exhibiting the positive temperature coefficient (PTC) effect [76,104–110]. Its bulk defect chemistry is largely similar to that of SrTiO3. The ferroelectric/paraelectric phase transition at 120 °C is in addition to the complex BaO-TiO2 phase diagram the major reason for not playing a similarly important role as a model material in this respect. While for capacitors a crucial property is the oxygen ion transport, which is known to be detrimental for the required long-term stability of such devices, in n-type (donor-doped) BaTiO3 large part of the research activity has been focussed on the role of the grain boundaries. Their resistive behavior has been under debate for long time and different mechanisms have been proposed over the years. Jonker suggested electrons to be trapped by oxygen diffusing along the gb [111]. Later, such a behavior was interpreted in terms of kinetic effects, namely the preferential oxidation of the grain surface layer resulting in the formation of a cation vacancy enriched diffusion layer at the grain boundaries (V00Ba [112]), whose extent can be far larger than the extent of a space charge layer. This is the consequence of the comparatively fast oxygen diffusion along the grain boundaries and the very sluggish oxygen diffusion in the bulk due to the low oxygen vacancy concentration in a donor doped perovskite. On the basis of transmission electron microscopy studies, Chiang and Takagi proposed the formation of acceptor defects in the gb core, resulting in a negative excess charge [102]. We note that a negative D/0 is the accepted reason of the positive temperature coefficient (PTC) effect [113–118]. This is usually explained in terms of the formation of acceptor states in the gb core, although their origin is still not fully understood. It has been suggested that these might stem either from the seg0000

regation of acceptor co-dopants into the core or the formation of cation vacancies (V00Ba or VTi ) during sintering [114], as it is known that the sintering conditions influence the defect compensation and, thus, the defect chemistry [119,120]. The strong decrease of er for temperatures above the tetragonal/cubic phase transition of BaTiO3 leads to an increase of the space charge potential (cf. Eq. (2.12) in Section 2.1) and correspondingly of the gb resistance as illustrated in Fig. 3.9. As far as acceptor doped BaTiO3 is concerned, the behavior is very similar to acceptor doped SrTiO3. The impedance spectra in Fig. 3.10a are a good example of the characteristic low-frequency semicircle of blocking grain boundaries (under oxidizing conditions). The thickness dgb of the depletion zone calculated from the experimental Cgb decreases with increasing

Fig. 3.9. (a) Impedance spectrum of 0.03% La-doped BaTiO3 ceramics (sintered in argon) at 25 °C. The semicircle is assigned to the blocking gb, the inset shows the bulk resistance (b) Temperature dependence of total  gb resistance, exhibiting a PTCR effect above the tetragonal/cubic phase transformation temperature. Reprinted from Ref. [121] with permission of John Wiley & Sons.

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==

Fig. 3.10. (a) Impedance spectra of BaTiO3 ceramics with different concentration of Mg Ti acceptor dopant. (b) The thickness dgb of the depletion zone obtained from the experimental impedance spectra decreases for increasing dopant concentrations. Reprinted from Ref. [122] with permission of John Wiley & Sons.

acceptor concentration as expected within the Mott Schottky model (Fig. 3.10b). The obtained space charge potentials of 0.5– 0.7 V are comparable to those found for SrTiO3 (Section 3.1.1) and BaZrO3 (Section 3.1.4). As far as nanocrystalline BaTiO3 is concerned, only a few studies addressed the importance of gb effects in such systems. Interestingly, nano BaTiO3 containing acceptor impurities revealed an enhanced p-type conductivity and a correspondingly lower activation energy, which were interpreted in terms of a reduced effective oxidation enthalpy compared to the microcrystalline case [123,124]. A full space charge analysis however is missing. 3.1.3. (Pb,La)(Zr,Ti)O3 Lead zirconate titanates (PZT) exhibit a variety of important functional properties (depending on the zirconium-totitanium ratio and dopant concentration) that make them suitable for piezoelectric and electro-optical applications [125]. Despite the large amount of studies on such compounds, a relatively small number of contributions have dealt with the defect chemistry of PZT and donor-doped PZT [126–138]. Quite interestingly, independent studies reported on p-type conductivity under oxidizing conditions not only in nominally undoped PZT [131,134] but even in La and Nd doped PZT [137,138]. Such findings clearly highlight the presence of a complex defect chemistry, in which the PbO release during sintering and the possible association of defects (particularly at low temperature) yield to defect compensation mechanisms far from the nominal situations. Large concentrations of Schottky defects (VO and V00Pb ) can occur far beyond an ideal solution model. In particular, the p-type conductivity can be explained by the formation of a large concentration of lead vacancies acting as acceptors and exceeding the donor concentration [137,138]. Moreover, impedance spectroscopy measurements revealed the rather blocking behavior of grain boundaries, which has been interpreted in terms of space-charge effects stemming from an excess positive charge residing in the grain boundary core [137]. We note that such an electric field would depress even more severely the concentration of mobile oxygen vacancies (owing to their double charge). This finding seems however not consistent with 18O diffusion experiments followed by SIMS analysis, according to which the grain boundaries exhibit an oxygen diffusion coefficient which is at least 5 orders of magnitudes higher than in the bulk [136]. 3.1.4. BaZrO3 Following the seminal work by Iwahara et al. on proton conducting perovskites [139,140], acceptor doped BaZrO3 (particularly Y-doped BaZrO3) has become over the years the material with the best combination of high bulk proton conductivity and excellent chemical stability under the required operation conditions [141–144]. Protonic carriers OHO are generated by dissociative incorporation of water into oxygen vacancies (cf. Ref. [145])

H2 O þ VO þ OxO 2OHO

ð3:1Þ

and has been shown to be mobile via phonon assisted proton hopping, i.e. the proton of a hydroxide ion is detached and transferred to a neighboring oxide ion rather than being transported by OH hopping [146]. The protonic defects have a lower migration barrier (typically 0.4–0.5 eV for the transfer of the proton to a neighboring O) compared to VO in perovskites (often in the range of 0.8–1 eV), leading to a higher ionic conductivity in particular at lower temperatures. At temperatures higher than 400–600 °C the dehydration gradually changes the materials from a proton to an oxide ion conductor. These materials

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10-2

bulk total

2

2

10-3

3

10-4

4

7 6

3

7

6

2 3

4

5

4

10-3

4

3 1 4

10-5

10-2 bulk grain boundary

σ / Scm

σ / Scm-1

2

(b)

1

3

-1

(a)

2

10-4

10-5

4

10-6

10-6 3

N2, 300 °C 20 mbar H2O

4

10-7 0.05

0.10

0.15

0.20

x in BaZr1-xYxO3-x/2

0.25

300°C, wet 5% H2/Ar 10-7 0.05

0.10

0.15

0.20

0.25

x in BaCe1-xSmxO3-x/2

Fig. 3.11. (a) Bulk and total (grain boundary) proton conductivity of Y-doped BaZrO3 as function of dopant concentration. Data from [154] = 1, [161] = 2, [160] = 3, [155,176] = 4, [156] = 5, [157] = 6, [158] = 7. (b) Bulk and grain boundary conductivity of Sm-doped BaCeO3, data taken from Ref. [159].

have found application as hydrogen sensors, electrochemical hydrogen pump, fuel cells and steam electrolyzers, see e.g. [147–150]. However, in spite of the remarkable bulk properties, grain boundaries hamper the proton migration substantially [141,151–153]. As demonstrated in Fig. 3.11, this blocking behavior of the grain boundaries is present in a similar way for barium zirconate materials as well as for barium cerates. The blocking character is most pronounced at low dopant concentrations (a further discussion of the dopant concentration dependence is given in Section 4.2). While for a given acceptor concentration the bulk conductivity of Y-doped BaZrO3 is higher, its grain boundaries are more strongly blocking (and grain sizes are typically lower) leading to a lower total conductivity compared to Sm-doped BaCeO3 (Fig. 3.11b). Dedicated studies have revealed that sintering conditions and extended annealing treatments can lead to a significant modification of the grain boundary properties even in the absence of undesired secondary phases [160,161]. Such observations point towards intrinsic properties to be the origin of the ionic blocking behavior of the grain boundaries. In the last few years, a number of independent studies have provided clear evidences that space-charge effect and specifically the presence of an excess positive charge at the grain boundary core induces the depression of the proton concentration in proximity of the grain boundaries and hence an increase of the local electrical resistivity [162–167]. Thus the grain boundary situation is

Fig. 3.12. (a) Systematic decrease of the depletion width d at the grain boundaries of BaZr1
xYxO3
x/2 with increasing dopant concentration x. Reproduced from Ref. [163] with permission of The Royal Society of Chemistry. (b) Bulk and grain boundary conductivity of BaZr0.9Y0.1O3
d ceramics measured over an extended pO2 range. Reprinted from Ref. [162], copyright 2010, with permission from Elsevier.

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closely related to the one discussed for acceptor doped SrTiO3. However, it is important to mention that DFT calculations performed for different boundary orientations revealed that both oxygen vacancies and protons tend to segregate into the grain boundary core both leading to an excess local positive charge and hence to the formation of a positive spacecharge potential (a more detailed discussion of the origin of the core charge will be given in Section 4) [168–171]. In particular for low dopant concentrations, the capacitance extracted from the grain boundary semicircle yields a thickness of the blocking layer that clearly exceeds the width of the structurally distorted grain boundary core (see e.g. [162,163,166]). Fig. 3.12a shows that the depletion width decreases systematically for increasing dopant concentration [163] as it is expected from Eq. (2.12) in Section 2.1 as long as the grain boundary core charge does not vary too much (see also Section 4.2). The observed space charge potentials decrease, from  0.9 V at 2% doping to 0.4 V at 20% doping (cf. Fig. 4.5 in Section 4.2). Overall, the space charge potentials in acceptor-doped BaZrO3 are larger than for acceptordoped SrTiO3 with comparable dopant concentration. Fig. 3.12b shows the bulk and grain boundary conductivity of polycrystalline BaZr0.9Y0.1O3
d measured over a large pO2 range [162]. While rbulk remains purely ionic over the whole range, the slight increase of electronic (n-type) conductivity at the lowest pO2 values indicates electron accumulation adjacent to the positively charged grain boundary core that decreases the blocking behavior for protons. Further evidence for space charge depletion of protons (and oxygen vacancies and electrons holes) adjacent to positively charged grain boundary cores comes from the observation that acceptor dopants accumulate in the grain boundary region when samples are sintered/annealed under conditions that allow for cation diffusion over a few nm [161,172–175]. It was found that this dopant accumulation (i) makes the grain boundaries less blocking (transition from pure Mott-Schottky case to frozen partial Gouy-Chapman case, Table 2.2) cf. Fig. 3.13a, (ii) the relative amount of dopant accumulation does not depend on the ion radius (Y3+ vs. Sc3+), i.e. it is not driven by size mismatch but by electrostatic attraction by the positive grain boundary core [176]. A high resolution TEM-EDX investigation of Sc-doped BaZrO3 in Fig. 3.13b shows that the dopants do not only accumulate in the space charge zone, but also enter the grain boundary core where they are most effective in directly compensating the positive core charge [177].

Normalized Intensity / a.u.

1.0

0.5

0.0

GB core

(a)

(b)

0

2

4

6

8

Distance / nm Fig. 3.13. (a) Bulk and grain boundary conductivity of BaZr0.94Y0.06O2.97. Blue symbols = as-prepared sample (compacted by spark plasma sintering, 5 min at 1600 °C), red symbols = additionally annealed sample (20 h at 1700 °C, allowing for dopant accumulation in the grain boundary region but not leading to grain growth). Reprinted with permission from Ref. [176]. Copyright 2012 American Chemical Society. (b) Accumulation of Sc3+ in the space charge zone and in the grain boundary core of BaZr0.94Sc0.06O2.97 annealed for 20 h at 1700 °C. Reprinted from Ref. [177], copyright 2014, with permission from Elsevier.

Fig. 3.14. Elemental composition of an individual grain boundary (position indicated by dashed line) in polycrystalline BaZr0.9Y0.1O2.95 from atom probe tomography, and resulting charge density (red = excess positive charge). The acceptor Y0Zr is accumulated in the grain boundary region at expense of Zr. Reprinted with permission from Ref. [178]. Copyright 2016 American Chemical Society.

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Very recently, the technique of laser assisted atom probe tomography could be applied to grain boundaries in BaZr1
xYxO3
x/2 [178]. This yields a high resolution three-dimensional picture of the elemental distribution (Fig. 3.14). Overall, this distribution is also in line with the model of positive grain boundary cores (accumulation of Y0Zr in the grain boundary region). These measurements also indicate that even along a single grain boundary the elemental distribution and thus the electrical properties exhibit some lateral inhomogeneities. A complete quantitative model (including extraction of the core charge and space charge potential) is however nontrivial: (i) the concentration of one charged defect – protons – cannot be measured, (ii) for 10% doping (and dopant accumulation at the boundary) the assumption of ideally dilute behavior of the defects is not necessarily valid any more, (iii) the sample is in a partially frozen-in situation (VO and protons are still mobile at intermediate T where the cation distribution is already frozen in), (iv) owing to the curved nature of many boundaries, a 3dimensional solution of the Poisson equation is required. Finally, the dominant role of space charge depletion zones for the electrical properties is evidenced by the influence of an applied dc bias on impedance spectra as well as in leading to nonlinear I-V curves. The applied dc voltage perceptibly modifies the barrier as soon as the voltage drop per grain boundary becomes comparable to the space charge potential. This requires either large bias values on very thin samples with large grains [165], or the ability to contact individual grain boundaries by microelectrodes as illustrated in the inset of Fig. 3.15 [166]. In the impedance spectra, the dc bias leads to a drastic decrease of the grain boundary semicircle (Fig. 3.15) while the bulk semicircle remains unchanged. The bias also moderately decreases the grain boundary capacitance, as expected within the space charge depletion model. In order to increase the total conductivity, numerous attempts have been made to increase the grain size of Y-doped BaZrO3. While the addition of sintering aids such as ZnO or NiO to BaZr1
xYxO3
x/2 powders allows to decrease the sintering temperature and/or increase the grain size, the resulting total conductivities are still clearly lower than the proper bulk conductivity (often, the use of sintering aids even decreases the bulk conductivity), see e.g. [152,179,180]. The ‘‘reactive sintering” approach, which proved to be very successful, is discussed in Section 4.4. The blocking behavior of the boundaries has naturally driven various groups towards the investigation of boundary-free epitaxial BaZrO3 thin films. The total conductivity of thin epitaxial BaZr0.8Y0.2O2.9 films on single crystalline MgO was found to match the corresponding bulk value and activation energy [181]. In Ref. [182], a high proton conductivity was claimed for epitaxial BaZr0.8Y0.2O2.9 films on single crystalline MgO, exceeding literature bulk values by about half an order of magnitude at 600 °C. However, the simultaneously higher activation energy raises some questions. Extremely high conductivity values (20 S/cm at 600 °C, i.e. exceeding that of 1 M aqueous HCl of 0.3 S/cm at 25 °C) were also claimed for 10 nm thin BaZr0.8Y0.2O2.9 films on single crystalline NdGaO3 substrates with a nominal compressive lattice mismatch of
9% (again with unexpectedly high activation energy) [183]. Such films were investigated by electrochemical strain microscopy in Ref. [184], and the presence of misfit dislocations between substrate and film was considered to be the reason of the enhanced interfacial protonic transport. XPS and TEM investigations indicated formation of extended defects as well as partial Ysubstitution for Ba, for which decreased proton migration barriers were obtained by DFT (opposite to the measured increased activation energy) [185]. On the other hand, molecular dynamics calculations for Y-doped BaZrO3 yielded only a moderate proton mobility increase (factor of two at 1000 K) for up to 1% compressive biaxial strain, and decreased mobility for tensile as well as higher compressive strain [186]. 3.1.5. (La, Sr)GaO3
d Acceptor-doped LaGaO3, in particular using Sr doping on the A-site and Mg on the B-site, exhibits a high oxygen ion conductivity and low electronic transference number [187,188]. The total (bulk + gb) conductivity reaches 0.08 S/cm at 700 °C.

Fig. 3.15. Impedance spectra measured without and with dc bias across an individual grain boundary in a large-grained BaZr0.98Y0.02O2.99 sample with the help of microelectrodes (circular features in the inset image) that can be contacted by Pt tips. The bias decreases the grain boundary semicircle, while the bulk arc (not well resolved here) remains unchanged. Adapted from Ref. [166] by permission of the PCCP Owner Societies.

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This high oxygen ion conductivity is related to a higher mobility of VO in the perovskite structure (similarly found also for SrTiO3 or (La,Sr)(Fe,Co)O3
d) compared to that in Y-doped zirconia. A blocking nature of the gb becomes obvious only at lower temperatures and/or low acceptor concentration, see e.g. [189,190]. The nature of the blocking effect was further investigated for a La0.99Sr0.01GaO2.995 sample [191,192]. TEM showed that the gb were free of secondary phases. Analyzing the gb resistance in terms of a space charge depletion zone yielded a space charge potential of about 0.4 V, which could consistently explain the measured pO2 and T dependences of ionic and electronic bulk and gb conductivity [191]. Further evidence of space charge depletion being the main contribution to the blocking gb nature was obtained from the fact that the diameter of the gb semicircle in impedance spectra decreases when a large DC bias is applied while the bulk arc remains unchanged [192]. Also for other large-bandgap perovskites (acceptor-doped LaAlO3, LaScO3, LaInO3) a blocking behavior of the grain boundaries for oxygen vacancies and/or holes was found, as shown in [193,194] and references therein. 3.1.6. Li3xLa2/3
xTiO3 Li3xLa2/3
xTiO3 perovskites with partial occupation of the A site by La3+, the much smaller Li+ and cation vacancies were identified as good lithium ion conductors [195,196]. While the bulk ionic conductivity amounts to 10
3 S/cm at room temperature, the total ionic conductivity is dominated by the much larger gb resistance (see e.g. [196–199]). The maximum bulk conductivity was found for the composition with x  0.1, i.e. Li0.3La0.57TiO3 (see e.g. [197,198]). Since these materials require high sintering temperatures in the range of 1300 °C easily leading to Li loss, the actual Li content has to be determined by chemical analysis. A recent high resolution TEM study showed that the blocking behavior of the gb is mainly related to a 2– 3 nm thick gb core at the majority of the gb with structure as well as composition (almost complete absence of Li and La) strongly differing from bulk [200]. However, the gb capacitance and the corresponding width of the low-conductive zone was not reported, thus no conclusion can be drawn whether such a gb core is additionally surrounded by space charge depletion zones. 3.2. Fluorites 3.2.1. Ceria Ceria is a material of great technological relevance for a number of important applications thanks to its ability of easily forming mobile oxygen vacancies. Unlike the previous examples, here the cation can easily undergo redox changes allowing for oxygen deficiency even without acceptor doping. This mixed conductivity nature is a major reason why ceria is such a good catalyst as it can show both redox and acid-base activity. CeO2 can also be substantially acceptor doped resulting in a large concentration of oxygen vacancies. Owing to the high mobility of these vacancies, rare-earth doped ceria is a prominent solid electrolyte for SOFC, but with the disadvantage of becoming n-type electronically conducting at the anode (and at T > 700 °C throughout the whole electrolyte) [201–204]. While its bulk defect chemistry was established in the seventies [205,206], the intrinsic electric properties of the grain boundaries was understood much later. Similar as for acceptor doped zirconia, the formation of insulating silica layers by segregation of SiO2 impurities in the starting powders at the grain boundaries led to large grain boundary resistances, see e.g. [207,208]. This could be overcome by increased purity of the powders; some additives were also found to act as Si scavenger (see e.g. [209]). Only after the first observations of highly electronically conducting grain boundaries in nanocrystalline nominally pure ceria (Fig. 3.16a) [210], a number of contributions realized that such a behavior stems from space-charge effects owing to positively charged grain boundary cores [20,23,211–214]. As illustrated in Fig. 3.16b, the positive space charge potential enhances the n-type electronic conductivity parallel to the gb owing to e0 accumulation, while simultaneously the transport of VO across the gb is impeded by strong carrier depletion. The occurrence of space-charge effects at the grain boundaries

Fig. 3.16. (a) Electrical conductivity as a function of oxygen partial pressure for microcrystalline (average grain size 5 lm) and nanocrystalline ceria ceramics (average grain size 10 nm, powder from inert gas condensation IGC or decomposition of acetates). Reproduced from Ref. [210] with permission of Springer. (b) Scheme emphasizing the contrary effects of the space charge potential on enhanced electron transport parallel to the gb (blue) and impeded VO transport perpendicular to the gb (red). Reproduced with permission from Ref. [23]. Copyright 2002, The Electrochemical Society.

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Fig. 3.17. Nyquist plot of the complex impedance as a function of different d.c. bias. In the inset a detail of the high frequency range corresponding to the bulk properties, which remains unaffected by the bias. Reproduced with permission from Ref. [215]. Copyright 2005, The Electrochemical Society.

Fig. 3.18. Total electrical conductivity measured at 280 °C and pO2 = 10
5 bar from Ce0.9Gd0.1O1.95 thin films having an average grain size of 30 (black symbols) and 10 nm (red symbols), respectively. The conductivity of the sample with smaller grain size exhibit a clear oxygen partial pressure dependence, which corresponds to an ionic transference number ti = 0.75. Reproduced from Ref. [216] by permission of the PCCP Owner Societies.

was further supported by bias-dependent impedance spectroscopy measurements carried out on 1 mol% Y2O3-doped CeO2 ceramics. Fig. 3.17 illustrates how the gb contribution to the complex impedance decreases as a function of the applied d.c. bias [215]. It is noteworthy that the presence of charged boundaries can lead to effective electric transport properties, which can considerably differ from the bulk situation. A good example in this sense is given by the measured n-type conductivity in acceptor doped ceria (namely Ce0.9Gd0.1O1.95) under oxidizing conditions and low temperature (Fig. 3.18) [216]. Independent evidences of the presence of a space charge situation at the gb have been provided by various experimental techniques. Under conditions of sufficient cation mobility (sintering, high-T annealing) the attraction by the positively charged core gives rise to an accumulation of acceptor dopants in the space charge zone and also in the gb core [217,218]. Also, positron annealing results of undoped and Gd-doped CeO2 were interpreted along the lines of the space charge model with positive core charge [219]. Atom probe tomography analysis for example revealed a perceptible segregation of defects [220] and even allowed for the determination of space-charge concentration profiles and potentials [221]. On the other hand, the Gd segregation and nanodomain formation observed at a very high dopant concentration of 25% might largely be caused by effects other than space charge [222]. Electrochemical strain microscopy (ESM) has been used to detect the lattice expansion of ceria under a controlled dc bias (Fig. 3.19) [223]. The grain boundaries exhibit a stronger volume response to the oscillating tip voltage. This was interpreted as result of small polaron (Ce3+) accumulation in the space charge zones, leading to larger volume effects by the redistribution of these polarons upon changing the tip voltage. Another series of ESM experiments carried out on YSZ single crystals [224] revealed also a perceptible volume change

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Fig. 3.19. Electrochemical strain microscopy detecting space charge in a polycrystalline Sm-doped ceria film. (a) Expansion of the lattice under positive tip potential, assigned to a local accumulation of polarons (Ce3+) causing the volume expansion. (b) Map of response magnitude for ±3 V perturbation, revealing increased response in the gb region. Reprinted from Ref. [223] with the permission of AIP Publishing.

Fig. 3.20. Measured compositional stresses Dr between reducing and oxidizing conditions for ceria films on Si substrates having different average grain size L. Reprinted from Ref. [225], copyright 2013, with permission from Elsevier.

(strain) under an electric bias. This was, however, interpreted in terms of oxygen vacancies generation and annihilation at the YSZ surface in contact with the ESM tip. Curvature measurements carried out on ceria thin films revealed nanocrystalline ceria to undergo rather severe and reversible stresses during oxidation/reduction cycling (Fig. 3.20). The experimental findings were interpreted in terms of space charge zones yielding substantial local strain effects owing to the point defect redistribution [225]. Interestingly, the electrical transport properties of the grain boundaries in donor-doped ceria have been barely investigated. The boundaries were found to hinder n-type conductivity either owing to the presence of a negative space charge potential or due to local strain effects, which might be related to the presence of oxygen interstitials [226]. 3.2.2. Acceptor doped zirconia The classic example of an oxide ion conductor, namely cubic acceptor-doped zirconia (typically 8 mol% Y2O3-doped) is still the material of choice for electrolytes in solid oxide fuel cells or electrolyzing cells. In spite of the bulk ion conductivity being lower than that of ceria, it confers a variety of excellent properties that are important for an electrolyte in a SOFC: (i) High ionic conductivity. (ii) Large electrolytic domain: unlike CeO2, zirconia does not become electronically conducting even at extremely low oxygen partial pressures (e.g. at 800 °C the electronic conductivity exceeds the ionic conductivity only for pO2 < 10
38 atm [227]). (iii) High chemical stability. (iv) High mechanical stability. (v) Tolerable thermal expansion properties. A disadvantage is that – similar to the situation in acceptor-doped ceria – the gb hamper ionic migration, particularly at low temperatures. The elucidation of the microscopic origin is difficult, as a high complexity is met: (i) The defect chemistry is far from ideal owing to the high dopant concentrations. (ii) Grain boundaries may be interrelated to pores (non-linear current lines). (iii) Segregation of dopants is an important issue. (iv) Impurity effects, in particular due to silica, are a serious problem. (v) Space charge effects occur in spite of the high doping concentrations because of the low free carrier concentration. (vi) Complex structural and compositional effects, e.g. occurrence of amorphous interphases additionally complicate the situation, see e.g. [228,229] and references therein. For highly pure acceptor-doped zirconia samples the blocking behavior was proven to result from space-charge effects due to an excess positive charge in the core of the grain boundaries [230,231], with /0 in the range of 0.25–0.3 V for

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Fig. 3.21. (a) Y accumulation in the gb region of YSZ. Reproduced from Ref. [238] with permission from Taylor & Francis. (b) Decrease of gb semicircle in 2 mol% Y2O3-doped ZrO2 under reducing conditions, note that the bulk resistance (inset) remains unchanged. Reproduced with permission from Ref. [240]. Copyright 2006, The Electrochemical Society.

8 mol% Y2O3-doped ZrO2 and a width of the blocking gb layer of 5 nm exceeding the structurally distorted zone observed in TEM. Isotope exchange experiments supported these findings by showing that oxygen diffusion is hindered at the grain boundaries of bicrystals [232] and nanocrystalline ceramics [233]. This has been supported by aberration corrected imaging of atomic occupancy in the grain boundary core, showing a relatively high concentration of oxygen vacancies and thus an excess positive charge in the grain boundary core [234]. Positron annihilation experiments on Y2O3-doped zirconia suggested the presence of a negatively charged region adjacent to the grain boundaries [235]. Accumulation of Y in the space charge zones (under sintering or high-T annealing conditions where the cations are sufficiently mobile) as observed e.g. in [217,236–239] (Fig. 3.21a) is also in line with the space charge model (see also Section 4.2). As discussed at the end of Section 2.7 this decreases the space charge potential [75]. Electron accumulation in the space charge zones – becoming important under reducing conditions – was found to decrease the gb resistance while rbulk remained unchanged and ionic (Fig. 3.21b) [240]. From the observation of an increased surface 18O/16O ratio in the vicinity of grain boundaries, an enhanced oxygen ion incorporation at gb of polycrystalline YSZ has been claimed [241], however a fit of the tracer profiles yielding effective tracer surface rate constants k⁄ and tracer diffusivities D⁄ (which will be space-dependent in the gb region) has not been given. While the bulk concentration of protons in zirconia is small and correspondingly no significant proton conductivity is found [242], nanocrystalline YSZ exhibited a significant and prevailingly protonic conductivity under humid conditions at low temperatures (T < 250 °C) [243–247]. A similar behavior was found also for other oxides (even undoped) [248–255]. The mechanism behind such unexpected proton transport properties has been under debate. Some publications suggest proton conductivity in the core of the grain boundaries [244–246,249,256], others claim the protonic transport to occur at open pores, cracks, and percolating intergranular channels, where water can be adsorbed [243,247,248,250,254,255] and space charge situations at the water/oxide may affect the local concentration of the mobile charge carriers [254]. Ab-initio calculations indicate that proton uptake is more favorable in gb of zirconia compared to bulk [257], and analyze proton insertion and resulting space charge scenarios at the ceria surface [258]. The dynamics of protons in adsorbed water layers on ZrO2 was investigated by ab initio molecular dynamics [259]. The success of the space charge model in explaining various resistive effects at grain boundaries even though the dopant concentration is very high (as in fluorite and perovksite based electrolytes) points towards a screening length that is larger than expected from Eqs. (2.7) and (2.8). This suggests an effective ‘‘neutralization” at high carrier concentrations as e.g. invoked by strong association. A similar effect is the poor screening in concentrated liquid electrolytes as recently debated in literature [260,261]. 3.3. Grain boundary transport properties of other structures 3.3.1. Corundum (a-Al2O3) Transport measurements in a-Al2O3 are extremely challenging because defect formation energies are high, defect concentrations correspondingly low and also migration barriers are high. Furthermore, even at high measurement temperatures it is difficult to completely rule out the presence of remaining frozen-in defects. In view of such low bulk mobilities it is not surprising that enhanced mobility along grain boundary pathways is to the fore. Results of single crystal bulk oxygen and aluminium tracer diffusion (both by a vacancy mechanism) are summarized and discussed in Refs. [262–264]; see also the references given there. D⁄O is found to be in the range of 10
20 cm2/s at 1500 °C with an activation energy 5.5 eV (all these measurements are assigned to the extrinsic regime, i.e. this activation energy represents the migration barrier). D⁄Al data are very limited. In Ref. [263] it was concluded from the available literature that D⁄Al clearly exceeds D⁄O, and has an activation energy lower by about 20%. Theoretical calculations had obtained a large range of migration barriers for oxygen vacancy migration bar-

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Fig. 3.22. Oxygen vacancy migration barriers (in eV) in bulk a-Al2O3 from DFT calculations; the transition O3-O4 is required for long range transport (O = red spheres, Al = gray). Al vacancy migration barrier values are also given. Reprinted with permission from Ref. [266]. Copyright 2013 by the American Physical Society.

Fig. 3.23. HRTEM image of a nominally undoped symmetrical R31 tilt Al2O3 bicrystal. In (b) the crystal structure is overlaid. Reprinted from Ref. [269], copyright 2007, with permission from Elsevier.

riers, with many values much below the experimental values. Recent pair potential Kinetic Monte Carlo [265] as well as ab initio calculations [266] indicate that these are low migration barriers which however allow only for exchange between one subgroup of oxygens, and that the barriers required for long-range transport are in the range of 5 eV close to the experimental data (Fig. 3.22). In diffusion experiments of polycrystalline samples, impurity segregation to the gb and its influence on the transport properties is a significant source of uncertainty. A moderate increase of hole conductivity (nominally undoped Al2O3, at 1500 °C) with decreasing grain size was tentatively interpreted by space charge effects, with the negative gb core charge caused by acceptor segregation [267,268]. Oxygen tracer diffusion experiments for nominally undoped bicrystals ([269,270], one example shown in Fig. 3.23) point towards accelerated transport along the gb by several orders of magnitude, but indicate also a strong dependence of this gb diffusivity on the actual gb orientation and structure. Closely related are the findings of fast O diffusion along dislocations created by deformation of single crystals [271,272]. While some aspects of this accelerated O transport are still under discussion, atomistic calculations indicate a contribution from decreased defect for 0Þ gb, segregation to the gb core was mation energies as well as increased mobility in the gb core. For R3(0 0 0 1) and R3ð1 0 1 found to be favorable for O as well as for Al vacancies [266]. The segregation energies were more negative for V000 Al compared  0Þ gb with lower interface energy. V segto VO , and also larger for the high-energy R3(0 0 0 1) gb compared to the R3ð1 0 1 O  1g twin gb in [273]. For the R3(0 0 0 1) gb core, the migration barriers of both V and regation was also found for the f1 0 1 O  V000 Al were lower than the bulk values, while the opposite trend was obtained for the low-energy R3ð1 0 1 0Þ gb [266]. An ab initio investigation showed that a decrease of the band gap in the structurally distorted gb region may switch the defect  chemical regime there from V000 Al dominated to VO dominated [274]. This could explain the experimental finding of two regimes of oxygen permeation depending on pO2. 3.3.2. Rock salt structured oxides (MgO, NiO) MgO is an example of an oxide with rocksalt structure. Owing to the dense and extremely stable structure, defects are realizable only at very high temperatures. Beyond that, Mg2+ is a very redox-stable ion which leads to a large band gap (7.8 eV) and only allows for p-type conductivity under realistic conditions. The conductivity of undoped MgO is low

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Fig. 3.24. Migration barriers of (a) Mg and (b) O vacancies at a {4 1 0} tilt gb in MgO obtained from pair potential calculations. The bulk migration barriers are for Mg vacancies 1.94 eV, oxygen vacancies 2.12 eV. Reprinted with permission from Ref. [283]. Copyright 1997 by the American Physical Society.

(10
10 S/cm at 1000 K); it is predominantly ionic at T below 1100 °C ([275] and references therein). Mg and O tracer diffusivities measured on nominally undoped single crystals differ by less than 2.5 orders of magnitude, with D⁄Mg > D⁄O [276,277]. Thus, the nature of the ionic conductivity (magnesium or oxygen defects) can change depending on the impurities and pO2 for nominally undoped MgO. Oxygen tracer diffusion measurements on polycrystalline MgO suggested a significantly enhanced O diffusivity along the gb [278]. Ni diffusion was found to be moderately accelerated parallel to gb in polycrystalline and bicrystal MgO samples, but the effect became unmeasurably small at T 1980 °C [279]. In a subsequent publication this enhanced gb diffusivity was related to impurity segregation [280]. From the Sc3+ segregation profile in the vicinity of grain boundaries in ceramic samples, a negative excess charge of the gb core was concluded [281]. Conductivity measurements on 26° tilt bicrystals at 800–1400 °C did not yield significant differences to the respective single crystal (neither across nor along the gb) [282]. Accelerated diffusion along gb was found in static lattice and molecular dynamics simulations (see e.g. [283– 285]). In [283], a lower formation energy was found for Mg and O vacancies in the core of a {4 1 0} tilt gb, with quite similar values for both sorts of vacancies. A similar observation was made in [285] for a number of gb orientations, thus these calculations do not directly allow a prediction of the gb core excess charge. In addition to increased defect concentrations, decreased defect migration barriers compared to bulk were found (Fig. 3.24), which are however partially compensated by defect association. Overall, an increased diffusivity along the gb is obtained from the combination of concentration and mobility effects. The formation of peculiar structures at the gb involving segregated impurities was emphasized in [286] based on highresolution TEM and ab initio calculations. An example for a rocksalt-structured oxide with smaller bandgap (3.6–3.8 eV) and partially occupied d orbitals is NiO. Under oxidizing conditions the conductivity of nominally undoped NiO is predominantly p-type electronic (see e.g. [287,288]); the electron holes are compensated by Ni vacancies. A blocking effect of the gb for the hole conductivity has not been reported so far. D⁄Ni ([289,290], mobile Ni vacancies) exceeds D⁄O [291,292] by more than 6 orders of magnitude at 1300 °C for nominally undoped single crystals. D⁄O was tentatively assigned to mobile O vacancies [292]. Higher D⁄O values reported in [293] might be less reliable because they were determined on the basis of 18O gas-phase analysis only, while in [291,292] the whole profile in the crystal was analyzed. Nickel tracer diffusion (see e.g. [290]) as well as chemical diffusion (Dd of single crystals [294–296] more than one order of magnitude lower than that of a polycrystalline sample [297]) was found to be accelerated parallel to the gb. This is also supported by simulations. For four different tilt gb, O as well as Ni vacancies were found to have similar segregation energies, in the range of
0.4 to
1.6 eV [298]. Decreased migration barriers are obtained for Ni vacancies in the gb core from molecular dynamics calculations [299,300]. Similar to the case of MgO, the increased gb diffusivity is rationalized mainly in terms of lower defect formation and migration energies rather than by pronounced space charge effects. 3.3.3. Rutile TiO2 Titania forms a number of polymorphs, among which rutile (the thermodynamically stable phase for large crystallites and elevated T) and anatase (stable phase only for small grains at low T [301]) are the most important. TiO2 has a vast number of applications ranging from application as white pigment to photocatalysis, active component in dye-sensitized solar cells and photoelectrochemical water splitting to memristive switching and potential use as anode material in lithium batteries (see e.g. [302–305]). Regarding grain boundary properties, we concentrate here on the stable rutile phase where bicrystals and polycrystalline ceramic samples can be studied conveniently. For rutile, values of the band gap in the range of 3.07–3.7 eV are reported [306–308]. Nominally undoped TiO2 typically contains traces of impurities such as Fe3+, Al3+ that act as acceptor dopants. However, TiO2 can easily be reduced and become n-type electronically conducting by heating in vacuum or presence of reducing gases. Acceptor dopants and/or excess elec trons are compensated by oxygen vacancies and/or Ti interstitials. Experimental and theoretical studies indicate that Tii 0000

tend to dominate at lower pO2 and VO at higher pO2 [309–313]. At very high pO2, the formation of VTi is hypothesized

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 0 bicrystals. Left: after oxidizing conditions, right: after annealing in dilute hydrogen. Reprinted with Fig. 3.25. STEM images for rutile R3(1 1 2)½1 1 permission from Ref. [319]. Copyright 2015 by Nature Publishing Group.

[314]. A recent DFT study suggests that protonic defects might also play an important role in the defect chemistry of TiO2  [315]. The mobility of Tii is found to be moderately higher than that of VO [310]. In a certain pO2 range this leads to complex stoichiometry relaxation kinetics after a pO2 step with two different time constants. Several groups investigated grain boundaries in bicrystals and polycrystalline rutile samples by TEM; unfortunately in most cases no direct comparison to the electrical properties is given. In [316,317], sign and magnitude of the space charge potential were extracted from the segregation of dopants, observed by TEM-EDX, in the space charge zones of ceramic samples (note that according to the definition in [316,317], a negative space charge potential corresponds to a positive excess gb core charge – opposite to the usual convention). Interestingly, in Al- and Nb codoped TiO2 samples the core charge switched from positive to negative when moving from effectively acceptor-doped to donor-doped compositions. Slightly blocking gb (i.e. a positive gb core charge) were also observed in nominally undoped rutile ceramic samples which contained about 65 ppm of acceptor impurities and had predominantly p-type electronic conductivity [318]. High-resolution TEM images  0 bicrystals showed that the actual structure of the gb core (Fig. 3.25) is modified by switching from oxidizof R3(1 1 2)½1 1 ing to reducing treatment [319]. In [320], impedance spectra of polycrystalline undoped TiO2 and TiO2 decorated with strongly oversized Y3+ at the grain boundaries were compared. While the undoped sample showed only a bulk semicircle, a low-frequency arc appeared for the Y-decorated samples indicating blocking behavior of the gb. Assuming that the bulk conductivity was n-type (however, no proof of such a pO2 dependence was given), this was seen as evidence for a negative excess charge in the gb core, and space charge potentials in the range of
0.6 to
0.8 eV were extracted. Strong Y-segregation to the gb core (as observed by TEM EDX [321]) was regarded as the origin of the negative core charge. Defect and dopant segregation energies to cores of different gb orientations were also derived from ab initio calculations. Donor (Nb5+) as well as acceptor (Al3+, Ga3+) dopants were found to segregate to the gb cores, with the segregation energy  tilt grain boundary, the being the more negative the higher the energy of the considered gb [322]. For the ½1 0 0 f0 5 1g 0000



energy of VO and Tii was found to be lower by up to 2 eV and 3 eV close to the gb core [323]. While also VTi and O00i have some segregation tendency towards the gb core, overall more energetically favorable positions were found for the positive

-1

[001] Pristine [001] disl - Pos 2 -AC [001] disl - Pos 2 -DC(Au-elec)

T = 550 °C

-1/6

-3

-1

-1

log(σ /Ω .cm )

-2

-4

-1/4 -5

-6

+1/4

-7 -30

-25

-20

-15

-10

-5

0

log( pO2/bar) Fig. 3.26. Conductivity of a nominally undoped rutile single crystal measured along the [0 0 1] direction. Squares: pristine sample (ac impedance), triangles: sample with dislocations formed by uniaxial compression at 1200 °C (ac impedance), stars: sample with dislocations – dc measurement with ion-blocking gold electrodes. Reprinted from Ref. [324] with permission of John Wiley & Sons.

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Norm. conductance Y

II

/S

7x10 6x10 5x10 4x10 3x10 2x10 1x10

-11 o

-11

190 C o 170 C o 150 C o 130 C o 110 C

-11 -11 -11

ion

-11

0 0

(a)

σion∞

II

ΔY

-11

200

400

600

Thickness / nm

800

(b)

Fig. 3.27. (a) Thickness dependence of the normalized ionic conductance of LiF films on single-crystalline TiO2 substrates. The positive intercept for zero thickness indicates an increase of ionic conductance by the interface to the substrate. (b) Schematic of charge carrier redistribution. Reprinted with permission from Ref. [325]. Copyright 2012 American Chemical Society.

defects at this specific grain boundary core. In summary, these calculations indicate that the segregation of native defects and/or of dopants lead to the excess gb charge. An interesting observation was made when free dislocations were created by mechanical deformation of nominally undoped rutile single crystals at elevated T [324]. The ac conductivity changes from p-type electronic (with the typical (pO2)1/4 dependence) to a higher, pO2 independent ac conductivity at lower pO2 as shown in Fig. 3.26. Most of the conduc tivity enhancement originates from ionic defects VO and Tii ; the increase of hole conductivity measured by ion-blocking dc measurements (stars) is much weaker. This indicates that the dislocation cores formed in this sample exhibit a negative  excess core charge, leading to an accumulation of mobile VO and Tii around the dislocations (‘‘one-dimensional doping”) which is the stronger the higher the positive charge of the defect. Let us briefly discuss here also an important heterostructure, TiO2/LiF, as there the ionic space charge effect could be predicted. TiO2 is known to tolerate a perceptible Li content. This follows from its function as intercalation anode, which implies that Li+ can be accommodated. In LiF, Li vacancies are well known defects with comparably low energy. As Li+ is considerably mobile, one reckons with a significant Li+ transfer from LiF to TiO2. This would lead to increased ion conductivity in LiF (V0Li ).  The Lii injection in TiO2 leads owing to the space charge principle to an increased p-type conductivity. In fact both phenomena have been observed (Fig. 3.27) [325]. This is not only fundamentally important, it is also of relevance for the use of TiO2 in Li batteries and potentially in photoelectrochemical cells.

3.3.4. Wurtzite (ZnO) Zinc oxide has a band gap of 3.3 eV, and at ambient conditions the wurtzite structure is the stable polymorph. Its main uses are photochemical/photoelectrochemical applications and varistors. Since the latter is directly related to gb space charge effects, we concentrate on these aspects here. Ref. [326] gives a comprehensive overview of ZnO varistors. The bulk defect chemistry of undoped ZnO is complex (see e.g. [327–329] and references therein). While it clearly behaves as an ntype semiconductor and has Zn excess relative to O, the nature of the ionic defect compensating the excess electrons is difficult to determine since oxygen vacancies and zinc interstitials lead to the same pO2 dependence of the conductivity (it may also vary depending on actual conditions). The following trends emerge on the basis of recent ab initio calculations [329– 331]: oxygen vacancies have a lower formation energy than zinc interstitials, but are deep donors (i.e. present as neutral VxO when the Fermi level is in the upper third of the band gap) while Zni are shallow donors. (Note that the absolute defects energies may depend sensitively on the chosen functional, e.g. hybrid vs. LDA + U [332]). Defect-chemical modelling on the basis of ab initio defect formation energies indicates that while neutral VxO exhibit the highest concentration, the excess electrons are compensated by a combination of VO , VO , and OHO , HO resulting from unintentional H doping [333]. Regarding the diffusion of ionic defects, the situation is also complex. The bulk Zn tracer diffusion coefficient (diffusion via Zni ) increases from 10
14 cm2/s at 1100 K to 10
9 cm2/s at 1700 K and is more than 2 orders of magnitude higher than O tracer diffusivity (3  10
18 cm2/s at 1100 K and 10
11 cm2/s at 1700 K), see [334,335] and the experimental references cited there. The O tracer diffusivity is suggested to have contributions from VO migration but also from an O interstitialcy mechanism (higher formation energy but lower migration barrier for O interstitials compared to VO ) [335]. For undoped ZnO ceramics as well as bicrystals, typically no pronounced space charge depletion effects at the gb are found (see e.g. [336–338]), indicating that the gb cores do not carry a large excess negative charge. The structures of a large number of well-defined ZnO gb have been investigated in undoped and doped bicrystals, see e.g. [337–339]. It was found that the gb structures can typically be represented by combinations of a few characteristic ‘‘structural units” [338–340] as exemplified in Fig. 3.28. These units differ in the number of 3-fold (under-coordinated) and 5-fold (over-coordinated) Zn atoms. For gb in undoped ZnO, ab initio calculations found increased or decreased Zn and O vacancy formation energies

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Fig. 3.28. (a) HRTEM image of a R49 gb in ZnO. (b) Structure model obtained from pair potential calculations. Arrays and b are ‘‘dislocation-like” with 5and 3-fold coordinated Zn, while array c has a ‘‘bulk-like” structure. Reprinted with permission from Ref. [340]. Copyright 2005 by the American Physical Society.

in the gb core compared to bulk depending on the actual gb orientation and conditions (Zn-rich or O-rich) [341]. These findings help to rationalize that undoped ZnO gb apparently have no strong tendency to develop a negative excess core charge. For the functioning as a varistor, the gb should be highly blocking at low voltages, and become nonblocking above a certain threshold voltage (see [326] and references therein). Such a switching can be realized when the blocking behavior is caused by space charge depletion zones, which become nonblocking under sufficiently large DC bias. For n-type ZnO this requires an excess negative charge at the gb cores. In practical varistors, the space charge potential amounts to about 0.8 V, and the highly nonblocking state is achieved when a voltage of about 3 V per gb is applied. For ZnO it is found that dopants such as praseodymium and cobalt lead to the desired high nonlinearity of the I-V curve, i.e. sufficiently large space charge effects. Pr and Co are found to segregate to the gb core (see e.g. [338]). While Co2+ is isovalent to Zn2+, Pr3+ is formally even a donor dopant. Nevertheless they favor the formation of acceptor defects in the gb core such as O interstitials or Zn vacancies [342]. This may be related to a modification of the gb core structure [339], and/or the tendency of these cations to acquire a higher coordination number than 4 and thus to promote a local oxygen excess. Also electronic trap states leading to a negative core charge are discussed. Varistors are typically prepared by liquid phase sintering with addition of Bi2O3 to obtain ZnO ceramics with the desired grain size (see [326] and references therein). However, the remaining Bi2O3 phase in the gb may also be important as a fast oxygen diffusion path for re-oxidation and formation of acceptor defects in the gb core after the high temperature sintering as indicated in [343]. 3.3.5. Apatite-related structures (La9.33(SiO4)6O2) Lanthanum silicates La9.33(SiO4)6O2 with cation deficient apatite structure were found to exhibit a perceptible oxygen ion conductivity. A number of compositions Ln9.33+x(SiO4)6O2+1.5x was investigated [344–349], with the highest total conductivities in the range of 6  10
2 S/cm at 700 °C [350]. Modelling showed that the mobile defects are oxygen interstitials (see e.g. [351,352]) while strongly Sr-doped apatites exhibit oxygen vacancies with a lower mobility than the interstitials. Owing to the anisotropic crystal structure, a higher conductivity is found along the c-direction in experiments [353] as well as in simulations ([351,352], curved migration path). Several studies found a blocking behavior of the grain boundaries in ceramic samples for the oxygen ion conductivity [346–348,354]. Based on the observation of the absence of secondary phases along the grain boundaries in TEM and a thickness of the blocking layer of 90 nm (extracted from the grain boundary capacitance, clearly exceeding the thickness of the structurally distorted gb core) the blocking behavior was assigned to a space charge depletion zone [354]. The extracted space charge potential amounts to
0.2 to
0.25 V. While an increased La/Si ratio was found for the gb cores, the detailed origin of the negative core charge could not be resolved so far. 3.3.6. Spinel (Li4Ti5O12) The spinel Li4Ti5O12 has gained interest as anode material in rechargeable lithium batteries, see e.g. [355–361]. The anode reaction is Li4Ti5O12 + 3Li ¢ Li7Ti5O12. While for battery applications typically porous electrodes with small Li4Ti5O12 grains and an electronically conducting binder are used, a few investigations deal also with the gb transport properties of ceramic Li4Ti5O12 samples. By impedance spectroscopy the gb were found to exhibit a moderately blocking behavior in Ref. [362]. However, the origin of this blocking (space charge depletion or other reasons) was not clarified. On the other hand, an increased ionic and electronic conductivity along the grain boundaries was hypothesized to develop in the early stages of the Li intercalation process in [363]. 3.3.7. Garnet (Li7La3Zr2O12) Li7La3Zr2O12 with a garnet type structure was identified as lithium electrolyte in 2007 [364]. Impedance spectra of polycrystalline samples exhibit a semicircle related to blocking grain boundaries, but the blocking character is not very pronounced. High conductivities close to 10
3 S/cm at 25 °C were found for the cubic polymorph, which can be stabilized in particular by slight Al or Y doping, see e.g. [365,366]. TEM showed that the gb were free of secondary phases and Al segre-

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Table 4.1 Summary of the main features of grain boundaries in different oxides under oxidizing conditions (for details see Section 3). These include the majority defects present in the gb core, the majority mobile charge carriers in the material, the charge core, typical space charge potential values and the effect of the gb towards the majority charge carrier. Note that for TiO2 and ZnO the solubility limit of dopants is in the range of 1% or lower, i.e. much smaller than for the other materials in the table. Material

Majority defect

Conductivity determining carrier

Sign of core charge

gb blocking for perpendicular transport?

Acceptor doped SrTiO3 Donor doped SrTiO3

VO V00Sr

Holes, VO Electrons

+


Yes Yes

Acceptor doped BaTiO3 Donor doped BaTiO3

VO V00Ba

Holes, VO Electrons

+


Yes Yes

Acceptor doped BaZrO3 Acceptor doped LaGaO3

VO VO

Protons, VO , holes VO

+ +

Yes Yes

Acceptor doped CeO2 Donor doped CeO2 Acceptor doped ZrO2

VO O00i VO

VO Electrons VO

+
+

Yes Yes Yes

Undoped ZnO Donor doped ZnO with decorated gb (varistors)

VO ,Zni Electrons,V00Zn

Electrons Electrons

0


No Yes

Acceptor doped TiO2 Undoped TiO2, Y decorated

VO ,Ti i VO ,Ti i

Holes, VO ,Ti i Electrons, holes,VO ,Ti i

+


Donor doped TiO2

VTi ,O00i

Electrons




Yes Depends on n- or p-type bulk conductivity Yes

0000

gation [367]. Despite the moderately blocking behavior of the gb for Li ion transport across the boundaries, sol-gel prepared nanocrystalline Li7La3Zr2O12 ceramics were found to exhibit a higher total conductivity compared to ceramics from solid state reaction with micrometer grain size (increase by a factor of 3 at 60 °C [368]). Interestingly, fine-grained Li7La3Zr2O12 samples exhibited a significantly smaller transfer resistance to metallic Li than large-grained samples (prepared by the same solid state reaction, but skipping an additional grinding step). This was interpreted as indication for an enhanced Li ion conductivity along the gb [369]. 4. Comprehensive discussion 4.1. Materials overview: similarities and differences Table 4.1 summarizes the transport properties of grain boundaries for many of the examples discussed in the previous chapter, together with the key characteristics of their bulk defect chemistry. While the table covers oxides of different structure types, it is striking that for most of these materials the grain boundaries exhibit a blocking effect with respect to the majority mobile carriers, and – as discussed in detail in Section 3 – this is typically caused by space charge depletion zones. The first important point to note is that all the examples in Table 4.1 refer to oxides with a band gap larger than 3 eV. In contrast, electronically and ionically conducting perovskites such as (La,Sr,Ba)(Mn,Fe,Co)O3
d used as SOFC cathode materials typically do not show blocking gb behavior. The transition between these limiting cases is nicely illustrated by the SrTi1
xFexO3
d solid solution series. While grain boundaries hinder the hole and oxygen vacancy conductivity for x  0.05, the gb semicircle vanishes in the impedance spectra for x 0.07 [370]. Remarkably, this effect is stronger than the decrease of the space charge potential with increasing acceptor concentration in BaZrO3 or BaTiO3 (see Sections 3.1.2 and 3.1.4); note that there the gb are still blocking for 10% doping. Such comparison between different perovskites is suggestive of two synergetic effects occurring in Fe-doped SrTiO3: on the one hand, the reduction of the gb blocking character is owed to the presence of Fe0Ti acceptors, on the other, to the formation of (partially occupied) Fe states within the SrTiO3 band gap [371,372]. It is to be expected that for materials with small band gap, i.e. with facile formation of electronic defects, the segregation of ionic defects to the gb core with concomitant buildup of a core charge (Section 4.2) occurs similarly as in the materials with large band gap. However, the easy formation of electronic defects due to the small band gap offers the possibility of compensating such an ionic defect accumulation directly in the core, resulting in a drastic decrease of the excess core charge and thus of the space charge effects. Another intriguing point in Table 4.1 is that the gb core charge typically seems to have the same sign as the bulk ionic majority defect (cf. signs in second and fourth column). In particular for perovskites (SrTiO3, BaTiO3) and fluorites (CeO2), the gb core charge typically changes from positive for acceptor doping (majority defect VO ) to negative for donor doping (majority defect cation vacancies, O interstitials). While the atomistic origin for the positive core charge of acceptor doped perovskites is quite well-understood (VO segregation caused by spatial requirements in the gb core, Section 4.2), the formation of the negative core charge in donor-doped perovskites and CeO2 is less investigated so far. The strong tendency for VO segregation to the gb core (and additionally proton segregation in Ba(Zr,Ce)O3 based proton conductors) makes attempts to

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decrease the blocking gb behavior in such electrolyte materials challenging. The relations between excess gb core charge, space charge potential, bulk dopant concentration and magnitude of carrier depletion are discussed in detail in Sections 2.1, 3.1.1, 3.1.4 and 4.2. Finally, the data in Section 3 show that for oxides such as TiO2, ZnO the sign of gb core charge can be switched comparably easily (e.g. by ‘‘decoration” of the gb). While again the detailed atomistic understanding is limited, this might be related to the following points: (i) TiO2 and ZnO have comparably small solubility limits for dopants, correspondingly small bulk ionic defect concentrations, and thus for the same driving force l0dop;bulk
l0dop;core a smaller core charge results (cf. Section 4.2); (ii) TiO2 and ZnO exhibit a complex bulk defect chemistry with perceptible contributions from more than one ionic defect (VO  and Zni for ZnO, VO and Tii for acceptor-doped TiO2). 4.2. Atomistic understanding of defect segregation to grain boundary cores The concentration of (charged) defects in the gb core is governed by the condition of equal electrochemical potential throughout the system

l~ j;core ¼ zj F/core þ l0j;core þ RT ln½defj core ¼ l0j;bulk þ RT ln½defj bulk

ð4:1Þ

which generally will lead to an accumulation or depletion in the gb core relative to the bulk (remember the convention /bulk = 0). This condition holds under conditions where the respective defect is sufficiently mobile (for species with low mobility such as dopants this may apply only to temperatures close to the sintering conditions). The gb core charge then results self-consistently from the distribution of all charged defects (where the resulting space charge potential limits further charge accumulation). In Eq. (4.1) one can recognize two driving forces for segregation of defects to the gb core: (i) an energetic driving force resulting from l0j;core being lower than l0j;bulk , and (ii) an entropic driving force corresponding to different RT ln½defj core and RT ln½defj bulk terms. When the doping level in the bulk is much larger than in the structurally different core then the compensating defect concentration in the bulk is forced to be much higher than it is in the electroneutrally thought core (in the hypothetical initial situation before equilibrating the defect concentrations according to Eq. (4.1)). Hence on forming the grain-grain contact, the mobile defects will segregate from the bulk to the boundary even if their local energy there is not lower than in the bulk (‘‘entropic driving force”). Note that such tendency would directly explain the finding that in the vast majority of cases the mobile majority defect is depleted in the space charge zone. In other words, the higher bulk carrier concentration forced by electroneutrality as a consequence of the frozen dopant concentration constitutes a driving force RT ln½defj core
RT ln½defj bulk to increase the concentration of this carrier in the gb core which is analogous to an osmotic pressure (e.g. for ½defj bulk ¼ 20%, ½defj core ¼ 0:1% this driving force amounts to 0.44 eV at 1000 K). Although it might not be the most likely case, this possibility should not be overlooked. Nonetheless the following considerations suggest that energetic effects are predominant at least in various cases. A specific case for energetic differences favoring core charge formation is the situation that a dopant has a very low bulk solubility limit, but a much higher solubility in the gb core with a ‘‘more relaxed” structure (corresponding to a strongly negative l0dop;bulk
l0dop;core ). Thus, at sintering conditions the dopants will segregate to the gb core (in particular if the nominal overall dopant concentration already exceeds the bulk solubility) and constitute the excess charge. This closely resembles the

Fig. 4.1. TEM image (bottom) and oxygen concentration (top) in the core of a R3{3 1 1} twin boundary of BaTiO3. Red = Ti, blue = O, green = Ba. From Ref. [373]. Reprinted with permission from AAAS.

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2nm

(a)

[001]

(010)

(b)

Fig. 4.2. (a) 0.05 mol% Fe-doped SrTiO3 bicrystal with symmetrical tilt grain boundary. Reprinted from Ref. [92] with permission of John Wiley & Sons. (b) Structure model, dislocation cores with partially occupied oxygen columns indicated by arrows (large solid circles: Sr, small solid circles: Ti, open circles: O). Reprinted from Ref. [94], copyright 2005, with permission from Elsevier.

situation of TiO2 ceramics in which the grains are decorated with strongly oversized Y0Ti acceptors [320] creating a negative core charge as discussed in Section 3.3.3. In particular for perovskites and fluorites that exhibit high bulk solubilities for acceptor dopants, one can expect also a comparable acceptor solubility in the gb core (see e.g. Fig. 3.13b). Correspondingly, for these cases the energetic driving force l0j;core
l0j;bulk is expected to be the main driving force for buildup of the core charge. For acceptor doped perovskites and flu-

orites, the energetic differences for VO should dominate such that possible segregation of the dopants is likely rather a consequence of the space charge potential than its origin. This VO segregation creating the core charge is discussed now in more detail for acceptor doped perovskites where the situation is best understood on the basis of experimental data as well as calculations. However, one has to keep in mind that the large number of possible gb configurations allows only exemplary situations to be studied, and generalizations have to be regarded with caution. Fig. 4.1 shows a TEM image of a R3{3 1 1} twin boundary of BaTiO3 together with the oxygen concentration in the gb core [373]. On average, only 68% of the O site in the gb core are occupied, which corresponds to a strong accumulation of oxygen vacancies. A similar situation is found for the array of edge dislocations forming the grain boundary in SrTiO3 tilt bicrystals (Fig. 4.2). While these observations emphasize the role of VO segregation to the gb core, also other structural features such as an increased Ti/Sr and Ti/O ratio in the core have been observed [374,375], which may coexist with VO accumulation or contribute to a positive core charge on its own. The energetics of ionic defect segregation to the gb core was furthermore investigated by semi-empirical (pair potential) as well as ab initio (density functional) calculations. The large number of possible configurations (orientation of the two crystal slabs and of the boundary plane, lateral displacement, different terminations, or other variation from bulk stoichiometry) represents a particular challenge [377]. Thus, the calculations rather give exemplary insight than a complete quantitative picture. Nevertheless these results give important information on the origin of the core charge, and largely agree with TEM results. Fig. 4.3 shows the segregation energies of different vacancies at gb cores in SrTiO3 [376]. Each of the configurations shown displays some sites of preferred VO segregation close to the gb core, possibly also some cation vacancy segregation sites. The calculations indicate that defect segregation is closely related to structural distortion occurring in the core region where differently oriented crystal structures merge. This leads e.g. to situations where two oxygen columns (or an O

Fig. 4.3. Energetics of vacancy segregation at differently oriented tilt grain boundaries in SrTiO3, calculated by DFT (red indicates preferred segregation sites). Reprinted with permission from Ref. [376]. Copyright 2011 by the American Physical Society.

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0.5 ΔE segr / eV

0.0 -0.5

. OHO

-1.0 -1.5 0.5

ΔEsegr / eV

0.0 -0.5 -1.0

.. VO

-1.5 0

(a)

1 2 3 layer from gb core

(b)

4

 1 0] tilt gb in BaZrO3 (Ba green, Zr blue, O red). Reprinted from Ref. [169] with permission of John Wiley & Sons. (b) Fig. 4.4. (a) Structure of (1 1 2)[1  1 0] tilt gb (solid symbols) and (2 1 0)[0 0 1] tilt gb (open symbols) in BaZrO3, data taken Segregation energies of VO and OHO towards the core of (1 1 2)[1 from Ref. [169].

and a cation column) come closer to each other than their optimum bulk distance, corresponding to an ‘‘overcrowded” situation (note that in perovskites, the oxide ions and A site cations are of similar size and together are arranged on the pattern of a cubic closest packing). It can be relaxed by leaving some ion sites partially empty, preferably of the large oxide ions and/ or A site cations, constituting the segregation driving force. For the bicrystal shown in Fig. 4.4, the excess positive charge density of the gb core corresponds to approximately 0.5 VO per unit cell height in the edge dislocation cores. The segregation energies of VO and A site vacancies V00A are often in a comparable range. Thus, one can rationalize that for acceptor doped SrTiO3 with a high bulk [VO ], VO segregation to the core dominates leading to a positive excess charge, while 0000

ρgb / ρbulk

10 5 10 4 10 3 10 2 10 1 at 300 °C, 20 mbar H 2O

10 0 0.00

(a)

0.05

0.10

0.15

Y concentration x

0.20

0.25

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0 0.00

(b)

0.05

0.10

0.15

0.20

gb core charge / Cm

BaZr1-xYxO3-x/2

10 6

space charge potential / V

10 7

-2

for donor-doped SrTiO3 with high bulk [V00Sr ] rather a negative core charge forms (VTi typically has the largest segregation energies, but low bulk concentrations even for donor doping). The pronounced VO segregation to the gb core for acceptor doped SrTiO3 was also confirmed by molecular dynamic calculations (based on pair potentials) [59]. The effect of different terminations corresponding to different Sr/Ti ratios in the gb core was explored computationally in [391]; while this modifies the cation vacancy segregation energies, always a strong VO segregation tendency is found. Similar to acceptor doped SrTiO3 and BaTiO3, proton conducting acceptor doped Ba(Zr,Ce)O3 perovskites exhibit positive gb cores and the corresponding proton depletion zone leads to blocking grain boundaries. Pair potential [378] as well as ab initio calculations [168,169,379–382] indicate that VO segregate to the gb core. In addition, also protons show a favorable segregation energy as demonstrated in Fig. 4.4 [169,378,380]. This might be one reason why the space charge potentials in BaZrO3 tend to exceed that of SrTiO3 with comparable acceptor concentration. The driving force for proton segregation is probably related to the fact that a protonic defect OHO is actually smaller than a regular oxide ion [383]. From the defect segregation energies, gb core charges and space charge potentials were estimated to be in the range of 0.5–0.7 V (10% acceptor concentration, humid conditions, 600 K) which is comparable to experimental results (Fig. 4.5a). The gb core charge densities extracted from the blocking behavior (Fig. 4.5a) correspond to 0.2–0.3 VO or 0.4–0.6 OHO per elementary cell in the gb core layer. It is interesting to note that the experimentally determined gb core charge varies by max. a factor of two when the acceptor concentration is increased from 0.02 to 0.2, while the decrease of the space charge potential changes the resistivity ratio (Eq. (2.21)) by more than a factor of 104 at 300 °C (Fig. 4.5b). Fig. 4.6 shows that the distorted structure at the gb core

0.0 0.25

Y concentration x

Fig. 4.5. Variation of (a) resistivity ratio (grain boundary vs. bulk), (b) space charge potential D/0 and gb core charge Qcore with the dopant concentration in proton conducting BaZr1
xYxO3. Dotted lines are guide for the eye only. Data taken from Refs. [165] triangle up, [163] triangle down, [162] diamond, [155,176] star; D/0 and Qcore calculated from Eqs. (2.23) and (2.12) within the Mott-Schottky model.

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Fig. 4.6. Calculated proton migration barriers at a R5(3 1 0)/[0 0 1] grain boundary in Y-doped BaZrO3 (Ba blue, Y yellow, Zr black, O red). The bulk value of proton transfer and reorientation barriers are 0.35 and 0.26 eV. Reprinted from Ref. [379], copyright 2012, with permission from Elsevier.

furthermore leads to increased proton migration barriers (which, however, can explain the blocking behavior only to a small degree; most of it originates from proton depletion in the space charge zones) [379]. Again, as mentioned also in Sections 2.6 and 4.3.1, in contrast to metals in many oxides the ion mobility in the structurally distorted gb core tends to be lower than in the bulk (some exceptions are found for materials with extremely low bulk mobility). Further interesting aspects emerge from comparing the blocking behavior in acceptor doped BaZrO3 and BaCeO3. Although a larger data basis of systematic investigations would be desirable, literature results suggest that for comparable acceptor concentration the space charge potentials in Y doped BaZrO3 (cf. Fig. 4.5a, D/0  0.5 V for 10% doping) tend to be higher than for Y or Gd doped BaCeO3 materials (D/0  0.35 V for 10% Y or Gd [384], D/0  0.3 V for 15% Gd [32]). Two effects might contribute to the higher space charge potentials: (i) owing to the smaller band gap of BaCeO3 (4 eV [385]) compared to BaZrO3 (5.3 eV [386]), the electron concentration will be higher, such that electrons accumulated in the core and space charge zone might contribute to the compensation of the positive core charge (cf. inversion situation, Fig. 2.3). (ii) BaCeO3 exhibits a much softer lattice (Young’s modulus of 154 GPa compared to 243 GPa for BaZrO3 [387]). This means that steric conflicts arising from the merging of the two crystallite lattices in the gb core can be resolved at a lower cost in terms of elastic energy, and that thus also the defect segregation energies and the resulting core charge are smaller. This interpretation is supported by lower VO and proton segregation energies from DFT calculations (40–50% lower for protons and 50– 70% lower for VO in BaCeO3) for structurally similar R3 grain boundaries in BaZrO3 and BaCeO3 [382,383]. Arguments in terms of elastic energy and materials elasticity (Young’s modulus) have been applied in order to estimate defect formation or migration enthalpies, see e.g. [388,389]. For donor doped perovskites, a negative gb core charge is expected. Calculations show that while VO often have the most negative core segregation energy, also cation vacancies on the A- and/or B-site exhibit a driving force to segregate into the core, see e.g. [376,390,391]. Since the bulk VO concentration in donor-doped perovskites is many orders of magnitude below that of cation vacancies, the cation vacancy segregation dominates and leads to an overall negative gb core.

Fig. 4.7. Concentration of VO and acceptor dopants for R5(3 1 0)/[0 0 1] gb (the position of the gb plane is indicated by A) from initial molecular dynamics (mainly equilibration of [VO ]) and hybrid Monte-Carlo/MD (full equilibration of all defects) simulations. (a) Zr0.8Y0.2O1.9, (b) Ce0.8Gd0.2O1.9. Reprinted from Ref. [398], copyright 2013, with permission from Elsevier.

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Also for acceptor doped fluorite-structured oxides the gb core structure has been investigated. For a YSZ bicrystal, STEMEELS with 0.5 nm probe beam size indicates a decreased O/(Zr + Y) ratio (i.e. increased VO concentration) in the gb core [392]. An increase of VO and Y0Zr concentrations in the gb core region was also found in [393]; however, the claimed negative excess core charge caused by electron trapping might require further validation from additional experiments. Very detailed information is obtained from the calculation of structures and energetics of gb cores. Static lattice calculations (e.g. [394,395]) indicate that VO , Y0Zr and in particular VO -Y0Zr associates segregate to the gb region. Effects of varying the gb orientation and the dopant element were investigated e.g. in [396,397]. Fig. 4.7 shows the acceptor dopant and VO concentrations profiles as obtained from molecular dynamics and hybrid Monte-Carlo/MD simulations for R5(3 1 0)/[0 0 1] gb in YSZ and GDC [398]. For the simulations an initial ‘‘modified initial gb structure” was derived that is more stable than the gb core obtained from simply connecting the slabs, but may nevertheless not represent the actual minimum of the grand canonical function. The temperature was chosen higher than typical sintering conditions. Together with potential inaccuracies of the pair potentials, this means the results have to be taken cum grano salis. The obtained dopant segregation trends are qualitatively in line with experimental results listed in [398]. From the comparison of YSZ and GDC (the latter having the smaller size mismatch of host cation and dopant but nevertheless a stronger dopant segregation) and the time evolution (fast VO segregation to the gb core followed by slower acceptor redistribution) it was concluded that the segregation is initiated by VO accumulating at the gb core, while the acceptor segregation mainly results from acceptor-VO interaction rather than from acceptor size mismatch, and cannot fully compensate the charge of the segregated VO . Since in electrolytes the bulk acceptor concentration is typically chosen as to match the maximum ion conductivity, a further acceptor accumulation at the gb is expected to decrease the conductivity. Thus, for YSZ and CGO the blocking character of the gb arises from a combination of VO space charge depletion as well as VO mobility decrease. This section can be summarized as follows: the structural distortion created by the grain boundary leads to preferential segregation of defects (often vacancies rather than interstitials, occasionally also dopants which exhibit a size or coordination mismatch in the host lattice) to the gb core. Since in ionic materials most defects are charged, this creates an excess charge of the gb core. As a consequence, an adjacent space charge zone is formed (which, depending on the material’s defect chemistry comprises depletion and/or accumulation of carriers) lowering the Gibbs energy of the system. Under sintering conditions also dopants can (partially) follow the driving force from the core charge and develop concentration profiles, which are then frozen-in under the typically lower measurement or application conditions. In particular for acceptor doped large-bandgap perovskites and fluorites, the accumulation of acceptor dopants in the space charge zone and in the gb core is a reaction to (partially) compensate the positive core charge created by the initial segregation of VO (and possibly protons) to the gb core. This process decreases the space charge potential, it is not the reason for the blocking character of the gb. 4.3. Effects on ionic transport 4.3.1. Blocking of ionic transport As discussed in the previous sections, in ionic solids the blocking effects of grain boundaries is often caused by carrier depletion in space charge zones. For a temperature-independent space charge potential, this effect is the stronger the lower the temperature (exponential contribution in cj ðxÞ ¼ cj;bulk e
zj F/ðxÞ=RT ) and the lower the carrier concentration (resulting in the Mott-Schottky model in a much larger depletion width). At 500 K, a space charge potential of 0.5 V decreases the concentration of singly positively charged carriers directly adjacent to the gb core by 5 orders of magnitude, and of doubly charged defects by 10 orders, very likely representing the predominant contribution to the overall blocking behavior of a grain boundary. Compared to this, a possible mobility decrease in the structurally distorted gb core appears less relevant. A different situation is met when low-conductive secondary phases form a continuous phase of perceptible thickness along the grain boundaries. 4.3.2. Fast O diffusion along grain boundaries While accelerated oxygen diffusion in nanocrystalline materials would be highly appealing for numerous electrochemical applications, the examples where a beneficial effect of grain boundaries on oxygen transport is unambiguously proven is rather limited (different to diffusion in metals, where space charge effects are absent and bulk diffusion is very sluggish). Typically, the examples come from materials which exhibit a very low bulk oxygen diffusivity. Keeping the discussion about space charge zones and their origin in mind, the scarcity of examples for fast transport along gb is not surprising. For ceria and zirconia, the analysis in Sections 3.2.1, 3.2.2 and 4.2 shows that the gb cores are accompanied by VO depletion zones. While VO are accumulated in the gb core, this does not guarantee that they are highly mobile along the gb core. Rather, the large segregation energy may lead to strongly bound VO in the gb core with high migration barrier along the core and a large energy required to escape from the core (see also discussion in Section 2.6). Indeed, 18O/SIMS measurements on nanocrystalline YSZ demonstrate no accelerated oxygen diffusion [233]. The enhanced total conductivity of nanocrystalline undoped and Gd doped CeO2 was shown to be electronic (electron accumulation in the space charge zones) [23]. Accelerated oxygen diffusion was found in ceramic La0.8Sr0.2MnO3+d (LSM) samples by 18O/SIMS (extended tails for large diffusion lengths with characteristic x6/5 scaling) [399], in which the extremely low bulk concentration (the material typically has oxygen excess which is accommodated by cation vacancies) leads to very low bulk D⁄ values. Relative to these extremely low values, grain boundaries may actually provide faster diffusion paths when the drastically increased VO con-

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Fig. 4.8. 18O concentration profile (cumulated 18O in a slice ranging from 0.9 to 1.2 lm depth, and depth profile. Red = higher concentration) of nominally donor-doped Pb(Zr,Ti)O3 ceramics after exchange for 4 h at 600 °C. Reprinted from Ref. [403] with permission of John Wiley & Sons.

centration outweighs possibly decreased VO mobility (cf. decreased mobility at dislocations in SrTiO3, Fig. 2.10). Fast O transport along gb was also evidenced for pore-free LSM films with varying lateral grain size by microelectrode impedance spectroscopy as well as SIMS [400–402]. Other examples of fast oxygen transport along grain boundaries come from donor doped perovskites (e.g. BaTiO3 and Pb (Zr,Ti)O3 PZT), which also exhibit extremely low bulk [VO ] owing to the donor doping [403–405]. The fast O transport along gb is illustrated in Fig. 4.8 for the example of a large-grained nominally donor doped ceramic PZT sample. It is obvious that the increased 18O concentration at the gb region penetrates deeply into the sample. For these materials, the authors concluded that the fast O diffusion mainly originates from VO accumulation space charge zones adjacent to negatively charged gb cores (cf. donor doped perovskites in Sections 3.1.1, 3.1.2 and 4.2), rather than concentration or mobility effects in the core. Accelerated O diffusion along gb was also found in undoped SrZrO3 ceramics [406], however without further discussion of its origin. 4.3.3. Accelerated cation diffusion along grain boundaries For a number of oxides with fluorite and perovskite (-related) structure, cation diffusion was found to be accelerated along grain boundaries (bulk diffusion is very slow in these materials, much slower than O diffusion). Both structures show cation diffusion via involvement of oxygen vacancies, although the mechanisms might be more complex than simple vacancy migration, see e.g. [407–409]. Strongly accelerated gb cation diffusion was observed in oxides with large band gap such as Y-doped ZrO2 as well as in La0.9Sr0.1Ga0.9Mg0.1O2.9 [410]. While for such oxides the presence of a positive gb core charge is well established (see Section 4.2) and therefore an accumulation of cation vacancies in the gb core would be expected, given the probably complex cation diffusion mechanism also other effects might play a role. Fast gb cation diffusion is found also in perovskites with mixed-valent transition metals for which no large gb core charges are expected, pointing towards modified defect formation energy and migration barriers in the gb core as the key factor. Examples are La1
xSrxFeO3 [411], LaMnO3 [412], Ba0.5Sr0.5Co0.8Fe0.2O3 [413], La0.6Sr0.4CoO3 [414] and La2NiO4 [415]. While in these materials the gb cation diffusivities are still orders of magnitude below oxygen diffusivity, the accelerated cation transport along the gb suffices to allow for changes in cation stoichiometry close to the surface (diffusion length few nm) already at moderate temperatures and annealing times [414]. 4.3.4. Mobility and strain effects (thin films) For epitaxial thin films, the lattice mismatch with the substrate and/or a second phase, if this is used for obtaining heterostructures, creates biaxial strains in the films, which may be partly released by the formation of higherdimensional defects (dislocations). The planar strain field is naturally followed by an opposite relaxation in the direction perpendicular to the film, possibly leading to anisotropic transport properties. Strain is expected to affect defect concentrations as well as mobilities, as summarized in recent reviews [416–420] (the effects at LaAlO3/SrTiO3 interfaces [77–79] are mainly based on electron redistribution and thus not covered here). Increasing ionic conductivity by strain effects on mobility appears particularly appealing because, for already highly doped electrolytes, the possibilities to increase carrier concentrations by space charge accumulation zones are naturally quite limited. From the experimental point of view, it is important to establish the thickness for which films are indeed coherently strained (the critical thickness can be smaller than 10 nm while complete relaxation by formation of extended defects may require 100 nm, see e.g. [421]). In particular for heterostructures, potential cation intermixing between adjacent layers during preparation has to be checked (e.g. during deposition by PLD, sputtering, or MBE, the growing layer may have much faster cation exchange with the layer beneath than expected from bulk diffusivities [422]). Further challenges are related to the possibility of separating the film conductivity from the substrate contribution (cf. also sample holder artefacts

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Fig. 4.9. Summary of literature conductivity data of thin YSZ films. (a) temperature dependence, (b) thickness dependence. 1 = [427], 3 = [429], solid line = bulk YSZ. Please see [419] for the full assignment of the labels to the original references. Reprinted from Ref. [419] with permission from Springer.

[423,424]), identifying its nature (ionic or electronic, possible substrate contributions), and distinguishing between effects from the bottom interface or the top surface and strain effects within the film. In pair potential or ab initio calculations, the strained structure should be tested for instability as indicated by the appearance of imaginary phonon frequencies (see e.g. [425]). The first prominent example of increased conductivity are thin YSZ films on MgO [426,427] among which epitaxial PLD films of 15 nm thickness show an increase in the pO2-independent conductivity by one order of magnitude at 700 °C and 2.5 orders at 400 °C compared to single-crystalline YSZ and YSZ-films thicker than 60 nm. However, a consistent atomistic explanation of the increase is hard to obtain (the 22% larger lattice constant of YSZ than MgO should rather lead to decreased rion by compressive strain). Increased conductivity for thin YSZ films on MgO was also found in two other studies: increase by one order of magnitude at 700 °C for 17 nm thick films prepared by evaporation [428]; increase by 3 orders at 400 °C for 58 nm thick films deposited by reactive sputtering, which exhibit a dislocation network at the interface accommodating most of the lattice mismatch [429]. On the other hand, epitaxial 12–50 nm thick YSZ films deposited on MgO by reactive sputtering showed conductivities below the single crystal values at 400–700 °C [430]. Fig. 4.9 summarizes some results of YSZ films on MgO. Differences in sample preparation may be important, microstructural investigations are not always sufficiently detailed with the consequence that the overall situation remains controversial. For RF-sputtered heterostructures comprising a 1 nm thick epitaxial YSZ film and a 10 nm SrTiO3 cover layer on SrTiO3, an even larger ionic conductivity enhancement by up to 8 orders of magnitude was claimed (relative to bulk YSZ, with rion being assigned exclusively to the 1 nm YSZ layer) [431]. A number of subsequent publications critically addressed some aspects of this study: the contribution of ionic and electronic conductivity by SrTiO3 depending on the temperature range [432,433], the electronic nature of the interfacial conductivity in PLD-prepared samples as evidenced by the pO2 dependence [434], physical limits of carrier mobility increase [435–438]. Computational studies further showed that the nominal lattice mismatch of 7% for ZrO2/SrTiO3 cannot be accommodated within the fluorite structure by elastic strain (development of columbite structure at low T, oxygen disorder at high T) [438–440]. A third set of experiments deals with multilayer heterostructures of YSZ with ceria or other fluorite-related materials, which is expected to lead to strained layers. Interestingly, in all these experiments the effect of strain on the ionic transport is rather limited. Stacks of 15 nm thick Gd-doped zirconia and ceria layers on sapphire showed an increase in rion by about one order of magnitude at 400 °C relative to bulk YSZ (however, the change relative to Ce1
xGdxO2
x/2 was not specified) [441]. Superlattices of YSZ and rare-earth oxides (quasi-coherent, but with lateral grain size in the range of 100 nm) yielded a decreased conductivity by a factor of 1.6 for
4% compressive mismatch and increased rion by 1.3 for tensile mismatch (560 °C, 15–20 nm individual layer thickness, about half of the nominal mismatch accommodated by extended defects) [435,442,443]. A stronger decrease of rion by about one order of magnitude at 365 °C was found for 5 nm GDC layers compressively strained in a superlattice with Er2O3 [444]. Epitaxial SDC/YSZ superlattices with 7.7 nm layer thickness and a nominal lattice mismatch of 6% yielded an increase of rion by about 1/2 to one order of magnitude relative to SDC and YSZ (400– 800 °C) [445]. On the other hand, superlattices of CeO2 and YSZ with layer thicknesses down to 5 nm showed no increased rion and 18O diffusivity [446]. Although dislocations were present, it was deduced from HRTEM that the YSZ layers were still under tensile strain. Again one can summarize that the magnitude of strain effects sensitively depends on the actual samples considered, and that it seems not to exceed one order of magnitude for 5 nm layer thickness. Recently, enhanced oxygen ion transport by 2 orders of magnitude compared to single crystals was claimed in YSZ-SDC nanopillars embedded in a SrTiO3 matrix on the basis of conductivity measurements (but without checking pO2 dependence) and electrochemical strain microscopy. Despite such an impressive improvement, no perceptible variation of the corre-

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Fig. 4.10. Enhancement of oxygen diffusivity in Y-doped ZrO2 under biaxial strain calculated from DFT migration barriers and Kinetic Monte Carlo simulation. Reproduced from Ref. [437] with permission of The Royal Society of Chemistry.

sponding ionic migration enthalpy compared to the bulk properties was observed. For this reasons, the authors interpreted the experimental findings in terms of an improved pre-exponential factor due to ‘‘structurally misfit vertical interfaces” [447,448]. Strain effects on oxide ion mobility in YSZ were investigated by various computational approaches (interatomic potentials and ab initio, static calculations and molecular dynamics) [437,438,449–452]. Generally, a decrease of the migration barrier is found for tensile biaxial strain. One important result is that for too large a tensile strain the structure becomes unstable (lattice distortions) and/or the migration barrier increases again. Fig. 4.10 shows a maximum increase of the oxygen diffusivity by 3.8 orders of magnitude at 400 K calculated for 4% tensile biaxial strain [437] (the improvement by >6 orders predicted in [449] suffers from an inappropriate combination of experimental and calculated migration barriers, see also [436]). From molecular dynamics calculations, a maximum acceleration by two orders of magnitude at 400 K was found [452]. Analogous results were obtained for oxygen migration in strained acceptor-doped ceria [436,452–454]. Recently, in situ strain measurements confirmed the beneficial correlation between the occurrence of tensile strain and the reduction of the ionic migration barrier [455]. It is interesting to note that under tensile biaxial strain the migration barrier decreases also in the direction perpendicular to the substrate, although the lattice becomes contracted in this direction. Furthermore, tensile strain tends to decrease the interaction with acceptor dopants. Computational studies were also performed for oxygen diffusion in perovskites. When the perovskites contain redoxactive transition metals, the oxygen vacancy concentration is variable, and tensile strain decreases not only the migration barrier but also the vacancy formation energy. On the other hand, the system reacts to applied tensile stress by increasing its vacancy concentration (chemical expansion) and thus decreasing the actual strain. Pair potential calculations for LaGaO3 yielded a decrease of the in-plane migration barrier from 0.8 eV to 0.55 eV at 4% tensile biaxial strain [456]. DFT calculations for the series of La-first row transition metal perovskites gave on average a decrease of the migration barrier by 0.066 eV per percent of biaxial tensile strain, which corresponds to an increase in diffusivity by one order of magnitude at 500 °C [457]. 18 O exchange on La0.8Sr0.2CoO3
d films on SrTiO3 (tensile strain) and LaAlO3 (compressive strain) showed an increase of D⁄ by about 1 order of magnitude at 300–400 °C for the tensile strain film compared to the compressed film [458]. So far it seems that effects due to elastic deformation are present but much less pronounced than expected. On the other hand, huge changes are possible by plastic deformation. As discussed in Section 3.3.3 for the example of TiO2 [324], the respective creation of dislocations with charged cores can lead to changes in carrier concentrations in the adjacent space charge zones by orders of magnitude. This leads to pronounced conductivity changes (together with potential strain effects on the mobility). 4.4. Approaches for grain boundary engineering The issue of decreasing the overall effect of blocking grain boundaries can be tackled by (i) increasing the grain size, or (ii) increasing the specific gb conductivity. Since methods of improving sintering and grain growth are covered in other literature (see e.g. [459,460]), we focus on exemplary approaches related to (ii). We also refrain from a detailed discussion of scavenging insulating SiO2 phases from the gb (see e.g. [461]), and approaches using core-shell particles and deliberately introducing secondary phases/heterocontacts. A number of attempts strive to ameliorate the gb conductivity by additives to decrease the positive gb core charge. The additives are either included already in the powder preparation process (e.g. in [462]), or the powder grain are impregnated with them prior to sintering attempting to locally modify the gb chemistry (e.g. [463,464,468] discussed below). Depending

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Fig. 4.11. (a) SEM image of BaZr0.9Y0.1O2.95 prepared by solid state reactive sintering (SSRS) at 1500 °C, (b) bulk and specific gb conductivity of BaZr0.9Y0.1O2.95 prepared by SSRS and by spark plasma sintering (SPS, 1700 °C) in humidified 5% H2/N2. Adapted from Ref. [179] with permission of The Royal Society of Chemistry.

on cation mobility under sintering conditions, solubility of the additives and driving forces for gb segregation (e.g. size mismatch) the additives will distribute between the grain interior and the gb region. Small amounts of transition metals (0.5–1%), in particular Fe and Co, were found to decrease the space charge potential of Ce0.99Gd0.01O1.995 from 0.55 V to 0.47 V, which corresponds to an increase of the gb conductivity by about one order of magnitude at 300–400 °C [462]. TEM/EELS showed that the transition metal cations accumulated in the gb region. A decrease of /0 and correspondingly higher specific gb conductivity was also found for cobalt additions into Ce1
xGdxO2
x/2 with higher Gd contents of 0.05  x  0.3 [463]. In a subsequent study, trivalent cations such as Yb, Y and Bi were added to the gb of undoped nanocrystalline ceria [464]. Interestingly, while the conductivity of the untreated nanocrystalline ceria was – as expected – clearly electronic under oxidizing conditions, the samples with ‘decorated’ grain boundaries exhibited ionic transport. Further analysis revealed, that during sintering the trivalent cations diffused towards the center of the grain creating a core-shell situation, in which the few nanometers thick and mesoscopic shell (see also Ref. [52]) was heavily doped and dominated the overall electric transport properties. Rather surprisingly, the decoration of gb of nanocrystalline ceria with boron lead instead to an increase of the n-type conductivity by a factor 10 due to a substantial increase of the space charge potential (about 0.17 V increase compared to the undecorated case). Complemented by TEM and EELS analyses, the experimental data pointed towards boron to segregate in the gb core as interstitial defects [465]. While in these approaches the additives were added already in the ceria powder preparation, in [466] Ni was diffused along the gb of nominally undoped CeO2 PLD films on MgO with columnar structure. The n-type electronic conductivity (as indicated by its pO2 dependence) measured parallel to the substrate decreased by Ni diffusion by half an order of magnitude, indicating that the gb core charge became less positive. Mg diffusion from a MgO substrate along the gb of nanocrystalline YSZ films seems to have a similar effect of increasing the gb conductivity [467]. For SrTiO3 nanoceramics with Rb decoration of the gb, impedance spectroscopy indicates that especially the gb core is affected by the presence of the oversized Rb cation. For small Rb concentrations a diminished space charge potential is found, 0  consistent with RbSr in the gb core, whereas at higher Rb addition Rbi seems to be the dominant species rather than the substitutional dopant [468]. For proton conducting BaZrO3 and BaCeO3-based electrolytes, a concept turned out to be successful which combines phase formation and sintering into a single step. A good total conductivity at significantly decreased sintering temperature was achieved by the so-called ‘‘reactive sintering” approach. The binary starting oxides/carbonates and a sintering aid (1% NiO proved most suitable) are ball-milled together, and this powder is directly sintered [179,469,470]. Transient formation of a liquid BaY2NiO5 phase lowers the sintering temperature and increases grain growth (Fig. 4.11a). Furthermore, the fact that the perovskite phase and the densification occur simultaneously may help to attenuate the ‘‘structural conflict” arising in the gb core (cf. Section 4.2) and thus diminish the segregation of VO to the core. BaZr0.9Y0.1O2.95 ceramics prepared by this method exhibit a perceptible decrease in the space charge potential of 0.27 V compared to 0.4 V for samples compacted by spark plasma sintering (SPS, Fig. 4.11b). However, it remains to be tested if this approach is successful also for other materials families.

5. Summary and conclusions Grain boundaries are extended defects, and their concentration is kinetically frozen-in from high temperature preparation conditions (similar to the concentration of dislocations). In that respect they set the boundary condition for constrained

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point defect equilibrium. As they represent a structurally distorted region, point defects (zero dimensional defects) experience a driving force to segregate into or away from this structurally different gb cores. In ionic solids this unavoidably creates a more or less pronounced excess charge of the gb core, which in turn leads to carrier redistribution in adjacent space charge zones. For numerous examples discussed in this review, these space charge effects constitute the main part of the transport behavior parallel and across grain boundaries. In the case of nanocrystalline ceramics with sufficiently small grain sizes, the space charge zones may overlap and undistorted bulk region vanish completely – then the material’s transport properties are exclusively determined by the interfacial effects. At hetero interfaces, the defect redistribution between the two different phases as well as with the structurally distorted core has to be considered. Atomistic insight into the driving forces for defect segregation into the gb cores is obtained from transmission electron microscopy as well as from pair potential and ab initio calculations. Strikingly, often vacancies of the largest ion (or of the two largest ions) in the lattice exhibit the largest segregation driving force, pointing towards sterical (geometrical) restrictions in the gb core. This motif helps to rationalize the typically positive gb core charges in acceptor-doped electrolytes with fluorite and perovskite structure (leading to blocking of the majority mobile carriers VO or protonic defects), and also helps to understand why attempts to decrease the core charge density (or even switch its sign) are so difficult. Nevertheless, also some strategies for grain boundary engineering can be derived from this, e.g. ‘‘decoration” with counter charges and deliberate introducing of more flexible (amorphous) layers in the boundary cores. Elastic strain effects are of influence on the transport behavior as well but have not shown to be of comparable significance as space charge effects. It is clear, though, that the realistic boundary is not ideally abrupt, rather structural modifications may penetrate into the grain. Yet, in many cases of interest the abrupt core-space charge model is a worthwhile approximation. This chapter of research to which our contribution is devoted, is not only exciting as it can explain a multitude of phenomena, but also as it gives directives of how to strategically improve properties by grain boundary engineering. Acknowledgements We thank B. Reichert and A. Fuchs for assistance with figures, copyrights, references, and W. Sigle for proofreading and helpful comments. Furthermore, we appreciate numerous valuable discussions with colleagues in the scientific community over many years. References [1] [2] [3] [4] [5] [6] [7] [8]

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