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Ionic transport in nano-sized systems
Solid State Ionics 175 (2004) 7 – 12 www.elsevier.com/locate/ssi
Ionic transport in nano-sized systems Joachim Maier* Max-Planck-Institut fu¨r Festkco¨rperforschung, Heisenbergstrage 1, 70569 Stuttgart, Germany
Abstract Ion transport in nano-sized systems is considered in a top-down approach. The equilibrium charge carrier concentrations are investigated as a function of the distance between neighboring interfaces and increased curvature of the grains. A variety of experimental examples involving accumulation, depletion and inversion layers are discussed, highlighting the significance of the nano-regime for Solid State Ionics. D 2004 Elsevier B.V. All rights reserved. Keywords: Nano-ionics; Ionic transport; Size effects
1. Introduction In recent years improved methods of preparing nanocrystalline or nanostructured materials as well as exciting findings of unusual electrical and electrochemical properties of such materials not only triggered the evolution of nanoelectronics and nano-electrochemistry, it also led to the appreciation of this size-regime for Solid State Ionics (bnano-ionicsQ) [1–14]. More detailed reports are given in Refs. [2,3,12–14]. Generally speaking, decrease of crystal size implies increase of the proportion of surfaces (interfaces) at the expense of bulk and usually implies an increased impact of the interfacial properties on the overall materials property (trivial size effects) (see, e.g., bricklayer model as described in Ref. [15]). If the spacing of the interface is very narrow, local properties may change as a function of distance and true size effects may occur. The three-dimensionality of the crystals is expressed by a mean curvature (1/r¯, where r¯ =R j r j a j /R j a j , distance of the surface plane j from the center, a j : its area) which increases if the sample size is reduced; crystallographically speaking we refer to edges large compared with a typical next neighbor distance. If we are not explicitly addressing edges and corners, their
* Tel.: +49 711 689 1720; fax: +49 711 689 1722. E-mail address: [email protected] (J. Maier). 0167-2738/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2004.09.051
influence then appears in the surface tension (c j ) and makes c j size dependent even if the interfacial structure remains unchanged. The increased mean curvature does not only lead to a size-dependent local chemical standard potential for the interfacial core (l s), it also leads to an increased internal pressure, which for a Wulff crystal amounts to p i =p+2c¯/r¯, where c¯ =R j c j a j /R j a j and p is the outer pressure [16]: lsk ðpi Þglsk ð pÞ þ
dlsk ðpi pÞglsk ð pÞ þ 2wvk: dp
ð1Þ
In Eq. (1) v k is the partial molar volume of the species k, w is the Wulff ratio c j /r j which is independent of j and then also equal to c¯/r¯ for the equilibrium morphology (the local excess energy of edges is neglected). The correction term in Eq. (1) easily accounts for the largely depressed melting point of tiny crystals [17] as well as for the e. m. f. values of electrochemical cells with nano-crystalline electrodes [18–22]. When we are addressing polycrystals, the morphology is not in equilibrium, only local equilibria may be reached. In the semi-quantitative approach taken below, we not only ignore anisotropies (i.e. c¯=c) whenever we deal with polycrystalline materials, we also use the bricklayer model constituted by identical cubic grains in a primitive arrangement. First of all we briefly consider charge carrier distribution in the bulk, by restricting to dilute conditions and treating
8
J. Maier / Solid State Ionics 175 (2004) 7–12
both ionic and electronic carriers by using benergy levelQ diagrams [23]. Then we proceed to charge carrier chemistry at planar interfaces and, finally, discuss effects due to curvature.
2. Bulk defect thermodynamics Let us consider the free energy of a system containing defects (excess particles such as interstitial ions or conduction electrons; or missing particles such as vacancies in the lattice or holes in the valence band). These particles introduced may possess a spectrum of energy levels, i.e. occupying different energy levels e. (If we speak of energy levels in the following, we more strictly mean local free energy levels since we include local entropy contributions (vibrational entropy).) If we cut out a differentially small (free) energy band in which we can assume a constant (free) energy eV for a single particle, the excess (free) energy therein upon introducing yNV non-interacting and randomly distributed defects is given by the binomial expression eVyN V kT lnð yZVÞ (yZV: number of available states in yN V that section) leading to a chemical potential of the form l ¼ lV þ RT ln
yN V=yZV 1 yN V=yZV
ð2Þ
with lV=N meV (N m: Avogadro’s number), i.e., to a Fermi Dirac type of expression due to the limited number of states/crystallographic positions for the electronic/ionic carriers. (Note that the problems of distributing NV defect over ZV sites of energy eV is statistically equivalent to distributing NV electrons over ZV quantum states of energy eV (see, e.g., Ref. [24]).) The bFermi-levelQ (generalized also to ionic carriers) corresponds to l/N m. The total particle number within the whole band of available states (with the upper and lower limits e H, e L) follows by integration as Z N ¼
eH F ðeVÞDðeVÞdeV
where l8 is identical to N me L supposed N¯ is identified with the integral according to Eq. (3) but the variable eV–e F in F being replaced by eV–e L. Eq. (4) holds generally for dilute randomly distributed carriers, be them electrons in the conduction band (D typically parabolic, N¯ =effective number of states in the conduction band), holes in the valence bands (D typically parabolic, N¯ =effective number of states in the valence band), interstitial ions in solids (D=delta function, N¯ =number of interstitial positions), vacancies in the lattice (D=delta function, N¯ =number of regular positions), or even excess or missing particles in aqueous solvents (H3O+, OH) (then N¯ may typically result from the integration of a Gaussian state density distribution corresponding to fluctuations in space and time; an analysis of different densities of states is given in Ref. [25]) If we deviate from dilute conditions, excess effects due to the restricting term in the denominator of Eq. (2) become important (exhaustibility of states/sites). This leads to an excess term in the chemical potential (l ex). In the case of single levels (D=delta function) we can just take Eq. (2) and write (l ex,c)=RTln(1N/Z). Note that this correction is explicitly (but strictly speaking falsely) introduced when considering chemical potentials of structure elements instead of building elements (since then the bchemical potentialQ of the regular species appears with the bBoltzmann-termQ RTln(1N/Z) in the balance because the number of regular and defective positions add up to Z). Whenever we deviate from dilute conditions we have to reckon with deviations from the ideal distribution and hence with interactions. The overall excess contribution (l ex) may nevertheless be split into configurational and non-configuration parts (l ex,c, l ex,o), only the latter being part of the benergy levelsQ. In addition, we have to add the electrical potential term zk F/ (which is important for interfaces; the mechanical term c¯/r¯ (see Eq. (1)) is contained in l8). Hence ex;c l˜ k ¼ ½lk8 þ ð2g¯=r¯ Þvk þ zk F þ ex;o k þ RT ln ckþ k :
ð5Þ
ð3Þ
eL
with D=dZ/ye and FuyN/yZ, the latter being given by Eq. (2). Without loss of generality let us consider excess particles. Assuming dilute conditions means that only the states of low energy are affected and the occupied levels are close to e L. In the case of excess electrons, e L is the lower edge of the conduction band and D is of a parabolic form, while in the case of interstitials D is a delta function referring to the single energy state e L with a high density of states (dilute conditions). Since for dilute situations |e Fe L|HkT and since the integral (Eq. (3)) only affects energy states e[e L, the deviation from unity in the denominator of Eq. (2) can be safely neglected; the integration then leads to the Boltzmann-form N ð4Þ l ¼ l8 þ RT ln ¯ N
The bracketed term corresponds to the (free) energy level used in the benergy levelQ diagrams. Structural inhomogeneities (e.g. elastic effects) are subsumed in l8 and/or l ex,o. If concentration measures (c) are used different from N/N¯ , the correction can be absorbed into l kex,c or, if dilute, also into the standard potential. Fig. 1 shows the benergy levelQ picture for two mixed conductors exhibiting (cation) Frenkel disorder in contact. The reader should ignore the middle part at the contact, which is to be considered below. (There the ionic Fermi-level analogue is termed bFrenkel-levelQ, + l˜ M uE FrenkeluN me Frenkel.) The diagrams for the ionic and the electronic excitations are coupled via lMþ þ le ¼ EFrenkel þ EFermi ¼ lM :
ð6Þ
(Note in this context that l M+=l i=l v and l e=l eV=l hS, S and i being short for interstitial building element (M i Vi)
J. Maier / Solid State Ionics 175 (2004) 7–12
Fig. 1. Three energy levels for the contact of two mixed conductors.
and v being short for the vacancy building element (VVMM M).) In Ref. [26] it is shown that these concepts can be used to construct a generalized acid–base concept of solids which is based on counting point defects to assess acidity/basicity whereby ionic associates play the role of internal acids and bases. But let us proceed to interfaces.
3. Interfacial defect chemistry Let us consider a flat interface and restrict to dilute interaction-free conditions. Fig. 1 refers to the equilibrium contact of two mixed ionic–electronic conductors if both cations and electrons are sufficiently mobile [27]. The constancy of the chemical potential of the component is not sufficient, rather the two Fermi-levels (ionic and electronic) have to be constant (which then implies a constant chemical potential of the neutral component) to satisfy the contact equilibrium. Let us consider the contact of MX with MXV, and assume substantial ionic defect concentrations (see, e.g., Fig. 2i). Let us also assume that the levels for metal vacancies and interstitial ions are such that there is a distinct transfer of cations in addition to the possible neutral M solubility. Naturally, the effect is restricted to the layers close to the interface in which deviations from electroneutrality can be tolerated. The changed defect concentrations correspond to bent levels owing to the zF/(x) term in Eq. (5) at constant bFermi-levelQ: yln ck ~ zk Fy/:
ð7Þ
(Cf. also Refs. [28–30]). At a homo-phase contact (grain boundary, see Fig. 2e) two chemically identical crystals meet. They are either insufficiently contacted (significant gas phase content) or sintered together with different orientations. The main space charge effect here is a charging via carrier segregation as a consequence of the different structure of the interfacial core.
9
This core layer may be approximately considered as a narrow phase with a given but different l8 value (l k8 contains a term which accounts for the excess surface energy and depends on ca k where a k is a partial molar area [16]). Of course in the previous situation (Fig. 2i) also a charge storage within the MX/MXV interface can occur. In all cases we may consider the situation as a composite of at least two structurally different regions, and a comparatively simple consideration is possible if we assume that l8 changes abruptly (see, e.g., the defect chemical calculation of the contact MX/MXV in Ref. [31]), which has been justified by atomistic calculations for some perovskites [32]. While in the bulk the electroneutrality condition gives a sufficient account of electrostatics, at the boundaries it has to be replaced by the more general Poisson’s equation R j z j c j aj2//e. The related concentration changes can be immense and seriously affect also the ionic conduction in solids [13,27,33]. While Fig. 1 gave a formal description in terms of energy level bending, Fig. 2 compiles a variety of examples from different areas of Solid State Ionics: Only a few of them shall be touched upon: At the contact of the ion conductor to an insulator (see Fig. 2a,b) there can be ion adsorption which leads to the well-understood phenomenon of heterogenous doping at which, e.g., one ion sort is internally adsorbed at the interface and vacancy defects are formed in the boundary zones. If such a surface active particle is dispersed into a matrix of low dielectric content (polymer, organic liquid) in which a salt is dissolved, an analogous effect must occur: As a large fraction of the salt is undissociated and present in the form of an ion pair (see Fig. 2d), the mechanism then consists in the adsorption of one species and setting free the counter ion (see Fig. 2d). This breaking up of an ion pair in a non-aqueous solvent has very recently been demonstrated by measuring the conductivity of a solution of LiClO4 in MeOH or THF at various contents of oxide particles and by measuring the zeta-potential. The overall conductivity increases with the acidity of the oxide and decreases with particle size [34]. The conductivity enhancement is the higher the lower the dielectric constant (more ion pairs), and vanishes for dilute solutions (no ion pairs). This indicates that the heterogeneous doping mechanism may be also viable for polymer electrolytes. In how far mobility effects have to be considered or are even predominant depends on the specific situation [35]. At the contact of two ion conductors there is a redistribution of ions comparable to an electron transfer at a semiconductor/semiconductor contact. Examples are twophase mixtures of AgI and AgBr [36], AgI and AgCl [37] (see Fig. 2i) as well as heterolayers of CaF2 and BaF2 (see Fig. 2j) [9]. Of course the interface itself offers the possibility of charge accumulation which is most important in the case of homojunctions (polycrystalline ion conductors) (see Fig. 2e) [27].
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J. Maier / Solid State Ionics 175 (2004) 7–12
Fig. 2. Examples of ionic charge transfer effects at equilibrium contacts.
Ref. [38] gives examples of depletion layer effects in oxides on electrical and chemical transport. Fig. 2f refers to an inversion layer to be discussed below. Fig. 2n is representative for the situation of a NEMCA experiment [39] where crucial for the catalytic oxidation of hydrocarbons is the spill-over layer of ionized oxygen (compensated by positive charges at the metal site). A recent thorough analysis in terms of (electro-)chemical potential of ions and electrons has been given in Ref. [40]. Also anomalous non-stoichiometries are possible as a consequence of interfacial charging. One relevant example is the mechanism of excess Li storage at the contact of an ion and an electron accepting system (e.g. Li2O/Ru) [22]. The small solubility of Li in Li2O is not due to the solubility of Li+, rather it is due to the difficulty to incorporate e. However, at the contact of Li2O with a (not too electropositive) metal, the storage can be as shown in Fig. 2l [21] with excess Li+ being accommodated at the Li2O side
counterbalanced by excess e at the metal side of the contact.
4. Size effects 4.1. Defect chemistry in the nano-regime 4.1.1. Examples Trivially but not unimportantly, the impact of the interfacial contribution on the overall property increases with decreasing grain size of a polycrystalline or heterophasic material. An impressive example of a quantitative conductivity alteration is nanocrystalline ceria the overall conductivity of which can be of n-type even if the bulk is ionically conducting [11,41,42]. The reason is that excess electrons are formed in the subsurface corresponding to a positive excess charge in the interfacial core. The origin of the excess charge is either due to a smaller deoxygenation
J. Maier / Solid State Ionics 175 (2004) 7–12
energy, i.e., easier formation of vacancies and excess electrons (see Fig. 2f), or to a grain boundary segregation of redox-active impurities. Another striking example in this context is the Tl+-conductivity of TlCl/Al2O3 even though we face an overwhelming anion conduction (Cl) in the bulk [43]. Most important are ionic transport effects at which the local interfacial situation depends on size because neighboring interfaces bperceive each otherQ (see Ref. [13]). A naturally appearing true size effect is the space charge overlap occurring if the sample thickness is smaller than the Debye length [44] (see Fig. 3, top l.h.s.). An elegant example is provided by the conductivity of ionic heterolayers composed of CaF2 and BaF2 [9] and another one by the overlap of the depletion zones around dislocation cores in low angle grain boundaries of SrTiO3 [33]. An interesting point is that a contact of thick and very thin (sub-Debye) layers (separated by an interface) is predicted to result in charge redistribution just like a contact of two chemically different layers (hetero-size charging, see Fig. 2o) [45]. A second non-trivial effect appears if the interfaces respond to the approach of neighboring interfaces by structural modifications (e.g., strained thin films) [13]. The range of such effects strongly depends on the mechanical properties. In soft materials the effects can be of rather short range; nevertheless, at spacings of the order of 1 nm or smaller we should generally expect serious structural alterations modifying then the l8 values as well as the mobilities which have been considered to vary step-function-like in the case of the abrupt structural
11
model: According to Eq. (5) we then have to extend Eq. (7) to ylnck ~ zk Fy/ ylk8:
ð8Þ
Note that in this notation, l8 also includes elastic effects. A further effect comes into play when the curvature of the particle becomes important. Then in the simplest case the chemical potentials have to be corrected by the Gibbs– Kelvin term (~c¯ /r¯ ) according to the interfacial pressure. One consequence is the charge redistribution expected at the contact of two chemically identical but differently curved particles (see Fig. 2p), which corresponds to a charge redistribution of two flat but chemically different particles (with the bchemical differenceQ Dl˚~c¯/r¯ ) [13]. According to Eq. (5): g¯ y lnck ~ zk Fy/ þ ð yl8Þ ð9Þ k 2d k : r¯ Already the space charge distribution becomes complex if the morphology deviates from quasi-one-dimensional distribution in a thin film or a spherical particle. In a bricklayer model there can be severe potential anomalies at edges and corners even if we ignore the different energetic behavior there [45]. Taking into account the specific energies of edges and corners is necessary for small sizes, and then finally when dealing with clusters rather than extended solids special configurations have to be considered that can be very different from the bulk structure. In such cases a bottom-up approach (i.e. from atomic to macro-size) is the treatment of choice rather than a top-down extrapolation (i.e. from macro-size to atomic size).
Fig. 3. L.h.s.: Top: Space charge effects decaying with k. Bottom: Structural effects which decay rapidly in the bulk. Within the indicated distance (S ) the structurally perturbed core (not shown, extension s) is perceived by the defects as far as l8 is concerned [13]. R.h.s.: Delocalized electrons perceive the boundaries much earlier (Dl8~L 2).
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J. Maier / Solid State Ionics 175 (2004) 7–12
As far as the electronic properties are concerned, a distinct difference should be made between narrow band and wide band semiconductors. Whereas in the former case the situation resembles the point defect chemical situations (see Fig. 3, l.h.s. bottom), in the latter the electrons are (owing to delocalization) effectively more extended, and changes in the l8 value of the electrons occur already at larger spacings since the electron perceives both boundaries already at moderately small size (see Fig. 3, r.h.s.). Hence, an area of particular interest with respect to size effects is expected to be the consideration of mixed electronic–ionic conductors. The consideration of interactions leads to a different point. In very small crystallites the mean distance between the charge carriers may be necessarily so small that interaction in form of associates (excitons, Frenkel pairs, color centers, etc.) have to be taken account of [4,13,46]. Considering the fact that attractive interactions lead to phase transformations this exhibits a route to boundary phase transformations in nano-crystals [10,13,47]. g¯ y lnck ~ zk Fy/ þ ð yl8k 2y k Þ y8;ex : r¯
ð10Þ
The list is far from being complete and is complicated by the necessity of implementing corrections such as charge discretization or gradient effects in order to obtain a more detailed insight [13]. The probably most important conclusion in this context is the statement that the spacing of interfaces offers a powerful degree of freedom for materials research. Quantitatively and qualitatively new transport properties may appear and the stability of the (metastable) morphology can be large enough to be of practical use. In this way, nanoionics offers the possibility of generating structurally and functionally complex materials with a high information content owing to the metastability of the higher-dimensional structure elements.
Acknowledgment The author acknowledges discussions with A. Bhattacharyya, R.A. De Souza, J. Fleig, S. Kim, E.A. Kotomin and I. Lubomirsky. He also thanks Ilan Riess for carefully reading the manuscript.
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J. Maier, Electrochemistry 68 (6) (2000) 395. J. Maier, Solid State Ionics 157 (2003) 327. J. Maier, Solid State Ionics 148 (3, 4) (2002) 367. J. Schoonman, Solid State Ionics 135–137 (2000) 5. H.L. Tuller, Solid State Ionics 131 (2000) 143.
[6] D.M. Kolb, Angew. Chem. 113 (2001) 1198. [7] J.-S. Lee, S. Adams, J. Maier, J. Electrochem. Soc. 147 (6) (2000) 2407. [8] Y.-M. Chiang, E.B. Lavik, I. Kosacki, H.L. Tuller, J.Y. Ying, J. Electroceramics 1 (1997) 7. [9] N. Sata, K. Eberman, K. Eberl, J. Maier, Nature 408 (2000) 946. [10] W. Puin, S. Rodewald, R. Ramlau, P. Heitjans, J. Maier, Solid State Ionics 131 (1, 2) (2000) 159. [11] S. Kim, J. Maier, J. Electrochem. Soc. 149 (10) (2002) J73. [12] P. Knauth, J. Schoonman (Eds.), Nanocrystalline Metals and Oxides. Selected Properties and Applications, Kluwer Academic Publishers, Boston, 2002. [13] J. Maier, Z. Phys. Chem. 217 (2003) 415. [14] J. Maier, Solid State Ionics 154–155 (2002) 291. [15] J. Maier, Ber. Bunsenges. Phys. Chem. 90 (1986) 26. [16] R. Defay, I. Prigogine, A. Bellemans, H. Everett, Surface Tension and Adsorption, Wiley, New York, 1960. [17] Ph. Buffat, J.-P. Borel, Phys. Rev., A 13 (1976) 2287. [18] Ch.P. Gr7f, U. Heim, G. Schwitzgebel, Solid State Ionics 131 (2000) 165. [19] A. Schrfder, PhD thesis, Stuttgart, in preparation. [20] K.T. Jacob, K.P. Jayadevan, R.M. Mallya, Y. Waseda, Adv. Mater. 12 (6) (2000) 440. [21] J. Jamnik, J. Maier, Phys. Chem. Chem. Phys. 5 (2003) 5215. [22] P. Balaya, H. Li, L. Kienle, J. Maier, Adv. Funct. Mater. 13 (8) (2003) 621. [23] J. Maier, Solid State Ionics 143 (2001) 17. [24] W. Gfpel, H.-D. Wiemhffer, Statistiche Thermodynamik, Spektrum Akademischer Verlag, Heidelberg, 2000. [25] R. Kirchheim, Prog. Mater. Sci. 32 (1988) 261. [26] J. Maier, Chem. Eur. J. 7 (22) (2001) 4762. [27] J. Maier, Prog. Solid State Chem. 23 (3) (1995) 171. [28] K.L. Kliewer, J.S. Koehler, Phys. Rev. 140 (4A) (1965) A 1226. [29] G. Gouy, J. Phys. 9 (1910) 457; D.L. Chapman, Philos. Mag. 25 (1913) 475. [30] R.B. Poeppel, J.M. Blakely, Surf. Sci. 15 (1969) 507. [31] J. Maier, Ber. Bunsenges. Phys. Chem. 89 (1985) 355. [32] E. Heifets, E.A. Kotomin, J. Maier, Surf. Sci. 462 (2000) 19; E. Heifets, R.I. Eglitis, E.A. Kotomin, J. Maier, G. Borstel, Phys. Rev., B 64 (2001) 23417. [33] R.A. De Souza, J. Fleig, R. Merkle, J. Maier, Z. Met.Kd. 94 (3) (2003) 218. [34] A. Bhattacharyya, J. Maier, Adv. Mater. 16 (2004) 811. [35] F. Croce, G.B. Appetecchi, L. Persi, B. Scrosati, Nature 394 (1998) 456; W. Wieczorek, Z. Florjanczyk, R. Stevens, Electrochim. Acta 40 (1314) (1995) 2251. [36] K. Shahi, J.B. Wagner, Appl. Phys. Lett. 37 (1980) 757. [37] U. Lauer, J. Maier, Ber. Bunsenges. Phys. Chem. 96 (1992) 111. [38] J. Maier, J. Eur. Ceram. Soc. 24 (2004) 1251. [39] C.G. Vayenas, S. Neophytides, Catalysis—Special Periodical Reports, Royal Society of Chemistry Cambridge, 1996, pp. 193 – 253. [40] I. Riess, C.G. Vayenas, Solid State Ionics 159 (2003) 313. [41] Y.-M. Chiang, E.B. Lavik, I. Kosacki, H.L. Tuller, J.Y. Ying, Appl. Phys. Lett. 69 (1996) 8. [42] A. Tschfpe, E. Sommer, R. Birringer, Solid State Ionics 139 (2001) 255; A. Tschfpe, Solid State Ionics 139 (2001) 267. [43] J. Maier, B. Reichert, Ber. Bunsenges. Phys. Chem. 90 (1986) 666. [44] J. Maier, Solid State Ionics 23 (1987) 59. [45] I. Lubomirsky, J. Fleig, J. Maier, J. Appl. Phys. 92 (2002) 6819. [46] E.L. Brus, J. Chem. Phys. 80 (1984) 4403. [47] N. Hainovsky, J. Maier, Phys. Rev., B 51 (22) (1995) 15789.
Ionic transport in nano-sized systems Joachim Maier* Max-Planck-Institut fu¨r Festkco¨rperforschung, Heisenbergstrage 1, 70569 Stuttgart, Germany
Abstract Ion transport in nano-sized systems is considered in a top-down approach. The equilibrium charge carrier concentrations are investigated as a function of the distance between neighboring interfaces and increased curvature of the grains. A variety of experimental examples involving accumulation, depletion and inversion layers are discussed, highlighting the significance of the nano-regime for Solid State Ionics. D 2004 Elsevier B.V. All rights reserved. Keywords: Nano-ionics; Ionic transport; Size effects
1. Introduction In recent years improved methods of preparing nanocrystalline or nanostructured materials as well as exciting findings of unusual electrical and electrochemical properties of such materials not only triggered the evolution of nanoelectronics and nano-electrochemistry, it also led to the appreciation of this size-regime for Solid State Ionics (bnano-ionicsQ) [1–14]. More detailed reports are given in Refs. [2,3,12–14]. Generally speaking, decrease of crystal size implies increase of the proportion of surfaces (interfaces) at the expense of bulk and usually implies an increased impact of the interfacial properties on the overall materials property (trivial size effects) (see, e.g., bricklayer model as described in Ref. [15]). If the spacing of the interface is very narrow, local properties may change as a function of distance and true size effects may occur. The three-dimensionality of the crystals is expressed by a mean curvature (1/r¯, where r¯ =R j r j a j /R j a j , distance of the surface plane j from the center, a j : its area) which increases if the sample size is reduced; crystallographically speaking we refer to edges large compared with a typical next neighbor distance. If we are not explicitly addressing edges and corners, their
* Tel.: +49 711 689 1720; fax: +49 711 689 1722. E-mail address: [email protected] (J. Maier). 0167-2738/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2004.09.051
influence then appears in the surface tension (c j ) and makes c j size dependent even if the interfacial structure remains unchanged. The increased mean curvature does not only lead to a size-dependent local chemical standard potential for the interfacial core (l s), it also leads to an increased internal pressure, which for a Wulff crystal amounts to p i =p+2c¯/r¯, where c¯ =R j c j a j /R j a j and p is the outer pressure [16]: lsk ðpi Þglsk ð pÞ þ
dlsk ðpi pÞglsk ð pÞ þ 2wvk: dp
ð1Þ
In Eq. (1) v k is the partial molar volume of the species k, w is the Wulff ratio c j /r j which is independent of j and then also equal to c¯/r¯ for the equilibrium morphology (the local excess energy of edges is neglected). The correction term in Eq. (1) easily accounts for the largely depressed melting point of tiny crystals [17] as well as for the e. m. f. values of electrochemical cells with nano-crystalline electrodes [18–22]. When we are addressing polycrystals, the morphology is not in equilibrium, only local equilibria may be reached. In the semi-quantitative approach taken below, we not only ignore anisotropies (i.e. c¯=c) whenever we deal with polycrystalline materials, we also use the bricklayer model constituted by identical cubic grains in a primitive arrangement. First of all we briefly consider charge carrier distribution in the bulk, by restricting to dilute conditions and treating
8
J. Maier / Solid State Ionics 175 (2004) 7–12
both ionic and electronic carriers by using benergy levelQ diagrams [23]. Then we proceed to charge carrier chemistry at planar interfaces and, finally, discuss effects due to curvature.
2. Bulk defect thermodynamics Let us consider the free energy of a system containing defects (excess particles such as interstitial ions or conduction electrons; or missing particles such as vacancies in the lattice or holes in the valence band). These particles introduced may possess a spectrum of energy levels, i.e. occupying different energy levels e. (If we speak of energy levels in the following, we more strictly mean local free energy levels since we include local entropy contributions (vibrational entropy).) If we cut out a differentially small (free) energy band in which we can assume a constant (free) energy eV for a single particle, the excess (free) energy therein upon introducing yNV non-interacting and randomly distributed defects is given by the binomial expression eVyN V kT lnð yZVÞ (yZV: number of available states in yN V that section) leading to a chemical potential of the form l ¼ lV þ RT ln
yN V=yZV 1 yN V=yZV
ð2Þ
with lV=N meV (N m: Avogadro’s number), i.e., to a Fermi Dirac type of expression due to the limited number of states/crystallographic positions for the electronic/ionic carriers. (Note that the problems of distributing NV defect over ZV sites of energy eV is statistically equivalent to distributing NV electrons over ZV quantum states of energy eV (see, e.g., Ref. [24]).) The bFermi-levelQ (generalized also to ionic carriers) corresponds to l/N m. The total particle number within the whole band of available states (with the upper and lower limits e H, e L) follows by integration as Z N ¼
eH F ðeVÞDðeVÞdeV
where l8 is identical to N me L supposed N¯ is identified with the integral according to Eq. (3) but the variable eV–e F in F being replaced by eV–e L. Eq. (4) holds generally for dilute randomly distributed carriers, be them electrons in the conduction band (D typically parabolic, N¯ =effective number of states in the conduction band), holes in the valence bands (D typically parabolic, N¯ =effective number of states in the valence band), interstitial ions in solids (D=delta function, N¯ =number of interstitial positions), vacancies in the lattice (D=delta function, N¯ =number of regular positions), or even excess or missing particles in aqueous solvents (H3O+, OH) (then N¯ may typically result from the integration of a Gaussian state density distribution corresponding to fluctuations in space and time; an analysis of different densities of states is given in Ref. [25]) If we deviate from dilute conditions, excess effects due to the restricting term in the denominator of Eq. (2) become important (exhaustibility of states/sites). This leads to an excess term in the chemical potential (l ex). In the case of single levels (D=delta function) we can just take Eq. (2) and write (l ex,c)=RTln(1N/Z). Note that this correction is explicitly (but strictly speaking falsely) introduced when considering chemical potentials of structure elements instead of building elements (since then the bchemical potentialQ of the regular species appears with the bBoltzmann-termQ RTln(1N/Z) in the balance because the number of regular and defective positions add up to Z). Whenever we deviate from dilute conditions we have to reckon with deviations from the ideal distribution and hence with interactions. The overall excess contribution (l ex) may nevertheless be split into configurational and non-configuration parts (l ex,c, l ex,o), only the latter being part of the benergy levelsQ. In addition, we have to add the electrical potential term zk F/ (which is important for interfaces; the mechanical term c¯/r¯ (see Eq. (1)) is contained in l8). Hence ex;c l˜ k ¼ ½lk8 þ ð2g¯=r¯ Þvk þ zk F þ ex;o k þ RT ln ckþ k :
ð5Þ
ð3Þ
eL
with D=dZ/ye and FuyN/yZ, the latter being given by Eq. (2). Without loss of generality let us consider excess particles. Assuming dilute conditions means that only the states of low energy are affected and the occupied levels are close to e L. In the case of excess electrons, e L is the lower edge of the conduction band and D is of a parabolic form, while in the case of interstitials D is a delta function referring to the single energy state e L with a high density of states (dilute conditions). Since for dilute situations |e Fe L|HkT and since the integral (Eq. (3)) only affects energy states e[e L, the deviation from unity in the denominator of Eq. (2) can be safely neglected; the integration then leads to the Boltzmann-form N ð4Þ l ¼ l8 þ RT ln ¯ N
The bracketed term corresponds to the (free) energy level used in the benergy levelQ diagrams. Structural inhomogeneities (e.g. elastic effects) are subsumed in l8 and/or l ex,o. If concentration measures (c) are used different from N/N¯ , the correction can be absorbed into l kex,c or, if dilute, also into the standard potential. Fig. 1 shows the benergy levelQ picture for two mixed conductors exhibiting (cation) Frenkel disorder in contact. The reader should ignore the middle part at the contact, which is to be considered below. (There the ionic Fermi-level analogue is termed bFrenkel-levelQ, + l˜ M uE FrenkeluN me Frenkel.) The diagrams for the ionic and the electronic excitations are coupled via lMþ þ le ¼ EFrenkel þ EFermi ¼ lM :
ð6Þ
(Note in this context that l M+=l i=l v and l e=l eV=l hS, S and i being short for interstitial building element (M i Vi)
J. Maier / Solid State Ionics 175 (2004) 7–12
Fig. 1. Three energy levels for the contact of two mixed conductors.
and v being short for the vacancy building element (VVMM M).) In Ref. [26] it is shown that these concepts can be used to construct a generalized acid–base concept of solids which is based on counting point defects to assess acidity/basicity whereby ionic associates play the role of internal acids and bases. But let us proceed to interfaces.
3. Interfacial defect chemistry Let us consider a flat interface and restrict to dilute interaction-free conditions. Fig. 1 refers to the equilibrium contact of two mixed ionic–electronic conductors if both cations and electrons are sufficiently mobile [27]. The constancy of the chemical potential of the component is not sufficient, rather the two Fermi-levels (ionic and electronic) have to be constant (which then implies a constant chemical potential of the neutral component) to satisfy the contact equilibrium. Let us consider the contact of MX with MXV, and assume substantial ionic defect concentrations (see, e.g., Fig. 2i). Let us also assume that the levels for metal vacancies and interstitial ions are such that there is a distinct transfer of cations in addition to the possible neutral M solubility. Naturally, the effect is restricted to the layers close to the interface in which deviations from electroneutrality can be tolerated. The changed defect concentrations correspond to bent levels owing to the zF/(x) term in Eq. (5) at constant bFermi-levelQ: yln ck ~ zk Fy/:
ð7Þ
(Cf. also Refs. [28–30]). At a homo-phase contact (grain boundary, see Fig. 2e) two chemically identical crystals meet. They are either insufficiently contacted (significant gas phase content) or sintered together with different orientations. The main space charge effect here is a charging via carrier segregation as a consequence of the different structure of the interfacial core.
9
This core layer may be approximately considered as a narrow phase with a given but different l8 value (l k8 contains a term which accounts for the excess surface energy and depends on ca k where a k is a partial molar area [16]). Of course in the previous situation (Fig. 2i) also a charge storage within the MX/MXV interface can occur. In all cases we may consider the situation as a composite of at least two structurally different regions, and a comparatively simple consideration is possible if we assume that l8 changes abruptly (see, e.g., the defect chemical calculation of the contact MX/MXV in Ref. [31]), which has been justified by atomistic calculations for some perovskites [32]. While in the bulk the electroneutrality condition gives a sufficient account of electrostatics, at the boundaries it has to be replaced by the more general Poisson’s equation R j z j c j aj2//e. The related concentration changes can be immense and seriously affect also the ionic conduction in solids [13,27,33]. While Fig. 1 gave a formal description in terms of energy level bending, Fig. 2 compiles a variety of examples from different areas of Solid State Ionics: Only a few of them shall be touched upon: At the contact of the ion conductor to an insulator (see Fig. 2a,b) there can be ion adsorption which leads to the well-understood phenomenon of heterogenous doping at which, e.g., one ion sort is internally adsorbed at the interface and vacancy defects are formed in the boundary zones. If such a surface active particle is dispersed into a matrix of low dielectric content (polymer, organic liquid) in which a salt is dissolved, an analogous effect must occur: As a large fraction of the salt is undissociated and present in the form of an ion pair (see Fig. 2d), the mechanism then consists in the adsorption of one species and setting free the counter ion (see Fig. 2d). This breaking up of an ion pair in a non-aqueous solvent has very recently been demonstrated by measuring the conductivity of a solution of LiClO4 in MeOH or THF at various contents of oxide particles and by measuring the zeta-potential. The overall conductivity increases with the acidity of the oxide and decreases with particle size [34]. The conductivity enhancement is the higher the lower the dielectric constant (more ion pairs), and vanishes for dilute solutions (no ion pairs). This indicates that the heterogeneous doping mechanism may be also viable for polymer electrolytes. In how far mobility effects have to be considered or are even predominant depends on the specific situation [35]. At the contact of two ion conductors there is a redistribution of ions comparable to an electron transfer at a semiconductor/semiconductor contact. Examples are twophase mixtures of AgI and AgBr [36], AgI and AgCl [37] (see Fig. 2i) as well as heterolayers of CaF2 and BaF2 (see Fig. 2j) [9]. Of course the interface itself offers the possibility of charge accumulation which is most important in the case of homojunctions (polycrystalline ion conductors) (see Fig. 2e) [27].
10
J. Maier / Solid State Ionics 175 (2004) 7–12
Fig. 2. Examples of ionic charge transfer effects at equilibrium contacts.
Ref. [38] gives examples of depletion layer effects in oxides on electrical and chemical transport. Fig. 2f refers to an inversion layer to be discussed below. Fig. 2n is representative for the situation of a NEMCA experiment [39] where crucial for the catalytic oxidation of hydrocarbons is the spill-over layer of ionized oxygen (compensated by positive charges at the metal site). A recent thorough analysis in terms of (electro-)chemical potential of ions and electrons has been given in Ref. [40]. Also anomalous non-stoichiometries are possible as a consequence of interfacial charging. One relevant example is the mechanism of excess Li storage at the contact of an ion and an electron accepting system (e.g. Li2O/Ru) [22]. The small solubility of Li in Li2O is not due to the solubility of Li+, rather it is due to the difficulty to incorporate e. However, at the contact of Li2O with a (not too electropositive) metal, the storage can be as shown in Fig. 2l [21] with excess Li+ being accommodated at the Li2O side
counterbalanced by excess e at the metal side of the contact.
4. Size effects 4.1. Defect chemistry in the nano-regime 4.1.1. Examples Trivially but not unimportantly, the impact of the interfacial contribution on the overall property increases with decreasing grain size of a polycrystalline or heterophasic material. An impressive example of a quantitative conductivity alteration is nanocrystalline ceria the overall conductivity of which can be of n-type even if the bulk is ionically conducting [11,41,42]. The reason is that excess electrons are formed in the subsurface corresponding to a positive excess charge in the interfacial core. The origin of the excess charge is either due to a smaller deoxygenation
J. Maier / Solid State Ionics 175 (2004) 7–12
energy, i.e., easier formation of vacancies and excess electrons (see Fig. 2f), or to a grain boundary segregation of redox-active impurities. Another striking example in this context is the Tl+-conductivity of TlCl/Al2O3 even though we face an overwhelming anion conduction (Cl) in the bulk [43]. Most important are ionic transport effects at which the local interfacial situation depends on size because neighboring interfaces bperceive each otherQ (see Ref. [13]). A naturally appearing true size effect is the space charge overlap occurring if the sample thickness is smaller than the Debye length [44] (see Fig. 3, top l.h.s.). An elegant example is provided by the conductivity of ionic heterolayers composed of CaF2 and BaF2 [9] and another one by the overlap of the depletion zones around dislocation cores in low angle grain boundaries of SrTiO3 [33]. An interesting point is that a contact of thick and very thin (sub-Debye) layers (separated by an interface) is predicted to result in charge redistribution just like a contact of two chemically different layers (hetero-size charging, see Fig. 2o) [45]. A second non-trivial effect appears if the interfaces respond to the approach of neighboring interfaces by structural modifications (e.g., strained thin films) [13]. The range of such effects strongly depends on the mechanical properties. In soft materials the effects can be of rather short range; nevertheless, at spacings of the order of 1 nm or smaller we should generally expect serious structural alterations modifying then the l8 values as well as the mobilities which have been considered to vary step-function-like in the case of the abrupt structural
11
model: According to Eq. (5) we then have to extend Eq. (7) to ylnck ~ zk Fy/ ylk8:
ð8Þ
Note that in this notation, l8 also includes elastic effects. A further effect comes into play when the curvature of the particle becomes important. Then in the simplest case the chemical potentials have to be corrected by the Gibbs– Kelvin term (~c¯ /r¯ ) according to the interfacial pressure. One consequence is the charge redistribution expected at the contact of two chemically identical but differently curved particles (see Fig. 2p), which corresponds to a charge redistribution of two flat but chemically different particles (with the bchemical differenceQ Dl˚~c¯/r¯ ) [13]. According to Eq. (5): g¯ y lnck ~ zk Fy/ þ ð yl8Þ ð9Þ k 2d k : r¯ Already the space charge distribution becomes complex if the morphology deviates from quasi-one-dimensional distribution in a thin film or a spherical particle. In a bricklayer model there can be severe potential anomalies at edges and corners even if we ignore the different energetic behavior there [45]. Taking into account the specific energies of edges and corners is necessary for small sizes, and then finally when dealing with clusters rather than extended solids special configurations have to be considered that can be very different from the bulk structure. In such cases a bottom-up approach (i.e. from atomic to macro-size) is the treatment of choice rather than a top-down extrapolation (i.e. from macro-size to atomic size).
Fig. 3. L.h.s.: Top: Space charge effects decaying with k. Bottom: Structural effects which decay rapidly in the bulk. Within the indicated distance (S ) the structurally perturbed core (not shown, extension s) is perceived by the defects as far as l8 is concerned [13]. R.h.s.: Delocalized electrons perceive the boundaries much earlier (Dl8~L 2).
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As far as the electronic properties are concerned, a distinct difference should be made between narrow band and wide band semiconductors. Whereas in the former case the situation resembles the point defect chemical situations (see Fig. 3, l.h.s. bottom), in the latter the electrons are (owing to delocalization) effectively more extended, and changes in the l8 value of the electrons occur already at larger spacings since the electron perceives both boundaries already at moderately small size (see Fig. 3, r.h.s.). Hence, an area of particular interest with respect to size effects is expected to be the consideration of mixed electronic–ionic conductors. The consideration of interactions leads to a different point. In very small crystallites the mean distance between the charge carriers may be necessarily so small that interaction in form of associates (excitons, Frenkel pairs, color centers, etc.) have to be taken account of [4,13,46]. Considering the fact that attractive interactions lead to phase transformations this exhibits a route to boundary phase transformations in nano-crystals [10,13,47]. g¯ y lnck ~ zk Fy/ þ ð yl8k 2y k Þ y8;ex : r¯
ð10Þ
The list is far from being complete and is complicated by the necessity of implementing corrections such as charge discretization or gradient effects in order to obtain a more detailed insight [13]. The probably most important conclusion in this context is the statement that the spacing of interfaces offers a powerful degree of freedom for materials research. Quantitatively and qualitatively new transport properties may appear and the stability of the (metastable) morphology can be large enough to be of practical use. In this way, nanoionics offers the possibility of generating structurally and functionally complex materials with a high information content owing to the metastability of the higher-dimensional structure elements.
Acknowledgment The author acknowledges discussions with A. Bhattacharyya, R.A. De Souza, J. Fleig, S. Kim, E.A. Kotomin and I. Lubomirsky. He also thanks Ilan Riess for carefully reading the manuscript.
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