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Ionic and Mixed Conductivity in Condensed Phases
Ionic and Mixed Conductivity in Condensed Phases 9
based on the ‘‘use’’ of ion-channel proteins as molecular tools for therapy of cardiac rhythm alteration. To date, cardiac electronic pacemakers have been widely used in the treatment of rhythm disorders. However, recent research suggests the possibility of developing therapeutic approaches based on ‘‘biological pacemakers’’ in which the ability of generating spontaneous cardiac electric activity could be restored by manipulating the level of ion channels expressed in a specific desired region of the heart. See also: Biomembranes; Membrane and Protein Aggregates.
PACS: 87.16.Uv; 87.19.Nn; 87.80.Jg; 87.17.Nn; 87. 64. Gb; 87.16.Dg Further Reading Ashcroft FM (2000) Ion Channels and Disease. San Diego: Academic Press. Armstrong CM (1975) Ionic pores, gates, and gating currents. Quarterly Reviews of Biophysics 7: 179–210. Colquhoun D and Hawkes AG (1981) On the Stochastic Properties of Single Ion Channels. Proceedings of the Royal Society of London. Series B 211: 205–235. DiFrancesco D (1993) Pacemaker mechanisms in cardiac tissue. Annual Review of Physiology 55: 455–472.
Doyle DA, Morais Cabral J, Pfuetzner RA, Kuo A, Gulbis JM, Cohen SL, Chait BT, and MacKinnon R (1998) The structure of the potassium channel: molecular basis of K þ conduction and selectivity. Science 280: 69–77. Hamill OP, Marty A, Neher E, Sakmann B, and Sigworth FJ (1981) Improved patch-clamp techniques for high-resolution current recording from cells and cell-free membrane patches. Pflu¨gers Archive: European Journal of Physiology 391: 85–100. Hille B (2001) Ion Channels of Excitable Membranes, 3rd edn. Sunderland, Massachussets, USA: Sinauer Associates, Inc. Hille B, Armstrong CM, and MacKinnon R (1999) Ion channels: from idea to reality. Nature Medicine 10: 1105–1109. Hodgkin AL and Huxley AF (1952) A quantitative description of membrane currents and its application to conduction and excitation in nerve. Journal of Physiology, London 117: 500–544. Hodgkin AL and Keynes RD (1955) The potassium permeability of a giant nerve fibre. Journal of Physiology, London 128: 61–88. Jiang Y, Lee A, Chen J, Ruta V, Cadene M, Chait BT, and MacKinnon R (2003) X-ray structure of a voltage-dependent K þ channel. Nature 423: 33–41. MacKinnon R (2003) Potassium channels. FEBS Letters 555: 62–65. Miller C (1985) Ion Channel Reconstitution. New York: Plenum Press. Neher E and Sakmann B (1992) The patch clamp technique. Scientific American 266: 44–51. Sakmann B and Neher E (1995) Single Channel Recording. New York: Plenum. Yellen G (1998) The moving parts of a voltage-gated ion channel. Quarterly Reviews of Biophysics 31: 239–295.
Ionic and Mixed Conductivity in Condensed Phases J Maier, Max-Planck-Institut fu¨r Festko¨rperforschung, Stuttgart, Germany & 2005, Elsevier Ltd. All Rights Reserved.
Introduction: Hopping Process While a quantum mechanical particle such as a quasifree-electron can tunnel through a barrier by interference of the states on both sides of the barrier (denoted by x and x0 ), that is, by seizing the hindrance in a wave-like way, the transport of a heavy particle such as an ion has to rely on the ensemble fraction with energy high enough to reach the final state (hopping). On the one hand, under favorable circumstances protons may also tunnel, while on the other hand electrons that strongly polarize their environment (large polarons) behave almost classically as ions. In most cases of interest (consider A as an ion with charge zA), the hopping of A from x to x0 ðx0 ¼ x þ DxÞ may be written as AðxÞ þ V 0 ðx0 Þ"VðxÞ þ A0 ðx0 Þ
½1
(The dash takes account of the possibility for A to have a different structural environment at x0 .) In cases where A denotes an ion on a regular site in a crystal, V is the vacancy that serves as a jump partner, while in cases where A refers to an excess ion sitting on interstitial sites, V denotes the vacant interstitial site. In crystals, V (in the first case) or A (in the second case) are usually dilute with the energy levels of these sites being well defined. The concentration of the respective reaction partner is then approximately constant and the corresponding rate equation for the hopping process is pseudo-monomolecular. In amorphous solids, the sites are less defined, which also holds for the spatial energy distribution; in polymers, melts, or liquids, these sites and energy distributions are time dependent. In these cases, one would better speak of smeared-out density of states, as far as thermal excitations are concerned. The rate constant for the hopping rate equation is decisively influenced by the activation free enthalpy. The parameters k and DGa , for example, for the forward reaction (index ,) related through (prefactor k0 is proportional to the attempt
10
Ionic and Mixed Conductivity in Condensed Phases
frequency G0) ,
,
k ¼ k0 exp
D Ga RT
! ½2
refer to the built-in part of the forward rate constant and its activation free energy, respectively. This expression has to be complemented by the factor , expða Df z F=RTÞ in order to take account of the applied electric field. The parameter Df is that part of the external voltage that drops from x to x0 and , the term a Df measures the portion of Df that is relevant for the forward reaction (distance between saddle point and initial state). The relations for the , , backward reaction are analogous ð a þ a ¼ 1Þ. This leads to the Butler–Volmer equation which describes the charge transfer from x to x0 for an asymmetric free-energy profile. In the homogeneous bulk region (and also in the space charge region sufficiently far from the interface, i.e.,,A ¼&A0 ), the built-in profile is , & symmetrical, that is, k ¼ k ¼ k, a ¼ 1=2 ¼ a and , & DG ¼ DG. The result for the rate represented by the current density ij of carrier j in x-direction ij ¼ 2kj cj zj F sinh C sj
zj FDf 2RT
Df Dx
½3
can be simplified to Ohm’s law (bottom part) for Df{RT=zj F; the conductivity sj of the carrier is proportional to its mobility uj (uj pkj ), (molar) concentration (cj) and (molar) charge (zjF) and given as sj ¼ zj Fuj cj
½4
Allowing also for a concentration variation during the process and generalizing to three dimensions, the Nernst–Planck equation is arrived at ij ¼ Dj zj Frcj sj rf
½5
where the diffusion coefficient (D) and conductivity (s) are related through the Nernst–Einstein equation. Close to equilibrium, this equation corresponds to the result obtained by linear irreversible thermodynamics, sj ij ¼ rm* j ½6 zj F which assumes the current to be proportional to the gradient in the electrochemical potential m* ¼ m þ zFf (chemical potential þ (molar charge electrical potential)). The latter relation is more general concerning the concentration range, while the former (as regards the diffusion term) is more general as far as
the magnitude of this driving force is concerned. In cases where the current of a given species is also generated by secondary driving forces (e.g., by rm* of a different particle), cross terms occur for which the Onsager–Casimir symmetry relations are valid. As regards the determination of the conductivity, the first task is to determine the carrier concentration as a function of the control parameters. This is feasible by solving the defect model for the system under consideration.
Equilibrium Charge Carrier Concentrations Pure Compounds and Dilute Bulk
Figure 1 displays the fundamental charge carrier formation reaction in an ‘‘energy level’’ diagram for a fluid (H2O) and an ionic crystal (AgCl). These levels refer to effective standard (electro) chemical potentials or – in the case of the ‘‘Fermi levels’’ that are positioned within the gap – full (electro) chemical potentials. (As long as the bulk is considered, zFf can be neglected and one can refer to m instead of m.) * As long as – in pure materials – the gap remains large compared to RT, the Boltzmann form of the chemical potential of the respective charge carrier (defect) is valid which has the form ! c j mj ¼ mj þ RT ln ½7 cj As outlined below, this relation holds largely independent of the charge carrier situation and the form of the energy level distribution. If the levels are smeared out or if there is a band of levels, mj refers to an effective level (i.e., band edges in nondegenerate semiconductors or an appropriately averaged energy when dealing with fluids or disordered solids). The coupling of the ionic level picture and the electronic level picture for AgCl (Figure 2) is established via the thermodynamic relation m* Agþ þ m* e ¼ mAg , and hence via the component potential which reflects the precise position in the phase diagram. The meaning of m becomes obvious, if for simplicity one considers an elemental crystal (with defect j). For the formation of Nj identical defects (among N regular positions), a local free enthalpy of Nj Dgj is required in the interaction-free case; including also the configurational contribution, the Gibbs energy of the defective crystal (GP refers to the perfect crystal) reads N G ¼ GP þ Nj Dgj kB T ln ½8 Nj
Ionic and Mixed Conductivity in Condensed Phases 11
AgCl
H2O
Ionic disorder H+ (in H3O+)
Null H2O
~ −H0+ − vacancy
H+ + |H|− H+ + OH−
e −(CB)
Ag+ (interst.) ~ 0 Agi
~ 0+ H − excess
~ += ~ Ag Agi ~ = −
~H+ = ~H+ − excess ~ + = − H − vacancy H+ (in H2O)
Electronic disorder
~e − = ~e ′ = ∋ F ~ = − h
V ′Ag
Ag+ (regular)
Null AgAg + V i
~0 V′Ag
ee−_(VB) (VB)
Ag + |Ag|′ Agi + V Ag ′
~ =∋ * e′ C
Null e VB + V CB
~* = ∋ − h V
e′ + h e′CB + V VB
~i ≡ i + zi F = 0i + RT In [i ] + ziF =~ 0i + RT In [i ]
Regular ionic level
Ag
~e −
Conduction band
Figure 2 Coupling of ionic and electronic levels in AgCl.
The configuration term describes the number of combinations of Nj elements out of N choices without repetition. The calculation of Dg , the free energy to form a single defect, requires an atomistic consideration. If one refers to ionic defects in crystals, it is essentially composed of lattice energy, polarization energy, and vibrational entropy terms. In more general terms, Nj is the number of defects and N the number of particles that can be made defective. Owing to the infinitely steep decrease of the configurational term with Nj, a minimum in G(Nj) (see Figure 3) is observed in the elemental crystal, if the chemical potential of j, that is, mj @G=@ðNj =Nm Þ, which follows as Nj mj ¼ mj þ RT ln N Nj Nj C mj þ RT ln N nj ¼ mj þ RT ln n
½9
vanishes (n N=Nm , nj Nj =Nm , Nm ¼ Avogadro’s number; mj ¼ Nm Dgj ). It is noteworthy that the
∆d G* = N d ∆d g* Contributions to the free enthalpy
Valence band
Interstitial ionic level ~ Ag+
Partial free energy of e −
Partial free energy of Ag +
Figure 1 Electronic and ionic disorder in ionic solids and water in ‘‘physical’’ (top) and ‘‘chemical’’ language (bottom).
G real − Gperfect 0 −TS cfg
^
Nd
Number of point defects (N = const)
Figure 3 Contributions to the free enthalpy of the solid by defect formation with a constant total number of sites.
strict result (top) which is formally valid for higher concentrations also is of the Fermi–Dirac type. This is due to the fact that one refers to combinations without repetition, that is, double occupancy is excluded and hence the sites are exhaustible similarly as is the case for quantum states in electronic problems. (Combinations with repetitions lead to Bose–Einstein distribution.) In a situation in which the levels are broadened to a more or less continuous zone (e.g., electrons in semiconductors and ions in fluids), the parameters Nj and mj of the Fermi–Dirac form have to be attributed to an infinitely small level interval. Denoting the partial free energy level by E, this integral ranges from Ej to Ej þ dEj and the total (molar) concentration
Ionic and Mixed Conductivity in Condensed Phases
follows from integration: DðEj Þ dEj nj ¼ 1 þ expðjE j mj j=RTÞ zone
½10
whereby the molar density of states D (Ej ) appears in the integrand. At equilibrium, mj is independent of Ej . Considering without the restriction of generality, the almost empty zone and the lower edge of this zone designated as Ej , one can neglect the unity in the denominator of the above equation for all levels more 0 distant from mj than Ej which leads to the Boltzmann expression nj Cn% j ðTÞexpðjE0j mj j=RTÞ. The effective value n% j is given by n% j ðTÞ ¼
jEj E0j j dEj DðEj Þexp RT zone
Z
Y
b
cj ðT; PÞ ¼ aj j PNj
½11
n% j being usually weakly temperature dependent. It is worth noting that, owing to the high dilution, changes of the parameters with occupation have been ignored (for electrons in semiconductors, this is called the ‘‘rigid band model’’). As anticipated above, the Boltzmann form of the chemical potential results with m representing an effective value. If n is identified with n% j , the effective value mj is given by E0j. As long as one refers to dilute conditions, considerable freedom of normalization is left with respect to the concentration measure. In ionic compounds, charged defects occur pairwise, the formation of which is subject to disorder reactions (and electroneutrality conditions). Then, only the reaction P combination of chemicalP potential vanishes,Pthat is, j nrj mj ¼ 0 (identical to j nrj m* j ¼ 0 since j nrj zj ¼ 0), where nrj is the stoichiometric coefficient of j in the respective disorder reaction r (cf. Figure 1, bottom). Boltzmann expressions can be written for the individual carriers, if the formation energies and the configuration entropies are considered to be independent. The application of chemical thermodynamics elegantly permits treatment of a variety of ionic and electronic disorder equations that simultaneously occur in a given solid. This is not only possible for the internal disorder equations (such as Schottky, Frenkel, anti-Frenkel, anti-Schottky, electron–hole formation) but also for external reactions such as the interaction of an oxide with the oxygen partial pressure and hence the stoichiometric variation. Under Boltzmann (i.e., dilute) and Brouwer (i.e., two majority carriers) conditions, an adequate expression is arrived at for any charge carrier concentration as a function of the control parameters (temperature T, doping content, component partial pressure P),
Kr ðTÞgrj
½12
r
(Nj ; grj ; aj ; bj being simple rational numbers; the total pressure is assumed to be constant.) For multinary compounds (i.e., m components) at complete equilibrium (m 1), component partial pressures are to be considered. This solution is only a sectional solution within a window in which the majority carriers situation (Brouwer condition) does not change. The power-law equation defines van’t Hoff diagrams characterized by X @ ln cj ¼ grj Dr H ½13 @1=RT r according to which, for not too extended T-ranges (i.e., Dr H , the reaction enthalpy of the defect reaction g, is constant), straight lines are observed in the ln cj versus 1/T representation, as well as Kro¨ger–Vink diagrams characterized by straight lines in the plots ln cj versus ln P with slopes @ ln cj ¼ Nj @ ln P
½14
Figure 4 displays the defect chemistry within the phase width of the oxide MO. At low oxygen pardd tial pressure (PO2 ), vacant oxygen ions ðVO Þ and
VO e′
log (|z| [defect z])
Z
namely
(±1/6)
(O) VO , Oi″ (±1/4)
Oi″
h
e′
N
I
Oi″ h
(±1/6) VO P
P(i )
log PO
2
= [Oi″ ] − [VO ]
T
12
[h ] − [e′] 2
M
MO
O
PO2 Figure 4 Internal redox and acid–base chemistry in a solid MO1þd within the phase width at a given temperature T (disorder in the M-sublattice assumed to be negligible).
Ionic and Mixed Conductivity in Condensed Phases 13 dd reduced states ðe0 Þ are in the majority (2½VO C½e0 c 00 d ½Oi ; ½h ) (N-regime) whilst oxygen interstitials (O00i ) and oxidized states ðhd Þ dominate for very high PO2 dd (2½O00i C½hd c½e0 ; ½VO ) (P-regime). At intermediate dd PO2 , usually ionic disorder prevails (½VO C 00 0 d ½Oi c½e ; ½h ) (I-regime); the alternative situation that electronic disorder predominates (½e0 C½hd c dd ½VO ; ½O00i ) is usually scarce. Exactly at the Dalton ðiÞ dd composition (i.e., at PO2 where 2½O00i ¼ 2½VO ¼ 0 d ½e ¼ ½h ), the ‘‘nonstoichiometry’’ d in MO1þd (i.e., dd ½O00i ½VO ¼ 12½hd 12½e0 ) is precisely zero. For . dc0, a p-conductor (mobility of h is usually much 00 greater than for Oi ) is referred to, for d{0, one refers (mobility of e0 is usually much greater than for dd VO ) to an n-conductor, while at dC0, mixed conduction is expected (only if the ionic concentrations are much larger than the electronic ones, pure ion conduction results). Usually, the entire diagram is not observed, as there are limits of experimentally realizing extreme PO2 values as well as limits with respect to the phase stability (formation of higher oxides or lower oxides, if not of the metal). Figure 5 gives three examples: SnO2 as an oxygendeficient material, La2CuO4 as an example of a
p-type conduction, and PbO as an example of I-regime exhibiting mixed conduction. (In PbO, most probably, the counterdefect to O00i is Pbdd i rather than dd VO ; the results, however, are not different then.) The slopes directly follow from simple mass action considerations. The model example of AgCl shall be considered in more detail. It also exhibits predominant ionic disorder, but now the decisive disorder reaction is the Frenkel reaction of the Ag sublattice (i.e., the for0 mation of silver ion vacancy, VAg , and interstitial d silver ion, Agi ). The role of PO2 is played by the chlorine partial pressure. The mass action law for the þ1=2 interaction with Cl2 reads ½Agdi PCl2 ½e0 ¼ constant, the mass action constant for the Frenkel reaction 0 ¼ KF and that for the band–band being ½Agdi ½VAg . 0 d 0 transfer ½e ½h ¼ KB. Since ½Agdi C½VAg c½e0 , ½h , it p ffiffiffiffiffiffi 1=2 0 follows as solution ½VAg ¼ ½Agdi ¼ KF , ½e0 pPCl2 , þ1=2 ½hd pPCl2 . Doping Effects
Dopants, as irreversibly introduced structure elements such as Cd2 þ substituting Ag þ to form the
I N
P
718°C
−2.0
745°C
−2.4 −2.8
653°C
ion
4.0
597°C
2 4 log(PO2)(Pa)
6
log PO
−3.6
(c)
2
−1.4 (a)
550°C
4.5 0
−3.2
800°C
−1.0 −0.6 −0.2
VO
log(PO2) (bar) SnO 2−
e′
(±1/6)
(O) VO , Oi″ (±1/4)
Oi″ N
1
1
eon
−1/4
log (|z| [defectz])
log (Ω −1 cm−1)
log
PbO (orh)
3.5
h
e′
I
0
Slope:
0
1/6
−1
−1
−2
−2 −4
−2 log(PO2) (bar)
Normalized thermopower
−1.6
−1/6
log (Ω −1 cm−1)
818°C
−log (Ω −1 cm−1)
−1.2
0
Oi″ h
(±1/6)
La2CuO4+
VO P
P(i )
(b)
logPO2
Figure 5 Three experimental examples of Kro¨ger–Vink diagrams in a pure oxide MO with ideal defect chemistry. (a) SnO2 as an n-type conductor, (b) PbO as a mixed conductor, and (c) La2CuO4 as a p-type conductor. (Reproduced with permission from Maier J (2003) Ionic and mixed conductors for electrochemical devices. Radiation Effects and Defects in Solids 158: 1–10; & Taylor and Francis.)
14
Ionic and Mixed Conductivity in Condensed Phases
defect CddAg , do not significantly influence the mass action laws, but appear in the electroneutrality relation, here 0 ½Agdi þ ½CddAg C½VAg
½15
Coupling the above relation with the Frenkel equa0 tion ½Agdi ½VAg ¼ KF leads to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C C2 0 ½VAg þ KF ½16 ¼ þ 2 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C C2 ½Agdi ¼ þ þ KF 2 4
½17
where C stands for ½CddAg .pffiffiffiffiffiffi Consider the Brouwer conditions again. For C{ K F, the intrinsic pffiffiffiffiffiffi pffiffiffiffiffiffiresult d ½VAg ¼ ½Agdi ¼ KF is obtained. For Cc KF, one arrives at d ½VAg ¼C
½18
½Agdi ¼ KF C1
½19
which are immediately obtained by neglecting ½Agdi from the electroneutrality relation. Power-law equations are then also valid for the electronic minority carriers in AgCl. Figure 6 displays the solutions for the ionic defect arrived at by the more accurate equations for both concentrations and conductivities. The different behaviors of defect concentrations and conductivities stem from the fact that the interstitials are more mobile than the vacancies. The introduction of the dopant increases the concentration of the oppositely charged defect and depresses the concentration of the counterdefect according to electroneutrality and mass action. This leads to a decreased ionic conductivity, before the counterdefect starts defining the overall conductivity. The conductivity minimum lies approxffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi imately at KF ui =uv . Figures 7 and 8 display the dependence of conductivity on temperature in pure samples and on the doping concentration at given T with regard to positive and negative doping. While the response to Cd2 þ doping (CddAg ) follows exactly the predictions (see Figures 7 and 8), the effect of S2– doping (S0Cl ) −1
−4 Dopant concentration = 10−5
[VA′g]
−7
−8
log (Ω−1cm−1)
logc (mol cm−3)
−5
−6
1.8
2.0
2.2
2.4
2.6
i 1.6
1.8
2.0
2.2
103 T
−1(K−1)
2.4
2.6
Total
v
T = 300 K T = 300 K
−8 [VA′g]
−11 [Agi]
−10 i
−12 −14 −13 −12 −11 −10 −9 −8 −7 −6
−14 −14 −13 −12 −11 −10 −9 −8 −7 −6 log dopant concentration
−9
−11
−13
(c)
−5
−7
−10
−12
v
(b)
log (Ω−1cm−1)
logc (mol cm−3)
−9
Total −4
−7 1.6
−7 −8
−3
−6
[Agi]
103 T −1(K−1)
(a)
Dopant concentration = 10−5
−2
(d)
log dopant concentration
Figure 6 Defect concentrations and conductivities calculated as a function of temperature for a fixed Cd-content ((a) and (b)) and as a function of Cd-content for a fixed temperature ((c) and (d)).
Ionic and Mixed Conductivity in Condensed Phases 15
−R
∂ In T ∂ 1/T
0
log T (Ω−1cm−1K)
It is clear that for doped samples under Brouwer and Boltzmann conditions, the concentration of any carrier is not only a function of P, T but also of C and reads ! ! Y bj Y Npj grj Mj Pp C Kr ðTÞ ½20 cj ðT; P; CÞ ¼ aj
Agi
2
≠
∆ F H 0/2 + ∆HAgi −R
∂ In T ∂ 1/T
≠
∆H V ′Ag
p
V ′
−2
Ag
+ CdCl2 5 −4
+ MnCl2
4 3 2
Nominally pure −6
1 1.6
2.0
2.4
2.8
3.2
3.6
103T −1(K−1) Figure 7 Experimental conductivity data for nominally pure and doped AgCl as a function of 1/T. Here log (sT) is plotted instead of log s to take account of the slight T-dependence of the prefactor. However, this does not alter the slope noticeably (DF H 0 : formation enthalpy of Frenkel defects; DHja : migration enthalpy). (Reproduced from Corish J and Jacobs PWM (1972) Ionic conductivity of silver-chloride single-crystals. Journal of Physics and Chemistry of Solids 33: 1799–1818, with permission from Elsevier.)
1.5
175°C
(/pure)
1.0
225°C
325°C 325°C 275°C
0.5 1000
400 [S] (ppm)
0
400
1000
Q N (PNj is replaced by p Pp pj in order to allow for multinary equilibria labeled by p). Since unlike (the in situ parameters) P and T, the variation of C requires a new high temperature preparation, C is designated as an ex situ parameter. Now this also predicts sectionally constant slopes in diagrams of the type log Cj versus log C according to @ ln cj ¼ Mj @ ln C
½21
For a simple defect chemistry, the sign of Mj is determined by (‘‘rule of heterogeneous doping’’) zj M j o0 zd
½22
which means that, for a positive (negative) doping, the concentration of all negative (positive) defects is increased and all positive (negative) defects are decreased (zd: charge number of dopant defect). A negative doping is also the reason for change in the slope from the intrinsic value 1/6 to the extrinsic value 1/4, in Figure 5a for SnO2, which directly results from the mass action law for oxygen in1=2 dd 0 2 corporation, that is, PO2 ½VO ½e ¼ constant (for 1=4 dd 2½VO CC, one obtains sCsn p½e0 pPO2 whereas 1=6 dd for 2½VO ¼ ½e0 , the result is spPO2 ).
175°C
225°C 275°C
r
2000
[Cd] (ppm)
Figure 8 The dependence of the conductivity increase brought about by the impurities on the S and Cd content of AgBr. The RHS solid curves are calculated according to the ideal defect chemistry. (Reproduced with permission from Teltow J (1950) Zur Ionen-, Elektronenleitung und Fehlordnung von Silberbromid mit Zusa¨tzen von Silber-, Cadmium- und Bleisulfid. Zeitschrift fu¨r Physikalischs Chemie 195: 213–224; & Oldenbourg Wissenschaftsverlag and Teltow J (1949) Zur Ionenleitung und Fehlordnung von Silberbromid mit Zusa¨tzen zweiwertiger Kationen. I. Leitfa¨higkeitsmessungen und Zustandsdiagramme. Annalen der Physik 6: 63–70; & Wiley-VCH.)
suffers from interaction effects (see below, the absence of a minimum in Figure 8, LHS, is in qualitative agreement with the fact that ½Agdi is increased).
High Charge Carrier Contributions — Interactions
The above solutions are valid for ionic and electronic carrier concentrations but require low concentrations. One way of correcting interactions is to introduce associates, that is, novel particles formed as the result of these interactions and responding differently to electrical fields or other driving forces. In this way, the scheme of defects is rescaled with the consequence of writing ideal chemical potentials (i.e., interaction free) for these to a better approximation. Examples are interactions of dopants with electronic or ionic carriers (these states appear as midgap states in Figure 1), associates between electronic carriers (excitons, Cooper-pairs), or between ionic carriers (such as vacancy pairs, interstitial pairs, and Frenkel associates). As an example, the association between 0 Cdd and VAg is mentioned that leads to a thermally
16
Ionic and Mixed Conductivity in Condensed Phases
activated process at low temperatures in the case of CdCl2 doped AgCl (lower than those shown in Figure 7). The more general approach is to include interactions via analytical corrections in the chemical potentials (mex) of the charge carriers. (On the level of the concentration, this leads to the activity coefficient as a correction factor.) If the high carrier concentration is a consequence of the high temperatures, such a procedure is indispensable. If it is a consequence of a high doping content, it may be replaced by or better combined with the association procedure. Interactions lead to entropic corrections as well. One of them is obviously due to the exhaustibility of sites (Fermi–Dirac correction). The more detailed account of deviations from a random distribution owing to interactions leads to very complex expressions (e.g., by the Mayer theory for real gases). Fortunately in the case of weak interactions, the neglect of such entropic corrections is a tolerable approximation. Then, besides possible local vibrational entropies, only energetic nonideality corrections need to be considered. The Debye–Hu¨ckel theory (considering each central ion embedded into a diffusive cloud of counter ions, the size being characterized by l) leads, in the first approximation, to an expression of the form mex j ¼ RT ln fj ¼
z2j F2 8peNm l
1=2
pcj
½23
that reasonably corrects the chemical potential for low defect concentrations. The Debye–Hu¨ckel theory proved particularly helpful for electrolytic solutions. At higher concentrations, the corrections become not only complicated but also individual. Here, an ad hoc model that is able to describe interactions in some simple binary crystals and estimate these interactions by assuming that the interaction energy (viz. the structure) can be assessed by a virtual Madelungsuperlattice of defects of the mean lattice constant pc1=3 is briefly discussed. The prefactor J in mex ¼ Jc1=3 is determined by the Madelung energy of the ground lattice, the Madelung constants of lattice defect, and ground lattice as well as the dielectric constant. The attractive interaction of oppositely charged carriers has the consequence that the formation becomes increasingly easier at high temperatures as soon as the carriers perceive each other in ionic crystals. The conductivity increases in an over-Boltzmann fashion (premelting) and eventually the avalanche effect leads to a phase transformation into the superionic (or molten) state. The order of the transition can be calculated by comparing the formation enthalpy, the formation entropy, and the interaction content.
Figure 9 shows the ‘‘thermal destiny’’ of a pure Frenkel disordered binary crystal from the perfect state at low T to the superionic state at high T. Boundary Layers (Heterogeneous Doping)
At boundary layers, electroneutrality is no longer fulfilled and space charge regions (width B Debye length) occur. The reason for this is the adjustment of structurally different regions. Not only are the electronic concentrations modified, but the mobile ionic charge carriers are also distributed according to the space charge field (both ‘‘Fermi levels’’ are positionally constant). Effects on the ionic conductivity can be immense. Double layers formed by the contact of electrolyte solution to an electrode or charged membrane surfaces in electrophysiology are well-established examples. Figure 10 depicts four prototype cases involving solid electrolytes: (1) the contact of an ionic conductor to an insulator, the surface of which is able to trap (internally absorb) ions, leading to the concept of heterogeneous doping (influence of conductivity by dispersing fine insulator particles characterized by a similar rule of heterogeneous doping as given above for homogeneous doping obtained by replacing the concentration and charge of the dopant defect by the surface charge density and its sign); (2) the contact of two ionic conductors leading to a charge transfer of ions from one conductor to the other (cf. ionic ‘‘p–n junctions’’); (3) charge can also be stored within that core of the boundary, that is, in a grain boundary which joins two chemically identical grains and the effects can be tuned by chemical modification of the boundary or by varying the misfit; and (4) the contact to a fluid, for example, gas phase, is relevant in this context due to the fact that, acid–base active gases may cause attractive or repelling interactions with mobile ions. Since in many mixed, even predominantly electronic conductors, ionic charge carriers may be in majority, the ionic redistribution effect may determine the overall electronic boundary concentrations (fellow-traveler effect). Figure 11 shows the bending of ionic and electronic energy levels and their intercorrelation. The determination of the bending effect requires treatment of the chemical thermodynamics of the contact problem and hence a knowledge of energy levels in the interfacial core. In the case of a strong interfacial effect (co: carrier concentration in the first layer adjacent to the interface), Ds8m
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2eRTco ¼ 2Fu L
½24
Ionic and Mixed Conductivity in Condensed Phases 17
(a)
Excited
Perfect
Weakly defective
T=0K
0 < T<< Tc
Weakly defective ∋ = ~o
Perfect
i
i
Ov ∋ reg = ~
Interstitial
Regular
Superionic
Heavily defective
T > Tc
T
Superionic
Heavily defective ~o + RT In y e i
i
∋ i (c ) ~ o − RT In y e − v v
0
∋i
∋ reg(c)
∋ reg
T >Tc
T
(b) Figure 9 The ‘‘thermal destiny’’ of an ionic crystal above absolute zero: a progressive development from a perfect to a superionic phase via an ideally defective and a heavily defective state. (a) Crystallographic defect picture and (b) corresponding ‘‘energy level’’ schemes. The indices v and i refer to the vacant regular and interstitial sites. (Reproduced with permission from Maier J and Mu¨nch W (2000) Thermal destiny of an ionic crystal. Zeitschrift fu¨r Anorganische and Allgemeine Chemie 626: 264; & Wiley-VCH.)
describes the excess mean conductivity of the mobile majority defect in a Gouy–Chapman accumulation layer, and Ag +
Ag +
Dr> m
Heterogeneous doping (a)
Ag +
Ag +
Grain boundary engineering (c)
V−i junction (b)
NH3
Ag +
Chemical sensor (d)
Figure 10 Four basic space charge situations involving ionic conductors (here silver ion conductor): (a) contact with an insulator, (b) contact with a second ion conductor, (c) grain boundary, and (d) contact with a fluid phase. (Reproduced with permission from Maier J (2002) Nanoionics and soft materials science. In: Knauth P and Schoonman J (eds.) Nanocrystalline Metals and Oxides. Selected Properties and Applications, Kluwer International Series in Electronic Materials: Science & Technology, vol. 7, pp. 81–110. Boston: Kluwer Academic; & Kluwer Academic/ Plenum Publishers.)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 eRT ¼ 2 3 L 2zd u z cN F4 c2o lnðco =cN Þ
½25
the excess mean resistivity of the mobile majority carriers across a single Mott–Schottky depletion layer situation (dopant with molar charge zdF and concentration cN). L is the distance perpendicular to the interface over which the measurement averages. Figure 12 shows heterolayers of CaF2 and BaF2 and the effect of the concentration of interfaces on the overall parallel conductance. In a polycrystalline material, bulk and boundary conductivities have to be superimposed according to the microstructural situation. An approximate relation that takes into account the conductivities across (>) and along (8) grain boundaries (s: # complex conductivity; s# 8 ; s# > refer to the local mean values) is s# m ¼
8 s# N s# > # 8gb s> gb þ bgb jgb s gb > s# > #N gb þ bgb jgb s
½26
(b8 ; b> refer to the portions of the volume fraction j that contribute to the path; in the simplest approach, b8 ¼ 2=3, b> ¼ 1=3.)
Ionic and Mixed Conductivity in Condensed Phases
Local partial free energy
Interstitial ionic level
~0 i
≈ ≈
~ + Ag
~ − 0v
Regular ionic level
≈ ≈
Valence band
~ − p0
Mi 0 Ag ~e −
~0n
Conduction band
log c k
18
V M′ e′ h
2
1 ≡ X /
0
Figure 11 Bending of ‘‘energy levels’’ and concentration profiles in space charge zones (x ¼ 0 refers to the interfacial edge). (Reproduced from (left) Maier J (2003) Defect chemistry and ion transport in nanostructured materials. Part II. Aspects of nanoionics. Solid State Ionics 157: 327 and (right) Maier J (2002) Nano-sized mixed conductors (Aspects of nano-ionics. Part III). Solid State Ionics 148: 367–374, with permission from Elsevier.)
101 0.95 eV
T (Ω −1cm−1 K)
100
16.2 nm
20 nm
10−1
50 nm 103 nm
10−2
250 nm
10−3
430 nm 0.72 eV
10−4 CaF2
BaF2
the point defect (i.e., point defect including the sphere of influence in which the structure is perceptibly modified), then the local standard chemical potential changes (Figure 13b). In the case of delocalized electrons, this may happen at much larger distances. These are only two examples of the nanosize effects on ion conduction. Amongst others, the effects of particle geometry or curvature are mentioned, by which approximately a Gibbs–Kelvin term (pg% =r%; g% : mean surface tension, r%: mean radius) is introduced to the chemical potential. Such effects do not only affect transport properties, they are also interesting as far as mass storage is concerned.
10−5 1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
10 3 T −1 (K−1) Figure 12 Variation of the parallel ionic conductivity of CaF2– BaF2 heterolayers as a function of temperature for different periods (spacings). (Reproduced with permission from Sata N, Eberman K, Eberl K, and Maier J (2000) Mesoscopic fast ion conduction in nanometre-scale planar heterostructures. Nature 408: 946–949; & Nature Publishing Group.)
Nano-Ionics
Nano-sized conductors can provide large ionic conductivity anomalies owing to the high concentration of interfaces. Moreover, the spacing of interfaces may be so narrow that they perceive each other. Figure 13a shows the space charge overlap occurring in materials when the distance of neighboring interfaces becomes smaller than the Debye length, with the structure being invariant (cf. also Figure 12). If the distance is smaller than the effective thickness of
Partial Equilibrium – Bridge between High-Temperature and Low-Temperature Situation Arbitrary deviations from equilibrium require a detailed kinetic treatment. The situations in which some carriers behave reversibly while others are completely frozen are briefly considered here. Such considerations are helpful when the bridge has to be spanned from high-temperature equilibrium (during preparation) to a low operational temperature. In fact, such a partial equilibrium situation was already tacitly assumed when doping effects were studied. At high enough temperatures, dopants become reversible and segregation equilibria become active. Generally speaking, freezing of carriers leads to a reshuffling of in situ to ex situ parameters (i.e., from the P-term to the C-term in the power-law equation [20]). The discrepancy between the equilibrium
Ionic and Mixed Conductivity in Condensed Phases 19 + + + + + + + +
− − −− −−−−
+ + + + + − −+ + − − − − −+ + − − − − −+ + +
(a)
(>> I )
(a)
°
(b)
°
(b)
l
Figure 13 (a) Space charge effects decaying with l. (b) Structural effects which decay rapidly in the bulk. Within the indicated distance (l ); the structurally perturbed core (not shown, extension s) is perceived by the defects as far as m is concerned. (Reproduced with permission from Maier J (2003) Nano-ionics: trivial and non-trivial size effects on ion conduction in solids. Zeitschrift fu¨r physikalische chemie 217: 415–436; & Oldenbourg Wissenschaftsverlag.)
concentration (usually fulfilled at high temperatures under preparation temperatures) and the partially frozen situation becomes apparent in the application of electroceramics (e.g., high-temperature superconductors and dielectrics) that are used at room temperature. Two aspects are important: (1) a reproducible preparation and (2) a detailed understanding of defect chemistry. For a systematic treatment of these complex problems, the reader is referred to the literature.
Mobility: From Low to High Concentrations For dilute concentrations, the mobility in ionic crystals is usually taken to be independent of the concentration. According to the hopping equation (uj pkj ), it follows as 2
uj pG0j ðDxÞ exp þ
DSa j R
! exp
DHja RT
! ½27
(c) Figure 14 Elementary jump mechanisms in crystals: (a) vacancy mechanism, (b) direct interstitial mechanism, and (c) (collinear or noncollinear), indirect interstitial mechanism (interstitialcy mechanism).
(G0 is the attempt frequency (cf. k0, i.e., the prefactor of k), Dx is the hopping distance; DH a and DSa denote activation enthalpy and entropy, respectively.) Figure 14 displays three fundamental hopping mechanisms. While the treatment of the ionic carrier concentrations is, in essence, very similar for liquids or solids, this is less so in the case of mobility. In solids, the activation thresholds of hopping transport are usually substantial (approximately several 100 meV), while in dilute liquid electrolytes, temperature effects act on carrier mobilities mainly through changes in the viscosity. In superionic solid conductors, activation thresholds are significantly less. At high concentrations, the jump partners in the hopping processes are important which formally leads to the appearance of a concentration factor (1 ðc=cmax Þ) in the mobility. Higher concentrations, however, affect carrier transport in such an individual way that no universal relation can be given and even the decomposition into charge carrier concentration and mobility becomes questionable. For not too strong interactions, a fairly general interaction effect is to be considered. Defect interactions lead to relaxation phenomena in that a particle
20
Ionic and Mixed Conductivity in Condensed Phases
that has made a jump from x to x0 does not directly create its equilibrium environment and has a higher tendency to jump back again. It is, in particular, the rearrangement of the other defects that determines the finite relaxation time. This mismatch situation causes back-jumps to be more likely, making the rate constants of the hopping equation time dependent and hence the conductivity frequency dependent. In liquids too, a relaxation effect occurs. Since the formation of the ion cloud does not immediately follow the mobile ion under regard, the charge cloud becomes asymmetrical and acts as a brake. The relaxation time is proportional to the viscosity. The occurrence of counter transport of differently charged ions which carry along these solvation spheres leads to friction, that is, to a second effect that diminishes the mobility, the so-called electrophoretic pffiffiffi effect. These phenomena explain the Kohlrausch’s c law, which is well established for strong electrolytes, and reads " # pffiffiffi A B Lc ¼ Lo c ½28 L þ 3=2 o 1=2 ðeTÞ ðeTÞ (A and B are constants, Lo Lc (c ¼ 0), e the dielectric constant, and Lc the equivalent conductivity, that is, conductivity per equivalent salt concentration.) Figure 15 gives some examples of the above phenomena.
150 KCl
140 1/3 LaCl3 AgNO3
Λ c /Λ*c
130
1/2 LaCl2 120 1/2 Na2So4 KIO3
110 1/2 NiSO4 0
2
4
1/2 MgSO4 6
8
Chemical Diffusion in Mixed Conductors The simultaneous presence of ionic and electronic carriers in so-called mixed conductors leads to the phenomenon of chemical diffusion (coupled transport of ions and electrons) and the possibility of changing stoichiometry. Technologically important examples for which this phenomenon plays a decisive role are insertion electrodes, electrochemical windows, permeation membranes, and conductivity sensors; more generally, chemical diffusion is important for all electroceramics, the properties of which depend on the detailed stoichiometry (such as in dielectrics or hightemperature superconductors). Stoichiometry changes also occur, if the electrodes applied in the electrochemical measurements are not completely reversible for both ions and electrons; in this case, the partial current density is given by s j ij ¼ ½29 i þ zj FDd rcj s Dd being the chemical diffusion coefficient that is determined by mobilities and concentrations of all charge carriers involved in the transport (i: total current density). If different internal reactions such as valence state changes occur for the conducting species, the equation has to be modified. The use of selectively blocking electrodes leads to the suppression of the current of one carrier type at the expense of a concentration polarization. This allows the separation of ionic and electronic conductivities and of the chemical diffusion coefficient. A major application of ion conductors is their use as electrolytes which nullifies gradients in the electrochemical potential of ions but supports gradients in the electrochemical potential of electrons. This gives rise to a cell voltage (e.m.f.) which is consequently proportional to the integrated gradient of the neutral component (cf. distance in Figure 2) and corresponds to the Nernst equation. In other words, differences in the chemical component potentials can be transformed into an electrical voltage (such as exploited in batteries, fuel cells, and sensors). In mixed conductors, a partial internal short-circuit occurs, causing a nonzero gradient in the electrochemical potential of the ions, which is proportional to the gradient in the electronic Fermi level. Hence, the cell voltage is diminished by a factor that depends on the proportion of electronic conductivity and consequently allows for its determination.
c /c* Figure 15 Equivalent conductivity for a variety of aqueous salt solutions vs. concentration (Lc and c denote unit values). (Reproduced with permission from Kortu¨m G (1972) Lehrbuch der Elektrochemie, 5th edn. Weinheim: VCH; & Wiley-VCH.)
See also: Conductivity, Electrical.
PACS: 72.60. þ g
Ionic Bonding and Crystals 21
Further Reading Allnatt AR and Lidiard AB (1993) Atomic Transport in Solids. Cambridge: Cambridge University Press. Funke K (1993) Jump relaxation in solid electrolytes. Progress in Solid State Chemistry 22: 111. Heifets E, Kotomin EA, and Maier J (2000) Semi-empirical simulations of surface relaxation for perovskite titanates. Surface Science 462: 19. Kirchheim R (1988) Hydrogen solubility and diffusivity in defective and amorphous metals. Progress in Materials Science 32: 261. Kortu¨m G (1965) Treatise on Electrochemistry, 2nd edn. Amsterdam: Elsevier. Kro¨ger FA (1964) Chemistry of Imperfect Crystals. Amsterdam: North-Holland. Lidiard AB (1957) Ionic conductivity. In: Flu¨gge S (ed.) Handbuch der Physik., vol. 20, p. 246. Berlin: Springer. Maier J (1993) Defect chemistry: composition, transport, and reactions in the solid state. Part I: Thermodynamics.
Angewandte Chemie (International Edition in English) 32: 313–528. Maier J (1995) Ionic conduction in space charge regions. Progress in Solid State Chemistry 23: 171. Maier J (2004) Physical Chemistry of Materials – Ions and Electrons in Solids. Chichester: Wiley. (Translation of Festko¨rper – Fehler und Funktion: Prinzipien der Physikalischen Festko¨rperchemie Teubner BG, Stuttgart). Mott NF (1974) Metal-Insulator Transitions. London: Taylor & Francis. Rickert H (1982) Electrochemistry of Solids, 1st edn. Berlin: Springer. Sasaki K and Maier J (1999) Low-temperature defect chemistry of oxides. I. General aspects and numerical calculations; II. Analytical reasons. Journal of Applied Physics 86: 5422–5443. Schmalzried H (1981) Solid State Reactions, 1st edn. Weinheim: VCH. Schmalzried H (1995) Chemical Kinetics of Solids, 1st edn. Weinheim: VCH.
Ionic Bonding and Crystals* M W Finnis, Queen’s University Belfast, Belfast, UK Published by Elsevier Ltd.
Introduction One can distinguish the models of ionic materials by their increasing degrees of sophistication, which are described briefly below. All contain parameters of one sort or another that have to be fitted either to experimental data or to the best available first-principles calculations. Following this introduction there is a more detailed discussion of how ionic models, in general, can all be related to the density-functional theory within the second-order perturbation theory. The Born model, the shell model, and the compressible ion model are three well-known types of increasing complexity. More recently, they have been joined by models including ionic polarizability up to the quadrupole level. The simplest ionic models are typified by constant nominal ionic charges. These are usually integers, although fractional charges have been used as additional fitting parameters. Variable charge models have also been developed, which allow charge transfer within a simple classical description of the equilibration of an electronic chemical potential. Finally, a class of models has been developed that includes most of the aforementioned as subclasses and also treats covalency, namely, the self-consistent tightbinding models. The scope of ionic models, especially when charge transfer can be included, is potentially *Adapted by permission of Oxford University Press.
very wide. Although so far they have been less developed and applied, self-consistent tight-binding models have potentially the widest scope, which embrace traditional textbook ionic materials such as NaCl, KF and all the other alkali halides, the insulating oxides with wide bandgaps such as MgO and Al2O3, perovskites such as SrTiO3, as well as oxides which are considered as less ionic, such as TiO2 and ZrO2. The simplest ionic model is the rigid ion model. It is sometimes called the Born model after its pioneer Max Born and his co-workers in Edinburgh, who over 50 years ago developed most of the mathematical description of the ionic model that is still being used (see Further reading section for more details). Within the context of the density-functional perturbation theory, one can think of it as a first-order model, as seen in the following section. The ions interact by the Coulomb interaction, in addition to which there is a short-range pairwise repulsion. Ions are not rigid, and the shell model was first developed by Dick and Overhauser in 1958 to include their polarizability. In its simplest version, a spherical shell of charge qs surrounds an ion of reduced charge DZ þ qs and is attached to it by a harmonic spring. The nominal ionic charge is DZ. The repulsion between ions is now modeled as a repulsion between the centers of these shells. The compressible ion model is an extension of the shell model in which the radius of the shells is a variable. The energy of a compressible ion is a function of this radius, which adjusts according to the environment of the ion. This model is most strongly motivated by the case of oxygen, as discussed further
based on the ‘‘use’’ of ion-channel proteins as molecular tools for therapy of cardiac rhythm alteration. To date, cardiac electronic pacemakers have been widely used in the treatment of rhythm disorders. However, recent research suggests the possibility of developing therapeutic approaches based on ‘‘biological pacemakers’’ in which the ability of generating spontaneous cardiac electric activity could be restored by manipulating the level of ion channels expressed in a specific desired region of the heart. See also: Biomembranes; Membrane and Protein Aggregates.
PACS: 87.16.Uv; 87.19.Nn; 87.80.Jg; 87.17.Nn; 87. 64. Gb; 87.16.Dg Further Reading Ashcroft FM (2000) Ion Channels and Disease. San Diego: Academic Press. Armstrong CM (1975) Ionic pores, gates, and gating currents. Quarterly Reviews of Biophysics 7: 179–210. Colquhoun D and Hawkes AG (1981) On the Stochastic Properties of Single Ion Channels. Proceedings of the Royal Society of London. Series B 211: 205–235. DiFrancesco D (1993) Pacemaker mechanisms in cardiac tissue. Annual Review of Physiology 55: 455–472.
Doyle DA, Morais Cabral J, Pfuetzner RA, Kuo A, Gulbis JM, Cohen SL, Chait BT, and MacKinnon R (1998) The structure of the potassium channel: molecular basis of K þ conduction and selectivity. Science 280: 69–77. Hamill OP, Marty A, Neher E, Sakmann B, and Sigworth FJ (1981) Improved patch-clamp techniques for high-resolution current recording from cells and cell-free membrane patches. Pflu¨gers Archive: European Journal of Physiology 391: 85–100. Hille B (2001) Ion Channels of Excitable Membranes, 3rd edn. Sunderland, Massachussets, USA: Sinauer Associates, Inc. Hille B, Armstrong CM, and MacKinnon R (1999) Ion channels: from idea to reality. Nature Medicine 10: 1105–1109. Hodgkin AL and Huxley AF (1952) A quantitative description of membrane currents and its application to conduction and excitation in nerve. Journal of Physiology, London 117: 500–544. Hodgkin AL and Keynes RD (1955) The potassium permeability of a giant nerve fibre. Journal of Physiology, London 128: 61–88. Jiang Y, Lee A, Chen J, Ruta V, Cadene M, Chait BT, and MacKinnon R (2003) X-ray structure of a voltage-dependent K þ channel. Nature 423: 33–41. MacKinnon R (2003) Potassium channels. FEBS Letters 555: 62–65. Miller C (1985) Ion Channel Reconstitution. New York: Plenum Press. Neher E and Sakmann B (1992) The patch clamp technique. Scientific American 266: 44–51. Sakmann B and Neher E (1995) Single Channel Recording. New York: Plenum. Yellen G (1998) The moving parts of a voltage-gated ion channel. Quarterly Reviews of Biophysics 31: 239–295.
Ionic and Mixed Conductivity in Condensed Phases J Maier, Max-Planck-Institut fu¨r Festko¨rperforschung, Stuttgart, Germany & 2005, Elsevier Ltd. All Rights Reserved.
Introduction: Hopping Process While a quantum mechanical particle such as a quasifree-electron can tunnel through a barrier by interference of the states on both sides of the barrier (denoted by x and x0 ), that is, by seizing the hindrance in a wave-like way, the transport of a heavy particle such as an ion has to rely on the ensemble fraction with energy high enough to reach the final state (hopping). On the one hand, under favorable circumstances protons may also tunnel, while on the other hand electrons that strongly polarize their environment (large polarons) behave almost classically as ions. In most cases of interest (consider A as an ion with charge zA), the hopping of A from x to x0 ðx0 ¼ x þ DxÞ may be written as AðxÞ þ V 0 ðx0 Þ"VðxÞ þ A0 ðx0 Þ
½1
(The dash takes account of the possibility for A to have a different structural environment at x0 .) In cases where A denotes an ion on a regular site in a crystal, V is the vacancy that serves as a jump partner, while in cases where A refers to an excess ion sitting on interstitial sites, V denotes the vacant interstitial site. In crystals, V (in the first case) or A (in the second case) are usually dilute with the energy levels of these sites being well defined. The concentration of the respective reaction partner is then approximately constant and the corresponding rate equation for the hopping process is pseudo-monomolecular. In amorphous solids, the sites are less defined, which also holds for the spatial energy distribution; in polymers, melts, or liquids, these sites and energy distributions are time dependent. In these cases, one would better speak of smeared-out density of states, as far as thermal excitations are concerned. The rate constant for the hopping rate equation is decisively influenced by the activation free enthalpy. The parameters k and DGa , for example, for the forward reaction (index ,) related through (prefactor k0 is proportional to the attempt
10
Ionic and Mixed Conductivity in Condensed Phases
frequency G0) ,
,
k ¼ k0 exp
D Ga RT
! ½2
refer to the built-in part of the forward rate constant and its activation free energy, respectively. This expression has to be complemented by the factor , expða Df z F=RTÞ in order to take account of the applied electric field. The parameter Df is that part of the external voltage that drops from x to x0 and , the term a Df measures the portion of Df that is relevant for the forward reaction (distance between saddle point and initial state). The relations for the , , backward reaction are analogous ð a þ a ¼ 1Þ. This leads to the Butler–Volmer equation which describes the charge transfer from x to x0 for an asymmetric free-energy profile. In the homogeneous bulk region (and also in the space charge region sufficiently far from the interface, i.e.,,A ¼&A0 ), the built-in profile is , & symmetrical, that is, k ¼ k ¼ k, a ¼ 1=2 ¼ a and , & DG ¼ DG. The result for the rate represented by the current density ij of carrier j in x-direction ij ¼ 2kj cj zj F sinh C sj
zj FDf 2RT
Df Dx
½3
can be simplified to Ohm’s law (bottom part) for Df{RT=zj F; the conductivity sj of the carrier is proportional to its mobility uj (uj pkj ), (molar) concentration (cj) and (molar) charge (zjF) and given as sj ¼ zj Fuj cj
½4
Allowing also for a concentration variation during the process and generalizing to three dimensions, the Nernst–Planck equation is arrived at ij ¼ Dj zj Frcj sj rf
½5
where the diffusion coefficient (D) and conductivity (s) are related through the Nernst–Einstein equation. Close to equilibrium, this equation corresponds to the result obtained by linear irreversible thermodynamics, sj ij ¼ rm* j ½6 zj F which assumes the current to be proportional to the gradient in the electrochemical potential m* ¼ m þ zFf (chemical potential þ (molar charge electrical potential)). The latter relation is more general concerning the concentration range, while the former (as regards the diffusion term) is more general as far as
the magnitude of this driving force is concerned. In cases where the current of a given species is also generated by secondary driving forces (e.g., by rm* of a different particle), cross terms occur for which the Onsager–Casimir symmetry relations are valid. As regards the determination of the conductivity, the first task is to determine the carrier concentration as a function of the control parameters. This is feasible by solving the defect model for the system under consideration.
Equilibrium Charge Carrier Concentrations Pure Compounds and Dilute Bulk
Figure 1 displays the fundamental charge carrier formation reaction in an ‘‘energy level’’ diagram for a fluid (H2O) and an ionic crystal (AgCl). These levels refer to effective standard (electro) chemical potentials or – in the case of the ‘‘Fermi levels’’ that are positioned within the gap – full (electro) chemical potentials. (As long as the bulk is considered, zFf can be neglected and one can refer to m instead of m.) * As long as – in pure materials – the gap remains large compared to RT, the Boltzmann form of the chemical potential of the respective charge carrier (defect) is valid which has the form ! c j mj ¼ mj þ RT ln ½7 cj As outlined below, this relation holds largely independent of the charge carrier situation and the form of the energy level distribution. If the levels are smeared out or if there is a band of levels, mj refers to an effective level (i.e., band edges in nondegenerate semiconductors or an appropriately averaged energy when dealing with fluids or disordered solids). The coupling of the ionic level picture and the electronic level picture for AgCl (Figure 2) is established via the thermodynamic relation m* Agþ þ m* e ¼ mAg , and hence via the component potential which reflects the precise position in the phase diagram. The meaning of m becomes obvious, if for simplicity one considers an elemental crystal (with defect j). For the formation of Nj identical defects (among N regular positions), a local free enthalpy of Nj Dgj is required in the interaction-free case; including also the configurational contribution, the Gibbs energy of the defective crystal (GP refers to the perfect crystal) reads N G ¼ GP þ Nj Dgj kB T ln ½8 Nj
Ionic and Mixed Conductivity in Condensed Phases 11
AgCl
H2O
Ionic disorder H+ (in H3O+)
Null H2O
~ −H0+ − vacancy
H+ + |H|− H+ + OH−
e −(CB)
Ag+ (interst.) ~ 0 Agi
~ 0+ H − excess
~ += ~ Ag Agi ~ = −
~H+ = ~H+ − excess ~ + = − H − vacancy H+ (in H2O)
Electronic disorder
~e − = ~e ′ = ∋ F ~ = − h
V ′Ag
Ag+ (regular)
Null AgAg + V i
~0 V′Ag
ee−_(VB) (VB)
Ag + |Ag|′ Agi + V Ag ′
~ =∋ * e′ C
Null e VB + V CB
~* = ∋ − h V
e′ + h e′CB + V VB
~i ≡ i + zi F = 0i + RT In [i ] + ziF =~ 0i + RT In [i ]
Regular ionic level
Ag
~e −
Conduction band
Figure 2 Coupling of ionic and electronic levels in AgCl.
The configuration term describes the number of combinations of Nj elements out of N choices without repetition. The calculation of Dg , the free energy to form a single defect, requires an atomistic consideration. If one refers to ionic defects in crystals, it is essentially composed of lattice energy, polarization energy, and vibrational entropy terms. In more general terms, Nj is the number of defects and N the number of particles that can be made defective. Owing to the infinitely steep decrease of the configurational term with Nj, a minimum in G(Nj) (see Figure 3) is observed in the elemental crystal, if the chemical potential of j, that is, mj @G=@ðNj =Nm Þ, which follows as Nj mj ¼ mj þ RT ln N Nj Nj C mj þ RT ln N nj ¼ mj þ RT ln n
½9
vanishes (n N=Nm , nj Nj =Nm , Nm ¼ Avogadro’s number; mj ¼ Nm Dgj ). It is noteworthy that the
∆d G* = N d ∆d g* Contributions to the free enthalpy
Valence band
Interstitial ionic level ~ Ag+
Partial free energy of e −
Partial free energy of Ag +
Figure 1 Electronic and ionic disorder in ionic solids and water in ‘‘physical’’ (top) and ‘‘chemical’’ language (bottom).
G real − Gperfect 0 −TS cfg
^
Nd
Number of point defects (N = const)
Figure 3 Contributions to the free enthalpy of the solid by defect formation with a constant total number of sites.
strict result (top) which is formally valid for higher concentrations also is of the Fermi–Dirac type. This is due to the fact that one refers to combinations without repetition, that is, double occupancy is excluded and hence the sites are exhaustible similarly as is the case for quantum states in electronic problems. (Combinations with repetitions lead to Bose–Einstein distribution.) In a situation in which the levels are broadened to a more or less continuous zone (e.g., electrons in semiconductors and ions in fluids), the parameters Nj and mj of the Fermi–Dirac form have to be attributed to an infinitely small level interval. Denoting the partial free energy level by E, this integral ranges from Ej to Ej þ dEj and the total (molar) concentration
Ionic and Mixed Conductivity in Condensed Phases
follows from integration: DðEj Þ dEj nj ¼ 1 þ expðjE j mj j=RTÞ zone
½10
whereby the molar density of states D (Ej ) appears in the integrand. At equilibrium, mj is independent of Ej . Considering without the restriction of generality, the almost empty zone and the lower edge of this zone designated as Ej , one can neglect the unity in the denominator of the above equation for all levels more 0 distant from mj than Ej which leads to the Boltzmann expression nj Cn% j ðTÞexpðjE0j mj j=RTÞ. The effective value n% j is given by n% j ðTÞ ¼
jEj E0j j dEj DðEj Þexp RT zone
Z
Y
b
cj ðT; PÞ ¼ aj j PNj
½11
n% j being usually weakly temperature dependent. It is worth noting that, owing to the high dilution, changes of the parameters with occupation have been ignored (for electrons in semiconductors, this is called the ‘‘rigid band model’’). As anticipated above, the Boltzmann form of the chemical potential results with m representing an effective value. If n is identified with n% j , the effective value mj is given by E0j. As long as one refers to dilute conditions, considerable freedom of normalization is left with respect to the concentration measure. In ionic compounds, charged defects occur pairwise, the formation of which is subject to disorder reactions (and electroneutrality conditions). Then, only the reaction P combination of chemicalP potential vanishes,Pthat is, j nrj mj ¼ 0 (identical to j nrj m* j ¼ 0 since j nrj zj ¼ 0), where nrj is the stoichiometric coefficient of j in the respective disorder reaction r (cf. Figure 1, bottom). Boltzmann expressions can be written for the individual carriers, if the formation energies and the configuration entropies are considered to be independent. The application of chemical thermodynamics elegantly permits treatment of a variety of ionic and electronic disorder equations that simultaneously occur in a given solid. This is not only possible for the internal disorder equations (such as Schottky, Frenkel, anti-Frenkel, anti-Schottky, electron–hole formation) but also for external reactions such as the interaction of an oxide with the oxygen partial pressure and hence the stoichiometric variation. Under Boltzmann (i.e., dilute) and Brouwer (i.e., two majority carriers) conditions, an adequate expression is arrived at for any charge carrier concentration as a function of the control parameters (temperature T, doping content, component partial pressure P),
Kr ðTÞgrj
½12
r
(Nj ; grj ; aj ; bj being simple rational numbers; the total pressure is assumed to be constant.) For multinary compounds (i.e., m components) at complete equilibrium (m 1), component partial pressures are to be considered. This solution is only a sectional solution within a window in which the majority carriers situation (Brouwer condition) does not change. The power-law equation defines van’t Hoff diagrams characterized by X @ ln cj ¼ grj Dr H ½13 @1=RT r according to which, for not too extended T-ranges (i.e., Dr H , the reaction enthalpy of the defect reaction g, is constant), straight lines are observed in the ln cj versus 1/T representation, as well as Kro¨ger–Vink diagrams characterized by straight lines in the plots ln cj versus ln P with slopes @ ln cj ¼ Nj @ ln P
½14
Figure 4 displays the defect chemistry within the phase width of the oxide MO. At low oxygen pardd tial pressure (PO2 ), vacant oxygen ions ðVO Þ and
VO e′
log (|z| [defect z])
Z
namely
(±1/6)
(O) VO , Oi″ (±1/4)
Oi″
h
e′
N
I
Oi″ h
(±1/6) VO P
P(i )
log PO
2
= [Oi″ ] − [VO ]
T
12
[h ] − [e′] 2
M
MO
O
PO2 Figure 4 Internal redox and acid–base chemistry in a solid MO1þd within the phase width at a given temperature T (disorder in the M-sublattice assumed to be negligible).
Ionic and Mixed Conductivity in Condensed Phases 13 dd reduced states ðe0 Þ are in the majority (2½VO C½e0 c 00 d ½Oi ; ½h ) (N-regime) whilst oxygen interstitials (O00i ) and oxidized states ðhd Þ dominate for very high PO2 dd (2½O00i C½hd c½e0 ; ½VO ) (P-regime). At intermediate dd PO2 , usually ionic disorder prevails (½VO C 00 0 d ½Oi c½e ; ½h ) (I-regime); the alternative situation that electronic disorder predominates (½e0 C½hd c dd ½VO ; ½O00i ) is usually scarce. Exactly at the Dalton ðiÞ dd composition (i.e., at PO2 where 2½O00i ¼ 2½VO ¼ 0 d ½e ¼ ½h ), the ‘‘nonstoichiometry’’ d in MO1þd (i.e., dd ½O00i ½VO ¼ 12½hd 12½e0 ) is precisely zero. For . dc0, a p-conductor (mobility of h is usually much 00 greater than for Oi ) is referred to, for d{0, one refers (mobility of e0 is usually much greater than for dd VO ) to an n-conductor, while at dC0, mixed conduction is expected (only if the ionic concentrations are much larger than the electronic ones, pure ion conduction results). Usually, the entire diagram is not observed, as there are limits of experimentally realizing extreme PO2 values as well as limits with respect to the phase stability (formation of higher oxides or lower oxides, if not of the metal). Figure 5 gives three examples: SnO2 as an oxygendeficient material, La2CuO4 as an example of a
p-type conduction, and PbO as an example of I-regime exhibiting mixed conduction. (In PbO, most probably, the counterdefect to O00i is Pbdd i rather than dd VO ; the results, however, are not different then.) The slopes directly follow from simple mass action considerations. The model example of AgCl shall be considered in more detail. It also exhibits predominant ionic disorder, but now the decisive disorder reaction is the Frenkel reaction of the Ag sublattice (i.e., the for0 mation of silver ion vacancy, VAg , and interstitial d silver ion, Agi ). The role of PO2 is played by the chlorine partial pressure. The mass action law for the þ1=2 interaction with Cl2 reads ½Agdi PCl2 ½e0 ¼ constant, the mass action constant for the Frenkel reaction 0 ¼ KF and that for the band–band being ½Agdi ½VAg . 0 d 0 transfer ½e ½h ¼ KB. Since ½Agdi C½VAg c½e0 , ½h , it p ffiffiffiffiffiffi 1=2 0 follows as solution ½VAg ¼ ½Agdi ¼ KF , ½e0 pPCl2 , þ1=2 ½hd pPCl2 . Doping Effects
Dopants, as irreversibly introduced structure elements such as Cd2 þ substituting Ag þ to form the
I N
P
718°C
−2.0
745°C
−2.4 −2.8
653°C
ion
4.0
597°C
2 4 log(PO2)(Pa)
6
log PO
−3.6
(c)
2
−1.4 (a)
550°C
4.5 0
−3.2
800°C
−1.0 −0.6 −0.2
VO
log(PO2) (bar) SnO 2−
e′
(±1/6)
(O) VO , Oi″ (±1/4)
Oi″ N
1
1
eon
−1/4
log (|z| [defectz])
log (Ω −1 cm−1)
log
PbO (orh)
3.5
h
e′
I
0
Slope:
0
1/6
−1
−1
−2
−2 −4
−2 log(PO2) (bar)
Normalized thermopower
−1.6
−1/6
log (Ω −1 cm−1)
818°C
−log (Ω −1 cm−1)
−1.2
0
Oi″ h
(±1/6)
La2CuO4+
VO P
P(i )
(b)
logPO2
Figure 5 Three experimental examples of Kro¨ger–Vink diagrams in a pure oxide MO with ideal defect chemistry. (a) SnO2 as an n-type conductor, (b) PbO as a mixed conductor, and (c) La2CuO4 as a p-type conductor. (Reproduced with permission from Maier J (2003) Ionic and mixed conductors for electrochemical devices. Radiation Effects and Defects in Solids 158: 1–10; & Taylor and Francis.)
14
Ionic and Mixed Conductivity in Condensed Phases
defect CddAg , do not significantly influence the mass action laws, but appear in the electroneutrality relation, here 0 ½Agdi þ ½CddAg C½VAg
½15
Coupling the above relation with the Frenkel equa0 tion ½Agdi ½VAg ¼ KF leads to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C C2 0 ½VAg þ KF ½16 ¼ þ 2 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C C2 ½Agdi ¼ þ þ KF 2 4
½17
where C stands for ½CddAg .pffiffiffiffiffiffi Consider the Brouwer conditions again. For C{ K F, the intrinsic pffiffiffiffiffiffi pffiffiffiffiffiffiresult d ½VAg ¼ ½Agdi ¼ KF is obtained. For Cc KF, one arrives at d ½VAg ¼C
½18
½Agdi ¼ KF C1
½19
which are immediately obtained by neglecting ½Agdi from the electroneutrality relation. Power-law equations are then also valid for the electronic minority carriers in AgCl. Figure 6 displays the solutions for the ionic defect arrived at by the more accurate equations for both concentrations and conductivities. The different behaviors of defect concentrations and conductivities stem from the fact that the interstitials are more mobile than the vacancies. The introduction of the dopant increases the concentration of the oppositely charged defect and depresses the concentration of the counterdefect according to electroneutrality and mass action. This leads to a decreased ionic conductivity, before the counterdefect starts defining the overall conductivity. The conductivity minimum lies approxffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi imately at KF ui =uv . Figures 7 and 8 display the dependence of conductivity on temperature in pure samples and on the doping concentration at given T with regard to positive and negative doping. While the response to Cd2 þ doping (CddAg ) follows exactly the predictions (see Figures 7 and 8), the effect of S2– doping (S0Cl ) −1
−4 Dopant concentration = 10−5
[VA′g]
−7
−8
log (Ω−1cm−1)
logc (mol cm−3)
−5
−6
1.8
2.0
2.2
2.4
2.6
i 1.6
1.8
2.0
2.2
103 T
−1(K−1)
2.4
2.6
Total
v
T = 300 K T = 300 K
−8 [VA′g]
−11 [Agi]
−10 i
−12 −14 −13 −12 −11 −10 −9 −8 −7 −6
−14 −14 −13 −12 −11 −10 −9 −8 −7 −6 log dopant concentration
−9
−11
−13
(c)
−5
−7
−10
−12
v
(b)
log (Ω−1cm−1)
logc (mol cm−3)
−9
Total −4
−7 1.6
−7 −8
−3
−6
[Agi]
103 T −1(K−1)
(a)
Dopant concentration = 10−5
−2
(d)
log dopant concentration
Figure 6 Defect concentrations and conductivities calculated as a function of temperature for a fixed Cd-content ((a) and (b)) and as a function of Cd-content for a fixed temperature ((c) and (d)).
Ionic and Mixed Conductivity in Condensed Phases 15
−R
∂ In T ∂ 1/T
0
log T (Ω−1cm−1K)
It is clear that for doped samples under Brouwer and Boltzmann conditions, the concentration of any carrier is not only a function of P, T but also of C and reads ! ! Y bj Y Npj grj Mj Pp C Kr ðTÞ ½20 cj ðT; P; CÞ ¼ aj
Agi
2
≠
∆ F H 0/2 + ∆HAgi −R
∂ In T ∂ 1/T
≠
∆H V ′Ag
p
V ′
−2
Ag
+ CdCl2 5 −4
+ MnCl2
4 3 2
Nominally pure −6
1 1.6
2.0
2.4
2.8
3.2
3.6
103T −1(K−1) Figure 7 Experimental conductivity data for nominally pure and doped AgCl as a function of 1/T. Here log (sT) is plotted instead of log s to take account of the slight T-dependence of the prefactor. However, this does not alter the slope noticeably (DF H 0 : formation enthalpy of Frenkel defects; DHja : migration enthalpy). (Reproduced from Corish J and Jacobs PWM (1972) Ionic conductivity of silver-chloride single-crystals. Journal of Physics and Chemistry of Solids 33: 1799–1818, with permission from Elsevier.)
1.5
175°C
(/pure)
1.0
225°C
325°C 325°C 275°C
0.5 1000
400 [S] (ppm)
0
400
1000
Q N (PNj is replaced by p Pp pj in order to allow for multinary equilibria labeled by p). Since unlike (the in situ parameters) P and T, the variation of C requires a new high temperature preparation, C is designated as an ex situ parameter. Now this also predicts sectionally constant slopes in diagrams of the type log Cj versus log C according to @ ln cj ¼ Mj @ ln C
½21
For a simple defect chemistry, the sign of Mj is determined by (‘‘rule of heterogeneous doping’’) zj M j o0 zd
½22
which means that, for a positive (negative) doping, the concentration of all negative (positive) defects is increased and all positive (negative) defects are decreased (zd: charge number of dopant defect). A negative doping is also the reason for change in the slope from the intrinsic value 1/6 to the extrinsic value 1/4, in Figure 5a for SnO2, which directly results from the mass action law for oxygen in1=2 dd 0 2 corporation, that is, PO2 ½VO ½e ¼ constant (for 1=4 dd 2½VO CC, one obtains sCsn p½e0 pPO2 whereas 1=6 dd for 2½VO ¼ ½e0 , the result is spPO2 ).
175°C
225°C 275°C
r
2000
[Cd] (ppm)
Figure 8 The dependence of the conductivity increase brought about by the impurities on the S and Cd content of AgBr. The RHS solid curves are calculated according to the ideal defect chemistry. (Reproduced with permission from Teltow J (1950) Zur Ionen-, Elektronenleitung und Fehlordnung von Silberbromid mit Zusa¨tzen von Silber-, Cadmium- und Bleisulfid. Zeitschrift fu¨r Physikalischs Chemie 195: 213–224; & Oldenbourg Wissenschaftsverlag and Teltow J (1949) Zur Ionenleitung und Fehlordnung von Silberbromid mit Zusa¨tzen zweiwertiger Kationen. I. Leitfa¨higkeitsmessungen und Zustandsdiagramme. Annalen der Physik 6: 63–70; & Wiley-VCH.)
suffers from interaction effects (see below, the absence of a minimum in Figure 8, LHS, is in qualitative agreement with the fact that ½Agdi is increased).
High Charge Carrier Contributions — Interactions
The above solutions are valid for ionic and electronic carrier concentrations but require low concentrations. One way of correcting interactions is to introduce associates, that is, novel particles formed as the result of these interactions and responding differently to electrical fields or other driving forces. In this way, the scheme of defects is rescaled with the consequence of writing ideal chemical potentials (i.e., interaction free) for these to a better approximation. Examples are interactions of dopants with electronic or ionic carriers (these states appear as midgap states in Figure 1), associates between electronic carriers (excitons, Cooper-pairs), or between ionic carriers (such as vacancy pairs, interstitial pairs, and Frenkel associates). As an example, the association between 0 Cdd and VAg is mentioned that leads to a thermally
16
Ionic and Mixed Conductivity in Condensed Phases
activated process at low temperatures in the case of CdCl2 doped AgCl (lower than those shown in Figure 7). The more general approach is to include interactions via analytical corrections in the chemical potentials (mex) of the charge carriers. (On the level of the concentration, this leads to the activity coefficient as a correction factor.) If the high carrier concentration is a consequence of the high temperatures, such a procedure is indispensable. If it is a consequence of a high doping content, it may be replaced by or better combined with the association procedure. Interactions lead to entropic corrections as well. One of them is obviously due to the exhaustibility of sites (Fermi–Dirac correction). The more detailed account of deviations from a random distribution owing to interactions leads to very complex expressions (e.g., by the Mayer theory for real gases). Fortunately in the case of weak interactions, the neglect of such entropic corrections is a tolerable approximation. Then, besides possible local vibrational entropies, only energetic nonideality corrections need to be considered. The Debye–Hu¨ckel theory (considering each central ion embedded into a diffusive cloud of counter ions, the size being characterized by l) leads, in the first approximation, to an expression of the form mex j ¼ RT ln fj ¼
z2j F2 8peNm l
1=2
pcj
½23
that reasonably corrects the chemical potential for low defect concentrations. The Debye–Hu¨ckel theory proved particularly helpful for electrolytic solutions. At higher concentrations, the corrections become not only complicated but also individual. Here, an ad hoc model that is able to describe interactions in some simple binary crystals and estimate these interactions by assuming that the interaction energy (viz. the structure) can be assessed by a virtual Madelungsuperlattice of defects of the mean lattice constant pc1=3 is briefly discussed. The prefactor J in mex ¼ Jc1=3 is determined by the Madelung energy of the ground lattice, the Madelung constants of lattice defect, and ground lattice as well as the dielectric constant. The attractive interaction of oppositely charged carriers has the consequence that the formation becomes increasingly easier at high temperatures as soon as the carriers perceive each other in ionic crystals. The conductivity increases in an over-Boltzmann fashion (premelting) and eventually the avalanche effect leads to a phase transformation into the superionic (or molten) state. The order of the transition can be calculated by comparing the formation enthalpy, the formation entropy, and the interaction content.
Figure 9 shows the ‘‘thermal destiny’’ of a pure Frenkel disordered binary crystal from the perfect state at low T to the superionic state at high T. Boundary Layers (Heterogeneous Doping)
At boundary layers, electroneutrality is no longer fulfilled and space charge regions (width B Debye length) occur. The reason for this is the adjustment of structurally different regions. Not only are the electronic concentrations modified, but the mobile ionic charge carriers are also distributed according to the space charge field (both ‘‘Fermi levels’’ are positionally constant). Effects on the ionic conductivity can be immense. Double layers formed by the contact of electrolyte solution to an electrode or charged membrane surfaces in electrophysiology are well-established examples. Figure 10 depicts four prototype cases involving solid electrolytes: (1) the contact of an ionic conductor to an insulator, the surface of which is able to trap (internally absorb) ions, leading to the concept of heterogeneous doping (influence of conductivity by dispersing fine insulator particles characterized by a similar rule of heterogeneous doping as given above for homogeneous doping obtained by replacing the concentration and charge of the dopant defect by the surface charge density and its sign); (2) the contact of two ionic conductors leading to a charge transfer of ions from one conductor to the other (cf. ionic ‘‘p–n junctions’’); (3) charge can also be stored within that core of the boundary, that is, in a grain boundary which joins two chemically identical grains and the effects can be tuned by chemical modification of the boundary or by varying the misfit; and (4) the contact to a fluid, for example, gas phase, is relevant in this context due to the fact that, acid–base active gases may cause attractive or repelling interactions with mobile ions. Since in many mixed, even predominantly electronic conductors, ionic charge carriers may be in majority, the ionic redistribution effect may determine the overall electronic boundary concentrations (fellow-traveler effect). Figure 11 shows the bending of ionic and electronic energy levels and their intercorrelation. The determination of the bending effect requires treatment of the chemical thermodynamics of the contact problem and hence a knowledge of energy levels in the interfacial core. In the case of a strong interfacial effect (co: carrier concentration in the first layer adjacent to the interface), Ds8m
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2eRTco ¼ 2Fu L
½24
Ionic and Mixed Conductivity in Condensed Phases 17
(a)
Excited
Perfect
Weakly defective
T=0K
0 < T<< Tc
Weakly defective ∋ = ~o
Perfect
i
i
Ov ∋ reg = ~
Interstitial
Regular
Superionic
Heavily defective
T > Tc
T
Superionic
Heavily defective ~o + RT In y e i
i
∋ i (c ) ~ o − RT In y e − v v
0
∋i
∋ reg(c)
∋ reg
T >Tc
T
(b) Figure 9 The ‘‘thermal destiny’’ of an ionic crystal above absolute zero: a progressive development from a perfect to a superionic phase via an ideally defective and a heavily defective state. (a) Crystallographic defect picture and (b) corresponding ‘‘energy level’’ schemes. The indices v and i refer to the vacant regular and interstitial sites. (Reproduced with permission from Maier J and Mu¨nch W (2000) Thermal destiny of an ionic crystal. Zeitschrift fu¨r Anorganische and Allgemeine Chemie 626: 264; & Wiley-VCH.)
describes the excess mean conductivity of the mobile majority defect in a Gouy–Chapman accumulation layer, and Ag +
Ag +
Dr> m
Heterogeneous doping (a)
Ag +
Ag +
Grain boundary engineering (c)
V−i junction (b)
NH3
Ag +
Chemical sensor (d)
Figure 10 Four basic space charge situations involving ionic conductors (here silver ion conductor): (a) contact with an insulator, (b) contact with a second ion conductor, (c) grain boundary, and (d) contact with a fluid phase. (Reproduced with permission from Maier J (2002) Nanoionics and soft materials science. In: Knauth P and Schoonman J (eds.) Nanocrystalline Metals and Oxides. Selected Properties and Applications, Kluwer International Series in Electronic Materials: Science & Technology, vol. 7, pp. 81–110. Boston: Kluwer Academic; & Kluwer Academic/ Plenum Publishers.)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 eRT ¼ 2 3 L 2zd u z cN F4 c2o lnðco =cN Þ
½25
the excess mean resistivity of the mobile majority carriers across a single Mott–Schottky depletion layer situation (dopant with molar charge zdF and concentration cN). L is the distance perpendicular to the interface over which the measurement averages. Figure 12 shows heterolayers of CaF2 and BaF2 and the effect of the concentration of interfaces on the overall parallel conductance. In a polycrystalline material, bulk and boundary conductivities have to be superimposed according to the microstructural situation. An approximate relation that takes into account the conductivities across (>) and along (8) grain boundaries (s: # complex conductivity; s# 8 ; s# > refer to the local mean values) is s# m ¼
8 s# N s# > # 8gb s> gb þ bgb jgb s gb > s# > #N gb þ bgb jgb s
½26
(b8 ; b> refer to the portions of the volume fraction j that contribute to the path; in the simplest approach, b8 ¼ 2=3, b> ¼ 1=3.)
Ionic and Mixed Conductivity in Condensed Phases
Local partial free energy
Interstitial ionic level
~0 i
≈ ≈
~ + Ag
~ − 0v
Regular ionic level
≈ ≈
Valence band
~ − p0
Mi 0 Ag ~e −
~0n
Conduction band
log c k
18
V M′ e′ h
2
1 ≡ X /
0
Figure 11 Bending of ‘‘energy levels’’ and concentration profiles in space charge zones (x ¼ 0 refers to the interfacial edge). (Reproduced from (left) Maier J (2003) Defect chemistry and ion transport in nanostructured materials. Part II. Aspects of nanoionics. Solid State Ionics 157: 327 and (right) Maier J (2002) Nano-sized mixed conductors (Aspects of nano-ionics. Part III). Solid State Ionics 148: 367–374, with permission from Elsevier.)
101 0.95 eV
T (Ω −1cm−1 K)
100
16.2 nm
20 nm
10−1
50 nm 103 nm
10−2
250 nm
10−3
430 nm 0.72 eV
10−4 CaF2
BaF2
the point defect (i.e., point defect including the sphere of influence in which the structure is perceptibly modified), then the local standard chemical potential changes (Figure 13b). In the case of delocalized electrons, this may happen at much larger distances. These are only two examples of the nanosize effects on ion conduction. Amongst others, the effects of particle geometry or curvature are mentioned, by which approximately a Gibbs–Kelvin term (pg% =r%; g% : mean surface tension, r%: mean radius) is introduced to the chemical potential. Such effects do not only affect transport properties, they are also interesting as far as mass storage is concerned.
10−5 1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
10 3 T −1 (K−1) Figure 12 Variation of the parallel ionic conductivity of CaF2– BaF2 heterolayers as a function of temperature for different periods (spacings). (Reproduced with permission from Sata N, Eberman K, Eberl K, and Maier J (2000) Mesoscopic fast ion conduction in nanometre-scale planar heterostructures. Nature 408: 946–949; & Nature Publishing Group.)
Nano-Ionics
Nano-sized conductors can provide large ionic conductivity anomalies owing to the high concentration of interfaces. Moreover, the spacing of interfaces may be so narrow that they perceive each other. Figure 13a shows the space charge overlap occurring in materials when the distance of neighboring interfaces becomes smaller than the Debye length, with the structure being invariant (cf. also Figure 12). If the distance is smaller than the effective thickness of
Partial Equilibrium – Bridge between High-Temperature and Low-Temperature Situation Arbitrary deviations from equilibrium require a detailed kinetic treatment. The situations in which some carriers behave reversibly while others are completely frozen are briefly considered here. Such considerations are helpful when the bridge has to be spanned from high-temperature equilibrium (during preparation) to a low operational temperature. In fact, such a partial equilibrium situation was already tacitly assumed when doping effects were studied. At high enough temperatures, dopants become reversible and segregation equilibria become active. Generally speaking, freezing of carriers leads to a reshuffling of in situ to ex situ parameters (i.e., from the P-term to the C-term in the power-law equation [20]). The discrepancy between the equilibrium
Ionic and Mixed Conductivity in Condensed Phases 19 + + + + + + + +
− − −− −−−−
+ + + + + − −+ + − − − − −+ + − − − − −+ + +
(a)
(>> I )
(a)
°
(b)
°
(b)
l
Figure 13 (a) Space charge effects decaying with l. (b) Structural effects which decay rapidly in the bulk. Within the indicated distance (l ); the structurally perturbed core (not shown, extension s) is perceived by the defects as far as m is concerned. (Reproduced with permission from Maier J (2003) Nano-ionics: trivial and non-trivial size effects on ion conduction in solids. Zeitschrift fu¨r physikalische chemie 217: 415–436; & Oldenbourg Wissenschaftsverlag.)
concentration (usually fulfilled at high temperatures under preparation temperatures) and the partially frozen situation becomes apparent in the application of electroceramics (e.g., high-temperature superconductors and dielectrics) that are used at room temperature. Two aspects are important: (1) a reproducible preparation and (2) a detailed understanding of defect chemistry. For a systematic treatment of these complex problems, the reader is referred to the literature.
Mobility: From Low to High Concentrations For dilute concentrations, the mobility in ionic crystals is usually taken to be independent of the concentration. According to the hopping equation (uj pkj ), it follows as 2
uj pG0j ðDxÞ exp þ
DSa j R
! exp
DHja RT
! ½27
(c) Figure 14 Elementary jump mechanisms in crystals: (a) vacancy mechanism, (b) direct interstitial mechanism, and (c) (collinear or noncollinear), indirect interstitial mechanism (interstitialcy mechanism).
(G0 is the attempt frequency (cf. k0, i.e., the prefactor of k), Dx is the hopping distance; DH a and DSa denote activation enthalpy and entropy, respectively.) Figure 14 displays three fundamental hopping mechanisms. While the treatment of the ionic carrier concentrations is, in essence, very similar for liquids or solids, this is less so in the case of mobility. In solids, the activation thresholds of hopping transport are usually substantial (approximately several 100 meV), while in dilute liquid electrolytes, temperature effects act on carrier mobilities mainly through changes in the viscosity. In superionic solid conductors, activation thresholds are significantly less. At high concentrations, the jump partners in the hopping processes are important which formally leads to the appearance of a concentration factor (1 ðc=cmax Þ) in the mobility. Higher concentrations, however, affect carrier transport in such an individual way that no universal relation can be given and even the decomposition into charge carrier concentration and mobility becomes questionable. For not too strong interactions, a fairly general interaction effect is to be considered. Defect interactions lead to relaxation phenomena in that a particle
20
Ionic and Mixed Conductivity in Condensed Phases
that has made a jump from x to x0 does not directly create its equilibrium environment and has a higher tendency to jump back again. It is, in particular, the rearrangement of the other defects that determines the finite relaxation time. This mismatch situation causes back-jumps to be more likely, making the rate constants of the hopping equation time dependent and hence the conductivity frequency dependent. In liquids too, a relaxation effect occurs. Since the formation of the ion cloud does not immediately follow the mobile ion under regard, the charge cloud becomes asymmetrical and acts as a brake. The relaxation time is proportional to the viscosity. The occurrence of counter transport of differently charged ions which carry along these solvation spheres leads to friction, that is, to a second effect that diminishes the mobility, the so-called electrophoretic pffiffiffi effect. These phenomena explain the Kohlrausch’s c law, which is well established for strong electrolytes, and reads " # pffiffiffi A B Lc ¼ Lo c ½28 L þ 3=2 o 1=2 ðeTÞ ðeTÞ (A and B are constants, Lo Lc (c ¼ 0), e the dielectric constant, and Lc the equivalent conductivity, that is, conductivity per equivalent salt concentration.) Figure 15 gives some examples of the above phenomena.
150 KCl
140 1/3 LaCl3 AgNO3
Λ c /Λ*c
130
1/2 LaCl2 120 1/2 Na2So4 KIO3
110 1/2 NiSO4 0
2
4
1/2 MgSO4 6
8
Chemical Diffusion in Mixed Conductors The simultaneous presence of ionic and electronic carriers in so-called mixed conductors leads to the phenomenon of chemical diffusion (coupled transport of ions and electrons) and the possibility of changing stoichiometry. Technologically important examples for which this phenomenon plays a decisive role are insertion electrodes, electrochemical windows, permeation membranes, and conductivity sensors; more generally, chemical diffusion is important for all electroceramics, the properties of which depend on the detailed stoichiometry (such as in dielectrics or hightemperature superconductors). Stoichiometry changes also occur, if the electrodes applied in the electrochemical measurements are not completely reversible for both ions and electrons; in this case, the partial current density is given by s j ij ¼ ½29 i þ zj FDd rcj s Dd being the chemical diffusion coefficient that is determined by mobilities and concentrations of all charge carriers involved in the transport (i: total current density). If different internal reactions such as valence state changes occur for the conducting species, the equation has to be modified. The use of selectively blocking electrodes leads to the suppression of the current of one carrier type at the expense of a concentration polarization. This allows the separation of ionic and electronic conductivities and of the chemical diffusion coefficient. A major application of ion conductors is their use as electrolytes which nullifies gradients in the electrochemical potential of ions but supports gradients in the electrochemical potential of electrons. This gives rise to a cell voltage (e.m.f.) which is consequently proportional to the integrated gradient of the neutral component (cf. distance in Figure 2) and corresponds to the Nernst equation. In other words, differences in the chemical component potentials can be transformed into an electrical voltage (such as exploited in batteries, fuel cells, and sensors). In mixed conductors, a partial internal short-circuit occurs, causing a nonzero gradient in the electrochemical potential of the ions, which is proportional to the gradient in the electronic Fermi level. Hence, the cell voltage is diminished by a factor that depends on the proportion of electronic conductivity and consequently allows for its determination.
c /c* Figure 15 Equivalent conductivity for a variety of aqueous salt solutions vs. concentration (Lc and c denote unit values). (Reproduced with permission from Kortu¨m G (1972) Lehrbuch der Elektrochemie, 5th edn. Weinheim: VCH; & Wiley-VCH.)
See also: Conductivity, Electrical.
PACS: 72.60. þ g
Ionic Bonding and Crystals 21
Further Reading Allnatt AR and Lidiard AB (1993) Atomic Transport in Solids. Cambridge: Cambridge University Press. Funke K (1993) Jump relaxation in solid electrolytes. Progress in Solid State Chemistry 22: 111. Heifets E, Kotomin EA, and Maier J (2000) Semi-empirical simulations of surface relaxation for perovskite titanates. Surface Science 462: 19. Kirchheim R (1988) Hydrogen solubility and diffusivity in defective and amorphous metals. Progress in Materials Science 32: 261. Kortu¨m G (1965) Treatise on Electrochemistry, 2nd edn. Amsterdam: Elsevier. Kro¨ger FA (1964) Chemistry of Imperfect Crystals. Amsterdam: North-Holland. Lidiard AB (1957) Ionic conductivity. In: Flu¨gge S (ed.) Handbuch der Physik., vol. 20, p. 246. Berlin: Springer. Maier J (1993) Defect chemistry: composition, transport, and reactions in the solid state. Part I: Thermodynamics.
Angewandte Chemie (International Edition in English) 32: 313–528. Maier J (1995) Ionic conduction in space charge regions. Progress in Solid State Chemistry 23: 171. Maier J (2004) Physical Chemistry of Materials – Ions and Electrons in Solids. Chichester: Wiley. (Translation of Festko¨rper – Fehler und Funktion: Prinzipien der Physikalischen Festko¨rperchemie Teubner BG, Stuttgart). Mott NF (1974) Metal-Insulator Transitions. London: Taylor & Francis. Rickert H (1982) Electrochemistry of Solids, 1st edn. Berlin: Springer. Sasaki K and Maier J (1999) Low-temperature defect chemistry of oxides. I. General aspects and numerical calculations; II. Analytical reasons. Journal of Applied Physics 86: 5422–5443. Schmalzried H (1981) Solid State Reactions, 1st edn. Weinheim: VCH. Schmalzried H (1995) Chemical Kinetics of Solids, 1st edn. Weinheim: VCH.
Ionic Bonding and Crystals* M W Finnis, Queen’s University Belfast, Belfast, UK Published by Elsevier Ltd.
Introduction One can distinguish the models of ionic materials by their increasing degrees of sophistication, which are described briefly below. All contain parameters of one sort or another that have to be fitted either to experimental data or to the best available first-principles calculations. Following this introduction there is a more detailed discussion of how ionic models, in general, can all be related to the density-functional theory within the second-order perturbation theory. The Born model, the shell model, and the compressible ion model are three well-known types of increasing complexity. More recently, they have been joined by models including ionic polarizability up to the quadrupole level. The simplest ionic models are typified by constant nominal ionic charges. These are usually integers, although fractional charges have been used as additional fitting parameters. Variable charge models have also been developed, which allow charge transfer within a simple classical description of the equilibration of an electronic chemical potential. Finally, a class of models has been developed that includes most of the aforementioned as subclasses and also treats covalency, namely, the self-consistent tightbinding models. The scope of ionic models, especially when charge transfer can be included, is potentially *Adapted by permission of Oxford University Press.
very wide. Although so far they have been less developed and applied, self-consistent tight-binding models have potentially the widest scope, which embrace traditional textbook ionic materials such as NaCl, KF and all the other alkali halides, the insulating oxides with wide bandgaps such as MgO and Al2O3, perovskites such as SrTiO3, as well as oxides which are considered as less ionic, such as TiO2 and ZrO2. The simplest ionic model is the rigid ion model. It is sometimes called the Born model after its pioneer Max Born and his co-workers in Edinburgh, who over 50 years ago developed most of the mathematical description of the ionic model that is still being used (see Further reading section for more details). Within the context of the density-functional perturbation theory, one can think of it as a first-order model, as seen in the following section. The ions interact by the Coulomb interaction, in addition to which there is a short-range pairwise repulsion. Ions are not rigid, and the shell model was first developed by Dick and Overhauser in 1958 to include their polarizability. In its simplest version, a spherical shell of charge qs surrounds an ion of reduced charge DZ þ qs and is attached to it by a harmonic spring. The nominal ionic charge is DZ. The repulsion between ions is now modeled as a repulsion between the centers of these shells. The compressible ion model is an extension of the shell model in which the radius of the shells is a variable. The energy of a compressible ion is a function of this radius, which adjusts according to the environment of the ion. This model is most strongly motivated by the case of oxygen, as discussed further