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Lattice-assisted proton motion in perovskite oxides
Solid State Ionics 136–137 (2000) 291–295 www.elsevier.com / locate / ssi
Lattice-assisted proton motion in perovskite oxides A.L. Samgin* High-Temperature Electrochemistry Institute, Ural Division of Russian Academy of Sciences, S. Kovalevskoy 20, Ekaterinburg, 620219, Russia
Abstract A new interpretation of properties of high-temperature protonic conductors based on the proton polaron idea is proposed. Mobility of protons in these materials is viewed as a result of phonon-activated jumps. A comparison with the experimental data is made. It follows that the proton polaron effect can be really large enough in these perovskite-structured solid solutions. 2000 Elsevier Science B.V. All rights reserved. Keywords: Small proton polaron; Perovskite; Protonic conductor Materials: BaCeO 3 ; SrCeO 3 ; SrZrO 3 ; SrTiO 3
1. Introduction
2. Polaronic picture
Rare earth doped SrCeO 3 , SrZrO 3 and BaCeO 3 are known to be protonic conductors at high temperatures (HTPC) [1–3]. This problem is promising for applications in fuel cells and other devices, however many aspects of proton behavior in the HTPC are still not fully understood. Reasoning from the large role of proton–lattice interaction, an approach in which the proton behavior in the HTPC is examined in terms of quasiparticles may be of interest. A consideration of quasiparticles analogous to the small polarons appears to be particularly important for protons forming O–H bonds in such perovskite oxides as SrZrO 3 . In this paper we propose an idealized picture for interpretation of properties of the HTPC based on the proton polaron idea.
In understanding of migration of such light ions as protons in a solid, it is of value to know in detail the role played by a thermal reservoir. A more rigorous treatment of the proton transfer mechanism in oxides requires the consideration of energy exchange with the surrounding cage [4,5]. A treatment based on the Kramers diffusion has been successfully used along this line for the HTPC [5]. This treatment assumes that after passing over the classical potential barrier the proton comes to equilibrium with the thermal bath. As a consequence, a correction factor to the absolute rate theory (ART) hopping rate has been obtained. We shall attempt to analyse the part of phonons on an other basis, in which the protonic defect is viewed as the small polaron, rather than the proton. We start from a formalism [6] which treats of the dynamical properties in terms of observables for the particle which may be considered to hop due to
*E-mail address: [email protected] (A.L. Samgin).
0167-2738 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S0167-2738( 00 )00406-9
292
A.L. Samgin / Solid State Ionics 136 – 137 (2000) 291 – 295
its interaction with the phonons. The transport properties in such model that takes account of only the essentials of the interactions governing the hopping motion of the particle, may be obtained, in the general case, from the Hamiltonian H 5 H1 1 H2
(1)
where H1 leads to tunneling, H2 couples the particle to a reservoir and governs the jump processes. There is no need to regard the reservoir as consisting of phonons. In some thermal conditions, the reservoir can be treated as the source of a classical thermally generated fluctuating field that acts on the particle. The main result of Ref. [6] is that, in this case, the gross phenomenological properties of Eq. (1) are equal to the corresponding ones for small polarons. Thus, the essential role of phonons is to provide a reservoir which interacts with carriers via H2 and brings them into thermal equilibrium [6]. Then, proton can be considered to hop at high enough temperatures from site to site due to an exchange of energy with phonons, because the hopping motion is the means whereby the energy of protons is dissipated. In view of this formal similarity, we have taken an interest in a description of proton behavior in the HTPC using directly the protonic polaron Hamiltonian which regards the reservoir as consisting specifically of such phonons, as OH-stretching mode and vibrations of the oxygen octahedron as a whole. The polaron Hamiltonian for protons in OHchains is given in Ref. [7]. However, application of this approach to some proton conducting oxides seems to be less formal than had been suspected. This is due to the fact that in H-bonded chains, the formation of small proton polarons can be really possible [7]. Fisher et al. have explained [8] the isotope effect and the infrared absorption of solid- state H-bonded substances by construction of a hydrogen quasiparticle analogous to the small polaron. Polaronic effects would be expected to occur in various H-bonded solids including doped crystals [8]. The protonic conductors are not infrequently considered as a type of ionic conductor. But, Eigen and De Maeyer have pointed out in fundamental work that the proton transport in H-bonded media is completely different from normal ionic migration
and corresponds more to electronic transport processes in semiconductors [9]. For this reason alone the protonic polaron approach has attracted considerable interest. We might expect to find a polaronic behavior in HTPCs. An interesting case exists for a large number of oxide compositions of BaCe 12x M x O 3 (M5trivalent rare earth). Although the dependence of the hole conduction on the dopant nature in an atmosphere free from hydrogen assumes a complex form, the same character of dependence was evidenced in hydrogen atmosphere [2]. This shows that protonic and electronic transport processes may have a common mechanism [10]. The electron defects in BaCeO 3 were found to be small polarons [11]. Thus, it is not unreasonable to expect that a proton polaron may be formed in ABO 3 oxides. The major requirements for any system, which should be susceptible to this approach, are that there be (1) a doubleminimum potential, (2) strong coupling, and (3) simple optical phonon spectrum [8]. We might expect to find them fulfilled in HTPCs. As an illustration, we shall use the simplest expression for hopping mobility of small proton polaron which can be written by analogy to the so-called Holstein molecular crystal model [12] for high temperature: Œ] p 2E m 5 ]ed 2skTd 23 / 2 " 23 / 2 u 21 v 21 / 2 J 2 exp ]] 2 kT (2)
S D
where the energy of activation is defined by expression
S D
"v E 5 2kTu 2 tanh ] 4kT
(3)
Here u 2 is the coupling constant between protons and lattice vibrations, d is the O–O distance, v is the frequency of phonons, e.g. frequency of O–H stretching mode or O–Ce–O bending mode in SrCeO 3 , J is the hopping integral depending on the resonance overlap for wave functions of the protons at two positions adjacent to lattice oxygens. This expression would suffice to describe some regular trends, such as temperature dependence of proton mobility. Let us use as activating system a phonon mode with the frequency v ¯ 6 3 10 13 s 21 . Using Eq. (3) and the fact that observed activation energy
A.L. Samgin / Solid State Ionics 136 – 137 (2000) 291 – 295
for SrCe 0.95 Yb 0.05 O 3 is found to be 0.63 eV [13], 2 one obtains u ¯ 30. Using dipole formation in SrCe 12xYb x O 3 [14], a similar strength of proton– lattice coupling may be determined. Let us suppose that J 5 3.2 3 10 221 J, this is in agreement with the inequality J/"v | kT. Mobilities of the protons in SrCe 0.95 Yb 0.05 O 3 estimated from Eq. (2) and the experimental data [3] are shown in Table 1. It is seen from Table 1 that the model values are in good agreement with the experimental results at T5600– 8508C. SrCeO 3 -type proton conductors have strong dependence of activation energy on lattice parameters. We invoke [15] a simple expression, which is a natural extension of the known Mott’s formula for polarons [16]:
S
1 1 E 5 kp ] 2 ] d rp
D
(4)
where k p is a proportionality constant, r p is the polaron radius. The observed activation energy for proton conduction in HTPCs decreases rapidly with d. These energies are found to be 0.63 eV at d53.02 ˚ and 0.51 eV at d53.11 A ˚ [13]. Substitution of A ˚ If d varies these values in Eq. (4) gives r p ¯3.45 A. ˚ to 3.02 A, ˚ those proton conductors have from 2.83 A the experimental values of E in the ratio 3:2 [13] to be consistent with the formula Eq. (4) for r p ¯3.45 ˚ A. The time between two proton jumps within the quasi one-dimensional O–H–O bonds can be in essence expressed as: 21 t |swm,m11 1 wm,m 21d 21 ¯ 2w m,m9
(5)
Table 1 Theoretical values of the proton mobilities m evaluated from Eq. (2) and corresponding experimental values [3] mexp in SrCe 12xYb x O 3 T (8C)
m (cm 2 s 21 V 21 )
mexp (cm 2 s 21 V 21 )
600 650 700 750 800 850 900
5.4310 26 7.9310 26 1.1310 25 1.4310 25 1.9310 25 2.3310 25 2.9310 25
5.4310 26 7.9310 26 1.2310 25 1.5310 25 1.9310 25 2.3310 25 3.4310 25
293
where w m,m9 is the proton hopping probability from site m to another m9. Then, Eq. (2) may be written as
S D
S D
e 2E m 5 ] d 2 w m,m9 5 m0 exp ]] kT kT
(6)
In a similar manner, the diffusion coefficient can be considered in a general way [10,17] D 5 d 2 w m,m9
(7)
Using Eq. (7), the corresponding time between jumps is found to be 3310 211 s for J 2 ¯ 10 241 J 2 , v 5 10 14 s 21 , u 2 5 20, and T56008C. This value is consistent with the value t ¯ 2.8 3 10 211 s that is obtained by the quasielastic neutron scattering study of SrCe 0.95 Yb 0.05 H 0.02 O 2.985 [18]. The conventional treatment gives the hopping rate, n, for a classical particle in terms of the ART as
S D
2U n 5 n0 exp ]] kT
(8)
where the attempt frequency n0 is of the order of typical phonon frequency, U is the height of the potential barrier [19]. For the OH-stretching mode, v 5 10 14 s 21 [5], we have "v ¯ kT at T56008C. Then, from Eqs. (2) and (6), we obtain Œ] p J2 2E ]] ] exp ]] w m,m9 5 kT 2u" 2 v
S D
(9)
Let U |E. With w m,m9 taken as n, the prefactor in that case is close to v for J ¯Œ]u"v as in the case of the ART. But, by virtue of the fact that in reality J , "v, we may expect that the proton prefactor is tangibly lower than the local phonon frequency. This is important to the HTPC, because this is in excellent agreement with observed facts that the prefactor for light ions in superionic conductors seems to be consistently lower by several orders of magnitude than n0 [19]. These anomalously low prefactors are explained in Ref. [19] by a treatment based on the Kramers diffusion due to a breakdown of the ART. The anomalous isotope effect in the HTPC is also explained in Ref. [5] by a correction to the ART. Thus, in our model this correction can be interpreted in terms of phonons, while these two approaches are not physically equivalent. As is evident from the foregoing, a polaron-like behavior of protons is shown by such oxides as
294
A.L. Samgin / Solid State Ionics 136 – 137 (2000) 291 – 295
doped SrCeO 3 which exhibit HTPC. We believe that in these materials polaronic effect is in reality large enough at such temperature conditions and governs in particular proton jumps on O–H–O bonds, as described by Eq. (2). However, in some oxides, such as SrTiO 3 , proton transport occurs at low temperatures predominantly by tunneling motion. With tunneling of protons as phenomenon, the situation is different in the general case. What we really need to know are the thermal conditions wherein the change from hopping to tunneling may take place in principle. Now that we have presented thermally-activated proton hopping, we turn to the formal treatment [6] which express the essential properties of small polarons in terms of phenomenological constants g and V representing thermally-activated and tunneling processes, respectively. It should be stressed once again that this treatment can be applied to the same system as the small polaron model independent of the microscopic properties of the interaction H2 between particle and reservoir. One important result of this heuristic model is that the temperature dependence of these constants is such that g , V for T ,T c and g . V for T .T c , where T c is a certain critical temperature [6]. Then, we may suppose that this temperature which defines at once the boundary of the proton jump process and the proton tunneling may be viewed as a phenomenological parameter of protonic jump process irrespective of whether the temperatures are appropriate for a classical treatment of reservoir. Then, as a rough guide, the value of T c in the polaron model may be accepted as correct for other perovskite proton conductors (e.g. for SrTiO 3 or SrZrO 3 ) independent of real tunneling mechanism. It is clear that this has nothing to do with coherent polaron tunneling. However, it is not inconceivable that at very low temperatures in some materials a proton behavior similar to polaron tunneling may be observed. To evaluate T c , consider SrTiO 3 . In this oxide, the proton is located in a site between the O–O ions ˚ as if it forms (this distance is evaluated to be 1.2 A) the hydrogen bond [20]. Using reasonable values for J and u 2 , one obtains T c close to "v / 2k. It follows that for phonon frequency 6310 13 s 21 , T c ¯217 K. Note, that T c appears to be sensitive to properties of the reservoir. Our formal estimate for T c correlates
with an another evaluation of T c for proton motion between potential wells on the H-bond in crystals with hydrogen bonds. In the last case this temperature can be written as [21]:
S D
U T c ¯ " ]]2 2mb
1/2
(10)
˚ and where b is the barrier width. Taking b51.2 A U 50.42 eV [20], we obtain T c ¯269 K. In Ref. [21], the distinctive range of T c is evaluated as T c ¯(1– 300) K for crystals with hydrogen bonds. This is in general agreement with the tunneling mechanism for proton motion observed in experiments with doped SrZrO 3 at 5 K,T ,300 K [22].
3. Concluding remarks The concept of proton polaron offers a new outlook on properties of the HTPC and yields predictions for hopping mobility, time between jumps, prefactor, dependence of activation energy on O–O distance and boundary temperature when tunneling predominates. These predictions are in agreement with the observed data. The model we consider may be regarded as a simplest illustration of polaron approach. This approach has much potential for yielding information about new proton conductors.
References [1] H. Iwahara, Solid State Ionics 86–88 (1996) 9. [2] A.V. Strelkov, A.P. Kaul, Yu.M. Kiselev, Yu.D. Tretjakov, in: The IX All-Union Conf. on Phys. Chem. and Electrochem. of Molten Salts and Solid Electrolytes, Sverdlovsk, USSR (Abstracts), 3(1) (1987) 217. [3] H. Uchida, H. Yoshikawa, T. Esaka, S. Ohtsu, H. Iwahara, Solid State Ionics 36 (1989) 89. [4] K.-D. Kreuer, A. Fuchs, J. Maier, Solid State Ionics 77 (1995) 157. [5] A.S. Nowick, A.V. Vaysleb, Solid State Ionics 97 (1997) 17. [6] G.L. Sewell, Phys. Rev. 129 (1963) 597. [7] V.V. Krasnogolovets, N.A. Protsenko, P.M. Tomchuk, Int. J. Quant. Chem. 33 (1988) 340. [8] S.F. Fisher, G.L. Hofacker, M.A. Ratner, J. Chem. Phys. 52 (1970) 1934. [9] M. Eigen, L. De Maeyer, Proc. R. Soc. 247A (1958) 505. [10] A.L. Samgin, Electrochimiya 35 (1999) 312. [11] T. He, P. Ehrhart, Solid State Ionics 86–88 (1996) 633. [12] Yu.A. Firsov (Ed.), Polarons, Nauka, Moscow, 1975.
A.L. Samgin / Solid State Ionics 136 – 137 (2000) 291 – 295 [13] T. Scherban, W. Lee, A. Nowick, Solid State Ionics 28 (1988) 585. [14] L. Zimmerman, H.G. Bohn, W. Schilling, E. Syskakis, Solid State Ionics 77 (1995) 163. [15] A.L.Samgin, in: The XI Conf. on Phys. Chem. and Electrochem. of Molten and Solid Electrolytes, Ekaterinburg, Russia (Abstracts), 2 (1998) 111. [16] I.G. Austin, N.F. Mott, Adv. Phys. 18 (1969) 41. [17] R.R. Dogonadze, A.A. Chernenko, Yu.A. Chizmadjev, Fiz. Tverdogo Tela 3 (1961) 3720. [18] R. Hempelmann, Ch. Karmonik, Th. Matzke, M. Cappadonia, U. Stimming, T. Springer, M.A. Adams, Solid State Ionics 77 (1995) 152.
295
[19] B.A. Huberman, J.B. Boyce, Solid State Commun. 25 (1978) 759. [20] N. Sata, K. Hiramoto, M. Ishigame, S. Hosoya, N. Niimura, S. Shin, Phys. Rev. B54 (1996) 15795. [21] M.I. Klinger, A.O. Asisyan, Fiz. Tech. Poluprovodnikov 13 (1979) 1873. [22] H. Yugami, S. Matsuo, M. Ishigame, Solid State Ionics 77 (1995) 195.
Lattice-assisted proton motion in perovskite oxides A.L. Samgin* High-Temperature Electrochemistry Institute, Ural Division of Russian Academy of Sciences, S. Kovalevskoy 20, Ekaterinburg, 620219, Russia
Abstract A new interpretation of properties of high-temperature protonic conductors based on the proton polaron idea is proposed. Mobility of protons in these materials is viewed as a result of phonon-activated jumps. A comparison with the experimental data is made. It follows that the proton polaron effect can be really large enough in these perovskite-structured solid solutions. 2000 Elsevier Science B.V. All rights reserved. Keywords: Small proton polaron; Perovskite; Protonic conductor Materials: BaCeO 3 ; SrCeO 3 ; SrZrO 3 ; SrTiO 3
1. Introduction
2. Polaronic picture
Rare earth doped SrCeO 3 , SrZrO 3 and BaCeO 3 are known to be protonic conductors at high temperatures (HTPC) [1–3]. This problem is promising for applications in fuel cells and other devices, however many aspects of proton behavior in the HTPC are still not fully understood. Reasoning from the large role of proton–lattice interaction, an approach in which the proton behavior in the HTPC is examined in terms of quasiparticles may be of interest. A consideration of quasiparticles analogous to the small polarons appears to be particularly important for protons forming O–H bonds in such perovskite oxides as SrZrO 3 . In this paper we propose an idealized picture for interpretation of properties of the HTPC based on the proton polaron idea.
In understanding of migration of such light ions as protons in a solid, it is of value to know in detail the role played by a thermal reservoir. A more rigorous treatment of the proton transfer mechanism in oxides requires the consideration of energy exchange with the surrounding cage [4,5]. A treatment based on the Kramers diffusion has been successfully used along this line for the HTPC [5]. This treatment assumes that after passing over the classical potential barrier the proton comes to equilibrium with the thermal bath. As a consequence, a correction factor to the absolute rate theory (ART) hopping rate has been obtained. We shall attempt to analyse the part of phonons on an other basis, in which the protonic defect is viewed as the small polaron, rather than the proton. We start from a formalism [6] which treats of the dynamical properties in terms of observables for the particle which may be considered to hop due to
*E-mail address: [email protected] (A.L. Samgin).
0167-2738 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S0167-2738( 00 )00406-9
292
A.L. Samgin / Solid State Ionics 136 – 137 (2000) 291 – 295
its interaction with the phonons. The transport properties in such model that takes account of only the essentials of the interactions governing the hopping motion of the particle, may be obtained, in the general case, from the Hamiltonian H 5 H1 1 H2
(1)
where H1 leads to tunneling, H2 couples the particle to a reservoir and governs the jump processes. There is no need to regard the reservoir as consisting of phonons. In some thermal conditions, the reservoir can be treated as the source of a classical thermally generated fluctuating field that acts on the particle. The main result of Ref. [6] is that, in this case, the gross phenomenological properties of Eq. (1) are equal to the corresponding ones for small polarons. Thus, the essential role of phonons is to provide a reservoir which interacts with carriers via H2 and brings them into thermal equilibrium [6]. Then, proton can be considered to hop at high enough temperatures from site to site due to an exchange of energy with phonons, because the hopping motion is the means whereby the energy of protons is dissipated. In view of this formal similarity, we have taken an interest in a description of proton behavior in the HTPC using directly the protonic polaron Hamiltonian which regards the reservoir as consisting specifically of such phonons, as OH-stretching mode and vibrations of the oxygen octahedron as a whole. The polaron Hamiltonian for protons in OHchains is given in Ref. [7]. However, application of this approach to some proton conducting oxides seems to be less formal than had been suspected. This is due to the fact that in H-bonded chains, the formation of small proton polarons can be really possible [7]. Fisher et al. have explained [8] the isotope effect and the infrared absorption of solid- state H-bonded substances by construction of a hydrogen quasiparticle analogous to the small polaron. Polaronic effects would be expected to occur in various H-bonded solids including doped crystals [8]. The protonic conductors are not infrequently considered as a type of ionic conductor. But, Eigen and De Maeyer have pointed out in fundamental work that the proton transport in H-bonded media is completely different from normal ionic migration
and corresponds more to electronic transport processes in semiconductors [9]. For this reason alone the protonic polaron approach has attracted considerable interest. We might expect to find a polaronic behavior in HTPCs. An interesting case exists for a large number of oxide compositions of BaCe 12x M x O 3 (M5trivalent rare earth). Although the dependence of the hole conduction on the dopant nature in an atmosphere free from hydrogen assumes a complex form, the same character of dependence was evidenced in hydrogen atmosphere [2]. This shows that protonic and electronic transport processes may have a common mechanism [10]. The electron defects in BaCeO 3 were found to be small polarons [11]. Thus, it is not unreasonable to expect that a proton polaron may be formed in ABO 3 oxides. The major requirements for any system, which should be susceptible to this approach, are that there be (1) a doubleminimum potential, (2) strong coupling, and (3) simple optical phonon spectrum [8]. We might expect to find them fulfilled in HTPCs. As an illustration, we shall use the simplest expression for hopping mobility of small proton polaron which can be written by analogy to the so-called Holstein molecular crystal model [12] for high temperature: Œ] p 2E m 5 ]ed 2skTd 23 / 2 " 23 / 2 u 21 v 21 / 2 J 2 exp ]] 2 kT (2)
S D
where the energy of activation is defined by expression
S D
"v E 5 2kTu 2 tanh ] 4kT
(3)
Here u 2 is the coupling constant between protons and lattice vibrations, d is the O–O distance, v is the frequency of phonons, e.g. frequency of O–H stretching mode or O–Ce–O bending mode in SrCeO 3 , J is the hopping integral depending on the resonance overlap for wave functions of the protons at two positions adjacent to lattice oxygens. This expression would suffice to describe some regular trends, such as temperature dependence of proton mobility. Let us use as activating system a phonon mode with the frequency v ¯ 6 3 10 13 s 21 . Using Eq. (3) and the fact that observed activation energy
A.L. Samgin / Solid State Ionics 136 – 137 (2000) 291 – 295
for SrCe 0.95 Yb 0.05 O 3 is found to be 0.63 eV [13], 2 one obtains u ¯ 30. Using dipole formation in SrCe 12xYb x O 3 [14], a similar strength of proton– lattice coupling may be determined. Let us suppose that J 5 3.2 3 10 221 J, this is in agreement with the inequality J/"v | kT. Mobilities of the protons in SrCe 0.95 Yb 0.05 O 3 estimated from Eq. (2) and the experimental data [3] are shown in Table 1. It is seen from Table 1 that the model values are in good agreement with the experimental results at T5600– 8508C. SrCeO 3 -type proton conductors have strong dependence of activation energy on lattice parameters. We invoke [15] a simple expression, which is a natural extension of the known Mott’s formula for polarons [16]:
S
1 1 E 5 kp ] 2 ] d rp
D
(4)
where k p is a proportionality constant, r p is the polaron radius. The observed activation energy for proton conduction in HTPCs decreases rapidly with d. These energies are found to be 0.63 eV at d53.02 ˚ and 0.51 eV at d53.11 A ˚ [13]. Substitution of A ˚ If d varies these values in Eq. (4) gives r p ¯3.45 A. ˚ to 3.02 A, ˚ those proton conductors have from 2.83 A the experimental values of E in the ratio 3:2 [13] to be consistent with the formula Eq. (4) for r p ¯3.45 ˚ A. The time between two proton jumps within the quasi one-dimensional O–H–O bonds can be in essence expressed as: 21 t |swm,m11 1 wm,m 21d 21 ¯ 2w m,m9
(5)
Table 1 Theoretical values of the proton mobilities m evaluated from Eq. (2) and corresponding experimental values [3] mexp in SrCe 12xYb x O 3 T (8C)
m (cm 2 s 21 V 21 )
mexp (cm 2 s 21 V 21 )
600 650 700 750 800 850 900
5.4310 26 7.9310 26 1.1310 25 1.4310 25 1.9310 25 2.3310 25 2.9310 25
5.4310 26 7.9310 26 1.2310 25 1.5310 25 1.9310 25 2.3310 25 3.4310 25
293
where w m,m9 is the proton hopping probability from site m to another m9. Then, Eq. (2) may be written as
S D
S D
e 2E m 5 ] d 2 w m,m9 5 m0 exp ]] kT kT
(6)
In a similar manner, the diffusion coefficient can be considered in a general way [10,17] D 5 d 2 w m,m9
(7)
Using Eq. (7), the corresponding time between jumps is found to be 3310 211 s for J 2 ¯ 10 241 J 2 , v 5 10 14 s 21 , u 2 5 20, and T56008C. This value is consistent with the value t ¯ 2.8 3 10 211 s that is obtained by the quasielastic neutron scattering study of SrCe 0.95 Yb 0.05 H 0.02 O 2.985 [18]. The conventional treatment gives the hopping rate, n, for a classical particle in terms of the ART as
S D
2U n 5 n0 exp ]] kT
(8)
where the attempt frequency n0 is of the order of typical phonon frequency, U is the height of the potential barrier [19]. For the OH-stretching mode, v 5 10 14 s 21 [5], we have "v ¯ kT at T56008C. Then, from Eqs. (2) and (6), we obtain Œ] p J2 2E ]] ] exp ]] w m,m9 5 kT 2u" 2 v
S D
(9)
Let U |E. With w m,m9 taken as n, the prefactor in that case is close to v for J ¯Œ]u"v as in the case of the ART. But, by virtue of the fact that in reality J , "v, we may expect that the proton prefactor is tangibly lower than the local phonon frequency. This is important to the HTPC, because this is in excellent agreement with observed facts that the prefactor for light ions in superionic conductors seems to be consistently lower by several orders of magnitude than n0 [19]. These anomalously low prefactors are explained in Ref. [19] by a treatment based on the Kramers diffusion due to a breakdown of the ART. The anomalous isotope effect in the HTPC is also explained in Ref. [5] by a correction to the ART. Thus, in our model this correction can be interpreted in terms of phonons, while these two approaches are not physically equivalent. As is evident from the foregoing, a polaron-like behavior of protons is shown by such oxides as
294
A.L. Samgin / Solid State Ionics 136 – 137 (2000) 291 – 295
doped SrCeO 3 which exhibit HTPC. We believe that in these materials polaronic effect is in reality large enough at such temperature conditions and governs in particular proton jumps on O–H–O bonds, as described by Eq. (2). However, in some oxides, such as SrTiO 3 , proton transport occurs at low temperatures predominantly by tunneling motion. With tunneling of protons as phenomenon, the situation is different in the general case. What we really need to know are the thermal conditions wherein the change from hopping to tunneling may take place in principle. Now that we have presented thermally-activated proton hopping, we turn to the formal treatment [6] which express the essential properties of small polarons in terms of phenomenological constants g and V representing thermally-activated and tunneling processes, respectively. It should be stressed once again that this treatment can be applied to the same system as the small polaron model independent of the microscopic properties of the interaction H2 between particle and reservoir. One important result of this heuristic model is that the temperature dependence of these constants is such that g , V for T ,T c and g . V for T .T c , where T c is a certain critical temperature [6]. Then, we may suppose that this temperature which defines at once the boundary of the proton jump process and the proton tunneling may be viewed as a phenomenological parameter of protonic jump process irrespective of whether the temperatures are appropriate for a classical treatment of reservoir. Then, as a rough guide, the value of T c in the polaron model may be accepted as correct for other perovskite proton conductors (e.g. for SrTiO 3 or SrZrO 3 ) independent of real tunneling mechanism. It is clear that this has nothing to do with coherent polaron tunneling. However, it is not inconceivable that at very low temperatures in some materials a proton behavior similar to polaron tunneling may be observed. To evaluate T c , consider SrTiO 3 . In this oxide, the proton is located in a site between the O–O ions ˚ as if it forms (this distance is evaluated to be 1.2 A) the hydrogen bond [20]. Using reasonable values for J and u 2 , one obtains T c close to "v / 2k. It follows that for phonon frequency 6310 13 s 21 , T c ¯217 K. Note, that T c appears to be sensitive to properties of the reservoir. Our formal estimate for T c correlates
with an another evaluation of T c for proton motion between potential wells on the H-bond in crystals with hydrogen bonds. In the last case this temperature can be written as [21]:
S D
U T c ¯ " ]]2 2mb
1/2
(10)
˚ and where b is the barrier width. Taking b51.2 A U 50.42 eV [20], we obtain T c ¯269 K. In Ref. [21], the distinctive range of T c is evaluated as T c ¯(1– 300) K for crystals with hydrogen bonds. This is in general agreement with the tunneling mechanism for proton motion observed in experiments with doped SrZrO 3 at 5 K,T ,300 K [22].
3. Concluding remarks The concept of proton polaron offers a new outlook on properties of the HTPC and yields predictions for hopping mobility, time between jumps, prefactor, dependence of activation energy on O–O distance and boundary temperature when tunneling predominates. These predictions are in agreement with the observed data. The model we consider may be regarded as a simplest illustration of polaron approach. This approach has much potential for yielding information about new proton conductors.
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