Non-stoichiometry, Fermi energy and work function of (La,Sr)MnO3. I. Theoretical model

Journal of Physics and Chemistry of Solids 62 (2001) 731±735

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Non-stoichiometry, Fermi energy and work function of (La,Sr)MnO3. I. Theoretical model T. Bak, J. Nowotny*, M. Rekas, C.C. Sorrell School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia Received 5 June 2000; accepted 15 August 2000

Abstract A model describing the effect of oxygen partial pressure on non-stoichiometry, in both cation and oxygen sublattices of oxide perovskite-type materials, such as (La,Sr)MnO3, and related changes in Fermi energy level, is derived. This model may be applied for evaluation of semiconducting properties of these materials from data of the deviation from stoichiometry. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Oxides; A. Surfaces; D. Semiconductivity; D. Electronic structure

1. Introduction Intensive research has been carried out on the characterisation of oxide materials, such as (La,Sr)MnO3 (LSM), (La,Sr)CoO3 (LSC), (La,Sr)FeO3 (LSF) and their solid solutions [1±4]. These materials, that appeared to be mixed conductors, have found an application as an air electrode for solid oxide fuel cells (SOFC's) [1±4]. From the viewpoint of this application, important properties of the electrode materials include their ability to conduct electricity via both electrons and ions (oxygen). The electronic conduction is closely related to chemical potential of electrons which also is termed as Fermi energy. Ionic conduction is related to oxygen non-stoichiometry. The purpose of the present paper is to establish a relationship between the two.

1. increase of the lattice oxygen content (decrease of oxygen non-stoichiometry), and 2. decrease of the Fermi energy level, EF, and increase of work function, f . Therefore, one should expect a close relationship between EF and non-stoichiometry. Performance of SOFC's requires that the reaction (1) takes place very fast resulting in effective transfer of mass and charge. Alternatively, however, increase of oxygen partial pressure, p(O2), results in the formation of oxygen chemisorbed species, such as O 2 O2…gas† 1 2e 0 ! 2O…ads† 2

2. Postulation of the problem The oxide electrode materials LSM, LSC and LSF have been applied as an air electrode. The main chemical reaction at this electrode is the transfer of oxygen from the gas phase into the oxide lattice O2…gas† 1 4e 0 ! 2O…lattice† 22

The reaction (1) represents oxygen incorporation resulting in the formation of negatively ionised oxygen species in the lattice. This reaction results, in consequence, in

…1†

* Corresponding author. Tel.: 161-2-9385-6465; fax: 161-29385-6467. E-mail address: [email protected] (J. Nowotny).

…2†

This process results in the formation of a chemisorptioninduced potential barrier retarding the process of charge transfer and oxygen incorporation into the lattice. Therefore, taking into account performance of SOFC's it is desired that the chemisorption-related surface barrier is reduced to a minimum. This may be achieved through 1. adjustment of performance temperature of SOFC's; 2. processing the surface of electrode materials. In order to assess the effect of performance or processing on

0022-3697/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0022-369 7(00)00230-4

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T. Bak et al. / Journal of Physics and Chemistry of Solids 62 (2001) 731±735

the chemisorption-related surface barrier it is essential to a mean for evaluation of this barrier. This evaluation requires understanding the impact of oxidation of the electrode material on its bulk semiconducting properties (determined by Fermi energy level in the bulk phase) and its surface semiconducting properties (determined by Fermi energy level at the surface and related work function data). Data describing the latter effect have been reported in Ref. [6]. Unfortunately, data on the Fermi energy of the perovskite-type oxide electrode materials at elevated temperatures are not available. Therefore, the purpose of this paper is to derive a model that can be used for evaluation of the Fermi energy level from the available data on oxygen non-stoichiometry [1±4]. Fermi energy is one of the most important quantity describing electronic structure of semiconductors and related defect chemistry [5]. For oxide materials the Fermi energy is determined by the concentration of both donor- and acceptor-type defects, such as anion and cation vacancies, and their ionisation degrees. These defects can be considered in terms of acceptor and donor states, respectively. Therefore, Fermi energy is a quantity which may be used in derivation of defect disorder models of non-stoichiometric compounds, if it is related to elevated temperatures at which a gas/solid equilibrium can be established [5]. In the equilibrium state the non-stoichiometry and related concentration of defects in oxide materials are determined by the parameters describing this equilibrium, such as temperature and oxygen partial pressure, p(O2). Concordantly, in equilibrium the Fermi energy, and related work function, could be related to both T and p(O2) [6]. The purpose of this paper is to derive a relationship between non-stoichiometry (caused by oxidation or reduction during high temperature processing) and Fermi energy for LSM, as an example. The purpose of the following paper [7] is to verify this model against experimental work function data for LSM, LSC and LSF [8]. 3. Mechanism of oxygen incorporation

000 z O2…gas† Y 3OO x 1 V 000 Mn 1 V Ln 1 6h

O2…ads† 1 e 0 Y O2…ads† 2

…6†

O2…ads† 2 1 e 0 Y O2…ads† 22

…7†

O…ads† 1 e 0 Y O…ads† 2

…8†

O…ads† 2 1 e 0 Y O22

…9†

Reactions (4) and (5) do not result in any electrical effect. Reactions (6)±(9) are accompanied by a charge transfer and, concordantly, lead to changes in surface potential. 4. Effect of temperature on oxidation and related electrical effects Mechanism of oxidation (or reduction), that is represented by the equilibria (1)±(9), depends essentially on temperature. The effect of temperature on the gas/solid equilibria for the oxygen/LSM system, can be considered within the following ranges: 1. high temperature range in which gas/solid equilibrium may be established very fast; 2. low temperature range in which the bulk phase is quenched and changes in oxygen partial pressure result in changes of chemisorption equilibria while the bulk phase is quenched; 3. intermediate temperatures in which changes of oxygen partial pressure results in changes of both bulk and surface composition. Reactions (1) and (4)±(9) may also be considered in terms of the changes of both Fermi energy, DEF, and work function, Df , using a simple ¯at band model of a semiconductor. In these considerations the effects at the oxygen/LSM interface will be considered in terms of the changes of both EF and f while knowledge of their absolute values is not required. 4.1. High temperature range

LSM is a non-stoichiometric compound involving defects in all three sublattices [9]. Its correct formula is (La12xSrx)MnO32d. In oxidising conditions, when oxygen partial pressure is higher than 10 25 Pa in the range 873± 1273 K, LSM is a p-type semiconductor [9]. Oxygen incorporation into the LSM lattice may be expressed by the following general reaction [10]: 3 2

Thus formed adsorbed species may be ionised according to the following reactions:

…3†

This reaction involves several steps, such as adsorption and dissociation O2…gas† Y O2…ads†

…4†

O2…ads† Y 2O…ads†

…5†

According to reaction (3) oxidation of LSM at elevated temperatures results in a decrease of the concentration of oxygen vacancies (that are donors) and the formation of both La and Mn vacancies (both are acceptors). Concordantly, oxidation leads to lowering the Fermi energy level from the EF1 level before oxidation to the EF2 level after oxidation and, consequently, results in increase of work function from f 1 to f 2 where the relation between the change of EF (DEF) and the work function change (Df ) is (Fig. 1) DEF ˆ 2Df

…10†

Isothermal work function changes of metal oxides as a function of oxygen partial pressure, p(O2), may be expressed

T. Bak et al. / Journal of Physics and Chemistry of Solids 62 (2001) 731±735

733

according to the following relation: 1 1 d…Df† ˆ mf kT d ln p…O2 †

…11†

where the parameter mf is related to defect chemistry of the overmost surface layer. Accordingly, mf may be used for veri®cation of defect disorder models for this layer. Fermi energy, EF, or chemical potential of electrons, is a basic quantity in the Fermi±Dirac (F±D) statistics 1 fn …E; T† ˆ E 2 EF 1 1 exp kT fp …E; T† ˆ 1 2 fn …E; T†

…12†

Fig. 1. Flat band model of p-type semiconductor illustrating the effect of oxygen partial pressure, p(O2), on the Fermi energy level and work function at high temperatures.

…13†

where fn(E,T ) and fp(E,T ) are the distribution functions describing the electrons and electron holes, respectively, and the Fermi energy [5]. The Fermi energy has a complex physical meaning. Its direct experimental determination is dif®cult. The purpose of this paper is to derive a relationship between EF and non-stoichiometry for LSM. This relationship then may be used for the determination of EF from nonstoichiometry data of this material [1±4]. 4.2. Low temperature range At lower temperatures (at which the gas/solid equilibrium cannot be reached due to the kinetic reason) oxidation of LSM results in a shift of equilibria (4)±(7) into the right, leading to an increase of the concentration of adsorbed and chemisorbed species while the bulk concentration of defects (donors and acceptors) remains unchanged. In this case oxidation results in the formation of a surface charge, related to the formation of chemisorbed oxygen species (reactions (6) and (7)), and related space charge while the Fermi energy of the bulk phase remains constant. This type of oxidation will lead to an increase of work function by the component f s that is related to band curvatures caused by the presence of the chemisorption-induced surface charge (Fig. 2). 4.3. Intermediate temperature range In this temperature range oxidation results in a change of both bulk non-stoichiometry (related EF level in the bulk phase) and a shift in oxygen chemisorption equilibria. The surface charge can be determined using WF measurements. It is dif®cult to distinguish between these two effects because both result in an increase of EF. However, both effects have different rates of appearance. Therefore, simultaneous monitoring of both bulk and surface electrical properties as a function of time is required in order to characterise the extent of these effects.

Fig. 2. Flat band model of p-type semiconductor illustrating the effect of oxygen chemisorption on the Fermi energy level, work function and surface potential barrier.

5. Basic relationships LSM exhibits defects within all three sublattices. Its defect disorder models should be considered within a low p(O2) Ð that corresponds to oxygen de®cit regime, and high p(O2) that corresponds to oxygen excess regime. 5.1. Oxygen de®cit regime The predominant defects of LSM in this regime are oxygen vacancies that are compensated by singly ionised Sr in the La sites. Accordingly, interaction between oxygen and LSM under low oxygen activity may be represented by the following equilibrium: O2 1 2VOzz 1 4e 0 Y 2OO

…14†

Then the chemical formula of LSM may be expressed by ‰…LaLa †12x …Sr 0La †x Š‰MnMn Š‰…OO †32d …VOzz †d Š

…15†

The chemical potential of oxygen and of oxygen vacancies may be expressed by respective relations

m…O2 † ˆ mo …O2 † 1 kT ln p…O2 †

…16†

d 32d

…17†

m…VOzz † ˆ mo …V O † 1 kT ln

where d denotes de®cit in the oxygen sublattice and mo …O2 †

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T. Bak et al. / Journal of Physics and Chemistry of Solids 62 (2001) 731±735

is a standard chemical potential of oxygen. Concordantly, equilibrium (14) assumes the form DG ˆ 2m…O2 † 2 2m…VOzz † 2 4m…e 0 † ˆ 0

…18†

000 0 DG ˆ m…V 000 Mn † 1 m…V La † 2 6m…e † 2

m…V 000Mn † ˆ mo …V 000Mn † 1 kT ln

m…e 0 † ˆ EF

m…V 000La † ˆ mo …V 000La † 1 kT ln

0

According to Eqs. (16)±(18) the term m (e ) may be expressed as 1 m…e 0 † ˆ 2 ‰mo …O2 † 1 2mo …VOzz †Š 4   kT d 2 ln p…O2 † 1 2 ln 32d 4

…20†

Assuming that 0

o

0

0

m…e † 2 m …e † ˆ Dm…e † ˆ DEF we obtain DEF ˆ 2

  kT d ln p…O2 † 1 2 ln 4 32d

Differentiation of Eq. (22) results in   1 1 6 1 ˆ2 11 mF 4 3 2 d md

…21†

…22†

…23†

where 1 2lnd ˆ md 2ln p…O2 †

…24†

and 1 1 2m…e 0 † ˆ mF kT 2ln p…O2 †

…25†

where mF is a parameter describing the effect of p(O2) on EF related to non-stoichiometry. 5.2. Oxygen excess regime The oxygen sublattice of the LSM perovskite-type structure is close packed and, therefore, it cannot accept additional oxygen in interstitial positions. Therefore, the apparent oxygen excess, which has been determined experimentally, may only be considered in terms of cation vacancies which are formed according to the following equilibrium [9]: 3 2

000 O2 1 6e 0 Y 3OO 1 V 000 Mn 1 V La

…26†

Then the chemical formula of (La12x,Srx)MnO31d may be expressed as ‰La…12x†=…11d=3† Sr…x=…11d=3†† …V 000 La †…d=3†=…11d=3† Š  ‰Mn…1=…11d=3†† …V 000 Mn †…d=3=…11d=3†† ŠO3 where d is related to the deviation from stoichiometry. The reaction (26) is in equilibrium when

…27†

m…O2 † ˆ 0

…28†

where

where m (e 0 ) is the chemical potential of electrons or Fermi energy …19†

3 2

d 3

d 3

…29† …30†

Combination of Eqs. (16) and (28)±(30) results in the following expression for Fermi energy:   1 o 000 3 m…e 0 † ˆ m …V Mn † 1 mo …V 000La † 2 mo …O2 † 6 2   kT d 3 1 2 ln 2 ln p…O2 † …31† 6 2 3 Concordantly, the changes of the Fermi energy vs changes of p(O2) may me expressed as DEF ˆ Dm…e 0 † ˆ m…e 0 † 2 mo …e 0 †   kT d 3 ˆ 2 ln 2 ln p…O2 † 6 2 3 Differentiation of Eq. (32) yields   1 1 3 1 ˆ 2 1 mF 6 2 md

…32†

…33†

Eqs. (22), (23), (32) and (33) are the key equations describing the relationship between non-stoichiometry and electrical properties (in terms of the Fermi energy) for LSM. These relations may also be applied for LSC, LSF and their solid solutions that represent a similar defect disorder model. 6. Conclusions A model describing the relationship between non-stoichiometry of LSM and its Fermi energy level is derived. This model may be applied for the determination of the effect of p(O2) on the changes of EF and related semiconducting properties in both oxygen excess and oxygen de®cit regimes at elevated temperatures at which direct experimental determination of EF is dif®cult. The model derived for the oxygen de®cit regime of LSM may also be applicable for both LSC and LSF. Concordantly, this model may be used for evaluation of semiconducting properties of perovskitetype oxide materials from non-stoichiometry data corresponding to the gas/solid equilibrium. References [1] M.H.R. Lankhorst, H.J.M. Bouwmeester, H. Verweji, Phys. Rev. Lett. 77 (1996) 2989. [2] H. Tagawa, J. Mizusaki, H. Takai, Y. Yonemura, H. Minamiue, T. Hashimoto, in: F.W. Poulsen, N. Bonanos, S. Linderoth, M. Mogensen, B. Zachau-Chrisiansen (Eds.),

T. Bak et al. / Journal of Physics and Chemistry of Solids 62 (2001) 731±735

[3] [4] [5] [6]

Proceedings of the 17th Riso International Symposium on Materials Science of High Temperature Electrochemistry Ceramics and Metals, Roskilde, Denmark, 1996, p. 437. M.H.R. Lankhorst, H.J.M. Bouwmeester, H. Verweij, Solid State Ionics 96 (1997) 21. J. Mizusaki, M. Yoshihiro, S. Yamauchi, K. Fueki, J. Solid State Chem. 58 (1985) 257. P. Kofstad, Non-stoichiometry, Electrical Conductivity and Diffusion in Binary Metal Oxides, Wiley, New York, 1972. J. Nowotny, M. Sloma, in: J. Nowotny, L.-C. Dufour (Eds.),

[7] [8] [9] [10]

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Surface and Near-Surface Chemistry of Metal Oxides, Elsevier, Amsterdam, 1988. T. Bak, J. Nowotny, M. Rekas, C.C. Sorrell, J. Phys. Chem. Solids 62 (2000) 737±742. S.P.S. Badwal, T. Bak, S.P. Jiang, J. Love, J. Nowotny, M. Rekas, C.C. Sorrell, E.R. Vance, J. Phys. Chem. 62 (2000) 723±729. J. Nowotny, M. Rekas, J. Am. Ceram. Soc. 81 (1998) 67. F.A. Kroeger, Chemistry of Imperfect Crystals, North Holland, Amsterdam, 1974.