Predicting metallic conductivity in oxides from simple chemical criteria

ARTICLE IN PRESS

Journal of Physics and Chemistry of Solids 68 (2007) 331–336 www.elsevier.com/locate/jpcs

Predicting metallic conductivity in oxides from simple chemical criteria Samir F. Matar,1, Guy Campet Institut de Chimie de la Matie`re Condense´e de Bordeaux, CNRS, University Bordeaux 1. Pessac, France Received 2 June 2006; received in revised form 20 September 2006; accepted 17 November 2006

Abstract The delocalization of d electrons in oxides can be described with a simple model taking into account the electronegativity and the chemical hardness. The metallic conductivity appears when (i) the electronegativity is high involving large conduction and valence bandwidths and (ii) the chemical hardness is low. From this we propose a delocalization criterion McritZ+2.8w whereby if Mcrit is larger than 18 eV a metallic conductivity should occur. The validity of this criterion is checked in actual oxide systems. We also discuss its limits in p-elements oxides with the help of electronic band structure calculations within the density functional theory framework. We propose that this simple wZ model can be used for the understanding of the electronic properties of different classes of oxides. r 2006 Elsevier Ltd. All rights reserved. Keywords: A. Oxides; C. ab initio calculations

1. Introduction Most of the oxides are insulators. However some of them, mostly d transition elements oxides, show a metallic conductivity. The discoveries of high temperature supraconductivity in cuprates and of colossal magnetoresistance in manganese oxides have enhanced interest of the scientific community for metallic oxides with the purpose of establishing trends of electronic properties within different classes of systems. These materials and their magnetic and electrical properties have been widely and extensively studied and one can mention the leading works of John Goodenough [1] who carried out a careful examination of the band structure of several systems but not in a generalized manner. Therefore, it is useful to look for simple criteria predicting the localization or the delocalization of electrons within a simple framework. A model was proposed by Zaanen, Sawatzky and Allen (ZSA) [2], based on two main parameters, namely the on site Coulomb parameter U mainly acting between d orbitals and the band width W. Torrance et al. [3] used the ZSA concept in the case of metallic oxides. An illustration of this model is Corresponding author. Fax: +33540002761. 1

E-mail address: [email protected] (S.F. Matar). http://www.m3pec.u-bordeaux1.fr/matar.

0022-3697/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2006.11.013

given in Fig. 1 where D is relevant to the charge transfer parameter. However, these values are hardly determined either from band-structure calculations or from experimental tools such as photoelectron spectroscopy [4]. Therefore, in this study we propose an original alternative analysis based on electronegativity w and chemical hardness Z which are more accessible form literature. 2. The vg (chi eta) model In our model we use electronegativity w and chemical hardness Z as defined by Parr and Pearson [5] and Parr and Chattaraj [6] as an illustrative but not limiting approach (Fig. 2). Chemical hardness Z represents half of the energy gap between the filled valence band of O(2p6) character and the first empty conduction band of cationic s-like character. In our model the d orbitals are not explicitly accounted for. Considering the case of d elements, the conduction band corresponds to the ns0 orbitals, if d orbitals are partially or totally occupied. Still in our model electronegativity represents the potential energy located between the bottom of the ns metal level and the top of the valence band. The numerical values for w and Z are calculated with the empirical model that was previously proposed by Portier et al. [7–10]. In this work, we use the

ARTICLE IN PRESS 332

S.F. Matar, G. Campet / Journal of Physics and Chemistry of Solids 68 (2007) 331–336

In these equations z is the formal oxidation number of the considered cation, r its ionic radius in A˚ and a is an atomic number depending term. From our introduction of w and Z our model is hence distinct from the ZSA and Torrance et al. ones. Note that the same parameters are implicitly present. Indeed, electronegativity w is related to the covalence of the metal-s oxygen-p bond and, as a result, to the bandwidth W: the higher is w, the larger is W. Chemical hardness Z is approximately equal to 12(DW) (Fig. 2) in the Torrance description. Let us recall that the wZ model does not account for the correlation parameter U. However, if w is high, i.e. W is large, and if simultaneously Z is weak, that is to say, D is also weak, the probability that partially filled d orbitals overlap with metal s orbitals or oxygen p orbitals is large, leading thus to metallic conductivity. 3. Results 3.1. d-elements oxides Fig. 1. Mott-Hubbard insulator (a) metallic oxide (b) (following Torrance [3]).

3.1.1. Binary oxides Metal oxides are located in a restricted domain of w–Z space. We explicit this from a scattered plot for a series of systems tabulated in Appendix A (Table A1). Fig. 3 shows the relevant points related with the oxide systems exhibiting metallic behavior. They are located in between lines which can be linearly parameterized as Z E2.8w+19 and ZE2.8w+18. Monoxides (or sub-oxides VO, TiO, etc.) are not taken into account due to their complex structure as well substoichiometry [23]. From Fig. 3, it is possible to define an empirical condition for metallic conductivity in a d transition metal oxide such as metallic character criterion parameter, Mcrit.: M crit: Z þ 2:8w.

Fig. 2. Schematic representation of electronegativity w and chemical hardness Z in an insulating oxide.

If Mcrit. is larger than 18 eV the oxide is metallic, if Mcrit. is lower than 18eV the oxide is insulator. In Appendix A 4.0

next formulae to derive overall system electronegativity woxide and chemical hardness Zoxide used throughout.

3.5 3.0

Value

Unit

Cationic electronegativity Absolute cationic electronegativity Cationic acid strength Cationic chemical hardness Oxide electronegativity Oxide chemical hardness

wionE0.274 z0.15 z r0.01 r+1+a |w|ionE2.976wion +0.612 ICPE0.434 Ln(z)0.87 Ln(r)1.38wion+2.07 ZionE3.745 ICP0.612

Pauling unit

eV

woxideE0.45|w|+3.36

eV

ZoxideE4.25ICP0.67

eV

eV Dimensionless

2.5 η (eV)

Property

2.0 1.5 1.0 0.5 0.0 5.0

5.2

5.4

5.6

5.8

6.0

6.2

6.4

6.6

6.8

7.0

χ (eV)

Fig. 3. w–Z chart for metal oxides meant to show the gathering of points between two limits (see text). Numerical data are listed in Appendix 1.

ARTICLE IN PRESS S.F. Matar, G. Campet / Journal of Physics and Chemistry of Solids 68 (2007) 331–336

333

Fig. 4. Variation of delocalization criterion Mcrit. with oxidation number z and ionic radius r (A˚).

(Table A2) we provide a whole set of Mcrit values for real and hypothetical oxide systems. Another feature of our approach is given for the variation of Mcrit with ionic radius r(A˚) and formal charge number, z. This is given in Fig. 4. From the gathering of the points it can be noticed that metallic conductivity occurs when the charge number is high (zX3) and the ionic radius is small (rp0.7 A˚). These characteristics correspond to a high electronegativity together with a low chemical hardness meaning that in our model the electron affinity is close to the work function. 3.1.2. Ternary oxides It is not possible to propose a straightforward criterion like Mcrit. for ternary oxides. However, we shall show that the ‘‘w Z’’ concept can help for explaining electronic properties of such oxides. When two binary oxides react to produce a ternary oxide, it is logical to assume that the electronegativities of the pristine compounds will tend to align. Let us examine the case of BaTiO3: the electronegativity of BaO is equal to 5.0 eV, that of TiO2 is 6.2 eV. One can assume that the electronegativity of BaTiO3 arises from the geometrical mean value which is 5.57 eV. On the contrary the calculation of the chemical hardness Z is more difficult. However, as 2Z represents the difference of energy between the top of the valence oxygen 2p band and the bottom of the ns conduction band, it is consistent to think that the chemical hardness of BaTiO3 is close to that of the component having the lowest 2Z band gap Z(TiO2) ¼ 1.5 eV; Z(BaO) ¼ 2.4 eV. Indeed in BaTiO3 and in TiO2, the band gap (2Z in this case) separates the oxygen 2p(6) orbitals from the 3d(0) empty orbital of tetravalent Ti4+. This assumption fits well with experimental data whereby Z(BaTiO3) ¼ 1.55 eV [11]. On this basis, it is possible to roughly approximate w and Z in metallic double oxides. As another example we can examine the case of SrMnO3 which is an anti-ferromagnetic insulator while MnO2 is a

Table 1 Mcrit, w,Z values for binary and ternary oxides discussed in the text Oxide

w (eV)

Z (eV)

Mcrit. (eV)

Property

MnO2 SrMnO3 V2O3 LaVO3 ‘‘CoO2’’ LiCoO2 PbO2 Tl2O3

6.18 5.62 5.74 5.56 6.47 5.48 5.90 5.60

2.14 2.14 2.68 2.68 1.60 1.87 1.72 2.01

18.95 17.43 18.31 17.80 19.20 16.79 17.76 17.24

Metal Insulator Metal/insulator Insulator Metal Insulator ? ?

metal. Without taking into account the structural disparity of both compounds, one can explain the differences in conductivity by the fact that the electronegativity of SrMnO3 is lower than that of manganese dioxide, which involves the bandwidth magnitude. This leads to the Mcrit. values in Table 1 which agree with the electronic characters of both systems. The same analysis can be done for V2O3 and LaVO3, whose computed Mcrit. values in Table 1 point to expected insulating and metallic characters, respectively. On the contrary PbO2 and Tl2O3 which is experimentally reported to be metallic are found in Table 1 as insulators. One has to invoke here the substoichiometric character known for these two systems (e.g. presence of both PbIII and PbIV in PbO2) (see Appendix A, Table A1). 3.1.3. Bronzes and intercalation compounds The case of bronzes and intercalation compounds is also interesting: when conduction electrons are provided by reduction then metallic conductivity appears. Bronzes of tungsten with alkali metals with general formulation Mx WO3 [12] are well known to be non-stoichiometric compounds; they have a quasi-bidimensional metallic character. The electrical properties of cathodic materials

ARTICLE IN PRESS S.F. Matar, G. Campet / Journal of Physics and Chemistry of Solids 68 (2007) 331–336

334

shape of the crystal potential in large voids. As a result, by inserting empty spheres into the respective unit cells we were able to keep the linear overlap of any two physical spheres below 7% and the overlap of any pair of physical and empty spheres below 19%. The calculations account for all electrons but the explicit basis set accounts for outermost valence states of each principal quantum number, i.e. s, p, d and f. Here we used ns (l ¼ 0), np (l ¼ 1) and (n1)d (l ¼ 2) orbitals. Such a limited valence set is compensated by adding lmax+1 states (here lmax ¼ 2) were added to the basis set to account for charge residues, which should be around 0.1 electron for ensuring convergence. The Brillouin zone sampling of the respective structures was done using an increased number of k points in the irreducible wedge. This way we were able to ensure convergence of our results with respect to the fineness of the k space grid. Self-consistency was achieved by employing an efficient algorithm for convergence acceleration [19]. The reader is referred to our former works [14,20] for details on the theory and on case studies of non-magnetic and magnetic oxide systems.

used in Li ion batteries can be also analyzed in the same way. LiCoO2 is an insulator [13 and therein cited works] and Mcrit. which is found smaller than the 18 eV critical value, fits with this observation. By deintercalation of lithium, phases with formula Li1–xCoO2 are obtained. For x40.5 i.e., for compositions close to those of the pseudooxides and hypothetical ‘‘CoO2’’, a metallic conductivity appears in agreement with the calculated Mcrit. value which is found much larger than 18 eV (Table A2). 3.2. The problem of p-elements oxides Most of the binary oxides of p-elements are insulators. However, some of them are degenerate semiconductors. This is the case of In2O3 when doped with SnO2 (ITO) which has studied by us [14] both experimentally and theoretically—see below. Two oxides, PbO2 and Tl2O3, are supposed to be metallic as discussed above but are found to be insulating from Table 1. This point would call for a complementary approach for assessing the problem. We here provide an interpretation in the framework of ab initio electronic structures of some p-metal oxides within the density functional theory (DFT) framework [15].

3.2.2. Rutile type PbO2 and other isochemical systems b-PbO2 is among the different crystal varieties of lead dioxide; it is used in a mixture with orthorhombic a form is acid lead high-energy batteries. The site projected DOS and the band dispersion curves are shown in Fig. 5. The energy is taken with respect to the Fermi level at the top of the valence band VB. All over the VB the DOS are relevant to Pb(6s) states (low lying O(2s) states are not shown) and Pb(6p) mixing with O(2p) states. The valence band is continued in the conduction band by low intensity extended DOS of antibonding states with a major s-character from lead. These DOS correspond to large dispersion bands as it can be seen on the right-hand side of Fig. 5 showing the band structure. This illustrates the

3.2.1. Computational method In this work we used the accurate augmented spherical wave ASW method in a scalar relativistic implementation due to the presence of heavy elements such as Pb [16,17]. The exchange-correlation effects in DFT framework have been accounted within a generalized gradient approximation (GGA) scheme according to Perdew, Burke and Ernzerhof functional [18]. Since this method uses the atomic sphere approximation (ASA), we inserted so-called empty spheres into the open crystal structures (here Rutile and Cubic perovskite types). These empty spheres are pseudo-atoms without a nucleus, used to model the correct

10

E-EF (eV)

5

0

-5

-10

Pb O

Γ

M

X

Z

Γ

A

Fig. 5. Density of states and band structure of b-PbO2 within GGA approximation.

R

Z

ARTICLE IN PRESS S.F. Matar, G. Campet / Journal of Physics and Chemistry of Solids 68 (2007) 331–336

335

8

-4

Ba Pb O

E-EF (eV)

4

-8 R

X

Γ

M

R

Γ

Fig. 6. Site projected DOS and band structure BaPbO3 perovskite in the GGA approximation.

weak metallic behavior. Another major feature is observed at the G point (center of the BZ) of the extension of the bands of oxygen-like states around and above the Fermi level which results in a hole like metallic system. While these observations are in agreement with the results of Heinemann et al. [21] obtained with a local density approximation (LDA) based calculations, we stress this result by the presentation in Fig. 6 of the DOS and the band structure of BaPbO3 perovskite system at experimental lattice volume, assuming a cubic perovskite lattice. There can be observed large similitudes with Fig. 5 and the same discussion as above: the extension of oxygen states above EF and the dispersion of the bands above EF. The feature of a compensated metal due to the simultaneous presence of a hole-like Fermi surface at R and along M–R and electron-like at G point is found within BaPbO3. Furthermore we should point out the flat band behavior in both figures. i.e., along the G-M-X-G directions in PbO2 and along M–R in BaPbO3. This flat band feature at the top of the VB in conjunction with the steep band behavior around the G point can be inscribed within the ‘‘scenario’’ [22] of superconductivity relating the presence of itinerant electrons (s-like dispersed bands) with local pairs (zero-slope bands) near the Fermi level. Whereas we reserve the development of these results to future works, we point to the potentiality of BaPbO3 as a high TC superconductor. The results obtained for Tl2O3 (cubic, Bixbyite-type structure, see [14] for details) point out to a semi-metal with similar features to above discussed systems. The holes and electrons in the valence and conduction bands occurring in the manner described above can also be found. The investigation of the p-metal oxides within the DFT point out to semi-metallic systems which are characterized by the property of compensated metals and a large dispersion of s-like states above the Fermi level. These

features are somehow complementary of the ‘‘wZ’’model devised above for d-electron systems. 4. Conclusion The delocalization of d electrons in oxides can be described with a simple model taking into account the electronegativity and the chemical hardness calculated on the basis of ionic radius and charge number. Indeed, this model expresses qualitative concepts concerning the conditions favorable to the delocalization of d electrons in the s band. Metallic conductivity appears when (i) the metal–oxygen or metal–metal distance is short (small ionic radii), (ii) the polarizing power of the d element is high (high charge number). The works of Pearson have been recently extended to mechanical aspects [24] and a presentation of the ‘‘maximal hardness’’ has been introduced. So it might reveal interesting to examine an extension of the wZ model in this direction in order to classify hard materials—see for instance the review work [25]. These developments are underway. Acknowledgments The authors are indebted and they dedicate this work to Dr. Josik Portier for his constructive ideas which largely helped to build the presented conceptual approach. Computations were carried out on the main frame computers of the M3PEC Mesocenter facilities of the University Bordeaux1 (http://www.m3pec.u-bordeaux1.fr) partially financed by the ‘‘Re´gion Aquitaine’’. Appendix A See Tables A1 and A2.

ARTICLE IN PRESS S.F. Matar, G. Campet / Journal of Physics and Chemistry of Solids 68 (2007) 331–336

336

Table A2 (continued )

Table A1 Numerical data used in Fig. 3 Oxide

woxide (eV)

Zoxide (eV)

Property [1,2]

Ti2O3 V2O3 NbO2 VO2 TaO2 MoO2 WO2 MnO2b RuO2 ReO2 RhO2 OsO2 IrO2 PtO2 CrO2 ReO3

5.51 5.74 5.98 6.03 6.05 6.11 6.12 6.18 6.21 6.23 6.24 6.25 6.27 6.29 6.30 6.84

3.57 2.68 1.97 2.32 1.67 1.57 1.46 2.14 1.34 1.20 1.33 1.08 1.03 0.97 1.46 0.00

M/I M M M/I M M M M M M M M M M M M

Table A2 Criterion for d electrons delocalization (Mcrit.) for oxides with cations in octahedral coordination Oxide

w (eV)

Z(eV)

Mcrit. (eV)

‘‘Ag2O3’’ ‘‘Au2O3’’ ‘‘Au2O5’’ ‘‘Cr2O5’’ ‘‘Cu2O3’’ ‘‘FeO’’ ‘‘FeO2’’ ‘‘Ir2O5’’ ‘‘Mo2O5’’ ‘‘Nb2O3’’ ‘‘Ni2O3 HS’’ ‘‘Ni2O3 LS’’ ‘‘NiO2’’ ‘‘Os2O5’’ ‘‘Os2O7’’ ‘‘OsO3’’ ‘‘Pd2O3’’ ‘‘PdO2’’ ‘‘Pt2O3’’ ‘‘Re2O5’’ ‘‘ReO2’’ ‘‘Ru2O3’’ ‘‘Ru2O5’’ ‘‘Ta2O3’’ ‘‘TaO2’’ ‘‘W2O5’’ ‘‘Zr2O3’’ Co2O3 HS Co2O3 LS CoO Cr2O3 CrO CrO2 CrO3 Cu2O CuO Fe2O3 HfO2 Ir2O3

6.08 5.89 6.56 6.58 6.29 5.75 6.33 6.54 5.97 5.62 5.93 5.96 6.28 6.53 7.11 6.82 5.91 6.24 5.97 6.46 6.17 5.90 6.48 5.52 5.78 6.09 5.36 6.01 6.05 5.71 5.97 5.64 6.26 6.91 5.72 5.95 6.05 5.80 5.97

0.50 0.90 0.35 0.84 0.81 1.05 0.85 0.47 2.68 2.69 1.99 2.14 1.84 0.46 1.10 0.27 1.23 1.09 1.27 0.74 1.30 1.67 0.77 3.10 2.72 2.12 2.75 1.58 1.82 1.40 1.75 1.43 1.40 0.14 0.05 0.42 1.21 2.46 1.38

17.05 16.91 18.20 18.74 17.92 16.7 18.08 18.25 18.9 17.97 18.12 18.34 18.91 18.23 18.2 18.28 17.29 18.05 17.50 18.31 18.08 17.72 18.38 18.12 18.44 18.68 17.3 17.93 18.28 16.9 17.98 16.8 18.43 18.93 15.5 16.6 17.66 18.25 17.60

Conductivity

I

? ? I I I M I I I I ?

Oxide

w (eV)

IrO2 Mn2O3 MnO MnO2 b MoO2 MoO3 NbO2 NiO OsO2 PdO PtO2 ReO3 Rh2O3 RhO2 RuO2 Ta2O5 Ti2O3 V2O3 VO2 WO2

6.24 5.81 5.49 6.14 5.70 6.24 5.87 5.65 6.24 5.65 6.25 6.75 5.92 6.21 6.18 6.05 5.86 5.78 6.07 5.82

Z(eV) 1.02 2.27 1.97 2.08 3.25 1.97 2.31 1.97 1.00 1.15 0.96 0.01 1.66 1.31 1.32 2.17 1.90 2.42 2.06 2.68

Mcrit. (eV)

Conductivity

17.99 18.06 16.9 18.77 18.75 18.95 18.29 17.3 17.97 16.5 17.97 18.37 17.76 18.20 18.12 18.6 17.84 18.14 18.56 18.49

M I I M M I I/M I M I ? M I M M I M/I I/M I/M M

The quoted oxides are hypothetical.

References [1] J.B. Goodenough, Metallic Oxides, Pergamon Press, 1971. [2] J. Zaanen, G.A. Sawatzki, J.W. Allen, Phys. Rev. Lett. 55 (1985) 418. [3] B. Torrance, P. Lacorre, C. Asavaroengchai, R. Metzger, Physica C 182 (1991) 351. [4] P.A. Cox, Transition Metal Oxides, Clarendon Press, Oxford, 1992. [5] R.G. Parr, R.G. Pearson, J. Am. Chem. Soc. 105 (1983) 7512. [6] R.G. Parr, P.K. Chattaraj, J. Am. Chem. Soc. 113 (1991) 1854. [7] J. Portier, G. Campet, J. Korean Chem. Soc. 41–8 (1997) 437. [8] J. Portier, G. Campet, J. Etourneau, B. Tanguy, Ecl. Quim, Sao˜ Paulo 17 (1992) 63–78. [9] J. Portier, G. Campet, J. Etourneau, B. Tanguy, J. Alloys Compd. 209 (1994) 285–289. [10] G. Campet, J. Portier, M.A. Subramanian, Mater. Lett. 58 (2004) 437. [11] H.H. Kung, H.S. Jarret, A.W. Sleight, L. Ferretti, J. Appl. Phys. 48 (1977) 2463. [12] G. Hollinger, P. Pertosa, J.P. Doumerc, F.J. Himpsel, B. Reihl, Phys. Rev. B 32 (1985) 1987. [13] M. Me´ne´trier, I. Saadoune, S. Levasseur, C. Delmas, J. Mat. Chem. 9 (1999) 1135. [14] S.F. Matar, A. Villesuzanne, G. Campet, J. Portier, Y. Saı¨ kali, CR Acad. Sci. Paris Ser. IIc 4 (2001) 367. [15] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133; P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [16] A.R. Williams, J. Ku¨bler, C.D. Gelatt Jr, Phys. Rev. B 19 (1979) 6094. [17] V. Eyert, Int. J. Quant. Chem. 77 (2000) 1007. [18] J. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [19] V. Eyert, J. Comp. Phys. 124 (1996) 271. [20] S.F. Matar, Progr. Solid State Chem. 31/4 (2004) 239–299. [21] M. Heinemann, H.J. Terpstra, C. Haas, R.A. de Groot, Phys. Rev. B 52 (1995) 11740. [22] S. Deng, A. Simon, J. Ko¨hler, Solid States Sci. 2 (2000) 31. [23] S. Bartkowski, M. Neumann, E.Z. Kurmaev, V.V. Fedorenko, S.N. Shamin, V.M. Cherkashenko, S.N. Nemnonov, A. Winiarski, Phys. Rev. B 56 (1997) 10656. [24] R.G. Pearson, J. Chem. Educ. 76 (1999) 267. [25] S.F. Matar, E. Betranhandy, M. Nakhl, M. Zakhour-Nakhl, N. Ouaini, Progr. Solid State Chem. 34 (2006) 21.