Microscopic and macroscopic dielectric description of mixed oxide thin films

Microscopic and macroscopic dielectric description of mixed oxide thin films F. J. Ferrer Centro Nacional de Aceleradores, C/Thomas A. Edison, 7, E-41092 Sevilla, Spain F. Yuberoa兲 Instituto de Ciencia de Materiales de Sevilla (CSIC-Universidad de Sevilla) C/Américo Vespucio 49, E-41092 Sevilla, Spain J. A. Mejías Departamento de Ciencias Ambientales, Universidad Pablo de Olavide, Ctra Utrera km 1, E-41013 Sevilla, Spain F. J. García-Lopez Centro Nacional de Aceleradores, C/Thomas A. Edison, 7, E-41092 Sevilla, Spain A. R. González-Elipe Instituto de Ciencia de Materiales de Sevilla (CSIC-Universidad de Sevilla) C/Américo Vespucio 49, E-41092 Sevilla, Spain 共Received 9 July 2007; accepted 31 August 2007; published online 30 October 2007兲 Compact Si–Ti–O and Si–Zr–O mixed oxide thin films are studied by optical characterization 共refractive index, band gap energy兲and local probes 共Auger parameter obtained by x-ray photoelectron spectroscopy兲. Interpretation of the obtained results is discussed in the framework of the classical dielectric theory that correlates the macroscopic refractive index to the microscopic electronic polarizability of each particular ion in the compound through the Lorentz-Lorenz relationship. Quantum mechanical cluster calculations have also been performed to support the correlations obtained between the experimental findings. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2801402兴 I. INTRODUCTION A way to precisely control the optical properties of oxide materials is to adjust the relative concentration of two single oxides mixed in a single phase.1–4The range of variation of the refractive index nand extinction coefficient kachieved in this way can be very wide, particularly if the two oxides mixed together have rather different nand band gap energies Eg. In the final mixed oxides, the actual values of the macroscopic quantities n,k, and Egare tightly connected with the local electronic structure of the materials, whereby the polarizability of the constituent ions, or the existence of specific electronic transitions plays a key role. In this context, it is of key importance to be able to distinguish between just mixed phases 共solid solution of pure single oxide phases兲or the formation of advanced materials with characteristic local and extensive properties. Information about microscopic electronic parameters such as binding energies of certain photoemission peaks, Auger parameters, extra-atomic relaxation energies, or the initial state energies of the system before the photoemission event can be gathered by means of x-ray photoemission spectroscopy 共XPS兲.5,6However, the establishment of relationships between these local electronic parameters and extensive optical properties such as nor Egof a given material has only been intended a few times in the literature.7,8In this context, the use of the so called Auger parameter5,9derived from the XPS measurements can be a powerful approximation that can justify the changes in the optical properties of the mixed oxide materials as a function of their composition. Besides, the use of quantum mechanical calculations with cluster models may provide an extra theoretical framework to account for the changes in their electronic parameters.10,11 In the present work, we study the connection between the optical properties and some electronic parameters derived from XPS analysis of mixed oxides materials 共Si–Ti–O and Si–Zr–O thin films with different M/Si proportions, M: Ti and Zr兲. These two systems have been chosen due to the large range of variation between SiO2and TiO2/ZrO2single oxides for their refractive index 共e.g., around 1.45 and 2.55/2.10 for bulk SiO2and TiO2/ZrO2, respectively兲12–14 and their band gap energies 共e.g., 8.5 and 3.2/5.0 eV for bulk SiO2and TiO2/ZrO2, respectively兲.4,15 A preliminary study for the Si–Ti–O mixed oxide thin films can be found in Ref. 16. II. EXPERIMENTAL Mixed oxides Si–Ti–O and Si–Zr–O thin films, with thicknesses around 100–300 nm, have been prepared by ion beam induced chemical vapor deposition17 共IBICVD兲and plasma enhanced chemical vapor deposition 共PECVD兲.18 Samples with different Si/Ti and Si/Zr atomic ratios were obtained by changing the relative partial pressures of the corresponding precursors of Si, Ti, and Zr. Thus, the Si–Ti–O a兲Author to whom correspondence should be addressed. Electronic mail: [email protected] JOURNAL OF APPLIED PHYSICS 102, 084112 共2007兲 0021-8979/2007/102共8兲/084112/7/$23.00 © 2007 American Institute of Physics102, 084112-1 Downloaded 01 Mar 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp thin films with more than 10% Ti were prepared at room temperature by IBICVD with 400 eV O2 +ions, using Si共C2H5O兲3Cl and TiCl4volatile precursors and O2 +ions. Si–Ti–O samples with less than 10% Ti were prepared by PECVD at 523 K.18 The Si–Zr–O thin films were also prepared by IBICVD at room temperature using 共CH3CH2O兲3SiH and Zr关O共CH2兲3CH3兴4volatile precursors and O2 ++Ar+400 eV ions. The films were amorphous and homogeneous in depth as determined by Rutherford backscattering spectrometry. This technique, together with x-ray fluorescence, was used to determine the composition of the films. Their bonding structure was examined by Fourier transform infrared spectroscopy 共FTIR兲and by x-ray absorption spectroscopy and their optical properties determined by UV-visible absorption spectroscopy and spectroscopic ellipsometry. A full account of the characterization can be found in Refs. 19 and 20. XPS spectra were recorded with VG-Escalab210 and Specs-Phoibos100 spectrometers and using an unmonochromatized Al K ␣ excitation source. As a reference for binding energy calibration, the C 1speak of the air adventitious carbon contaminating the surface of samples was taken at a value of 284.6 eV. All the samples presented some charging displacement in peaks positions of about 2–4 eV. The samples were introduced in the chamber and examined without any additional cleaning treatment. Auger parameters5,9of silicon ␣ ⬘ Si, titanium ␣ ⬘ Ti and zirconium ␣ ⬘ Zr cations in the films were defined as ␣ ⬘ Si =BE共Si2p兲+KE共SiKVV兲, ␣ ⬘ Ti =BE共Ti2p3/2兲+KE共TiLMV兲, ␣ ⬘ Zr =BE共Zr3d5/2兲+KE共ZrLM4,5M4,5兲, where BE and KE refer to the binding energy of the photoemitted peak and kinetic energy of the Auger transition in parenthesis. Reflection electron energy loss spectroscopy 共REELS兲 measurements were recorded in the VG spectrometer using a primary electron beam of 1500 eV that was supplied with a LEG62 electron gun from VG. For the quantum mechanical calculations, we have considered cluster models similar to those used previously by some of us.10,11 To account for the nearest environments of the Si, Ti, and Zr atoms, we use small clusters in which the Si and M 共M=Zr, Ti兲are connected through one or two oxygen atoms, and then the coordinations of both Si and M are completed with OH or H2O fragments. The resulting structures are optimized by means of Density functional theory21 共DFT兲calculations using the B3LYP exchangecorrelation potential22 and 6-31G** basis set23 on all atoms except for Zr for which SDD basis set and core pseudopotential24 are used. The electronic excitation energies are then calculated using time dependent DFT-B3LYP theory25 and including the lower 40 singlet excited states. For the calculation of the relaxation energies, we make use of the Z+1 approximation, in which the effect of the core hole formation on the valence electrons is simulated increasing by one the atomic number of the metal and adding a positive charge to the cluster. The extra-atomic relaxation energy is calculated as the difference between the relaxation energy of the cluster with Ti, Zr, or Si, and the energy of the Ti+4,Zr +4,orSi +4 free ions. In these calculations a 6–31G* basis set is used for P and V and SDD including core pseudopotential for Nb. All the calculations are made with the GAUSSIAN03 program.26 III. RESULTS A. Valence band photoemission, low loss electron energy losses, and band gap energy XPS provides a view of the occupied valence band density of states of the analyzed mixed oxide materials. Figures 1共left兲and 2共left兲show the evolution of the valence band spectra of these Si–M–O 共M: Ti/Zr兲mixed oxides from the situation of SiO2共bottom兲to that of TiO2and ZrO2共top兲, respectively. Note that the onset of the valence band is progressively shifted to lower binding energies as the content on Ti/Zr in the mixed oxides is increased. On the other hand, REELS provides a clear view of the electronic transitions from the occupied states at the top of the valence band to the unoccupied states at the bottom of the conduction band. Figures 1共right兲and 2共right兲show the evolution of the REELS spectra of these Si–M–O 共M: Ti/Zr兲 mixed oxides from the situation of SiO2共bottom兲to that of FIG. 1. XPS valence band 共left兲and REELS 共right兲spectra of Si–Ti–O thin films. FIG. 2. XPS valence band 共left兲and REELS 共right兲spectra of Si–Zr–O thin films. 084112-2 Ferrer et al. J. Appl. Phys. 102, 084112 共2007兲 Downloaded 01 Mar 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp TiO2and ZrO2共top兲, respectively. In this case, transitions appear inside the wide SiO2band gap 共⬎8eV兲as soon as the weakest amount of Ti/Zr is incorporated in the mixed oxides. Besides, the onset of these energy losses is progressively shifted to lower energy losses as the content on Ti/Zr in the mixed oxides is increased. As it is well known, the onset of these energy losses is a measure of the band gap energy Egof the analyzed mixed oxide material.27 Figure 3shows the gap Egof the Si–Ti–O and Si–Zr–O mixed oxide thin films. The evolution profile of Egis characterized by a sharp decay from the value of SiO2关i.e., ⬎8eV共Ref. 27兲兴 to that of a sample with ⬃2% Ti or 15% Zr with Egof approximately 4 and 6 eV, respectively. This sharp decrease is followed by a smoother decrease to reach the value of pure TiO2关i.e., 3.2 eV 共Refs. 13 and 15兲兴 and pure ZrO2关i.e., 5.0 eV 共Refs. 15,28, and 29兲兴. B. Refractive index Figure 4shows the refractive index nat ␭=550 nm in the Si–Ti–O and Si–Zr–O mixed oxide thin films. Note that for both mixed oxides nincreases steadily with the amount of Ti or Zr in the films from n=1.45 for SiO2to n=2.35 for TiO2or n=1.95 for ZrO2. C. Core level photoemission Figure 5共Fig. 6兲shows a series of Si 2pand Ti 2p共Si 2p and Zr 3d兲photoemission spectra for the Si–Ti–O 共Si–Zr–O兲 mixed oxide thin films. It is interesting that the binding energy of the Si 2p, and Ti 2p/Zr 3dpeaks changes with the amount of Ti/Zr in the films. A similar effect is found when looking to the kinetic energies of the corresponding Ti LMV, Zr LMM, and Si KVV Auger peaks 共not shown兲. As a result, the Auger parameters5,9of silicon ␣ ⬘ Si, titanium ␣ ⬘ Ti, and zirconium ␣ ⬘ Zr cations in the films change to a large extent when comparing the values of the different samples. An advantage of using the Auger parameter instead of the BE of photoemission peaks to distinguish between different mixed oxide films is that the former is not affected by charging effects on the samples. Plots of ␣ ⬘ Si and ␣ ⬘ Ti/ ␣ ⬘ Zr as a function of the percentage of Ti and Zr in the Si–Ti–O and Si– Zr–O mixed oxide films are reported in Figs. 7and 8, respectively. In Fig. 7we observe that both ␣ ⬘ Ti and ␣ ⬘ Si increase by ⬃2.0 eV when going from pure SiO2to pure TiO2 samples. Similarly, Fig. 8shows that ␣ ⬘ Zr and ␣ ⬘ Si increase by ⬃1.5 eV when going from pure SiO2to pure ZrO2 samples. From the binding energies of the Si 2p,Ti2p, and Zr 3d and the corresponding Auger parameters, we have deterFIG. 3. Band gap energy Egof Si–Ti–O 共bold dots兲and Si–Zr–O 共open squares兲mixed oxide thin films as a function of the percentage of Ti/Zr. FIG. 4. Refractive index nat ␭=550 nm for Si–Ti–O 共bold dots兲and Si– Zr–O 共open squares兲mixed oxide thin films as a function of the percentage of Ti/Zr. FIG. 5. Si 2p共left兲and Ti 2p共right兲photoemission spectra of Si–Ti–O mixed oxide thin films with different contents of Ti. FIG. 6. Si 2p共left兲and Zr 3d共right兲photoemission spectra of Si–Zr–O mixed oxide thin films with different contents of Zr. 084112-3 Ferrer et al. J. Appl. Phys. 102, 084112 共2007兲 Downloaded 01 Mar 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp mined the maximum variation in the binding energy of photoemitted peak ⌬BEmax and the Auger parameter ⌬ ␣ ⬘ max for each Si–M–O 共M: Ti, Zr兲as ⌬BEmax =BE共M–O–Si兲−BE共M–O–M兲, ⌬ ␣ ⬘ max = ␣ ⬘共M–O–Si兲− ␣ ⬘共M–O–M兲, where ␣ ⬘共M–O–M兲and BE共M–O–M兲are the Auger parameter and binding energy of the selected core level of the M4+ ions in bulk MO2, while ␣ ⬘共M–O–Si兲and BE共M–O–Si兲are the Auger parameter and binding energy of the selected core level of the M4+ ion for the samples measured with lowest M concentration. Thus, relevant electronic parameters such as the change of extra-atomic relaxation energy of the photoholes ⌬Rea 共evaluated as half of ⌬ ␣ ⬘ max兲or the change in the initial state energy of the system ⌬␧ before photoemission 共evaluated as the addition of ⌬BEmax+⌬Rea兲6can be calculated. Note that these values are representative of the different characteristics of M4+ ions diluted in a SiO2matrix and inaMO 2oxide. A similar analysis can be performed for the Si4+ ions defining ⌬BEmax,⌬ ␣ ⬘ max ⌬Rea, and ⌬␧ with respect to the data obtained from pure SiO2. A summary of these values obtained from the Si–Ti–O and Si–Zr–O systems is reported in Tables Iand II, respectively. D. Quantum mechanical calculations with cluster models In previous publications we have shown the possibility to account for the variations of extra-atomic relaxation energies ⌬Rea and band gap energies Egof transition metal cations in metal oxides, when their structure changes from a M–O–M to a M–O–M⬘local environment, by means of quantum mechanical 共QM兲calculations with clusters models.10,30 Similar calculations have been carried out here with clusters that model the local environments of Si, Ti, and Zr in the investigated mixed oxide thin films. Schematic representations of the clusters used in the QM calculations are drawn in Fig. 9. Tetrahedral 共t兲and octahedral 共o兲coordinated M4+ cations are considered connected either by 1 or 2 bridging oxygen atoms. The results of these QM calculations are reported in Tables III and IV. These tables include calculated Egenergies, defined as the energy difference between the highest occupied molecular orbital and lowest unoccupied molecular orbital of these clusters. They also contain relaxation energies in the final states Rea by assuming that a core electron has been extracted from the Si 2p,Ti2p,orZr 3dlevels 共the energy of free Si4+ →P5+, free Ti4+ →V5+, and free Zr4+→Nb5+ has been subtracted to all the values given in the table兲. Details about the used approximation can be found in Ref. 10 These tables also report on the differences in the relaxation energies in the final state according to the actual local environment of the cations 共i.e., depending on the type of clusters兲. The differences have been calculated by assuming the clusters Si共t兲-Si共t兲-1, Ti共o兲-Ti共o兲, and Zr共o兲Zr共o兲共see Fig. 9and Tables III and IV兲as representative of pure SiO2, TiO2, and ZrO2, respectively. The calculated values can be directly compared with the measured values of ⌬Rea reported in Tables Iand II. It appears that the calculated and measured values of ⌬Rea show a good correlation with, in some cases, quantitative agreement. Thus, for example, according to the QM calculations, larger difference in relaxation energy at the Si4+ sites are expected for the Si–Ti–O system 共1.0 eV兲than for the Si–Zr–O system 共0.7 eV兲, that is supported by the experimental findings. IV. DISCUSSION A. Electronic parameters and composition of the films The values of the experimentally determined electronic parameters 共cf. Tables Iand II兲correlate with the differences TABLE I. ⌬BEmax,⌬ ␣ ⬘ max,⌬Rea,and⌬␧ of Ti4+ and Si4+ cations in Si– Ti–O mixed oxide thin films obtained from XPS measurements 共all energies in eV兲. System ⌬BEmax 共eV兲⌬ ␣ ⬘ max 共eV兲⌬Rea 共eV兲⌬␧ 共eV兲 Si–Ti–O Ti4+ 共Ti 2p兲 2.1 −1.7 −0.85 1.25 Si4+ 共Si 2p兲 −1.9 1.8 0.9 −1.0 TABLE II. ⌬BEmax,⌬ ␣ ⬘ max,⌬Rea, and ⌬␧ of Zr4+ and Si4+ cations in Si– Zr–O mixed oxide thin films obtained from XPS measurements 共all energies in eV兲. System ⌬BEmax 共eV兲⌬ ␣ ⬘ max 共eV兲⌬Rea 共eV兲⌬␧ 共eV兲 Si–Zr–O Zr4+ 共Zr 3d5/2兲 0.4 −0.9 −0.45 −0.05 Si4+ 共Si 2p兲 −1.7 1.4 0.7 −1.0 FIG. 8. Auger parameter of Si4+ ions ␣ ⬘ Si 共bold squares兲and Zr4+ ions ␣ ⬘ Zr 共open squares兲in Si–Zr–O thin films as a function of the Zr content. FIG. 7. Auger parameter of Si4+ ions ␣ ⬘ Si 共bold dots兲and Ti4+ ions ␣ ⬘ Ti 共open circles兲ions in Si–Ti–O thin films as a function of the Ti content. 084112-4 Ferrer et al. J. Appl. Phys. 102, 084112 共2007兲 Downloaded 01 Mar 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp in the electron density distribution at the cations, their actual coordination, and the differences in the polarizabilities of the environment. Referring to the local environment of the Ti4+ ions in Si–Ti–O mixed oxides, the obtained negative values of the corresponding ⌬ ␣ ⬘ max and ⌬Rea indicate that relaxation of the photoholes at the Ti sites is more favorable 共i.e., more energy is released兲when the surrounding matrix is TiO2-like than when it is SiO2-like. This is confirmed by the QM cluster calculations reported in Table III. The opposite occurs for the photoholes at the Si sites that relax more easily when the Si4+ ion is diluted within the TiO2matrix. A similar conclusion can be met for the case of the Si–Zr–O mixed oxide thin films from Tables II and IV. The QM cluster calculations reported in Tables III and IV provide additional evidences about the importance of changing the local coordination of the Ti4+ and Zr4+ cations when they are present in small concentrations in the films. Previous spectroscopic studies by FTIR and x-ray absorption near edge structure spectroscopies of the Si–Ti–O mixed oxide films have revealed that for concentrations of Ti below 30%, the majority of the Ti4+ ions in the glass films present a tetrahedral coordination. A similar situation is expected for Zr in the Si–Zr–O films.31 According to the calculations in Tables III and IV,⌬Rea values of the different cations are affected by both the presence of Si–O–M cross linking structures 共M:TiorZr兲and the local coordination around them. In this regard, the presence of a tetrahedral coordination, particularly when M4+ can be bonded through two oxygen 关i.e., clusters Ti共t兲-Si共t兲-2 and Zr共t兲-Si共t兲-2兴present the maximum ⌬Rea values. Data in Tables Iand II also show that differences in ⌬␧ have an opposite sign when Si4+ and Ti4+/Zr4+ cations are diluted within a network of, respectively, TiO2/ZrO2and SiO2. In previous works, we have studied the evolution of the electronic parameters of oxide cations when a subnanometric thin film of one oxide is deposited on the surface of another oxide.6,32 In general, the observed differences in ⌬␧ were attributed to variations of the Madelung potential at the cation sites due to the different ionic character of the M–O–M⬘bonds. A similar interpretation can be accepted here. Thus, in samples where few percents of Si ions are distributed within a network of TiO2/ZrO2, silicon will always have titanium/zirconium atoms as second neighbors. In this structure, the linking oxide ions will be rather ionic and contribute through a change in the Madelung potential to decrease the initial state energy of silicon with respect to the value in SiO2. A reverse effect can be assumed for the titanium which, for the diluted conditions, will be surrounded by silicon as second neighbors. Here, a change from fourfold to sixfold coordination can be an additional reason for the observed effects.4,10 The existence of this kind of mixed structures where silicon and titanium/zirconium ions are linked by oxide ions with different electronic characteristics that in the bulk oxides has been previously evidenced by FTIR.4,19,33 Intermediate species can be tentatively attributed to oxygen ions bonded to silicon and titanium/zirconium 共i.e., Si–O–Ti and Si–O–Zr bond structures兲. The existence of such bonding structures has been previously evidenced by FTIR for the Si–Ti–O thin films19,33 and Si–Zr–O.34 B. Band gap and composition of the films Band gap energies of the different samples have been determined by both optical and REELS measurements. The distribution of electron density of the valence band of the different samples has been also reported in Figs. 1and 2. The Egvalues obtained by optical and REELS analysis are equivalent and define a tendency characterized by a sharp decrease from the value of SiO2to that of the sample with 2% Ti. Then, it follows a smooth linear decay up to reach the situation of TiO2. The appearance of band states induced by the presence of titanium ions in the system is responsible for this type of behavior. A narrowing of the electron density TABLE III. Egand Rea obtained from QM cluster calculation 共see Fig. 9兲. The energy of free Ti+4→V+5 or free Si+4→P+5 has been subtracted to all the Rea values given in the table. The corresponding ⌬Rea are indicated in parenthesis. Cluster Eg共eV兲 Rea 共⌬Rea兲共eV兲 Ti4+→V5+ Rea 共⌬Rea兲共eV兲 Si4+→P5+ Si共t兲-Si共t兲-1 7.72 62.22共0.00兲 Si共t兲-Si共t兲-2 7.31 62.19共−0.03兲 Ti共t兲-Si共t兲-1 5.40 64.48共−1.33兲62.57共0.35兲 Ti共t兲-Si共t兲-2 4.73 64.10共−1.7兲62.61共0.39兲 Ti共o兲-Si共t兲4.59 64.49共−1.32兲63.24共1.02兲 Ti共o兲-Ti共o兲anatase structure 3.08 65.81共0.00兲 TABLE IV. Egand Rea evaluated by QM cluster calculation 共see Fig. 9兲.The energy of free Zr+4→Nb+5 or free Si+4→P+5 has been subtracted to all the Rea values given in the table. The corresponding ⌬Rea are indicated in parenthesis. Model Eg共eV兲 Rea 共⌬Rea兲共eV兲 Zr4+→Nb5+ Rea 共⌬Rea兲共eV兲 Si4+→P5+ Si共t兲-Si共t兲-1 7.72 62.22共0.00兲 Zr共t兲-Si共t兲-1 6.49 48.33共−1.24兲62.62共0.40兲 Zr共t兲-Si共t兲-2 5.18 47.78共−1.79兲62.86共0.64兲 Zr共o兲-Si共t兲-1 5.99 48.35共−1.22兲62.95共0.73兲 Zr共o兲-Zr共o兲monoclinic 5.6 49.57共0.00兲 FIG. 9. 共Color online兲Schematic representation of the Si–M 共M: Si, Ti, Zr兲 clusters used in the quantum mechanical calculations. 共t兲refers to tetrahedral coordination, 共o兲refers to octahedral coordination. 084112-5 Ferrer et al. J. Appl. Phys. 102, 084112 共2007兲 Downloaded 01 Mar 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp distribution in the valence band can be also deduced from the x-ray photoemission spectra reported in Figs. 1and 2. The narrowing of the band gap as the concentration of Zr/Ti in the films increases has been also reproduced by the QM calculations with cluster models 共see Tables III and IV兲. In fact, the calculated Egdata in Tables III and IV, although not always in a good quantitative agreement with the experimental band gaps, clearly evidence that the band gap energy of pure SiO2drops drastically when Si–O–M 共M: Ti, Zr兲 bond structures are present in the system. C. Correlation between local electronic parameters and macroscopic optical properties The classical theory of dielectrics35,36 provides the way to correlate extensive and local dielectric properties of a given material. The microscopic electronic polarizability ␣ e correlates the local field at a given position and the dipole moment induced by this local field. It is a measurement of the ability of the electronic clouds around the ions to move as a response to the local electric fields to form permanent dipoles. The microscopic electronic polarizability ␣ eis related to the macroscopic refractive index nby the general refractivity formula, ␣ e=W ␳ f共n兲,共1兲 where Wis the molecular weight in g/mol, ␳ is the density in g/cm3, and f共n兲is a function of the refractive index that takes the general form f共n兲=n2−1 4 ␲ +b共n2−1兲,共2兲 where bis a dimensionless constant. In the case of point dipole ions and cubic symmetry without overlap of the electron distribution, b=4 ␲ /3 so f共n兲=共n2−1兲/共n2+2兲, and Eq. 共1兲takes the well-known Lorentz-Lorenz form.37 Introducing covalency 共i.e., overlapping between electron distributions兲 leads to lower values of bas it is reported for silicates, for which b=1.2–1.3.36 In any case, the dependence of the electronic polarizability with the refractive index is a smooth increasing function of n. In a previous publication on Al–Ti–O thin films, we correlated the Auger parameter of Ti with the optical properties of the films, evidenced by the values of nas the Ti/Al ratio varied.8Such a correlation between an extensive magnitude such as nand a local prove 共i.e., the Auger parameter兲is possible through the Eqs. 共1兲and 共2兲. In particular, it is found that, to a good approximation, ␣ eis proportional to f共n兲 =共n2−1兲/共n2+2兲8共i.e., Lorentz-Lorenz approximation兲. In our case, the changes in electronic parameters of Ti/Zr and Si can be related to the optical properties of the thin films in the same manner. In particular, it has been shown that the refractive index nof the Si–Ti–O and Si– Zr–O mixed oxide films increase with the content of Ti and Zr, respectively 共see Fig. 4兲. This increase runs in parallel to the Auger parameter as the mixed oxide composition changes 共cf. Figs. 7and 8兲. The main contribution to the electronic polarizability in mixed oxide systems comes from the oxygen ions.38 Changes in extra-atomic relaxation energies of a given cation in an oxide 共note that oxygen atoms are always their nearest neighbors兲show up as variations of the Auger parameter. This is the reason why the Auger parameter is so sensitive to the electronic polarizability of the system. The effect is as strong at the Si sites as at Ti/Zr sites 共similar ⌬ ␣ ⬘ max observed兲for the two systems studied. Thus, variations in the Auger parameter of cations in mixed oxide system, disregarding the type of cation, correlate qualitatively with changes in the electronic polarizabilities of the oxygen anions in the corresponding compounds. Figure 10 shows 共n2−1兲/共n2+2兲as a function of the Ti/Zr content in the films 共ntaken at 550 nm兲. It is apparent that in both cases this function varies in a similar way with the percentage of Ti/Zr, thus indicating that the two curves are generated by similar physical phenomena. It is only for Ti/Zr content larger than 60% that they differ significantly from each other. As mentioned before, ␣ e关and hence 共n2 −1兲/共n2+2兲兴 is related with the ease by which the electron cloud around a given ion can be reorganized to create a local electronic dipole. Meanwhile, differences in Auger parameter with respect to a reference compound are a measure of the different relaxation energies of the photoholes once the photoemission event has taken place.6In dielectric materials, the main mechanism for the screening of the photoholes is the formation of local dipoles in the surrounding medium.30 The ease with which these local dipoles are formed around the photoemitting atom depends on ␣ eand hence would be connected with the value of 共n2−1兲/共n2+2兲. An apparent misfit in the plots of Fig. 7is the low value of ␣ ⬘ Ti for the samples with the minimum content of Ti. This deviation from the linear relationship defined by the other samples can be related with the actual local structure around Ti in each case. It has been proven by x-ray absorption spectroscopy that the Ti ions in the 2%–10% samples are fourfold coordinated9and that the coordination increases up to six with the percentage of this element. It has been previously reported6,30 that the changes in the Auger parameter of oxides are the result of two contributions, a larger one due to the nearest coordination sphere and a second one due to the polarization of the rest of the matrix. We attribute the loss of linearity of the two samples with a small Ti content to this FIG. 10. 共n2−1兲/共n2+2兲as a function of the Ti 共bold dots兲or Zr 共open squares兲content in the mixed oxide thin films. 084112-6 Ferrer et al. J. Appl. Phys. 102, 084112 共2007兲 Downloaded 01 Mar 2010 to 161.111.180.191. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp fourfold coordination. Recently, Moretti et al.39 have identified four coordinated titanium ions in titanosilicate compounds. The Auger parameter of these ions was much smaller than that of six-coordinated Ti4+, a similar tendency than that found in our thin films. Figure 11 shows the correlation between the electronic polarizability, through the 共n2−1兲/共n2+2兲function, and the ␣ ⬘ Si for Si–Ti–O and Si–Zr–O mixed oxides. We observe that there exists a quasilinear correlation between these two magnitudes, disregarding the type of mixed oxide considered. In fact, it is remarkable that larger range of variation in electronic polarizability is correlated to larger range of variation of the Auger parameter. Besides, the steeper initial variation of the Si–Ti–O system might be related to the larger difference for this system between extra-atomic relaxation energies observed in the QM calculation between Si共t兲-M共t兲 and Si共t兲-M共o兲clusters. V. CONCLUSIONS Variation of local electronic parameters obtained by standard XPS analysis as the Auger parameter, extra-atomic relaxation energies, or binding energy of core levels in mixed oxide thin films can be easily correlated to extensive optical properties as the refractive index within the framework of the classical theory of dielectrics. It is shown that the Auger parameter in mixed oxides correlates linearly with the electronic polarizability obtained from the refractive index through the Lorentz-Lorenz relationship. In particular it is shown that the Auger parameter of Si in Si–Ti–O and Si– Zr–O mixed oxide thin films can be used as a way to asses their refractive index. 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