A Generalized Semiempirical Approach to the Modeling of the Optical Band Gap of Ternary Al-(Ga, Nb, Ta, W) Oxides Containing Different Alumina Polymorphs

A generalization of the modeling equation of optical band gap values for ternary oxides, as a function of cationic ratio composition, is carried out based on the semiempirical correlation between the differences in the electronegativity of oxygen and the average cationic electronegativity proposed some years ago. In this work, a novel approach is suggested to account for the differences in the band gap values of the different polymorphs of binary oxides as well as for ternary oxides existing in different crystalline structures. A preliminary test on the validity of the proposed modeling equations has been carried out by using the numerous experimental data pertaining to alumina and gallia polymorphs as well as the crystalline ternary Ga(1–x)AlxO3 polymorphs (α-Ga(1–x)AlxO3 and β-Ga(1–x)AlxO3) covering a large range of optical band gap values (4.50–8.50 eV). To make a more rigorous test of the modeling equation, we extended our investigation to amorphous ternary oxides anodically formed on Al-d-metal alloys (Al-Nb, Al-Ta, and Al-W) covering a large range of d-metal composition (xd-metal ≥ 0.2). In the last case, the novel approach allows one to overcome some difficulties experienced in fitting the optical band gap dependence from the Al-d-metal mixed anodic oxide composition as well as to provide a rationale for the departure, at the lowest d-metal content (xd-metal < 0.2), from the behavior observed for anodic films containing higher d-metal content.


INTRODUCTION
Band gap modeling of simple and complex oxides is of paramount importance for the application of such materials in numerous and important applications pertaining to different fields such as solar energy conversion (photoelectrochemical and photovoltaic solar cells, photocatalysis, and electrocatalysis), microelectronics (high-k, high-band-gap materials and resistive random access memory (ReRAM)), and active and passive anticorrosive coatings. 1−7 Owing to this, the band structure calculation of solids and, notably, of oxides has been one of the most investigated subjects of materials science in the past 20 years or more. 8 In spite of the rewarding achievements obtained in computing numerous solid-state properties of different materials, the prediction of band gap values of materials is still a formidable task also for the most advanced quantum mechanical methods rooted on density functional theory (DFT). 1,8−15 Apart from the theoretical difficulties and computing cost in predicting reliable band gap values for different materials, a further complication arises when a comparison of E g values, obtained by using theoretical models, is carried out for an assigned material. 9,10 This task is made further complicated by the large variability of experimental band gap values reported in the literature for a given material. The variability in the experimental E g values, not always accounted for the comparison, can originate from different sources pertaining to the different experimental techniques (electron energy loss spectroscopy (EELS), variable angle spectroscopy ellipsometry (VASE), diffuse reflectance spectroscopy (DRS), photocurrent spectroscopy (PCS), etc.) used to extract the E g value as well as in the data treatment model employed in deriving the band gap values. 16−19 Moreover, the assumption of identical composition and structure should have to be carefully checked, especially in the presence of differences in the method of fabrication of the investigated material.
This aspect is particularly intriguing in the case of thin films of oxides obtained by using different growth techniques or grown in different conditions but with the same technique. In the case of anodic oxide films grown by anodizing in different solutions, two common sources of variability can be attributed to the differences in the crystallinity degree of the anodic films as well as to the possible incorporation, into the oxide film, of chemical species derived from reacting species lying in solution. 20 −26 In a previous work, 27 we have shown that it is possible to correlate semiempirically the optical band gap values of numerous crystalline binary oxides to their composition by means of a general expression as where χ M and χ O are the electronegativities (ENs) of metal and oxygen, respectively, in Pauling's scale. A and B have been determined by a linear best fitting procedure of experimental E g values vs (χ O − χ M ) 2 .
In the case of pure s,p and d-metal oxides, two different interpolating straight lines were derived with for s,p-metal oxides. An analogous correlation has been shown to hold also for AlN, GaN, and InN compounds and their alloys. 28 Equations 1 and 2 were able to fit, with a reasonable accuracy and with some discussed exceptions, 27 the E g values of many f,d-metal oxides (III−XII groups and TM-oxides), while eqs 1 and 3 were able to reproduce with good accuracy the experimental E g values of sp-metal oxides (II and XIII−XV groups in the periodic table of the elements).
More recently, 29−32 in an attempt to perform photocurrent spectroscopy (PCS), a more quantitative technique for the in situ characterization of surface passive films and corrosion layers grown on metallic alloys, we tried to extend the proposed correlations valid for pure binary oxides to regular (sp-sp; d-d) and nonregular (sp-d) ternary oxides. We are aware 32 that the most critical aspects of our semiempirical approach, especially when extended to multinary oxides, are related to the following: 32,33 a The lack of any dependence of estimated E g on the nature of optical transitions (direct or indirect) determining the optical band gap values. b The limited ability of eq 1 in accounting for the different optical band gap values of different polymorphs by assuming for the electronegativity value, χ M , of each cation metal an uncertainty range equal to ±0.05 around the χ M 0 value reported in Pauling's scale 34 (χ M = χ M 0 ± 0.05). We have shown 29,30,32 that the semiempirical approach initially proposed for pure oxides is able to model the optical band gap behavior of more complex multinary oxides, as a function of their composition, once some further refinements to the initial approach are introduced, which account, at least partially, for the criticisms previously reported in the literature. 33 In this paper, we will discuss how to extend the initial correlation to the case of different polymorphs by treating extensively some Al-based ternary oxides containing both spand d-metal cations. In particular, we will discuss some technically relevant systems like Ga x Al (1−x) O 3 ternary oxides, finding numerous applications as wide band gap materials in the field of high-power electronic devices and solar-blind ultraviolet (UV) photodetectors. 35−39 Al-Ga mixed oxides, moreover, are the basis for the fabrication of rare earth (RE)-a l u m i n u m g a r n e t ( Y 3 ( A l x G a 1 − x ) 5 O 1 2 ( Y A G G ) , Lu 3 (Al x Ga 1−x ) 5 O 12 (LuAGG), and Gd 3 (Al x Ga 1−x ) 5 O 12 (GAGG)), which, doped with various trivalent RE-cations, are widely used as optical materials in technologically relevant applications including light-emitting diodes (LEDs), plasma display panels, and scintillator materials used in medical imaging techniques. 40−42 As for the high-band-gap (>3 eV) ternary amorphous oxides, apart from their importance in the field of corrosion protection, the Al-d-metal amorphous mixed oxides are currently under investigation for their possible use in the field of electrical and electronic engineering as high-k materials in microelectronics 3,4 and in advanced electrolytic capacitors. 43 Ta-based ternary amorphous oxides, with high thermal stability and suitable refractive index, have been used and are under current investigation as high-resolution light mirrors in interferometric detectors, including those of recent gravitational waves in LIGO experiments. 44−48 The choice, moreover, allows a rigorous test of the proposed approach owing to the fact that the involved alumina polymorphs span a very large energy range, from about 4.50 eV to about 9.0 eV, of band gap values.

THE BASIS OF THE GENERALIZED SEMIEMPIRICAL
APPROACH In previous works, 27,32 we presented the details of our semiempirical approach from which eqs 1−3 were derived. According to this, we were able to derive the following relationships for the two parameters A and B of eq 1 where, according to Phillips' suggestion, 49 the parameter E I is assumed "to vary with the hybridization configuration, i.e., with different atomic coordinations in different crystal structures", D M-M and D O-O are the enthalpies of dissociation (bond strengths) of M 2 and O 2 in the gaseous phase, and R MOy is the repulsive term of the MO y binary oxide. By neglecting the London components in the repulsive term, we can write In the ionic limit, U bond coincides with the Madelung energy, U M , while in the covalent limit, a quantum mechanical calculation of the bonding energy is required. U lattice represents the experimental or calculated lattice energies of ionic solids. 50 According to the theory, a dependence from the inverse of the square of the anion−cation distance, d M-O , has been assumed for R MOy . 51 The problem of the variability of E g values in the presence of different polymorphs was left unsolved in the initial work, but it has been recently afforded 32 in an attempt to extend the original approach to the case of nonregular ternary oxides containing both sp-and d-metal cations. We have shown that in several cases, the band gap value of different polymorphs of s,p-and d-metal oxides could be fitted by using eq 1 with fixed A and B values and by spanning χ in the entire range of χ M values (χ M = χ M 0 ± 0.05) accepted in deriving Pauling's scale of electronegativity. 34 As evidenced (see Table IV of our previous work), 32  In the case of alumina polymorphs, band gap values ranging from 5.54 eV to about 6.41 eV are obtained by using eq 1 with the A sp and B sp above reported and a χ Al = 1.5 ± 0.05 value. Both of these values do not cover the range of band gap values reported in the literature for the different alumina polymorphs (see Table 1). To extend the limits of eq 1, it appears necessary to take into account, for different polymorphs, possible variations of A and B values around the experimental ones derived according to eq 1. According to Phillips's suggestion, E I can be assumed to be dependent on the crystalline structure so that by keeping the variability of χ within the uncertainty range of Pauling's scale, it seems reasonable to hypothesize a variation in the A values (see eq 4a) with the nature of different polymorphs too. Moreover, according to eq 5, different values of the repulsive term R MOy and then of B values could be associated to different MO y polymorphs as a consequence of their different crystalline structure. To check the validity of the previous assumptions, we will carry out the modeling of the optical band gap of ternary systems as a function of their composition with the aim of illustrating how to face with the problem of the band gap values of different polymorphs and in the presence of E g,opt data spanning a range of experimental values well above that estimated according to eqs 1−3 once the accepted range (±0.05 unit) of variability in χ M is also accounted for.
With this aim, we identified several ternary systems of variable composition xMO y + (1 − x)AlO 1.5 , covering all the ranges of experimental and theoretically estimated optical gap values of alumina polymorphs, 52−65 as the most challenging test to our assumptions. However, for the sake of brevity, in this work, we will discuss the results pertaining to a few ternary oxides containing, apart from Al, both sp (Ga) and d-metals (Ta, Nb, and W). 43,66,67 Both crystalline ternary polymorphs and amorphous ternary oxides will be taken into account to test the ability of the proposed approach to model reliably the experimentally derived band gap values.

MODELING THE OPTICAL BAND GAP OF TERNARY OXIDE SYSTEMS
In a previous work, 32 we derived a generalized modeling equation of the optical band gap of ternary oxides by extending the results previously obtained for regular (sp,sp-or d,dmetals) mixed oxides to the nonregular (s,p-TM metals) mixed oxides. For nonregular ternary oxides, the band gap value, E g,nr , of mixed oxide was derived from an extension of eq 1 and by using as a starting equation the following relationship g,nr 1 g,1 av 2 g,2 av g,2 av 1 g,1 av g,2 av χ χ with x i and χ i representing the cationic fraction (in at %) and Pauling's electronegativity parameter, respectively, of each metal M i present in the mixed oxides. E g,1 and E g,2 represent the band gap values of pure binary oxides assumed to follow one of the two previous correlations (sp-sp or d-d-metal) according to eqs 1−3. After substitution of eqs 7a−7c in eq 6 and rather simple algebraic manipulations, we got the equation for a generic ternary oxide system as where E g,2 is the band gap of the pure oxide corresponding to x 1 = 0 and Equations 8a−8d represent, however, more general relationships useful for modeling ternary systems where one or both pure oxides exist as different polymorphs spanning a large range of optical band gap values, which cannot be estimated with the initial correlation (see eq 1) by keeping constant the A and B parameters reported in eqs 2 and 3. Moreover, a more general expression is now obtained for the bowing coefficient, which could help fit a larger class of mixed semiconductor alloys, changing their crystallographic structure as a function of the composition. To reduce the arbitrariness in the derivation of the A and B parameters for the different alumina polymorphs, we will assume as a reference for the different polymorphs the experimental E g,opt values estimated by optical techniques, whenever available, or the E g,th theoretical values recently reported by Peintinger et al. 52 This choice is supported by the fact that the investigated polymorphs are quite numerous and the E g values are, in many cases, in good agreement with those experimentally obtained by different research groups using also different techniques.
As mentioned above, in the presence of different polymorphs, the value of the two parameters A and B, necessary to estimate the E g value according to eq 1, can differ from those derived from the best fitting procedure. 27 This can be attributed to the fact that in different crystalline structures, both the coefficient E I and the repulsive energy term R MOy can assume different values for different polymorphs.
As for Al 2 O 3 , a band gap value of 6.3 ± 0.1 eV was used for alumina, as derived from photocurrent spectroscopy measurements of a thick amorphous anodic alumina film. 27,68,69 It is encouraging that an indirect band gap value of 6.2 eV has been derived very recently by Peintinger et al. for the γ-Al 2 O 3 polymorph 52 as well as by Liu et al. for amorphous alumina. 70 It is known that anodic alumina crystallizes as γ-Al 2 O 3 at not too high temperature 71 (450°C) or under an electron beam of a transmission electron microscope during extended observation. 72 However, the band gap values of α,κ,θ,η-Al 2 O 3 are outside the range of values accessible with the proposed correlation so that a search for the right values of A (α,κ,θ,-Al2O3) and B (α,κ,θ,-Al2O3) is necessary if we want to fit the ternary systems α-(Ga (1−x) Al x ) 2 O 3 and β-(Ga (1−x) Al x ) 2 O 3 ) for which E g,opt values between 7.80 eV (α-phase) and 6.50 eV (β-phase), respectively, have been reported in the literature 73,74 in the presence of Al at % content into the oxide well below 100% (x ≈ 0.70−0.80) (see also Figures 1 and 2).
To fit the series of experimental and/or theoretical data for both ternary polymorph systems, we need to obtain further information on the possible changes in A and B values for the different gallia and alumina polymorphs. As mentioned above, by looking into the expression of B reported in eq 4b, it shows that the only term affecting the B parameter can be traced out to the repulsive energy term R MOy usually assumed to depend from the inverse of the square of the anion−cation average distance. 51 Regarding the different alumina and gallia polymorphs, α-Al 2 O 3 and α-Ga 2 O 3 display the lowest volume per formula unit 52,61,70,75 so that it seems reasonable attributing the highest (absolute value) B sp value of −2.71 to this polymorph. Moreover, by assuming as a proxy of anion−cation average distance the cubic root of the unit formula volume, we corrected the B term of each polymorph, B i , by the ratio r v = (V α /V i ) 2/3 . According to this, and by using eqs 4b and 5, each B i value, for a different polymorph, was obtained through the relationship where B i , V i , B α , and V α represent respectively the B parameter and the unit formula volume of the ith-polymorph and of the α-phase. As for the repulsive energy term (R (α-MOy) /y), the values of 8 and 7.27 eV were estimated from the literature data 76 for α-alumina and β-gallia, respectively.
According to this, whenever reliable unit volume data were at our disposal and in the presence of appreciable variation in the volume ratio (r v > 2%) of the different polymorphs, we carried out a correction to the B i parameter value reported in eqs 2 and 3 for s,p-metals and d-metal oxides, respectively. Moreover, the A term of each polymorph was adjusted as a function of the repulsive term, through eq 1, by using the estimated B values according to eq 10a and by keeping constant, as a first choice, the EN value χ M 0 of the metal for all polymorphs. Values of A obtained according to this procedure seem to be more physically acceptable than the choice of fitting the experimental data reported in Figures 1 and 2 (see below) by assuming a constant electronegativity parameter for both βphase and α-phase as a constraint and leaving both B and A as free variables.
As for the gallia polymorphs (α,β,γ,δ,κ/ε-Ga 2 Ο 3 ), for which reliable optical band gap values (5.20−5.30 eV for α and 4.40− 4.90 eV for the β,γ,δ,κ-phase) (see Table 2) have been reported in the literature, 70 we like to stress that almost all band gap values (between 4.70 and 5.50 eV) can be derived through eq 1 by using the A and B parameters reported for s,poxides (eq 3) and an EN parameter χ Ga = 1.60 ± 0.05 as reported in Pauling's scale. 34   Inorganic Chemistry pubs.acs.org/IC Article can be derived by keeping the χ Ga value within the uncertainty limit of Pauling's scale of electronegativity.
In fitting the experimental optical band gap data of crystalline oxide, we will use for different polymorphs the B values reported in Tables 1 and 2 derived according to eq 10a. In the case of amorphous phases or in the absence of information on the value of unit formula volume, V i , a similar approach as devised in eq 10b, but based on the data density ρ of the anodic film or other noncrystalline phases, 71,77,78 was adopted to get the B parameter as follows where ρ α and ρ i are the densities of α and i-phase, respectively.
In Table 1, the E g value for five polymorphs of Al 2 O 3 , theoretically derived from DFT methods 52 and covering all the ranges of expected band gap values, are reported. In the same table, we reported the A and B values, the latter of which was estimated through eqs 10a and 10b and the former through eq 1 by using the calculated B and the chosen theoretical or experimental E g value.
We stressed above that γ-Al 2 O 3 (E g = 6.20 eV) is the one that follows eq 1 by using the average A and B values of the semiempirical correlation valid for s,p-oxides. After correction of the B value according to eqs 10a and 10b (B = −2.30 eV), a value of E g = 6.20 eV is obtained for such a polymorph with χ Al = 1.52 and A = 2.17.
It is evident that the decrease in B value is now compensated, at a fixed A parameter, by the small variation in the electronegativity parameter well within the confidence limits of Pauling's scale of electronegativity (≤0.05). At fixed χ Al = 1.50, a very small variation in the value of A (2.15 ± 0.025) is sufficient to get a very nice agreement between estimated (6.30 ± 0.1 eV) 69 and band gap values of γ-Al 2 O 3 reported by Peintinger et al. 52 As for θ-Al 2 O 3 , a good agreement is found between the DFT-estimated E g value (6.57 70 −6.90 eV) and that one obtained (E g,opt = 6.70 eV) according to eq 1 by using the new value of B (−2.22 eV) and a slightly higher A value (2.23) by keeping χ Al = 1.50 constant. The last E g,opt value is the experimental optical band gap of θ-Al 2 O 3 obtained by Tauc plot analysis in the hypothesis of indirect optical transitions (see below). We like to stress that if we assume for θ-Al 2 O 3 the DFT-based value 52 of Table 1 (E g,th = 6.90 eV), a value of 2.17 for the parameter A is still compatible with the E g value of 6.90 eV provided that a value of χ Al = 1.45, still within the accepted EN variability, is assumed for Al in this polymorph.
From this short analysis of the data, it shows that eq 1 is able to fit the band gap values of the two most common alumina polymorphs (γ,θ-Al 2 O 3 ) by using the average A value for s,pmetal oxides together with the χ 0 Al value reported in the classical Pauling's book and in agreement with the confidence limits of Pauling's scale of electronegativity.
As for the other two polymorphs (α,κ-Al 2 O 3 ) reported in Table 1, showing the highest band gap values, we derived   Table  2). A possible rationale for this finding could be related to the smaller difference in electronegativity between oxygen and Ga (Δχ (Ga) = 1.9 vs Δχ (Al) = 2.0), which could account for the smaller range of experimental E g values reported for the different gallia polymorphs (see Table 2). Among the polymorphs considered in Table 1, the most intriguing aspects are those related to the η-phase, for which a DFT-derived E g value of 4.40 eV has been reported. 52 As discussed in the literature, η-alumina, with respect to the other polymorphs, is a less ordered and more defective crystalline phase for which no experimental E g value has been reported up to know, at our best knowledge. The lower E g value reported for this polymorph, with respect to all other polymorphs, should agree with the suggestions made by different authors 59,60,77,78,94 For α-Ga 2 O 3 , if we assume the most common reported value 96 of 5.30 eV for E g,opt , a value of χ Ga = 1.578, very close to Pauling's value of 1.60, is obtained from eq 1 by using the average A and B values of s,p-oxides. On the other hand, by assuming the value of χ Ga = 1.60, a value of A = 2.22 (+2.3%) is obtained by keeping E g and B constant. As for the β-Ga 2 O 3 polymorph, owing to the large ranges of band gap values reported in the literature, the more detailed discussion underlying the choice of E g value will be presented in the following section.
3.2. Fitting of the Pseudoregular Ternary sp-sp Oxide Band Gap. 3.2.1. α-(Ga (1−x) Al x ) 2 O 3 Ternary System. Although eqs 8a−8d have been derived for nonregular (sp,d-metal mixed oxide) ternary oxides, they can be used also for "pseudoregular" ternary systems where one or both binary oxides, although belonging to the same correlation, are present as a polymorph whose band gap value could be different with respect to that calculated through eq 1.
In Figure 1, we report the experimental data (blue full circles) of direct optical band gap values, 73 E g,dir , derived for crystalline films of corundum-like polymorphs (α- The experimental E g,opt values of pure oxides, E g2 = E g,α-Ga2O3 and E g1 = E g,α-Al2O3 , have been fixed as 5.30 eV 96 and 8.50 eV, 53 respectively. This last value is in good agreement with recent experimental values, obtained by the REELS technique (8.
To fit the experimental data set points by using eqs 8a−8d, we need to derive the A and B values of Tables 1 and 2 by using the parameter discussed above and reported in captions. Very good accordance is reported with the experimental data, demonstrating the reliability of our approach.
We like to stress that the bowing coefficient of the fitting curve related to experimental data (b = 0.287 eV, see eq 11a) is much smaller than the theoretical values recently reported by Weng et al. 98 (b = 1.60) and Peelaers et al. 99 (b = 1.87), derived by using for α-(Ga (1−x) Al x ) 2 O 3 the E g (x Al ) values calculated by DFT techniques, but it is in very good agreement with the S q value (0.243) calculated according to eq 8c. Although further investigations are necessary on these aspects, before reaching final conclusions, the very nice agreement between experimental and theoretically estimated E g and bowing coefficient values is a strong support in favor of the suggested approach. It also shows that eq 11b can be reliably used to estimate the E g values of α-(Ga  101 Still, different values have been reported in the case of E g values estimated by using ellipsometry or the REELS technique. 74,101,114−121 In some cases, nearly identical E g values have been reported by using REELS or the Tauc method of determination of band gap, but large scattering of data has been reported from the same laboratory. 119,120 No experimental data, unfortunately, have been reported in the literature for the band gap of single crystal θ-Al 2 O 3 , at our best knowledge, while different theoretical band gap values have been calculated by using DFT-based techniques. According to more recent works, direct band gap values of 6.57 95  The band gap of these films was derived from REELS (orange triangles) or optical techniques through the Tauc plot (azure circles) in the hypothesis of indirect optical transitions. 101 The band gap values, obtained by the REELS technique, of polycrystalline powders grown by solution combustion synthesis 74 are also reported in Figure 2a  On the other hand, the data represented by triangle and diamond symbols in Figure 2a pertain to PLD films and crystalline powders, respectively, for which the band gap values have been obtained by the same REELS technique. 74,116 The E g,REELS vs x Al data set (dark blue symbols in Figure 2b) pertaining to both polycrystalline powders and PLD films will be assumed as the "reference data set" owing to the larger interval of composition covered (0.11 ≤ x ≤ 0.81) as well as to the good reproducibility of the REELS data derived from different laboratories and for samples grown by different techniques.
For the considered experimental data, the equation of the best fitting line is with E g,1 = 6.60 eV (x = 1) in the hypothesis of a linear fit. By assuming a second-order fitting equation, the best fitting line is with E g,1 = 6.70 eV. We have to mention that the large differences in the linear terms observed in the previous fitting lines must be attributed to the fact that no constraints were imposed to the band gap values, E g1 and E g2 , of pure oxides. The linear fitting equation is almost coincident with that reported by Krueger et al. 74 and Fares et al., 116 while the parabolic fitting gives, once again, a bowing coefficient much smaller than that reported by Wang et al. 98 (b = 1.0 eV/x 2 ) and Van de Walle et al. 99 (b = 0.93). As for θ-Al 2 O 3 , the dispersion in the DFT-calculated E g,ind values is smaller than for E g,dir , but the DFT-estimated values (E g,ind = 6.90−7.25 eV) 52,98,99 are appreciably higher than the value of θ-Al 2 O 3 band gap (6.60−6.70 eV) obtained by extrapolation to x = 1 from eqs 12a and 12b, pertaining to the "reference data set". As mentioned above, the E g,REELS values in the low x region (x Al ≤ 0.35) are quite close to the indirect optical band gap values derived from Tauc plots so that it seems reasonable to use as an end point in the fitting process (E g2 in eq 8a) an E g value of β-Ga 2 O 3 obtained by the same (REELS and Tauc plot) techniques. If we fix the E g2 value for β-Ga 2 O 3 by assuming a value of 4.75 eV, i.e., an average value of experimental E g data reported in the literature for E g,ind of β-Ga 2 O 3 , the fitting lines equations now become A change in the bowing parameter from b = 0.3623 eV/x 2 to b = 0.241 eV/x 2 is now recorded going from eq 12b to eq 12d, with a consequent better agreement in the linear terms of eq 12c (S l = 1.8624 eV/x) and eq 12d (S l = 1.668 eV/x). A value of E g,ind = 6.67 eV is obtained for θ-Al 2 O 3 by extrapolating to x = 1, in good agreement with the frequently reported REELS band gap values of 6.8 ± 0.2 eV measured for ALD Al 2 O 3 films. 82,115,123 This value is in good agreement also with that estimated by Peintinger et al. and is reported in Table 1.
The fitting of experimental E g,REELS data was carried out according to eqs 8a−8d (see red squares in Figure 2b Tables 1 and 2) for β-gallia and θ-alumina, respectively. The value of A parameter for β-gallia (A β-Ga2O3 = 1.96) was derived according to eq 1 by using the band gap value of 4.75 eV for βgallia. The theoretical points calculated according to eqs 8a−8d by using A θ-Al2O3 = 2.23 nicely fits the experimental data point, providing the following theoretical equation From eq 12e, a value of E g1 = 6.69 eV for pure θ-Al 2 O 3 was obtained by extrapolating to x = 1. This value is in very good agreement with the REELS band gap value (6.80 ± 0.2 eV) estimated for pure Al 2 O 3 ALD films 123 and with the value of about 7.05 eV estimated for a very thin film of θ-Al 2 O 3 obtained by thermal treatment at 1200 K of an initial amorphous film. 82 As for the bowing parameter value, a good agreement is observed between the experimental one (0.241 eV/x 2 ) and theoretical one (0.1278 eV/x 2 ). As for the A values of the two polymorphs, a negligible increase (+3%) for θ-Al 2 O 3 and a more appreciable decrease (−10%) for β-Ga 2 O 3 are observed with respect to the average A value (2.17) of s,p-metal oxides.
To stress the validity of the chosen fitting strategy, consisting in the use of a reduced number of experimental data set but covering the largest compositional range, where the metastable Inorganic Chemistry pubs.acs.org/IC Article monoclinic structure of β-(Ga (1−x) Al x ) 2 O 3 exists, we reported in the Supporting Information ( Figure S1) the fitting equation derived by using all experimental data sets of E g,exp vs x Al values including all samples and band gap values regardless of the method of preparation and technique of E g measurement. In the case of E g optical values, the E g,dir data, derived from the Tauc method, were used, although no relevant change occurred in the fitting equation if E g,ind data were used. A value of the bowing parameter very near 0.46 was obtained in both cases. It is worth noting that the E g values, estimated by using the fitting equation of theoretical data, differ by less than 0.1 eV from those estimated by using the fitting equation derived in Figure S1 in all ranges of composition exploited (0 ≤ x Al ≤ 0.81). In the Supporting Information, we report also the fitting of the experimental data set 101 Figure 2 is able to fit the experimental data.

Fitting of the Nonregular Amorphous Ternary Oxide Band Gap.
In previous sections, we tested the ability of our semiempirical approach in modeling the band gap of crystalline ternary oxides by considering also the influence of different polymorphs in determining the band gap of pseudoregular ternary oxides. In this section, we will take into account how to extend our approach to the case of amorphous nonregular ternary oxides. With this aim, we will take advantage of our previous studies on the photoelectrochemical characterization of anodic oxide films grown on pure transition metals (TMs) and mixed Al-TM (TM = Nb, Ta, W, Ti, etc.) alloys playing an important role in passivity studies 43,66,67,124,125 as well as in microelectronics 4,126 and near-infrared interferometry equipment. 44 In previous works, in the absence of a general semiempirical approach to the modeling of nonregular amorphous ternary systems, we proposed on a purely heuristic basis the fitting of the E g,opt vs x Al data points of amorphous oxides anodically grown on magnetron-sputtered Al-TM alloys based on the use of eq 1 modified as where χ M,av = x Al χ Al + x TM χ TM and from which the value of A to be used in the bowing equation of mixed ternary oxide was derived. Equation 13b follows from eqs 8a−8d whenever A 1 = A 2 = A and B 1 = B 2 = B. As previously reported, 67 a good agreement was observed, for the am-(Al x Nb (1−x) ) 2 O (5−2x) ternary system, between the experimental E g,opt data and those estimated according to eq 13b by using χ Al = 1.50 and χ Nb = 1.615 as electronegativity values and A = 2.145. Analogously, for Al-Ta alloys, eq 13b was able to fit also the data of E g,opt vs the Al content once the values of χ Al = 1.478 and χ Ta = 1.505 were substituted in eq 13b, together with the value of E g,Ta2O5 measured for passive films grown on pure magnetron-sputtered Ta. On the other hand, in the case of passive films grown on Al-W, 66 a value of A = 1.35 with χ Al = 1.50 and χ W = 1.70 was able to fit the experimental E g,opt vs x Al data points. In spite of the good accordance between experimental and theoretically derived E g values, 43,66,67 a close inspection of eq 13b evidences its limits with respect to the general approach on which eqs 8a−8d are based. In fact, eq 13b provides a constant identical band gap value for mixed sp,d-metal oxides having the equal EN parameter (χ i = χ j ) in all ranges of composition including the pure oxides, at x = 0 and x = 1, at variance with the results of eqs 2 and 3. This inconvenience does not occur with eqs 8a−8d, in agreement with the general approach outlined above, and more importantly, we do not need to derive the new values of A to be used in eq 13b to estimate the E g,opt values of nonregular ternary mixed oxides. With respect to the crystalline systems investigated in Section 3.2, in the case of Al-TM passive films, grown by anodizing on their respective bulk metals or magnetronsputtered alloys, amorphous pure or mixed oxides were formed, provided that the anodizing process is stopped before the onset of electrical breakdown phenomena when a crystallization process can occur. It has been shown for different amorphous passive films on valve metals (Al, Ta, Nb, W, Ti, etc.) and their alloys 43,66,67 that during the anodization process, both anodizing current and final voltage, despite the fact the final anodizing voltage is kept below the onset of the electrical breakdown process, affect sensibly the measured optical band gap values of the amorphous anodic film. 127,128 Usually, amorphous thicker films, grown at a constant growth rate, or films grown at lower growth rates display lower optical band gap values with respect to thinner films or films grown at higher anodizing rates. 43,66,67 In previous works, 31,32,43,66,67,128,129 we suggested as a possible rationale for these findings a different degree of amorphousness as a function of the different anodizing parameters. Our suggestion was in agreement with the model of electronic density of state (DOS) in amorphous semiconductors described in the classical Mott−Davis book 130 showing how a different degree of lattice disorder affects the DOS in the vicinity of the conduction and valence band edges. 128 This suggestion is supported by older and very recent structural studies showing the striking similarity existing in a short range order between amorphous and crystalline valvemetal oxides. 44−47,72,131−135 This is a rather special situation occurring in stoichiometric amorphous materials where the only crystallographic defect is the absence of a medium-longrange crystalline order, while a short-range order still exists. When this occurs, the mobility gap, E mg , is given by the distance in energy between the conduction band mobility edge (E CM ) and valence band mobility edge (E VM ) in the sense of Mott−Davis. 130 E mg is usually derived from the photocurrent spectra and Tauc plot analysis in the hypothesis of indirect (nondirect in the case of amorphous materials) optical transitions. We have shown that for amorphous anodic films grown on valve metals 31,43,128,136,137 (Ta, Nb, W, Ti, etc.), in not incorporating an electrolyte solution, the difference between the mobility gap and the optical band gap of the crystalline counterpart, ΔE am = (E mg − E g,cryst ), can reach a value in the order of 0.3−0.4 eV. According to Mott−Davis, such a difference is a measure of the influence of lattice disorder on the measured optical band gap values of amorphous materials and, when it occurs, an exponential tail (Urbach tail) in the measured photocurrent, for photon energies lower than the mobility gap value (hν < E mg ), could be a signature of the existence of an exponential DOS Inorganic Chemistry pubs.acs.org/IC Article distribution of localized states at energies below the mobility edge.
In the Mott−Davis model of an amorphous semiconductor (SC), the ΔE am term is essentially positive or zero, but we are aware that this term can turn to a negative value if, apart from the absence of long-range crystalline order, intense optical transitions originated by a distribution of vacant/filled electronic states (DOS distribution) within the mobility gap are occurring. In this case, the presence of other possible crystallographic defects as well as of impurities and/or nonstoichiometry has been suggested in the literature. 22−25,69,124,138−140 It is important to note that with respect to the previous approach, 31,32 the relationship usually employed for amorphous oxide will be modified, in the following, by introducing the term ΔE am into the B value so that for an amorphous TM anodic film, eq 1 used in the fitting procedure becomes As far as we know, an experimental E g value for η-Al 2 O 3 has not yet been reported in the literature, but as reported in the literature, η-Al 2 O 3 is a defective spinel structure 52,83, 141 showing the lowest crystallinity and estimated E g,opt value between all the metastable alumina phases. According to this, η-alumina, with its largest unit formula volume, will be assumed as the reference structure for the calculation of the B parameter in the case of amorphous alumina for which largely variable densities (ρ = 2.35−3.60 g cm −3 ) have been reported in the literature. 94, 131 These findings suggest that in the case of amorphous alumina, B values equal or lower than that reported in Table 1 for η-Al 2 O 3 (−2.25 eV) must be expected.
In Table 1, we reported also possible values of A and B parameters of amorphous alumina (am-Al 2 O 3 ) obtained by using eq 1 and band gap values measured by EELS or optical methods of thin and thick alumina films grown by PVD, spray pyrolysis, or aluminum anodizing in a barrier film, forming electrolytes. For amorphous alumina films grown by the PVD technique (ALD and reactive magnetron sputtering), a rather large range of ρ am values (2.32−3.77 g cm −3 ) has been reported in the literature 92 77,146,147 For anodic passive films grown up to 8 V on Al metal in aqueous solutions of variable pH values (8 < pH < 14), 68 band gap values in the range of 3.0−5.0 eV have been measured using the PCS technique. In the absence of direct information on the nature of passivating layers and on the basis of Pourbaix electrochemical equilibria diagrams, the lowest measured band gap (3.0 eV) was attributed to the possible presence of an external amorphous Al(OH) 3 phase, while the higher extrapolated E g values (4.5−5.0 eV) were associated to a less hydrated, possibly oxyhydroxide γ-AlOOH phase. More recently, the anodic photocurrent spectra displaying an indirect band gap value around 4.6 eV were reported in the literature 65 for an as-formed very thin (≃3 nm) alumina film grown up to about 1 V above the equilibrium potential in a quasi-neutral aqueous solution on a pure (99.999%) Al sample. A slight decrease in the measured band gap value (about 4.0 eV) was recorded after longer polarization times at the same anodic potential. The lowering of the band gap could be related to the formation of an external hydrated aluminum oxide (see above), while it is tempting to attribute the highest band gap value initially measured to the possible formation of a defective anhydrous alumina barrier layer having a structure similar to the η-phase described in Peintinger et al.'s work.
In a previous DFT study, E g values as low as 2.5−3.0 eV have been reported for amorphous alumina; 77 such lower values have been attributed to the underestimation of the band gap usually observed in DFT methods, while the lowering of band gap in amorphous alumina has been associated mainly to a downshift of the conduction band of about 2.5 eV going from crystalline α-Al 2 O 3 to am-Al 2 O 3 . In a recent DFT-based study, Liu et al. 70 have reported a band gap value for amorphous alumina equal to 6.2 eV for a film with a density equal to 3.2 g cm −3 , in good agreement with the value reported for an anodic thick film for which a density of 3.3 g cm −3 value has been reported. 93 As for thick amorphous barrier-type anodic oxide films grown in aqueous solutions, E g values around 6.3 eV have been reported in the literature. 68,69 To further clarify the aspects related to the presence of amorphous alumina units in ternary systems and to highlight the use of eqs 8a−8d in nonregular ternary oxide systems, we will discuss shortly the (sp,d In previous works, 43,66,67 we reported the dependence of E g,opt values of these systems as a function of the starting metal alloy compositions obtained by magnetron cosputtering of pure metals (x Al ≤ 100 at %). In the following, we will show how previous results pertaining to the ternary s,pd-metal ternary oxides, challenging the initial correlation, can be now nicely fitted in the frame of the general approach outlined above by using eqs 8a−8d.
3.3.1. Amorphous (Nb (1−x) Al x ) 2 O (5−2x) . In Figure 3, we report the experimental E g,opt data set points for anodic films grown on Nb-Al magnetron-sputtered alloys 67  As for η-Al 2 O 3 , the B value of −2.25 eV and the A value of 1.67 obtained according to eq 1 and by using E g,opt = 4.40 eV were used as reported in Table 1. As for amorphous niobia (am-Nb 2 O 5 ), we assumed ΔE am = 0.15 eV obtained as the difference between the optical band gap (3.35 eV) measured for the thick (about 20−30 nm) niobia anodic film and the band gap (3.20 eV) of the crystalline niobia (cr-Nb 2 O 5 ) phase recently reported in the literature for the H-Nb 2 O 5 crystalline material. 149,150 According to eq 1, a value of 1.302 is now derived for A H-Nb2O5 , which is slightly (−3.2%) lower than the average 1.35 value reported for d-metal oxides. We like to stress, however, that the experimental data could be also fitted by keeping A = 1.35 for H-Nb 2 O 5 but assuming for χ Nb a value of 1.634, instead of the usual 1.60, still within the uncertainty range of Pauling's scale. It can be noted that a very good fitting of the experimental data is obtained with an EN value of 1.50 for Al and A and B values reported above. The band gap value of 4.40 eV, estimated by Peintinger et al. for η-Al 2 O 3 , is slightly higher (0.07 eV) than that derived by fitting the experimental data according to eq 15a.
To stress the validity of this new approach, we report also in the same figure the theoretical points (white squares in Figure  3) of the crystalline ternary system derived from the previous ones by assuming ΔE am = 0, i.e., B am = B cryst = −1.50 eV, together with the experimental E g value of 3.63 eV derived from the data reported in the literature 148 for the polycrystalline AlNbO 4 sample. This band gap value, as reported by the authors, was obtained by extrapolating linearly to zero the optical absorption spectra of the AlNbO 4 sample. A more accurate extrapolation to the constant baseline of the spectrum, in the long-wavelength region, provides a band gap value of 3.71 eV. This last value compares very favorably with the value of 3.76 eV obtained in Figure 3 for the crystalline ternary system of composition x Al = x Nb = 0.5. These findings support the initial assumption on the use of η-Al 2 O 3 as a partner of amorphous ternary oxides in transition metal aluminates (TMaluminates), at least as long as the concentration of the TM partner cation remains above a "limiting threshold" of about 20%. The new approach suggests that for Al content up to around 80 at %, the structure of the ternary system am-(Nb (1−x) Al x ) 2 O (5−2x) can be imagined as a mixed oxide solid solution where both metallic cations are engaged in structural unities 46,72,132,135 where the bond structure is very similar to that experienced by the same cations in their amorphous (am-Nb 2 O 5 ) or strongly disordered (η-Al 2 O 3 ) structure.
For Al content around 90 at % (or higher), the optical band gap of the ternary oxide reaches values much higher than those  (5−2x) . The approach previously described for ternary am-(Nb (1−x) Al x ) 2 O (5−2x) oxides was employed to fit the experimental data of the optical band gap of amorphous mixed oxides am-(Ta (1−x) Al x ) 2 O (5−2x) (0 ≤ x ≤ 0.81) grown on magnetron-sputtered Ta-Al alloys anodized up to 10 V vs Hg/HgO in a borate buffer (pH = 8) solution. 43 The fitting of the experimental data was carried out on a heuristic basis by using the semiempirical correlation previously described for the similar am-(Nb ( To fit according to eqs 8a−8d the experimental data, we used now, as starting values, the A and B parameters of d-metal oxides, 1.35 and −1.50 eV, respectively, with χ Ta = 1.50, 34 which provide a band gap value for crystalline Ta 2 O 5 (E g2 ) equal to 3.90 eV. This last value is in very good agreement with the reported band gap values of crystalline orthorhombic β-Ta 2 O 5 , 151 obtained by thermally annealing the CVD amorphous Ta 2 O 5 film in air, as well with an E g value of 3.95 eV 152 reported for a thicker stabilized anodic oxide film grown on an electropolished (99.9% purity) Ta rod in a 0.5 M H 2 SO 4 solution. This value is also in agreement with the band gap value of Ta 2 O 5 calculated by DFT techniques with the HSE06 hybrid functional. 153 To take into account the amorphous nature of anodic films grown on magnetron-sputtered Ta and Ta-Al alloys, 154,155 we assumed a value of ΔE am = 0.375 eV. Such a value accounts, according to eq 14a, for the difference in E g value between the crystalline Ta 2 O 5 (E g = 3.90 eV) and the larger band gap value (4.275 eV) measured for anodic films grown in a borate buffer solution up to 10 V. 43 As for the alumina polymorph partner in ternary am-(Ta (1−x) Al x ) 2 O (5−2x) , a value of E g = 4.47 eV is obtained by extrapolating to x = 1 the quadratic best fitting line of the experimental data points (see Figure 4) The theoretical E g values for the am-(Ta (1−x) Al x ) 2 O (5−2x) system were derived by using eqs 8a−8d with the parameters reported in Table 1, slightly modified by the best fitting procedure, for the two pure binary oxides. The theoretical data points were obtained by using for amorphous Ta from which a band gap value of 4.47 eV is derived for a pure alumina film (x = 1). The quadratic and cubic terms in eq 17 are zero (see eqs 8c and 8d) owing to the assumption that χ Ta = χ Al = 1.50. The band gap values of hypothetical ternary cr-(Ta (1−x) Al x ) 2 O (5−2x) mixed oxides are also reported (white squares in Figure 4) from which the band gap value of a crystalline AlTaO 4 (x = 0.5) phase can be derived. The optical gap value of 4.30 eV pertaining to a vitreous 54Al 2 O 3 -46Ta 2 O 5 sample 156 is about 0.08 eV higher than the corresponding crystalline phase and about 0.08 eV lower than the corresponding amorphous anodic film of equal composition. It seems reasonable to attribute such difference to a different value (0.20 eV) of ΔE am term of this sample owing to the different degree of amorphousness of vitreous Ta 2 O 5 (E g,v-Ta2O5 = 4.10 eV) with respect to the anodic film.
Although the general trend of E g,opt vs x Al for Ta-Al anodic films parallels the behavior described above for anodic oxides grown on the Al-Nb system, new features arise when Ta substitutes Nb metal in the sputtered alloys. With respect to the case of Al-Nb alloys, we evidence that a negative bowing coefficient (b = S q = −0.065) is now derived in the fitting procedure owing to the fact that the measured band gap of anodic oxides show a clear tendency, at higher Al (x ≃ 0.5) contents, toward a constant limiting E g,opt value of 4.45 eV very near the theoretical one (E g,th = 4.40 eV) derived for η-Al 2 O 3 . More importantly, such a limiting value, appearing for Al content around 60 at % for thicker oxide films (≃15 nm), is anticipated to Al content around 40 at % for thinner films (≃8−9 nm) grown up to 5 V vs Hg/HgO under identical conditions (see Figure S3).
As for anodic films grown on Al-Ta alloys, the data of Figure  4 suggest that owing to the smaller difference in the band gap value between am-Ta 2 O 5 and η-Al 2 O 3 , the Al composition range exploitable before the ternary oxide reaches a band gap value equal to that one of η-alumina is now more limited so that for Al content near about 40 at %, in the case of a thinner (5 V) film, or 60 at % for a thicker (10 V) film, the ternary system reaches a band gap value equal to that one of ηalumina.
The data of oxide films grown to 5 and 10 V indicate that such a limiting band gap value, frozen up to around 80 at % of Al content, can be maintained as long as the concentration of the TM cationic partner is sufficient to freeze a disordered lattice structure similar to that of η-alumina. At lower concentration (<10 at %) of the TM cation into the ternary oxide, as mentioned before for the Al-Nb alloy, band gap values larger than that estimated by means of eq 17 are expected owing to the absence of any photocurrent signal under illumination with photons of 5.2 eV. These findings agree with the results reported in the literature 54 Finally, we like to mention that as expected from the generalized approach, a nice fitting of the experimental E g,opt data 136 of anodic films grown on Ta-Nb alloys was observed (see Figure S4) by using for am-(Nb (1−x) Ta x)2 O 5 the identical parameters employed for TM oxides in fitting the E g,opt values of ternary oxides grown on Al-Ta and Al-Nb magnetronsputtered alloys.
3.3.3. Amorphous (W (1−x) Al 2x )O 3 . The last system we shall shortly discuss is represented from the ternary am-Inorganic Chemistry pubs.acs.org/IC Article (W (1−x) Al 2x )O 3 anodic films grown on Al-W alloys of various compositions for which a detailed PCS characterization has been reported in a previous work. 66 In Figure 5, the E g,opt values as a function of the Al composition (obtained by RBS analysis) 66 into the mixed oxide film are reported.
The theoretical fitting points have been obtained by means of eqs 8a−8d by assuming for a-WO 3 a value of ΔE am = 0.35 eV with χ W = 1.70 and the average values of d-metal oxide for A and B parameters (1.35 and −1.5 eV, respectively). As for η-Al 2 O 3 , the values reported in Table 1 were used for A (1.667) and B (−2.25 eV), in agreement with the value of E g = 4.40 eV and by assuming an EN value for χ Al = 1.50. For x Al > 71 at %, the best fitting parabolic equation provides E g values between 4.08 eV (x = 0.8) and 4.40 eV at variance with the experimental results. In fact, for the concentration of x W , into the ternary oxide, equal to 21 at %, we were not able to measure any photocurrent as previously noted for Ta-Al and Nb-Al at around the same concentration of TM cations into the films.
The experimental results on all three investigated amorphous ternary systems suggest that when the concentration of the cationic partner decreases below a critical threshold (about 10−20 at %), the η-alumina structure becomes unstable, under anodizing conditions, by changing to a different metastable structure having a larger band gap value. A candidate polymorph could be γ-alumina, which is the more common phase under ordinary conditions and displays a disordered lattice structure used as the starting structure to derive the more disordered η-phase. 52,71,83

CONCLUSIONS
A generalization of the semiempirical approach to the modeling of the band gap of ternary oxides has been carried out, taking into consideration previous criticisms to the limited ability of the initial approach in accounting for the role of oxide polymorphisms in determining the band gap of mixed oxides. We have shown that the influence of the different crystalline structures plays a role in determining the optical band gap of different polymorphs by affecting the two parameters A and B of our semiempirical approach. The B parameter of each polymorph has been assumed to depend on the unit formula volume or density, while the A parameter, derived by means of the initial correlation between the band gap and difference in electronegativity between oxygen and average cationic electronegativity, changes owing to changes in the average strength of the oxygen−metal bond.
A re-examination of experimental data of pseudoregular (sp,sp mixed oxides) and nonregular (sp,d mixed oxides) systems seems to validate the proposed generalization of our semiempirical model. The investigated systems allowed also one to test the proposed generalization in a rather large range of band gap values as those reported for alumina polymorphs by including also the metastable η-Al 2 O 3 polymorph for which the proposed band gap value of E g (4.4 eV) derived by DFT techniques was used to fit the band gap dependence, E g vs x Al at %, of amorphous mixed oxides grown on Al-TM (Nb, Ta, and W) alloys of various compositions. From the study of these last amorphous systems, it shows that a minimum concentration threshold of the TM cation, in the order of 20 at % into the oxide, seems necessary to keep a short-range order between Al 3+ cations and O 2− anions similar to that reported for η-Al 2 O 3 . The concentration threshold of the TM cation is quite near the percolation threshold foreseen for the formation of a delocalized distribution of electronic states (formation of a conduction band) in random distributed Ni x Mg (1−x) O mixed oxide. 140 At lower TM cation concentration (around 10 at %), a deviation from the theoretical fitting line was observed in all amorphous nonregular ternary systems investigated, probably owing to the formation of a mixed ternary oxide having a different short-range order between Al 3+ and O 2− anions, more similar to that reported for other metastable alumina phases of a larger band gap doped with a very low (<10 at %) concentration of TM ions. 54,157,158 In our case, the formation of amorphous mixed ternary oxides with short-range order Al-O between Al 3+ and O 2− anions similar to β,γ-Al 2 O 3 seems reasonable and is in agreement with the photocurrent spectroscopy experimental findings. Further studies on these aspects could be very useful for the possible use of these mixed oxides in the field of high-k materials. Although only three alumina polymorphs were involved in this work, it will be shown in a future paper how other alumina polymorphs have to be used for fitting the optical band gap values of different crystalline alumina-TM ternary and quaternary oxides.  Figure S2); fitting of E g vs x Al for Al-Ta mixed oxide thin films ( Figure S3); fitting of E g,opt values for Nb-Ta mixed oxide values ( Figure S4); fitting parameters for the different Al-(Ga, Nb, Ta, W) oxide systems studied (