Review—Photocurrent Spectroscopy in Corrosion and Passivity Studies: A Critical Assessment of the Use of Band Gap Value to Estimate the Oxide Film Composition

A critical assessment of the Photocurrent Spectroscopy (PCS) Technique for the semi-quantitative characterization of passive film and corrosion layers composition is carried out. We take into account more than three decades of PCS usage as “in-situ” analytical technique and related results as well as the criticism of the underlying semi-empirical correlation relating the measured optical bandgap (Eg) to the passive film composition. The discrepancies between the experimental data, gathered by PCS measurements, and Eg estimates originating from recently developed Density Functional Theory based modeling of solid state properties are stressed with particular emphasis on the case of anodic passive film grown on technologically important alloys (Fe-Cr and stainless steels). The extension of this correlation to mixed oxides and its use for relating the oxide composition to the bandgap values is critically reviewed by comparing the predicted Eg of mixed oxides with the experimental values. Suggestions on how to account for different bandgap values of oxide polymorphs and how to correlate the Eg values to the composition of mixed s,p-d-metal oxides are presented and discussed on the basis of experimental results reported in the literature. On the basis of this assessment, the ability of PCS in providing quantitative information on the composition of passive film and corrosion layer is generally confirmed. © The Author(s) 2017. Published by ECS. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI: 10.1149/2.1621712jes] All rights reserved.

Photocurrent Spectroscopy (PCS) is currently employed for the characterization of solid-state properties of semiconducting and insulating materials, since the knowledge of their bandgap is a prerequisite to any possible application in different fields such as solar energy conversion (photoelectrochemical and photovoltaic solar cells, photocatalysis) and microelectronics (high-k, high band-gap materials). 1,2 In the last 20-30 years an increasing number of electrochemists working in the field of corrosion have been attracted by this technique owing to its versatility and ability to scrutinize "in-situ" corrosion layers and passive films having semiconducting or insulating behavior. In previous as well as in very recent works [3][4][5][6][7] we have shown that PCS is able to provide detailed information on characteristic energy levels of passive film/electrolyte junctions (flatband potential: U FB ; internal photoemission threshold: E th ; bandgap value: E g ). Such information is necessary for a deeper understanding of the possible mechanisms of charge transfer (electrons and ions) at the metal/corrosion layer/electrolyte interface. Further advantages stem from the fact that the PCS technique does not require particular surface finishing control, it can be used both on large areas as well as in microscopic regions of the electrode, and can reach high sensitivity by using a lock-in amplifier coupled to a mechanical light chopper to detect the photocurrent signal. According to this, numerous experimental studies have shown how it is possible to scrutinize corrosion layers and passive films of very limited thickness grown on metals and alloys (Fe, Cr, Ni and SS) of relevant industrial interest. 3,4,8 In an attempt to make more quantitative the characterization of corrosion layer and passive film by PCS, in previous works [3][4][5][6]8,9 we have shown that it is possible to correlate semi-empirically the E g values of numerous binary oxides to their composition by means of the following expressions: 9 E g = 2.17(χ M − χ O ) 2  where χ M and χ O are the electronegativity parameters of metal and oxygen, respectively, on the Pauling scale. Eqs. 1a and 1b were able * Electrochemical Society Member. z E-mail: francesco.diquarto@unipa.it to fit with a reasonable accuracy the E g values of sp-metal oxides (2 and 13-15 groups in the periodic table of the elements), whilst Eq. 1b was able to reproduce with good accuracy the experimental E g values of many d-metal oxides (3-12 groups). In the original paper, 9 the E g values of passive films oxides and of most common crystallographic structures have been used by neglecting the differences in the bandgap values of different polymorphs. When possible, E g values derived from PCS studies by means of Tauc's treatment of data have been used. As for the electronegativity values of different elements, the values of Pauling scale, 10 integrated with other data 11,12 for Mg and Sn in different oxidation states, have been used in previous studies 9 and in the following.
In several successive works we extended Eqs. 1a and 1b to ternary oxides, M m N n O o , using an average electronegativity defined as the arithmetic mean between those of partner metals, χ M and χ N , each weighted according to their corresponding atomic fraction. This procedure allowed correlation of the bandgap values measured by PCS with the mixed oxide composition, as confirmed with several mixed oxides of known composition. [3][4][5][6]8,9 In this work, we will discuss the limits of validity of Eqs. 1a and 1b for binary and ternary oxides spanning a large range of bandgap values (from ∼1 eV to ∼9 eV) by discussing data reported in the literature pertaining to ternary mixed oxides prepared by different techniques and including anodic passive films grown on metallic alloys. Moreover, we will compare the bandgap values of mixed d-d metal oxides, of large interest in corrosion studies, theoretically derived by Density Functional Theory (DFT), to those estimated by PCS. Possible explanations for the difference between experimental and theoretical data will be presented and discussed in order to highlight the limits and possibilities of the PCS technique in providing information on the chemical composition of passive films and corrosion layers. Finally, a generalization of Eq. 1a and 1b to the case of ternary s,p-d mixed oxides will be presented and discussed by using the literature data of mixed Zn x Mg 1-x O oxides covering all the range of composition (0 < x < 1) and both wurtzite (WZ) and rock-salt (RS) crystalline structures.

Theoretical Background
Following previous works of Manca 13 and Vijh, [14][15][16] we suggested to write the following relationship for the bandgap of inorganic C672 Journal of The Electrochemical Society, 164 (12) C671-C681 (2017) compounds: 9 where H eq is the heat of atomization per equivalent, in the case of polyatomic compounds, and U R is an energy term including repulsive and London components. 13 Moreover, we modified Eq. 2a for a polyatomic oxide MO y by using for H eq the Pauling equation 10 for the average single bond energy D M-O and re-writing Eq. 2a as: 9  17 introducing the parameter E I assumed "to vary with hybridization configuration, i.e., with different atomic coordination in different crystal structures" in order to take into account the possible influence of different type of bonds on the "extraionic energy" as defined in Pauling's classical book. After substitution of the Pauling equation in Eq. 2a it was possible to get a general expression for the bandgap of semiconducting and insulating compounds as: where χ an and χ M are Pauling electronegativity values of the oxygen (or generally speaking of the most electronegative non-metallic element) and metal, respectively, present in the compound. The term: contains the average bond energies of the diatomic molecules in the gas phase (1/2(D A-A + D M-M )) and the repulsive term R MOy = (|U| bond-|U| lattice ). In the ionic limit, U bond coincides with the Madelung energy, while in the covalent limit a quantum mechanical calculation of the bonding energy is required. 18 In spite of the widely discussed limitations, the use of Pauling equation is still very attractive owing to its simplicity, reliable estimate of the heats of formation per equivalent of inorganic compounds, assumed as a measure of the bond strength in solids, and the possibility to relate this classical chemical concept with quantum mechanical derived concepts and parameters. 19,20 Moreover, owing to the periodic trends of the electronegativity values in the Pauling scale, strictly related to the periodic table of chemical elements, the use of Eq. 2c allows for easy identification of the expected general trend in the optical bandgap of inorganic semiconductor compounds. We would like to stress this aspect as one of the most rewarding results of our previous studies on pure sp-metal and d-metal oxides together with the possibility to tailor, to some extent, the bandgap values of mixed oxides. As assumed in previous works, [3][4][5][6]8,9 in so far as the term is constant, Eq. 2c predicts that the bandgap of semiconducting compounds is linearly dependent on the square of the difference of electronegativity between the most electronegative element (oxygen in the case of oxides) and the metallic element. According to this, Eq. 2c was fitted according to a general expression as: where A and B have been determined by a linear best fitting procedure of experimental E g values vs (χ an − χ M ). 2 In the case of pure s,p and d-metal oxides, two different interpolating straight lines were derived with:  9 is about 50% higher than  3,4 In this case, however, we have to mention that the most common E g quoted in literature is a direct band gap ∼0.7-0.9 eV higher than the previous one. 22 Although still controversial, the most recent E g value (2.70 eV) for single crystal c-In 2 O 3 23 seems to support our initial assumption of In 2 O 3 based on the d-metal correlation. We will come back on these aspects in the following.

Regular (sp,sp or d,d-metals) mixed oxides.-
We previously attempted 9 to extend Eqs. 1 to mixed oxides (M m N n O o ) by replacing χ M with an average electronegativity, χ av , parameter defined as the arithmetic mean between the electronegativity of partners metals (χ M and χ N ), each weighted by the corresponding atomic fraction as: [4b] The selection of A and B is straightforward when M and N are both d-metals or s,p-metals (regular) mixed oxides, whilst the problem of coefficients selection arises for s,p-d metals (non-regular) mixed oxides (see below). We also suggested that if the difference of electronegativity (in the Pauling scale) between the partner metals in the mixed oxide is higher than ∼0.5, the bandgap of the mixed oxide was very close to that of the oxide of the most electronegative metal (lower bandgap oxide, see Table I). 9 Although, initially, this observation seemed limited to the s,p-d mixed oxides more recent data confirm that analogous results can be expected also for regular solutions (d-d metal or s,p-s,p metal mixed oxides) when a large difference in the electronegativity values of the cations is observed (see Table  II). This finding is in agreement with the results of Eng et al., 24 where the bandgap value of a series of MTiO 3 (M = Ca, Sr) mixed oxides is reported to be not significantly influenced by the presence (and concentration as reported in Table III) of Sr and Ca ions, therefore, described as "cation spectators". The small influence of the cation spectator on the optical bandgap value, in such mixed oxides, has been attributed to the steric action of the different cation spectator through local distortion of bond angle and lengths. Our observation allows prediction of which metallic element combination could originate such a behavior, although it is not yet clear if the threshold value of χ cat , marking the onset of such a behavior, is equal or slightly different for regular (≥ 0.7) and non-regular (≥ 0.5) mixed oxides. In the case of mixed semiconducting alloys, Zunger and coworkers [25][26][27] discussed the role of the composition in determining the values of physical parameters of a generic semiconductor alloy M (1-x) N x X and observed that: "in absence of composition induced structural phase transitions in alloys or electronic direct-to-indirect bandgap crossover the physical properties P(x) were traditionally assumed to be simple continuous functions of the composition". In the case of the optical bandgap value, such a function assumes the form of the traditional bowing equation, represented by a linear term plus a small quadratic correction term, which can be written as: 27 [5] where b represents the so-called bowing coefficient. Zunger and coworkers noted that for regular mixed cations semiconductor alloys the bowing parameter assumes usually small values well below 1 eV. In a previous work, 28 we have shown that in the case of regular mixed nitrides following Eq. 4 it is possible to get an expression analogous to Eq. 5 by simple algebraic manipulation of Eq. 4. By following the same procedure in the case of regular mixed oxides it is possible to derive from Eqs. 1a, 1b the following relationship for the composition dependence of the bandgap value of mixed regular oxides: 29 where E g,M (or E g,N ) is the bandgap of pure MO (or NO) oxide, x N (or x M ) is the cationic fraction (0 ≤ x ≤ 1) of N (or M) into the oxide. Obviously, from Eq. 4 it follows that for x N = 1 Eq. 6a gives the bandgap of pure NO oxide. After algebraic manipulation, Eq. 6a can be rewritten in a manner similar to the bowing equation usually reported for semiconductor alloy as: Table III. Comparison between the experimental bandgap and E g,th estimated according to Eq. 7 of mixed oxides. χ is the difference of electronegativity of the partners metals in the oxide, while E g,th is the bandgap estimated according to Eqs. 1 using χ of the more electronegative metal, (χ c2 ). 24 Mixed  where the linear term is now: In Eq. 6 A is the coefficient derived by Eq. 4, therefore A = 1.35 or A = 2.17 for regular d,d metals and s,p-s,p metals mixed oxides, respectively.
In previous papers, [30][31][32] we have shown that conditions such as those mentioned above in the paper of Zunger et al. can be encountered in practice in the case of passive films grown on valve-metals alloys where, in suitable electrolytic solutions, the oxide composition maintains the same cation ratio present in the metallic alloy. This is reported in Figures 1-3 for d,d-metals mixed oxides, where the measured optical band gaps as a function of the alloys composition are reported for passive films grown on Ti-Zr and Ta-Nb alloys as well as for physically deposited (Gd x Y (1-x) ) 2 O 3 . [30][31][32][33][34] The amorphous nature of the passive film on Nb-Ta alloys and the absence of any indirectto-direct optical transition crossover in the case of crystalline films grown on Ti-Zr alloys, allowed testing of the suggestion of Zunger and coworkers 25,26 for the regular semiconducting alloys reported above. Figures 1 and 2 show the dependence of experimental E g values of thin anodic passive films vs alloys composition for Ti-Zr and Ta-Nb  mixed oxides. [30][31][32][33] In these figures the experimental optical bandgap values were obtained by PCS, whilst the theoretical data were calculated by using Eq. 4a with the average electronegativity parameter value of the cations calculated according to Eq. 4b. The cation ratio in the anodic film, derived from RBS analysis, was estimated to be almost coincident to that in the alloys. 35,36 As shown in Figure 2 experimental E g values of anodic oxides, grown on Nb-Ta sputtered alloys, follows the expected trend as a function of the oxide composition but with values slightly higher (about 0.1 eV) than the values of crystalline counterparts. As discussed in literature, 4,5,32 a larger (mobility) gap with respect to the E g of crystalline counterparts is expected for stoichiometric amorphous semiconductors. In the absence of stoichiometric deviations, the disordered structure of the a-SC induces the localization of electronic states close to the conduction and/or valence band edges, as described by the density of states (DOS) model proposed by Mott-Davis for amorphous semiconductors. [37][38][39] A further support to our approach to the estimate of the bowing parameter comes out from the data of Figure 3 pertaining to mixed (Y x Gd 1-x ) 2 O 3 oxides for which a good agreement between experimental and theoretical values is also observed by assuming for the two d-metal cations A = 1.35 and B = −1.5 (see Eq. 3b) and an electronegativity parameter of 1.20 and 1.27 for Y and Gd respectively, in quite good agreement with the Pauling scale and uncertainty limits. The value of 1.27 for χ Gd was derived from Eq. 3a considering an optical E g value for Gd 2 O 3 pure oxide of 5.22 eV. 34 Eq. 6c shows that for regular mixed oxides the bowing coefficient b is always positive or zero, whilst the linear term can be positive or negative depending on the sign of the term (χ M -χ N ). The linear term will be positive (negative . According to our correlations, lower bandgap materials have higher Pauling electronegativity parameters and vice versa. Positive b values are expected for regular mixed oxides together with a linear dependence of the bandgap as a function of the composition, as long as the difference of electronegativity of the cationic partners present into the alloy is small ( χ ∼ 0.2 for regular s,p-or d-metals mixed oxides).

Critical Issues in the Field
Composition dependence of optical band gap in mixed oxides.-Regular (sp,sp or d,d-metals) mixed oxides.-The previous correlations are mainly based on a semi-empirical approach and thus subjected to the criticism reported in literature, 40 where the authors discuss the limits of the ability to predict the expected E g values of different oxide polymorphs, as well as the lack of any dependence of  Table IV) which can be fitted within the proposed correlations by assuming small changes in the parameter B and by leaving unchanged the coefficient A of each correlation. This suggestion derives from the fact that according to Eqs. 2c and 3 the term B: includes the repulsive term R, which can be assumed dependent on the crystallographic structure of each polymorph through the difference (|U| bond-|U| lattice ). We are aware that reasonable changes in the R values cannot always account for the differences in the reported or estimated bandgap values of several polymorphs, 41 and that changes in the A value of the employed correlation can be also necessary (see below for α-(Ga x Al 1-x ) 2 O 3 ). We would like to stress, however, that predicting the E g of pure and mixed amorphous semiconducting compounds is not an easy task even for the more sophisticated quantum-mechanical based DFT techniques and it is not yet, extensively, used to determine the E g,opt values as a function of the composition of passive films or corrosion layers having amorphous or strongly disordered nature. 37,38 This aspect must be taken into account owing to the fact that there are few techniques at our disposal able to correlate the gathered in-situ information on the solid state properties (E g , flatband potential, U FB , and internal photoemission threshold, E th ) of passive films and corrosion layers with their chemical composition, as a function of chemical and electrical variables employed in the experiments.
A theoretical support to the use of Eqs. 1 can be found in seminal papers of Zaanen, Sawatsky, and Allen 42,43 who suggested that the E g of transition-metal compounds (i.e. oxides, sulfides and halides) is of charge-transfer type, i.e. E gap ∝ , provided that the charge transfer energy, , is less than the d-d coulomb and exchange interactions term U, used in the theory for the calculations of band-structure in transition metals compounds. In these works it was suggested that the term is directly related to the electronegativity of the anion and the Madelung potential of the solid. The semi-empirical correlation proposed for the dependence of E g,opt values on the nature of the partner involved in the solid agrees, in principle, with the suggestion by Zaanen-Sawatsky-Allen as long as the lowest measured optical gap value is related to an inter-atomic charge transfer process. However, lower E g values in transition metal oxides 42,43 are foreseen by their model as long as the charge transfer term is larger than the d-d coulomb repulsion and exchange interactions term U used in the theory for the calculations of band-structure. In the case of oxides, the charge transfer process frequently occurs from oxygen p orbital, forming the valence band, to the conduction band made mainly from the orbital of the cationic counterparts. According to this it can be generally inferred that E g values of pure or mixed oxides not in agreement with the proposed correlations could be rationalized by assuming the onset of intra-atomic transitions owing to the fact that the electronic exchange and d-d coulomb interaction term U becomes smaller than the charge transfer term . Recent theoretical estimates of mixed Fe-Cr oxides E g values 44-48 between 1.5 eV and 1.70 eV for iron-chromium mixed oxides containing a variable concentration of Cr in between 25 and 80 at.%, seem to support the Zaanen-Sawatsky-Allen suggestion. Further complications to this scheme can be expected in the presence of passive films containing transition metal cations with d-band partially filled originating from possible intra-atomic d-d optical transitions. For this purpose, we need to compare some recent theoretical results pertaining to the bandgap of mixed iron-chromium oxides with the experimental data derived from PCS investigations of passive films grown on SS and Fe-Cr alloys.
As previously reported, and in agreement with the proposed correlations, increasing optical bandgap values are expected for passive film grown on metallic alloys obtained by alloying pure iron (χ Fe +2 = 1.8 and χ Fe +3 = 1.9) with increasing amount of chromium (χ Cr 3+ = 1.6), at not too high electrode potential, i.e. in a potential range in which chromium keeps its usual Cr +3 oxidation state. Actually, such an expectation has been confirmed in the literature by several investigators, 8,49-60 who characterized passive films grown on ironchromium alloys or both ferritic and austenitic stainless steel (SS) by using PCS. We have to mention that in the case of SS other ex-situ analytical techniques (such as: XPS, EXAFS etc.) have confirmed the presence in the passive films of Cr 3+ enrichment with respect to the base alloys, where at least 13% of Cr metal is present. 8,51,52,[61][62][63][64][65][66][67][68] In Table V we report some E g values measured by PCS on Fe-Cr alloys as well as on ferritic and austenitic SS. Table V indicates that in the all range of electrode potentials, where a passive film is present on the metallic alloys, E g values higher than 1.90 eV were usually measured. This value is typically measured for passive films grown on pure iron, 69 but it was also measured in the case of films grown on ferritic SS exposed to a chloride containing solution at high potential. 50 Under these severe conditions, it is likely that localized corrosion processes occur producing an excess of Fe 3+ ions in solution with consequent precipitation on the SS surface of iron(III) oxide or oxy-hydroxide. According to the literature 70 several iron oxides and oxy-hydroxides polymorphs have such a bandgap value. Almost all passive films on SS display indirect or non-direct (if amorphous) optical bandgap values larger than 2.0 eV and, in the case of austenitic SS, a bandgap value at around 2.4 eV 3,4,49,50 was detected independently from the electrode potential. Moreover, E g values be-tween 2.0 and 3.0 eV were measured for the passive films grown on Fe-Cr alloys in different electrolytic environments and at different electrode potentials. [51][52][53][54][55][56][57][58][59] Ex-situ analytical techniques revealed a Cr(III) content ranging between 20 and 35%. [51][52][53] All these data suggest that passive films with a concentration of Cr between 10 and 40 at% or higher than that present in the starting alloy (≥13% in SS), are formed on Fe-Cr alloys, at different Cr concentration, but all these films display bandgap values larger than that of pure Fe 2 O 3 . According to this, bandgap values larger than 1.90 eV have been attributed to the presence of a passive film containing a Cr-rich Cr x Fe y O z mixed oxide.
In line with these results, it is not clear how to reconcile the E g values of mixed transition metal oxides (TMO) containing partially occupied d-level (d (10-n) -TMO, with n =0) based on DFT theoretical estimates with the experimental data obtained by PCS on passive films grown on mixed Cr-Fe alloys and/or SS. However it is also possible to suppose that in the case of passive films, due to their amorphous nature and/or extremely thin thickness, the theoretical predictions are not followed since excited electronic carriers near the mobility gap or at lower photon energy recombine easily so that the PCS measured bandgap values pertain to the higher energy optical transitions of the mixed oxides. In any case, it appears urgent to clarify such aspects in order to have a correct interpretation of the PCS data for such mixed oxides of extreme relevance for a correct interpretation of the corrosion behavior of these technologically important alloys. 71 In order to test the general validity of Eq. 4a and Eqs. 6 in predicting the composition dependence of E g values also in the case of s,pmetals mixed oxides, and in agreement with the general rules of the regular semiconductor alloys above mentioned, in Figure 4a we report the optical bandgap of mixed α-(Ga x Al 1-x ) 2 O 3 72 as a function of the square of the difference of electronegativity between oxygen and the average electronegativity of cations derived according to Eq. 4a. As indicated in Figure 4a, E g linearly depends on (χ O -χ av ) 2 by assuming χ Ga = 1.61 and χ Al = 1.475, in agreement with the Pauling scale and within the accepted uncertainty. Moreover, from the data shown in Figure 4b, it is evident that the dependence on composition of the band gaps of mixed α-(Ga x Al (1-x) ) 2 O 3 can be nicely fitted according to the following expression:  Figure 4b). In the case of α-Ga 2 O 3 , the bandgap reported in the literature is around 5.30 eV in agreement with the value reported previously 73 and with the E g values derived by using Eq. 3b valid for s,p-metal oxides (see also Table IV). It is of some interest to stress that the linear correlation reported in Figure 4a is able to provide also a value of bandgap for α-In 2 O 3 (corundum structure) equal to 3.40 eV (χ In = 1.70 in the Pauling scale) in quite good agreement with the bandgap value of 3.6 eV reported in the literature for In 2 O 3 by assuming direct optical transitions as for mixed α-(Ga x Al (1-x) ) 2 O 3 . According to the literature 22 the value of E g = 3.6 eV is not the fundamental bandgap but the lowest direct optical bandgap value of α-In 2 O 3 . Moreover, recently, it has been experimentally shown that the fundamental gap for single crystal cubic-In 2 O 3 is indirect with a value of ∼2.70 eV. 23 This last value well compares with the experimental value reported for plasma grown polycrystalline In 2 O 3 21 better interpolated by the d-metal correlation as, initially, suggested previously. 9 In fact, by assuming for In the Pauling electronegativity value χ In = 1.70 ± 0.05, an E g value of 2.87 ± 0.24 eV is obtained from d-metal correlation, whilst a bandgap value of 4.30 ± 0.385 eV is derived according to the s,p-metal correlation, quite far from the values experimentally measured for direct and indirect transitions.
In the case of Al 2 O 3 , the E g values experimentally measured (or derived by DFT studies) for different polymorphs span a quite large range of energies 41 well outside the range of energies (5.5-6.40 eV) derived by using Eq. 3b and the uncertainty in the electronegativity parameter of the cation (χ Al = 1.5 ± 0.05). According to this, we assumed for aluminum a value of χ = 1.475 which provides, by using Eq. 3b, a value of E g (Al 2 O 3 ) = 6.20 eV. This last value is in good agreement with the experimental E g,opt values (6.0-6.2 eV) measured for amorphous alumina anodic barrier film 74 and Atomic Layer Deposition films 75 as well as with the theoretical estimated value of the spinel phase γ-Al 2 O 3 . 41 As mentioned above, a limited variation of the E g values can be easily accommodated by using the range of uncertainty embodied in the Pauling scale of electronegativity, whilst the large variation in the E g values of the different polymorphs of Al 2 O 3 requires, in our opinion, a different approach which takes into account possible changes in the value of A and B.
In the next section, pertaining to the non-regular mixed oxides, we will presents a more general approach for the extension of Eq. 3 also to the case of regular mixed oxides but with a E g value of a polymorph (in this case α-Al 2 O 3 ) deviating largely from the value expected by the corresponding correlation.
Non-regular (s,p-d metals) mixed oxides and estimates of polymorphs E g .-In the case of ternary oxide containing cationic elements belonging both to sp-metal and d-metal groups (non-regular mixed oxides), the choice of the coefficients A and B to be used in Eq. 3 becomes undetermined. In order to overcome such a difficulty and in agreement with the model previously suggested for regular mixed oxides, we here generalize Eq. 4 to mixed non-regular systems by assuming that for mixed oxides the bandgap value can be written as: where: with χ av = x 1 χ 1 + x 2 χ 2 in which x i and χ i representing the cationic fraction (in at%) and electronegativity parameter, respectively, of each metal M i present in the mixed oxides. E g,1 and E g,2 represent the bandgap values of pure oxides assumed to follow the previous correlations according to Eq. 3. Eq. 7 after simple algebraic manipulations can be rewritten as: where: As can be easily checked, Eq. 7 contains Eq. 6 as particular case as soon as we assume that the bandgap of pure oxide M 1 O y1 and M 2 O y2 belongs to the same correlation (A 1 = A 2 ; B 1 = B 2 ). From a more physical point of view, Eq. 7 (as well as Eq. 4) can be considered a chemical approach to the estimation of bandgap values of semiconductor alloys based on the virtual crystal approximation (VCA) model 76,77 with averaged electronegativity of the cationic group. A comparison between experimental E g values and that calculated by using Eq. 7 for several mixed oxides is reported in Table VI. In order to get a test on the validity of the suggested approach in predicting the composition dependence of bandgap of non-regular mixed oxides a detailed analysis of the composition dependence of E g values of the Zn (1-x) Mg x O system has been carried out. The choice of this system was suggested by the fact that numerous experimental data (E g values and cationic ratios) are available from the literature in almost all the range of composition corresponding to the different stable polimorphs WZ-ZnO (wurtzite) for x Mg ≤ 0.40at% and RS-MgO (rock salt) for x Mg ≥ 0.45 at%. [78][79][80][81][82][83][84] Moreover, very recently, a metastable wurtzite phase containing 51at% of Zn with an optical bandgap of 4.49 eV has been reported 83 and it will be also taken into account in the fitting procedures discussed below.
We have to mention that, as for the pure oxides, there are not experimental data for the bandgap values of RS-ZnO and WZ-MgO under ambient conditions, although experimental E g,opt values ranging between 2.50-2.80 eV have been reported for the optical bandgap of RS-ZnO under high-pressure conditions 85,86 or as nano-crystals. Furthermore, a very recent study on the electronic band structure of MgO, based on DFT techniques, reports values of 6.06 eV and 6.2 eV for WZ-MgO and ZB-MgO (zincblend structure), respectively. 87 The zinc-magnesium mixed oxides system presents further advantages owing to the fact that: In Figures 5a and 5b we report the optical bandgap values of mixed (Zn (1-x) Mg x )O 78-84 oxides both in the RS as well as in WZ phase as a function of the square of the difference of electronegativity of oxygen and of the average electronegativity of cations (χ av = 1.615 × x Zn + 1.315 × x Mg ). By direct visual inspection it is evident that the mixed oxides belonging to different crystallographic systems can be much better linearly fitted, separately, as confirmed by the improved R 2 value of the two interpolating lines (R 2 = 0.985 for RS and   88 No experimental data for such a phase have been reported in the literature but this extrapolated value is also in relatively good agreement with the direct bandgap value of 6.06 eV theoretically derived from a recent DFT study. 86 It is also worth noting that a linear dependence of E g from the composition (x Mg ) of Zn (1-x) Mg x O ternary oxides was reported according to Eq. 8: 88 E g = 3.35 + 2.33x Mg in eV [8] where only the linear term is present in Eq. 8 whilst the bowing parameter is missing, in spite of the large difference in χ values of the two cations, probably owing to the short composition range fitted 88 (see below). From Eq. 8a E g value of 5.68 eV is derived for WZ-MgO phase in very good agreement with the value above reported ( Figure  5b). Analogously, it is possible to derive an E g value of 2.74 eV for ZnO in the rock salt structure (RS-ZnO) by extrapolating the interpolating line of RS-Zn x Mg (1-x) O system to the value of (χ O -χ av ) 2 = (3.5-1.625) 2 , i.e. χ av ≡ χ Zn (see Figure 5b). Such a value is in quite good agreement with the experimental indirect optical bandgap values reported in the literature [85][86][87] for RS-ZnO under high pressure conditions (E g = 2.5 ± 0.15 eV) and with that (2.80 eV) reported for RS-ZnO nanocrystals embedded in MgO matrix. 89 By using the uncertainty range of χ Zn values around the value assumed for WZ-ZnO, we estimate a range of possible E g values for RS-ZnO equal to 2.74 ± 0.34 eV. We have to mention that indirect bandgap values ranging between 1.1 and 4.5 eV have been calculated by DFT for RS-ZnO. 87 The data shown in Figures 5a and 5b seem to confirm the general validity of the correlation between E g and the square of the difference of Pauling electronegativity between oxygen and the cationic average electronegativity initially proposed 9 also for mixed sp,d metal oxides. However the A values, obtained from the fitting lines for WZ-Zn x Mg 1-x O (1.803) and RS-Zn x Mg 1-x O (3.673) in Figure 5b, display no direct relationship with the initial A values derived for pure s,pmetal (2.17) and d-metal (1.35) oxides correlations, 9 so that it is impossible to predict the composition dependence of mixed s,p-d,d ternary oxides also in presence of previous knowledge of the E g values of pure s,p-metal and d-metal oxides.
On the other hand, according to Eqs. 7a-7d it should be possible to predict the bandgap value of mixed s,p-d metal oxides as a function of composition if E g values of pure oxides are in agreement with the E g values calculated by the correlations valid for s,p and d-metal oxides. According to this, we report in Figure 6a the best fitting of experimental data, derived from different authors, [78][79][80][81][82][83][84] of Zn (1-x) Mg x O films in the whole range of composition. All data have been used regardless of the crystallographic system in which each alloy is stable owing to the fact that E g values of pure oxides can be calculated Journal of The Electrochemical Society, 164 (12) C671-C681 (2017) C679 by d,d and s,p-metal oxides correlations as above mentioned and in agreement with the derivation of Eq. 7.
We like to stress that: a) theoretical data calculated by using Eq. 7 (green triangles in Figure 6a) are close to experimental values of mixed oxides in both crystallographic systems (rock-salt structure up to 50at% in Mg content for stable phases and wurtzite structure with Mg content ≤ 20at%) by using as electronegativity parameters Data shown in Figure 6a support the usefulness of the use of Eq. 7 in predicting, with reasonable accuracy, the optical bandgap values of non-regular mixed oxides provided that the bandgap value of both pure oxides can be expressed by means of Eq. 3. A rapid inspection of the bowing terms as expressed according to Eq. 6c (regular mixed oxides) and 7c (non-regular mixed oxides) indicates that for nonregular mixed oxides the quadratic term can assume positive as well as negative values and that bowing coefficients much larger than those expected for regular mixed oxides can be obtained also in the presence of a constant value of χ = (χ 1 -χ 2 ) term. This last statement can be easily checked by comparing the S q term given by Eq. 7c (S q = 1.142) with the analogous term of Eq. 6c (S q = 0.360) calculated by using the A value (3.41) derived from fitting procedure shown in Figure 5a. It is clear that the large difference in the quadratic (bowing) term stems out from the fact that, with respect to the regular mixed oxides, a new contribution to the bowing parameter is now appearing in Eq. 7c: 2(A 1 − A 2 )(χ O − χ 2 )(χ 2 − χ 1 ) which accounts for the difference in the A values of the two original correlations and for the difference of electronegativity of the two cations. This fact can help to explain why in mixing different semiconducting alloys larger values of the bowing term are usually observed in alloys having different anionic partners but the same metallic cation. Different A values and larger difference in the electronegativity values of the anionic partner (the more electronegative atom) cooperate in magnifying the bowing coefficient. 76 We will go into the details on this point in a forthcoming paper. The difference between the S q term (1.142) and the bowing coefficient, b (1.271), reported in Figure 6a, must be attributable to the interpolating quadratic law used to compare the bowing equations of the experimental data and of the theoretical data calculated according to Eq. 7.
As previously noted, an improved fitting of the experimental data (E g vs x Mg ) is clearly obtained by separating the mixed oxides for each crystallographic system in which they are stable (RS and WZ for Mg and Zn rich phase respectively). In Figure 6b we report the experimental E g values as a function of Mg composition for mixed oxides crystallizing in wurtzite structure including also the metastable Mg richer phases (x Mg = 51at%). The theoretical E g values as a function of the Mg content were obtained by using Eq. 7a with a value of E g(WZ-ZnO) = 3.35 eV, very close to that reported in literature for single crystal WZ-ZnO. The resulting equation is: We have to mention that the indirect bandgap values of E g,RS-ZnO estimated by DFT span a very wide range of values (0.75-5.5 eV) with a more recent one equal to 3.93 eV. 87 From the quadratic fitting of experimental data point (including 2.8 eV for RS-ZnO) we derive a bowing parameter b exp = 1.19 whilst a value of b th = 1.35 is obtained from the fitting of theoretical points derived by means of Eq. 9a.
The importance of the use of Eq. 7 in predicting the bandgap values of mixed oxides can be further appreciated by comparing the theoretical E g values obtained by means of Eq. 7 with the experimental data of α-(Ga 1-x Al x ) 2 O 3, already discussed (Figure 4c), in order to extract the A and B parameters to be used in Eq. 3 for getting a theoretical bandgap value for α-Al 2 O 3 polymorph closer to the experimental value (8.5eV) by using also a value of χ Al = 1.5 ± 0.05 equal to that one reported by Pauling. We have to recall that the E g value for Al 2 O 3 calculated by using the s,p-metals correlation (Eq. 3b) and a value of χ Al = 1.475 is equal to 6.2 eV quite near to γ-Al 2 O 3 .
In this case we take advantage of the fact that the bandgap value of α-Ga 2 O 3 agrees nicely with the value estimated by using the correlation valid for s,p-metal oxide. In fact, by using the χ Ga = 1.60 ± 0.05 and the usual values for A s,p (2.17) and B s,p (−2.71), we derive a value of E g (Ga 2 O 3 ) = 5.12 ± 0.40 eV in agreement with the values reported in literature for α-Ga 2 O 3 (see Table IV). By following the same approach described above for the non-regular mixed oxides, we obtain the A and B values for α−Al 2 O 3 by the best fitting procedure and by using Eqs. 7b and 7c to derive the unknown A 1 and B 1 values pertaining to α-Al 2 O 3 . In Eqs. 7a-7c E g,2 , A 2 and B 2 represent, respectively, the E g values of α-Ga 2 O 3 and the values of A (2.17) and B (−2,70) typical of the s,p-metal oxides.
The results of such a procedure are reported in Figure 4c showing that a good agreement between theoretical E g values and experimental data was obtained by using the values of A α-Al 2 O 3 = 2.50 and B α-Al 2 O 3 = −2.0 and by keeping a constant to 1.475 the electronegativity parameter, χ Al , of aluminum. The increase in the value of A, about 15% larger for α-Al 2 O 3 , would agree with the suggestion by Phillips reported above that the extra-ionic energy unit E I can change "with different atomic coordinations in different crystal structures". The parameter B is expected to change too, due to possible changes in the repulsive term contained in Eq. 2. Further investigations and more experimental data are necessary before reaching a deeper understanding of these aspects.

Future Perspectives
A critical assessment of the use of PCS in passivity studies to characterize the film composition has been presented aimed to clarify the limits of the semi-empirical correlation proposed by the present authors. It has been shown that in many cases the proposed correlation is able to provide quantitative information on the composition of mixed regular oxides for sp-metals as well as of mixed TMO with d 0 electronic configuration. It has been shown that some of the criticism pertaining to a chemical approach to the estimate of optical bandgap of different polymorphs can be accommodated within the theoretical framework underlying the proposed correlations. In particular, it has been suggested that changes in the empirically derived A and B parameters of the proposed correlations are able to fit the changes in E g values reported for different polymorphs of s,p and d-metal oxides by keeping almost constant, or within the Pauling's claimed uncertainty, the electronegativity value of the metallic partners. We have shown that, by neglecting very minor changes in the electronegativity parameter, the semi-empirical correlation proposed is able to predict the bandgap value of different MgO, Al 2 O 3 and ZnO polymorphs. Further investigations on the composition dependences of mixed oxides with different crystallographic structures are mandatory before reaching any definitive conclusion on the dependence of A and B parameters from the details of oxides crystallographic structure. These studies are particularly welcome in the case of oxides systems where the different polymorphs span a very large range of bandgap values (Al 2 O 3 , MgO, etc.).
We have also shown that the same semi-empirical approach can be used to predict the dependence of E g,opt values as a function of composition for mixed oxides as well as to derive the dependence of the bowing coefficient for both regular and non-regular mixed oxides. In the absence of change in the nature of optical transitions, the reported equation could help to explain the origin of the large difference in the bowing coefficient in presence of cationic partner having large difference in the electronegativity values as well as the large bowing coefficient values, usually observed, in presence of nonregular mixed oxide (or semiconducting alloys).
A physical approach based on the more recent models of DFT and quantum mechanical techniques, can provide a more reliable approach to the estimation of the optical bandgap of different crystalline polymorphs. However, such an approach, which is unavoidable in any attempt to put on physically sound basis any theory of electronic properties of materials or to get physical insights on any deviations from observed experimental regularities, is rather discouraging from a practical point of view in providing general indications on the role that the chemical nature of the oxides components plays in determining the optical bandgap value as well as on how it changes with changing oxide composition.
Further theoretical and experimental studies on well characterized systems, possibly including mixed oxides of TM with partially full d-orbital, are necessary before reaching final conclusions on the validity of the semi-empirical approach described above, and the PCS technique to extract reliable information on mixed oxides as a function of their composition. This is particularly true for very complex systems such as those represented by very thin anodic passive films on metallic alloys of large industrial interest (SS, Fe-Cr alloys etc.). However, we are confident that the efforts toward a generalization of the initial semi-empirical correlations will extend a more quantitative use of PCS technique beyond corrosion studies.