Chapter 8 Print ISBN: 978-81-976007-2-2, eBook ISBN: 978-81-976007-8-4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings ∗ Juan Carlos Alonso-Huitrón a and Augusto Garcı́a-Valenzuela b DOI: https://doi.org/10.9734/bpi/cmsdi/v4/1269 Peer-Review History: This chapter was reviewed by following the Advanced Open Peer Review policy. This chapter was thoroughly checked to prevent plagiarism. As per editorial policy, a minimum of two peer-reviewers reviewed the manuscript. After review and revision of the manuscript, the Book Editor approved the manuscript for final publication. Peer review comments, comments of the editor(s), etc. are available here: https://peerreviewarchive.com/review-history/1269 ABSTRACT In this work a comprehensive model for the optical transmission as a function of wavelength and thickness of ZnO :Al films deposited on glass substrates by ultrasonic spray pyrolysis, is developed. The mathematical expression developed for the transmission of the transparent conducting film on a transparent substrate, considers: 1) the interference effects of multiple specular reflections of coherent light from the front and the back of the flat-parallel-sided interfaces film-air and filmglass substrate, 2) the contribution of free carrier concentration (electrons in the conduction band due to Al doping) to the weak absorption in the visible and nearinfrared range, 3) the Urbach tail absorption edge at the low wavelength region ( < 400 nm ), 4) the effect of surface diffuse scattering of light originated by the roughness of these interfaces on the specular reflection and transmission coefficients. The wavelength dependence of the coefficients of reflection and transmission, and the absorption coefficient of the ZnO :Al film in the low absorptionvisible region (400-800 nm), were calculated from the formulas derived for the refractive index and extinction coefficient by using a LorentzDrude expression to separate the contribution of the bound-electrons and free-electrons, respectively, to the complex dielectric function. The carrier concentration and dc-electrical a Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México. Apartado Postal 70-360, Coyoacán 04510, Distrito Federal, México. b Instituto de Ciencias Aplicadas y Tecnologı́a, Universidad Nacional Autónoma de México. Apartado Postal 70-186, Coyoacán 04510, Distrito Federal, México. *Corresponding author: E-mail: alonso@unam.mx; Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings conductivity of the ZnO : Al films were measured using Hall effect and currentvoltage measurements in the van der Pauw configuration. The optical transmission of the films, in the range of wavelengths from 190 to 1100 nm, was measured using an Uv-Vis spectrometer. The fitting of the semi-empirical formula for the optical transmission with the experimental transmission spectrum for each film was good and the effects of the different parameters involved in the model was evidenced. The formulas derived here for the optical transmission can be used for a more precise determination of previously defined figures of merit for these type of films for their use as transparent conductive electrodes as a function of thickness of ZnO : Al. The correctness of the figures of merits considered and the usefulness of the model for selecting the optimal thickness for a transparent conductive contact was discussed. Keywords: Optical transmission; theoretical modeling; transparent conductive coatings; Zinc oxide; Al-doped; figure of merit. 1 INTRODUCTION Recently, aluminum-doped zinc oxide ( ZnO : Al or AZO ) thin films deposited by different techniques, have received much attention as transparent conductive coatings (TCCs) for a wide variety of optoelectronic devices such as electroluminescent flat panel displays, solar cells, ultraviolet sensors, etc. [1]-[13]. For optimal applications of these films as TCCs the optical transmission should be as high as possible but at the same time the sheet resistance should be as low as possible. A common parameter that has been used to evaluate the quality of diverse TCCs deposited on transparent substrates (such as borosilicate, corning or vitreous silica slides), is the figure of merit defined originally by Fraser and Cook [14] as: T (1) Rs where T is the average transmission in the visible wavelength range ( 400 − 800 nm ) and RS is the sheet resistance defined by Rs = σ1o l , where σo is the dc electrical conductivity in Ω−1 cm−1 and l is the thickness of the TCC in cm. According to this definition, the ideal TCC should have a figure of merit with a maximum value. However, since both parameters, T and RS , depend on the TCC thickness an important question to solve has been whether there is an optimal film thickness for which a maximum figure of merit occurs, and how it can be calculated. An attempt to solve this question was made by Haacke [15], by using the Beer’s law for expressing the optical transmission of a TCC film in its simplest form as: TB = e−αl , where α is the optical absorption coefficient in cm−1 . In this case the figure of merit was also expressed as a function of l in a simple form as: FB = σo le−αl . According to this formula it is easily found that the figure of merit of a TCC with a given σo and α achieves a maximum value at lm = 1/α, and for this thickness the sheet F = 162 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings resistance is Rs = α/σo and the transmission is TB (l = lm ) = e−1 = 0.37. As can be seen, the use of the simplest formula for the optical transmission (TB = e−αl ), in Eq. 1 for the figure of merit predicts that the maximum figure of merit occurs at a film thickness which reduces the optical transmission to only 37%, which is unacceptable for most of the applications of a TCC. For example for a TCC with a value of α = 4 × 102 cm−1 and σo = 102 Ω−1 cm−1 , the
thickness for which a maximum figure of merit FB max = (σo /α) 0.37 = 0.092Ω−1 is obtained is: lmax = 2.5 × 10−3 cm = 25000 nm, and although this TCC has a very low sheet resistance; Rs = 4Ω/ square, it is very thick and it has also a very low transmission (0.37). In order to solve this problem Haacke redefined the figure of merit by [15]: FH = T 10 Rs (2)
In this case using the same Beer’s formula for the optical transmission ( TB = e−αl , 1 the film thickness which maximizes FHB = σo le−10αl is now, lmax = 10α , and the transmittance for this thickness is TB = e−0.1 = 0.90. Thus for the same TCC with α = 4 × 102 cm−1 and σo = 102 Ω−1
cm−1 , the thickness to obtain σo (0.9)10 = 8.71 × 10−3 Ω−1 ), the maximum new figure of merit FHB max = ( 10α −4 is lmax = 2.5 × 10 cm = 2500 nm, the corresponding sheet resistance is, Rs = 40Ω/ square., and the transmittance should be of 90%. However, this prediction is not realistic because in the practice this TCC is still thick and the optical transmission for a TCC with this thickness is typically below 80%. On the other hand, the new maximum figure of merit FHB max is one order of magnitude lower than the original maximum figure of merit: FB max . Thus, although the redefined figure of merit has been used in some works to evaluate TCCs [1, 5, 8, 9, 16, 17], it seems artificial and unsatisfactory for determining the optimal thickness of a suitable TCC. In order to compare different TCCs, independently of film thickness, other definitions of the figure of merit have been made for TCCs with very small thickness and very low optical absorption, in terms only of the electrical conductivity and the absorption coefficient [18, 19, 20, 13]. However, this figure of merit is not valid for thicker or more absorbing films, and it does not allow adjusting the optimal thickness for a specific application of the TCC. It is worth to mention that in most of the works where the figure of merit has been calculated or predicted theoretically, it has been implicitly assumed that α is independent of σo . However, according to the DrudeLorentz model, the optical absorption in a TCC is related with the dc-electrical conductivity and/or the free carrier concentration. As an important motivation for the present work, the use of the simple Beer’s formula ( TB = e−αl ) in the original definition of the figure of merit has given rise to paradox results, because it does not express the real dependence of the optical transmission with film thickness. The original definition of the figure of merit given 163 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings by eq. (1) can be considered adequate and can be used in the practice to evaluate the figure of merit of thin and/or thick TCCs, whenever the optical transmission be calculated in a more rigorous form, as a function of the TCC thickness. So, here a comprehensive semi-empirical model has been developed for the optical transmittance ( T ) of TCC of ZnO :Al films deposited on glass, by ultrasonic spray pyrolysis. For the calculation of T, this model considers the multiple reflections at the three interfaces (air-coating, coating-substrate, substrate-air) and includes: the interference effects of multiple reflections at the coating interfaces, the dispersion formulas for the refractive index of the film and substrate, the effect of free electrons concentration and the roughness of the film surfaces [13]. 2 MODEL FOR THE SPECULAR OPTICAL TRANSMITTANCE Since the optical transmittance of the films was measured under normal incidence, the optical configuration of parallel plates shown in Fig. 1 was used to model the transmission coefficient through the ZnO :Al film (TF ), and the total transmission coefficient (T ) through the whole system ZnO : Al/ glass substrate ( T ). As shown in Fig. 1, the refractive index of the incident medium (air) is n0 = 1, the refractive index of the transparent glass substrate is real and it is denoted by ng = ng (λ), and the complex refractive index of the film is denoted by ñ = ñ(λ) = n(λ) + iκ(λ), where the real part, n(λ) is called the refractive index and κ(λ) is the extinction coefficient. Since the ZnO : Al film has a high transparency in the visible range (400 -800 nm), in this range the absorption is weak, and therefore |κ(λ)|  |n(λ)| Fig. 1. Optical configuration of a ZnO : Al thin film on a thick finite transparent glass substrate. Reprinted with permission from [13] @ Optica Publishing Group Requirements and applications Assuming that the thickness l of the ZnO : Al film is smaller than the coherence length of the light, the interference between multiple reflections inside the film is important [21]. Since the film is deposited on the glass substrate, we have 164 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings to include the effect of the transparent substrate (αS = 0) in the transmission. The large thickness of the substrate implies the incoherent limit, in which there is no interference among the multiple reflecting beams [21]. Thus, considering the interference effects in the addition of the electric field of the beams transmitted after multiple reflections through the film, and adding the intensities of the multiple reflected beams through the substrate, the transmission through the film/substrate optical system is (see the appendix A) [21]: !   T3 T1 T2 e−αl T = (3) 1/2 1/2 1 − R20 R30 1 − 2R1 R2 cos Φe−αl + R1 R2 e−2αl where α is the absorption coefficient of the film, Φ = 4πnl/λ is the phase shift due to a round-trip of the light wave in the film, λ is the vaccum wavelength of the light, and n is the real part of the complex refractive index of the film. R1 and R2 are the internal (inside the film) specular reflectances at the front (1) and back (2) interfaces (assumed ideally flat), respectively (see Fig. 1). R20 and R30 are, the specular reflectances inside the substrate, at the substrate-film and substrate-air interfaces, respectively. According to the Fresnel equations, these specular reflectances for weakly absorbing media are expressed as (see the appendix): R1 = r12  = 2 (4) 2  n − ng n + ng  2 ng − n = ng + n R2 = r22 = R20 = r202 n − no n + no R30 = r302 =  ng − no ng + no (5) (6) 2 (7) The coefficients of transmission at the three interfaces are expressed as: T1 = T2 = T3 = n 2 n t1 = no no  ng 2 ng t2 = n n  n0 2 n0 t2 = ng ng  2no no + n 2 2n n + ng 2 (8) 2ng no + ng (9) 2 (10) It is worth to mention that the rigorous expression (3) is similar to that used in the past for the determination of the thickness and the optical constants of transparent 165 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings and weakly absorbing films on transparent substrates [22]-[26]. However in this work the transmission curve predicted by equation 3 was fitted with the experimental transmission obtained for different ZnO :Al films, by introducing the values for the optical constants as a function of wavelength, measured and/or obtained from the models described below, and fitting the film thickness and other parameters such as film roughness and damping constant. The dependence of the refractive index of the glass substrate with the wavelength was obtained from the experimental spectrum for the interference-free transmission of the glass substrate alone in the absence of film, using the formula [25, 27]: 1 ng = + Ts  1/2 1 −1 Ts2 (11) The refractive index n and extinction coefficient κ of the ZnO : Al film was calculated in terms of the real (1 ) and imaginary (2 ) parts of the complex optical dielectric function ˆ = 1 + i2 , using the well known formulas [21]:
1/2 1/2 1  n = √ 1 + 21 + 22 2
1/2 1/2 1  κ = √ −1 + 21 + 22 2 (12) (13) According to the Drude-Lorentz model the complex optical dielectric function
can in the be expressed as a function of the angular frequency of the light ω = 2πc λ following form [6, 21, 28] : ˆ(ω) = ˆb + δˆ f (14) 2 Ne X fj ˆb = b1 + ib2 = 1 + 2 0 mo j ωoj − ω 2 − iγj ω δˆ f = δf1 + iδf2 = − ne e2 o m (ω 2 + iγω) (15) (16) Thes expressions separate explicitly the dielectric function, ˆb , including only the
b boundelectrons ˆ of the ZnO lattice with resonant frequencies ωoj and oscillator strengths fj , and the contribution ( δˆ f ) of free-electrons (electrons in the conduction band). In the region of transparency, far from the resonant frequencies, ωoj , the imaginary part, b2 , can be considered null and the real part of the optical dielectric function due 166 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings to bound electrons, b1 , can be expressed in the form of a Sellmeier equation [21]. Thus for wavelengths far from the absorption edge of ZnO : Al (λabs ≈ 360 nm ) we can express b2 = 0  2 b1 = nb = A + (17) 2 2 Bλ Dλ + 2 λ2 − C 2 λ − E2 (18) with the values of the A, B, C, D and E parameters given in Table 1 [29, 30]. Table 1. Fitting parameters of Sellmeier model for the refractive index of ZnO thin films [29, 30] A 2.0065 B 1.5748 ×106 C(nm) 1.0 × 106 D 1.5868 ×106 E(nm) 270.63 On the other hand, the explicit expressions for the real and imaginary parts of the free electron contribution are: ne e2 o m (ω 2 + γ 2 ) (19) ne e2 γ o mω (ω 2 + γ 2 ) (20) δf1 = − δf2 = where m is the electron mass, o is the vacuum permittivity, and γ = τ1 is the damping rate or damping constant, where τ is the free carrier scattering time or relaxation time. Some recent works have shown that the value of the damping constant depends mainly on the frequency or wavelength of the light, but there is also a dependence on the carrier concentration [6, 31]. For example, for a ZnO : Al film with a carrier concentration of
3.62×1020 cm−3 , the damping constant
increases from 1.78 × 1014 Hz 944 cm−1 to 2.85 × 1014 Hz 1515 cm−1 as the wavelength range of the light increases from the visible region to the infrared region [6]. For larger wavelengths, in the range of terahertz, as the carrier concentration increase from 5.9 × 1017 cm−3 to 4.0 × 1019 cm−3 the damping constant increase from 9.2 × 1013 Hz to 7.04 × 1014 Hz [31]. Thus, substituting the equations (17)-(20) in (14), the real and imaginary parts of the complex optical dielectric function of the ZnO : Al films can be expressed as: 167 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings 1 (λ) = b1 + δf1 = A + Bλ2 Dλ2 ne e2 + − λ2 − C 2 λ2 − E 2 o m (ω 2 + γ 2 ) 2 (λ) = b2 + δf2 = ne e2 γ o mω (ω 2 + γ 2 ) (21) (22) For the evaluation of these expressions as a function of the wavelength or angular frequency of the light, the concentration of electrons in the ZnO : Al films was measured by Hall effect, as mentioned in the experimental part. The expressions (21) and (22) were used in the formulas (12) and (13) to obtain the wavelength dependence of the refractive index n(λ) and extinction coefficient κ(λ) of the ZnO : Al films. The refractive index was substituted in equations (4) and (5) to calculate the wavelength dependence of the reflection coefficients, R1 and R2 , for internal specular reflection at the film-air and film-susbstrate interfaces, respectively, which appear in formula (3) for the transmission through the film/substrate optical system. For the absorption coefficient of the ZnO : Al film, that appears in the same formula (3) two contributions were considered. The first contribution was that proportional to the extinction coefficient due to free electrons 2κ(λ)ω 4πκ(λ) = c λ where κ(λ) is the extinction coefficient obtained from the formula (13). αf (λ) = (23) The second contribution was to include the absorption edge of the ZnO : Al film, and it was through an Urbach rule of the absorption edge coefficient, expressed in terms of the energy of the photons (~ω) and the energy band gap (Eg ) of the ZnO : Al film as: αU = αo eb(~ω−Eg )/kB Ta = αo e(~ω−Eg )/EU , where αo and b are fitting parameters, kB is the Boltzmann constant and Ta is the absolute temperature in kelvin degrees [21, 32]. It must be pointed out that, physically, EU , is the Urbach energy which is equal to the energy width of the absorption edge and αo is the convergence value of the absorption coefficient when ~ω = Eg [32]. The exponential increase of the absorption coefficient below the absorption edge is explained by transitions between the tails of density of states in the valence band and the conduction band and the shape and size of these tails depend on the presence of different type of disordering. In terms of the photon wavelength and the absorption edge wavelength which is related to the energy band gap by: Eg (eV) = 1240/λg ( nm), the Urbach absorption coefficient is commonly expressed as [30, 33, 34] :  1240·β αU (λ) = αo e where αo and β = 1/EU are fitting parameters. 168 1− 1 λ λg  (24) Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings The band gap of the ZnO : Al films was measured from the absorption edge in the experimental transmission curve (Texp ) using the formula for direct interband absorption in a direct band gap semiconductor: 1 αexp = − ln (Texp ) ∝ (~ω − Eg )1/2 (25) d where d is the thickness of the ZnO : Al film and ~ω is the photon energy. Thus, the absorption coefficient α(λ) used in equation (3) was finally expressed by the formula:   1− 1 1240·β λ 4πκ(λ) λg (26) + αo e λ At this point it is important to mention that for the deduction of equation (3) it was implicitly assumed that the surfaces of the film and substrate are perfectly smooth. However, according to the AFM and SEM images obtained for various ZnO : Al films (see Figs. 3 and 4), these films have a rough surface, with an average roughness in the range from 15 to 20 nm. As it was shown in an elderly work the specular reflectance of a rough surface is reduced with respect to that of a perfectly smooth surface of the same material [35, 13]. Since then several models have been developed to obtain expressions relating the roughness, σs , of a plane surface to the specular reflectance and transmittance at normal incidence, for different magnitudes of the roughness compared with the wavelength, λ [36, 37]. Based on these models, in order to include the effect of the surface roughness of the ZnO :Al films in the total transmission, the specular reflectances at normal incidence of the perfectly smooth surfaces at the front and the back of the films, R1 and R2 , given by formulas (4) and (5), were multiplied by a surface scattering factor, to obtain the specular reflectance at the film rough surfaces, expressed as [36]: α(λ) = αf + αU = 2 R1rs = R1 e−(2(2πnσs ) 2 R2rs = R2 e−(2(2πnσs ) /λ2 ) 2 /λ ) (27) (28) In these equations it is assumed that σs is the surface roughness at macroscopic level measured by the rms value of the irregularity heights and also that σs  λ. In similar way the specular transmittance at the film rough surfaces was expressed as [36, 37]:   1 2πσ − 2 ( s1 (nef f −1))2 /λ2 T1rs = T1 e T2rs = T2 e   2 − 1 2πσs2 (nef f −1)) /λ2 2( 169 (29) (30) Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings In these equation the rough surface of the film was modeled as a thin homogeneous layer with a thickness that is twice the rms roughness of the surface and with an effective refractive index intermediate to the indices of the two adjacent optical media, and in this case (filmair), given by [37]:  nef f = n2 + 1 2 1/2 (31) Thus, substituting the expressions (27)-(31) in equation (3), the final expression for the transmittance, including the effect of the roughness, is: !   T3 T1rs T2rs e−αl T = (32) 1/2 1/2 1 − R20 R30 1 − 2R1rs R2rs cos Φe−αl + R1rs R2rs e−2αl 3 EXPERIMENTAL The ZnO : Al films modeled in this work were deposited on glass substrates by ultrasonic spray pyrolysis at atmospheric pressure, using the same home-made system, precursor solution and preparation conditions given elsewhere [7, 13]. In this case, the substrate temperature was fixed at 350◦ C. A series of ZnO : Al films with different thickness were obtained using different deposition times varying in the range from 5 to 15 min. A double beam PerkinElmer 35Uv-Vis spectrometer was used to measure the optical transmission of the films, in the range of wavelengths from 190 to 1100 nm, with a resolution of 1 nm. Xrays diffraction (XRD) measurements were made for determining the crystalline structure of the films, using a Bragg-Brentano Rigaku ULTIMA IV diffractometer with an X-ray source of CuKa line ( 0.15406 nm ), at a grazing beam configuration (incidence angle of 1◦ ). The surface morphology of the films was explored by atomic force microscopy (AFM) and scanning electron microscopy (SEM) using a JEOL JSPM-4210 scanning probe microscope and a JEOL 7600F field emission scanning electron microscope (FESEM), respectively [13]. The carrier concentration and electrical conductivity of the films were measured at room temperature by Halleffect in the van der Pauw configuration, using a Ecopia HMS3000 system, applying a magnetic field of 0.540 T and a current of 1.0 mA. 4 RESULTS AND DISCUSSION Fig. 2 shows the transmission spectra of six ZnO : Al films with different thickness, which were ordered with decreasing thickness (decreasing deposition time) and named AZO1, AZO2, AZO3, AZO4, AZO5 and AZO6, respectively. As can be seen from this figure all the transmission spectra show maxima and minima whose number decreases as the deposition time and thickness of the films decreases [13]. 170 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings Fig. 2. Experimental optical transmission spectra for ZnO : Al thin films with different thickness deposited on glass substrates. The thickness of the films was decreased from sample AZO1 to sample AZO6, by decreasing the deposition time. Reprinted with permission from [13] @ Optica Publishing Group [13] The thickness d of the films was preliminary estimated using the well-known formula obtained from the conditions for the constructive and destructive interference of 1 λ2 the multiple reflections of light in the film [22]-[25]: d = 2(λ1 n(λλ1 )−λ , where 2 n(λ2 )) λ1 and λ2 are the wavelengths corresponding to two consecutive maxima (λM ) or minima (λm ) of the optical transmission [13]. The thickness of the films can be also calculated from consecutive maxima (λM ) and minima (λm ), using the factor 4 instead of the factor 2 in the denominator of the previous formula. For the measurements of the thickness of AZO q films the Sellmeier expression were used 2 2 for the refractive index, n(λ) = nb = A + λ2Bλ + λ2Dλ , given in equation −C 2 −E 2 (18), with the values of the parameters listed in Table 1. As it has been shown in some works [25], this formula is not very accurate because in practice it is very sensitive tovariations or non-uniformities in the refractive index and thickness of the films, which give rise to some dispersion in the vales of d. Due this the following 171 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings procedure to estimate the average thickness ( dav ) for each sample was used. First, the average thickness diM m was obtained from the list of wavelengths for consecutive maxima and minima in all the spectral range from 190 to 1100 nm. Then the average thicknesses diM and dim from the list of wavelengths for consecutive maxima or consecutive minima, respectively was obtained. The average thickness, dav , for each sample was the average of, diM m , diM and dim , and all these thicknesses are shown in Table 2. Table 2. List of estimated thicknesses by the interference formula, along with the carrier concentration, ne , conductivity, σ, and sheet resistance, Rsheet, and band gap, Eg, experimentally measured for each sample Sample diMm (nm) dim (nm) dim (nm) dav (nm) ne
cm−3 σ (Ωcm)−1 Rsheet Ω AZO1 AZO2 AZO3 AZO4 AZO5 AZO6 1059 972 768 608 431 258 1056 947 758 600 437 256 1027 897 726 567 422 - 1047 939 751 592 430 257 2.7 × 1020 3.02 × 1020 2.54 × 1020 2.6 × 1020 2.22 × 1020 2.14 × 1020 2.08 × 102 1.98 × 102 1.46 × 102 1.24 × 102 7.9 × 101 1.44 × 102 45.4 51.9 91.2 136.2 293 269 Egg (eV) (band gap) 3.41 3.42 3.43 3.40 3.41 3.45 Table 2 also shows the carrier concentration, ne , conductivity, σ, and sheet resistance, Rsheet , of the films, measured by Hall effect using the van der Pauw configuration. It must be pointed out that the average thickness, dav , was used as the input thickness required for the Hall measurements. The values of the energy band gaps, Eg , of the films listed in Table 2 were calculated by taking the plot of (αd )2 vs ~ω obtained from the equation (25) using the experimental transmittance, Texp , and the average film thickness d = dav . The AFM and SEM analysis showed that all the samples have similar morphology. For example, Fig. 3 shows the AFM micrographs and profiles of samples AZO1 and AZO6. The AFM profiles of these samples show similar surface heights differences of around 20 nm. Fig. 4 show a cross section FESEM micrograph with a certain rotation and inclination of the AZO1 sample, where it is clearly seen that there is roughness at the top surface and at the bottom surface of the film. Using the values of carrier concentration, band gap, and average thickness of each film, given in Table 2, the experimental transmission curves along with the modeled transmission using equation (3) were plotted and started to improve the fitting by changing the values of the thickness l appearing in equation (32) around the corresponding average thickness, dav , and adjusting the values of the Urbach parameters, αo and β. Although the fitting was tried using different constant values of the damping constant, γ, the best fitting was achieved expressing the damping constant as a linear function of the wavelength, as: 172 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings Fig. 3. AFM images and profiles of sample AZO1 (upper) and sample AZO6 (lower). Reprinted with permission from [13] @ Optica Publishing Group [20] Fig. 4. Cross section FESEM micrograph with a certain rotation and inclination of sample AZO1 173 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings γ = ξλ (33) where ξ was a fitting parameter. Fig. 5 shows the experimental and the best modeled optical transmission for all the samples, using the formula (32), which includes the effect of the roughness at the two interfaces of the films, σs1 and σs2 . As can be seen, the modeled transmission curve fits quite well with the experimental transmission curve, above all in the range of visible wavelengths (400 − 800 nm ). Fig. 5. Experimental (AZO) and the best modeled (TAZO) optical transmission using Eq. (32) for samples AZO1 to AZO6 [20] Table 3 shows the values of all the parameters which gave rise to the best fitting. Comparing the thicknesses for the fitting, listed in Table 3, with the average thicknesses, dav , listed in Table 2, the fitting thickness for samples AZO1-AZO5 was found to be around 2 − 4% lower than the corresponding average thickness, and for the thinnest sample AZO6, l is 9% lower than dav . 174 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings Table 3. Values of thickness and parameters used in equation (3) for the theoretical optical transmission to get the best fitting with the experimental transmission curves. The average thickness estimated by the interference formula, the electron concentration and conductivity experimentally measured for each sample is also listed Sample Average Thickness (nm) Thickness l ( nm) ne (×
1020 cm−3
σ 102 −1 (Ωcm) ξ(×
1011 Hznm−1 αo (×
10−3 nm−1
β eV −1 λg (nm) σs1 ( nm) nm) AZO1 AZO2 AZO3 AZO4 AZO5 AZO6 1047 939 751 592 430 257 1024 900 735 575 424 233 2.7 3.02 3 2.6 2.22 2.88 2.08 1.98 2.54 1.24 1.53 2.84 2.8 2.8 2.8 3.5 4.8 4.8 2.5 2.5 3 2.5 3 3.2 10 363 20 18 10 363 20 18 10 363 20 18 10 364 21 20 9.2 364 16 16 10 364 16 16 Recently, this comprehensive model for the optical transmission and the theoretical expression for the transmittance given by the Equation (3), has also been used to calculate the thickness of NiO :Li films, used as p-type transparent conductive contacts in inorganic optoelectronic devices [38]. 4.1 Effect of the Different Parameters on the Fitting of the Modeled Transmittance Curves In this section the effect that the different parameters have on the fitting of the modeled transmission curves compared with the experimental ones was shown. Fig. 6 a) shows the experimental and the best modeled optical transmission for sample AZO3, meanwhile Fig. 6 b) shows the experimental and the modeled transmission using the corresponding average thickness measured for this sample given in Table1 (dAv = 751 nm), instead of the thickness of l = 735 nm, which gives the best fitting. As can be seen from Fig. 6 b), a change of 2% in the thickness shifts the wavelength positions of the maxima and minima for the modeled curve with respect to the experimental one. This misfit is expected since the thickness l enters in the formula for the modeled transmittance, which includes the contribution 175 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings of both, the free electrons and the bound electrons in the dependence of the refractive index of the films with respect to the wavelength, meanwhile the thickness dAv is calculated considering only the contribution of the bound electrons (only the Sellmeier formula). As Fig. 6 c) shows, the effect of neglecting (ne = 0) the contribution of the free carriers to the refractive index of the films is to increase the modeled transmission with respect to the experimental transmission, mainly in the infrared region, and also to shift the wavelength position of the maxima and minima. This is also expected since ne = 0 implies that there is no optical absorption in the visible and infrared region due to free carriers and only remains the absorption edge in the ultraviolet region due to interband absorption. ne = 0, also implies a change in the refractive index of the film and the consequent shift in the maxima and minima of interference. Fig. 6. Experimental optical transmission for sample AZO3 and for the same sample: a) the best modeled transmission, b) the modeled transmission using l = dAv = 751 nm, and c) the modeled transmission neglecting (ne = 0) the contribution of the free carriers [20] Figs. 7 a) and 7 b) show the effect of using a fixed value for the damping constant, γ, of 5 × 1013 Hz and 3 × 1014 Hz, respectively, instead of using the linear dependence 176 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings given by equation (33) in the modeled transmission for sample AZO3, which gives rise to the value of 5.34 × 1013 Hz for λ = 190 nm and 3.08 × 1014 Hz for λ = 1100 nm. As can be seen from Figs. 7 a) and 7 b) a low fixed value of the damping constant gives rise to an increase in the modeled transmission with respect to the experimental transmission, mainly in the infrared region, meanwhile a high fixed value decreases the modeled transmission in the infrared region and even in the visible region. Fig. 7 c) shows the effect of removing the roughness from the modeled curve. As can been from this figure, neglecting the roughness, the visibility of the modeled curve (the difference between transmission percentage between maxima and minima) increases with respect to that of the experimental curve. So, as expected, the effect of the roughness is to decrease the visibility of the interference pattern in the transmission curve, mainly in the visible region. It is worth to mention that the rms roughness values listed in Table 3 , which gave the best fitting of the visibility of the films, are in good agreement with the roughness observed in the AFM profiles of the samples (see Figs 3.) Fig. 7. Experimental optical transmission for sample AZO3 and the modeled optical transmission for the same sample using: a) γ = 5 × 1013 Hz, b) γ = 3 × 1014 Hz, and c) σs = 0, i.e. neglecting the effect of the surface roughness [20] 177 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings Fig. 8 a) shows the effect of changing the value of parameter β in the Urbach rule formula to β = 5eV−1 , instead of the value of this parameter for the best fitting of the experimental transmission curve of sample AZO3, which, as shown in Table 3 is: β = 10eV−1 . As can be seen from this figure, the decrease in the value of this parameter widens the absorption edge in such a way that the AZO film starts to absorb light at a larger wavelength, or equivalently at a lower energy. This is well expected since the Urbach energy is: EU = β1 , so, a decrease in the parameter β means an increase in the energy width of the absorption edge, which in turn means a widening of the tails of the density of states above the valence band and below the conduction band [32]. Consequently, the optical absorption due to electron transitions between these tails occurs at lower energies or larger wavelengths. As Fig. 8 b) shows a decrease in the value of the parameter αo , from αo = 3 × 10−3 mm−1 (best fit) to αo = 1 × 10−3 nm−1 , in the Urbach rule formula has the effect of increasing the optical transmission, just in the region of the absorption edge. This is directly explained by the fact that αo is the value of the absorption coefficient when ~ω = Eg [32], and a decrease in the value of this coefficient implies an increase in the optical transmittance a wavelengths close to λg (nm) = 1240/Eg (eV). On the other hand, from this last relation between λg and the band gap Eg , it is evident why an increase in the absorption wavelength parameter, from λg = 363 nm (best fit) to λg = 370 nm, shifts the transmittance curve to larger wavelengths, as observed in Fig. 8 c). 4.2 Figure of Merit and Criteria to Choose The Best TCC In this section the rigorous expression (32) was used for the transmission of a ZnO : Al thin film on a glass substrate, with the parameters for the best fit for sample AZO3, to calculate and/or predict the average optical transmission, in the visible range from 400 to 800 nm, as a function of thickness, assuming thicknesses in the range from 50 nm to 10000 nm. Using the average value of the absorption coefficient in the same visible range, ᾱ = 3.45×10−5 nm−1 , the Beer’s transmittance as a function of thickness ( TB = e−ᾱl ) was calculated. Along with this the sheet resistance as a function of thickness, Rs = 1/σo l, was calculated assuming that the dc-electrical conductivity, σo , of the film remains constant in the value measured for sample AZO3 listed in Table 1, σo = 146Ω−1 cm−1 . Fig. 9 shows the plot of the modeled optical transmission, T , the Beer’s transmission, TB , and the sheet resistance as a function of thickness [13]. As can be seen, the sheet resistance decreases hyperbolically with thickness, meanwhile both transmissions, T and TB decrease almost linearly with increasing the thickness in the range of 50 to 10000 nm. This cuasi-linear decrease is because in this range of thicknesses the factor ᾱl is small. However, the modeled transmission is around 20% lower than the unrealistic Beer’s transmission TB . and the latter decreases with o lower slope than the former [13]. 178 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings Fig. 8. Experimental optical transmission for sample AZO3 and the modeled optical transmission for the same sample using: a) β = 5eV−1 , b) αo = 1 × 10−3 nm−1 , and c) λg = 370 nm [20] Fig. 9. Plots of the modeled optical transmission, T , the Beer’s transmission, TB , and the sheet resistance, RS , as a function of thickness 179 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings Fig. 10. Plots of original figures of merit  using the modeled trasnmisssion ( F = RTs ) and the Beer’s transmission FB = TRBs ), and the redefined Figures T 10 10 of merit by Haacke, FH = TRs and FHB = RBs . Reprinted with permission from [20] @ Optica Publishing Group [20] Based on the definitions given by equations (1) and (2) for the original figure of merit and the redefined Figure of merit by Haacke, respectively, these figures of merit as a function of thickness were calculated, using both, the modeled trasnmisssion and the Beer’s transmission. Fig. 10 shows the plots of these figures of merit, and as can be seen the original figures of merit, F = RTs and FB = TRBs , are far from reaching the maximum value, for the region of thickness considered in the plot. This was expected since based on the discussion given in the introduction, this 1 maximum value should be reached for a thickness equal to l = ᾱ1 = 3.45×10−5 nm−1 ≈ 28985 nm. This shows clearly that the criteria of choosing a thin film of ZnO :Al with this thickness, as the best TCC is unrealistic, even using the modeled transmission. On the other hand, as the same Fig. 10 shows, the redefined figures 10 T 10 of merit, FH = TRS and FHB = RBS , do reach the maximum value of ∼ 3×10−3 Ω−1 and 2 × 10−2 Ω−1 , respectively, for a thickness l ≈ 2900 nm [20]. Although this thickness is much lower, it is still questionable to choose this as the thickness for a ZnO : Al thin film to be the best TCC. As Fig. 9 shows, in spite of the fact that this film would have a very low sheet resistance of ∼ 17Ω, its transmission would be T ≈ 0.75, which do not meet the general criteria of T ≥ 0.80 to be considered a good TCC [39]. As Fig. 9 shows, in order to meet this criteria the thickness of the film should be: l ≈ 1000 nm, and this film would have a sheet resistance 180 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings of ∼ 50Ω, and a figure of merit of 1.6 × 10−2 Ω−1 . From this analysis it can be concluded that the criteria of choosing the thickness of the best TCC film as that to get the maximum of the original or the redefined figure of merit, is not a realistic criterion. The alternative method proposed in this work to determine the thickness of a ZnO : Al film, for itsapplication as a good TCC, is to select this thickness from the plots shown in Fig 10 of the sheet resistance and the average transmission calculated from our modeled transmission, in the region of thicknesses where this average transmission is equal or higher than 80%. From these plots it is clearly seen that in order to obtain TCCs with sheet resistances between 100 and 50Ω, the thickness of the ZnO : Al films must be in the range from 500 to 1000 nm, and the figure of merit of this TCCs have values in the range from 8.1 × 10−3 to 1.16 × 10−2 Ω−1 . Fig. 11. Plots of sheet resistance and average transmission calculated from the modeled transmission, in the region of thicknesses where this average transmission is equal or higher than 80%. Reprinted with permission from [13] @ Optica Publishing Group [13] 5 CONCLUSIONS A comprehensive expression for the optical transmission of thin films of ZnO : Al deposited by ultrasonic spray pyrolysis on glass substrates have been developed, with the purpose of calculating their figure of merit and their optimal thickness as TCCs. The modeled transmission considers the effect of the free carrier concentration, measured by Hall effect, in the optical absorption of the films, as well as the Urbach absorption edge, the interference effects of multiple specular 181 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings reflections and roughness of the film’s surfaces. Our results show that by fitting the model presented here to the experimental transmission curves for films, very precise values of the thickness and rms roughness of the films can be obtained, as well as the values of some important optical and electronic parameters such as the band gap and width of the energy band tails associated to the conduction and valence bands. DISCLAIMER (ARTIFICIAL INTELLIGENCE) Author(s) hereby declare that NO generative AI technologies such as Large Language Models (ChatGPT, COPILOT, etc) and text-to-image generators have been used during writing or editing of manuscripts. ACKNOWLEDGMENTS We want to thank to the undergraduate students, Benito Juárez Garcı́a, Jesús Gonzalez Gutierrez and Martha Judith Rivera Medina, for the preparation and characterization of the films analyzed in this work. Thanks are extended to Dr. Carlos David Ramos Vilchis by the technical assistance with the maintenance of some thin film deposition equipment. Recent research activities related to this work have received partial financial support from project PAIIT-UNAM number IN111022. COMPETING INTERESTS Authors have declared that no competing interests exist. REFERENCES [1] Acosta M, Mendez-Gamboa J, Riech I, Acosta C, Zambrano M. AZO/Ag/AZO multilayers electrodes evaluated using a photonic flux density figure of merit for solar cells applications,” Superlattices Microstruct. 2019;127:49-53. DOI: 10.1016/j.spmi.2018.03.018. [2] El Hamali SO, Cranton WM, Kalfagiannis N, Hou X, Ranson R, Koutsogeorgis DC. Enhanced electrical and optical properties of room temperature deposited Aluminium doped Zinc Oxide (AZO) thin films by excimer laser annealing. Opt. Lasers Eng. 2016;80:45-51. DOI:10.1016/j.optlaseng.2015.12.010. [3] Singh S. Al doped ZnO based metal-semiconductor-metal and metalinsulatorsemiconductor-insulator-metal UV sensors,” Optik (Stuttg). 2016;127(7):3523-3526. DOI: 10.1016/j.ijleo.2016.01.012. 182 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings [4] Zhai CH, et al. Effects of Al Doping on the properties of ZnO Thin Films Grown by Atomic Layer Deposition,” Nanoscale Res. Lett. 2016;11:407-414 DOI: 10.1021/jp2023567. [5] Gençyılmaz O, Atay F, Akyü I. Deposition and ellipsometric characterization of transparent conductive Al-doped ZnO for solar cell application. J. Clean Energy Technol. 2016;4(2):90-94. DOI: 10.7763/JOCET.2016.V4.259. [6] Romanyuk V, et al. Optical and electrical properties of highly doped ZnO : Al films deposited by atomic layer deposition on Si substrates in visible and near infrared region. Acta Phys. Pol. A. 2016;129(1):A36-A40. DOI: 10.12693/APhysPolA.129.A36. [7] Rivera MJ, et al. Low temperature-pyrosol-deposition of aluminum-doped zinc oxide thin films for transparent conducting contacts,” Thin Solid Films, vol. 2016;605:108-115. DOI: 10.1016/j.tsf.2015.11.053. [8] Grilli ML, Sytchkova A, Boycheva S, Piegari A. Transparent and conductive Aldoped ZnO films for solar cells applications,” Phys. Status Solidi Appl. Mater. Sci. 2013;2109(4):748-754. DOI: 10.1002/pssa.201200547. [9] Sytchkova A, et al. Radio frequency sputtered Al:ZnO-Ag transparent conductor: A plasmonic nanostructure with enhanced optical and electrical properties. J. Appl. Phys. 2013;114(9). DOI: 10.1063/1.4820266 [10] Ellmer K. Past achievements and future challenges in the development of optically transparent electrodes,” Nat. Photonics. 2012;6(12):809-817. DOI: 10.1038/nphoton.2012.282. [11] Cisneros-Contreras IR, López-Ganem G, Sánchez-Dena O, Wong YH, Pérez-Martı́nez AL, Rodrı́guez-Gómez A. Al-Doped ZnO Thin Films with 80% average transmittance and 32 ohms per square sheet resistance: A genuine alternative to commercial high-performance Indium Tin Oxide. Phys. 2023;5(1):45-58.// DOI: 10.3390/physics5010004. [12] Sarma BK, Rajkumar P. Al-doped ZnO transparent conducting oxide with appealing electro-optical properties - Realization of indium free transparent conductors from sputtering targets with varying dopant concentrations,” Mater. Today Commun. September 2019;23:100870. DOI: 10.1016/ j.mtcomm.2019.100870. [13] Juárez-Garcı́a B, González-Gutiérrez J, Rivera-Medina MJ, Garcı́aValenzuela A, Alonso-Huitrón JC. Requirements and applications of accurate modeling of the optical transmission of transparent conducting coatings. Appl. Opt. 2019;58(19):5179. DOI: 10.1364/ao.58.005179. [14] Fraser DB, Cook HD. Highly conductive, transparent films of sputtered in[sub 2-x]Sn[sub x]O[sub 3-y]. J. Electrochem. Soc. 1972;119(10):13681374. DOI: 10.1149/1.2403999. 183 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings [15] Haacke G. New figure of merit for transparent conductors. J. Appl. Phys. 1976;47(9):4086-4089. DOI: 10.1063/1.323240. [16] Sivaramakrishnan K, Theodore ND, Moulder JF, Alford TL. The role of copper in ZnO/Cu/ZnO thin films for flexible electronics. J. Appl. Phys. 2009;106(6):0635101-0635108. DOI: 10.1063/1.3213385. [17] Wu HW, Yang RY, Hsiung CM, Chu CH. Influence of Ag thickness of aluminum-doped ZnO/Ag/ aluminum-doped ZnO thin films. Thin Solid Films. 2012;520(24):7147-7152. DOI: 10.1016/j.tsf.2012.07.124. [18] Gordon RG. Criteria for choosing transparent conductors. MRS Bull. 2000;25(8):52-57. DOI: 10.1557/mrs2000.151. [19] Jain VK, Kulshreshtha AP. Indium-tin-oxide transparent conducting coatings on silicon solar cells and their figure of merit. Sol. Energy Mater. 1981;4:151158. [20] Cisneros-Contreras IR, Muñoz-Rosas AL, Rodrı́guez-Gómez A. Resolution improvement in Haacke’s figure of merit for transparent conductive films. Results Phys. August 2019;15. DOI: 10.1016/j.rinp.2019.102695. [21] Fox M. Optical properties of solids, second Ed. Oxford University Press; 2010. [22] Manifacier JC, Gasiot J, Fillard JP. A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E. 1976;9(11):1002-1004. DOI: 10.1088/0022-3735/9/11/032. [23] Goodman AM. Optical interference method for the approximate determination of refractive index and thickness of a transparent layer.,” Appl. Opt. 1978;17(17):2779-2787. DOI: 10.1364/AO.17.002779. [24] Cody GD, Wronski CR, Abeles B, Company E. Optical characterization of amorphous silicon hydride films. Sol. Cells, 1980;2(3):227-243. [25] Swanepoel R. Determination of the thickness and optical constants of amorphous Ge-Se-Bi thin films. J. Phys. E Sci. Instrum. 1983;16:1-10. DOI: 10.1080/14786430902835644. [26] Márquez E, et al. Optical characterization of H-free a-Si layers grown by rfmagnetron sputtering by inverse synthesis using matlab: Tauc-lorentzurbach parameterization. Coatings. 2021;119(11):1-26. DOI: 10.3390/coatings11111324. [27] Wang MD, et al. Determination of thickness and optical constants of ZnO thin films prepared by filtered cathode vacuum arc deposition. Chinese Phys. Lett. vol. 2008;25(2):743-746. DOI: 10.1088/0256-307X/25/2/106. 184 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings [28] Rakić AD, Djurišić AB, Elazar JM, Majewski ML. Optical properties of metallic films for vertical-cavity optoelectronic devices. Appl. Opt. vol. 1998;37(22):5271-5283. DOI: 10.1364/AO.37.005271. [29] Sun XW, Kwok HS. Optical properties of epitaxially grown zinc oxide films on sapphire by pulsed laser deposition. J. Appl. Phys. 1999;86(1):408-411. doi: 10.1063/1.370744. [30] Khelladi NB, Sari NEC. Optical properties of ZnO thin film. Adv. Mater. Sci. 2013;13(1):21-29. DOI: 10.2478/adms-2013-0003. [31] Tang J, et al. Determination of carrier concentration dependent electron effective mass and scattering time of n- ZnO thin film by terahertz time domain spectroscopy. J. Appl. Phys. 2014;115(3). DOI: 10.1063/1.4861421. [32] Studenyak I, Kranj M, Kurik M. Urbach rule in solid state physics. Int. J. Opt. Appl. 2014;49(3):76-83. DOI: 10.5923/j.optics.20140403.02. [33] Khoshman JM, Kordesch ME. Optical characterization of sputtered amorphous aluminum nitride thin films by spectroscopic ellipsometry. J. Non. Cryst. Solids. 2005;351:40-42, pp. 3334-3340. DOI: 10.1016/j.jnoncrysol.2005.08.009. [34] Capan R, Chaure NB, Hassan aK, Ray aK. Optical dispersion in spun nanocrystalline titania thin films. Semicond. Sci. Technol. 2004;19:198-202. DOI: 10.1088/0268-1242/19/2/012. [35] Bennet HE, Porteus PO. Relation between surface roughness and specular reflectance at normal incidence. J. Opt. Soc. Am. 1961;51(2):123-128. [36] Nagy PB, Adler L. Surface roughness induced attenuation of reflected and transmitted ultrasonic waves. The Journal of the Acoustical Society of America. 1987;82(1):193-197. DOI: 10.1121/1.395545. [37] Carniglia CK, Jensen DG. Single-layer model for surface roughness. Appl. Opt. 2002;41(16):3167-3171. DOI: 10.1364/AO.41.003167. [38] López-Lugo VH, Garcı́a-Hipólito M, Rodrı́guez-Gómez A, AlonsoHuitrón JC. Fabrication of Li-Doped NiO thin films by ultrasonic spray pyrolysis and its application in light-emitting diodes. Nanomaterials. 2023;13(1). DOI: 10.3390/nano13010197. [39] Huang M, Hameiri Z, Gong H, Wong WC, Aberle AG, Mueller T. Novel hybrid electrode using transparent conductive oxide and silver nanoparticle mesh for silicon solar cell applications. Energy Procedia. 2014;55:670-678. DOI: 10.1016/j.egypro.2014.08.043. 185 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings APPENDIX A The expression (3) for the transmission T through the system air/film/glass substrate /air shown in Fig 1 is derived in this appendix. It is assumed that the refractive index of air is real (n0 = 1), the refractive index of the film is complex (ñ = n(λ) + iκ(λ)), and the refractive index of the glass substrate is real ng = ng (λ). The spatial component of the electric field of one plane wave propagating in the x direction (at normal incidence of the light beam with the film surface) inside the film of thickness l can be expressed as: Ef = Etf eik̃x (A1) where 0 ≤ x ≤ l, Etf is the electric field of the transmitted wave at the interface 1 air-film (x = 0) and ω 2π = ñ c λ where λ is the wavelength of the light in air vacuum. k̃ = ñ (A2) According to Fresnel’s relations, at normal incidence, the amplitude transmission coefficients at the air-film (1) and film-substrate (2) are, respectively:  t̃1 =  t̃2 = E0t1 E01  E0t2 E02  = 2no no + ñ (A3) = 2ñ ñ + ng (A4) and the corresponding amplitude reflection coefficients are:   E0r1 no − ñ r̃1 = = E01 no + ñ   E0r2 ñ − ng r̃2 = = E02 ñ + ng Let us assume that the coherence length of light exceeds the thickness, l, of the film and thus, the interference effects of the multiply reflected beams are important. So considering multiple reflections, the electric field of the first, second, and the m-th beam transmitted through the film (from the air toward the glass substrate) are, respectively: 2π Etf 1 = E0 t̃1 t̃2 eiñ λ l = E0 t̃1 t̃2 eiQ̃ h i Etf 2 = E0 t̃1 t̃2 eiQ̃ r̃1 r̃2 ei2Q̃ 186 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings h i Etm = E0 t̃1 t̃2 eiQ̃ r̃1 m−1 r̃2 m−1 ei2(m−1)Q̃ where Q̃ = ñ 2π l λ Adding all the terms from m = 1 to m → ∞ and redefining the index k = m − 1 ∞  iX k h ET = E0 t̃1 t̃2 eiQ̃ r̃1 r̃2 ei2Q̃ k=0 Then using q̃ = r̃1 r̃2 ei2kQ̃ < 1 ∞  ∞ k X X r̃1 r̃2 ei2Q̃ = q̃ k = k=0 k=0 t̃ = 1 1 = 1 − q̃ 1 − r̃1 r̃2 ei2Q̃ ET t̃1 t̃2 eiQ̃ = E0 1 − r̃1 r̃2 ei2Q̃ Separating the real part and the imaginary parts of Q̃, Q̃ = Φ + iα/2 2 where α= 4πκ λ and, Φ= 4πn l. λ Therefore, t̃ = t̃1 t̃2 e−αl/2 e−iΦ/2 1 − r̃1 r̃2 eiΦ e−αl and thus, 187 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings |t̃|2 = t̃1 2 t̃2 |1 − r̃1 r̃2 2 e−αl eiΦ e−αl |2 In the region of low absorption |κ(λ)|  |n(λ)| n0 − n r̃1 ∼ = r1 = n0 + n2 n − ng r̃2 ∼ = r2 = n + ng 2no t̃1 ∼ = t1 = no + n and t̃2 ∼ = t2 = 2n n + ng Therefore |t̃|2 = |t1 |2 |t2 |2 e−αl 1 − 2r1 r2 cos Φe−αl + r12 r22 e−2αl On the other hand, the intensity or flux of energy of a plane electromagnetic wave in any non-magnetic material is equal to the magnitude of the Poynting vector: ~ 2 ~m = |E|| ~ H| ~ = nm |E| Im = S µo c where nm is the real part of the refractive index of the non-magnetic material. Now, the transmittance through the film Tf is the ratio of the transmitted intensity to the incident intensity, that is, Tf = ng ET2 ng 2 = |t̃| n0 E02 n0 Finally, substituting the expression for |t̃|2 and using R1 = r12 and R2 = r22 yields Tf = e−αl ng |t1 |2 |t2 |2 1/2 1/2 n0 1 − 2R1 R2 cos Φe−αl + R1 R2 e−2αl 188 Chemical and Materials Sciences - Developments and Innovations Vol. 4 A Comprehensive Model for the Optical Transmission for Determining the Optimal Thickness and Figure of Merit of Al-Doped ZnO Films as Transparent Conducting Coatings On the other hand, the transmittance from air to the film is, T1 =  n 2 n t1 = no no 2no no + n 2 whereas the transmittance from the film to the substrate is, T2 =  ng 2 ng t2 = n n 2n n + ng 2 Thus, the transmittance from the air to the glass through the film is, Tf = T1 T2 e−αl 1− 1/2 1/2 2R1 R2 cos Φe−αl + R1 R2 e−2αl Since the thickness of the glass substrate is too large, considering incoherent multiple reflections, the transmission through the glass substrate with αg = 0 is T = Tf T3 1 − R2 R3 Where T3 = n0 ng  2ng no + ng 2  2 and R3 = r32 = ng − 1 ng + 1 ————————————————————————————————————————– © Copyright (2024): Author(s). The licensee is the publisher (B P International). DISCLAIMER This chapter is an extended version of the article published by the same author(s) in the following journal. Applied Optics. 2019 Jul 1;58(19):5179-86. Available:https://opg.optica.org/ao/abstract.cfm?uri=ao-58-19-5179: :text=A%20comprehensive%20model%20for%20the, substrates%20by%20ultrasonic%20spray%20pyrolysis. Peer-Review History: This chapter was reviewed by following the Advanced Open Peer Review policy. This chapter was thoroughly checked to prevent plagiarism. As per editorial policy, a minimum of two peer-reviewers reviewed the manuscript. After review and revision of the manuscript, the Book Editor approved the manuscript for final publication. Peer review comments, comments of the editor(s), etc. are available here: https://peerreviewarchive.com/review-history/1269 189