Energy gap measurements based on enhanced absorption coefficient calculation from transmittance and reflectance raw data

The absorption coefficient plays an important role in studying and characterizing semiconducting materials. It is an important parameter to study the mechanism of photons absorption within the structure of the studied material. Thus, it helps to study the several types of charge carrier transport along with the energy band structure and its defects. In literature, a formula was reported to precisely calculate the absorption coefficient from raw data of transmittance and reflectance of electromagnetic radiation. However, the reported formula has several issues limiting its validity in the literature. In this paper, we provide a more mathematically accurate form of this equation to precisely obtain the absorption coefficient from the raw data, by considering the total internal reflection at the different interfaces. Moreover, the equation is tested by simulated data and is applied to study the optical characteristics of a single-component epoxy resin from its transmittance and reflectance raw data.


Introduction
The absorption coefficient is an essential parameter to study the optical characteristics of semiconducting materials.This parameter plays a significant role in understanding the absorption of photons within material layers, generation/combination of excitons between energy bands, defects in energy bands, and tailing of bands edges.The absorption coefficient can be expressed as a function of transmittance and reflectance of photons when interacting with a material, or as a function of the energy of incident photons as presented by Tauc [1][2][3][4].
To express the absorption coefficient T R , ( ) a as a function of transmittance T ( ) l and reflectance R , ( ) l the general formula of T ( ) l when the radiation is transmitting through a film of thickness x can be used.The existence of interfaces within the film structure is considered along with the total internal reflection at each interface [5].This formula is given by: In literature, the absorption coefficient is calculated from the raw T of a semiconducting film of thickness x using the Bouguer-Beer-Lambert law, which neglects the reflectance as mentioned in the equation [6,7]: Moreover, this equation (1) was used by El-Fadl et al [8] to obtain α [5,9].However, they used the approximation where the value of x a is too large.This approximation has the form: In this paper, we present the full form of equation (3) as derived from equation (1) by considering the total internal reflection of electromagnetic waves inside the film.A simulation prepared from the transmittance range of 0.01-1 and reflectance range of 0.01-0.25 is used to compare the results with other forms found in the literature.Moreover, a thin film of thickness 0.01 cm was prepared from the UPR4 single-component epoxy resin [10][11][12].The transmittance and reflectance obtained from this film were used to study part of the optical characteristics of this type of epoxy resin as a practical application of the newly derived equation, mainly the energy gap of the allowed direct and indirect transitions and Urbach tailing characteristics.Moreover, a model of the energy gap measurements is proposed for further future applications to study the cold field emission characteristics of composite tungsten-UPR4 electron sources.

Methodology
In this section, we provide the derivation of the new formula and the theoretical interpretation of .
a Moreover, the experimental setup to prepare the epoxy-thin film and to measure T and R is presented for the purpose mentioned before in section 1.

Theoretical interpretation
Consider the case of a thin film of thickness x, where an incident light of intensity I 0 is directed to its surface.We start with the case where the film does not reflect light, then the transmitted part of the light is expressed by the Bouguer-Beer-Lambert law [6,7]: In equation (4), I is the transmitted intensity of light and m is the attenuation coefficient, which describes the photon absorption and scattering events that occurred through the film.The transmittance of light is the ratio between the transmitted to incident amount of light T I I .0 = / Using the definition of T in equation (4), for ideal cases where the reflectivity is ignored (figure 1(a)), yields the following equation to calculate the extinction coefficient: The extinction coefficient is a linear combination of the absorption a and scattering s coefficients .( ) m a s = + Moreover, if a ratio of the incident light was reflected by the amount RI 0 at the air-film interface without considering any internal reflections (figure 1(b)), the traversed part of the incident light at this interface is reduced by where R is the reflectivity of the film.In this case, the transmitted intensity from the film is given by I -Thus, m is extended to a form that considers a single reflection event on the air-film interface as follows: If only one internal reflection is considered at the second film-air interface, and considering that the value of x m is large enough to neglect any other internal reflections (figure 1(c)), then the internal reflected part will be given by R Thus, for this situation, m is given by: However, if multiple internal reflections are considered inside the film, then the total transmitted part of the light is eventually attenuated (figure 1(d)).The total transmitted intensity n internal reflections is given by the following series: Solving the infinite series in equation (8) yields a more generalized form of equation (1) by considering the linear attenuation coefficient m instead of .
a The solution provides the following form to measure the transmittance through the film: and following algebraic arrangements to get: which is always positive.Thus, equation (10) has two real solutions given by: In equation (11), the solution with the negative sign for the square root can be neglected since it provides a non-defined solution for .
a Furthermore, we substitute the value of y x exp ( ) m = into equation (11), apply the natural logarithm, and divide by x to get the corrected form for the attenuation coefficient function Finally, applying the definition m a s = + to the left side of equation (12) and applying some algebraic arrangements to the right-side yields to: 13) provides a precise calculation methodology for finding the absorption coefficient of the thin film being studied.In the case where s can be neglected, then equation ( 14) can be reduced to the following form: Equation ( 14) is then combined with the Tauc equation to measure the direct/indirect optical energy gaps to express the generation/combination of excitons in semiconductors.Tauc equation is given by: where h is the plank constant, v is the frequency of radiation, E g is the optical energy gap, A is a proportional parameter, and n is a constant to define the type of electron/hole transition during the generation/combination of excitons (n 2 = for allowed direct transitions and n 1 2 = / allowed indirect transitions).Moreover, the linear form of the Urbach empirical equation can be used to obtain the Urbach band tailing energy E U as follows [13,14]:

Experimental setup
The epoxy thin film was prepared by curing the epoxy resin at 180 °C for 4 h.The epoxy resin was placed on a glass substrate which was tilted at 45°during the curing process, which is important to prevent shrinking the resin and prepare a flat thin film.
Ocean optics JAZ-3 channels spectrometer was used to obtain T and R. The instrument is supplied with a halogen VIS/NIR source along with halogen-deuterium extension for UV, and the tested wavelength range was 200-1000 nm [15,16].

Results and discussion
In literature, other forms of equation (14) were derived from equation (1) by Vahalová et al [17][18][19][20].The Vahalová form of a formula is given by: Following the Vahalová group results, another form of equation (17) was found in literature derived by Hassanien et al [21][22][23], which has the following form: Note that equations (17) and (18) have a few mathematical mistakes in their presentation of a that can be summarized as missing some terms and negative signs.First to mention is the absence of the negative sign for the term R in the nominator of equation (17).Missing the sign leads to a non-defined solution of , a which was solved by the authors by applying absolute value for the entire fraction inside the logarithm function, which helps to keep the values of the fraction valid from the mathematical point of view and to produce real values of .a In the second point, the term R 2 is missing in the dominator of equation (17).Eliminating this part produces fraction values always larger than one, which produces only positive values from the natural logarithmic function until it reaches a critical point where a changes its behavior when R is constant.
Finally, producing positive values of the logarithmic function leads to the need to neglect the negative sign in the exponent of Otherwise, the obtained range of a values is always negative as can be seen in figure 2(a) in comparison to the results from equation (14).However, applying the absolute value for the function again solves the problem as can be seen in figure 2(b).
An advantage point can be seen in figure 2(b) in which equations ( 17) and (18) still provide nearly identical values as equation (14), which of course related to the changes made to their structure.However, the form of equations ( 17) and (18) should not be considered as they are not based on correct theoretical explanation.It is important to mention that R was set to 0.032 as the average value of all reflectance values to obtain figure 2.
From the practical point of view, the obtained results from the epoxy thin film are presented in figure 3. The transmittance and reflectance were measured as functions of photons wavelength , l and presented as functions of the photon's energy h .
n The results were obtained at different places from the sample surface and the average values are presented in figures 3(a) and (b).We focus on ultraviolet radiation because the aim of presenting these practical results is to evaluate the optical energy gap of allowed transitions and Urbach tailing energy for this type of epoxy resin.The results from T and R were then used to evaluate a using equation ( 14) as a function of h , n Moreover, values of a were multiplied by photon energies to acquire the two forms of Tauc plots using equation (15).The direct and indirect energies were then calculated from the xaxis intercept point of the linear parts of figures 3(d) and (e) respectively.Furthermore, equation ( 16) was used to acquire the band tailing energy for the thin film being studied, which is important for studying possible defects of the valence and conduction bands.Thus, the curves presented in figure 3(f) are related to the linear part of the ln ( ) a vs hn plot, where the reciprocal of the slope presents the value of E 1 slope.U = / To present the efficiency of equation (14), the obtained values of , a and the related Tauc and Urbach plots, are compared to the results obtained from equation (5) (Bouguer-Beer-Lambert law) and equation (7) (the approximation where x a is too large).The results showed semi-identical behavior between equations ( 7) and ( 14), with very small deviations from equation (5), which is related to the absence of R in equation (5).Starting with figure 3(d), the analysis shows that the direct optical energy gap is E 3.240 g dir = eV as measured by equation (5), while it was E 3.245 g dir = as obtained from equation (14).These values are within the range of optical energy gap values for epoxy resins [24,25].Furthermore, the indirect optical energy gap is found to be E 3.199 g indir = eV using equation ( 14) and E 3.174 g indir = eV using equation (5), which were calculated from figure 3(e).
Epoxy resins have an amorphous structure, which causes them to have some defects in their density of states.Moreover, this single-component epoxy resin is cured thermally which also affects the defects in energy bands and the phonon distributions within its structure [26].Such materials have localized states inside their energy gap that were obtained from the extension of the energy bands into the energy gap.These defects are caused by the impurities that can be found inside the epoxy adhesive structure, which in turn is caused by the dipping of the curing agent in the case of the single component epoxy resins.For example, doping the semiconductors could lead to a decrease in the energy gap of the semiconductor, but concurrently, it increases the defect's energy width because of the increase of impurities inside the material structure [26,27].
From figure 3(f), the straight lines of the figure have slopes of 3.484 eV -1 when a is measured from equations (5) and 3.559 eV −1 when it is measured from equation (14).These slope values lead to energy band edge tailings of E 1 slope 0.287 U = @ / eV and 0.281 eV respectively.This energy tailing is responsible for transitions from the valence band to some states lower than the conduction band bottom, some state higher than the valence band top to the conduction band bottom, or even between energy the localized energy states of the two bands.Table 1 presents a summary of the obtained results.
The direct energy gap is related to direct electric transitions related to an allowed absorption of photons between the top of the valence band and the bottom of the conduction band, which is presented by the a b  transition in figure 4(a).The locations of these two extrema are aligned as head-to-valley at the same momentum value in the momentum hk space.
Moreover, the direct energy gap is related to indirect electric transitions of allowed absorption of photons between the top of the valence band and a shifted bottom of the conduction band in the momentum space, as presented by the a c d   transition in figure 4(a).This type of transition is related to the phonon-assisted transition between the two bands, which is necessary to satisfy the conservation of momentum and energy.An electron may absorb a photon, but it could lose some energy during the direct transition (a c  ) that may cause an incomplete creation/combination of an exciton.In such cases, an electron-phonon interaction could be created which assists the electron in traveling to an adjacent bottom of the conduction band (c d  ) accomplishing both conservation rules.Furthermore, this type of epoxy resin shows a wide range of delocalized states inside the band gap as seen in figure 4(b).The low value of band tailing between the conduction and valence band is related to a high ratio of defects in the band structure, which is caused by the amorphous structure of epoxy resins.

Conclusions
In this paper, we introduced a formula to precisely calculate the absorption coefficient from raw transmittance and reflectance data.The introduced equation considers the scattering coefficient corrections to the absorption coefficient as presented in equation (13).Moreover, an approximation for the cases where the scattering coefficient can be neglected is introduced in equation (14).
A practical application for equation ( 14) was applied to thin films of epoxy resin to show the validity of the newly derived equation.The results of studying the thin film were obtained by calculating the absorption coefficient from the transmittance and reflectance of ultraviolet radiation without approximations.The allowed direct and indirect transitions within its structure were studied by calculating the direct and indirect optical energy gaps from Tauc plots.These have values of 3.245 eV for the allowed direct transition and 3.199 eV for the allowed indirect transition.Moreover, the energy band tailing energy was found to have a value of 0.59 eV.The measured values are close to the related values of different pure epoxy resins and can be useful in several optical applications and some electrical applications in field electron emission studies.Moreover, the results were compared to the Bouguer-Beer-Lambert law, and the results are presented in table 1.
To support the findings of this study, a Microsoft Excel datasheet template was prepared to calculate the absorption coefficient, optical allowed direct and indirect energy gaps, and Urbach band tailing energy.The file is attached as supplementary material along with the practical findings in this research.Researchers who would like to use it need only to copy their raw data to the template and the required information will appear instantly.

Figure 1 .
Figure 1.Systematic diagram to show the different types of transmission processes.In this figure, (a) presents the ideal transmission process without the consideration of light reflection at any interface.(b) presents the case when considering a single reflection at the air-film interface.(c) presents the case when considering a single reflection at the air-film interface in addition to a single internal reflection at the film-air interface.(d) presents the generalized case when considering multiple internal reflections and the attenuation caused by the absorption, scattering, and internal reflection of the incident beam.

Figure 2 . 2 a
Figure 2. The absorption coefficient as a function of transmittance at fixed reflectance value of 0.032.In this figure, a presents the obtained values from equation (14) presented by 1 a and the negative of equation (17) presented by , 2 a and b presents the obtained values from equation (14) presented by 1 a and equation (2) after neglecting the negative sign presented by 2 a .

Figure 3 .
Figure 3.The absorption coefficient parameters and related Tauc and Urbach plots.(a) Transmittance T spectra. (b) Absolute reflectance R spectra. (c) Absorption coefficient. (d) Tauc plot for evaluating the direct optical energy gap. (e) Tauc plot for evaluating the indirect optical energy gap.(f) Urbach plot to evaluate the Urbach band tailing energy.All are measured as functions of photons energy hn .

Figure 4 .
Figure 4.The direct and indirect electric transitions and bands in UPR4 epoxy.(a) Energy diagram in the momentum k space to present the allowed direct electric transitions when absorbing a photon of energy hn and the allowed indirect electric transitions when absorbing a photon of energy h h n n ¢ < then interacting with a phonon of energy h .w (b) Energy diagram in the density of states space to express the defects localized states caused by the energy tailing of E U .