First-principles Calculations of Optical Energy Loss Functions for 30 Compound and 5 Elemental Semiconductors

The energy loss function (ELF) describes the interaction between electrons and matter in solids. Electron inelastic mean free paths (IMFPs), which are important and basic parameters for describing electron inelastic scattering in matter, have been calculated theoretically from the ELFs of materials. These are essential for understanding quantitative surface spectroscopies, such as Auger electron and X-ray photoelectron spectroscopy. However, the optical constants or the ELFs of most compounds are unknown in the 10−50 eV energy range, where electron-solid interactions are strong, owing to experimental difficulty. In this study, the ELFs and the optical constants were calculated for 35 inorganic semiconductors [AgBr, AgCl, AgI, AlAs, AlN, AlSb, cubic-BN (c-BN), hexagonal-BN (h-BN), CdS, c-CdSe, h-CdSe, CdTe, C (diamond), GaAs, GaN, GaP, GaSb, GaSe, Ge, InAs, InP, InSb, PbS, PbSe, PbTe, Se, Si, c-SiC, h-SiC, SnTe, Te, c-ZnS, h-ZnS, ZnSe, and ZnTe] in a wide energy range (from 0.1 eV to 1 MeV) using first-principles calculations with FEFF and WIEN2k. The resulting 35 ELFs were evaluated using two sum rules, the f-sum rule and the Kramers-Kronig sum rule, resulting in average relative errors of 1.6% and 0.05%, respectively. The calculated ELFs for InAs, GaAs and InSb agreed well with the experimental ELFs obtained from the transmission electron energy loss spectroscopy (EELS) experiments and the reflection EELS experiments. The resulting database of the ELFs and the optical constants for 35 compound semiconductors was concluded to be accurate and useful for understanding the inelastic scattering processes of semiconductors with respect to IMFPs and electron stopping powers. All detailed data are available in the materials data repository provided by the National Institute for Materials Science (https://doi.org/10.34968/nims.1434).


I. INTRODUCTION
The energy loss function (ELF) is a fundamental physical quantity describing the interaction between electrons and matter in solids and is essential for quantitative evaluation of inelastic scattering during electron transport in solids [1]. The electron inelastic mean free paths (IMFPs) and electron stopping powers (SPs), which are important fundamental parameters in actual surface analysis by X-ray photoelectron spectroscopy (XPS), Auger electron spectroscopy (AES), and electron energy loss spectroscopy (EELS), have been calculated theoretically from the ELFs of materials [2]. In particular, the IMFP is a fundamental physical quantity used for surface quantification, representing the surface sensitivity and matrix effect of an element. Therefore, many studies have been conducted to determine the IMFPs for various materials, including elemental solids, and inorganic and organic compounds [2−11]. However, the number of the IM-FPs determined remains insufficient, both in terms of the number of target materials and the applicable energy range.
In particular, the number of the ELFs of compounds remains limited, which is a bottleneck for the precise quantitative evaluation of their surfaces. Therefore, ELF measurements have been conducted for various materials and substances, but most are limited in their energy range and have accuracy problems.
Usually, the ELF can be obtained from optical constants, namely, the refractive index (n) and extinction coefficient (k). These optical constants have been measured for various materials and are summarized by Palik in a three-volume handbook, which is widely used in various studies [12−14]. However, optical constant data in the energy range of 10−50 eV, which is most important for IMFP and SP calculations, are lacking for many materials owing to the difficulty of the measurement, such as the requirement of synchrotron radiation. In particular, the measurement data are not available for most compound materials for optical constants or ELFs in this energy range, with some exceptions including GaAs and InP. In contrast, although transmission electron microscopy and reflection EELS have been used to measure optical constants or ELFs, fewer materials have been measured owing to the difficulty of sample preparation and analysis.
Therefore, we obtained a database of ELFs or optical constants for various compound semiconductors in the wide energy range from 1 eV to 1 MeV using first-principles calculations. The resulting database for the ELFs would aid the calculation of IMFPs, SPs, effective attenuation lengths (EALs), mean escape depths (MEDs), and other important physical quantities for the evaluation and the measurement of these materials by XPS, EELS, AES, and transmission electron microscopy (TEM).
The ELF is a function of energy loss (ω) and momentum transfer (q) and is given as the imaginary part of the reciprocal of the complex dielectric function, ( , ). Using the ELF and Born approximation, the differential inelastic scattering cross section, per atom or molecule, for energy loss ω and momentum transfer q in an infinite medium is given by Eq. (1) [15,16]: where E is the electron energy and Im[−1 ( , ) ⁄ ] is the ELF, which is now defined by the complex dielectric function, ( , ). Hartree atomic units were used in Eq. (1), in which m e = e 2 = ℏ = 1, where m e is the electron rest mass, e is the elementary charge, and ℏ is the reduced Planck constant.
In the limit of zero momentum transfer, the probability of inelastic scattering is proportional to the ELF, Im(−1/ ) = , where ε 1 and ε 2 are the real and imaginary parts of the complex dielectric constant ε. ε 1 and ε 2 are simply related to the conventional optical constants, the refractive index (n), and the extinction coefficient (k) by 1 = 2 − 2 and 2 = 2 [12]. Therefore, energy loss functions are needed for various materials in a wide energy range (typically from 1 eV to 200 keV), but optical data are lacking for most materials in the 10−50 eV energy range, where electron-solid interactions are strong, because of experimental difficulties, as stated above [11,17,18]. Accordingly, optical data were calculated in a wide energy range (1 eV to 1 MeV) using first-principles calculations with FEFF 8.2 [19] and WIEN2k [20], and a database of optical energy loss functions and optical constants were obtained. The target materials selected were mainly compound semiconductors for which little optical data were available. A database of the energy loss functions was then prepared for the following 35 materials between 1 eV and 1 MeV: AgBr, AgCl, AgI, AlAs, AlN, AlSb, c-BN, h-BN, CdS, CdSe, CdSe, CdTe, C (diamond), GaAs, GaN, GaP, GaSb, GaSe, Ge, InAs, InP, InSb, PbS, PbSe, PbTe, Se, Si, c-SiC, h-SiC, SnTe, Te, c-ZnS, h-ZnS, ZnSe, and ZnTe.

II. CALCULATION
This section describes the method used to calculate optical constants from first principles in a wide energy region. The imaginary part (ε 2 ) of dielectric function = 1 + 2 was calculated from the WIEN2k and FEFF codes. Combining ε 2 values obtained from each approach, ε 2 was obtained over a wide loss energy range (from 0.1 eV to 1 MeV). The real part of the dielectric function (ε 1 ) was then calculated using the Kramers-Kronig relation. Using ε 1 and ε 2 , the refractive index (n), the extinction coefficient (k), and the ELF (Im[−1 ( ) ⁄ ]) were finally obtained.
A. Dielectric functions from first-principles calculations 1

. WIEN2k
WIEN2k calculates the electronic structure of solids based on the all-electron approach of DFT, named the full-potential linearized augmented plane wave (FLAPW) method. WIEN2k version 08.02 was used in this study. The generalized gradient approximation (GGA), as parameterized by Perdew, Burke, and Ernzerhof, was adopted for the exchange-correlation potential [21]. The spin polarization was not calculated because our target materials were not magnetic. For hexagonal crystals, optical constants under an electric field perpendicular to the c-axis of the crystal were calculated. The imaginary part of the dielectric function, ε2, was calculated from optically allowed transition amplitudes of photon energies between 0.1 and 136 eV. Owing to computational limitations, the available loss energy is known to be less than 100 eV [20,22].

FEFF
FEFF 8.2 is an automated program for calculating X-ray absorption spectra using an ab initio all-electron, real-space relativistic Green function [19]. This code gives photoabsorption cross-sections from any absorption edge for all elements in a compound. Only information on the crystal structure is needed to obtain the absorption spectra. This allows ε 2 to be easily calculated, as shown below. These calculations were applied to photon energies between 10 eV and 1 MeV.
FEFF allows us to compute absorption spectra with three cards for each absorption edge (K, L 1 , L 2 , L 3 , M 1 , M 2 , ⋯) of an atom. We used the XANES card near the absorption edge, the EXAFS card for tens to hundreds of eV from the absorption edge, and the FPRIME card beyond that. In all cases, the RGRID card was set to 0.01 to improve the calculation accuracy at high photon energies. Although the FPRIME card gives the imaginary part of the atomic scattering factor, f2(ω), the XANES and EXAFS cards give the optical absorption spectra μ(ω), which can be converted to f 2 (ω) using Eq. (2): where α = 1/137.0359895 is the fine-structure constant. We connected f 2 (ω) values obtained from the XANES, EXAFS, and FPRIME cards, and constructed f 2 (ω) in a wide photon energy range from a single absorption edge. For the outer-shell orbital contribution to valence electrons, we confirmed that it was better to use FPRIME cards only. In the same manner, f 2 (ω) was obtained from all the absorption edges of the same atom. By combining them, f 2 (ω) was obtained for one atom. As each absorption edge has a different energy grid of absorption spectral data, the data needed to be interpolated when added together. Next, the imaginary part of the atomic scattering factor f 2 (ω) per molecule was calculated for a compound with a composition represented by A A B B . In this case, by labeling the imaginary part of the atomic scattering factor of an atom X, as f 2,X (ω), the scattering factor f 2 (ω) per molecule can be simply calculated using Eq. (3): The imaginary part of the dielectric function ε 2 (ω) is calculated using f 2 (ω) [23]: where ω is the photon energy (eV), ρ is the mass density of a material (g cm −3 ), and F W is the molecular weight of the compound (g mol −1 ). A dielectric function has not previously been created by connecting the results of WIEN2k and FEFF calculations. Although Rehr and coworkers [24−26] have calculated the dielectric function in the wide loss energy range using only FEFF, the accuracy of the calculation below 50 eV was questionable, especially around the prominent energy loss region.

B. Determination of energy loss function
We calculated ε 2 using two methods, WIEN2k for 0.1−136 eV and FEFF from 10 eV to 1 MeV. For example, Figure 1 shows ε 2 as a function of energy loss ω for InAs and PbS below 100 eV, with ε 2 from WIEN2k and FEFF overlapping well at 30 and 50 eV, respectively. Similarly, for all 35 materials, the ε 2 values overlapped well at a certain loss energy between 30 and 80 eV, allowing them to connect at the loss energy. Table 1 shows the connection energies of the 35 materials in this study. Finally, a dataset of ε 2 calculated by WIEN2k and FEFF for each compound was prepared for photon energies from 0.1 eV to 1 MeV.
The real part of the dielectric function ε 1 was calculated from ε 2 using the Kramers-Kronig relation [27]: where P is the Cauchy principal value. The actual integration range of Eq. (6) was from 0.1 eV to 1 MeV. The algorithm used to numerically calculate the Kramers-Kronig relation is described in Appendix.
The optical ELF was then calculated from the data sets of ε 1 and ε 2 with the following Eq. (7): From the complex dielectric function 1 + 2 , the complex refractive index, + , where n is the refractive index and k is the extinction coefficient, can be obtained easily. Notably, erroneous conversion often occurs between the complex dielectric function and complex refractive index (Henke [23] reports inaccurate expressions). Using the relation between the complex dielectric function and complex refractive index, 1 + 2 = ( + ) 2 , mutual conversion was performed as follows: where | | = � 1 2 + 2 2 . Only structural information of the crystal lattice was needed for first-principles calculation of the dielectric function. Table 2 shows crystal information for the inorganic compounds used in the WIEN2k [20] and FEFF [19] calculations [28]. Crystal information for all but eight materials [c-CdSe, C (diamond), Ge, Se, Si, h-SiC, Te, and h-ZnS] are taken from Ref. 11.

III. RESULTS
The results calculated using WIEN2k and FEFF for the 35 semiconductors are shown in Figures 2−6: Figure 2 shows ELFs, Figure 3 shows the refractive index (n), Figure 4 shows the extinction coefficients (k), Figure 5 shows the real part of the dielectric function (ε 1 ), and Figure 6 shows the imaginary part of the dielectric function (ε 2 ). All numerical data can be obtained from the materials data repository pro-vided by the National Institute for Materials Science [29]. Herein, the ELF results, which are strongly related to surface analysis, are described.
A. ELFs  excitation energy. Absorption edges related to the excitation of inner-shell electrons were observed, from which the constituent elements of a material can be identified. The physical properties of a material mainly originate from the difference in shape and intensity of the ELF below 100 eV. A prominent sharp peak in the ELF, related to a pseudo-plasmon excitation, was observed for GaAs, GaSb, InAs, Si, and other materials. In these cases, the pseudo-plasmon energies [ p = 28.8( / ) 1/2 , where N v is the number of valence electrons per molecule, ρ is the bulk density (g cm −3 ), and M is the molecular weight] agreed with the peak position of the ELF data. However, materials with broad ELF shapes, such as AgBr and AgCl, were dominated by single-electron excitations rather than pseudo-plasmon excitations.
Some materials were missing ELF and ε 2 data at very high loss energy due to data processing in FEFF, including AlN, c-BN, h-BN, C (diamond), Si, c-SiC, and h-SiC. The ELFs in this range were sufficiently small, so that the influence on the results is negligible.

B. Evaluations of ELFs
The internal consistency of our ELF data was checked for each compound using the oscillator-strength or the f-sum rule and a limiting form of the Kramers-Kronig integral (or the KK-sum rule) [5,30]. The f-sum can be evaluated as the total effective number of electrons per molecule, Z eff , contributing to inelastic scattering using Eq. (12): where Δ = ℏ , p = (4 a 2 / ) 1/2 , a = a / is the density of atoms, a is the Avogadro number, ρ is the mass density, and M is the molecular weight. The minimum and maximum energy losses in Eq. (12), ΔE min and ΔE max , were 0.1 eV and 1 MeV, respectively. The f-sum for ε 2 is similar, as shown in Eq. (13): In the limit Δ max → ∞ , both eff and eff, 2 are expected to approach Z, the total number of electrons per molecule. The KK-sum can be expressed as shown in Eq. (14): where Re[1 (Δ min ) ⁄ ] ≃ −2 (Δ min ) when Δ min is sufficiently small. In the limit Δ max → ∞ , eff → 1 is expected. Table 3 lists the errors in the f-sum and KK-sum rules for each material, which are the differences between the computed values of eff and eff , and the expected values (total number of electrons per molecule and unity, respectively). The error in eff ranged from −2.9% to +1.7%, and the mean absolute percentage error (MAPE) was 1.6%. The MAPE is defined as shown in Eq. (15): where True is a true datum and Calc a calculated datum for the i-th material. Most materials (65%) had an error of less than 2.0%. Table 3 shows that eff, 2 obtained with the f-sum rule for ε 2 agreed well with eff obtained with the f-sum rule for the ELF. The error in eff ranged from −0.43% to +0.06%, and the MAPE was 0.05% for all 35 materials. Unsurprisingly, the KK-sum was almost exactly constant because Eq. (13) is derived from the causality itself.  Table 3 shows that both the f-sum and KK-sum errors are sufficiently small, with the KK-sum error being particularly small.

IV. DISCUSSION
In this section, several our calculated results are compared with the experimental data. Figure 7 shows the ELFs calculated by WIEN2k and FEFF for InAs, PbS, GaAs, and InSb together with the ELFs obtained from the transmission EELS experiments by von Festenberg [31], those analyzed by Chen et al. [32], those obtained from the measured optical constants by Palik [12−14], and those obtained from the reflection EELS experiments by Jin et al. [17]. For InAs in Figure 7(a), the calculated ELF agreed with the ELF of von Festenberg. The plasmon peak position obtained from the calculated ELF was 14.2 eV, which agreed well with the value of 13.8 eV obtained from the Festenberg ELF. Furthermore, the microstructure found in the measured range of 18−22 eV agreed well with the microstructure obtained by calculation. On the other hand, we see a large difference between our ELF and the ELF calculated from optical constants by Palik. As noted by Palik [12−14], the InAs sample of Ehrenreich and Philipp probably had a ~2-nm-thick native oxide on its surface. Therefore, it is unsurprising that the InAs ELF in Figure 7(a) from the Palik optical data differed considerably from the other ELFs. In particular, the plasmon peak from the Palik data was at 13.0 eV and was much weaker than the peaks for other ELFs in Figure 7(a).

A. Comparison of ELFs with experimental data
For PbS in Figure 7(b), as a whole the calculated ELF did not coincide well with the ELF obtained from the experimental data of Palik. However, the three peaks at 15.2, 20.6, and 22.7 eV in the experimental data agreed well with the calculated peak positions of 15.4, 19.6, and 21.6 eV. The calculated ELF also had small peaks at 6.8 and 7.8 eV, which corresponded well with the measured shoulder structure at 6−10 eV. The differences between the peak heights and peak widths for two ELFs are not known exactly, but they are thought to be originated from surface oxidation and crystal imperfection, since the sample used in the experiment was a natural crystal. Furthermore, the sum rule errors for our ELF for PbS are less than 3% as shown in Table 3. Then, our calculated ELF for PbS is superior to the experimental ELF of Palik. Figure 7(c) shows that the GaAs ELF from WIEN2k for energy losses up to 50 eV is in excellent agreement with the ELF determined by Jin et al. from angle resolved EELS data using target factor analysis method. It is also in good agreement with the ELF obtained by Chen et al. over 0 eV to 25 eV. On the other hand, our ELF is very different from the ELF that was calculated from experimental optical constants tabulated by Palik. The details of the comparison between our ELF and Palik ELF were figured in our previous paper [11].
The ELF for InSb from the WIEN2k is shown in Figure  7(d) and is in good agreement with the ELF determined by von Festenberg [31] from transmission-EELS experiments. The ELF from the optical data of Palik shows a plasmon peak at 12.0 eV that is much weaker than the plasmon peak found in the ELFs from WIEN2k and von Festenberg. The optical constants tabulated by Palik for InSb were obtained from the reflectivity measurements of Philipp and Ehrenreich [33] in the energy range from 6 to 24 eV. Their InSb sample was prepared and handled in the poor vacuum conditions, and it is then likely that the InSb sample had a surface oxide.
We conclude that the ELFs of InAs, PbS, GaAs, and InSb obtained from WIEN2k are superior to the ELFs obtained from published optical data [12−14]. We also point out that the resulting 35 ELFs gave excellent sum rule results, as shown in Table 3. Therefore, we could conclude that a sufficiently practical database of the ELFs for compound semiconductors has been constructed.

B. Band gap energies
When calculating IMFPs from ELFs with the full Penn algorithm using the Boutboul approach, the values of the band gap are known to significantly influence the resulting IMFP values in the low energy region [11]. In contrast, it is generally difficult to measure the band gap energies of nonconductors precisely, resulting in their values obtained from the literature varying widely [10]. Band gap energies were then determined from the ELFs calculated with WIEN2k for compounds and compared with the measured values (median values in the literature). Figure 8 shows the resulting band gaps obtained with WIEN2k for 35 materials as a function of measured band gap energies. The solid line shows the case where the measured ( g mes ) and calculated band gap values ( g WIEN2k ) are equal. Major parts of band gaps calculated with WIEN2k (27 of the 35 materials) were smaller than the measured band gaps. This tendency was consistent with the fact that the band gaps obtained by DFT calculations using the local density approximation [34] or GGA [21] tend to be smaller than measured band gaps [34].
The dotted line indicates the curve fit results of band gaps obtained from WIEN2k, which represents g WIEN2k = 0.88 g mes (±0.06 g mes ). The root-mean-square differences for measured and calculated band gaps was 1.0 eV. Therefore, the differences between the calculated and measured band gap energies were not large. This indicated that there was no problem in the calculation of IMFPs above 50 eV when using the calculated band gap energy obtained from WIEN2k [11].

V. CONCLUSION
The dielectric function and optical constants for 35 semiconductors in the photon energy range from 0.1 eV to 1 MeV were calculated using first-principles calculation pro-grams WIEN2k and FEFF. The imaginary parts of the dielectric functions of the low-loss energy side and high-loss energy side were calculated using WIEN2k and FEFF, respectively, and were connected at a certain loss energy between 30 and 80 eV. The connection energies were determined for each material. The real part of the dielectric function was calculated from the imaginary part of the dielectric function using the Kramers-Kronig relation. The optical ELF, refractive index, and extinction coefficient were also calculated. All values determined here can be obtained from the materials data repository provided by the National Institute for Materials Science [29].
The ELFs calculated by WIEN2k and FEFF for InAs, PbS, GaAs, and InSb were compared with the experimental ELF data. We found that the resulting ELFs are in good agreement with the measured ELFs that obtained from the transmission EELS experiments by von Festenberg [31], analyzed by Chen et al. [32], and obtained from measured optical constants by Palik [12−14] and from the reflection EELS experiments by Jin et al. [17,18]. On the other hand, our ELFs are very different from the ELFs that were calculated from experimental optical constants tabulated by Palik. We also compared the band gap energies for 35 materials that were determined from the ELFs calculated by WIEN2k with the measured values. We found that the calculated band gaps for 27 of the 35 materials were smaller than the measured band gaps, and this was consistent with the fact that band gaps obtained through DFT calculations using the local density approximation tend to be smaller than the real band gaps. However, the difference between the calculated and measured band gaps was not so large; the curve fit results of band gaps obtained from WIEN2k, which represents g WIEN2k = 0.88 g mes (±0.06 g mes ). The root-mean-square difference for measured and calculated band gaps was 1.0 eV.
We also point out that the resulting 35 ELFs gave excellent sum rule results; the mean absolute percentage error of the f-sum and the KK-sum were 1.6% and 0.05%, respectively. Therefore, we can conclude that a sufficiently practi-cal database of the ELFs for compound semiconductors has been constructed.  [12−14]. Green dashed line represents the ELF measured by Festenberg et al. [31] or the ELF as analyzed by Chen et al. [32]. The Kramers-Kronig relation relates the real and imaginary parts of a complex dielectric function and is expressed as a Cauchy principal-valued integral. Although the integrated function has a singularity, the integral converges if we avoid the singularity properly. Given the discrete data {( , )} =1 , we can calculate the Kramers-Kronig integral S(xj) as follows: Here, we assume that y(x) is a polyline connecting each discrete data point: { } =1 need not be equally spaced. When the integral range is independent of the singularity, = , i.e., 1 ≤ ≤ −1 or +1 ≤ ≤ , the integral of each interval can be calculated as follows: For the remaining integration range −1 ≤ ≤ +1 sandwiching the singularity = , two intervals are integrated together to prevent divergence of the integrals: