Simulation of infrared spectra of trace impurities in silicon wafers based on the multiple transmission–reflection infrared method

The content of trace impurities, such as interstitial oxygen and substitutional carbon, in silicon is crucial in determining the mechanical and physical characteristics of silicon wafers. The traditional infrared (IR) method is adopted as a normal means to analyse their concentration at home and abroad, but there are two problems. The first problem is the poor representativeness of a single local sampling point because the impurity distribution in a solid sample is not as uniform as that in a liquid sample. The second problem is that interference fringes appear in the infrared spectra of the sample due to the thin wafer (≤ 300 μm thick). Based on this, controversial issues existed regarding the measured trace impurity concentrations between wafer manufacturers and solar cell assembly businessmen who used silicon sheets made by the former. Therefore, multiple transmission-reflection (MTR) infrared (IR) spectroscopy was proposed to solve the problems mentioned above. In the MTR setup, because light passes through different parts of the silicon chip several times, multiple sampling points make the final result more representative. Moreover, the optical path is lengthened, and the corresponding absorbance is enhanced. In addition to amplification of weak signals, the MTR-IR method can eliminate interference fringes via the integrating sphere effect of its special configuration. The signal-to-noise ratio of the corresponding spectrum is considerably improved due to the aforementioned dual effects. Thus, the accuracy and sensitivity of the detection method for trace impurities in silicon chips are greatly increased. In this study, silicon wafers were placed in the MTR setup, and then, their relative properties at room temperature were investigated. The corresponding theoretical calculation and simulation of infrared spectra of silicon chips were provided. This affords an optional method for the semiconductor material industry to analyse trace impurities in their chips.


Scientific Reports
| (2021) 11:1254 | https://doi.org/10.1038/s41598-020-80883-0 www.nature.com/scientificreports/ infrared spectrometry 10 , can remove some interference fringes; however, it takes a longer time and consumes much power in the process of refrigeration, and its window materials can cause loss of light. At present, the nondestructive infrared (IR) method, which can keep the chip intact, is commonly used to analyse the content of interstitial oxygen and substitutional carbon in silicon at home and abroad [11][12][13][14] . However, the distribution of interstitial oxygen and substitutional carbon in solid silicon samples is heterogeneous, unlike that in liquid solution. In addition, this traditional IR method only has one sampling point for each measurement, so the number of sampling points is insufficient to represent the real condition. Some commercial disputes based on the above reasons arose between silicon chip manufacturers (e.g., Wacker Company, Hemlock Company, OCI Company, and Golden Concord-Zhongneng Silicon Industry Company) and their downstream users, that is, solar cell module manufacturers (e.g., Yingli Company, Trina Solar Company, Hanwha Solarone Company, and Ja Solar Company) who usually used the traditional IR method.
Silicon wafers 100-300 μm thick are often used to fabricate ultrathin solar cells. However, the traditional IR technique cannot measure silicon wafers with a thickness of 100-300 μm, which is less than the wavelength of incident light, thereby resulting in severe interference fringes in infrared spectra. The interference fringes strongly cover the absorption peaks.
To solve these problems, multiple transmission-reflection (MTR) infrared (IR) spectroscopy was proposed. As shown in Fig. 1, incident light enters and is transmitted through a silicon chip and re-enters it via reflection from golden mirrors placed on both sides of the silicon slice.
The first advantage of the MTR-IR method is that the number of sampling points can be increased to a dozen times that in the traditional IR method, so the final result is more representative. Second, this MTR-IR method can eliminate interference fringes in spectra caused by thin wafers, making it suitable for analysing the thinner wafers (≤ 300 μm thick) often used by all solar cell module manufacturers, such as Yingli Company, Trina Solar Company, Hanwha Solarone Company, and Ja Solar Company. Third, the MTR-IR method can increase the optical path ten times; thereby, the same enhancement occurs for the absorbance signal. Thus, the MTR-IR technique can decrease the limit of detection of impurities in silicon wafers with the same thickness by an order of magnitude compared with the traditional IR method. Fourth, due to the integrating sphere effect generated by the special MTR-IR structure, when transmitting light and reflecting light with the same phase meet, their wave crests and troughs are added and offset, which can eliminate interference fringes in spectra. Therefore, the signal-to-noise ratio of the infrared spectra can be greatly improved by this method.
Professor Xiao's group successfully applied MTR-IR technology in qualitative and quantitative measurements of ultrathin organic, macromolecular, and biological molecules on silicon surfaces at room temperature, in which the absorbance ratio of s-to p-polarized light that they obtained could be used to calculate the monolayer molecule group orientation on the surface relative to the substrate [15][16][17][18][19] . Several teams abroad used the MTR-IR method for scientific research in various fields. Prof. Spencer investigated the tribological and other characteristics of organic molecules and polymer brushes on the surface of silicon wafers by using the MTR-IR method as the main tool 20,21 . He also studied the composition and orientation of various functional molecules fabricated by different film-making methods 22,23 . Application of the MTR-IR method in the determination of molecular orientation can also be used to explore the effect of chromophore orientation on the photoelectric conversion efficiency of solar cells and organic light-emitting diode (OLED) devices.
MTR-IR technology has a bright future. In view of the manpower and financial resources, this study mainly focused on MTR-IR analysis experiments and theoretical simulations of infrared spectra of silicon wafers.

Experimental
Silicon wafer data. Double-side-polished silicon wafers (<100>-oriented, n-type, B-doped, 2000 (CZ) and 8000 (FZ) Ω cm resistivity, and 200 μm, 300 μm and 500 μm thick from Fujian Hanchen New Materials Company Limited, China) were cut into rectangular shapes (15 mm × 50 mm) for infrared analysis. CZ silicon wafers were used as samples and FZ as references to analyse the content of interstitial oxygen and substitutional carbon in silicon.  (1:1:1, V:V:V) for 30 min, cooling to room temperature, rinsing with deionized water, and storage in deionized water. Silicon chips were immersed in 1% HF for 5 min to remove the native silicon oxide layer and dried using N 2 gas before infrared analysis.
Analysis. The optical equipment was designed for adaption to any commercial FTIR spectrometer, which was a Bruker Tensor 27 in our case. A Brewster incidence angle of 74°, a deuterium triglyceride sulfate detector and scan times of 100 at 4 cm −1 resolution were used for analysis over the middle infrared wavenumber range from 400 to 4000 cm −1 in the MTR-IR method. The entire analysis procedure was carried out according to SEMI MF 1188-2007 and SEMI MF 1391-2007 13,14,24,25 . Each silicon sample was analysed successively four times by the MTR-IR and traditional IR methods by slightly relocating the Si wafer each time to analyse different sampling points. Therefore, for each silicon sample, four different sampling points were analysed by the traditional IR method, whereas 40 different sampling points were analysed by the MTR-IR method with N = 10.

Simulation theory
Interference elimination function of an integrating sphere 26 . The MATLAB program for calculation of an integrating sphere can be seen in the supporting information. When original light passes through a silicon sheet once, the attenuation factor U (Eq. (2)) of light caused by silicon wafer absorption, reflectance R (Eq. (3)), and transmittance T (Eq. (4)) are given in the following formulas 26 In Fig. 2, n refers to the real part of the complex refractive coefficient of silicon; κ denotes its imaginary part; b represents the thickness of the silicon wafer, cm; n 1 and n 2 correspond to the refractive index of air, with n 1 = n 2 ; θ 1 and θ 2 specify the incidence angles of light from air to silicon and from silicon to air, respectively; r 1 and r 2 denote the Fresnel reflection coefficients at the first and second interfaces, respectively; and t 1 and t 2 refer to the Fresnel transmission coefficients at the first and second interfaces, respectively. φ 1r and φ 2r denote the reflection phase changes at the first and second interfaces, respectively. N Z indicates the projection of n onto the z axis. K is the angle between the normal line of the isophase surface of light and z axis. σ is the wavenumber, cm −1 . δ' www.nature.com/scientificreports/ represents the phase thickness of the silicon wafer. T is the transmittance, %; R is the reflectance, %; and U is the amplitude attenuation factor. Figure 3 shows that peaks and troughs periodically appear in the reflection R and transmission T profiles, in which the crest of the R curve corresponds to the trough of T, and vice versa. After the addition of R and T, their crests and troughs cancel out, and a curve that is almost a straight line is obtained. In the MTR setup, when the original light reaches the surface of a silicon chip, it splits into reflected and transmitted parts. The reflected light and transmitted light are reflected by gold mirrors on both sides of the silicon chip and re-enter it. At this point, the wave crest of reflected light correspondingly offsets the trough of transmitted light, and vice versa, thereby eliminating the interference fringes generated by the thin silicon chip. Thus, the quality of the IR spectrum acquired is improved. This phenomenon is the same as reflection light meeting transmission light in an integrating sphere, that is, reducing interference and increasing absorption 27 . However, there is a slight difference between the integrating sphere and the MTR setup. Spherical reflection appears in the former, whereas planar reflection appears in the latter. In addition to the increased optical path length, enlargement of the absorption peak in the MTR-IR spectrum was also achieved due to the energy brought into the sample by the light reflected by gold mirrors. In the MTR-IR method, the interference fringes arising from a thin silicon wafer are eliminated for two reasons: the use of the Brewster incidence angle and the integrating sphere effect of the MTR geometry.
Simulation and calculation of the infrared spectra of s-polarized light passing through a thin silicon wafer in the MTR-IR method. MATLAB programs for the simulation calculation of the infrared spectra of s-and p-polarized light based on the MTR-IR method are shown in the Supporting Information. In Fig. 4, when s-polarized light enters the silicon chip at the Brewster incidence angle, R s (reflection coefficient of s-polarized light on the surface of a silicon wafer) is equal to 0.71.
where T S Si indicates the total transmittance of s-polarized light through the silicon wafer (including multiple transmissions within the silicon wafer). R S Si represents the total reflectance of s-polarized light on the surface of the silicon wafer (including multiple reflections within the silicon wafer). δ denotes the phase change caused by reflection in the absorbing medium. n is the refractive index of silicon. σ specifies the wavenumber, cm −1 .  Fig. 4, when the transmission time N is equal to 1 or 2, the thin-film interference effect is considered according to thin-film optical theory. Fig. 4, when N is greater than or equal to 3, the film thickness is calculated according to the thick-film optical principle, and the interference effect of the thin film is disregarded. Figure 5 exhibits the step-by-step calculations, in which the accumulative total intensity of the output s-polarized light can be obtained. When the incident light impinges on the silicon interface, it splits into reflection and transmission components and returns to the interface after reflection on gold mirrors. The split ray intensity is multiplied by T Si , R Si , or R Au each time as light reaches these interfaces. After each step, the horizontal path length of the ray is increased by L down or L up (Fig. 5). The next calculation will be stopped when L is greater than L Si or when the total intensity is less than 10 −4 . The sum of all terminal intensities of each branch in Fig. 5 corresponds to the accumulative total intensity of the output s-polarized light.

Simulation of thick-film optics for s-polarized light. In
Simulation and calculation of the infrared spectra of p-polarized light passing through a thin silicon wafer based on the MTR-IR method. In Fig. 6, when p-polarized light enters a silicon wafer at the Brewster incidence angle, R p (the reflectivity of p-polarized light impinging on the silicon surface) is equal to 0. The image was created using AutoCAD 2014 (simplified Chinese), http://win.xmjfg .com/pg/246.html. Simulation of thin-film optics for p-polarized light. In Fig. 6, when N is equal to 1 or 2, the interference effect of the thin film is considered.
Simulation of thick-film optics for p-polarized light. In Fig. 6, when N is greater than or equal to 3, thick-film theory is applied without considering the interference effect of the thin film.  Results and discussion Figure 7 shows two simulated infrared spectra. The peak height of p-polarized light is higher than that of s-polarized light owing to the energy loss of reflection when s-polarized light reaches the silicon surface. Numerous interference fringes appear in the spectrum of s-polarized light, particularly at the absorption bands, due to thin-film interference from the silicon wafer with a 200 μm thickness. Figure 8 shows eight curves. The heights of the simulated absorption bands increase in proportion to N (from 1 to 8) when using p polarization. No interference fringes are observed, unlike with s-polarized light. Figure 9 shows four curves. There are no evident differences between the sample and simulated infrared spectra for p-polarized light, which are almost the same. However, the sample and simulated infrared spectra  (2) All of the p-polarized light passes totally through the silicon wafer at the Brewster incidence angle, and no reflection appears. Thus, all p-polarized rays reach the signal detector, and the maximum absorption peak is acquired. In contrast, 29% of the s-polarized light enters the silicon chip at the Brewster incidence angle because 71% of it is reflected. Therefore, less than 29% of the s-polarized light is obtained by the signal detector, and its minor absorption band is obtained.

Conclusion
In this paper, the calculation of the content of interstitial oxygen and substitutional carbon in crystalline silicon was theoretically deduced based on the MTR-IR method, and the corresponding infrared spectra were simulated. By comparing the simulated spectra to the sample spectra, there are no differences between them, proving that the proposed principle model for calculation is appropriate. This theoretical simulation is not only applicable to the measurement of trace impurities in crystalline silicon but also suitable for quantitative characterization of trace impurities in other semiconductors by the MTR-IR method. This work might provide experimental operation and theoretical simulation support for research on the influence of the chromophore orientation and structure of organic molecules on the photoelectric conversion efficiency at room temperature and changes in interfacial characteristics, such as the limited dipole moment, charge separation, and photoelectric transmission at low temperature. It could be helpful for scientists in understanding the principles and mechanisms of solar cells and LED technologies, which is made easier by the proposed method.
Received: 29 July 2020; Accepted: 29 December 2020 License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.