Uniqueness of parameter estimates obtained from fitting free carrier absorption data of silicon wafers

Electronic transport parameters, i.e., the carrier bulk lifetime (τ), diffusion coefficient (D), and front and rear surface recombination velocities (S₁ and S₂), are important parameters to characterize semiconductors and provide useful information for device fabrication. Until now, various non-destructive and non-contact diagnostic methodologies, such as photoconductance decay (PCD),^1,2 photoluminescence (PL),^3,4 photothermal radiometry (PTR),^5,6 free carrier absorption (FCA),^7–11 and photocarrier radiometry (PCR),^12–15 have been developed to determine the electronic transport properties of silicon wafers.

In all these techniques, the electronic transport properties are simultaneously determined by fitting the experimental data to corresponding rigorous theoretical models via a multi-parameter fitting procedure. The process is in fact an inverse problem in physics. Practically, the presence of various measurement errors and instrumental noises does not allow all the above-mentioned parameters to be measured reliably and uniquely. To improve the accuracy of measurement results, the sensitivities of signals to each transport parameter have been analyzed in detail, and several methods have also been proposed recently.^16–22 Huang et al.^19,20 proposed to combine PCR and FCA to improve the accuracy of multi-parameter estimation for characterization of silicon wafers. Song et al.^21,22 investigated and optimized the uniqueness and reliability ranges of the above-mentioned transport parameters through the combination of heterodyne lock-in carrierography (HeLIC) and PCR measurements. However, the above-mentioned methods will increase not only the measurement cost but also the complexity of the measurement process, which may introduce extra measurement errors.

As is well known, the uniqueness of fitted results is closely related to the number of free parameters.¹⁸ Therefore, the fitted results will significantly depend on the fitting initial values when there are many free parameters to be extracted. In this paper, taking the determination of the electronic transport parameters by modulated FCA (MFCA) as an example, a novel data process method is proposed to improve the uniqueness of the fitted transport parameters. After describing the theory model, the sensitivity coefficient of the parameters (τ, D, S₁, and S₂) is defined and analyzed in detail. Moreover, the dependence of the fitted results on the initial values and the number of free parameters by the conventional multi-parameter fitting procedure are also discussed. Finally, we propose the mean square variance graph or map method and verify its feasibility by both simulation and experiment.

The MFCA measurement principle has been described previously^19,20 and can be briefly introduced as follows: A pump beam with photon energy higher than the bandgap of the sample irradiates the front surface of the sample; excess carriers are produced due to the absorption of photon energy. A probe beam with photon energy lower than the bandgap of the sample irradiates the material at the same position. Due to the free carrier absorption, the intensity of the transmitted probe beam will change. The excess carrier concentration depends on the electronic transport parameters of the sample. Therefore, the carrier transport parameters can be obtained by recording the change in the transmitted probe beam and using a multi-parameter fitting process. If the intensity of the pump beam is periodically modulated, such as a square wave, the excess carrier concentration also has the same periodicity and a phase delay with respect to the pump beam. The transmitted probe beam can be usually expressed as⁸ 

Here, const is constant. ω = 2πf is the angular frequency. f and λ are the modulation frequency and the wavelength of the pump beam, respectively. J₀(ξ) is the first kind of the zero order Bessel function. r₀ is the pump-to-probe separation. The effective beam size is defined as $a=a12+a22$⁠, where a₁ and a₂ are the size of the pump and the probe beam, respectively. $ΔÑξ,ω,λ$ can be found in the  Appendix.

A lock-in amplifier is often used to demodulate the measured MFCA signal, and the amplitude and phase signal are

and

It can be seen from Eq. (1) that MFCA signals depend on the electronic transport parameters of the sample. The parameters can be determined simultaneously by using the multi-parameter fitting procedure. However, when all four transport parameters mentioned above, i.e., τ, D, S₁, and S₂, are set as free parameters, the question of uniqueness of the parameters’ fit to experimental data should be taken into account.²³ 

Based on the above-mentioned theoretical model, the sensitivities of the MFCA signal to individual electronic transport parameters are investigated first. Considering that the amplitude signal is usually self-normalized in the experiments, both the simulated and experimented amplitude data are self-normalized in the following sections. The sensitivity coefficients of the amplitude and phase signals are defined as¹⁶ 

where P represents the electronic transport parameters, i.e., τ, D, S₁, and S₂. Three different cases are considered in the simulations: case 1 for long lifetime samples (τ = 10 ms), case 2 for moderate lifetime samples (τ = 100 µs), and case 3 for short lifetime samples (τ = 5 µs). The other carrier transport parameters used are D = 12.5 cm²/s, S₁ = 10 m/s, and S₂ = 10 m/s. The sample thickness is 500 μm, the effective beam size is 100 μm, and the pump power is 50 mW. The pump beam wavelength is 660 nm, and the absorption coefficient of crystalline silicon is 3.04 × 10⁵ m⁻¹ at this wavelength. In the following simulations, we focus on an n-type silicon wafer with moderate lifetime (case 2). The simulation results for the other two cases are shown in the supplementary material. For p-type wafers, an analogous manner can be used.

Figure 1 shows the sensitivity coefficients of the simulated MFCA amplitude and phase to the electronic transport parameters. From Fig. 1, it can be seen that the phase has a larger sensitivity coefficient for all four electronic transport parameters than that of the amplitude. The dependence of the amplitude and phase sensitivity coefficient on the modulation frequency is similar. The sensitivity coefficients of the amplitude and phase to τ are negative and have maximum sensitivity in the immediate frequency range. For D and S₁, the sensitivity coefficients increase with increasing modulation frequency. The sensitivity coefficient to S₂ is close to zero in the whole modulation frequency range shown in Fig. 1. This is because only a few carriers could diffuse to the rear surface and recombine. Moreover, the amplitude and phase sensitivity coefficients of all parameters are close to zero at the low modulation frequency end, indicating that each parameter has little effect on the signal in this region. In addition, with the decrease in the carrier lifetime from 10 ms to 5 µs, the amplitude and phase sensitivity coefficients of τ have a significant improvement, as shown in Fig. 1 and Fig. S4 in the supplementary material. However, when the carrier lifetime continues to decrease to 0.01 µs, the measurement sensitivity of all parameters drops toward a very small value, and the above-mentioned transport parameters cannot be determined accurately. Increasing the upper limit of the modulation frequency may be an alternative method for this problem.

Due to the different frequency dependences of the sensitivity coefficient, in order to obtain all the above-mentioned electronic transport parameters simultaneously, a multi-parameter fitting procedure with a suitable objective function is usually needed. In the process of multi-parameter fitting, a mean square variance commonly used in many studies^7,14,18,19

is minimized via a least-squares procedure. Here, A_T(ω_i), A_E(ω_i), Φ_T(ω_i), and Φ_E(ω_i) are the MFCA amplitude and phase as a function of the modulation frequency of the pump beam. The subscripts T and E denote the calculated and experimental values, respectively. N is the total number of data points.

The initial values of the free parameters are usually selected according to the existing research findings before fitting. However, when the number of the free parameters increases, the different initial values may result in different fitted results. Table I shows the influence of initial values on the fitted results when all four parameters mentioned above are free parameters to be fitted. There are five groups of results fitted by using different initial values. Considering the very low noise in our experiments shown in Fig. 4, unless explicitly stated otherwise, the measurement errors of the simulation amplitude and phase are assumed to obey a normal distribution with a standard deviation of ±0.1%. As can be seen from Table I, using the true value as the initial value will result in the smallest mean square (2.16 × 10⁻⁶) and the most accuracy fitted results. The error (<3%) mainly comes from the noise. However, if the initial value deviates from the true value, the error will increase obviously, indicating the fitted results are significantly affected by the initial value. Note that even there are great differences between the fitted results and the true values, the mean square variance has little change. In other words, the ability to unambiguously determine the four transport parameters mentioned above is limited due to lack of uniqueness. In addition, the error of the rear surface recombination velocity is largest due to the low sensitivity coefficient, as shown in Fig. 1.

To solve the above-mentioned problems, researchers usually change the initial value many times and then select the results with the smallest mean variance as the best-fit results. This method can improve the accuracy of the measurement results in a certain extent. However, the combination of initial values is many, and it is still difficult to determine whether the results selected are the best-fit results, which results in the low accuracy of the fitted parameters. As we all know, the uniqueness of fitted results is related to the number of free parameters, and it provides us an access to improve the uniqueness by reducing the number of free parameters. Here, S₂ is first set to its true value, and τ, D, and S₁ are free parameters. The fitted results are shown in Table II. Although the selected initial values are different, most of the fitted results are close to the real values with low errors. Only when the initial value of the lifetime deviates seriously from the true value, the fitted results have large errors.

The fitted results with only τ and S₁ as the free parameters have also been calculated and shown in Table III. Here, D and S₂ are also set to the true value. The results show that when the number of free parameters reduces to two, i.e., τ and S₁, the fitted results almost do not depend on the initial value in a large range and are well consistent with the true values. Therefore, the accuracy of the fitted parameters is greatly improved.

From the simulation results, we see that for all three cases mentioned above and both here and in the supplementary material, the fewer the free parameters, the less the influence of the initial values on the fitted results, and the better the uniqueness of the parameter estimates. However, reducing the number of the free parameters usually means that the other parameters need to be accurately known in advance, which has been used in some studies.^7,18 However, this method may increase the complexity of the measurement process and decrease the accuracy of the measurement results finally. Taking the fit of τ and S₁ of the long lifetime sample as an example, if there are 10% errors in D and S₂, the fitted results will become τ = 104.00 µs and S₁ = 9.10 m/s, respectively, which is much greater than the effect of the noise added shown in Table III.

As discussed in Sec. III B, the determination of S₂ and D by independent experiments may increase the complexity of the experimental measurement and the uncertainty of the measurement results. To solve this problem, a method based on the mean square variance graph or map is proposed. In this method, only two or three parameters are set to the free parameters, and the other parameters are scanned to seek the minimum mean variance, by which all four transport parameters can be extracted without other independent experiments. Due to only two or three free parameters, the uniqueness of the fitted parameters is improved, as shown in Table III. The scanning range of S₂ and D can be given approximately according to the literature. For crystalline silicon wafers, the surface recombination velocity is usually below 10³ m/s, which depends on the surface passivation quality. Recently, the surface recombination velocity <1 cm/s has been achieved by using advanced passivation processes.²⁴ The diffusion coefficient is generally lower than 12.5 cm²/s in n-type silicon and less than 35 cm²/s in p-type silicon.²⁵ Taking the n-type silicon wafer as an example, the scanning ranges of S₂ and D are set to 0.01–10³ m/s and 5–15 cm²/s, respectively.

Figure 2 shows the simulated mean square variance when the different S₂ and D values are used. When one of the parameters is scanned, the other three parameters are set as free parameters. Figure 2(a) shows the mean square variance during scanning S₂. For comparison, the fitting results with and without noise are given at the same time. When the rear surface recombination velocity is equal to its true value, S₂ = 10 m/s, both the variance and the errors of the fitted parameters are the lowest. Considering the effect of the noise, the smallest mean square variance increases to 2.13 × 10⁻⁶ with S₂ = 9.0 m/s, τ = 95.22 µs, D = 13.1 cm²/s, and S₁ = 9.41 m/s. As the S₂ value gradually deviates from the true value, the mean square variance increases gradually. Figure 2(b) further analyzes the dependence of the mean square variance on the diffusion coefficient. The fitted results of the transport parameters are D = 13.1 cm²/s, τ = 95.35 µs, S₂ = 9.01 m/s, and S₁ = 9.42 m/s, which are similar to the results of scanning S₂.

According to the results in Table II, reducing the free parameter can improve the accuracy of the fitting results. However, because the true value of each parameter in the actual measurement is unknown, if the initial value is deviated significantly from the true value, the fitted results still have the probability of falling into the local optimum. When the number of free parameters is reduced to two, the fitted results are nearly not affected by the initial values; therefore, the accuracy is greatly improved, as shown in Table III. Figure 3 further shows the mean square variance with scanning D and S₂ simultaneously for the long lifetime sample. It can be clearly seen that the minimum mean variance occurs near the position with the true parameter values. As D and S₂ deviate from the true values, the mean square variance increases. Figure 3(b) also gives the mean square variance map with the noise. As we expect, the variance increases due to the noise. The best-fit values of D and S₂ at the minimum variance are 13 cm²/s and 9.12 m/s, respectively, which deviate slightly from the true values. The other parameters and mean square variance are τ = 96.09 µs, S₁ = 9.52 m/s, and var = 2.13 × 10⁻⁶, which are consistent with the results shown in Fig. 2. In the area surrounded by the red curve, the variance is smaller than 1.01 × var_min. Similar phenomena also appear in the other two cases, and the corresponding results are given in the supplementary material.

To investigate the influence of different noise levels on the simultaneous determination of the transport parameters by the proposed method, the simulated data with different measurement noise levels are used to extract the transport parameters via multi-parameter fitting. The fitting procedure is repeated 20 times, and the fitted results are statistically analyzed for each fitted parameter. As an example, Table IV shows the statistics of the fitted results for the long lifetime sample. It can be seen that the mean square variances and the uncertainties for each fitted parameter increase with increasing noise level, indicating that reducing the experimental errors is conducive for further improving the measurement accuracy of the transport parameters.

An experiment was performed to determine the electronic transport parameters of a silicon wafer. The MFCA experimental setup has been described previously.^18,19 In brief, photo-excitation was provided by a periodically intensity-modulated 660 nm semiconductor laser with an incident power of ∼50 mW. The photo-generated excess carriers are monitored by a 1560 nm probe laser beam. The transmitted probe beam is detected with an InGaAs photodetector with the spectral response range from 700 to 1800 nm. A lock-in amplifier was used for the MFCA amplitude and phase measurements. The sample used in the experiment is a (100)-oriented n-type c-Si wafer with 1–10 Ω⋅cm resistivity, 500 ± 10 μm thickness, and a polished front surface.

Figure 4 shows the amplitude and phase of the MFCA signals in the modulation frequency range of 0.6–50 kHz. The fitted results by using the traditional and proposed data processing methods are also given. The experimental data and all theoretical fits agree well at all frequencies for both amplitude and phase. However, the fitted results by different methods have a great difference, as shown in Table IV. When the four transport parameters are fitted simultaneously, the best-fit results depend on the initial values significantly despite the similar variance, which is consistent with the theoretical expectations shown in Table I.

Similar to the simulations, Fig. 5 shows the mean square variances with S₂ and D being scanned separately. The minimum mean square variance is 5.74 × 10⁻⁵ with S₂ = 15.4 m/s and D = 11.2 cm²/s regardless of which parameter is scanned. Other parameters are estimated to be as follows: τ = 96.02 µs, S₁ = 24.80 m/s, and S₂ = 15.43 m/s in the former and τ = 95.97 µs, D = 11.2 cm²/s, and S₁ = 24.78 m/s in the latter, respectively. There is very little difference between these results. If both D and S₂ are scanned simultaneously, a mean square variance map can be obtained, as shown in Fig. 6. The minimum mean square variance is 5.74 × 10⁻⁵ with D = 11.3 cm²/s and S₂ = 15.14 m/s correspondingly. The best-fit results of the τ and S₁ are 96.70 µs and 24.64 m/s, respectively. In addition, the mean square variance does not change significantly in a certain range, especially near the best-fit results. Combined with the simulation results in Fig. 3, we believe that the experimental noise caused by the stability of the measurement instruments and the environments may be main causes. Of course, the influence of the reflectivity change of the probe laser should also not be ignored. However, these influences can be reduced by using low noise instruments and simultaneously measuring the reflective and transmission signals.

Table V lists the results of the carrier transport parameters obtained by the traditional method and the proposed method. It can be seen that when different initial values are used in the traditional method, the fitted results have great discrepancies. Even the fitted parameters are may be close to that by using the proposed method if the suitable initial values are sometimes chosen, such as the results with the mean square variance 5.96 × 10⁻⁵; it is in fact difficult to confirm that the minimum mean square variance has been obtained unless the more initial values are used, so the reliability of the fitted results is doubtful. On the contrary, the proposed method is less affected by the initial values, and the uniqueness of the fitted results can be improved. In addition, the nearly same electronic transport parameters are obtained no matter which parameter is scanned by the proposed method.

Although the proposed method can improve the uniqueness of the fitted parameters, some limitations should also be noticed. The scanning step of the parameter should be as small as possible if we want to get more accurate results, but it is time-consuming. The calculation time can be reduced by using a large step in the large parameter range and a small step in the parameter range of interest. In addition, if there are more parameters to be fitted for samples, such as ion implantation or annealing wafers,^26,27 the improvement in the uniqueness of the parameters by the proposed method may be limited.

In this article, we first discuss the uniqueness of the carrier transport parameter estimates in the traditional MFCA measurements. The results show that when the number of parameters to be fitted is large, the fitted results are greatly affected by the initial values. With the decrease in the number of free parameters, the influence decreases gradually. Based on this phenomenon, a mean square variance graph or map method to improve the uniqueness of the parameter estimates is then proposed by scanning one or two of the parameters and fitting the remaining parameters. Any one of the four carrier transport parameters in the method does not need to be measured independently by other methods in advance. Both the simulation and experimental results confirm the feasibility of the proposed data processing method to improve the uniqueness of the parameter estimates in the MFCA measurements, which is also expected to be applied to other measurement techniques such as photocarrier radiometry, photothermal radiometry, and modulated optical reflection.

See the supplementary material to access the simulation results of the other two cases.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61704132 and 61905187), the National Natural Science Foundation in Shaanxi Province of China (Grant No. 2020JQ-815), the Scientific Research Program Funded by Shaanxi Provincial Education Department (Grant No. 20JS059), and Xi’an Technological University Foundation Key Projects (Grant No. XGPY200206).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

$ΔÑξ,ω,λ$ in Eq. (1) can be obtained by solving the three-dimensional carrier transport equation with the boundary conditions in Hankel space,⁸ 

Here, α is the absorption coefficient of the sample to the pump beam. η is the quantum conversion efficiency. P is the incident light power. h is the Planck constant. c is the speed of light in the vacuum. R is the reflectivity of the sample surface to the pump light,