A method and device for comparing secondary ion mass spectrometry analysis curves

Abstract

The present invention provides a method and device for comparing secondary ion mass spectrometry analysis curves, which are used to diagnose whether there is an abnormality in ion implantation, and include the following steps: S1, finely adjust the machine platform; S2, in the first decision-making stage, the machine platform performs self-check of the ion implantation dose deviation. If the ion implantation dose deviation is less than 1%, go to step S3. If the ion implantation dose difference is greater than or equal to 1%, go to step S1; S3, perform abnormal point detection and elimination; S4, perform curve smoothing processing and model diagnosis; S5, calculate relevant non-parametric quantification indexes and compare the secondary ion mass spectrometry analysis curves according to these indexes; S6, if the indexes meet the threshold, the ion implantation for this time is normal; if the threshold is not met, the ion implantation for this time is abnormal, and go to step S1. It solves the problems in the prior art that the comparison parameters of secondary ion mass spectrometry analysis curves are too subjective and the analysis of ion implantation results is incomplete, effectively realizes early detection and guides practitioners to formulate solutions in time, and reduces the influence of deviations.

Description

Technical Field

[0001] The present invention relates to the field of semiconductor manufacturing, and particularly to a method and apparatus for comparing secondary ion mass spectrometry analysis curves. Background Art

[0002] In recent years, with the rapid growth of the demand for sub-nanometer depth resolution and higher sensitivity, secondary ion mass spectrometry (SIMS) has received increasing attention. SIMS bombards the surface of a sample with a high-energy primary ion beam, causing the sample surface to absorb energy and generate secondary ions. By collecting and analyzing these secondary ions with a mass analyzer, a spectrum of information about the sample surface can be obtained. Compared with other surface analysis techniques, such as scanning electron microscopy (SEM), energy dispersive spectroscopy (EDS), Auger electron spectroscopy (AES), and X-ray photoelectron spectroscopy (XPS), its sensitivity is very high, ranging from parts per million (PPM) to parts per billion (PPB). In addition, SIMS also has some special advantages, especially in depth and imaging resolution. These capabilities enable the analysis of all elements and isotopes, including hydrogen, as well as the analysis of compound components and molecular structures, which cannot be detected by SEM and XPS. Figure 1 FIG. 1 is a schematic diagram of the working principle of SIMS. A primary ion beam with an energy range of 250 eV to 30 keV bombards the sample surface and generates ionized secondary ions, which are then accelerated by an electric field and analyzed by a mass spectrometer.

[0003] For reliability assessment and semiconductor process control, the quantitative analysis of impurities is of crucial importance. From the perspective of reliability assessment, accurate SIMS profiling and comparison are essential to ensure consistent reliability performance. For example, in the release of new devices and process transfers, sufficient redundancy is crucial. Credible early assessment for these scenarios relies on the robust analysis and comparison of SIMS depth profiles (SDPs). If a non-conforming SDP is not identified after ion implantation (IMP), it may take until the late (possibly one month later) wafer acceptance test (WAT) and / or subsequent chip probing (CP) / final test (FT) to detect these negative impacts. Obviously, it is quite bad to detect non-conforming products only until the reliability test. On the other hand, this also means that the in-line, WAT, CP, FT, and reliability performance can comprehensively reflect the SDP performance, and these results must be relied on to further verify and optimize the SDP benchmark.

[0004] From the perspective of semiconductor process control, SIMS plays an important role in surface analysis. For example, it supports process development, monitoring, and troubleshooting. In the microelectronics field, samples are usually flat, and substances are usually of low density and can only be detected by SIMS. Device formation starts from IMP or dopant diffusion on the wafer to a specific pattern defined by a mask. SIMS can determine the junction depth by obtaining a dopant profile that matches the dopant atom density of the substrate, which is particularly important for the junction depth (<10 nm) of current ultra-shallow implant devices. In all of the above applications, based on the ion dose and distribution, the SDP is compared with the theoretical or benchmark profile for determination.

[0005] For Figure 2 the SDP, the "Y" axis represents the concentration (usually represented on a logarithmic scale) and the "X" axis represents the depth, which can intuitively reflect the change of ion concentration with the sample depth. So when the ion implanter injects ions into the split, and then by simultaneously analyzing and comparing the split with the blank (BL). Standards with known depth, concentration, or dose created by IMP are used for quantitative calibration of each sample. SIMS has evolved from a semi-quantitative method to a dose measurement technique, with a reproducibility of relative standard deviation of impurity and matrix element measurements <1% for quadrupole, magnetic sector, and time-of-flight analyzers.

[0006] The application of SIMS in depth profiling has become mainstream in the semiconductor industry. Generally, the SDP of amorphous layers follows a Gaussian distribution. However, there are several different interactions between the implanted ions and the wafer surface. Some ions have enough energy so that almost all of them are implanted into the wafer; or they are reflected by the wafer when the energy is very weak. Channeling effect refers to the situation where ions move parallel to the crystal axis or plane and show an abnormally deep penetration into the crystal lattice. Channeling effect needs to be avoided as much as possible during the IMP process because it causes the implanted ions to get out of control and deviate from the Gaussian distribution. In this case, although the energy is precisely set, a small angular deviation will make a significant difference. Figure 2 It reveals that there are significant differences in SDP for two different implantation angles, 0° and 1°. Quantification at the interface (shallow depth) and in the bulk state (especially at deeper depths) may be difficult. At depths less than 200 nm, due to particle interference caused by contamination, the SDP with low mass resolution is unreliable. However, at depths greater than 800 nm, the detection ability of SDP is restricted by the detection limit.

[0007] IMP is a low-temperature process that introduces precisely controlled impurities into semiconductors, which provides more flexibility than the diffusion process. Ion implanters are usually divided into high current (HC), medium current (MC), high energy (HE), and low energy (LE). The typical performance indicators of beamline ion implanters include uniformity, repeatability, reliability, in-situ angle control for precise pure doping placement, high throughput, and contamination control. Each implantation can be achieved by controlling the beam current, implantation energy, angle, and doping material of the IMP process, all of which are crucial for the performance and reliability of the device. In most cases, ions will be implanted onto the wafer at a certain tilt angle to reduce the influence of channeling effect. The common practice is to implant at a 7° tilt angle to avoid doping channels, although due to the shadow of the photoresist, a 7° tilt may not be the best. By taking advantage of channeling, a 0° tilt can be used in association with a lower energy to achieve the target depth and reduce damage compared to tilted implantation. Taking the HE implanter as an example, due to the conical angle effect, batch tools may cause differential channeling, which may lead to large variations in the doping distribution on the wafer, especially for 0° implantation. On the other hand, single-wafer HE implanters are not affected by this conical angle effect and can provide uniform 0° implantation. These characteristics of single-wafer implanters are particularly beneficial for CMOS image sensor (CIS) applications, which require deep implantation and have more stringent requirements for wafer uniformity. In summary, a slight angular deviation will have a significant impact on SDP, and the implantation angle is a key factor in the IMP process.

[0008] In addition to the injection energy and angle mentioned above, the degree of equipment degradation also seriously affects SDP. For example, the wear and degradation of the beamline part cause the hardware to gradually deviate from the original settings. Therefore, regularly verifying the implantation angle and the accuracy of the ion gauge can minimize process variations. Usually, measuring the thermal-wave (TW) signal and the sheet resistances (Rs) are used to monitor the deviation of the implantation angle relative to the crystal axis. However, TW has a sensitivity curve, which makes it difficult to use under certain injection conditions. To effectively use this tool, a calibration table for the operation of specific process settings must be adopted. Even at the same injection angle, the Rs values between two specific ion implanters may vary significantly. Theoretically, any IMP anomaly can be found by comparing the SIMS results with a benchmark.

[0009] For a long time, SDP comparison has mainly relied on the subjective evaluation of practitioners, without any meaningful metrics based on statistics or physics. As Figure 2 shown, a 1° angle difference results in significantly different SDPs. Further in Figure 3a , according to the comments of practitioners, the SDP of the sample for identification ("Split") matches well with the reference sample ("Baseline"). In Figure 3b , the IMP dose change of the sample to be identified is within the limit range, but the SDP match is worse than that in Figure 3a ; in this case, it is difficult to judge whether the SDP in Figure 3b is acceptable. In Figure 2 , there is a significant difference between the SDPs of the sample to be measured ("Split") and the reference sample ("Baseline"). Therefore, it is relatively easy to visually identify ion implanter anomalies from the SDP in Figure 2 .

[0010] Because of this, there is clearly a need for a quantifiable and more meaningful metric to compare SDP. To obtain a relatively reliable and accurate judgment for comparing two or more SDPs, great efforts have been made by predecessors. Document [1] (CN104237279A) proposed a judgment method for calculating the deviation rate to compare SDP. It uses the Gaussian smoothing method to reduce the signal-to-noise ratio (S / N), and then performs point-to-point normalization, as Figure 4 shown.

[0011] Then this method excludes the noise area, divides the remaining area into three equal parts, and calculates their respective slopes. Finally, the largest of the absolute values of these slopes is designated as the deviation rate. Although the deviation rate method is straightforward, it has some doubts and is too subjective in some parameter settings. Specifically, it is described as follows:

[0012] (1) Curve smoothing is quite important and is also the basis for subsequent analysis. However, there is a lack of thorough verification of curve smoothing in the deviation rate method, and even outliers are not excluded. Subsequently, the normalized curve inevitably jumps up and down, especially at shallower and deeper depths.

[0013] (2) Since the function of SDP has not been accurately obtained, the authors of the deviation rate method have to choose the so-called point-to-point normalization. However, it depends on the sampling frequency of SDP, and a lot of information will be lost.

[0014] (3) The process of obtaining the deviation rate value is subjective and unscientific. On the one hand, the normalized curve does not necessarily conform to a linear function. In addition, if two SDPs are parallel and there are significant differences, the deviation rate method does not work.

[0015] (4) For various different technical platforms, the deviation rate threshold should actually be different rather than a fixed value of 1.4.

[0016] (5) The number of segments into which the acceptable region is divided also lacks theoretical support. In addition, the determination of the acceptable region (covering 95% of the effective dose) also needs to be further discussed in depth.

[0017] (6) Finally and most importantly, practical experience shows that the deviation rate method may lead to incorrect conclusions. For example, Figure 3b for the SDP in Figure 3b , its deviation rate = 1.35 < 1.4. That is to say, through the deviation rate method, this SDP is considered comparable to the Baseline (because its deviation rate < 1.4). However, the devices from this split ( Figure 5 the SDP in Summary of the Invention

[0018] The object of the present invention is to provide a method and device for comparing secondary ion mass spectrometry analysis curves for diagnosing whether there are abnormalities in ion implantation, so as to solve the problems in the prior art that the comparison parameters of secondary ion mass spectrometry analysis curves are too subjective and the analysis of ion implantation results is not comprehensive, effectively realizing early detection and guiding practitioners to formulate solutions in a timely manner and reducing the influence of deviations.

[0019] The present invention provides a method for comparing secondary ion mass spectrometry analysis curves for diagnosing whether there are abnormalities in ion implantation, including the following steps:

[0020] S1, Fine-tuning machine

[0021] S2, The first decision-making stage. The machine performs self-check on the ion implantation dose deviation. If the ion implantation dose deviation is less than 1%, go to step S3. If the ion implantation dose difference is greater than or equal to 1%, go to step S1;

[0022] S3, Perform outlier detection and rejection;

[0023] S4, Perform curve smoothing and model diagnosis;

[0024] S5, Calculate relevant non-parametric quantitative indicators and compare the secondary ion mass spectrometry curves according to these indicators;

[0025] S6, If the said indicator meets the threshold, the ion implantation for this time is normal; if it does not meet the threshold, the ion implantation for this time is abnormal, go to step S1.

[0026] Optionally, the non-parametric quantitative indicator is "percentage difference D", and its calculation formula is where G A (x) and G B (x) are two given SDP curve smoothing functions, a represents the shallow region depth value, and b represents the deep region depth value.

[0027] Optionally, the threshold is defined as 0.276%, that is, when D <= 0.276%, the SDP curves of the sample to be measured and the standard sample are comparable, and the injection for this time is normal. When D > 0.276%, there are significant differences between the SDP curves of the sample to be measured and the standard sample, and the injection for this time is abnormal.

[0028] Optionally, the non-parametric quantitative indicator is the Kolmogorov-Smirnov K-S statistic DKS, and its formula is D KS = max|F(x) - G(x)|, where F(x) and G(x) represent the empirical cumulative distribution functions of two SDP curves.

[0029] Optionally, the method is not restricted by the sample distribution and the size of the overall sample size; when the sample dimension is greater than 1, the maximum absolute difference between the empirical cumulative distribution functions of the sample in all quadrants can be found under any possible sorting.

[0030] Optionally, when the dimension of the sample is 2, n sample points are given in the two-dimensional space, and 4n of the planes defined by all pairs (Xi, Yj) are calculated 2The empirical cumulative distribution function in the quadrant is calculated, and the maximum absolute difference between the empirical distribution functions in all quadrants is calculated, where Xi and Yj are the coordinates of any point pair in the given sample. The counting step can be performed by a brute-force algorithm that sweeps through each quadrant for each point in the sample to determine whether the point is within it.

[0031] Optionally, if the significance level a = 0.05, the threshold is defined as 0.09. That is, when DKS <= 0.09, the SDP curves of the sample to be tested and the standard sample are comparable, and the ion implantation is normal this time; when D > 0.09, there are significant differences between the SDP curves of the sample to be tested and the standard sample, and the implantation is abnormal this time.

[0032] Optionally, the outlier detection can adopt the method of non-linear regression diagnosis, including finding the conditional probability of another random variable Y on an independent variable X and using backward search to distinguish outliers; first, a regression model is constructed using all the data, and then the observations with the largest errors are continuously or simultaneously excluded from the model. When the model is assumed to be correct, the true residuals always have a standard normal distribution.

[0033] Optionally, the curve smoothing process can adopt the neural network method. The form of the neural network method is a multi-layer perceptron, and model diagnosis is performed. The model diagnosis can adopt the residual worm plot. When all observations fall within the "accepted" region between two elliptical curves, that is, approximately 95% of the points fall between the two elliptical curves and no specific shape is detected in the observations, it is determined that the overall fit of the model is good.

[0034] Optionally, when the number of past data samples is less than 30, the non-parametric quantization index uses the K-S statistic DKS.

[0035] Optionally, the outlier detection can also adopt the box plot method, the density-distance-based method or the projection pursuit-based method.

[0036] The present invention provides a device for comparing secondary ion mass spectrometry analysis curves, which is used to diagnose whether there is an abnormality in ion implantation, including:

[0037] A machine tool fine-tuning unit for fine-tuning the machine tool;

[0038] A decision-making unit for the machine tool to perform self-check on the ion implantation dose deviation. If the ion implantation dose deviation is less than 1%, the machine tool is fine-tuned again. If the ion implantation dose difference is greater than or equal to 1%, the next step is entered;

[0039] An outlier detection unit for performing outlier detection and elimination;

[0040] A curve smoothing unit for performing curve smoothing processing and model diagnosis;

[0041] A quantization index calculation unit for calculating relevant non-parametric quantization indexes;

[0042] A comparison unit for comparing secondary ion mass spectrometry analysis curves according to the indexes calculated by the quantization index calculation unit. If the indexes meet the threshold, the current ion implantation is normal; if they do not meet the threshold, the current ion implantation is abnormal, and the fine-tuning machine is restarted.

[0043] Optionally, the non-parametric quantization index is "percentage difference D", and its calculation formula is where G A (x) and G B (x) are two given SDP curve smoothing functions, a represents the shallow region depth value, and b represents the deep region depth value.

[0044] Optionally, the threshold is defined as 0.276%, that is, when D <= 0.276%, the SDP curves of the sample to be measured and the standard sample are comparable, and the current implantation is normal; when D > 0.276%, there are significant differences between the SDP curves of the sample to be measured and the standard sample, and the current implantation is abnormal.

[0045] Optionally, the non-parametric quantization index is the Kolmogorov-Smirnov K-S statistic DKS, and its formula is D KS = max|F(x) - G(x)|, where F(x) and G(x) represent the empirical cumulative distribution functions of two SDP curves.

[0046] Optionally, the device is not limited by the sample distribution and the size of the overall sample size; when the sample dimension is greater than 1, the maximum absolute difference between the empirical cumulative distribution functions of the sample in all quadrants can be found under any possible sorting. Optionally, when the dimension of the sample is 2, n sample points are given in the two-dimensional space, the empirical cumulative distribution functions in the 4n2 quadrants of the plane defined by all pairs (Xi, Yj) are calculated, and the maximum absolute difference between the empirical distribution functions in all quadrants is calculated, where Xi and Yj are the coordinates of any pair of points in the given sample, and the counting step can be performed by a brute-force algorithm, which sweeps through each quadrant for each point in the sample to determine whether the point is in it.

[0047] Optionally, if the significance level a = 0.05, the threshold is defined as 0.09, that is, when DKS <= 0.09, the SDP curves of the sample to be measured and the standard sample are comparable, and the current ion implantation is normal; when D > 0.09, there are significant differences between the SDP curves of the sample to be measured and the standard sample, and the current implantation is abnormal.

[0048] Optionally, the outlier detection may adopt the method of non-linear regression diagnosis, including finding the conditional probability of another random variable Y on an independent variable X and using reverse search to distinguish outliers; first, a regression model is constructed using all the data, and then, the observations with the largest errors are excluded from the model continuously or simultaneously. When the model is assumed to be correct, the true residuals always have a standard normal distribution.

[0049] Optionally, the curve smoothing process may adopt the neural network method. The form of the neural network method is a multi-layer perceptron, and model diagnosis is performed. The model diagnosis may adopt the residual worm diagram. When all the observations fall within the "accepted" region between two elliptical curves, that is, approximately 95% of the points fall between the two elliptical curves and no specific shape is detected in the observations, it is determined that the overall fitting of the model is good.

[0050] Optionally, when the sample size of past data is less than 30, the non-parametric quantization index uses the K-S statistic D KS 。

[0051] Optionally, the outlier detection may also adopt the box plot method, the density-distance-based method or the projection pursuit-based method. Brief Description of the Drawings

[0052] Figure 1 is a schematic diagram of the working principle of SIMS;

[0053] Figure 2 is a typical SDP curve graph based on different injection angles (identification sample: 1°, reference sample: 0°);

[0054] Figure 3(a) is an SDP curve graph with good matching between the identification sample and the reference sample, and Figure 3(b) is an SDP curve graph with poor matching between the identification sample and the reference sample;

[0055] Figure 4 is a division graph of the regional coordinates in the K method;

[0056] Figure 5 is a leakage current degradation graph at saturation of two groups of samples in Figure 3(b);

[0057] Figure 6 is a flowchart of a comparison method for secondary ion mass spectrometry analysis curves provided by the present invention;

[0058] Figure 7 is an outlier detection graph based on the box plot method;

[0059] Figure 8 is an outlier detection graph based on the density-distance method;

[0060] Figure 9Graph for outlier detection based on the projection pursuit method;

[0061] Figure 10 Box plot of residuals before and after removing outliers;

[0062] Figure 11 Graph for outlier detection based on the regression diagnosis method;

[0063] Figure 12 Graph for comparing SDP curves using different smoothing methods;

[0064] Figure 13 Graph for visual expression of the neural network model;

[0065] Figure 14 SDP curve graph smoothed based on the neural network method;

[0066] Figure 15 Residual graph of the SDP curve of the reference sample in Fig. 3(a);

[0067] Figure 16 Worm graph of the residuals of the SDP curve of the reference sample in Fig. 3(a);

[0068] Figure 17 Graph for comparing SDP curves based on the NRC method; Specific implementation mode

[0069] The following further elaborates on the method and device for comparing secondary ion mass spectrometry analysis curves provided by the present invention in conjunction with the accompanying drawings and specific embodiments.

[0070] First, please refer to Figure 6 , a method for comparing secondary ion mass spectrometry analysis curves provided by the present invention includes:

[0071] S1, fine-tune the machine tool;

[0072] S2, the first decision-making stage, the machine tool performs self-checking on the ion implantation dose deviation. If the ion implantation dose deviation is less than 1%, go to step S3; if the ion implantation dose difference is greater than or equal to 1%, go to step S1;

[0073] S3, perform outlier detection and elimination;

[0074] S4, perform curve smoothing processing and model diagnosis;

[0075] S5, calculate relevant non-parametric quantification indicators and compare the secondary ion mass spectrometry analysis curves according to these indicators;

[0076] S6, if the said indicator meets the threshold, the ion implantation for this time is normal; if it does not meet the threshold, the ion implantation for this time is abnormal, and go to step S1.

[0077] The following combines Figures 7 - 17 to elaborate in detail on the above steps of the present invention.

[0078] A. Outlier detection.

[0079] SDP provides compositional and depth information. However, the identification of the interface (shallow layer) can be difficult. On the one hand, the interface is usually very thin and usually marks the proximity of different materials, which may have relatively different secondary ion yields. On the other hand, surface contamination can be a serious problem because particles on the original surface can also cause secondary ion signals. The secondary ion signals may be deeply embedded in the analyzed sample and mask the true impurity distribution. In statistics, these spurious signals that obscure the true impurity distribution are regarded as. Statistically, an outlier is an observation that is so discrepant from other observations that it raises suspicions that it was generated by a different mechanism. Outliers can seriously mislead predictions and make inferences from the global dataset less precise. Therefore, identifying and removing outliers from true observations is an important and necessary task in statistical analysis.

[0080] Most classical methods for outlier detection are based on some statistical distribution, and outliers are identified as those points with a low probability of occurrence. However, the distribution of SDP is not yet clear. Taking the "Split" SDP in Figure 3b as an example. Four different non-parametric methods are introduced to accurately identify outliers.

[0081] 1) Tukey method (box plot).

[0082] Although outliers are relatively rare and uncommon, their importance is very high compared to other observations, making their detection very crucial. In practical applications, it is easy to detect outliers in univariate samples by simply using some appropriate graphs for visualization, such as box plots. A box plot is a graphical display used to depict a set of numerical data through quartiles without making any assumptions about the underlying statistical distribution. The width of the box, defined as the interquartile range (IQR), is equal to the third quartile (Q3, 3 rd quartile) minus the first quartile (Q1, 1 st quartile). The IQR represents the dispersion and skewness in the data. If a point is below Q1 - 1.5×IQR or above Q3 + 1.5×IQR, it is identified as an outlier.

[0083] Similarly, for bivariate data such as SDP, a simple approach is to set reasonable interval breaks and then convert the variable type of depth from a continuous variable to a categorical variable. Outliers at each categorical level are marked as points outside the whiskers of the box plot. For Figure 3b the "Split" SDP in Figure 7 , the variable type of depth has been changed to a categorical variable and the corresponding intervals of depth are set to 10. As

[0084] shown, a series of box plots are drawn. Clearly, almost all outliers (represented by large dots) occur at shallow and deep depths, corresponding to the interface bulk profile. These findings are consistent with the physical explanations elaborated earlier.

[0085] 2) Density-Distance (DD)-Based Method.

[0086] Although outliers can be easily detected through box plots, the selection of interval breaks is subjective. Therefore, a density-distance (DD)-based method is considered to avoid this subjective selection. This method requires calculating the distances between data observations with geometric interpretations, and moreover, it is specifically developed for multivariate outlier detection. This method also does not need to consider the data distribution pattern, which is also a key advantage of it. The classical method for detecting outliers is to calculate the distances between each data point and its neighbors, and then those points with significantly lower density than their neighbors are assumed to be outliers. The Local Outlier Factor (LOF) is based on the concept of local density; the location is given by the k nearest neighbors, and their distances are used to estimate the density. By comparing the local density of an object with the local density of its neighbors, regions with similar densities and those points with much lower density than their neighbors can be identified. These points are considered outliers. To determine the range of the k nearest neighbors, the k value used in calculating LOF is set to 7. The outliers identified by the LOF algorithm are marked as boxes in Figure 8 .

[0087] 3) Projection Pursuit (PP)-Based Method.

[0088] The computational overhead of the above LOF cannot be ignored because it needs to calculate all the distances between data points. In addition, although LOF makes no assumptions about the distribution, it cannot be considered a purely non-parametric method because its results are very sensitive to the choice of k, which still needs to be specified (empirically k = 7). Projection pursuit (PP) is a statistical technique used to find linear combinations of variables so that the corresponding data can exhibit clustering. Its framework is formulated as an optimization problem whose objective is to find the projection index. The outliers identified by the PP algorithm are marked as boxes in Figure 9 as shown below.

[0089] 4) Methods based on regression diagnostics.

[0090] Regression analysis involves finding the conditional probability of another random variable Y on an independent variable X. Reverse search is used to distinguish outliers. First, a regression model is constructed using all the data. Then, the observations with the largest errors are excluded from the model either sequentially or simultaneously. When the model is assumed to be correct, the true residuals always have a standard normal distribution. Figure 10 The quantile-quantile (QQ) normal plots of the residuals before (left) and after (right) removing the outliers are shown below.

[0091] Obviously, the normal QQ plot is approximately linear (intercept 0 and gradient 1) after removing the outliers; the outliers are marked as boxes in Figure 11 as shown below.

[0092] Although detecting outliers using box plots is simple and straightforward, the choice of the interval is subjective and uncertain. LOF is an unsupervised outlier detection method that calculates the local density deviation of a given data point relative to its neighbors without making any assumptions about the distribution. However, it is very sensitive to the user-defined parameter k, the number of neighbors. PP is a method for finding linear combinations of variables so that the data can provide clusters related to these variables; however, it cannot model complex generative models and is highly computationally intensive. Based on this, it is recommended to use non-linear regression diagnostics to detect outliers.

[0093] B. Curve smoothing: Non-linear regression.

[0094] Accurately estimating the potential function of SDP is the most crucial and fundamental issue for reliable quantitative analysis. Usually, a parametric form is proposed based on prior knowledge. However, when these strict parametric assumptions are inaccurate, they may lead to biased estimates. Due to the strong flexibility of non-parametric smoothing methods, several non-parametric methods were compared to obtain the best characteristic function of SDP. These non-parametric smoothing models are Cubic Spline Interpolation (CSI), locally weighted scatterplot smoothing (LOWESS), and NNs; their smoothing results are as Figure 12 shown.

[0095] NNs provide a flexible fitting method for non-linear regression models. They are over-parametrized non-linear statistical models, which makes them very flexible and thus can approximate any smoothing function. They can find high-order interactions in the explanatory variables, which is very difficult for traditional regression methods. As Figure 12 shown, the characteristics of SDP (especially in the shallow part) can only be fully characterized by NNs. Due to the over-parametrization of NNs, they are difficult to interpret compared to more traditional smoothing methods.

[0096] The most common form of NNs is the multi-layer perceptron (MLP), which consists of an input, an output, and possibly one or more hidden layers. The input units pass their inputs to the units in the first hidden layer or directly to the output units. Each unit in the hidden layer adds a constant (called "bias") to the weighted sum of its inputs and calculates the result of the activation function φ h . Then it is passed to the hidden units or output units in the next layer.

[0097] Empirical studies have shown that compared with NNs with one or two hidden layers, multi-layer networks usually do not perform better and often perform worse. Therefore, the method provided by the present invention starts from the simplest but most common NN structure, which only contains one hidden layer (to avoid overfitting in univariate regression) and 10 nodes. As Figure 13 , where the thicker lines indicate that the coefficients have larger values. In Figure 13 , I1 and O1 are the input and output terms respectively; H1 to H10 are the hidden nodes. The activation functions are set as the hyperbolic tangent function in the hidden layer and the exponential function in the output layer. The input is represented as x i , for an MLP with one hidden layer, its output t k can be obtained from formula (1):

[0098] t k = φ0(αk +∑ j→k w jk φ h (α j +∑ i→j w ij x i )) (1).

[0099] Among them, α k represents the k non-linear neurons, and w jk represents the weight delay.

[0100] If there is only one output node, then k is equal to 1. The weights can be determined by optimizing some appropriate criterion functions, for example, minimizing the sum of the squared errors of the predictor variables or maximizing the log-likelihood of the data under the assumption that the distribution of the response variable can be assumed.

[0101] The structure of the MLP enables it to fit very general non-linear functional relationships between the input and output. Research shows that an NN with sufficient hidden units can approximate any functional relationship. However, overfitting can be a serious problem in this framework. Usually, early stopping optimization or more use of regularization techniques is used to penalize the optimization criterion. By adding a penalty term to the optimization criterion, the estimation of the standard weights will shrink, which is also called the shrinkage method. The smoothing penalty in equation (2) is commonly used in the shrinkage method:

[0102]

[0103] In the NN literature, this process is also called "weight decay"; in Figure 13 , two penalty nodes B1 and B2 are specified. The use of weight decay seems to help both the optimization process and avoid overfitting. The adjustment parameter k can be selected by cross-validation, and for a fixed number of hidden units, minimize formula (2) to obtain the weight estimate.

[0104] Such as Figure 14 , based on its diagnostic plot,[[]] Figure 2 and the SDP in Figure 3 are well smoothed. As expected, the NN can best depict the SDP behavior. The main advantage of the normalized quantile residuals is that regardless of the distribution of the response variable, when the model is assumed to be correct, the true residuals always have a standard normal distribution. The residuals perform well. Because Figure 15 in the first two plots of the residuals in[[[]]], the fitted values of depth and the fitted values and indices of the index show random scatter around a horizontal line at a mean of 0; while the kernel density estimate of the residuals is approximately normal, and the QQ plot is approximately linear (intercept 0, gradient 1). In addition, its R 2 is also very high, at 0.994.

[0105] The worm plot of the residuals is as[[[]]Figure 16 as shown, to identify the region of the explanatory variable within which the model cannot adequately fit the data (referred to as "model violation"). This is a diagnostic tool for examining the residuals for different ranges (by default non - overlapping) of one or two explanatory variables. Since all observations fall within the "accepted" region (where approximately 95% of the points lie between the two elliptical curves) within the two elliptical curves and no specific shape is detected at the points, as Figure 16 shown, the model seems to fit well overall.

[0106] Determination of C.SDP comparison.

[0107] The comparison method of SDP can be quantified by statistical methods or by normalized ratio comparison (NRC).

[0108] 1) NRC method.

[0109] To better compare two profiles given by the functions G A (x) and G B (x), where x represents depth, the difference ratio can be written as the following expression by concentration normalization.

[0110]

[0111] Obviously, in formula (3), if Ratio(x) = 1 holds, these two profiles can be identified as perfectly matching. Therefore, the problem of characterizing the difference between these two profiles can be simply explained as how to judge the equality. To obtain a quantitative index of curve coincidence, the integral value per unit depth of the shaded region enclosed by the horizontal line (Ratio(x) = 1), called the "percentage difference", can be calculated by formula (4):

[0112]

[0113] where a represents the lower limit of integration and b represents the upper limit of integration.

[0114] The method based on formula (4) is called by statistical methods or by normalized ratio comparison (NRC).

[0115] As Figure 17 shown, the respective Ratio(x) functions of SDP( Figure 3a , 3b and 2) are plotted by different line types (dotted line, dashed line and solid line respectively). According to the process requirements, the lower limit and the upper limit are set to 20nm and 800nm respectively. At the same time, the percentage difference value D is marked in Table I and Figure 17is also present in the upper left corner. Based on a lot of experimental data, the threshold of the percentage difference is defined as 0.276% to judge whether two SDPs are comparable. From common sense, the smaller the percentage difference, the greater the likelihood of the two samples. Therefore, Figure 3a the SDPs in

[0116] TABLE I

[0117] THE CALCULATED VALUES BASED ON NRC

[0118]

[0119] 2) Kolmogorov–Smirnov test

[0120] The Kolmogorov-Smirnov (KS) test is distribution-free, can be widely applied and is not restricted by the size of the overall sample size. These are especially important for the comparison of two SDPs because their distributions are basically unknown.

[0121] Generally, the KS test can be used to compare a sample with a reference probability distribution, or to compare two samples. The two-sample KS test is more useful because it is sensitive to the differences in the location and shape parameters of the empirical cumulative distribution functions (CDFs) of the two samples. It quantifies the distance between the empirical distributions of the two samples. The null distribution of this statistic is calculated under the null hypothesis that the samples are drawn from the same distribution in the two-sample case. The KS statistic, D KS is defined as D KS = max|F(x) - G(x)|, where F(x) and G(x) represent the CDFs of the two samples.

[0122] Extending the KS statistic to multi-dimensional space is a very challenging task. In a one-dimensional sample, the empirical distribution only changes at the observation points, and the univariate KS statistic is obtained by evaluating the distance between the empirical and theoretical distribution functions at these points. However, when the dimension is greater than 1, the empirical distribution function jumps at infinitely many points. Peacock introduced the idea of making the statistic independent of any specific ordering by finding the maximum difference between the CDFs under any possible ordering. Given n points in a two-dimensional space, it is necessary to calculate the CDFs in the 4n i ,Y j quadrants of the plane defined by all pairs (X 2 ), where X i and Y jare the coordinates of any pair of points in a given sample. Then, the maximum absolute difference between the CDFs in all quadrants is calculated. The counting step can be performed by a brute-force algorithm that sweeps through each quadrant for each point in the sample to determine if the point lies within it.

[0123] Here, Peacock's idea is utilized to implement the comparison of two SDPs. Figure 2 And the calculated statistical values for Figure 3 are in Table II. If α = 0.05 is used, the threshold can be calculated as 0.090 using the KS approximation. Thus, for KS statistic values below the threshold, the null hypothesis that they are comparable is not rejected. As the conclusion in Table II, Figure 3a the SDPs in Figure 2 are comparable, while the others (i.e., Figure 17 and 3b) are not. The conclusion is consistent with the NRC (

[0124] TABLE II

[0125] THE CALCULATED KS STATISTIC VALUES

[0126]

[0127] In summary, both the NRC and KS tests can handle SDP comparison well. In practice, their choice depends on the actual situation. In particular, when the previous database is not large enough (<30), the KS test is better.

[0128] The precision characteristics (energy, dose, and angle) of IMP can directly affect device performance, and thus it is very important for semiconductor manufacturing. The present invention provides a new method to improve the previous K method, which is a quantifiable index of a purely non-parametric method without making any subjective assumptions, can handle various shapes, and realizes SDP comparison. Some outlier determination methods are introduced to exclude outliers, and non-linear regression diagnosis is recommended here. The most flexible neural network SDP non-linear regression is used to smooth the SDP curve. In fact, the four different outlier identification methods are very useful references for industrial practitioners and beneficial to almost all statistical data analysis. Finally, quantifiable indexes for SDP comparison can be obtained by means of two non-parametric methods, KS and NRC. When the previous database contains fewer than 30 records, the KS method may be a better choice. These easily understandable and confirmable methods help with early monitoring and guide practitioners to complete corresponding solutions to reduce the impact of bias.

[0129] Obviously, those skilled in the art can make various modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalent technologies, then the present invention is also intended to include these modifications and variations.

Claims

1. A method for comparing secondary ion mass spectrometry analysis curves, used for diagnosing whether there is an abnormality in ion implantation, characterized in that, Including the following steps: S1, Fine-tune the machine tool; S2, In the first decision-making stage, the machine tool conducts self-verification of the ion implantation dose deviation. If the ion implantation dose deviation is less than 1%, go to step S3; if the ion implantation dose difference is greater than or equal to 1%, go to step S1; S3, Conduct outlier detection and elimination; S4, Conduct curve smoothing processing and model diagnosis; S5, Calculate relevant non-parametric quantitative indicators and compare the secondary ion mass spectrometry analysis curves according to these indicators; S6, If the said indicators meet the threshold, the ion implantation is normal; If the threshold is not met, the ion implantation is abnormal, go to step S1; The non-parametric quantization index is the "percentage difference D", and its calculation formula is , where , G A (x) and G B (x) are two given SDP curve smoothing functions, a represents the shallow zone depth value, and b represents the deep zone depth value; Or, The non-parametric quantization index is the Kolmogorov-Smirnov K-S statistic D KS , and its formula is D KS = max|F(x) - G(x)|, where F(x) and G(x) represent the empirical cumulative distribution functions of two SDP curves.

2. The method for comparing secondary ion mass spectrometry analysis curves according to claim 1, characterized in that, When the non-parametric quantitative indicator is "percentage difference D", the threshold is defined as 0.276%, that is, when D <= 0.276%, the SDP curves of the sample to be tested and the standard sample are comparable and the injection this time is normal; when D > 0.276%, there are significant differences between the SDP curves of the sample to be tested and the standard sample, and the injection this time is abnormal.

3. The method for comparing secondary ion mass spectrometry analysis curves according to claim 1, characterized in that, The non-parametric quantization index is the Kolmogorov-Smirnov K-S statistic D KS When, the method is not restricted by the sample distribution and the size of the overall sample size; when the sample dimension is greater than 1, the maximum absolute difference between the empirical cumulative distribution functions of the sample in all quadrants can be found under any possible sorting.

4. The method for comparing secondary ion mass spectrometry analysis curves according to claim 3, characterized in that, The non-parametric quantization index is the Kolmogorov-Smirnov K-S statistic D KS When KS , when the dimension of the sample is 2, n sample points are given in a two-dimensional space, and 4n of the planes defined by all pairs (Xi, Yj) are calculated 2 quadrants of the empirical cumulative distribution function, and calculate the maximum absolute difference between the empirical distribution functions in all quadrants, where Xi and Yj are the coordinates of any pair of points in the given sample, and the counting step is performed by a brute-force algorithm that sweeps through each quadrant for each point in the sample to determine whether the point is in it.

5. The method for comparing secondary ion mass spectrometry analysis curves according to any one of claims 1, 3 and 4, characterized in that, The non-parametric quantization index is the Kolmogorov-Smirnov K-S statistic D KS When, if the significance level a = 0.05, the threshold is defined as 0.09, that is, when D KS <= 0.09, the SDP curves of the sample to be tested and the standard sample are comparable, and the ion implantation is normal this time; when D > 0.09, there are significant differences between the SDP curves of the sample to be tested and the standard sample, and the implantation is abnormal this time.

6. The method for comparing secondary ion mass spectrometry analysis curves according to claim 1, characterized in that, The outlier detection adopts the method of non-linear regression diagnosis, including finding the conditional probability of another random variable Y on an independent variable X and using backward search to distinguish outliers; first, build a regression model using all data, and then continuously or simultaneously exclude the observations with the largest errors from the model. When the assumed model is correct, the true residuals always have a standard normal distribution.

7. The method for comparing secondary ion mass spectrometry analysis curves according to claim 1, characterized in that, The curve smoothing processing adopts the neural network method. The form of the neural network method is a multi-layer perceptron and model diagnosis is carried out. The model diagnosis adopts the residual worm plot. When all observations fall within the "accepted" area between two elliptical curves, that is, 95% of the points fall between the two elliptical curves and no specific shape is detected in the observations, it is determined that the overall fitting of the model is good.

8. The method for comparing secondary ion mass spectrometry analysis curves according to claim 1, characterized in that, When the number of past data samples is less than 30, the non-parametric quantization index uses the K-S statistic D KS .

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