Soft constraint-based depth analysis inversion method for surface analysis
By introducing a joint probability function of soft constraint and regularization terms into XPS, the problems of overfitting and hard constraint bias in XPS are solved, and more accurate and stable element concentration depth distribution analysis is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- UNIV OF SCI & TECH OF CHINA
- Filing Date
- 2025-12-15
- Publication Date
- 2026-05-12
AI Technical Summary
In X-ray photoelectron spectroscopy (XPS), conventional least squares fitting is prone to overfitting, which amplifies signal noise infinitely, leading to inaccurate and unstable analysis results. Hard constraints can also cause bias when the physical prior is inaccurate.
A soft constraint term is introduced into the joint probability function to reduce the hard dependence on precise physical priors through a penalty term. The joint probability function is established by combining the regularization term and the residual term, and iterative calculations are performed to determine the depth distribution of element concentration.
It improves the accuracy and stability of the deep profiling inversion algorithm, reduces the bias of the analysis results, and enhances the reliability of the results under poor signal-to-noise ratio conditions.
Smart Images
Figure CN122020354A_ABST
Abstract
Description
Technical Field
[0001] This disclosure relates to the field of surface analysis technology, and in particular to a soft-constraint-based depth profiling inversion method for surface analysis. Background Technology
[0002] Surface analysis techniques are crucial for obtaining information on the composition, chemical state, and structure of material surfaces, and are widely used in semiconductors, thin films, catalysis, and other fields. Among numerous surface analysis techniques, X-ray photoelectron spectroscopy (XPS) is a commonly used one. This technique utilizes X-rays to excite the inner-shell electrons of atoms on the sample surface and analyzes the energy distribution of the emitted photoelectrons, thereby quantitatively analyzing the elemental composition and their chemical valence states.
[0003] In XPS, elemental information at different depths of a sample is obtained by measuring photoelectron signals at different emission angles. The least squares method is typically used to fit the angular distribution of the emitted electrons, and depth information is obtained through residual analysis. The angle-depth mapping is essentially an ill-conditioned inverse Laplace transform process, requiring a depth profiling inversion algorithm. In conventional least squares methods, if only the residual term is used as the criterion, signal noise may be amplified infinitely in the algorithm, leading to severe overfitting. Therefore, a regularization term (smoothing prior) can be introduced to form a joint probability function with the residual term, and minimizing this function effectively suppresses overfitting. However, simply introducing a smoothing prior often fails to guarantee the accuracy and stability of the analysis results.
[0004] To further enhance the accuracy and stability of the algorithm, physically meaningful priors can be introduced as constraints. Strict physical priors are typically used for these constraints, known as hard constraints. However, incomplete accuracy of the physical priors can lead to biases in the analysis results.
[0005] To address the aforementioned issues, this invention introduces a soft constraint term into the regularized inversion algorithm. This involves incorporating physical priors as a penalty term into the joint probability function, reducing the rigid dependence on precise physical priors and thereby improving the accuracy and stability of the deep profiling algorithm. Summary of the Invention
[0006] In view of this, in order to solve the aforementioned technical problems, this disclosure provides a soft-constraint-based depth profiling inversion method for surface analysis, the technical solution of which is as follows:
[0007] This invention provides a soft-constraint-based depth profiling inversion method for surface analysis, comprising the following operations:
[0008] S1: Based on the measured peak angle distribution of the X-ray photoelectron spectrum, the measured apparent concentration distribution of elements in the material is obtained;
[0009] S2: Preset the depth distribution of element concentration in the material and calculate the peak angle distribution of the photoelectron spectrum;
[0010] S3: Based on the calculated photoelectron spectrum peak angle distribution, the calculated apparent concentration distribution of elements in the material is obtained;
[0011] S4: Based on the measured apparent concentration distribution and the calculated apparent concentration distribution, determine the residual term, and introduce a regularization term and a soft constraint term to establish a joint probability function; and
[0012] S5: The function value of the joint probability function is used as a criterion for iterative calculation to finally determine the depth distribution of element concentration in the material.
[0013] According to an embodiment of this disclosure, the photoelectron spectral peak angle distribution is as follows: :
[0014]
[0015] Where θ is the photoelectron emission angle. It represents the depth distribution of elemental concentrations in the material, where z is the material depth and λ is the electron inelastic mean free path. It is a constant that includes the instrument transfer function, geometric factor, and atomic photoionization cross section.
[0016] According to an embodiment of this disclosure, the joint probability function is:
[0017]
[0018] in For residual terms; For regularization terms, For regularization parameters; This is a soft constraint term. These are the weighting factors. The residual term is:
[0019]
[0020] in, To measure the apparent concentration distribution, To calculate the apparent concentration distribution, j represents the types of elements in the material, and k represents different emission angles. This represents the variance of the data.
[0021] According to embodiments of this disclosure, the soft constraint term is a priori with physical meaning. Optionally, the soft constraint term is the concentration and normalization of different elements at the same depth.
[0022] According to an embodiment of this disclosure, based on the measured X-ray photoelectron spectrum peak angle distribution, the peak area is divided by the F constant and normalized to obtain the measured apparent concentration distribution of elements in the material.
[0023] According to an embodiment of this disclosure, the depth distribution of element concentration in a preset material is used to calculate the peak angle distribution of the photoelectron spectrum, and the calculated apparent concentration distribution of elements in the material is obtained using the calculated peak angle distribution of the photoelectron spectrum.
[0024] According to embodiments of this disclosure, residual terms are determined based on the measured apparent concentration distribution and the calculated apparent concentration distribution, and a joint probability function is established by introducing regularization terms and soft constraint terms.
[0025] According to embodiments of this disclosure, the function value of the joint probability function is used as a criterion for iterative calculation to minimize the function value, thereby determining the depth distribution of element concentration in the material. Attached Figure Description
[0026] The above and other objects, features and advantages of this disclosure will become clearer from the following description of embodiments of this disclosure with reference to the accompanying drawings, in which:
[0027] Figure 1 This is a schematic diagram of the soft-constraint-based depth profiling inversion method for surface analysis according to an embodiment of the present disclosure.
[0028] Figure 2 This is a schematic diagram of a material sample according to an embodiment of the present disclosure.
[0029] Figure 3 This is a schematic diagram of the apparent concentration angular distribution of different elements in a material sample from 0° to 60° in an embodiment of this disclosure.
[0030] Figure 4 This is a schematic diagram of the depth distribution of material sample concentration obtained by the soft-constraint-based depth profiling inversion method for surface analysis according to embodiments of this disclosure.
[0031] Figure 5 This is a schematic diagram showing the depth distribution of element concentrations in a material sample with and without soft constraints under the condition of introducing 5% Gaussian noise according to an embodiment of this disclosure.
[0032] Figure 6 This is a schematic diagram showing the depth distribution of element concentrations in a material sample with or without soft constraints under the condition of introducing 10% Gaussian noise according to an embodiment of this disclosure. Detailed Implementation
[0033] This disclosure provides a depth profiling inversion algorithm for surface analysis, used to obtain accurate and stable depth distribution results of sample element concentrations from the angular distribution information of emitted photoelectrons.
[0034] This disclosed soft-constraint-based depth profiling inversion method for surface analysis is based on X-ray photoelectron spectroscopy (XPS). It utilizes X-rays to excite the inner-shell electrons of atoms on the sample surface and analyzes the energy distribution of the emitted photoelectrons, thereby quantitatively analyzing the elemental species and their chemical valence states. The sensitivity depth is typically within 10 nanometers. In XPS, the angle between the emitted photoelectron and the sample surface normal is defined as the emission angle. By measuring the photoelectron signals at different emission angles, elemental information at different depths of the sample is obtained: the larger the emission angle, the shallower the detection depth; conversely, the smaller the emission angle, the deeper the detection depth. The least squares method is typically used to fit the angular distribution of the emitted electrons, and depth information is obtained through residual analysis. The angle-depth mapping relationship is essentially an ill-conditioned inverse Laplace transform process, which must rely on the depth profiling inversion method. In conventional least squares methods, if only the residual term is used as the criterion, signal noise may be infinitely amplified in the inversion method, leading to severe overfitting. To address this, the soft-constraint-based deep profiling inversion method for surface analysis in this application introduces a regularization term (smoothing prior) and a residual term to jointly establish a joint probability function, effectively suppressing overfitting by minimizing it. However, simply introducing a smoothing prior is insufficient to guarantee the accuracy and stability of the analysis results in many cases. Therefore, to further enhance the accuracy and stability of the algorithm, this application also introduces a suitable soft constraint term, incorporating the physical prior as a penalty term into the joint probability function. This reduces the problem of large biases in analysis results caused by a rigid dependence on precise physical priors, thereby improving the accuracy and stability of the deep profiling algorithm.
[0035] To make the objectives, technical solutions, and advantages of this disclosure clearer, the following detailed description is provided in conjunction with specific embodiments and the accompanying drawings.
[0036] In this disclosure, a soft-constraint-based depth profiling inversion method for surface analysis is provided, such as... Figure 1 As shown, the analysis method includes the following operations:
[0037] S1: Based on the measured peak angle distribution of the X-ray photoelectron spectrum, the measured apparent concentration distribution of elements in the material is obtained;
[0038] S2: Preset the depth distribution of element concentration in the material and calculate the peak angle distribution of the photoelectron spectrum;
[0039] S3: Based on the calculated photoelectron spectrum peak angle distribution, the calculated apparent concentration distribution of elements in the material is obtained;
[0040] S4: Based on the measured apparent concentration distribution and the calculated apparent concentration distribution, determine the residual term, and introduce a regularization term and a soft constraint term to establish a joint probability function; and
[0041] S5: The function value of the joint probability function is used as a criterion for iterative calculation to finally determine the depth distribution of element concentration in the material.
[0042] According to embodiments of this disclosure, the X-ray photoelectron spectrum peak angle distribution of the elements in the material is obtained by measuring the photoelectron signals generated by the X-ray excitation of elements in the material.
[0043] Furthermore, the apparent concentration distribution of elements in the material is determined by normalizing the peak angle distribution of the measured photoelectron spectrum.
[0044] According to embodiments of this disclosure, the photoelectron spectral peak angle distribution function for:
[0045]
[0046] Where θ is the photoelectron emission angle. λ represents the depth distribution of elemental concentrations in the material, z is the depth of the material, and λ is the electron inelastic mean free path. It is a constant that includes the instrument transfer function, geometric factor, and atomic photoionization cross section.
[0047] According to formula (1), obtaining the depth distribution of elemental concentration from the photoelectron spectrum peak angle distribution is an ill-conditioned inverse Laplace transform, which requires a depth profiling algorithm to obtain the mapping relationship. The soft-constraint-based depth profiling inversion method for surface analysis in this invention is a soft-constraint-based regularized depth profiling inversion method. It first presets the depth distribution of elemental concentration in the material, then uses formula (1) to calculate the photoelectron spectrum peak angle distribution based on the preset elemental concentration depth distribution, and further uses the calculated photoelectron spectrum peak angle distribution to obtain the calculated apparent concentration distribution. Calculate the apparent concentration distribution The apparent concentration distribution of elements in the material determined based on the measured peak angle distribution of X-ray photoelectron spectra. Determine the residual term, and introduce regularization and soft constraint terms to establish a joint probability function, Q:
[0048]
[0049] in For residual terms; For regularization terms, For regularization parameters; This is a soft constraint term. The weighting factor is the residual term. for:
[0050]
[0051] in, To measure the apparent concentration distribution, To calculate the apparent concentration distribution, j represents the types of elements in the material, and k represents different emission angles. It represents the variance of the data and measures the reliability of the data.
[0052] It should be noted that all physical priors can be used as soft constraints, which can be selected and set according to the actual application.
[0053] During iterative calculations, the value of the joint probability function is used as a criterion to perform iterative calculations to minimize the function value, thereby determining the depth distribution of element concentrations in the material.
[0054] Preferably, the regularization term R is selected as a Tikhonov regularization term based on the concentration slope, i.e.: The soft constraint term L selects the element concentration at the same depth and the normalized physical prior as the soft constraint term, that is: .
[0055] Specifically, with Figure 2 Taking the three-layer test sample shown as an example, the top layer is element A, the middle layer is a compound BC2 composed of elements B and C, and the bottom layer is element D. The top layer of the sample is 1.1 nm, the middle layer is 1.0 nm, and the bottom layer is the substrate. The characteristic binding energy of element A is 72.6 eV, the characteristic binding energy of element B is 228.0 eV, the characteristic binding energy of element C is 161.5 eV, and the characteristic binding energy of element D is 99.4 eV. The angular distribution of the measured apparent concentration of different elements in the sample obtained by Monte Carlo simulation is shown below. Figure 3 As shown. Further, the depth distribution of element concentration in the sample is preset, and the calculated photoelectron spectrum peak angle distribution of the elements in the material is obtained based on formula (1), thereby obtaining the calculated apparent concentration distribution of the elements in the material. Then, the residual term is determined based on the measured apparent concentration distribution and the calculated apparent concentration distribution, and a joint probability function is established by introducing a regularization term and a soft constraint term. The function value of the joint probability function is used as a criterion for iterative calculation to determine the final depth distribution result of element concentration in the material, such as Figure 4 As shown, the black, red, green, and blue solid lines represent the depth distributions of elements A, B, C, and D obtained by the algorithm, respectively. Based on the depth distributions, the top layer (A) has a thickness of 1.1 nm, and the middle layer (BC2) has a thickness of 1.0 nm, which matches the actual thickness. Integrating the depth distributions of element B and element C yields relative concentrations of 33.7% and 66.3%, respectively, with a deviation of less than 0.5%, consistent with the 1:2 component ratio of the two elements in compound BC2. This demonstrates that the surface analysis method disclosed herein possesses good accuracy and stability.
[0056] Furthermore, to assess the impact of the soft constraint term on the robustness of the results, taking element A as an example, 5% and 10% Gaussian noise were added to the measurement of the apparent concentration distribution, respectively. (No soft constraints) and (With soft constraints) 50 iterations of optimization were performed in both cases, and the depth distribution results of element A concentration are shown in Figures 5 and 6, respectively. The horizontal axis represents element depth in nanometers, and the vertical axis represents element relative concentration in at. Figure 5 Parts (a) and (b) show the depth distribution results of element A concentration with and without soft constraints under 5% noise conditions, respectively. The depth distribution curve of the average concentration of element A after 50 iterations (black solid line) and its average standard deviation (shaded area near the solid line) are shown. Figure 6 Parts (a) and (b) show the depth distribution results of element A concentration with and without soft constraints under 10% noise conditions, respectively. The depth distribution curve of the average concentration of element A after 50 iterations (black solid line) and its mean standard deviation (shaded area near the solid line) are also shown. It can be seen that, under both noise levels, the addition of soft constraints makes the obtained average depth distribution curve closer to the actual depth distribution, and the standard deviation is also significantly reduced. This indicates that the soft constraint term significantly improves the stability and reliability of the obtained depth distribution results of element concentration.
[0057] This disclosure introduces a soft-constraint-based depth profiling inversion method for surface analysis. This method incorporates a soft-constraint term, introducing physical priors as a penalty term into the joint probability function. This reduces the rigid dependence on precise prior knowledge, thereby improving the accuracy and stability of the depth profiling results and avoiding biases caused by incomplete prior knowledge. The introduction of a new constraint term into the joint probability function reasonably limits the parameter space, facilitating rapid searching for the global optimum and enabling reasonable results even with poor signal-to-noise ratios. This soft-constraint-based depth profiling inversion method can be used with XPS depth profiling technology to obtain the depth distribution of elemental concentrations on material surfaces, and can also be applied to Auger electron spectroscopy surface analysis. It has important applications in semiconductors, thin films, catalysis, and other fields. Furthermore, it can be extended to other analytical techniques that require obtaining mapping relationships through ill-conditioned inverse Laplace transforms.
[0058] The embodiments of this disclosure have been described in detail above with reference to the accompanying drawings. It should be noted that implementations not illustrated or described in the drawings or the main text of the specification are forms known to those skilled in the art and are not described in detail. Furthermore, the definitions of the various elements and methods described above are not limited to the specific structures, shapes, or methods mentioned in the embodiments, and those skilled in the art can easily modify or substitute them.
[0059] The specific embodiments described above further illustrate the purpose, technical solutions, and beneficial effects of this disclosure. It should be understood that the above descriptions are merely specific embodiments of this disclosure and are not intended to limit this disclosure. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this disclosure should be included within the protection scope of this disclosure.
Claims
1. A soft-constraint-based depth profiling inversion method for surface analysis, characterized in that, include: S1: Based on the measured peak angle distribution of the X-ray photoelectron spectrum, the measured apparent concentration distribution of elements in the material is obtained; S2: Preset the depth distribution of element concentration in the material and calculate the peak angle distribution of the photoelectron spectrum; S3: Based on the calculated photoelectron spectrum peak angle distribution, the calculated apparent concentration distribution of elements in the material is obtained; S4: Determine the residual term based on the measured apparent concentration distribution and the calculated apparent concentration distribution, and introduce regularization term and soft constraint term to establish a joint probability function; as well as S5: The function value of the joint probability function is used as a criterion for iterative calculation to finally determine the depth distribution of element concentration in the material.
2. The soft-constraint-based depth profiling inversion method for surface analysis according to claim 1, characterized in that, The peak angle distribution of the X-ray photoelectron spectrum is as follows: : Where θ is the photoelectron emission angle. It represents the depth distribution of elemental concentrations in the material, where z is the material depth and λ is the electron inelastic mean free path. It is a constant that includes the instrument transfer function, geometric factor, and atomic photoionization cross section.
3. The soft-constraint-based depth profiling inversion method for surface analysis according to claim 1, characterized in that, The joint probability function is: in For residual terms; For regularization terms, For regularization parameters; This is a soft constraint term. This is the weighting factor.
4. The soft-constraint-based depth profiling inversion method for surface analysis according to claim 1 or 3, characterized in that, The residual term is: in, To measure the apparent concentration distribution, To calculate the apparent concentration distribution, j represents the types of elements in the material, and k represents different emission angles. This represents the variance of the data.
5. The soft-constraint-based depth profiling inversion method for surface analysis according to claim 1, characterized in that, By measuring the peak angle distribution of X-ray photoelectron spectra of different elements, the depth distribution of element concentration can be accurately obtained.
6. The soft-constraint-based depth profiling inversion method for surface analysis according to claim 1, characterized in that, The soft constraint can be any prior that has physical meaning.
7. The soft-constraint-based depth profiling inversion method for surface analysis according to claim 1, characterized in that, The depth distribution of element concentration in the material is determined by using the value of the joint probability function as a criterion for iterative calculation to minimize the function value.