Electronic material surface modification method based on fractional Fourier transform

Abstract

The invention discloses an electronic material surface modification method based on fractional Fourier transform, and relates to the technical field of electronic digital data processing. The method comprises the following steps of: 1, acquiring an atomic-scale discrete image of a material surface by using a high-resolution scanning tunneling microscope; 2, performing fractional Fourier transform on the continuous function form to obtain spectrum distribution, and extracting spectrum features; 3, defining a material surface characteristic vector, wherein the material surface characteristic vector comprises a plurality of surface characteristic elements; based on the spectrum features, constructing a nonlinear dynamic model for describing surface characteristic evolution based on a surface characteristic vector; and 4, solving the solution of the nonlinear kinetic model under the constraint of the minimum objective function to obtain modified objective parameters. According to the method, through high-resolution data acquisition, flexible frequency domain analysis and accurate kinetic model optimization, high-precision and comprehensive control over the surface characteristics of the material is achieved.

Description

Technical Field

[0001] The present disclosure relates to, but is not limited to, the field of electronic digital data processing technology, and in particular to a method for surface modification of electronic materials based on fractional Fourier transform. Background Art

[0002] With the rapid development of electronic materials and semiconductor technology, the demand for material surface modification is increasing. The properties of material surfaces, such as roughness, anisotropy, conductivity, and surface energy, play a vital role in their performance in practical applications. For example, in micro-electromechanical systems (MEMS) and nanoelectronic devices, the flatness and conductivity of the material surface directly affect the performance and reliability of the device. However, existing material surface modification technologies still face many challenges in practical applications.

[0003] In the prior art, commonly used material surface modification methods include physical treatment methods, chemical treatment methods and composite treatment methods. Physical treatment methods mainly include mechanical polishing, ion beam etching and laser treatment. These methods change the morphology and structure of the material surface by physical means, thereby improving its surface properties. For example, mechanical polishing can improve the surface finish by removing microscopic unevenness on the surface of the material. However, this method easily introduces mechanical stress, causing defects inside the material, affecting its performance and reliability. Although ion beam etching can achieve high-precision surface processing, its equipment is complex, costly, and inefficient when processing large-area materials. Laser treatment can locally heat and melt the surface of the material through a laser beam to change its surface structure, but this method has a high thermal sensitivity to the material and is prone to thermal damage.

[0004] Although the existing technology has made certain progress in material surface modification, there are still some problems that need to be solved. First, it is difficult to guarantee the processing accuracy and uniformity of materials with existing methods, especially when processing large-area, high-complexity materials, local performance inconsistencies are prone to occur. Secondly, the modification effect of existing methods on materials has certain limitations, and it is difficult to simultaneously meet the optimization requirements of multiple surface properties (such as roughness, anisotropy, conductivity, and surface energy). In addition, the existing methods are prone to introduce mechanical stress, thermal damage or chemical contamination during the processing process, affecting the performance and reliability of the materials. Summary of the invention

[0005] The present disclosure provides a method for surface modification of electronic materials based on fractional Fourier transform, which achieves high-precision and comprehensive control of material surface characteristics through high-resolution data acquisition, flexible frequency domain analysis and precise kinetic model optimization.

[0006] In order to solve the above problems, the technical solution of the present invention is achieved as follows:

[0007] The electronic material surface modification method based on fractional Fourier transform performs the following steps:

[0008] Step 1: Use a high-resolution scanning tunneling microscope to obtain an atomic-level discrete image of the material surface, and convert the obtained discrete image into a continuous function form through interpolation and filtering;

[0009] Step 2: Perform fractional Fourier transform on the continuous function form to obtain the spectrum distribution and extract the spectrum features;

[0010] Step 3: Define the material surface characteristic vector, which includes multiple surface characteristic elements; based on the frequency spectrum characteristics, construct a nonlinear dynamic model based on the surface characteristic vector to describe the evolution of surface characteristics, and define the objective function of the nonlinear dynamic model;

[0011] Step 4: Solve the solution of the nonlinear dynamic model under the constraint of minimizing the objective function to obtain the modification target parameters, and use the modification target parameters to modify the surface of the electronic material.

[0012] Furthermore, the surface characteristic elements include at least: surface roughness, surface anisotropy, conductivity and surface energy.

[0013] Furthermore, the continuous function form f(x,y) in step 1 is expressed by the following formula:

[0014]

[0015] Among them, h ij Indicates the height of the (i,j)th sampling point, x i and j Respectively represent the horizontal and vertical coordinates of the (i, j)th sampling point; Δx and Δy represent the horizontal sampling interval and vertical sampling interval respectively; there are N*M sampling points in total, N is the upper limit of the horizontal axis sampling, and M is the upper limit of the vertical axis sampling; W ij (x,y) is the adaptive weight function, defined as:

[0016]

[0017] Among them, x is the horizontal axis coordinate variable; y is the vertical axis coordinate variable; σ ij is a smoothing parameter based on adaptive adjustment of local curvature; sinc is the sine Cardinal function;

[0018] Furthermore, the smoothing parameter σ is adaptively adjusted based on the local curvature. ij Use the following formula to express it:

[0019]

[0020] in, Indicates the height h at the (i,j)th sampling point ij Laplace operator; σ0 is the basic smoothing parameter, which represents the default smoothing degree when there is no local curvature influence; a is the scale factor, which controls the influence of curvature on the smoothing parameter; b is the exponential parameter, which determines the nonlinear degree of curvature influence.

[0021] Furthermore, in step 2, a two-dimensional fractional Fourier transform is performed on the continuous function, and the obtained spectrum distribution is expressed using the following formula:

[0022]

[0023] Among them, K α (x,u) is the horizontal axis Fourier kernel function; K β (y,v) is the vertical axis Fourier kernel function; u is the horizontal axis variable of the two-dimensional fractional Fourier transform; v is the vertical axis variable of the two-dimensional fractional Fourier transform; α and β are the horizontal axis fractional order parameter and the vertical axis fractional order parameter, respectively, both in the range of (0,1), which control the characteristics of the transformation; Ψ α,β (u,v)=λ(|u| α-1 +|v| β-1 )·exp(-μ(u 2 +v 2 )) is the nonlinear correction term, λ is the nonlinear correction strength; μ is the nonlinear correction exponential parameter; is the weight function; σ is the exponential parameter of the weight function; ω is the modulation parameter of the weight function; is the normalization factor, expressed using the following formula:

[0024]

[0025] Where Γ() is the gamma function.

[0026] Furthermore, the horizontal axis Fourier kernel function K α (x,u) is expressed using the following formula:

[0027]

[0028] Vertical axis Fourier kernel function K β (y,v) is expressed using the following formula:

[0029]

[0030] Among them, l is the imaginary number sign; sgn() is the sign function.

[0031] Furthermore, in step 3, the material surface characteristic vector is defined as S(t):

[0032]

[0033] Where, t is time; R a (t) is the surface roughness at time t; A(t) is the surface anisotropy at time t; σ(t) is the conductivity at time t; γ(t) is the surface energy at time t; based on the spectral feature F α,β (u,v), construct the following nonlinear dynamic model:

[0034]

[0035] Among them, k 1,1 =1.2×10 -3 , k 1,2 =0.5, k 1,3 =0.5; k 2,1 =1.8×10 -2 , k 2,2 =0.3; k 3,1 =0.21, k 3,2 =0.7; k 4,1 =0.14, k 4,2 =0.5; θ is the temperature; ∈ is a positive number to prevent division by zero; T is the set total time.

[0036] Furthermore, the objective function J of the nonlinear dynamics model defined in step 3 is expressed using the following formula:

[0037]

[0038] Wherein, ∥·∥ represents the L1 norm; λ1 is the preset first regularization coefficient; λ2 is the preset second regularization coefficient; S target is the target surface characteristic vector.

[0039] Furthermore, the specific process of solving the solution of the nonlinear dynamic model under the constraint of minimizing the objective function in step 4 to obtain the modified target parameters includes:

[0040] The Lagrangian function including high-order nonlinear terms and coupling terms is constructed by the following formula:

[0041]

[0042] Among them, Λ(t) is the Lagrange multiplier vector, and then we get the following partial differential equation:

[0043]

[0044] The high-order numerical method is combined with the Jacobi iteration method to solve the above partial differential equations to obtain the specific solution of S(t) and determine the modified target parameter S target .

[0045] The surface modification method of electronic materials based on fractional Fourier transform of the present invention has the following beneficial effects: the present invention obtains atomic-level discrete images of the material surface through a high-resolution scanning tunneling microscope, thereby ensuring high precision and high resolution of the surface characteristic data. STM can provide subatomic resolution, capture the microstructure characteristics of the material surface, and ensure the accuracy and reliability of the data used in subsequent processing. In contrast, traditional physical and chemical treatment methods usually have the problem of insufficient resolution in the data acquisition stage, and it is difficult to fully capture the subtle characteristics of the material surface. Secondly, the present invention uses fractional Fourier transform to extract the spectral characteristics of the material surface, providing a flexible and accurate frequency domain analysis tool. Fractional Fourier transform can not only realize arbitrary angle rotation on the time-frequency plane, but also capture the multi-scale and multi-directional characteristics of the material surface structure. This flexibility enables the transformation to fully reflect the complex characteristics of the material surface and provide more abundant spectral information than traditional Fourier transform. Through the extraction and analysis of spectral characteristics, the present invention can accurately describe the periodicity, symmetry and local defects of the material surface, and provide key data support for subsequent surface modification. Another significant beneficial effect of the present invention is to construct a nonlinear dynamic model that describes the evolution of material surface characteristics. The model comprehensively considers multiple properties such as surface roughness, surface anisotropy, conductivity and surface energy, and realizes comprehensive control and optimization of material surface properties through precise mathematical description and optimization methods. Traditional surface modification methods can often only optimize a single property, while the method of the present invention can simultaneously consider multiple surface properties, ensuring the coordination and balance between the various properties during the modification process, thereby achieving overall performance improvement. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 A schematic flow chart of a method for surface modification of electronic materials based on fractional Fourier transform provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0047] In order to make the technical problems, technical solutions and beneficial effects to be solved by the present disclosure clearer and more understandable, the present disclosure is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present disclosure and are not used to limit the present disclosure.

[0048] Example 1: Reference Figure 1 , a method for surface modification of electronic materials based on fractional Fourier transform, the method performing the following steps:

[0049] Step 1: Use a high-resolution scanning tunneling microscope to obtain an atomic-level discrete image of the material surface, and convert the obtained discrete image into a continuous function form through interpolation and filtering;

[0050] Step 1 is to use a high-resolution scanning tunneling microscope to obtain an atomic-level discrete image of the material surface, and convert the obtained discrete image into a continuous function form through interpolation and filtering. The principle of this step is that through advanced scanning tunneling microscope technology, the arrangement and distribution of atoms on the surface of electronic materials can be captured at extremely high resolution. The working principle of scanning tunneling microscope (STM) is based on the quantum tunneling effect, that is, when the STM probe approaches the surface of the material, due to the very small distance between the probe and the surface, electrons can tunnel through this narrow space to form a tunneling current. The intensity of this current is closely related to the distance between the probe and the surface atoms. Therefore, by measuring the change in tunneling current, a three-dimensional image of the material surface can be constructed to obtain detailed atomic-level information on the surface. These images are discrete in nature, that is, they are composed of independent pixels, each pixel corresponding to an atom or its surrounding electron cloud. However, discrete images may have problems of discontinuous information and incomplete data during subsequent processing, so these discrete data need to be converted into a continuous function form. This step is usually achieved through interpolation and filtering technology. Interpolation refers to filling data between discrete points to make the image smoother and more continuous. Common interpolation methods include linear interpolation, spline interpolation, etc., through which new data points can be generated between the original discrete data points to make the image more continuous. Filtering is used to remove noise in the image and improve the quality of the image. Noise is usually a random error caused by equipment limitations or external environmental influences, which will affect the clarity and accuracy of the image. By applying appropriate filtering techniques, such as Gaussian filtering, median filtering, etc., the interference of noise on the image can be effectively reduced, thereby obtaining a more accurate and clear image in the form of a continuous function. In the present invention, the purpose of converting a discrete image into a continuous function form is to provide a more stable and accurate basis for the subsequent fractional Fourier transform. Fractional Fourier transform needs to process continuous signals. By interpolating and filtering, discrete image data is converted into a continuous function form, which can enable fractional Fourier transform to better extract the spectral characteristics of the image. These spectral characteristics are crucial for describing the microstructural characteristics of the material surface and can provide important parameters and basis for subsequent surface modification.

[0051] Step 2: Perform fractional Fourier transform on the continuous function form to obtain the spectrum distribution and extract the spectrum features;

[0052] Fractional Fourier transform (FrFT) is a generalized form of Fourier transform, which can not only reveal the frequency components of the signal, but also provide more degrees of freedom and flexibility between time and frequency. The traditional Fourier transform is an important tool in signal processing, but its application is limited to converting the signal from the time domain to the frequency domain, while the fractional Fourier transform can rotate at any angle between the two, thereby providing richer information. In the present invention, the purpose of applying the fractional Fourier transform is to more finely analyze the spectral characteristics of the material surface image. By performing FrFT on the image in the form of a continuous function, the multi-scale and multi-directional spectral components contained in the image can be revealed, which are crucial for understanding and describing the microstructural characteristics of the material surface. A key advantage of FrFT is that it can adjust the order of the transformation to observe the spectral distribution of the signal at different angles. This flexibility enables FrFT to capture the characteristics of the material surface at different scales, providing detailed and accurate spectral information for subsequent surface modification. The spectral distribution is obtained by inputting the material surface image in the form of a continuous function into the kernel function of FrFT for integration operation, thereby obtaining the spectral components at different angles. By adjusting the order of FrFT, the signal representation at any position between the time domain and the frequency domain can be obtained respectively. This representation method can not only reflect the overall spectral characteristics of the material surface, but also highlight the intensity and distribution of certain specific frequency components. This is of great significance for identifying and characterizing the microstructural characteristics of the material surface. For example, the enhancement of certain frequency components may correspond to specific atomic arrangements or periodic structures on the material surface, and these structures are the key factors affecting the material performance. The process of extracting spectral features involves further analysis and processing of the spectral distribution obtained by FrFT. These spectral features may include the amplitude, phase, main frequency components of the spectrum, and the relative intensity between different frequency components. Through the comprehensive analysis of these features, a comprehensive material surface characteristic vector can be constructed, which describes the microstructure and physical properties of the material surface in detail. In the present invention, these spectral features are not only used to describe the current state of the material surface, but also used as input parameters to construct subsequent nonlinear dynamic models. The model is used to predict the evolution path of the material surface, thereby guiding specific surface modification measures. The application of fractional Fourier transform here reflects the innovation of the present invention in processing and analyzing complex signals. Compared with traditional Fourier transform, FrFT provides richer spectral information and can more accurately reveal the details of the material surface structure. This meticulous spectrum analysis capability enables the present invention to achieve higher accuracy and effect in the material surface modification process. The spectral features extracted by FrFT provide an important parameter basis for the subsequent modification steps, making the modification process more scientific and efficient, and ultimately achieving the purpose of optimizing the material surface performance.

[0053] Step 3: Define the material surface characteristic vector, which includes multiple surface characteristic elements; based on the frequency spectrum characteristics, construct a nonlinear dynamic model based on the surface characteristic vector to describe the evolution of surface characteristics, and define the objective function of the nonlinear dynamic model;

[0054] First, defining the material surface characteristic vector is the basis of this step. The material surface characteristic vector is a multidimensional vector, and each dimension represents a specific surface characteristic. Common surface characteristics include surface roughness, surface anisotropy, conductivity, and surface energy. Surface roughness describes the microscopic unevenness of the material surface, which directly affects the material's friction, wear, and lubrication properties. Surface anisotropy describes the differences in the physical and chemical properties of the material in different directions, which has an important impact on the material's mechanical properties and processing properties. Conductivity is the ability of a material to conduct current under the action of an electric field, which is particularly important for electronic materials. Surface energy is the surface free energy per unit area, which affects the wettability and adhesion properties of the material. These characteristics determine the performance and behavior of the material in practical applications to a certain extent. In order to fully characterize the surface state of the material, the characteristic vector not only includes these traditional physical properties, but also integrates key information extracted from spectral features. By analyzing the material surface image by fractional Fourier transform (FrFT), multi-scale and multi-directional spectral components can be extracted, such as the amplitude and phase of specific frequency components. These spectral features can reflect the periodicity and symmetry of the material surface structure, as well as local defects and inhomogeneities. Therefore, the material surface characteristic vector is a comprehensive description, which includes both macroscopic physical properties and microstructural information, providing rich basic data for subsequent nonlinear dynamic modeling. After defining the material surface characteristic vector, the next step is to build a nonlinear dynamic model that describes the evolution of these characteristics. The nonlinear dynamic model is a complex mathematical framework used to simulate the changes in material surface characteristics under different conditions. Unlike linear models, nonlinear dynamic models can capture the complex interactions and nonlinear relationships between material properties, thereby providing more accurate and reliable predictions. In the present invention, the nonlinear dynamic model describes how the surface characteristics evolve under time and external influences based on the material surface characteristic vector and the information extracted from the spectral features. This model not only takes into account the inherent properties of the material, but also includes the effects of external factors such as temperature, pressure and chemical environment on the material surface. For example, the changes in the material surface characteristic vector at different time points can be described by a nonlinear dynamic equation. This equation combines the intrinsic properties of the material and the influence of external conditions, and can predict how the material surface characteristics will change and evolve under different modification conditions. In this way, the modification process of the material surface can be accurately simulated and controlled, thereby achieving the expected modification goals. In order to make the nonlinear dynamic model more effective and accurate in practical applications, an objective function must be defined. The objective function is used to evaluate the output of the model, that is, whether the changes in the surface properties of the material during the modification process have achieved the expected effect. In the present invention, the objective function not only includes the absolute values ​​of the surface properties, but also takes into account the relative relationship between the properties and the overall optimization goal.For example, if the goal is to increase the conductivity of the material and reduce the surface roughness, the objective function will take both properties into account and comprehensively evaluate their changes through a weighted approach. By optimizing the objective function, the optimal modification path and parameters can be determined to achieve the best modification effect on the material surface. The definition of the objective function can be concretized as an optimization problem, in which we hope to minimize or maximize a specific performance indicator by adjusting external parameters. In order to find the specific parameters for achieving the best modification, numerical optimization methods such as gradient descent, genetic algorithms, etc. can be used. These methods are able to find the optimal solution in a complex parameter space, thereby ensuring that the modification effect of the material surface reaches the expected goal.

[0055] Step 4: Solve the solution of the nonlinear dynamic model under the constraint of minimizing the objective function to obtain the modification target parameters, and use the modification target parameters to modify the surface of the electronic material.

[0056] Specifically, in the previous steps, a nonlinear dynamic model has been established and an objective function has been defined. The model and the objective function together describe the evolution of the material surface properties during the modification process and the expected goals. In this step, the model is solved to minimize the objective function in order to find the optimal modification path and parameters. This process usually involves complex numerical calculations and optimization algorithms, because nonlinear dynamic models may have multiple local optimal solutions, and finding the global optimal solution requires sophisticated algorithm design and a large amount of computing resources. In the specific solution process, a variety of optimization algorithms can be used, including gradient descent, genetic algorithm, simulated annealing algorithm, etc. These algorithms iteratively approach the optimal solution by continuously adjusting model parameters. For example, the gradient descent method continuously adjusts the parameters to reduce the value of the objective function by calculating the gradient of the objective function relative to the model parameters; the genetic algorithm gradually optimizes the parameters by simulating natural selection and genetic variation; the simulated annealing algorithm finds the optimal solution by simulating the energy state changes in the physical annealing process. The selection of a suitable algorithm should not only consider its convergence speed and computational efficiency, but also its dependence on the initial value and its ability to avoid falling into the local optimum. Once the optimal modification target parameters are found, these parameters are used in the actual material surface modification operation. Modification operations may include chemical treatment, physical treatment, or a combination of the two. For example, chemical treatment may involve etching, depositing, or doping the surface of a material using specific chemical reagents to change its surface properties; physical treatment may include heat treatment, plasma treatment, or laser treatment to adjust the microstructure and properties of the material surface. These modification operations need to be precisely controlled according to the optimal parameters obtained to ensure that the modification effect is consistent with expectations. When implementing the modification operation, a series of precision equipment and strict process control may be required. For example, chemical vapor deposition (CVD) equipment can be used to deposit a uniform thin film on the surface of a material, and the thickness and composition of the film can be precisely adjusted by controlling the temperature, pressure, and gas flow of the gas phase chemical reaction; plasma treatment equipment can be used to etch or activate the surface of a material, and the microstructure of the surface can be adjusted by controlling the energy and treatment time of the plasma; laser treatment equipment can locally modify the surface of a material through precise laser beam scanning, thereby achieving complex surface structure adjustment. The modification operation is continuously performed so that the final adjusted surface properties need to reach the modification target parameters.

[0057] Embodiment 2: Surface property elements include at least: surface roughness, surface anisotropy, conductivity and surface energy.

[0058] Specifically, surface roughness refers to the microscopic unevenness of the material surface, which directly affects the friction, wear, lubrication and adhesion of the material. Surfaces with higher roughness usually have larger surface areas, which can increase the contact area between the material and other substances, thereby improving its adhesion. However, high roughness may also lead to stress concentration on the surface of the material, thereby reducing its mechanical strength and wear resistance. In electronic materials, surface roughness also affects the electrical properties of electronic components. For example, rough surfaces may cause changes in electron mobility, which in turn affects the performance and reliability of components. Therefore, in the process of surface modification, by regulating the surface roughness, the mechanical and electrical properties of the material can be optimized to meet the needs of specific applications. Surface anisotropy describes the differences in the physical and chemical properties of a material in different directions. This anisotropy is usually determined by the crystal structure of the material and the arrangement of surface atoms. In many electronic material applications, surface anisotropy is a key factor because it affects the electrical conductivity, thermal conductivity, optical properties, etc. of the material. For example, in optoelectronic devices, the anisotropy of the material surface may affect the absorption and scattering properties of light, thereby affecting the efficiency of the device. In semiconductor devices, anisotropy affects the current flow path and electron mobility, which in turn affects the performance of the device. Therefore, by adjusting the anisotropy of the material surface through surface modification technology, its performance in specific applications can be significantly improved. Conductivity refers to the ability of a material to conduct current under the action of an electric field, which is particularly important for electronic materials. Conductivity is affected by the intrinsic properties of the material (such as carrier concentration and mobility) as well as the surface state. For example, surface roughness and contaminants can scatter carriers, thereby reducing the conductivity of the material. In the surface modification process, the conductivity of the material can be improved by cleaning and smoothing the surface, doping with specific elements, or forming a protective layer, thereby improving the performance of electronic devices. For example, in semiconductor devices, optimizing conductivity through surface modification technology can improve the switching speed and power efficiency of the device. Surface energy is the surface free energy per unit area, which affects the wettability, adhesion, and chemical reactivity of the material. High surface energy usually means that the surface has strong adsorption capacity and high chemical reactivity, which is of great significance for the surface modification of materials. For example, in the manufacturing process of electronic materials, materials with high surface energy are more likely to form strong bonds with other materials, thereby improving the overall performance of the composite material. In addition, surface energy also affects the performance of materials in terms of wetting and antifouling. For example, by reducing the surface energy, the material can have better hydrophobicity and antifouling properties, thereby improving its durability and stability in harsh environments.

[0059] Embodiment 3: The continuous function form f(x, y) in step 1 is expressed by the following formula:

[0060]

[0061] Among them, hij Indicates the height of the (i,j)th sampling point, x i and j Respectively represent the horizontal and vertical coordinates of the (i, j)th sampling point; Δx and Δy represent the horizontal sampling interval and vertical sampling interval respectively; there are N*M sampling points in total, N is the upper limit of the horizontal axis sampling, and M is the upper limit of the vertical axis sampling; W ij (x,y) is the adaptive weight function, defined as:

[0062]

[0063] Among them, x is the horizontal axis coordinate variable; y is the vertical axis coordinate variable; σ ij is a smoothing parameter that is adaptively adjusted based on the local curvature; sinc is the sine Cardinal function.

[0064] Specifically, the discrete sampling point data h on the material surface ij Represents the height information of the surface at a specific point. These data are usually obtained through high-resolution scanning tunneling microscopy (STM). STM can provide subatomic resolution and capture the fine structure of the material surface. However, these data are discrete, and direct use may result in information loss or discontinuity, so they need to be converted into a continuous function form f(x,y) through interpolation. In the formula f(x,y), the sinc function plays a key role. The sinc function is defined as It has ideal low-pass filter characteristics and can effectively reconstruct continuous signals from discrete data. Interpolation through the sinc function can ensure that the reconstructed signal will not produce spectrum aliasing when the sampling interval is Δx and Δy. This means that the sinc function interpolation can achieve a smooth transition of the signal while maintaining high resolution, avoiding data breaks and discontinuities. In addition, the adaptive weight function W is introduced in the formula ij (x,y), which is another innovation of this method. The weight function is defined as Among them, σ ij It is a smoothing parameter that is adaptively adjusted based on the local curvature. The weight function is to give a larger weight to the area close to the sampling point, and the weight is smaller to the area farther away from the sampling point. This weighting method ensures that the change of surface height can transition naturally and smoothly during the interpolation process, so as to better reflect the actual structural characteristics of the material surface. The introduction of the adaptive weight function makes it possible to adjust the smoothing parameter σ in areas with large curvature changes. ij , providing higher interpolation accuracy. For example, in areas with tiny protrusions or depressions on the surface of the material, the weight function can be reduced by ij , so that the interpolation result is closer to the actual height change; in the smoother area, by increasing σ ijTo achieve an overall smooth transition. This adaptive adjustment mechanism allows the interpolation process to retain details while avoiding over-smoothing, thereby accurately describing the microstructure of the material surface. Through this interpolation and weighting method, the formula f(x, y) realizes the transformation from discrete sampling points to continuous function form, providing high-quality input data for the subsequent fractional Fourier transform. The fractional Fourier transform can convert these continuous surface height data into the frequency domain and extract the spectral characteristics of the surface structure. These spectral characteristics are key parameters in the subsequent surface modification process. By analyzing and optimizing these characteristics, the optimal modification path and parameters can be determined to achieve precise control and optimization of the material surface. In the specific surface modification process, these extracted spectral characteristics will be input into the nonlinear dynamic model, and the modification target parameters will be obtained by solving the model and minimizing the objective function.

[0065] Example 4: Smoothing parameter σ based on local curvature adaptive adjustment ij Use the following formula to express it:

[0066]

[0067] in, Indicates the height h at the (i,j)th sampling point ij Laplace operator; σ0 is the basic smoothing parameter, which represents the default smoothing degree when there is no local curvature influence; a is the scale factor, which controls the influence of curvature on the smoothing parameter; b is the exponential parameter, which determines the nonlinear degree of curvature influence.

[0068] Specifically, the Laplace operator It represents the second-order derivative of the surface height at that point, reflecting the curvature of the material surface at that point. Where the curvature is large, the surface height changes dramatically, and where the curvature is small, the surface is relatively flat. By introducing the Laplace operator, the influence of local curvature on the smoothing parameter can be reflected in the formula. High curvature areas require smaller smoothing parameters to retain the details of height changes; while low curvature areas require larger smoothing parameters to achieve smooth transitions and reduce the impact of noise. The basic smoothing parameter σ0 provides a smooth benchmark value, which represents the default smoothing degree when there is no local curvature influence. The scale factor a and the exponential parameter b determine the degree of influence and nonlinear characteristics of the curvature on the smoothing parameter. The scale factor a controls the extent to which the curvature affects the smoothing parameter, while the exponential parameter b determines the nonlinearity of this influence. A larger a value makes the influence of curvature on the smoothing parameter more significant, and a smaller b value makes this influence more nonlinear. This adaptive adjustment mechanism enables the interpolation process to dynamically adjust the smoothing parameter, thereby providing higher interpolation accuracy in high curvature areas and achieving smooth transitions in low curvature areas. This method not only retains the details of the material surface, but also effectively suppresses noise and irregularities, improving the overall quality of the data. In the context of the present invention, this precise interpolation method provides high-quality input data for subsequent fractional Fourier transforms. Through fractional Fourier transforms, these continuous surface height data can be converted to the frequency domain to extract the spectral characteristics of the surface structure. These spectral characteristics are key parameters in the surface modification process. By analyzing and optimizing these characteristics, the optimal modification path and parameters can be determined. Specifically, the modification target parameters include surface roughness, surface anisotropy, conductivity and surface energy. Surface roughness affects the friction, wear and lubrication properties of the material; surface anisotropy determines the differences in the physical and chemical properties of the material in different directions; conductivity is the ability of the material to conduct current under the action of an electric field, which is particularly important for electronic materials; and surface energy affects the wettability and adhesion properties of the material. These characteristics determine the performance and reliability of the material in practical applications. Through precise interpolation and spectral features extracted by fractional Fourier transform, a nonlinear dynamic model can be constructed to describe the evolution of the surface properties of the material. This model combines the intrinsic properties and external conditions of the material, and can predict how the surface properties of the material will change under different modification conditions. By optimizing the objective function, specific modification parameters are found to make the target parameters (such as surface roughness, anisotropy, conductivity and surface energy) reach the best state. These modification parameters are used for specific surface modification operations, including chemical treatment, physical treatment, etc. Chemical treatment may involve etching, deposition or doping the surface of the material with specific chemical reagents to change its surface properties; physical treatment may include heat treatment, plasma treatment or laser treatment to adjust the microstructure and properties of the material surface. By precisely controlling these modification operations, it is ensured that the modification effect is consistent with the expected goal, thereby achieving optimization of the material surface.

[0069] Embodiment 5: In step 2, a two-dimensional fractional Fourier transform is performed on the continuous function, and the obtained spectrum distribution is expressed using the following formula:

[0070]

[0071] Among them, K α (x,u) is the horizontal axis Fourier kernel function; K β (y,v) is the vertical axis Fourier kernel function; u is the horizontal axis variable of the two-dimensional fractional Fourier transform; v is the vertical axis variable of the two-dimensional fractional Fourier transform; α and β are the horizontal axis fractional order parameter and the vertical axis fractional order parameter, respectively, both in the range of (0,1), which control the characteristics of the transformation; Ψ α,β (u,v)=λ(|u| α-1 +|v| β-1 )·exp(-μ(u 2 +v 2 )) is the nonlinear correction term, λ is the nonlinear correction strength; μ is the nonlinear correction exponential parameter; is the weight function; σ is the exponential parameter of the weight function; ω is the modulation parameter of the weight function; is the normalization factor, expressed using the following formula:

[0072]

[0073] Where Γ() is the gamma function.

[0074] Specifically, the key to the two-dimensional fractional Fourier transform is that it can achieve arbitrary angle rotation on the time-frequency plane, and by adjusting the fractional order parameters α and β of the horizontal and vertical axes, it provides richer spectral information than the traditional Fourier transform. In the formula, K α (x,u) and K β (y,v) represent the Fourier kernel functions of the horizontal and vertical axes respectively. These kernel functions can flexibly adjust the angle and depth of frequency domain analysis by controlling the fractional order parameters of the transformation. The value range of the fractional order parameters α and β is (0,1). By selecting appropriate values, more detailed and diverse surface characteristic information can be captured in the frequency domain. For example, for some material surfaces with periodic structures, specific fractional order transformations may make these periodic features more prominent, thereby helping to identify and characterize the microstructure of the material. In addition, the nonlinear correction term Ψ introduced in the formula α,β(u,v) further enhances the ability of the transform, enabling it to more accurately describe the complex characteristics of the material surface. The nonlinear correction term consists of two parts: one is a nonlinear function related to the horizontal and vertical frequency variables |u| and |v|, and the other is an exponential decay function. This combination ensures that the spectral distribution can maintain appropriate frequency domain characteristics while taking into account nonlinear effects, thereby providing more accurate and useful spectral information. The introduction of nonlinear correction terms enables the transform to not only capture the linear characteristics of the material surface, but also reflect its nonlinear behavior, which is particularly important for the accurate characterization of complex material surfaces. The weight function W(x,y) also plays an important role in the two-dimensional fractional Fourier transform. The weight function provides an adaptive weighting method by combining Gaussian distribution and sinusoidal modulation, so that the transform can have different weighting effects in different regions. Specifically, the Gaussian distribution term in the weight function This ensures that the weight of the central area is larger, while the weight of the peripheral area gradually decreases, thereby achieving centralized processing of the signal. The sinusoidal modulation term (1+sin(ωx)sin(ωy)) introduces spatial frequency modulation, allowing the transformation to adapt more flexibly to different surface characteristics. This adaptive weighting mechanism helps to improve the flexibility and accuracy of the transformation, enabling it to better reflect the true characteristics of the material surface. Normalization factor The introduction of ensures the standardization and consistency of the transformation results. The normalization factor is implemented by the gamma function Γ, which plays a role of regulation and standardization in the calculation process, making the transformation results under different fractional-order parameters comparable. This is of great significance for analyzing and comparing the characteristics of different material surfaces, especially when optimizing the modification target parameters. Through this complex two-dimensional fractional Fourier transform method, detailed spectral characteristics of the material surface can be obtained. These characteristics include not only the linear characteristics that can be captured by traditional Fourier transform, but also the complex behaviors introduced by nonlinear correction and adaptive weights. These rich spectral information provides key input parameters for subsequent nonlinear dynamic models. In the nonlinear dynamic model, these spectral features will be used to describe the evolution of material surface properties, and determine the optimal modification parameters by solving the optimization objective function. Specifically, by analyzing and optimizing the spectral characteristics, precise control of material surface roughness, anisotropy, conductivity and surface energy can be achieved. For example, surface roughness affects the friction and wear properties of the material. By adjusting the modification parameters, the surface smoothness can be optimized, thereby improving the durability of the material; surface anisotropy affects the physical and chemical properties of the material in different directions. By optimizing the modification parameters, the best performance of the material in a specific application direction can be achieved; conductivity is particularly important for electronic materials. By adjusting the surface treatment process, the electrical properties of the material can be significantly improved; surface energy affects the wettability and adhesion of the material. By precisely controlling the surface energy, the surface functionality of the material can be improved.

[0075] Example 6: Horizontal Axis Fourier Kernel Function K α (x,u) is expressed using the following formula:

[0076]

[0077] Vertical axis Fourier kernel function K β (y,v) is expressed using the following formula:

[0078]

[0079] Among them, l is the imaginary number sign; sgn() is the sign function.

[0080] Specifically, the horizontal axis Fourier kernel function K α (x,u) and the vertical axis Fourier kernel function K β (y,v) form, |x| α-1 and |y| β-1 is the power factor that controls the weight of the transform at different locations. The fractional order parameters α and β adjust the characteristics of the transform so that it can analyze signals at different scales. Smaller α and β emphasize low-frequency components, while larger α and β highlight high-frequency components, which is essential for capturing details and structural changes on the surface of the material. and By introducing information in the complex domain, the transform contains a detailed description of the phase and amplitude. This complex exponential term ensures that the Fourier transform can capture the complexity of the signal, not just its amplitude information. In this way, the Fourier transform can provide richer spectral information about the material surface, including characteristics such as the periodicity, symmetry, and local defects of the surface structure. The fractional Fourier transform can achieve flexible rotation in the frequency domain by adjusting the α and β parameters, so as to analyze the characteristics of the material surface at different angles and scales. This flexibility enables the transform to capture the diversity and complexity of the material surface. For example, when analyzing the surface of a material with a periodic structure, specific α and β values ​​can highlight the periodic features and make them more obvious in the spectrum diagram. This is crucial for identifying and characterizing the microstructure of the material. In addition, the fractional Fourier transform also combines nonlinear correction terms to further enhance the ability of the transform. These nonlinear correction terms include nonlinear functions of the frequency variables |u| and |v| and exponential decay functions, which ensure that the spectral distribution can maintain appropriate frequency domain characteristics while taking into account nonlinear effects. The introduction of nonlinear correction terms enables the transformation to not only capture the linear characteristics of the material surface, but also reflect its nonlinear behavior, which is particularly important for the accurate characterization of complex material surfaces. The weight function W(x,y) also plays an important role in the two-dimensional fractional Fourier transform. The weight function provides an adaptive weighting method by combining Gaussian distribution and sinusoidal modulation, so that the transformation can have different weighting effects in different regions. Specifically, the Gaussian distribution term in the weight function This ensures that the weight of the central area is larger, while the weight of the peripheral area gradually decreases, thereby achieving centralized processing of the signal. The sinusoidal modulation term (1+sin(ωx)sin(ωy)) introduces spatial frequency modulation, allowing the transformation to adapt more flexibly to different surface characteristics. This adaptive weighting mechanism helps to improve the flexibility and accuracy of the transformation, enabling it to better reflect the true characteristics of the material surface. Normalization factor The introduction of ensures the standardization and consistency of the transformation results. The normalization factor is implemented by the gamma function Γ, which plays a role of regulation and standardization in the calculation process, making the transformation results under different fractional-order parameters comparable. This is of great significance for analyzing and comparing the characteristics of different material surfaces, especially when optimizing the modification target parameters. Through this complex two-dimensional fractional Fourier transform method, detailed spectral characteristics of the material surface can be obtained. These characteristics include not only the linear characteristics that can be captured by traditional Fourier transform, but also the complex behaviors introduced by nonlinear correction and adaptive weights. These rich spectral information provides key input parameters for subsequent nonlinear dynamic models. In the nonlinear dynamic model, these spectral features will be used to describe the evolution of material surface properties, and determine the optimal modification parameters by solving the optimization objective function. Specifically, by analyzing and optimizing the spectral characteristics, precise control of material surface roughness, anisotropy, conductivity and surface energy can be achieved. For example, surface roughness affects the friction and wear properties of the material. By adjusting the modification parameters, the surface smoothness can be optimized, thereby improving the durability of the material; surface anisotropy affects the physical and chemical properties of the material in different directions. By optimizing the modification parameters, the best performance of the material in a specific application direction can be achieved; conductivity is particularly important for electronic materials. By adjusting the surface treatment process, the electrical properties of the material can be significantly improved; surface energy affects the wettability and adhesion of the material. By precisely controlling the surface energy, the surface functionality of the material can be improved.

[0081] Example 7: In step 3, the material surface characteristic vector is defined as S(t):

[0082]

[0083] Where, t is time; R a (t) is the surface roughness at time t; A(t) is the surface anisotropy at time t; σ(t) is the conductivity at time t; γ(t) is the surface energy at time t; based on the spectral feature F α,β (u,v), construct the following nonlinear dynamic model:

[0084]

[0085] Among them, k 1,1 =1.2×10 -3 , k 1,2 =0.5, k 1,3 =0.5; k 2,1 =1.8×10 -2 , k 2,2 =0.3; k 3,1 =0.21, k 3,2 =0.7; k 4,1=0.14, k 4,2 =0.5; θ is the temperature; ∈ is a positive number to prevent division by zero; T is the set total time.

[0086] Specifically, the constructed nonlinear dynamic model describes the temporal variation of these characteristics and is combined with the spectral features F α,β (u,v) interaction. The dynamic equations in the model describe the complex relationship between surface properties through various nonlinear functions. For example, for the surface roughness R a (t), the kinetic equation contains a square term R a (t) 2 , which reflects the nonlinear growth trend of roughness. This trend indicates that the surface roughness may show an accelerating growth behavior over time. In addition, the equation also includes A(t)sin(F α,β (u,v)) term, which shows the effect of the periodic interaction between surface anisotropy and spectral characteristics on the roughness. By introducing the exponential decay term exp(-F α,β (u,v) / θ), the model also considers the regulatory effect of temperature on the change of roughness. The temperature parameter θ controls the intensity of this regulatory effect. The higher the temperature, the faster the exponential term decays and the more significant the change in roughness. For the change of surface anisotropy A(t), the model uses the cubic term A(t) 3 and surface roughness R a (t) relationship, through The term reflects the complex nonlinear relationship between the two. This indicates that anisotropy may increase significantly in high roughness areas. This relationship reveals the inhibitory or promoting effect of surface roughness on the development of anisotropy, especially when the microstructure changes significantly. In addition, the model also uses the logarithmic term log(1+σ(t))cos(F α,β (u,v)), combined with the conductivity and spectrum characteristics, describes the effect of conductivity changes on surface anisotropy. The conductivity σ(t) affects the anisotropy smoothly through a logarithmic function, indicating that its effect on anisotropy is more significant in the low conductivity range. For the conductivity σ(t), the model uses the term σ(t)·(F α,β (u,v)+t·cos(πt / T)) reflects the influence of spectral characteristics and time periodicity on conductivity. This means that conductivity is not only directly affected by spectral characteristics, but also related to time-related periodic changes. The time periodic term cos(πt / T) represents the fluctuation behavior of conductivity within a specific time period, which may be related to the periodic changes of the electronic structure of the material or the external electric field. In addition, the model also includes The term describes the combined effect of surface anisotropy and roughness on conductivity. The anisotropy term A(t) combines the change in the second-order derivative of conductivity to show how changes in surface structure affect the conductive path, while the roughness term R a (t) reflects the dynamic regulation of conductivity by microscopic surface structure through the modulation of spectral characteristics. For surface energy γ(t), the model is The term describes how surface roughness and anisotropy affect surface energy through the cumulative effect of time. This exponential term reflects the complex change trend of surface energy over time and the interaction with other characteristics. The time parameter T represents the set total time. Through the cumulative effect of time, the change of surface energy shows a nonlinear growth or decay behavior. In addition, the model also includes The term reflects the regulatory effect of conductivity and spectral characteristics on surface energy. This term describes how conductivity regulates surface energy under the influence of specific spectral characteristics, especially under the influence of high conductivity and high spectral characteristics, the change in surface energy will be more significant. Through these kinetic equations, the model can capture the complex changes in material surface properties over time and reveal the interactions between these properties. By analyzing and optimizing these equations, precise control of material surface properties can be achieved, thereby optimizing the modification target parameters. Specifically, by adjusting the parameter k in the model 1,1 ,k 1,2 ,k 1,3 ,k 2,1 ,k 2,2 ,k 3,1 ,k 3,2 ,k 4,1 , and k 4,2 , the surface roughness, anisotropy, conductivity and surface energy of the material can be precisely controlled to meet the needs of specific applications.

[0087] Example 8: The objective function J of the nonlinear dynamic model defined in step 3 is expressed using the following formula:

[0088]

[0089] Wherein, ∥·∥ represents the L1 norm; λ1 is the preset first regularization coefficient; λ2 is the preset second regularization coefficient; S target is the target surface characteristic vector.

[0090] Specifically, the objective function J consists of three main parts, namely the error between the feature vector and the target feature vector, the first-order derivative regularization term, and the second-order derivative regularization term.

[0091] ∥S(t)-S target ∥ 2Represents the actual surface characteristic vector S(t) and the target surface characteristic vector S target The purpose of this part is to make the actual surface characteristics gradually approach the preset target value, so as to achieve the expected material surface modification effect. By minimizing this error term, the surface roughness R a (t), surface anisotropy A(t), conductivity σ(t) and surface energy γ(t) gradually approach the set ideal values ​​during the optimization process. Part II is a second-order derivative regularization term, which is used to suppress the sharp fluctuations of surface characteristics over time and ensure smoothness during the optimization process. By minimizing the sum of squares of the second-order derivatives, the characteristic vector can be prevented from undergoing unstable and drastic changes during the optimization process, thereby improving the stability of the model. This regularization term is particularly important for surface modification of electronic materials because it can effectively reduce high-frequency oscillations caused by noise and other irregularities, ensuring that the evolution of surface characteristics is smooth and controllable. Part III is a first-order derivative regularization term, which controls the rate of change of surface characteristics over time and ensures the continuity and consistency of the change process. By minimizing the sum of squares of the first-order derivatives, the speed of change of surface characteristics can be smoothed to prevent too fast or too slow changes during the optimization process, thereby ensuring the controllability and practical operability of the model. The introduction of regularization terms not only improves the robustness of the model, but also effectively balances the contradiction between approximation accuracy and smoothness. In the optimization process, the selection of regularization coefficients λ1 and λ2 is crucial. Larger λ1 and λ2 values ​​will strengthen the regularization effect and smooth the changes in surface characteristics, but may cause the optimization results to deviate from the target value; smaller λ1 and λ2 values ​​will weaken the regularization effect and improve the approximation accuracy of the target value, but may introduce instability and noise. Therefore, the selection of appropriate regularization coefficients is the key to ensuring the optimization effect, which needs to be debugged and verified in experiments and practical applications. This optimization process is particularly important in the surface modification method of electronic materials based on fractional Fourier transform. Through fractional Fourier transform, the spectral characteristics of the material surface can be obtained, which reflect the complexity and diversity of the surface structure. In the optimization process, these spectral features are used as input parameters to guide the model to adjust the surface properties. Specifically, the spectral feature F α,β (u,v) provides information about the material surface at different scales and directions, and by combining it with the dynamic model, the evolution of surface properties can be more accurately controlled.

[0092] Example 9: The specific process of solving the solution of the nonlinear dynamic model under the constraint of minimizing the objective function in step 4 to obtain the modified target parameters includes:

[0093] The Lagrangian function including high-order nonlinear terms and coupling terms is constructed by the following formula:

[0094] Among them, Λ(t) is the Lagrange multiplier vector, and then we get the following partial differential equation:

[0095]

[0096] The high-order numerical method is combined with the Jacobi iteration method to solve the above partial differential equations to obtain the specific solution of S(t) and determine the modified target parameter S target .

[0097] Specifically, the Lagrangian function The construction of is the core of the whole process. It combines the objective function J with the dynamic constraints, so that the optimization process must not only approach the target characteristics, but also meet the dynamic laws of the evolution of the material surface characteristics. In the Lagrangian function, Λ(t) is the Lagrangian multiplier vector, which is used to introduce dynamic constraints to ensure the rationality and feasibility of the optimization process. By By performing partial differentiation, we can obtain the corresponding partial differential equations. These equations describe the influence of the error term and regularization term in the objective function on S(t), as well as the change law of the material surface characteristics over time. Specifically, the partial differential equations ensure that the actual surface characteristic vector S(t) is consistent with the target characteristic vector S target The gap between them is as small as possible, and the regularization term is used to suppress high-frequency oscillations and drastic changes to ensure the smoothness and stability of the optimization process. In this optimization process, high-order numerical methods and Jacobi iteration methods are used to solve these partial differential equations. High-order numerical methods can improve the calculation accuracy and ensure the accuracy of the solution, while the Jacobi iteration rule is used to gradually approximate the solution of the equation. Through repeated iterations and adjustments, the optimal solution that meets all constraints can be found, that is, the path of the change of the material surface characteristic vector S(t) over time. In the optimization process, the error term and regularization term of the objective function play a key role. The error term ensures that the actual characteristic vector S(t) is consistent with the target characteristic vector S target The gap between them is as small as possible, so as to achieve the expected modification target. The regularization term ensures the smoothness and stability of the optimization process by suppressing high-frequency oscillations and drastic changes. This regularization treatment is particularly important for complex material surface characteristics. The characteristic vector S(t) obtained by optimization provides detailed modification target parameters, including surface roughness R a(t), surface anisotropy A(t), conductivity σ(t) and surface energy γ(t). These parameters are the key to achieving material surface modification. By precisely controlling these parameters, the performance and reliability of the material can be significantly improved. In practical applications, this optimization process is combined with the spectral characteristics of the fractional Fourier transform, which enables it to capture subtle changes in the surface properties of the material. This combination not only provides rich spectral information and a solid data foundation for material surface modification, but also demonstrates great potential in theoretical and practical applications. In this way, precise control of the surface properties of the material can be achieved, thereby significantly improving the performance and reliability of the material in practical applications.

[0098] The preferred embodiments of the present disclosure are described above with reference to the accompanying drawings, but the scope of the present disclosure is not limited thereby. Any modification, equivalent substitution and improvement made by those skilled in the art without departing from the scope and essence of the present disclosure shall be within the scope of the present disclosure.

Claims

1. A method for surface modification of electronic materials based on fractional Fourier transform, characterized in that: The method performs the following steps: Step 1: Use a high-resolution scanning tunneling microscope to obtain an atomic-level discrete image of the material surface, and convert the obtained discrete image into a continuous function form through interpolation and filtering; Step 2: Perform fractional Fourier transform on the continuous function form to obtain the spectrum distribution and extract the spectrum features; Step 3: Define the material surface characteristic vector, which includes multiple surface characteristic elements; based on the frequency spectrum characteristics, construct a nonlinear dynamic model based on the surface characteristic vector to describe the evolution of surface characteristics, and define the objective function of the nonlinear dynamic model; Step 4: Solve the solution of the nonlinear dynamic model under the constraint of minimizing the objective function to obtain the modification target parameters, and use the modification target parameters to modify the surface of the electronic material.

2. The method for surface modification of electronic materials based on fractional Fourier transform according to claim 1, characterized in that: Surface property elements include at least: surface roughness, surface anisotropy, conductivity and surface energy.

3. The method for surface modification of electronic materials based on fractional Fourier transform according to claim 2, characterized in that: The continuous function form f(x,y) in step 1 is expressed by the following formula: Among them, h ij Indicates the height of the (i,j)th sampling point, x i and j Respectively represent the horizontal and vertical coordinates of the (i, j)th sampling point; Δx and Δy represent the horizontal sampling interval and vertical sampling interval respectively; there are N*M sampling points in total, N is the upper limit of the horizontal axis sampling, and M is the upper limit of the vertical axis sampling; W ij (x,y) is the adaptive weight function, defined as: Among them, x is the horizontal axis coordinate variable; y is the vertical axis coordinate variable; σ ij is a smoothing parameter that is adaptively adjusted based on the local curvature; sinc is the sine Cardinal function.

4. The method for surface modification of electronic materials based on fractional Fourier transform according to claim 3, characterized in that: The smoothing parameter σ is adaptively adjusted based on the local curvature ij Use the following formula to express it: in, Indicates the height h at the (i,j)th sampling point ij Laplace operator; σ0 is the basic smoothing parameter, which represents the default smoothing degree when there is no local curvature influence; a is the scale factor, which controls the influence of curvature on the smoothing parameter; b is the exponential parameter, which determines the nonlinear degree of curvature influence.

5. The method for surface modification of electronic materials based on fractional Fourier transform according to claim 4, characterized in that: In step 2, the continuous function is subjected to a two-dimensional fractional Fourier transform, and the resulting spectrum distribution is expressed using the following formula: Among them, K α (x,u) is the horizontal axis Fourier kernel function; K β (y,v) is the vertical axis Fourier kernel function; u is the horizontal axis variable of the two-dimensional fractional Fourier transform; v is the vertical axis variable of the two-dimensional fractional Fourier transform; α and β are the horizontal axis fractional order parameter and the vertical axis fractional order parameter, respectively, both in the range of (0,1), which control the characteristics of the transformation; Ψ α,β (u,v)=λ(|u| α-1 +|v| β-1 )·exp(-μ(u 2 +v 2 )) is the nonlinear correction term, λ is the nonlinear correction strength; μ is the nonlinear correction exponential parameter; is the weight function; σ is the exponential parameter of the weight function; ω is the modulation parameter of the weight function; is the normalization factor, expressed using the following formula: Where Γ() is the gamma function.

6. The method for surface modification of electronic materials based on fractional Fourier transform according to claim 5, characterized in that: Horizontal axis Fourier kernel function K α (x,u) is expressed using the following formula: Vertical axis Fourier kernel function K β (y,v) is expressed using the following formula: Among them, l is the imaginary number sign; sgn() is the sign function.

7. The method for surface modification of electronic materials based on fractional Fourier transform according to claim 6, characterized in that: In step 3, the material surface characteristic vector is defined as S(t): Where, t is time; R a (t) is the surface roughness at time t; A(t) is the surface anisotropy at time t; σ(t) is the conductivity at time t; γ(t) is the surface energy at time t; based on the spectral feature F α,β (u,v), construct the following nonlinear dynamic model: Among them, k 1,1 =1.2×10 -3 , k 1,2 =0.5, k 1,3 =0.5; k 2,1 =1.8×10 -2 , k 2,2 =0.3; k 3,1 =0.21, k 3,2 =0.7; k 4,1 =0.14, k 4,2 =0.5; θ is the temperature; ∈ is a positive number to prevent division by zero; T is the set total time.

8. The method for surface modification of electronic materials based on fractional Fourier transform according to claim 7, characterized in that: The objective function J of the nonlinear dynamics model defined in step 3 is expressed as follows: Wherein, ∥·∥ represents the L1 norm; λ1 is the preset first regularization coefficient; λ2 is the preset second regularization coefficient; S target is the target surface characteristic vector.

9. The method for surface modification of electronic materials based on fractional Fourier transform according to claim 8, characterized in that: The specific process of solving the solution of the nonlinear dynamic model under the constraint of minimizing the objective function in step 4 to obtain the modified target parameters includes: The Lagrangian function including high-order nonlinear terms and coupling terms is constructed by the following formula: Among them, Λ(t) is the Lagrange multiplier vector, and then we get the following partial differential equation: The high-order numerical method is combined with the Jacobi iteration method to solve the above partial differential equations to obtain the specific solution of S(t) and determine the modified target parameter S target .

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