A method and system for quantitatively measuring surface charge of dielectric material based on KPFM-AM mode

Abstract

This invention discloses a method and system for quantitative measurement of surface charge on dielectric materials based on the KPFM-AM mode, relating to tribocatalysis technology, triboelectric nanogenerators, and other fields. By deeply analyzing the probe vibration characteristics driven by an electric field, this invention derives the correlation between surface potential and surface charge density. Based on this, a series of finite element simulations are conducted using COMSOL software to simulate the actual KPFM measurement process: on the one hand, verifying the applicability of the theoretical formula in the scenario of an infinitely large uniformly charged surface; on the other hand, exploring the influence of finite-sized or non-uniformly charged surfaces on the surface potential measurement results. Simultaneously, simulation analysis is performed on the surface potential distribution of point charges, and a fitting formula based on the Lorentz function is proposed for accurate calculation of the surface potential distribution. Finally, based on the above theory and simulation results, a dedicated program is developed that can inversely calculate the surface charge distribution from the surface potential distribution, providing important theoretical support and practical tools for the efficient and accurate characterization of the surface electrical properties of dielectric materials.

Description

Technical Field

[0001] This invention relates to the field of tribocatalysis technology, specifically to a method and system for quantitatively measuring the surface charge of dielectric materials based on the KPFM-AM mode. Background Technology

[0002] Triboelectricity (also known as contact electrification) is the phenomenon of charge transfer occurring when the surfaces of objects come into contact and separate. It plays a crucial role in industrial and scientific applications. In recent years, triboelectric nanogenerators (TENGs) based on this principle have achieved highly efficient conversion of mechanical energy into electrical energy, while tribocatalysis technology has also shown great promise in the field of pollutant degradation due to its environmental and energy-saving advantages. Dielectric materials, due to their poor conductivity and slow charge dissipation, easily accumulate large amounts of surface charge, making them the core application materials for these two technologies. The performance of TENGs and tribocatalysis directly depends on the total amount and spatial density distribution of the accumulated charge on the dielectric surface. Therefore, precise and reliable measurement methods are urgently needed to determine the amount and density distribution of the dielectric surface charge.

[0003] Currently, while Faraday cups can accurately measure the total surface charge, they cannot resolve the spatial distribution of the charge. Therefore, obtaining the charge density distribution by measuring the surface potential has become the mainstream approach. The Kelvin method, a widely used surface potential characterization technique, has been continuously developed since its inception in 1898 for measuring the work function of metals. In 1991, the Kelvin probe force microscope (KPFM), combined with atomic force microscopy (AFM), achieved microscopic measurement of the surface work function. With its extremely high spatial resolution, KPFM is widely used to measure the surface potential and charge distribution of dielectric surfaces. Researchers have used it to investigate the effects of factors such as friction speed, electric field, temperature, and humidity on triboelectricity. To quantify the surface charge density of dielectric samples, a capacitor model has been proposed. The charge density is determined by measuring the surface potential, dielectric film thickness, and relative permittivity using KPFM. However, this model needs further verification. Because the charge in dielectric materials cannot move freely, the surface potential distribution is uneven, and the measurement results are affected by various factors such as surface charge distribution, film parameters, and probe distance. In particular, the applicability of the model needs to be tested when the charge distribution is uneven.

[0004] Therefore, a quantitative measurement method and system for the surface charge of dielectric materials based on the KPFM-AM mode is proposed to solve the above problems. Summary of the Invention

[0005] In view of this, the technical problem to be solved by the present invention is to propose a quantitative measurement method and system for surface charge of dielectric materials based on the KPFM-AM mode. The method explores the influence of sample thickness, relative permittivity and surface charge distribution on the measured surface potential through finite element simulation, and obtains the analytical formula for the surface potential distribution induced by point charge by fitting. This enables the goal of inferring the actual charge density distribution from the measured potential distribution, and provides an accurate charge measurement method for fields such as TENG and tribocatalysis.

[0006] To achieve the above objectives, the present invention provides the following technical solution: a method and system for quantitative measurement of surface charge of dielectric materials based on KPFM-AM mode, the specific steps of which include;

[0007] S1. Set up the KPFM measurement system, place the dielectric sample to be tested on the sample stage and ground the bottom surface of the sample. The dielectric sample to be tested has a structure in which a dielectric film is coated or deposited on a conductive substrate. Define the sample parameters, including the dielectric film thickness. d Relative permittivity ε r ;

[0008] S2, Set KPFM Measurement Parameters: Select KPFM-AM mode. The instrument will apply a bias voltage containing DC and AC components to the probe. The frequency of the AC component is obtained by calibrating the resonant frequency of the probe used. Adjust and record the distance between the probe and the sample surface. h And remain stable;

[0009] S3, Surface Potential Measurement: The probe is controlled to scan a preset area on the sample surface. By controlling parameters such as the probe's free amplitude and target amplitude, the probe operates within the gravitational range during the morphology scanning stage, thereby obtaining the surface potential distribution of the sample. V M ;

[0010] S4, Contact Potential Difference Calibration: Measure the contact potential difference between the probe and the sample stage or conductive substrate. V cpd The surface potential measured in step 3 V M Perform offset correction and eliminate V cpd The impact;

[0011] S5, Surface Charge Density Calculation: Based on a parallel-plate capacitor model with a dielectric, a theoretical relationship is established between the surface charge density of an infinitely large uniformly charged sample and the corrected surface potential. For finite-sized or non-uniformly charged sample surfaces, the Lorentz function is used to fit the point charge-induced surface potential distribution and determine its fitting parameters. The sample surface is then divided into several micro-charged regions. Using a MATLAB curve fitting program, the surface potential distribution, sample and probe parameters are input, and residuals and R are applied. 2 With higher-order differential regularization constraints, the surface charge density distribution is iteratively output.

[0012] Preferably, the iteration termination condition of the MATLAB curve fitting program described in S5 is that the residual is less than a preset threshold and R0 2 ≥0.99, and combined with higher-order differential regularization methods to limit fluctuations in the fitting results that do not conform to physical laws, wherein S5 specifically includes:

[0013] S5.1, Theoretical relationship established based on parallel plate capacitor model: When the sample surface is a uniformly charged infinitely large surface, the surface charge density... σ S With the corrected surface potential V M The specific formula is as follows:

[0014]

[0015] In the formula, This is expressed as the sample thickness. Expressed as the surface charge density of the sample; Expressed as vacuum permittivity, ε 0 = 8.854 × 10 -12 F / m; It is expressed as the relative permittivity of the sample.

[0016] S5.2, When the surface charge distribution of the sample is non-uniform, the specific formula expression for describing the surface potential distribution induced by point charge using the Lorentz function is as follows:

[0017]

[0018] In the formula V M (x) For position x The surface potential measured at [location] A The fitting constant represents the total area under the curve. w Let be the fitting constant representing the full width at half maximum (FWHM). x 0 represents the position of a point charge;

[0019] S5.3 Determining the Lorentz function parameters through numerical fitting. Aand w ,in A The specific formula is expressed as follows:

[0020]

[0021] in Expressed as the charge of a point charge; This is expressed as sample thickness; Expressed as vacuum permittivity, ε 0 = 8.854 × 10 -12 F / m; It is expressed as the relative permittivity of the sample.

[0022] Given w The dimension is length, assuming w Follow the following format:

[0023]

[0024] in C and E Two dimensionless coefficients are used; the fitting results of the simulation data are obtained. E The value is approximately 0.5; for simplified calculation, we use [the value here]. E =0.5, w The specific formula is expressed as follows:

[0025]

[0026] In the formula, This is expressed as sample thickness; This is expressed as the height between the probe and the sample;

[0027] S5.4 divides the sample surface into several micro-charged regions. By integrating the potential contribution of point charges in each region, and combining this with the MATLAB curve fitting program, the surface potential distribution, sample parameters, and probe height are input. h Apply residual constraints, R 2 Constraints and higher-order differential regularization constraints are used to iteratively output the surface charge density distribution. σ(x) The fitting formula used in the program can be expressed as:

[0028]

[0029] In the formula, σ(x) Represents the discrete distribution of surface charge density. This indicates the boundary location of each microscopic region; by summing the surface potential distributions generated in each region, the surface potential distribution of the entire region can be obtained.

[0030] A quantitative measurement system for surface charge of dielectric materials based on KPFM-AM mode includes a KPFM measurement module, a data calculation module, and a control and display module;

[0031] The KPFM measurement module includes an atomic force microscope body, a conductive probe, a cantilever beam assembly, and a bias voltage generation module. The bias voltage generation module can output a composite bias voltage containing DC and AC components, and the cantilever beam assembly can realize the vibration of the probe under the action of an electric field and collect displacement signals.

[0032] Data calculation module: Includes a built-in surface charge density calculation model and MATLAB curve fitting program, inputting calibrated potential data, sample parameters, and probe height from the signal processing unit. h Output surface charge density distribution σ(x) ;

[0033] Control and display module: used to set measurement parameters, control the coordinated operation of each unit, and display the surface potential image and charge density distribution results in real time.

[0034] Preferably, the KPFM measurement module further includes a sample carrying unit and a signal processing unit;

[0035] The sample carrying unit includes a grounded sample stage for fixing the dielectric sample to be tested. The sample stage enables horizontal movement and positioning of the sample. The signal processing unit performs Fourier transform, spectral analysis, and interpolation on the probe displacement signal to determine the potential of the measured surface. V M and complete the contact potential difference V cpd Calibration;

[0036] Data calculation module: Includes the surface charge density calculation model and MATLAB curve fitting program described in claim 1, and inputs the calibrated potential data and sample parameters output by the signal processing unit. d , ε r and probe height h Output surface charge density distribution σ(x) ;

[0037] Control and display module: used to set measurement parameters, control the coordinated operation of each unit, and display the surface potential image and charge density distribution results in real time.

[0038] Preferably, the cantilever length of the conductive probe is on the order of micrometers, the radius of curvature of the probe tip is no greater than 50 nm, the sampling frequency of the signal processing unit is no less than 1 kHz, and the displacement measurement accuracy is no less than 0.1 nm.

[0039] Compared with existing technologies, the present invention provides a method and system for quantitative measurement of surface charge of dielectric materials based on KPFM-AM mode, which has the following advantages:

[0040] 1. Based on the KPFM measurement of the structure of dielectric samples, this invention establishes a "parallel plate capacitor with dielectric" model and derives the theoretical formula between the measured surface potential and surface charge density of an infinitely large uniformly charged surface.

[0041] 2. This invention innovatively uses the finite element simulation method to simulate the measurement process of the KPFM-AM mode, clearly and intuitively revealing the measurement principle of the KPFM-AM mode;

[0042] 3. For non-uniform charged surfaces, the relationship between the potential distribution and surface charge density of the non-uniform charged surface is revealed by simulating the surface potential distribution.

[0043] 4. This invention uses the Lorentz function to fit the surface potential distribution induced by point charge, which is beneficial for calculating the surface potential distribution;

[0044] 5. This invention combines the Lorentz function, which is used to describe the distribution of surface potential induced by point charge, and proposes a calculation method that fits the surface charge density distribution obtained by back-deriving the surface potential distribution measurement results, thereby realizing the quantitative measurement of surface charge density distribution. This is of great significance for research in fields such as triboelectricity and tribocatalysis. Attached Figure Description

[0045] Figure 1 middle

[0046] (a) represents the two-dimensional model of the KPFM measurement process used in COMSOL according to the present invention;

[0047] (b) shows the vibration curves of the probe under different DC bias voltages in the simulation results;

[0048] (c) represents the probe vibration spectrum obtained by performing a Fourier transform on the displacement curve in (b);

[0049] (d) represents the relationship between the fundamental frequency (500Hz) amplitude and the DC bias voltage;

[0050] Figure 2 middle

[0051] (a) shows the surface potential distribution image of a 10μm×10μm region on the sample surface, and the surface potential distribution images measured at different probe heights before and after contact charging;

[0052] (b) shows the surface potential distribution of the same cross section in each image of Figure (a) and the surface potential distribution obtained by simulation under the corresponding conditions;

[0053] Figure 3 This is a schematic diagram of a parallel-plate capacitor model with a dielectric based on the KPFM measurement process of this invention.

[0054] Figure 4 middle

[0055] (a) shows the curves of the simulated potential of an infinitely large uniformly charged surface versus the surface charge density at different sample thicknesses; probe height h =100nm;

[0056] (b) shows the variation curves of the simulated potential of an infinitely large uniformly charged surface with sample thickness under different surface charge densities, and the probe height. h =100nm;

[0057] (c) shows the curves of the simulated potential of an infinitely large uniformly charged surface as a function of surface charge density at different probe heights.

[0058] (d) represents the dimensionless relationship between the simulated potential of an infinitely large uniformly charged surface and the width of the charged region at different probe heights. The dashed lines in the four figures represent the calculated theoretical values.

[0059] Figure 5 middle

[0060] (a) represents the dielectric sample at different probe heights. d =300nm, ε r Point charge ( =2.5) σ P =5×10 -11 Simulated values ​​of the surface potential distribution above (C / m);

[0061] (b) represents the dielectric sample at different cantilever positions. d =300nm, ε r Point charge ( =2.5) σ P =5×10 -11 The simulated value of the surface potential distribution above (C / m), where "Lfet" indicates that the probe cantilever is located on the left side of the probe tip, and "Right" indicates the right side position;

[0062] (c) is represented as dielectric samples of different thicknesses. ε r Point charge ( =2.5) σP =5×10 -11 Simulated values ​​of the surface potential distribution above (C / m), with the probe height set to h=25nm during simulation;

[0063] (d) shows a comparison between the simulated surface potential distribution and the curve fitted using the Lorentz function. Dotted symbols represent simulation results, and line symbols represent fitted curves; in the simulation, point charges are all located at coordinate 0.

[0064] Figure 6 middle

[0065] (a) represents a dielectric sample. d =300nm, ε r The simulation results show the surface potential distribution above a point charge (=2.5), where the charge range is 2.5 × 10⁻⁵. -11 C / m to 1×10 -10 C / m, probe height fixed at h =25nm;

[0066] (b) is the curve obtained by dividing the simulated surface potential value under different point charge values ​​in (a) by the point charge value;

[0067] (c) is represented by using the Lorentz function for a point charge ( σ P =5×10 -11 The coefficients obtained by fitting the potential distribution induced by C / m) A Follow d / ε r A changing function, with the probe height fixed. h =25nm, the dashed line represents the point passing through zero. A The linear fitting was performed;

[0068] (d) represents the application of the Lorentz function to point charges for different sample thicknesses. σ P =5×10 -11 The coefficients obtained by fitting the potential distribution induced by C / m) A With probe height h Changes;

[0069] (e) represents the application of the Lorentz function to a point charge ( σ P =5×10 -11 The coefficients obtained by fitting the potential distribution induced by C / m) w With sample thickness d The probe height is fixed at the change. h =25nm;

[0070] (f) represents the application of the Lorentz function to point charges for different sample thicknesses. σ P =5×10 -11 The coefficients obtained by fitting the potential distribution induced by C / m) w With probe height h Changes;

[0071] Figure 7 middle

[0072] (a) shows the comparison between the simulated value of the surface potential distribution and the calculated value of the fitting formula for a step change in charge distribution; and the surface charge density is calculated using a fitting program, and the original surface charge density is compared with the calculated surface charge density.

[0073] (b) represents a comparison of the surface potential distribution and surface charge density distribution for a sinusoidal function;

[0074] Figure 8 This diagram illustrates the flow chart of a method for quantitatively measuring the surface charge of dielectric materials based on the KPFM-AM mode according to the present invention. Detailed Implementation

[0075] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0076] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0077] For an example, please refer to... Figures 1 to 8 As shown:

[0078] To address the problems mentioned in the technical solutions, this application provides a method for quantitatively measuring the surface charge of dielectric materials based on the KPFM-AM mode, specifically including:

[0079] A simulation model simulating the actual KPFM measurement process was built in COMSOL (see...). Figure 1 (a)). In this model, the dielectric sample is placed at the bottom, and the probe is located at the free end of the cantilever beam, above the sample. The other end of the cantilever beam is fixed and constrained, allowing it to vibrate under the influence of an electric field. The bias voltage applied to the probe contains a DC component and an AC component with a frequency set to ω=500Hz. The bottom surface of the dielectric sample is grounded.

[0080] According to the measurement principle of KPFM, the amplitude of the AC bias applied to the probe only affects the probe's vibration amplitude, not its vibration frequency. Therefore, it does not affect the potential measurement results. In the simulation, we set the AC bias to a fixed amplitude of 1V. Based on the simulation results, we obtained the displacement of the probe tip over time under different DC bias voltages, such as... Figure 1 As shown in (b). Performing a Fourier transform on these vibration curves yields the amplitudes at different vibration frequencies, as shown... Figure 1 As shown in (c), and the change of amplitude at the fundamental frequency (500Hz) with DC bias, as follows: Figure 1 As shown in (d). Then, by interpolation, the DC bias voltage corresponding to the minimum fundamental frequency amplitude was obtained, which is exactly the surface potential measured in KPFM.

[0081] To verify the effectiveness of the above model, this method used KPFM to measure the surface potential on the charged dielectric surface and compared it with the simulation data obtained from the model. In the experiment, a silicon dioxide coated sheet was selected as the sample. The silicon dioxide film thickness was 300 nm, the relative permittivity was 3.9, and the silicon substrate was grounded. First, the surface potential of a 10 μm × 10 μm area on the sample surface was measured as the background potential (see...). Figure 2 (a) Then, the surface of the central 2μm × 2μm region is scanned using the probe tip in a tapping mode. Parameters need to be set to ensure the probe operates within the repulsive region. During the scan, contact electrification occurs due to the contact-separation action between the probe and the sample surface, resulting in charge transfer and a certain amount of charge on the sample surface. Subsequently, surface potential images of the same 10μm × 10μm region are acquired at different probe heights, with the probe heights set in the following order: 300nm - 400nm - 500nm - 600nm - 600nm - 500nm - 400nm - 300nm, as shown below. Figure 2 As shown in (a).

[0082] Measurement results show that, after electrification, the surface potential distribution in the central 2μm × 2μm region differs significantly from the initial surface due to contact electrification. To compare with experimental results, we performed simulation calculations using the same parameters as the experiments. In the simulation, the initial surface charge density was obtained by matching the simulation results with the initial measured potential at a probe height of 300nm, while also considering the charge density decay over time. The comparison results show that the simulated potential distribution at different probe heights highly agrees with the experimental measurements, verifying the effectiveness of the finite element simulation model based on the KPFM measurement process.

[0083] In terms of theoretical model construction, to obtain the relationship between the dielectric surface charge density and the measured surface potential, this scheme proposes a parallel-plate capacitor model with a dielectric based on the KPFM measurement process. In this model, the probe and sample stage are both considered as infinitely large conductive plates, while the dielectric sample is constructed as an infinitely large thin film with a thickness of [missing information]. d The relative permittivity is ε r The dielectric sample has a uniformly distributed charge on its surface, and its surface charge density is... σ S The distance between the upper plate and the sample is... h A bias voltage is applied between the probe and the sample stage, as shown in the following formula:

[0084] (1)

[0085] In the formula, This is represented as the bias voltage applied to the probe; This represents the DC component. This is expressed as the amplitude of the AC component. This is expressed as the frequency of the AC bias voltage;

[0086] like Figure 4 As shown, through electrostatic field analysis, the charge density induced on the lower surface of the probe plate due to the sample surface charge and the bias voltage between the probe and the sample stage can be expressed as:

[0087] (2)

[0088] In the formula, ε0 It is the vacuum permittivity, with a value of 8.854 × 10⁻¹² F / m. The electric field force on the probe... Fel The expression can be represented as:

[0089] (3)

[0090] in S Let represent the area of ​​the upper electrode of the probe. The frequency of the electric field is located at . The components can be decomposed into:

[0091] (4)

[0092] Under the influence of an electric field, the probe vibrates, and the amplitude of its fundamental frequency component... A ω and F ω It is proportional. Then, according to the KPFM measurement mechanism, the surface potential is measured. V M equal to DC bias voltageV dc .when A ω It is zero. Therefore, we can conclude that:

[0093] (5)

[0094] In the formula, This is expressed as the sample thickness. Expressed as the surface charge density of the sample; Expressed as the vacuum permittivity, its value is 8.854 × 10⁻⁶. -12 F / m; It is expressed as the relative permittivity of the sample.

[0095] In addition to the electrostatic potential difference caused by the sample surface charge and the applied bias voltage, the potential difference between the probe and the sample stage also includes the contact potential difference caused by the difference in work function. This contact potential difference can be expressed as:

[0096] (6)

[0097] In the formula ， This is expressed as contact potential difference; Represented as probe work function; Represented as the sample work function; Expressed as electron charge, with a value of -1.6 × 10⁻⁶. -19 C.

[0098] in, This can be considered as part of the DC bias. Therefore, the total bias voltage can be expressed as: This means that, considering the contact potential difference, the measured surface potential can be expressed as:

[0099] (7)

[0100] As shown in formula (7), when there is no charge distribution on the surface of the dielectric sample, the surface potential can be measured simply by offsetting the contact potential difference with a bias voltage. V M equal -V cpd When there is a charge on the surface of the dielectric sample, the DC bias voltage needs to be canceled out simultaneously. V cpd And the potential difference caused by surface charge. In this case, the measured surface potential is related not only to the charge density on the dielectric sample surface, but also to... V cpd Correlation. This means that measurement results will differ when using different probes. However, V cpdDuring the measurement process, this is a measurable constant that only causes a parallel shift in the measured potential distribution, and can therefore be easily eliminated. Therefore, for simplicity, the interaction between the probe and the sample stage is omitted in subsequent simulations. V cpd .

[0101] Due to the significant difference in shape between the probe and the infinitely large plate, the applicability of the parallel-plate capacitor model in actual KPFM measurements needs to be verified. Therefore, a two-dimensional model (such as...) is used. Figure 1 (a) As shown, simulations were performed on an infinitely large uniformly charged surface with surface charge densities ranging from 25 μC / m² to 100 μC / m². The dielectric material used here was SiO₂, with a thickness ranging from 100 nm to 500 nm, and a relative permittivity of [missing value]. εr =3.9.

[0102] like Figure 4 As shown in (a) and 4(b), the measured surface potential is directly proportional to both the surface charge density and thickness of the sample, which is in perfect agreement with the theoretical formula (5). This indicates that the capacitor model proposed in this paper can be applied to KPFM measurements on infinitely large uniformly charged surfaces. In actual experiments, it is usually necessary to adjust the probe height to obtain higher accuracy or resolution. Therefore, the influence of probe height on the measurement results was further investigated. The results show that, in the case of an infinitely large uniformly charged surface, the measured potential is independent of the probe height (see [reference needed]). Figure 4 (c)).

[0103] An infinitely large, uniformly charged surface represents an idealized scenario that rarely occurs in actual experiments. Therefore, a series of studies on dielectric thin films (…) are needed. d =100nm, ε r Simulations were performed on a surface charge density of 2.5 μm, with charged region widths ranging from 0.1 μm to 300 μm. In these simulations, the probe tip was positioned above the center of the charged region, and the probe height varied from 25 nm to 500 nm. In all simulations, the surface charge density was set to a constant. For ease of comparison, a dimensionless potential was used. It is expressed as the ratio of the measured potential to the potential calculated by the theoretical formula (5). Figure 4 (d) gives the results for different probe heights. The relationship between the width of the charged region and its variation shows that the width of the charged region has a significant impact on the measured surface potential. As the width of the charged region increases, the measured potential initially rises rapidly, then gradually increases, eventually approaching the theoretical value. For charged regions with smaller widths, the measured surface potential deviates significantly from the theoretical value. When the width is 0.1 μm, the measured surface potential is less than 10% of the theoretical value. The results also indicate that although the measured surface potential on an infinitely large charged surface is independent of the probe height, for a finite charged surface, the probe height can significantly affect the measurement results. As the probe height increases, the measured surface potential decreases significantly, especially when the width of the charged region is small. When the width is less than 1 μm, increasing the probe height from 25 nm to 500 nm results in a change in the measured surface potential exceeding five times. Since both the width of the charged region and the probe height significantly affect the measurement of the surface potential, the surface charge distribution and probe height should be considered when calculating the surface charge density from the measured surface potential.

[0104] Therefore, if the ideal formula provided by the parallel capacitor model (i.e., formula (5)) is applied to calculate the potential measurement results at different probe heights on the same surface and at the same location, different surface charge densities will be obtained. These findings indicate that the parallel capacitor model is not suitable for calculating the surface charge density of a finite charged region. Therefore, the influence of the width of the charged region on the potential measurement needs further investigation.

[0105] According to existing research, the charge density at a point on a dielectric surface is calculated solely from the surface potential measured at that point. However, according to electrostatic theory, the measured potential at a specific location on a dielectric surface depends not only on the amount of charge at that location but also on the surrounding surface charge. To obtain the measured surface potential distribution from a given surface charge density distribution, it is necessary to analyze the surface potential distribution induced by a point charge (line charge in a two-dimensional model). In the current two-dimensional model, we first conducted numerical simulations. The results show that the measured potential distribution above the point charge exhibits an approximately single-peak curve distribution, and the height and width of this peak curve are significantly affected by factors such as probe height and dielectric layer thickness (see...). Figure 5 As probe height increases, peak height decreases while peak width increases (see...). Figure 5 (a) and (b)). Meanwhile, as the sample thickness increases, both peak height and peak width show an increasing trend (see...). Figure 5 (c)). Furthermore, we observed a slight spatial asymmetry in the measurement curves at higher probe heights (see...). Figure 5 (a)). This asymmetry stems from the influence of the relative positions of the probe cantilever (see...). Figure 5(b) To simplify the analysis, this asymmetry will be ignored in subsequent studies. By fitting and comparing the simulated surface potential distribution results using various functions such as the Gaussian function and the Voigt function, we found that the measured potential distribution of a point charge can be accurately described by the Lorentz function, the mathematical expression of which is as follows:

[0106] (8)

[0107] in V M (x) For position x The surface potential measured at [location] A is a fitting constant, representing the total area under the curve. w is the fitting constant, representing the half-width at half-maximum; x 0 represents the position of a point charge (see...) Figure 5 (d)).

[0108] First, fix the probe height. h Simulations were performed at a wavelength of 25 nm, with a point charge of 2.5 × 10⁻⁶. -11 C / m to 1×10 -10 C / m, dielectric sample thickness d =300nm, relative permittivity ε r =2.5 (see) Figure 6 (a)). The results show that the surface potential V M (x) It is strictly proportional to the point charge (see Figure 6 (b)). In order to obtain V M The analytical expression of a point charge was used to simulate a point charge. σ P =5×10 -11 C / m) in different d , h and ε r The surface potential distribution under different conditions was determined. Subsequently, the potential distribution was fitted according to formula (8) to obtain the results under different conditions. A and w The values ​​are shown. The results indicate that the coefficients... A and d / ε r Roughly proportional and almost unaffected h The impact (see) Figure 6 (c) and 6(d)). The above analysis shows that, A Specifically, it can be represented as

[0109] (9)

[0110] Point charge σ P Located in C / m, B It represents a dimensionless coefficient.

[0111] At a surface charge density of σ S The expression for an infinitely large uniformly charged surface can be obtained through integral formula (8), and its expression can be expressed as follows:

[0112] (10)

[0113] At the same time, the measured potential of an infinitely large surface with uniform charge must also satisfy formula (5). This indicates that formula (5) must be consistent with formula (10), thus yielding B=1.

[0114] Furthermore, the fitting results also show that, w Follow h and d Changes, but with ε r Basically irrelevant (see Figure 6 (e) and Figure 6 (f)). In view of w The unit should be m, we assume w Follow the following format:

[0115] (11)

[0116] in C and E These are two dimensionless coefficients. The fitting results to the simulation data yield... E The value is approximately 0.5. To simplify calculations, we use [the value is not specified here]. E =0.5, further fitting yields:

[0117] (12)

[0118] This is basically consistent with the fitting result of w (see Figure 6 (e) and 6(f)). Therefore, the surface potential distribution of point charges in the two-dimensional model is as follows:

[0119] (13)

[0120] To verify the accuracy of the above formula (13), a two-dimensional model was used to simulate the surface potential distribution of charged surfaces with two different surface charge density distributions, and the corresponding surface potential distribution was calculated using formula (13). The surface potential distributions obtained by the two methods were almost identical, which indicates that the formula for calculating the surface potential distribution of point charges has high accuracy.

[0121] Obtaining the surface charge density distribution from the measured surface potential distribution is crucial. Typically, the measured surface potential distribution is a set of discrete values. Based on parameters such as the measurement frequency and range, the surface can be divided into several uniformly charged micro-regions according to this set of discrete values. Therefore, the surface potential distribution can be expressed as:

[0122] (14)

[0123] in, σ(x) Represents the discrete distribution of surface charge density. This indicates the boundary position of each microscopic region. The surface potential distribution of the entire region can be obtained by summing the surface potential distributions generated in each region; while the surface potential distribution of each region is obtained by integrating the formula (13) for that region.

[0124] Based on formula (14), a curve fitting program using MATLAB was developed. This program requires input of the sample's surface potential distribution, thickness, relative permittivity, and probe height. During the iteration process, residual constraints, R-squared constraints, and higher-order differential regularization constraints are applied to ultimately output the corresponding surface charge distribution.

[0125] against Figure 7 The two simulated surface potential distributions shown are used to fit the surface charge distribution using this program. The resulting surface charge density distribution closely matches the two given surface charge density distributions (see...). Figure 7 Therefore, the method described above for obtaining surface charge density through reverse calculation is effective. Figure 7 As shown, the derived surface charge density distribution is closer to a slowly varying charge density distribution (i.e., a sinusoidal distribution, see...). Figure 7 (b) is a given surface charge density distribution. However, for rapidly changing distributions (i.e., step distributions, see [reference]). Figure 7 (a) The derived distribution deviates significantly from the given distribution.

[0126] In this invention, we investigated the fundamental differences in measuring metallic and dielectric samples using KPFM-AM mode. By modeling a parallel-plate capacitor with a dielectric, we proposed a theoretical formula that correlates the measured surface potential with the surface charge density of the dielectric material. Subsequently, we validated this formula using finite element analysis in COMSOL. This formula can calculate the surface potential of an infinitely large, uniformly charged dielectric film surface based on the surface charge distribution and sample characteristics. Then, to address the applicability of this formula to finite-sized or non-uniformly charged surfaces, we proposed a fitting formula based on the Lorentz function, which can calculate the surface potential distribution induced by point charges. Finally, based on the surface potential distribution, we developed a program to calculate the surface charge distribution. This paper improves upon the current difficulty in quantifying measurement results in KPFM-based dielectric surface potential measurements, providing convenience for the quantitative calculation of charge transfer.

[0127] It should be noted that, in this document, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0128] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for quantitatively measuring the surface charge of dielectric materials based on the KPFM-AM mode, characterized in that, The specific steps include: S1. Set up the KPFM measurement system, place the dielectric sample to be tested on the sample stage and ground the bottom surface of the sample. The dielectric sample to be tested has a structure in which a dielectric film is coated or deposited on a conductive substrate. Define the sample parameters, including the dielectric film thickness. d Relative permittivity ε r ; S2, Set KPFM measurement parameters: Select KPFM-AM mode. The instrument will apply a bias voltage containing DC and AC components to the probe. The frequency of the AC component is obtained by calibrating the resonant frequency of the probe used. Adjust and record the distance between the probe and the sample surface. h ; S3, Surface Potential Measurement: The probe is controlled to scan a preset area on the sample surface. By controlling the probe's free amplitude and target amplitude parameters, the probe operates within the gravitational range during the morphology scanning stage, thereby obtaining the sample surface potential distribution. V M ; S4, Contact Potential Difference Calibration: Measure the contact potential difference between the probe and the sample stage or conductive substrate. V CPD The surface potential measured in step 3 V M Perform offset correction and eliminate V CPD The impact; S5, Surface charge density calculation: The theoretical relationship between the surface charge density of a uniformly charged infinite sample and the modified surface potential was established based on the parallel plate capacitor model with dielectric. For finite or non-uniformly charged sample surface, Lorentz function was used to fit the surface potential distribution induced by point charges and determine the fitting parameters. Then the sample surface was divided into several micro-charged regions. Combined with MATLAB curve fitting program, the surface potential distribution, sample and probe parameters were input and residual R 2 and high-order differential regularization constraints, the surface charge density distribution was iteratively output.

2. The method for quantitative measurement of surface charge of dielectric materials based on KPFM-AM mode according to claim 1, characterized in that, The iteration termination condition for the MATLAB curve fitting program described in S5 is that the residual is less than a preset threshold and R0 = 0. 2 ≥0.99, and combined with higher-order differential regularization methods to limit fluctuations in the fitting results that do not conform to physical laws, wherein S5 specifically includes: S5.1, Establishing theoretical relationships based on the parallel-plate capacitor model: When the sample surface is an infinitely large uniformly charged surface, the surface charge density... σ S With the corrected surface potential V M The specific formula is as follows: In the formula, This is expressed as the sample thickness. Expressed as the surface charge density of the sample; Expressed as vacuum permittivity, ε 0 = 8.854 × 10 -12 F / m; It is expressed as the relative permittivity of the sample; S5.2, When the sample surface has a non-uniform charge distribution, the specific formula expression for describing the surface potential distribution induced by point charge using the Lorentz function is as follows: In the formula V M (x) For position x The surface potential measured at [location] A The fitting constant represents the total area under the curve. w Let be the fitting constant representing the full width at half maximum (FWHM). x 0 represents the position of a point charge; S5.3 Determining the Lorentz function parameters through numerical fitting. A and w ,in A The specific formula is expressed as follows: in Expressed as the charge of a point charge; This is expressed as sample thickness; Expressed as vacuum permittivity, ε 0 = 8.854 × 10 -12 F / m; It is expressed as the relative permittivity of the sample; Given w The dimension is length, assuming w Follow the following format: in C and E Two dimensionless coefficients are used; the fitting results of the simulation data are obtained. E The value is approximately 0.5; for simplified calculation, we use [the value here]. E =0.5, w The specific formula is expressed as follows: In the formula, This is expressed as sample thickness; This is expressed as the height between the probe and the sample; S5.4 divides the sample surface into several micro-charged regions. By integrating the potential contribution of point charges in each region, and combining this with the MATLAB curve fitting program, the surface potential distribution, sample parameters, and probe height are input. h Apply residual constraints, R 2 Constraints and higher-order differential regularization constraints are used to iteratively output the surface charge density distribution. σ(x) The fitting formula used by the program can be expressed as: In the formula, σ(x) Represents the discrete distribution of surface charge density. This indicates the boundary location of each microscopic region; by summing the surface potential distributions generated in each region, the surface potential distribution of the entire region can be obtained.

3. A quantitative measurement system for the surface charge of dielectric materials based on the KPFM-AM mode, applicable to the quantitative measurement method for the surface charge of dielectric materials based on the KPFM-AM mode as described in any one of claims 1-2, characterized in that, It includes a KPFM measurement module, a data calculation module, and a control and display module; The KPFM measurement module includes an atomic force microscope body, a conductive probe, a cantilever beam assembly, and a bias voltage generation module. The bias voltage generation module can output a composite bias voltage containing DC and AC components, and the cantilever beam assembly can realize the vibration of the probe under the action of an electric field and collect displacement signals. Data calculation module: Includes a built-in surface charge density calculation model and MATLAB curve fitting program, inputting calibrated potential data, sample parameters, and probe height from the signal processing unit. h Output surface charge density distribution σ(x) ; Control and display module: used to set measurement parameters, control the coordinated operation of each unit, and display the surface potential image and charge density distribution results in real time.

4. The quantitative measurement system for surface charge of dielectric materials based on KPFM-AM mode according to claim 3, characterized in that, The KPFM measurement module also includes a sample carrying unit and a signal processing unit; The sample carrying unit includes a grounded sample stage for fixing the dielectric sample to be tested. The sample stage enables horizontal movement and positioning of the sample. The signal processing unit performs Fourier transform, spectral analysis, and interpolation on the probe displacement signal to determine the potential of the measured surface. V M and complete the contact potential difference V CPD Calibration.

5. The quantitative measurement system for surface charge of dielectric materials based on KPFM-AM mode according to claim 3, characterized in that, The cantilever length of the conductive probe is on the order of micrometers, the radius of curvature of the probe tip is no greater than 50 nm, the sampling frequency of the signal processing unit is no less than 1 kHz, and the displacement measurement accuracy is no less than 0.1 nm.

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