A method for decoupling the topography and resonance frequency of contact resonance atomic force microscope

Abstract

The invention relates to a method for decoupling the morphology and resonance frequency of a contact resonance atomic force microscope, comprising step S1, preparing a microscope, a probe, a phase-locked amplifier, a sample to be tested, a sapphire sheet and a PC host computer; step S2, calibrating probe parameters; step S3, building a resonance frequency measurement device; step S4, fixing the sample; step S5, detecting the resonance frequency, and then collecting the test results of the morphology and resonance frequency of the surface of the sample to be tested and transmitting them to the PC host computer; step S6, processing data, calculating the tilt angle corresponding to the morphology result; drawing the corresponding relationship between the resonance frequency and the tilt angle, and then using a probe-sample contact model to solve the ratio of lateral stiffness to normal stiffness; then substituting into the probe-sample contact model, restoring each pixel point of the test result to an indentation modulus, and completing the decoupling of the morphology coupling.

Description

Technical Field

[0001] The invention belongs to the technical field of precision measurement, and in particular relates to a method for decoupling the morphology and resonance frequency of a contact resonance atomic force microscope. Background Art

[0002] With the continuous advancement of ultra-precision machining technology, various ultra-precision components are widely used in important fields related to the national economy and people's livelihood, such as aerospace, quantum communications, new energy, and biomedicine. For example, optical crystal free-form surfaces for laser-induced nuclear fusion, microlens arrays for high-resolution earth observation systems, and nanostructured optical films for photovoltaic energy cells. In addition to strict requirements on machining geometric accuracy, ultra-precision components are also required to have extremely high structural integrity and service reliability. The surface and sub-surface defects of ultra-precision components are one of the key factors affecting their service reliability. High-precision measurement of the surface and sub-surface defects of ultra-precision components is not only a key requirement to ensure the quality of ultra-precision machining, but also an important frontier development direction in the field of micro-nano measurement.

[0003] Research shows that in addition to the inherent defects of the material itself, the complex mechanical, physical and chemical interaction between the tool and the material during ultra-precision machining is the main reason for inducing defects in the workpiece. The defective areas caused by the machining process mainly exist on the surface and sub-surface of the workpiece. Among them, surface defects are easy to detect and can also be removed through subsequent ultra-precision polishing and other processes; while sub-surface defects are hidden below the surface and are difficult to detect and remove, which is a serious hidden danger affecting the performance and reliability of ultra-precision components. For example: For the primary mirror of the aerospace telescope used in the aerospace observation system, even extremely small sub-surface defects are easy to further expand and cause mechanical stress concentration in the extreme environment of deep space with large temperature differences, eventually leading to mirror distortion, seriously affecting the imaging quality and the safety of the aerospace system. The formation mechanism and morphological characteristics of sub-surface defects generated during ultra-precision machining are different from traditional machining technologies. For example, in the ultra-precision cutting process of brittle materials such as silicon and germanium, due to the combined influence of inherent material properties such as high hardness, low toughness, easy brittle cracking, as well as scale effects and crystal anisotropy, material extrusion phase transition, lattice slip dislocation and other phenomena are prone to occur, resulting in sub-surface damage. Specifically, they manifest as sub-surface defects such as micro-nanoscale fractures, cracks, dislocations, twin structures, etc., and their scales can be as large as several microns, hundreds of nanometers, or as small as tens of nanometers or even several nanometers.

[0004] In order to meet the sub-surface detection needs of ultra-precision machining components, micro-nano-scale sub-surface defect detection has become a key technical hotspot. Common detection methods include electron microscopy, ultrasound and other methods. Using a transmission electron microscope, sub-nanometer sub-surface defect detection can be achieved by extracting cross-sections of the sample to be tested for transmission detection. However, this method requires that the sample used for detection is thin enough, so the sample needs to be sliced ​​and thinned by focused ion beam. On the one hand, this method will cause irreversible damage to the sample. On the other hand, the method of focused ion beam sampling and thinning is not only time-consuming, but also has high costs and operating conditions. Ultrasound, as the most common defect detection method, is widely used in various fields such as medicine and industry. The ultrasonic method can analyze the ultrasonic echo signal to realize micron-level sub-surface defect detection, and can achieve high-precision three-dimensional positioning of defective materials. The resolution of this method is determined by the Rayleigh limit and increases with the increase of ultrasonic drive frequency. However, the existing GHz ultrasonic drive detection system cannot achieve nanometer-level high-precision detection.

[0005] In order to meet the needs of non-destructive sub-surface defect detection with nano-level precision, ultrasonic atomic force microscopy (UAFM) combining ultrasonic vibration with atomic force microscopy came into being. UAFM reflects the sub-surface defect information of materials by driving the atomic force microscope probe with nano-level amplitude from kHz to MHz and tracking the changes in the probe working frequency. This method gives the atomic force microscope with nano-level lateral resolution the ability to characterize sub-surface defects, while also retaining extremely high detection resolution. For samples with nano-level flat surfaces, UAFM can detect nano-level defect structures buried under hundreds of nanometers or even microns, and is a very promising sub-surface defect detection technology for ultra-precision components. However, this method will cause coupling between surface morphology and resonant frequency signals when there are morphological fluctuations on the sample surface, which weakens the scope of application of this method to a certain extent. Decoupling of surface morphology and resonant frequency information is an urgent need for UAFM to achieve high-accuracy detection of ultra-precision components with surface morphology, such as metasurface lenses. In addition, the decoupling of surface morphology and material modulus information can also achieve high-precision detection of material micro-areas and improve the adaptability of material characterization.

[0006] The prior art CN201711249013.2 also discloses a method for exciting probe vibration in an atomic force microscope. This method can scan and image the sample surface, and the accuracy of the imaging results needs to be further improved. Summary of the invention

[0007] The purpose of this technical solution is to provide a method for decoupling the contact resonance atomic force microscope morphology and resonance frequency, which can achieve the decoupling of surface morphology and material modulus information.

[0008] The purpose of this technical solution is achieved in this way:

[0009] A method for decoupling the morphology and resonance frequency of a contact resonance atomic force microscope, characterized in that it specifically comprises the following steps:

[0010] Step S1, prepare equipment: including a microscope, a probe, a lock-in amplifier, a sample to be tested, a sapphire sheet and a PC host computer; the microscope is an atomic force microscope, and the probe is an atomic force microscope probe;

[0011] Step S2, probe parameter calibration: install the probe on the atomic force microscope, and use a sapphire sheet to calibrate the probe sensitivity and determine the cantilever optical lever sensitivity constant a; then calibrate the probe stiffness by the thermal noise method;

[0012] Step S3, build a resonance frequency measurement device: connect the response signal (Vertical signal) of the four-quadrant photodetector of the atomic force microscope to the signal input port of the phase-locked amplifier. The response signal is the difference between the upper plate voltage and the lower plate voltage of the four-quadrant photodetector, which is used to reflect the longitudinal deflection of the probe. Connect the excitation signal output port of the phase-locked amplifier to the piezoelectric ceramic drive interface of the atomic force microscope probe. The excitation signal is used to drive the probe to generate mechanical vibration; output the frequency signal of the phase-locked amplifier to the PC host computer;

[0013] Step S4, fixing the sample: fixing the sample on the atomic force microscope stage by vacuum adsorption, and then moving it to the atomic force microscope detection area;

[0014] Step S5, resonance frequency detection: record the set value Setpoint of the atomic force microscope, which is the working deflection of the probe cantilever, and is used to multiply the probe sensitivity described in step S2 to calculate the contact force, which is generally 20-100nN; then insert the needle and contact the sample, use a phase-locked amplifier to drive the probe with a voltage of 10-50mV in the range of 10nm, then scan the probe resonance frequency through the phase-locked amplifier, select the second-order resonance peak and perform contact resonance frequency scanning after phase locking, collect the test results of the surface morphology and resonance frequency of the sample to be tested and transmit them to the PC host computer;

[0015] Step S6, data processing, importing the test data into MATLAB software, the Chinese name of which is Matrix Lab, a commercial mathematical software produced by MathWorks, USA; using the software to calculate the tilt angle corresponding to the morphology result; and plotting the resonance frequency. The corresponding relationship with the tilt angle is then solved using the probe-sample contact model to obtain the ratio of the lateral stiffness to the normal stiffness. ; then Substitute the probe-sample contact model and restore each pixel point of the test result to the indentation modulus to complete the decoupling of morphology coupling.

[0016] Preferably, in step S6, the probe-sample contact model equates the normal and tangential contact between the probe tip and the sample to and Two springs, the probe is installed at a fixed tilt angle due to the design of the atomic force microscope , the probe cantilever a is equivalent to the length The cantilever beam has a probe tip length of , the theoretical modeling expression formula of the tilted cantilever is:

[0017]

[0018] in:

[0019]

[0020]

[0021]

[0022] In formula (1) is the wave number, which is given by the resonant frequency:

[0023]

[0024] In formula (2): , where is the Young's modulus of the cantilever, is the moment of inertia of the cantilever section, , is the cantilever width, is the cantilever thickness, is the cantilever cross-sectional area; is the cantilever nth order circular frequency, and the resonant frequency The conversion relationship is as follows:

[0025]

[0026] In formula (1) is the cantilever stiffness, which is obtained by the thermal noise method in step S2 ; is the contact stiffness between the probe and the sample, calculated as follows:

[0027]

[0028] In formula (4) is the probe contact radius, which can be obtained from electron microscope photos. is the reduced modulus of the probe and sample, calculated as follows:

[0029]

[0030] in: and are the indentation modulus of the sample and the probe, respectively; and General indentation modulus Calculation formula for indentation modulus It is a parameter used to characterize the elastic properties of materials through indentation testing, reflecting the stiffness or deformation resistance of the material when subjected to local stress.

[0031]

[0032] In the formula , are Young’s modulus and Poisson’s ratio of the corresponding materials, respectively;

[0033] The contact force in formula (4) The following partial calculations are used:

[0034]

[0035] The displacement stiffness of the cantilever caused by the torque is and angular stiffness As shown below:

[0036]

[0037] In formula (1) is the cantilever lateral stiffness and normal stiffness The ratio,

[0038]

[0039] Here According to formula (4), Related to the shear modulus of the sample, reduced shear modulus The calculation is as follows:

[0040]

[0041] in and are the shear modulus of the sample and probe, respectively, and Universal cut modulus The formula for the cut modulus According to the Young's modulus of the material and Poisson's ratio Calculation of relationships;

[0042] .

[0043] Preferably, in step S6, the ratio of lateral stiffness to normal stiffness is solved by the relationship between the angle and the resonant frequency. The method is as follows:

[0044] According to formulas (2) and (3), the measured contact resonance frequency is converted into wave number, and formula (1) is rewritten as follows:

[0045]

[0046] Formula (8) is used to calculate the displacement stiffness and angular stiffness of the cantilever caused by the torque, and it is substituted into formula (7) to solve for different angles under the set cantilever deflection voltage;

[0047] The corresponding contact force; in formula (5) and formula (9) Set as a variable, combine formula (7) and formula (4) to get the probe-sample contact stiffness, then combine this formula with (12) and fit it through the constrained linear least squares fitting algorithm to solve The minimum condition that can meet the actual test results ; From this solution, the ratio of lateral stiffness to normal stiffness is obtained For subsequent compensation.

[0048] Preferably, in step S6, each pixel of the test result is restored to the indentation modulus by the following method:

[0049] The measured frequencies are converted into wave numbers using formulas (2) and (3). The corresponding angle also needs to be calculated. For any point, the ratio of the lateral stiffness to the normal stiffness calculated above is Substituting into formula (1), we can solve it according to the angle and wave number to obtain ;Will Substitute into formula (4), where the contact force Formula (7) is used to calculate the angle, from which the reduced modulus of each point can be inferred According to formula (5), the indentation modulus of each point can be restored, thereby realizing the decoupling of surface morphology and resonance frequency detection.

[0050] Compared with the prior art, the technical solution has the following outstanding and beneficial technical effects:

[0051] The method for separating the influence of morphology information on resonance frequency in contact resonance detection of atomic force microscope provided by the present invention is based on the Euler-Bernoulli beam equation, the Hertz contact model, and the cantilever stiffness model in different directions for analysis, and the relationship between the cantilever deflection angle, the applied contact force, and the resonance frequency offset is clarified; after solving the lateral to normal stiffness ratio for unknown material testing, the established model is used to achieve high-precision decoupling of the morphology and contact resonance frequency of the position material with morphology sample. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a schematic diagram of the influence of the separated morphological information on the resonance frequency of the present invention.

[0053] Figure 2 is a schematic diagram of the probe-sample contact model.

[0054] Figure 3 This is a graph showing the decoupling results of the surface morphology and contact resonance frequency of a graphene oxide sample with rectangular groove defects. DETAILED DESCRIPTION

[0055] The specific implementation of the technical solution is further described in detail below with reference to the accompanying drawings.

[0056] like Figure 1-3 As shown, a method for decoupling the morphology and resonance frequency of a contact resonance atomic force microscope is characterized in that it specifically includes the following steps:

[0057] Step S1, preparing equipment: including an atomic force microscope, an atomic force microscope probe, a lock-in amplifier, a sample to be tested, and a sapphire sheet;

[0058] Step S2, probe parameter calibration: install the probe on the atomic force microscope, and use a sapphire sheet to calibrate the probe sensitivity and determine the cantilever optical lever sensitivity constant a; then calibrate the probe stiffness by the thermal noise method;

[0059] Step S3, build a resonance frequency measurement device: connect the Vertical signal of the four-quadrant photodetector of the atomic force microscope to the signal input port of the phase-locked amplifier. The Vertical signal is the difference between the upper plate voltage and the lower plate voltage of the four-quadrant detector, which is used to reflect the longitudinal deflection of the probe. Connect the excitation signal output port of the phase-locked amplifier to the piezoelectric ceramic drive interface of the atomic force microscope probe, output and collect the frequency signal of the phase-locked amplifier for subsequent processing;

[0060] Step S4, fixing the sample: fixing the sample on the atomic force microscope stage by vacuum adsorption, and then moving it to the atomic force microscope detection area;

[0061] Step S5, resonance frequency detection: record the Setpoint of the atomic force microscope, that is, set the cantilever working deflection, which is used to multiply the sensitivity described in step S2 to calculate the contact force. The common contact force is set to 20-100nN, and then insert the needle; after contacting the sample, use a phase-locked amplifier to drive the probe with a voltage of 10-50mV in the range of 10nm, and then scan the probe resonance frequency through the phase-locked amplifier, select the second-order resonance peak and perform contact resonance frequency scanning after phase locking, and collect the test results of morphology and resonance frequency;

[0062] Step S6, data processing, importing the test data into MATLAB software, the Chinese name of which is Matrix Lab, a commercial mathematical software produced by MathWorks, USA; using the software to calculate the tilt angle corresponding to the morphology result; and plotting the resonance frequency. The corresponding relationship with the tilt angle is then solved using the probe-sample contact model to obtain the ratio of the lateral stiffness to the normal stiffness. ; then Substitute the probe-sample contact model and restore each pixel point of the test result to the indentation modulus to complete the decoupling of morphology coupling.

[0063] Preferably, in step S6, the probe-sample contact model equates the normal and tangential contact between the probe tip and the sample to and Two springs, the probe is installed at a fixed tilt angle due to the design of the atomic force microscope , the probe cantilever ax is equivalent to the length The cantilever beam has a probe tip length of , the theoretical modeling expression formula of the tilted cantilever is:

[0064]

[0065] in:

[0066]

[0067]

[0068]

[0069] In formula (1) is the wave number, which is given by the resonant frequency:

[0070]

[0071] In formula (2): , where is the Young's modulus of the cantilever, is the moment of inertia of the cantilever section, , is the cantilever width, is the cantilever thickness, is the cantilever cross-sectional area; is the cantilever nth order circular frequency, and the resonant frequency The conversion relationship is as follows:

[0072]

[0073] In formula (1) is the cantilever stiffness, which is obtained by the thermal noise method in step S2 ; is the contact stiffness between the probe and the sample, calculated as follows:

[0074]

[0075] In formula (4) is the probe contact radius, which can be obtained from electron microscope photos. is the reduced modulus of the probe and sample, calculated as follows:

[0076]

[0077] in: and are the indentation modulus of the sample and the probe, respectively; and General indentation modulus Calculation formula for indentation modulus It is a parameter used to characterize the elastic properties of materials through indentation testing, reflecting the stiffness or deformation resistance of the material when subjected to local stress.

[0078]

[0079] In the formula , are Young’s modulus and Poisson’s ratio of the corresponding materials, respectively;

[0080] The contact force in formula (4) The following partial calculations are used:

[0081]

[0082] The displacement stiffness of the cantilever caused by the torque is and angular stiffness As shown below:

[0083]

[0084] In formula (1) is the cantilever lateral stiffness and normal stiffness The ratio,

[0085]

[0086] Here According to formula (4), Related to the shear modulus of the sample, reduced shear modulus The calculation is as follows:

[0087]

[0088] in and are the shear modulus of the sample and probe, respectively, and Universal cut modulus The formula for the cut modulus According to the Young's modulus of the material and Poisson's ratio Calculation of relationships;

[0089] .

[0090] Preferably, in step S6, the ratio of lateral stiffness to normal stiffness is solved by the relationship between the angle and the resonant frequency. The method is as follows:

[0091] According to formulas (2) and (3), the measured contact resonance frequency is converted into wave number, and formula (1) is rewritten as follows:

[0092]

[0093] Formula (8) is used to calculate the displacement stiffness and angular stiffness of the cantilever caused by the torque, and it is substituted into formula (7) to solve for different angles under the set cantilever deflection voltage;

[0094] The corresponding contact force; in formula (5) and formula (9) Set as a variable, combine formula (7) and formula (4) to get the probe-sample contact stiffness, then combine this formula with (12) and fit it through the constrained linear least squares fitting algorithm to solve The minimum condition that can meet the actual test results ; From this solution, the ratio of lateral stiffness to normal stiffness is obtained For subsequent compensation.

[0095] Preferably, in step S6, each pixel of the test result is restored to the indentation modulus by the following method:

[0096] The measured frequencies are converted into wave numbers using formulas (2) and (3). The corresponding angle also needs to be calculated. For any point, the ratio of the lateral stiffness to the normal stiffness calculated above is Substituting into formula (1), we can solve it according to the angle and wave number to obtain ;Will Substitute into formula (4), where the contact force Formula (7) is used to calculate the angle, from which the reduced modulus of each point can be inferred According to formula (5), the indentation modulus of each point can be restored, thereby realizing the decoupling of surface morphology and resonance frequency detection.

[0097] by Figure 3 For example, the sample under test is graphene oxide with rectangular groove defects. Figure 3 a1 in the figure is the morphology collected from the sample surface. Figure 3 a2 in is the resonance frequency during the sample surface morphology acquisition process; Figure 3 a3 in is the indentation modulus of the sample surface before decoupling, Figure 3 a4 in is the indentation modulus after sample surface decoupling.

[0098] The above shows and describes the basic principle and main features of the technical solution and the advantages of the technical solution. The technicians in this industry should understand that the technical solution is not limited by the above embodiments. The above embodiments and the description only illustrate the principle of the technical solution. Without departing from the spirit and scope of the technical solution, the technical solution will have various changes and improvements, which fall within the scope of the technical solution to be protected. The scope of protection claimed by the technical solution is defined by the attached claims and their equivalents.

Claims

1. A method for decoupling contact resonance atomic force microscope morphology and resonance frequency, characterized in that: The specific steps include: Step S1, prepare equipment: including a microscope, a probe, a lock-in amplifier, a sample to be tested, a sapphire sheet and a PC host computer; the microscope is an atomic force microscope, and the probe is an atomic force microscope probe; Step S2, probe parameter calibration: install the probe on the atomic force microscope and use a sapphire sheet to calibrate the probe sensitivity; then calibrate the probe stiffness by the thermal noise method; Step S3, building a resonance frequency measurement device: connecting the response signal of the four-quadrant photodetector of the atomic force microscope to the signal input port of the phase-locked amplifier; the response signal is the difference between the upper plate voltage and the lower plate voltage of the four-quadrant photodetector, and the response signal is used to reflect the longitudinal deflection of the atomic force microscope probe; connecting the excitation signal output port of the phase-locked amplifier to the piezoelectric ceramic drive interface of the atomic force microscope probe, and the excitation signal is used to drive the probe to generate mechanical vibration; Step S4, fixing the sample: fixing the sample on the atomic force microscope stage by vacuum adsorption, and then moving it to the atomic force microscope detection area; Step S5, resonance frequency detection: record the atomic force microscope setpoint, which is the working deflection of the probe cantilever, and the working deflection is used to multiply the probe sensitivity in step S2 to calculate the contact force; insert the needle and contact the sample, use the phase-locked amplifier to drive the probe with a voltage of 10-50mV in the range of 10nm, and then scan the phase-locked amplifier to search for the probe resonance frequency , select the second-order resonance peak and perform contact resonance frequency scanning after phase locking, collect the morphology results of the surface of the sample to be tested; compare the morphology results and the resonance frequency The signal is transmitted to the PC host computer; Step S6, data processing: The PC host computer calculates the tilt angle of the probe based on the topography results , and plot the resonant frequency With tilt angle The corresponding relationship between the lateral stiffness and the normal stiffness of the probe is then solved using the probe-sample contact model. ; then Substitute the probe-sample contact model and restore each pixel point of the test result to the indentation modulus to complete the decoupling of morphology coupling.

2. A method for decoupling contact resonance atomic force microscope morphology and resonance frequency according to claim 1, characterized in that: In step S6, the probe-sample contact model equates the normal and tangential contact between the probe tip and the sample to and Two springs, the probe is installed at a fixed tilt angle due to the design of the atomic force microscope , the probe cantilever is equivalent to a length The cantilever beam has a probe tip length of , the theoretical modeling expression formula of the tilted cantilever is: ； in: ； ； ； In formula (1) Refers to the wave number, which is determined by the resonant frequency It turns out that: ； In formula (2): , where is the Young's modulus of the cantilever, is the moment of inertia of the cantilever section, , is the cantilever width, is the cantilever thickness, is the cantilever cross-sectional area; is the cantilever n-order circular frequency, and the resonant frequency The conversion relationship is as follows: ； In formula (1) is the cantilever stiffness, which is obtained by the thermal noise method in step S2 ; is the contact stiffness between the probe and the sample, calculated as follows: ； In formula (4) is the probe contact radius, which can be obtained from electron microscope photos. is the reduced modulus of the probe and sample, calculated as follows: ； in: and are the indentation modulus of the sample and the probe, respectively, and General indentation modulus Calculation of ; Indentation modulus It is a parameter used to characterize the elastic properties of materials through indentation testing, reflecting the stiffness or deformation resistance of the material when subjected to local stress. ； In the formula , are Young’s modulus and Poisson’s ratio of the corresponding materials, respectively; The contact force in formula (4) The following partial calculations are used: ； a is the sensitivity constant of the cantilever optical lever obtained by calibrating the probe sensitivity using a sapphire sheet in step S2; The displacement stiffness of the cantilever caused by the torque is and angular stiffness As shown below: ； In formula (1) is the cantilever lateral stiffness and normal stiffness The ratio, ； Here According to formula (4), Related to the shear modulus of the sample, reduced shear modulus The calculation is as follows: ； in and are the shear modulus of the sample and probe, respectively, and Universal cut modulus The formula for the cut modulus According to the Young's modulus of the material and Poisson's ratio Calculation of relationships; 。 3. A method for decoupling contact resonance atomic force microscope morphology and resonance frequency according to claim 2, characterized in that: In step S6, the ratio of lateral stiffness to normal stiffness is solved by the relationship between the angle and the resonant frequency. The method is as follows: According to formulas (2) and (3), the measured contact resonance frequency is converted into wave number, and formula (1) is rewritten as follows: ； Formula (8) is used to calculate the displacement stiffness and angular stiffness of the cantilever caused by the torque, and it is substituted into formula (7) to solve for different angles under the set cantilever deflection voltage; The corresponding contact force; in formula (5) and formula (9) Set as a variable, combine formula (7) and formula (4) to get the contact stiffness between the probe and the sample, then combine this formula with (12) and fit it through the constrained linear least squares fitting algorithm to solve The minimum condition that can meet the actual test results ; From this solution, the ratio of lateral stiffness to normal stiffness is obtained For subsequent compensation.

4. The method for decoupling contact resonance atomic force microscope morphology and resonance frequency according to claim 3, characterized in that: In step S6, each pixel of the test result is restored to the indentation modulus in the following manner: The measured frequencies are converted into wave numbers using formulas (2) and (3). The corresponding angle also needs to be calculated. For any point, the ratio of the lateral stiffness to the normal stiffness calculated above is Substituting into formula (1), we can solve it according to the angle and wave number to obtain ;Will Substitute into formula (4), where the contact force Formula (7) is used to calculate the angle, from which the reduced modulus of each point can be inferred According to formula (5), the indentation modulus of each point can be restored, thereby realizing the decoupling of surface morphology and resonance frequency detection.

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