Atomic force microscope bistable detection and regulation method

Abstract

The invention mainly provides a bistable detection and regulation method for an atomic force microscope. The method comprises the following steps: establishing a nonlinear dynamic model of an atomic force microscope probe; setting parameters according to AFM experiment conditions; performing numerical solution on the kinetic equation by adopting a four-order Runge-Kutta method to obtain a steady-state amplitude and phase response curve of the probe under different probe-sample distances; the kick behavior in the amplitude and phase response curve is analyzed, the occurrence interval of the bistable region is recognized, and kick represents irreversible transition or sudden change of the amplitude and the phase; and outputting a regulation and control suggestion, and adjusting at least one parameter to enable the system to work in a non-bistable interval. According to the technical scheme, through nonlinear dynamic modeling, multi-parameter collaborative optimization and intelligent feedback regulation and control, remarkable beneficial effects are shown in the aspects of bistable suppression and imaging quality improvement of the atomic force microscope in different scenes.

Description

Technical Field

[0001] The present invention relates to the field of atomic force microscopy (AFM) imaging technology, and specifically to a bistable detection and control method based on nonlinear dynamic modeling and parameter optimization, which is used to improve the accuracy and stability of AFM nanoscale characterization. Background Art

[0002] In nanoscale characterization using atomic force microscopy (AFM), the dynamic interaction between the probe and the sample often induces complex nonlinear phenomena, among which bistability has attracted much attention due to its significant interference with imaging quality.

[0003] When the probe approaches or departs from the sample, a sudden change in the system's energy potential leads to irreversible transitions in amplitude and phase, forming hysteresis loops in the loading and unloading curves. This dynamic abruptness not only causes artifacts in surface topography and loss of resolution, but also leads to systematic deviations in the quantification of key parameters such as adhesion and elastic modulus. Although existing research has recognized the existence of bistability, its microscopic mechanism and critical transition conditions still lack a clear theoretical definition. The multi-solution characteristics of the amplitude-phase response in dynamic modes and their correlation with imaging distortion also require further investigation. Summary of the Invention

[0004] To solve the above problems, the present invention provides a method for detecting and controlling the bistability of an atomic force microscope, comprising the following steps:

[0005] Step 1: Establish a nonlinear dynamic model of the atomic force microscope probe. The model is based on a mass-spring-damper system, taking into account the nonlinear effects of adhesion and repulsion between the probe and the sample. The dynamic equation is:

[0006]

[0007] Where m is the equivalent mass, c is the damping coefficient, k is the cantilever stiffness, and F tip-sample is the probe-sample interaction force, including van der Waals force and DMT contact force, F drive For external motivation;

[0008] Step 2: Set the parameters according to the AFM experimental conditions, including free amplitude A0, cantilever stiffness k, probe radius R, quality factor Q, material elastic modulus E, Hamaker constant H, and surface energy γ;

[0009] Step 3: The fourth-order Runge-Kutta method is used to numerically solve the dynamic equation to obtain the steady-state amplitude and phase response curves of the probe at different needle-sample spacings;

[0010] Step 4: Analyze the sudden jump behavior in the amplitude and phase response curves to identify the occurrence interval of the bistability region. The sudden jump is manifested as an irreversible transition or mutation in amplitude and phase;

[0011] Step 5: Output control suggestions, and make the system work in the non-bistable range by adjusting at least one parameter, including: increasing the free amplitude A0, increasing the cantilever stiffness k, reducing the probe radius R, or increasing the quality factor Q.

[0012] Furthermore, the probe-sample interaction force includes the following forms:

[0013] When the distance z>D, the spherical-surface van der Waals force model is used:

[0014]

[0015] When the distance z≤D, the DMT contact force model is introduced to compensate for the short-range repulsive force:

[0016]

[0017] Where H is the Hamaker constant, R is the probe radius, and E * is the equivalent elastic modulus, and a0 is the minimum interaction distance between atoms.

[0018] Furthermore, the phenomenon that the phase change exceeds a preset angle when a sudden jump occurs is used as a criterion for the existence of bistability, and the phase sudden drop interval is used to calibrate the critical point of the bistability.

[0019] Furthermore, the preset angle is 30°.

[0020] Furthermore, step 5 further includes performing separate scanning simulations on the free amplitude A0, cantilever stiffness k, probe radius R and quality factor Q, and material elastic modulus based on the control variable method to analyze their effects on the amplitude and range of the bistability mutation.

[0021] Furthermore, based on the identification of the bistable region, the AFM amplitude is set to the non-bistable working range to achieve stability control of the phase response and suppression of morphological artifacts during the imaging process.

[0022] Furthermore, step 4 of identifying the bistable interval includes:

[0023] Step 4.1: As the probe-sample distance changes, record the simulated response curve of the probe amplitude versus distance.

[0024] Step 4.2: Identify a multi-value segment with upper and lower turning points on the response curve, and determine the distance range corresponding to the multi-value segment as a bistable interval;

[0025] Step 4.3: Use the distance between the upper and lower turning points as a criterion for quantifying the hysteresis behavior and for analyzing or optimizing the excitation amplitude and frequency settings during the imaging process.

[0026] Furthermore, the method further includes setting the following control strategy in the feedback control module of the atomic force microscope: setting a target interval for probe amplitude feedback control according to the bistability interval range predicted by the simulation;

[0027] Adjust the gain parameter in the feedback loop to prevent the probe operating point from falling into the bistability range;

[0028] When the probe signal change rate exceeds a preset threshold, the feedback delay mechanism is triggered or the feedback sensitivity is reduced to suppress system transitions.

[0029] Furthermore, the method further includes the step of inversely identifying system parameters using the simulation model, including: obtaining actual probe amplitude or phase response curves of the AFM system under different operating parameters;

[0030] A simulation model is constructed that contains the parameters to be identified, including cantilever stiffness, damping coefficient, driving frequency and equivalent elastic modulus. By fitting the simulation results with the actual response data, the objective function is calculated and the error is minimized to achieve system parameter inversion.

[0031] Therefore, through nonlinear dynamic modeling, multi-parameter collaborative optimization, and intelligent feedback control, this invention demonstrates significant benefits in suppressing bistability and improving imaging quality in atomic force microscopes. For high-resolution imaging of hard materials, the combination of a high-stiffness cantilever and a high free amplitude effectively suppresses the amplitude jumps and phase distortions caused by bistability, meeting the requirements for nanoscale characterization of precise structures such as semiconductor devices and crystal surfaces.

[0032] In the dynamic monitoring of soft materials, the low-stiffness cantilever and dynamic amplitude adjustment strategy significantly reduce probe contact damage. Combined with scanning rate optimization, high-fidelity capture of dynamic processes such as biological cell surface fluctuations can be achieved, providing key technical support for the study of the mechanical properties of living cells.

[0033] For complex material systems, the combination of multi-parameter inversion algorithm and intelligent database effectively solves the problem of resolving heterojunctions of nanocomposite materials, constructs a "detection-inversion-control" closed-loop system, and significantly improves the efficiency of high-throughput detection.

[0034] Overall, this technical solution breaks through the imaging stability and resolution bottlenecks caused by the bistability effect through cross-scale dynamic modeling and scenario-based parameter decoupling, providing a universal technical framework for the quantitative analysis of atomic force microscopy in fields such as materials science and life sciences, and has significant engineering application value and technological innovation. DETAILED DESCRIPTION

[0035] In order to make the technical problems, technical solutions and beneficial effects to be solved by the present invention more clearly understood, reference may be made to the embodiments to further illustrate the technical solutions of the present invention. It should be understood that the embodiments described herein are only used to explain the technical solutions or principles of the present invention and are not intended to limit the scope of protection of the present invention.

[0036] The present invention provides a method for detecting and controlling the bistability of an atomic force microscope, comprising the following steps:

[0037] Step 1: Establish a nonlinear dynamic model of the atomic force microscope probe. The model is based on a mass-spring-damper system, taking into account the nonlinear effects of adhesion and repulsion between the probe and the sample. The dynamic equation is:

[0038]

[0039] Where m is the equivalent mass, c is the damping coefficient, k is the cantilever stiffness, and F tip-sample is the probe-sample interaction force, including van der Waals force and DMT contact force, F drive is the external excitation force; the probe-sample interaction force includes the following forms;

[0040] When the distance z>D, the spherical-surface van der Waals force model is used:

[0041]

[0042] When the distance z≤D, the DMT contact force model is introduced to compensate for the short-range repulsive force:

[0043]

[0044] Where H is the Hamaker constant, R is the probe radius, and E * is the equivalent elastic modulus, and a0 is the minimum interaction distance between atoms.

[0045] Step 2: Set the parameters according to the AFM experimental conditions, including free amplitude A0, cantilever stiffness k, probe radius R, quality factor Q, material elastic modulus E, Hamaker constant H, and surface energy γ.

[0046] Step 3: The fourth-order Runge-Kutta method is used to numerically solve the dynamic equation to obtain the steady-state amplitude and phase response curves of the probe at different needle-sample spacings.

[0047] Step 4: Analyze the outburst behavior in the amplitude and phase response curves to identify the occurrence interval of the bistability region. The outburst is manifested as an irreversible transition or mutation in amplitude and phase. The phenomenon of a phase change exceeding 30° when the outburst occurs is used as a criterion for the existence of bistability. The phase drop interval is used to calibrate the bistability critical point. The steps for identifying the bistability interval also include Step 4.1: Recording the simulated probe amplitude versus distance response curve during the probe-sample distance change; Step 4.2: Identifying multi-value segments with upper and lower inflection points on the response curve and determining the distance range corresponding to the multi-value segment as the bistability interval; Step 4.3: Using the distance between the upper and lower inflection points as a criterion for quantifying hysteresis behavior and using it to analyze or optimize the excitation amplitude and frequency settings during the imaging process.

[0048] Furthermore, based on the identification of the bistable region, the AFM amplitude can be set to be in the non-bistable working range to achieve stability control of the phase response and suppression of morphological artifacts during the imaging process.

[0049] Step 5: Output control suggestions. Adjust at least one parameter to make the system operate in the non-bistability range, including increasing the free amplitude A0, improving the cantilever stiffness k, reducing the probe radius R, or increasing the quality factor Q. Based on the control variable method, separate sweep simulations are performed for the free amplitude A0, cantilever stiffness k, probe radius R, quality factor Q, and material elastic modulus to analyze their effects on the amplitude and range of the bistability mutation.

[0050] Furthermore, the following control strategies can be set in the feedback control module of the atomic force microscope: (1) set the target range of the probe amplitude feedback control according to the bistable range predicted by simulation; (2) adjust the gain parameter in the feedback loop to prevent the probe operating point from falling into the bistable range; (3) trigger the feedback delay mechanism or reduce the feedback sensitivity when the probe signal change rate exceeds the preset threshold to suppress system transitions.

[0051] In addition, the method can further include the step of using the simulation model to inversely identify the system parameters, including: obtaining the actual probe amplitude or phase response curve of the AFM system under different operating parameters; constructing a simulation model containing the parameters to be identified, including cantilever stiffness, damping coefficient, driving frequency and equivalent elastic modulus; and calculating the objective function and minimizing the error by fitting the simulation results with the actual response data to achieve system parameter inversion.

[0052] Example 1 Bistability suppression of hard material imaging based on parameter scanning

[0053] Model construction and parameter setting The DMT contact model is used to construct the nonlinear dynamic equation of the probe-sample interaction:

[0054]

[0055] The parameter configuration is as follows:

[0056] Probe parameters: free amplitude A0 = 50 nm, high amplitude weakens the surface force gradient through kinetic energy advantage, cantilever stiffness k = 40 N / m, matches the stiffness of hard materials and suppresses the indentation depth, probe radius R = 10 nm, taking into account atomic-level resolution and wear resistance, quality factor Q = 300 (typical high Q value in air, improving force detection sensitivity).

[0057] Sample parameters: Silicon elastic modulus E* = 135 GPa, Hamaker constant H = 2.5 × 10 -19 J, molecular equilibrium distance z0 = 0.38nm.

[0058] Numerical simulation and bistability identification, the fourth-order Runge-Kutta method (RK4) is used to solve the dynamic equations, and the adaptive step size is set from 10 -8 Encrypted to 10 -10 , capturing the sudden jump event when the probe approaches the sample.

[0059] Result analysis: The amplitude-spacing curve shows that at the normalized spacing z c A sudden amplitude jump occurs near / A0=0.9, dropping from 0.8A0 to 0.3A0, and the phase difference drops from 120° to 60°, thus determining that the bistable range is z c / A0∈[0.85,0.95], the hysteresis loop area is quantified as E dis =1.2×10 -18 J.

[0060] Parameter optimization and control were performed to set the normalized spacing corresponding to the amplitude to 1.1 (non-contact state), away from the bistable range, and to avoid the hysteresis effect of the loading / unloading curve.

[0061] Feedback control is adjusted, and the integral gain is reduced from 0.8 to 0.5 to reduce the overshoot response of the feedback system to the sudden jump signal; the differential damping is increased to 0.8 to suppress the high-frequency noise of the cantilever vibration.

[0062] Experimental verification shows that the step structure on the surface of the silicon wafer is imaged, and the results show clear atomic-level step edges, a lateral resolution of 2nm, no step-like artifacts, and a 60% reduction in the hysteresis loop area of the amplitude-spacing curve.

[0063] Through the synergistic effect of high-rigidity cantilever and high free amplitude, the bistable interference in hard material imaging is effectively suppressed, and the resolution is improved to the nanometer level, meeting the precision characterization needs of semiconductor devices, crystal surfaces, etc.

[0064] Example 2: Bistable Adaptive Control of Dynamic Imaging of Soft Materials

[0065] Model adjustment and parameter setting: The modified DMT model is used, and the surface deformation parameter δ = 2 nm is introduced. The dynamic equation is:

[0066]

[0067] The parameter configuration is as follows:

[0068] Probe parameters: free amplitude A0 = 20 nm, reducing contact force and protecting soft material structure; cantilever stiffness k = 2 N / m, matching the elasticity of soft materials and improving sensitivity; probe radius R = 20 nm, blunt tip to reduce puncture damage; quality factor Q = 50, low Q value in liquid environment, adapting to fluid damping.

[0069] Sample parameters: PDMS elastic modulus E = 1 MPa, surface energy γ = 0.02 J / m 2 , the liquid bridge capillary force coefficient β = 0.5.

[0070] Bistable detection and dynamic feedback, through the phase-locked amplifier to extract the phase signal in real time, when the phase difference suddenly drops A sudden drop from 110° to 80° indicates the probe has entered a bistability region. Dynamic amplitude adjustment triggers a feedback loop to automatically increase A0 to 30nm, suppressing adhesion-driven sudden jumps by increasing kinetic energy. The scan rate is reduced to 0.5μm / s, and the data acquisition time per pixel is increased from 10ms to 50ms to stabilize the phase signal.

[0071] It has been verified that when imaging membrane fluctuations of living cells (HeLa cells), the phase artifacts caused by bistability are reduced by 75%, the periodic fluctuations of the cell membrane with a frequency of 0.1-1Hz are accurately captured, the adhesion force measurement error is reduced from ±20% to ±8%, and the time resolution is improved to 100ms.

[0072] Through the synergistic effect of low-stiffness cantilever, dynamic amplitude adjustment and scanning rate optimization, the bistability interference in soft material imaging is effectively suppressed, achieving high-fidelity recording of the dynamic processes of biological samples.

[0073] Example 3 Multi-parameter inversion and intelligent database-assisted bistability prediction

[0074] 1. Experimental data acquisition and model inversion

[0075] (1) Broadband scanning preprocessing: perform an initial scan on an unknown material (such as graphene / PDMS composite film), obtain the amplitude-phase curve, and extract key feature points: sudden contact point: z c =15nm, amplitude A=12nm; phase dip value:

[0076] (2) Parameter inversion model: Construct an objective function including cantilever stiffness k, Hamaker constant H, and surface energy γ:

[0077]

[0078] By optimizing the parameters through genetic algorithm, we can get k = N / m, H = 1.8×10 -19 J,γ=0.035J / m 2 .

[0079] 2. Intelligent database matching and parameter recommendation

[0080] (1) Database construction: Pre-store parameter combinations and bistability intervals of typical materials (such as silicon, PDMS, and graphene) to form a “material-parameter-bistability boundary” mapping relationship.

[0081] (2) Matching and recommendation: The inversion parameters are input into the database and matched to the optimized parameters of the graphene / PDMS composite membrane: A0 = 35 nm, k = 20 N / m, R = 15 nm, Q = 200; the bistability interval prediction z c / A0∈[0.88,0.98].

[0082] 3. Closed-loop feedback and parameter iteration

[0083] Based on the recommended parameter imaging, if the phase fluctuation detected exceeds ±10°, the secondary parameter solution will be automatically called, such as increasing Q to 250, to improve system stability and form a "detection-inversion-control" closed loop.

[0084] Experience has shown that in composite film imaging, the area of the bistable hysteresis loop is reduced by 50%, the phase contrast is improved by 40%, the boundary between the graphene sheet and the PDMS substrate is clearly distinguishable, and the lateral resolution reaches 5 nm.

[0085] By combining multi-parameter inversion with an intelligent database, the efficiency of bistability prediction and parameter optimization in complex material testing is significantly improved, avoiding the time loss of traditional trial-and-error methods and making it suitable for high-throughput nanomaterial characterization.

[0086] The present invention is based on the bistability phenomenon in the AFM dynamic mode, constructs a nonlinear dynamic model of the probe-sample interaction, and reveals the critical conditions and energy transition paths of the sudden jump behavior. By analyzing the dynamic equilibrium relationship between the adhesion force gradient and the cantilever stiffness, the formation mechanism of the bistability hysteresis loop is elucidated; combined with numerical simulation methods, the co-evolution law of the amplitude and phase with the probe-sample spacing is systematically studied, and the interference mechanism of the multi-solution characteristics of the bistability region on the imaging signal is analyzed. The study further explores the intrinsic relationship between the frequency response truncation phenomenon and the morphology-phase coupling effect, and finally proposes a bistability suppression strategy based on parameter optimization. Therefore, the present invention not only provides theoretical support for understanding the nonlinear dynamic characteristics of the AFM dynamic mode, but also lays a methodological foundation for parameter selection and artifact elimination for high-precision imaging.

[0087] The above shows and describes the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions are merely preferred examples of the present invention and are not intended to limit the present invention. Various changes and improvements may be made to the present invention without departing from the spirit and scope of the present invention. Such changes and improvements fall within the scope of the present invention. The scope of protection claimed in the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for detecting and controlling the bistability of an atomic force microscope, characterized in that: The steps include: Step 1: Establish a nonlinear dynamic model of the atomic force microscope probe. The model is based on a mass-spring-damper system and considers the nonlinear effects of adhesion and repulsion between the probe and the sample. The dynamic equation is: Where m is the equivalent mass, c is the damping coefficient, k is the cantilever stiffness, and F tip-sample is the probe-sample interaction force, including van der Waals force and DMT contact force, F drive For external motivation; Step 2: Set the parameters according to the AFM experimental conditions, including free amplitude A0, cantilever stiffness k, probe radius R, quality factor Q, material elastic modulus E, Hamaker constant H, and surface energy γ; Step 3: numerically solve the kinetic equation using the fourth-order Runge-Kutta method to obtain the steady-state amplitude and phase response curves of the probe at different needle-sample spacings; Step 4: Analyze the sudden jump behavior in the amplitude and phase response curves to identify the occurrence interval of the bistability region. The sudden jump is manifested as an irreversible transition or mutation of the amplitude and phase; Step 5: Output control suggestions, and make the system work in the non-bistable range by adjusting at least one parameter, including: increasing the free amplitude A0, increasing the cantilever stiffness k, reducing the probe radius R, or increasing the quality factor Q.

2. The method for detecting and controlling the bistability of an atomic force microscope according to claim 1, wherein: The probe-sample interaction force includes the following forms: When the distance z>D, the spherical-surface van der Waals force model is used: When the distance z≤D, the DMT contact force model is introduced to compensate for the short-range repulsive force: Where H is the Hamaker constant, R is the probe radius, and E * is the equivalent elastic modulus, and a0 is the minimum interaction distance between atoms.

3. The method for detecting and controlling the bistability of an atomic force microscope according to claim 1, wherein: The phenomenon that the phase change exceeds a preset angle when a sudden jump occurs is used as a criterion for the existence of bistability, and the phase sudden drop interval is used to calibrate the bistability critical point.

4. The method for detecting and controlling the bistability of an atomic force microscope according to claim 1, wherein: The preset angle is 30°.

5. The method for detecting and controlling the bistability of an atomic force microscope according to claim 1, wherein: The step 5 further includes performing separate scanning simulations on the free amplitude A0, the cantilever stiffness k, the probe radius R and the quality factor Q, and the material elastic modulus based on the control variable method to analyze their effects on the amplitude and range of the bistability mutation.

6. The method for detecting and controlling the bistability of an atomic force microscope according to claim 1, wherein: Based on the identification of the bistable region, the AFM amplitude is set in the non-bistable working range to achieve stability control of the phase response and suppression of morphological artifacts during the imaging process.

7. The method for detecting and controlling the bistability of an atomic force microscope according to claim 1, wherein: The step 4 of identifying the bistable interval includes: Step 4.1: As the probe-sample distance changes, record the simulated response curve of the probe amplitude versus distance. Step 4.2: Identify a multi-value segment with upper and lower turning points on the response curve, and determine that the distance range corresponding to the multi-value segment is a bistable interval; Step 4.3: The distance between the upper and lower turning points is used as a criterion for quantifying the hysteresis behavior and used to analyze or optimize the excitation amplitude and frequency settings during the imaging process.

8. The method for detecting and controlling the bistability of an atomic force microscope according to claim 1, wherein: The method further includes setting the following control strategy in the feedback control module of the atomic force microscope: According to the bistable range predicted by simulation, the target range of probe amplitude feedback control is set; Adjust the gain parameter in the feedback loop to prevent the probe operating point from falling into the bistability range; When the probe signal change rate exceeds a preset threshold, the feedback delay mechanism is triggered or the feedback sensitivity is reduced to suppress system transitions.

9. The method for detecting and controlling the bistability of an atomic force microscope according to claim 1, wherein: The method further includes the steps of using the simulation model to perform inverse identification of system parameters, including: Obtain the actual probe amplitude or phase response curve of the AFM system under different operating parameters; A simulation model is constructed that contains the parameters to be identified, including cantilever stiffness, damping coefficient, driving frequency and equivalent elastic modulus. By fitting the simulation results with the actual response data, the objective function is calculated and the error is minimized to achieve system parameter inversion.

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