AI 中文总结
研究外差式开尔文探针力显微镜中电容梯度描述，建立非截断泰勒级数描述并证明其收敛性，推导相关系数及截断标准，扩展至双峰运动，为静电力分量和多模力显微镜中CG动力学提供通用框架。
AI 中文摘要
开尔文探针力显微镜（KPFM）通过导电尖端与表面之间的静电力探测局部表面电位变化，该力取决于电位差和尖端 - 表面电容梯度（CG）。在外差式KPFM中，振荡尖端通常通过将偏置调制电场与CG的一阶截断泰勒级数展开相结合来处理。虽然方便，但这种处理限于定义不明确的小振幅范围，且级数的收敛性未解决。本文建立了超出此近似的CG动力学和由此产生的静电力的严格频谱描述。我们制定了CG的非截断泰勒级数描述，并证明了其在单峰和双峰运动中基于现实哈德利电容模型的收敛性。在单峰情况下，展示了傅里叶级数和泰勒级数描述之间的等效性，推导了主导傅里叶系数的显式表达式，并引入了取代通常小振幅范围定性概念的阶截断标准。然后将形式主义扩展到双峰运动，推导了控制静电相互作用的静态、第一本征模和第二本征模分量的有效CG系数。数值模拟证实了基于泰勒的系数向傅里叶系数的收敛，并支持两种配置中的截断范围层次结构。这项工作为描述开环外差实验中的静电力分量和原子力显微镜可观测物建立了形式基础，并为涉及非线性机电耦合和频率转换的多模力显微镜中的CG动力学提供了一个通用框架。
英文摘要
Kelvin probe force microscopy (KPFM) probes local surface-potential variations through the electrostatic force between a conductive tip and a surface, which depends on the potential difference and the tip-surface capacitance gradient (CG). In heterodyne KPFM, the oscillating tip is usually treated by combining a bias-modulated electric field with a first-order truncated Taylor-series expansion of the CG. Although convenient, this treatment is limited to a poorly defined small-amplitude regime and leaves the convergence of the series unresolved. Here, we establish a rigorous spectral description of the CG dynamics and of the resulting electrostatic force beyond this approximation. We formulate a non-truncated Taylor-series description of the CG and prove its convergence for a realistic Hudlet-based capacitance model in both monomodal and bimodal motion. In the monomodal case, we show the equivalence between Fourier-series and Taylor-series descriptions, derive explicit expressions for the dominant Fourier coefficients, and introduce order-truncation criteria that replace the usual qualitative notion of a small-amplitude regime. We then extend the formalism to bimodal motion and derive the effective CG coefficients governing the static, first-eigenmode, and second-eigenmode components of the electrostatic interaction. Numerical simulations confirm the convergence of the Taylor-based coefficients toward the Fourier coefficients and support the truncation-regime hierarchy in both configurations. This work establishes the formal basis for describing electrostatic force components and AFM observables in open-loop heterodyne experiments and provides a general framework for CG dynamics in multimode force microscopy involving nonlinear electromechanical coupling and frequency conversion.
Comments51 pages, 10 figures, 23 references, a supplementary information file is available
