Erratum: “Sub-10 nm spatial resolution for electrical properties measurements using bimodal excitation in electric force microscopy” [Rev. Sci. Instrum. 92, 023703 (2021)]

3D Imaging and Quantitative Subsurface Dielectric Constant Measurement Using Peak Force Kelvin Probe Force Microscopy

“…Although these methods can achieve higher scan rates, the complex dynamics interplay between electrostatic forces and mechanical vibrational amplitudes (specifically in single‐scan modes) constitutes an important source of artefacts, hence require careful selection of amplitude ratios and applied voltages to avoid interference between channels. [ 49 ] In addition, operating in the intermittent contact mode provides very little control over the contact forces, leading to significant degradation in the stability of the measurements. However, it should be emphasized that imaging the static phase in PF KPFM as described in this work corresponds to parallel imaging in what is usually called EFM mode by measuring the variation in the second‐harmonic signal.…”

“…The dependencies of the capacitance derivatives on the tip‐sample separation distance have been extensively studied and demonstrated by analytical, numerical, and experimental works. [ 47,49,50,54,55,60 ] Therefore, to fit the spectra in Figure 3d, we used the expression for the capacitance between the spherical tip apex at the end of a truncated cone and a dielectric film of thickness d as given by its first‐order gradient: [ 61 ] $$$$\begin{equation}\frac{{\partial C}}{{\partial z}}\ \left( {z,\ {\varepsilon }_r,\ d} \right) = \ 2\pi {\varepsilon }_0\left[ {\frac{{R\tilde{R}}}{{\left( {z + \frac{d}{{\varepsilon _r^{eq}}}} \right)\left( {z + \frac{d}{{\varepsilon _r^{eq}}} + R\ \sin \theta )} \right)}} + \kappa \times \left( {\ln \left( {\frac{H}{{z + \frac{d}{{\varepsilon _r^{eq}}} + \tilde{R}}}} \right) - 1 + \frac{{R{{\cos }}^2\theta /\sin \theta }}{{z + \frac{d}{{\varepsilon _r^{eq}}} + \tilde{R}}}} \right)} \right]\end{equation}$$$$where, $$\tilde{R} = \ R( {1 - sin\theta } )$$, R is the radius of the spherical tip apex, θ is the half‐angle of the cone opening, $$\varepsilon _r^{eq}$$ is the equivalent relative dielectric constant of the entire dielectric film under the tip, and κ = (ln [tan θ/2]) −1 . The expression of the second‐order derivative of the capacitance was determined using Matlab symbolic calculations and fitted to the experimental spectra, as shown in Figure…”

“…The dependencies of the capacitance derivatives on the tip-sample separation distance have been extensively studied and demonstrated by analytical, numerical, and experimental works. [47,49,50,54,55,60] Therefore, to fit the spectra in Figure 3d, we used the expression for the capacitance between the spherical tip apex at the end of a truncated cone and a dielectric film of thickness d as given by its first-order gradient: [61] 𝜕C 𝜕z…”

“…[47] In this configuration, the monitored mechanical oscillations of the cantilever during the second scan (lift mode) are influenced by the variations of the electric field in which it vibrates. Its dynamics are easily simulated by the well-known spring-mass model in the driven damped oscillator, which is commonly used in the fundamental understanding of the tapping mode [48][49][50] (S1, Supporting Information). Therefore, the resonance frequency of the oscillating cantilever depends on the variations of the electrostatic force gradient, which is dominant at a distance from the sample's surface.…”

“…It is worth noting here that the majority of AFM dielectric constant measurement methods reported in the literature are based on measuring the first‐order derivative of the tip‐sample capacitance ($$\frac{{\partial C}}{{\partial z}}$$). The advantages of measuring the second‐order derivative in terms of improving the spatial resolution and the accuracy of measurements have been extensively discussed in the literature [ 47,49,50,54,55 ] (Figure S1, Supporting information).…”