Numerical study of the lateral resolution in electrostatic force microscopy for dielectric samples

We present a study of the lateral resolution in electrostatic force microscopy for dielectric samples in both force and gradient modes. Whereas previous studies have reported expressions for metallic surfaces having potential heterogeneities (Kelvin probe force microscopy), in this work we take into account the presence of a dielectric medium. We introduce a definition of the lateral resolution based on the force due to a test particle being either a point charge or a polarizable particle on the dielectric surface. The behaviour has been studied over a wide range of typical experimental parameters: tip-sample distance (1-20) nm, sample thickness (0-5) μm and dielectric constant (1-20), using the numerical simulation of the equivalent charge method. For potential heterogeneities on metallic surfaces expressions are in agreement with the bibliography. The lateral resolution of samples having a dielectric constant of more than 10 tends to metallic behaviour. We found a characteristic thickness of 100 nm, above which the lateral resolution measured on the dielectric surface is close to that of an infinite medium. As previously reported, the lateral resolution is better in the gradient mode than in the force mode. Finally, we showed that for the same experimental conditions, the lateral resolution is better for a polarizable particle than for a charge, i.e. dielectric heterogeneities should always look 'sharper' (better resolved) than inhomogeneous charge distributions. This fact should be taken into account when interpreting images of heterogeneous samples.     

A resolution study for electrostatic force microscopy on bimetallic samples using the boundary element method

Electrostatic force microscopy (EFM) is a special design of non-contact atomic force microscopy used for detecting electrostatic interactions between the probe tip and the sample. Its resolution is limited by the finite probe size and the long-range characteristics of electrostatic forces. Therefore, quantitative analysis is crucial to understanding the relationship between the actual local surface potential distribution and the quantities obtained from EFM measurements. To study EFM measurements on bimetallic samples with surface potential inhomogeneities as a special case, we have simulated such measurements using the boundary element method and calculated the force component and force gradient component that would be measured by amplitude modulation (AM) EFM and frequency modulation (FM) EFM, respectively. Such analyses have been performed for inhomogeneities of various shapes and sizes, for different tip-sample separations and tip geometries, for different applied voltages, and for different media (e.g., vacuum or water) in which the experiment is performed. For a sample with a surface potential discontinuity, the FM-EFM resolution expression agrees with the literature; however, the simulation for AM-EFM suggests the existence of an optimal tip radius of curvature in terms of resolution. On the other hand, for samples with strip-?and disk-shaped surface potential inhomogeneities, we have obtained quantitative expressions for the detectability size requirements as a function of experimental conditions for both AM-?and FM-EFMs, which suggest that a larger tip radius of curvature is moderately favored for detecting the presence of such inhomogeneities.     

Analytical Solutions of the Frequency Shifts of Several Modes in Dynamic Force Microscopy Subjected to AC Electrostatic Force

Kelvin probe force microscopy is currently used to measure contact potential difference and topography of sample on a nanometer scale, based on the behaviors of the first two resonant frequencies. In this study, an analytical method to determine the frequency shifts of several modes in dynamic force microscopy subjected to AC electrostatic force is proposed. The numerical results determined by the proposed method are close to the experimental ones. If the amplitude and the frequency of the AC voltage are suitable, a large frequency shift and a linear relation between the frequency shift and the tip-sample distance will be obtained. In other words, properly controlling the AC voltage will increase the accuracy of measuring the sample's topography. Moreover, it is found that even without an optimized compensation dc voltage, increasing the first frequency shift and decreasing the tip radius and the half open angle of the tip can increase the accuracy of measuring a step height. Finally, it is found that when the oscillation amplitude and the tip-sample distance are constant, increasing the negative dc voltage will increase the accuracy of the measurement of contact potential by using the frequency shift. In addition, an assessment of the conventional perturbation method is presented and the two methods are compared.     

Anomalous response of supported few-layer hexagonal boron nitride to DC electric fields: a confined water effect?

We use electric force microscopy (EFM) to study the response of supported few-layer hexagonal boron nitride (h-BN) to an electric field applied by the EFM tip. Our results show an anomalous behavior in the dielectric response of h-BN atop Si oxide for different bias polarities: for a positive bias applied to the tip, h-BN layers respond with a larger dielectric constant than the dielectric constant of the substrate, while for a negative bias, the h-BN dielectric constant appears to be smaller. Based on ab initio calculations, we propose that this behavior is due to a water layer confined between the Si oxide substrate and h-BN layers. This hypothesis was experimentally confirmed by sample annealing and also by a comparative analysis with h-BN on a non-polar substrate.     

Domain wall observation and dielectric anisotropy in PZN–PT by SPM

The voltage drops on the surfaces in PZN–20% PT and PZN–30% PT were investigated using the electrostatic force microscopy (EFM). It was found that the voltage drop per unit length in the a-domain is larger than the one in the c-domains on the sample surface with the ferroelastic domain wall structure, implying that the dielectric constant perpendicular to the spontaneous polarization is larger than the one parallel to the spontaneous polarization in the vicinity of the morphotropic phase boundary (MPB). Dielectric anisotropy in the vicinity of the MPB was discussed.     

耐电晕聚酰亚胺薄膜表面电荷特性

为了解微纳米尺度下聚合物绝缘材料表面电荷生成、发展规律和机理,利用电场力显微镜（electrostatic force microscope,EFM）研究了两种聚酰亚胺薄膜的表面电荷生成及其衰减特性.采用EFM的导电探针在聚酰亚胺薄膜表面注入电荷,并对产生的电荷进行原位表征,结果表明原始（100 HN）和耐电晕（100 CR）两种聚酰亚胺薄膜上电荷生成和衰减特性不同.耐电晕薄膜被注入的表面电荷数量少且注入后衰减较快,其衰减服从指数规律,衰减时间常数为19.9 min;原始薄膜被注入的电荷量较多,衰减时间常数为48.1 min.分析表明,耐电晕薄膜中由于掺杂了Al2O3成分,使得材料的介电常数提高、电阻率下降.介电常数提高使得金属-电介质界面势垒提高,增加了电荷注入难度,导致表面电荷数量少;电阻率下降使得电荷消散速度加快.     

A nonlinear bilayer beam model for an interfacial crack in dielectric bimaterials under mechanical/electrical loading

A bilayer beam model is extended to study the fracture behavior of dielectric interfacial cracks. In this model, a semi-infinite crack with an original opening value is oriented along the interface between two dielectric layers which are under mechanical/electrical loading. Taking into account the effect of the electrostatic traction on the interfacial crack, a nonlinear analytical solution is derived, along with also a developed finite element analysis method where a special constitutive equation for the capacitor element in ANSYS is utilized to simulate the electrostatic tractions. Both the analytical and numerical solutions predict the same results which further show that the elastic and dielectric mismatches can play a significant role in the interfacial cracking behavior under mechanical and electrical loading. Furthermore, the electrostatic tractions may cause hysteresis loops in the curve of crack opening versus applied mechanical displacement or versus applied electric voltage. An applied mechanical load is the driving force for the interfacial cracking, while an applied electric field retards it.     

Mechanical manifestations of rare atomic jumps in dynamic force microscopy

The resonance frequency and the excitation amplitude of a silicon cantilever have been measured as a function of distance to a cleaved KBr(001) surface with a low-temperature scanning force microscope (SFM) in ultrahigh vacuum. We identify two regimes of tip-sample distances. Above a site-dependent critical tip-sample distance reproducible data with low noise and no interaction-induced energy dissipation are measured. In this regime reproducible SFM images can be recorded. At closer tip-sample distances, above two distinct atomic sites, the frequency values jump between two limiting curves on a timescale of tens of milliseconds. Furthermore, additional energy dissipation occurs wherever jumps are observed. We attribute both phenomena to rarely occurring changes in the tip apex configuration which are affected by short-range interactions with the sample. Their respective magnitudes are related to each other. A specific candidate two-level system is also proposed.     

Mapping of local electrical properties in epitaxial graphene using electrostatic force microscopy

Local electrical characterization of epitaxial graphene grown on 4H-SiC(0001) using electrostatic force microscopy (EFM) in ambient conditions and at elevated temperatures is presented. EFM provides a straightforward identification of graphene with different numbers of layers on the substrate where topographical determination is hindered by adsorbates. Novel EFM spectroscopy has been developed measuring the EFM phase as a function of the electrical DC bias, establishing a rigorous way to distinguish graphene domains and facilitating optimization of EFM imaging.     

A technique for dielectric measurement of cylindrical objects in arectangular waveguide

In this paper, the inverse scattering problem of a homogeneous dielectric post in a rectangular waveguide is considered. A novel inversion algorithm, based on the method of moments and eigen analysis, for computation of the dielectric constant of the post (ϵ) from the measured voltage reflection coefficient is introduced. In this method the integral equation for the polarization current induced in the dielectric post is cast into a matrix equation, and then the contribution of ϵ to the resulting reflection coefficient is expressed explicitly using the eigen analysis. It is shown that the dielectric constant can be obtained from the solution of a complex polynomial function which in turn can be obtained numerically using the conjugate gradient method. Practical aspects of dielectric measurement using this technique are discussed. The HP-8510 network analyzer is used to measure the reflection coefficient of dielectric posts in an X-band waveguide sample holder. Metallic and known dielectric posts are used to determine the accuracy of the dielectric measurement technique     

Liquid crystal alignment in electro-responsive nanostructured thermosetting materials based on block copolymer dispersed liquid crystal

Novel well-defined nanostructured thermosetting systems were prepared by modification of a diglicydylether of bisphenol-A epoxy resin (DGEBA) with 10 or 15?wt% amphiphilic poly(styrene-b-ethylene oxide) block copolymer (PSEO) and 30 or 40?wt% low molecular weight liquid crystal 4'-(hexyl)-4-biphenyl-carbonitrile (HBC) using m-xylylenediamine (MXDA) as a curing agent. The competition between well-defined nanostructured materials and the ability for alignment of the liquid crystal phase in the materials obtained has been studied by atomic and electrostatic force microscopy, AFM and EFM, respectively. Based on our knowledge, this is the first time that addition of an adequate amount (10?wt%) of a block copolymer to 40?wt% HBC-(DGEBA/MXDA) leads to a well-organized nanostructured thermosetting system (between a hexagonal and worm-like ordered structure), which is also electro-responsive with high rate contrast. This behavior was confirmed using electrostatic force microscopy (EFM), by means of the response of the HBC liquid crystal phase to the voltage applied to the EFM tip. In contrast, though materials containing 15?wt% PSEO and 30?wt% HBC also form a well-defined nanostructured thermosetting system, they do not show such a high contrast between the uncharged and charged surface.     

Three-dimensional dynamic force spectroscopy measurements on KBr(001): atomic deformations at small tip-sample separations

Three-dimensional dynamic force spectroscopy measurements were carried out above KBr(001) at low temperature in order to investigate the distance dependence of the tip-sample interactions. In particular, the recorded 3D frequency shift data as well as the extracted interaction force and potential energy fields were analysed with respect to influences of tip and/or sample deformations. We found that a postprocessing correction of the observed deformations significantly modifies the magnitude of the extracted interaction forces and also the image contrast.     

Characterization of trapping force on metallic mie particles

Transverse trapping force on three types of metallic Mie particles (gold, nickel, and silver) is measured for different values of the numerical aperture of an objective used for trapping. The experimental results are compared with those calculated with a modified ray-optics model. It is found that, unlike the situation for a trapped dielectric particle, the maximum transverse trapping efficiency for a trapped metallic particle is increased with the numerical aperture of the trapping objective. After consideration of radiometric force, which is caused by the heating effect, and spherical aberration, which is induced by the refractive-index mismatch, the measured results agree well with the theoretical prediction. The magnitude of the radiometric force is approximately ten times stronger than the maximum transverse trapping force.     

Simplified description of optical forces acting on a nanoparticle in the Gaussian standing wave

We study the axial force acting on dielectric spherical particles smaller than the trapping wavelength that are placed in the Gaussian standing wave. We derive analytical formulas for immersed particles with relative refractive indices close to unity and compare them with the numerical results obtained by generalized Lorenz-Mie theory (GLMT). We show that the axial optical force depends periodically on the particle size and that the equilibrium position of the particle alternates between the standing-wave antinodes and nodes. For certain particle sizes, gradient forces from the neighboring antinodes cancel each other and disable particle confinement. Using the GLMT we compare maximum axial trapping forces provided by the Gaussian standing-wave trap (SWT) and single-beam trap (SBT) as a function of particle size, refractive index, and beam waist size. We show that the SWT produces axial forces at least ten times stronger and permits particle confinement in a wider range of refractive indices and beam waists compared with those of the SBT.     

Effects of nonlinear forces on dynamic mode atomic force microscopy and spectroscopy

In this paper, we describe the effects of nonlinear tip-sample forces on dynamic mode atomic force microscopy and spectroscopy. The jumps and hysteresis observed in the vibration amplitude (A) versus tip-sample distance (h) curves have been traced to bistability in the resonance curve. A numerical analysis of the basic dynamic equation was used to explain the hysteresis in the experimental curve. It has been found that the location of the hysteresis in the A-h curve depends on the frequency of the forced oscillation relative to the natural frequency of the cantilever.     

Synthesis of an inhomogeneous dielectric axisymmetric lens

The problem of determining the index of refraction as a function of the distance from the axis of symmetry of an inhomogeneous dielectric lens is considered for the input and output wave fronts. An analytical method of obtaining a solution for a layered lens is proposed, which is based on the geometric optics approximation and the assumption that the index of refraction within each layer is constant. It is demonstrated that this method is applicable to both layered and gradient lenses (provided that the number of layers is sufficiently large).     

High spatial resolution Kelvin probe force microscopy with coaxial probes

Kelvin probe force microscopy (KPFM) is a widely used technique to measure the local contact potential difference (CPD) between an AFM probe and the sample surface via the electrostatic force. The spatial resolution of KPFM is intrinsically limited by the long range of the electrostatic interaction, which includes contributions from the macroscopic cantilever and the conical tip. Here, we present coaxial AFM probes in which the cantilever and cone are shielded by a conducting shell, confining the tip-sample electrostatic interaction to a small region near the end of the tip. We have developed a technique to measure the true CPD despite the presence of the shell electrode. We find that the behavior of these probes agrees with an electrostatic model of the force, and we observe a factor of five improvement in spatial resolution relative to unshielded probes. Our discussion centers on KPFM, but the field confinement offered by these probes may improve any variant of electrostatic force microscopy.     

Surface potential domains on lamellar P3OT structures

In this work the electrostatic properties of poly(3-octylthiophene) thin films have been studied on a nanometer scale by means of electrostatic force microscopy and Kelvin probe microscopy (KPM). The KPM images reveal that different surface contact potential domains coexist on the polymer surface. This result, together with additional capacitance measurements, indicates that the potential domains are related to the existence of dipoles due to different molecular arrangements. Finally, capacitance measurements as a function of the tip-sample bias voltage show that in all regions large band bending effects take place.     

Charge trapping in carbon nanotube loops demonstrated by electrostatic force microscopy

Electronic devices made from carbon nanotubes (CNTs) can be greatly affected by substrate charges, which, for instance, induce strong hysteresis in CNT field effect transistors. In this work, electrostatic force microscopy (EFM) is employed to investigate single-walled nanotubes grown by chemical vapor deposition on SiO2 substrates. We demonstrate the use of this technique to gain quantitative information on the substrate charges. It is found that charge pools with densities around 10(-8) C/cm2 can be trapped inside nanotube loops for extended periods of time, showing that nanotubes can act as confining barriers for substrate charges. The trapped charges can be removed by scanning probe manipulation.     

Force gradient sensitive detection in lift-mode Kelvin probe force microscopy

We demonstrate frequency modulation Kelvin probe force microscopy operated in lift-mode under ambient conditions. Frequency modulation detection is sensitive to force gradients rather than forces as in the commonly used amplitude modulation technique. As a result there is less influence from electric fields originating from the tip's cone and cantilever, and the recorded surface potential does not suffer from the large lateral averaging observed in amplitude modulated Kelvin probe force microscopy. The frequency modulation technique further shows a reduced dependence on the lift-height and the frequency shift can be used to map the second order derivative of the tip-sample capacitance which gives high resolution material contrast of dielectric sample properties. The sequential nature of the lift-mode technique overcomes various problems of single-scan techniques, where crosstalk between the Kelvin probe and topography feedbacks often impair the correct interpretation of the recorded data in terms of quantitative electric surface potentials.