Characterizing Dielectric Permittivity of Nanoscale Dielectric Films by Electrostatic Micro-Probe Technology: Finite Element Simulations

Abstract

Finite element simulations for detecting the dielectric permittivity of planar nanoscale dielectrics by electrostatic probe are performed to explore the microprobe technology of characterizing nanomaterials. The electrostatic force produced by the polarization of nanoscale dielectrics is analyzed by a capacitance gradient between the probe and nano-sample in an electrostatic detection system, in which sample thickness is varied in the range of 1 nm–10 μm, the width (diameter) encompasses from 100 nm to 10 μm, the tilt angle of probe alters between 0° and 20°, and the relative dielectric constant covers 2–1000 to represent a majority of dielectric materials. For dielectric thin films with infinite lateral dimension, the critical diameter is determined, not only by the geometric shape and tilt angle of detecting probe, but also by the thickness of the tested nanofilm. Meanwhile, for the thickness greater than 100 nm, the critical diameter is almost independent on the probe geometry while being primarily dominated by the thickness and dielectric permittivity of nanomaterials, which approximately complies a variation as exponential functions. For nanofilms with a plane size which can be regarded as infinite, a pertaining analytical formalism is established and verified for the film thickness in an ultrathin limit of 10–100 nm, with the probe axis being perpendicular and tilt to film plane, respectively. The present research suggests a general testing scheme for characterizing flat, nanoscale, dielectric materials on metal substrates by means of electrostatic microscopy, which can realize an accurate quantitative analysis of dielectric permittivity.

A gradient electromechanical theory for thin dielectric curved beams considering direct and converse flexoelectric effects

“…Our simulations show that this value increases drastically for smaller thickness of flexoelectric materials and surpasses the coupling coefficient of piezoelectric materials for very thin structures (thickness of the order of a few micron). Along with the mechanical size effects, our model is able to capture electrical size effects at lower scales, as observed by Ren and Sun [19]. The finite element framework developed in this work is generalized and its application to sensor and actuator device design has been demonstrated through various examples.…”

“…The value of mechanical length scale parameters (l 0 = l 1 = l 2 ) are taken as 2 µm [63]. Ren and Sun [19] observed an enhanced electric permittivity of the material at microscale (similar to mechanical stiffness). The electric length scale parameters (l e 0 , l e 1 ) are thus defined to incorporate lower scale effects through a higher-order permittivity term and their values are chosen phenomenologically to accurately predict the flexoelectric material behavior.…”

“…Abdollahi et al [1] found that neglecting converse flexoelectricity results in an incorrect estimation of piezoelectric coefficient using piezoresponse force microscopy (PFM). Moreover, analogous to the mechanical size effects that result in a stiffer material at microscale, recent studies have demonstrated an increase in the electrical permittivity of the material at smaller scales [19]. Motivated by these recent studies, here we aim to develop a generalized two-way coupled size-dependent flexoelectric framework, that can explain these observation and enable modeling of a wider range of actuator and sensor devices by incorporating both direct and converse flexoelectric effects.…”

“…Furthermore, electric field based formulations have been used to analyze flexoelectric beams [37][38][39][40] for different trasnducer applications. However, the existing electric field-strain gradient flexoelectric theories do not consider the higher-order effects due to electrical field gradient in the constitutive relations and also ignore the electrical size effects that contribute to an increase in electric permittivity of the material at lower scales [19]. Recent experimental studies suggest the need for a generalized gradient electroelastic theory that can corroborate the experimental observations and enable design of a wide range of sensor and actuator devices.…”

Magnetic Force Probe Characterizations of Nanoscaled Ferromagnetic Domains: Finite-Element Magnetostatic Simulations

“…If the magnetic coercivity of probe is significantly lower compared with the magnetism of ferromagnetic sample, the probe magnetic moment will be influenced by the magnetic field from sample, while if the condition is reversed, the probe will dominate the ferromagnetic states of sample so that the detected magnetic signal of MFM cannot correctly represent magnetic characteristics of ferromagnetic nanomaterials [25]. For our models, we use numerical calculation method (finite-element difference method) to accurately solve stray magnetostatic fields without any approximations on probe geometry and magnetization which is more reliable than any analytical models [27]. It is indicated that the magnetic probe and nanosample of our MFM model are consistent to present a magnetic force that is accurately linear in dependence on magnetization intensity of sample, as shown in Figure 4a.…”