Scanning microwave impedance microscopy (sMIM) has become a powerful tool for nanoscale characterization, utilizing microwave frequencies to probe the material properties of diverse systems with remarkable spatial resolution. This review offers an in-depth analysis of the foundational principles, technological advancements, and broad applications of sMIM. By harnessing near-field microwave interactions between a sharp metallic probe and the sample, sMIM enables simultaneous acquisition of both real (resistive) and imaginary (capacitive) components of the reflected signal, providing detailed insights into the local permittivity and conductivity of materials at the nanoscale. We address critical challenges, including impedance matching, probe–sample interactions, and the influence of environmental factors such as surface water layers and meniscus formation on resolution and contrast. Recent advancements in finite element modeling and the application of lumped-element circuit models have further enhanced the precision of signal interpretation, enabling more accurate analysis of complex systems. This review highlights sMIM’s wide-ranging applications, from material science and semiconductor diagnostics to biological systems, showcasing its ability to perform non-destructive, high-resolution imaging down to the single-digit nanometer scale. These capabilities position sMIM as an indispensable tool for advancing future innovations in nanotechnology and related fields.

Scanning probe microscopies have revolutionized the field of nanometrology by providing insights into the nanoworld with unprecedented resolution. These methods are increasingly crucial in addressing modern challenges in materials characterization, as highlighted by recent advancements in physics and material science methodologies.1 

The conceptual breakthrough behind scanning probe microscopy that allowed for assessing different properties was the force feedback mechanism connected to the scanning platform used in scanning tunneling microscopy (STM),2 leading to an evolution of new families of probe modalities that continue to reveal unexpected insights into the nanoworld.3–7 While the progression from qualitative to quantitative assessments of images still poses challenges, recent literature is abundant with exquisite examples probing various material properties formerly thought inaccessible to scanning probe techniques (see, for example, Refs. 8 and 9).

A relatively new member of the family is the so-called scanning microwave impedance microscopy (sMIM).7,10–14 Its key operational principle is the use of microwaves evanescent field as a probe, operating in the near-field regime, to extract the real (in-phase, or loss, or resistive component) and imaginary (π/2 out-of-phase, or dispersive, or capacitive component) components of the reflected microwaves.

Operating at the intersection of optics and electronics, microwaves provide unique data insights that promise to revolutionize scanning probe microscopy. This review covers the foundational principles and experimental prerequisites necessary for the qualitative interpretation of these techniques, which are critical for modeling and conducting a scanning microwave impedance microscopy (sMIM) experiment. The discussion will begin with a brief overview of relevant historical developments, followed by methods and methodologies for modeling and signal processing in sMIM systems.

Near-field imaging traces its origins back to Synge, who proposed that an aperture system with a 10 nm radius hovering over a surface at a height of 10 nm could be used to image features at sub-wavelength resolution.15 Actual imaging of objects using microwaves was first demonstrated by Soohoo in 1962,16 and further refined by Ash and Nichols in 197217 to demonstrate a wavelength-to-resolution figure of merit of 60. Ash employed an aperture-based system that collected the reflected microwaves as samples were scanned under the illumination source. These seminal works, together with the theory and proposal developed by Synge, are today considered the origin of near-field microwave microscopy. Aperture-based schemes, however, also have signal detection limits that were described by Bethe upon demonstration that the transmitted energy flux would inversely scale to the sixth power of the aperture size.18 

A significant development took place with the transition from an aperture-based system to apertureless systems. Bryant developed a non-contact method employing the open end of a transmission line for measuring surface resistivity that established the basis of apertureless schemes.19 The coaxial transmission line was made an integral number of half wavelengths corresponding to the measurement frequencies, and its simplicity was key to establishing a quantitative approach based on a lumped element description of the tip–surface interaction.

The key concept of sMIM is to map in-phase and out-of-phase components of the reflected microwaves on a sample surface, which can later be related to conductance, loss, or resistivity, and capacitance, permittivity, or dispersion, respectively. From a lumped element perspective, as one measures the in-phase and quadrature (i.e., π/2 out of phase) components of the reflected microwaves, the common approach is to make a parallel association of reactance and conductance, as discussed in later sections. In Bryant’s experiment, a 50 Ω coaxial cable probe with a central conductor tapered to a diameter of 2a = 1 mm was lapped down to provide maximum capacitive coupling to the sample. The inner conductor, thus mounted, was positioned at a distance of about 1 μm from the surface, creating a capacitive coupling to the surface that could be represented by a series association of this coupling capacitor with the parallel combination of spreading resistance Rs and capacitance Cs of the sample:20 
(1)
(2)

Their calibration protocol included the variation of the frequency to properly match the λ/2 electrical length of the system until the real part of the measured impedance was independent of the probe-sample spacing.

In 1989, Fee et al. proposed that an open-ended transmission line could, itself, serve as the scanning probe. By chemically etching and attaching a tungsten tip to the end of the coaxial line, they demonstrated a resolution better than 30 μm.21 In 1993, Tabib-Azar et al. proposed a compact means of connecting a tip to a microstripline resonator capacitively coupled to a feedline, which, through proper probe construction, was able to couple magnetic or electric fields.3 Vlahacos showed that the resolution was limited by the tip radius and sample distance.22 Later, Gao et al.23 showed that the resolution could be 100 nm by bringing the tip in contact with the sample, and up to two orders of magnitude smaller than the tip radius for insulating samples of moderate relative permittivities, ɛr = 50.24 

Although remarkable, these experiments suffered from tip wear caused by contact abrasion with the surface, which led to frequent, unstable variations in tip–surface impedance that consequently disturbed the amplitude and phase of the reflected microwaves during the course of a scan. A subsequent major improvement in the technique came about when Atomic Force Microscopy (AFM) compatible tips were designed with coplanar waveguide structures,3,25 enabling constant force operation by conventional feedback loops. Later, through the implementation of shielded probes and probe-sample modeling by Finite Element Method (FEM),26 sMIM advanced from a purely qualitative evaluation of systems modeled by lumped elements to a more quantitative analytical technique. Also noteworthy are Quartz-tuning force implementations, which can provide some advantages for low-temperature operation and lower stray capacitances, as shown by Huang et al.27 More recently, a significant breakthrough that has advanced the field is the development of a Johnson-noise-limited, cancellation-free embodiment.28 This advancement integrates conventional monolithic silicon cantilever probes with cancellation-free architecture, achieving Johnson-noise-limited, drift-free operation with a spatial resolution of 15 nm. This work has basically allowed for the use of conventional tips as well as enabled the possibility of absolute rather than relative assessment of the materials’ properties, both essential for expanding the usage of the technique and its adoption by a wider user base.

The sensing strategy bears significant importance for imaging, ranging from scalar detection of reflected amplitude or phase3,17,22 tracking of resonant frequency and quality factor,23,24,29 and heterodyne detection that provides the in-phase and quadrature signals of the reflected microwaves in vector format.19,21,26 These latter embodiments provided the current solutions that we shall be covering in this paper and will be described more thoroughly in subsequent sections.

Meanwhile, as the field continues to advance, the adaptation of clever modulation and spectroscopic schemes,10,30 the demonstration of spatial resolutions as small as a single nanometer,13,27,31 and the application of the technique to areas outside materials science, such as biology,32,33 all suggest that the possibilities of sMIM are far from exhausted.

Accurate modeling of tip–sample interactions in sMIM is essential for interpreting experimental results. Several challenges must be addressed, primarily involving the distribution of the electric field generated by a microwave source with wavelengths of ∼0.1 m. This field is guided by a metallic tip with a radius of only a few nanometers and positioned <1 nm from a surface with specified conductivity and complex permittivity. A recurring issue in these systems is that imperfections in the tip, such as asperities or foreign objects, can introduce artifacts into the imaging process.14 Moreover, rough surfaces can lead to geometry-dependent capacitance variations, which often dominate the variations in the electronic properties of the substrate. These effects complicate the interpretation of images and the accuracy of the measurements (see, for example, the detailed work on bacteria for compensation of topography effects32).

For instance, at microwave frequencies around 3 GHz, water with its high dielectric permittivity significantly influences both experimental results and computational models.12,34 Therefore, understanding the field distribution into the substrate is critical, as it directly determines the resolution limits of the system. To explore this, different tip–substrate configurations were modeled, providing insights into how these fields propagate.

Given the complexity of the geometries involved, analytical solutions tend to oversimplify the problem, making numerical simulations essential.12,35 Both 2D and 3D numerical models are utilized to accurately describe the system. In scenarios where a 2D axisymmetric model is applicable, satisfactory results can be achieved with less computational cost compared to a full 3D model, which is more suited to complex configurations.36 

One of the most effective approaches for simulating a microwave impedance microscope (MIM) is the finite element method (FEM). FEM allows for the division of the problem into smaller, more manageable elements, known as finite elements, which are interconnected through common points or nodes. This method offers an approximate solution to differential equations by focusing on the values at the boundaries,37 representing a powerful tool for solving the intricate, multi-dimensional field distribution challenges inherent to sMIM systems.12,37

1. Physical description of the system

The ideal tip for scanning probe applications is typically modeled as a cone with a specific angle, a parameter determined largely by the fabrication process. Commercially available tips often possess a nominal radius of curvature around 50 nm. However, this idealized geometric model rarely remains intact during practical use, particularly after the tip undergoes an approach or scan. Various factors such as surface debris, electrostatic interactions with nanometer-sized particles, tip wear, surface asperities, contamination, and the ubiquitous presence of water contribute to deviations from this simplified geometry. These factors affect both the tip and its interaction with the surface. A graphical representation of these potential scenarios is schematically shown in Fig. 1.

FIG. 1.

Graphical representation of selected scenarios for the modeling of a sMIM experiment: (a) tip above the surface with native layer on both; (b) ideal tip connected to the surface by a water meniscus; (c) tip with a metallic asperity connected to the surface by a water meniscus; (d) tip with an attached dielectric (ɛr = 4) 5 nm diameter nanoparticle connected to the surface by a water meniscus.

FIG. 1.

Graphical representation of selected scenarios for the modeling of a sMIM experiment: (a) tip above the surface with native layer on both; (b) ideal tip connected to the surface by a water meniscus; (c) tip with a metallic asperity connected to the surface by a water meniscus; (d) tip with an attached dielectric (ɛr = 4) 5 nm diameter nanoparticle connected to the surface by a water meniscus.

Close modal

In scanning probe microscopy (SPM), the standard protocol involves the approach of the tip toward the sample surface by monitoring forces between the tip and the surface. This permits the precise control of the tip height relative to the surface. This is typically achieved by selecting a set point corresponding to a force in the range of a few nanonewtons (nN). The tip–surface interaction can be categorized into two main regimes: attractive and repulsive. In the attractive regime, capillary forces from the native water layer on the surface dominate the interaction, pulling the tip toward the surface. This native water forms a water bridge or meniscus.38–42 In contrast, the repulsive regime signifies physical contact between the tip and the surface.

In the specific context of scanning microwave impedance microscopy (sMIM), z-scans, first introduced by Bryant and Gunn,19 are utilized to calibrate the phase of the measurement. As the tip approaches the surface, finer details of the electromagnetic interaction that provide critical information can only be extracted through appropriate modeling techniques.

A simplified analytical model that captures the tip–surface interaction, particularly focusing on changes in capacitance as the tip moves closer during a z-scan, has been proposed by Hudlet et al.43 This model is instrumental in understanding the capacitive response as the tip approaches the sample, allowing for a more accurate interpretation of experimental data in sMIM applications:
(3)

In the context of modeling, the cylindrical geometry can serve as a canonical approximation to contrast with FEM simulations. As shown in Fig. 2(a), the geometry and the applied boundary conditions12 along with the bias utilized to calculate the tip–substrate impedance. Figures 1(b) and 1(c) also highlight and present the capacitance results evaluated by Eq. (3) and through a two-dimensional (2D) COMSOL model with axial symmetry. These capacitance values are normalized by the change in capacitance over a range from 1 μm to 1 nm.10,44

FIG. 2.

(a) A 2D axisymmetric model is employed in place of a full 3D representation for computational efficiency. Admittance is measured between the tip and the sample edges under a 3 GHz signal applied to the tip. A small gap is used to simulate the non-Ohmic contact nature of the system. (b) The solid black and dashed red lines represent the capacitance and tip deflection force, respectively, as functions of the tip–sample distance during the approach. The orange line with solid circles and dashed black lines correspond to the analytical and FEM models of the tip–surface capacitance for a dielectric substrate, based on Eq. (3). The blue dashed lines and solid circles depict the FEM results incorporating the tip, surface, and a water meniscus with varying thicknesses (1–6 nm) and radii (3 and 6 nm). The lower x-axis tracks the tip–sample distance, terminating with tip pull at 8 nm, where capillary forces dominate. During retraction, the tip snaps off at 60 nm. The shaded blue region indicates the operational regime of sMIM, dominated by the influence of a water meniscus. In the absence of force curves, only COMSOL results align with the analytical models.44 

FIG. 2.

(a) A 2D axisymmetric model is employed in place of a full 3D representation for computational efficiency. Admittance is measured between the tip and the sample edges under a 3 GHz signal applied to the tip. A small gap is used to simulate the non-Ohmic contact nature of the system. (b) The solid black and dashed red lines represent the capacitance and tip deflection force, respectively, as functions of the tip–sample distance during the approach. The orange line with solid circles and dashed black lines correspond to the analytical and FEM models of the tip–surface capacitance for a dielectric substrate, based on Eq. (3). The blue dashed lines and solid circles depict the FEM results incorporating the tip, surface, and a water meniscus with varying thicknesses (1–6 nm) and radii (3 and 6 nm). The lower x-axis tracks the tip–sample distance, terminating with tip pull at 8 nm, where capillary forces dominate. During retraction, the tip snaps off at 60 nm. The shaded blue region indicates the operational regime of sMIM, dominated by the influence of a water meniscus. In the absence of force curves, only COMSOL results align with the analytical models.44 

Close modal

The comparison between analytical models and FEM results shows a strong correlation in capacitance changes down to 50 nm, beyond which deviations begin to appear.44 These discrepancies are particularly relevant given that sMIM operates in close proximity to the surface—either in contact or in attractive mode. As a result, simple analytical models become inadequate at very small distances, and deviations must be further examined using FEM. Moreover, the presence of particles and/or water in the gap between the tip and the surface adds complexity to the problem, creating challenges for analytical models that could accurately improve the understanding and interpretation of imaging results in more realistic scenarios.

2. Water meniscus modeling

Atomic force microscopes (AFM) have been extensively employed to investigate capillary force interactions and water menisci dynamics at the nanoscale due to their remarkable experimental versatility.38–41,45–49 AFM enables the control of various parameters such as the tip and substrate materials, tip geometry, force, velocity, as well as environmental conditions like temperature and humidity. Having a handle on these parameters enables a detailed examination of tip–surface interactions under diverse hydrophilicity conditions.48,50

Modeling the behavior of water at the surface, particularly its influence on interactions such as the microwave response, can become highly complex.39,41 Electrical measurements of the effect of water meniscus surrounding tips in contact mode have been previously reported, exhibiting a parasitic contribution for sub-10 nm scale devices.51 On attractive mode operating conditions, however, menisci can have radii as small as 1 nm,45 and hence, whether as a parasitic element or a coupling element, proper modeling is essential for a quantitative assessment of the measured quantities.

A key question in such modeling is how much does the presence of water affect the distribution of the electromagnetic fields in typical sMIM experiments. To simplify these models, particularly those that assume the interaction between a spherical particle and a surface, a simplified profile for the water meniscus can be proposed. Generally speaking, the total radius of curvature r, of the meniscus can be given by
(4)
where r2 represents the radius of the AFM tip and r1 is the radius of negative vertical curvature of the meniscus, which is often approximated by a nodoid shape.52 This simplified geometric assumption allows researchers to estimate the meniscus profile more easily and predict its influence on capillary forces, as shown in Fig. 3.
FIG. 3.

Schematic representation of the AFM tip in contact with the sample surface, illustrating the formation of a nanometric water meniscus at the tip–sample interface. The meniscus forms due to capillary condensation under ambient conditions, creating a highly localized water bridge with a radius of curvature of ∼1 nm. This water meniscus acts as a focal point for the electric displacement field, significantly enhancing field concentration at the air–water boundary.

FIG. 3.

Schematic representation of the AFM tip in contact with the sample surface, illustrating the formation of a nanometric water meniscus at the tip–sample interface. The meniscus forms due to capillary condensation under ambient conditions, creating a highly localized water bridge with a radius of curvature of ∼1 nm. This water meniscus acts as a focal point for the electric displacement field, significantly enhancing field concentration at the air–water boundary.

Close modal
When it is in thermal equilibrium, this radius is given by the Kelvin equation, which has been recently shown to hold down to 1 nm dimensions:53 
(5)
where γ is the surface tension, v0 is the molecular volume, k is the Boltzmann constant, T is the temperature of the measurement system, and the ratio p/ps is the relative humidity. To calculate the profile, one has to take into account the contact angles θ1 and θ2, which are materials dependent and represent the boundary conditions for meniscus formation for a given tip and substrate, respectively.

This approach to modeling allows for the assessment of nanoscale interactions with minimal computational complexity, making it more feasible to account for the presence and behavior of water menisci in various environmental conditions.

3. Contrast mechanisms and spatial resolution

Given that sMIM involves the measurement of both the real and imaginary parts of reflected microwaves, FEM modeling allows us to independently verify these parameters as they vary with the sample’s electrical properties, such as conductivity and permittivity. It has been shown by many authors that to understand the contrast for both real and imaginary parts of the reflected microwave signal, the first step is to map a family of curves varying the surface permittivity and conductivity.7,12,14,54,55 In our approach, we calculated the real and imaginary components of the admittance as functions of the sample’s permittivity and conductivity. The results, obtained from a model incorporating the water meniscus, showed that with the increase of the sample’s permittivity, the influence of both real and imaginary components on conductivity diminishes, and the real part is a non-monotonic function of the sample’s conductivity (illustrated in Fig. 4). However, solving the inverse problem is required, namely, the derivation of the material’s electrical properties from the microwave reflection measurements. This process requires a rigorous combination of experimental measurements and theoretical modeling, taking into account the interactions between the probe, sample, environmental factors, as well as tip geometry and potential artifacts.

FIG. 4.

(a) Real and (b) imaginary part of the admittance as a function of the conductivity and permittivity of the sample.

FIG. 4.

(a) Real and (b) imaginary part of the admittance as a function of the conductivity and permittivity of the sample.

Close modal

Comprehensive simulations are required to best understand the mechanisms contributing to the spatial resolution of sMIM systems. These simulations were designed to model the electric displacement field, considering the geometry of the probe tip, the thickness of the sample film, and the dielectric properties (ɛr) of both the film and the underlying substrate, including the water meniscus at the tip, as well as the adventitious water layer at the surface.50 In particular, the water meniscus has an important role in the electric field distribution given its high relative permittivity, thus concentrating the field and significantly enhancing the resolution of sMIM at nanometric scales.44 

Using FEM in the COMSOL simulation tools, a 1-nm-thick layer of adventitious water was incorporated, along with a modeled meniscus having a 1 nm radius of curvature. The results demonstrated a concentration of the electric displacement field at the air–water boundary of the meniscus, leading to effective confinement of the field to a nanometer size region, as depicted in Figs. 5(b) and 5(c).

FIG. 5.

Distribution of the D field between the tip and the surface. (a) A 1 nm gap separating the tip and a metal surface, and (d) a dielectric surface. (b) Contact through a 1-nm-thick water meniscus bridging the gap between the tip and a metal surface and (e) a dielectric surface, and (c) direct contact between the tip and a metal surface and (f) a dielectric surface. Dmax = 2.5 × 10−3 C/m2.

FIG. 5.

Distribution of the D field between the tip and the surface. (a) A 1 nm gap separating the tip and a metal surface, and (d) a dielectric surface. (b) Contact through a 1-nm-thick water meniscus bridging the gap between the tip and a metal surface and (e) a dielectric surface, and (c) direct contact between the tip and a metal surface and (f) a dielectric surface. Dmax = 2.5 × 10−3 C/m2.

Close modal

4. Field distribution in realistic experimental conditions

All scanning probe microscopy modalities require sharp tips to extract the utmost spatial resolution. Nevertheless, the scanning process itself creates several issues pertaining to the tip integrity since most of the time there is physical contact between the tip and surface. Here we examine the electric field distribution due to changes in the tip geometry due to wear and tear, modeled by a 5 nm radius metallic sphere representing an asperity, and debris picked up by the tip during the scanning process, modeled by a 5 nm diameter dielectric sphere, both at the tip end. A control tip is used for reference, with a tip radius of 50 nm. The key question is how these configurations influence the field distribution in three distinct scenarios: (1) prior to meniscus formation at a 2 nm tip-surface separation, (2) during meniscus formation with a 1 nm thickness, and (3) at direct contact, considering both insulating and metallic substrates.

Analyzing these scenarios is crucial for assessing the electromagnetic field distribution around the tip and within the sample, ultimately determining the system’s resolution. Figures 57 illustrate the outcomes for all possible combinations. From a resolution perspective, the results suggest that the resolution progressively improves for metallic substrates as the system transitions from non-contact with no water meniscus to direct contact with a water meniscus surrounding the tip.51 The best resolution is obtained when operating in the attractive mode, with the water meniscus concentrating the field into a small spatial region.34,56,57

FIG. 6.

Distribution of the D field between the tip with a 5 nm radius metallic asperity and the surface. (a) A 1 nm gap separating the tip and a metal surface and (d) a dielectric surface (d). (b) Contact through a 1-nm-thick water meniscus bridging the gap between the tip and a metal surface and (e) a dielectric surface, and (c) direct contact between the tip and a metal surface and (f) a dielectric surface. Dmax = 2.5 × 10−3 C/m2.

FIG. 6.

Distribution of the D field between the tip with a 5 nm radius metallic asperity and the surface. (a) A 1 nm gap separating the tip and a metal surface and (d) a dielectric surface (d). (b) Contact through a 1-nm-thick water meniscus bridging the gap between the tip and a metal surface and (e) a dielectric surface, and (c) direct contact between the tip and a metal surface and (f) a dielectric surface. Dmax = 2.5 × 10−3 C/m2.

Close modal
FIG. 7.

Distribution of the D field between the tip with a dielectric (ɛ = 4) 5 nm hydrophilic sphere attached to the tip end and the surface. (a) 1 nm gap separating the tip and a metal surface and (d) a dielectric surface (d). (b) Contact through a 1-nm-thick water meniscus bridging the gap between the tip and a metal surface and (e) a dielectric surface, and (c) direct contact between the tip and a metal surface and (f) a dielectric surface. Dmax = 2.5 × 10−3 C/m2.

FIG. 7.

Distribution of the D field between the tip with a dielectric (ɛ = 4) 5 nm hydrophilic sphere attached to the tip end and the surface. (a) 1 nm gap separating the tip and a metal surface and (d) a dielectric surface (d). (b) Contact through a 1-nm-thick water meniscus bridging the gap between the tip and a metal surface and (e) a dielectric surface, and (c) direct contact between the tip and a metal surface and (f) a dielectric surface. Dmax = 2.5 × 10−3 C/m2.

Close modal

Theoretical and experimental research by Knoll and Keilmann highlighted the effect of tip geometry and materials on the near-field signal, supporting the notion that highly conducting substrates improve resolution when the tip operates in the attractive mode.58 According to Figs. 57, an increase in conductivity generates a concentration of the D field, which suggests greater resolution. A notable observation is the effect of the particle at the tip apex—whether metallic or insulating. The presence of an insulating particle significantly reduces the field intensity near the tip, affecting the resolution, as well as the signal sensitivity since it acts as a series element.

For insulating substrates, the field distribution becomes more diffuse unless the tip is in close proximity or a water meniscus is present. With water’s relative permittivity (ɛr) around 80 at frequencies below 3 GHz, the concentrating effect of the water persists even when the substrate is conductive, provided that the substrate’s ɛr remains lower than that of water. These results illustrate the interplay between tip geometry, material properties, and environmental factors in defining sMIM’s ultimate resolution.

5. Surface impedance and reflected microwave signal modeling: A lumped element perspective

In a typical scanning microwave impedance microscopy (sMIM) experiment, both the real and imaginary components of the reflected microwaves are measured. The reflected microwave signal, S11 = Γ, is related to the load impedance Ztip, observed from a matched transmission line with characteristic impedance Z0 (usually ensured by impedance matching circuitry), as follows:
(6)
The boundary conditions for this system, as illustrated in Fig. 2(a), account for voltages referenced to the ground plane, which is represented by the substrate. From these boundary conditions, we can evaluate the two components of the impedance: the real and imaginary parts, which are described by the conductance (G) and capacitance (C), respectively. The impedance at the tip can be expressed as
(7)
where R is the resistance (real part) and X is the reactance (imaginary part). The corresponding admittance (Ytip) is given by the inverse of the impedance:
(8)
In this expression, G represents the conductance and B is the susceptance, which is related to capacitance by
(9)
(10)
where ω = 2πf is the angular frequency and f is the system test frequency. These equations form the foundation of sMIM, where a vector analysis of the reflected signal Γ provides information about both the real (conductance, G) and imaginary (capacitance, C) components of the impedance at the tip.

Several authors have proposed the modeling of G and C in terms of the reflected microwave response [Eqs. (6)(10)]. To connect the lumped element approach to FEM modeling analysis, it is essential to have these models consistent with the boundary conditions shown in Fig. 2(a), which account for variations in surface conductivity.

Figure 8 illustrates the calculated G and C from the simulated real and imaginary components of the reflected microwave signal for three different tip geometries: ideal conical tip, tip with a metallic asperity, and dielectric nanoparticle at the tip end, as described in Sec. III A 4 and Figs. 57. These results were taken under three distinct conditions: non-contact with a z = 1 nm gap, contact through a 1 nm radius water meniscus (1 nm thick), and direct contact with the surface. Figure 8 shows how the surface conductivity sample affects the values of G and C.

FIG. 8.

Conductance (G) and capacitance (C) for three different tip geometries: ideal conical tip (— line), tip metallic asperity (-- dashed), and dielectric nanoparticle at the tip end (⋯ dotted) for the three conditions, non-contact with z = 1 nm gap, contact via meniscus, and direct contact.

FIG. 8.

Conductance (G) and capacitance (C) for three different tip geometries: ideal conical tip (— line), tip metallic asperity (-- dashed), and dielectric nanoparticle at the tip end (⋯ dotted) for the three conditions, non-contact with z = 1 nm gap, contact via meniscus, and direct contact.

Close modal

When examining low-conductivity (σ) substrates, capacitance (C) and conductance (G) exhibit the lowest values across all cases. This phenomenon can be understood by modeling the substrate as part of a capacitor in series with the tip impedance. In contrast, for high-conductivity substrates, the tip impedance is effectively shorted by direct contact with the substrate, leading to an increase in capacitance. The conductance in this regime is heavily influenced by whether the tip is in direct contact with the substrate. In intermediate conductivity regimes, both real and imaginary components of the sMIM signal become significant, corresponding to the condition where maximum contrast is observed. As shown in Fig. 8, the range of maximum contrast spans approximately from 10−2 to 102 S/m in surface conductivity, and this range can be adjusted by modifying the test signal frequency (f).

From the previous FEM analysis for a variety of tip geometries, surface interactions, and substrate conductivities, we can envisage a more complex lumped element analysis that can help in identifying and de-embedding parasitics from the experiment. Lai et al.26 and Jones et al.59 proposed different equivalent circuits, the former consisting of a tip capacitively coupled to the surface and the latter where the tip touches the surface, as seen in the insets of Figs. 9(d) and 10(d).

FIG. 9.

Lumped element model representation of a capacitively coupled tip–substrate system, illustrating key aspects of the system’s behavior. In (a), a numerical simulation of the sMIM system is shown, with a water meniscus between the tip and substrate. Panels (b) and (c) present the capacitance and conductance for 1 and 2 nm water meniscus, respectively, considering the current through the meniscus–water film interface (b) and the total current (c). Finally, (d) compares the results with the lumped element circuit model from Ref. 19. The model captures the impact of parasitic effects that arise from the shank of the tip cone (c), with currents outside the entire tip, resulting in no dependence on the meniscus radius and over two orders of magnitude larger capacitance than the case where we only monitor the current through the water meniscus (b).

FIG. 9.

Lumped element model representation of a capacitively coupled tip–substrate system, illustrating key aspects of the system’s behavior. In (a), a numerical simulation of the sMIM system is shown, with a water meniscus between the tip and substrate. Panels (b) and (c) present the capacitance and conductance for 1 and 2 nm water meniscus, respectively, considering the current through the meniscus–water film interface (b) and the total current (c). Finally, (d) compares the results with the lumped element circuit model from Ref. 19. The model captures the impact of parasitic effects that arise from the shank of the tip cone (c), with currents outside the entire tip, resulting in no dependence on the meniscus radius and over two orders of magnitude larger capacitance than the case where we only monitor the current through the water meniscus (b).

Close modal
FIG. 10.

Lumped element model of a capacitively coupled tip–substrate system. (a) and (c) The FEM results for both an ideal tip and a tip with asperity, considering the native water layer that surrounds the tip when it contacts the surface. (b) The FEM results for varying contact areas in the ideal case, as well as the tip asperity’s dependence on surface conductivity. (d) The Jones model,59 which captures some aspects of the asperity results and exhibits the relationship between water volume and contact area.

FIG. 10.

Lumped element model of a capacitively coupled tip–substrate system. (a) and (c) The FEM results for both an ideal tip and a tip with asperity, considering the native water layer that surrounds the tip when it contacts the surface. (b) The FEM results for varying contact areas in the ideal case, as well as the tip asperity’s dependence on surface conductivity. (d) The Jones model,59 which captures some aspects of the asperity results and exhibits the relationship between water volume and contact area.

Close modal

Lai’s26 model of a coupling capacitor, as depicted in Fig. 9 and based on Bryant’s work,19 provides a sufficiently simple yet accurate representation of the meniscus details when compared to FEM results. However, caution must be exercised in understanding what is being modeled vs what is being measured. A comparison of calculated values considering only the current through the meniscus–water film interface [Fig. 9(b)] vs the total current [Fig. 9(c)] reveals the significant effect of tip shank, where current through the entire tip can substantially exceed by two orders of magnitude the case for current through the meniscus only, for both capacitive and resistive components. Therefore, only the model that integrates current through the water meniscus exhibits a clear dependence on the meniscus radius and aligns well with the lumped element circuit results [Fig. 9(d)] assuming the water meniscus as the coupling capacitor. While the water meniscus between the tip and substrate [Fig. 9(a)] captures a large portion of the displacement field, reaching maximum values in that confined space, the region is relatively small compared to the tip size. The lumped element model can effectively describe the system’s behavior over several orders of magnitude for conductivity (σ), provided it is correctly interpreted and applied.

Jones et al.59 proposed a model in which the tip makes direct contact with the substrate, offering an alternative, plausible explanation for certain scanning conditions, especially when the tip is operated in the repulsive mode. The validity of this model can be assessed by comparing it with FEM results. Figures 10(a) and 10(c) display the FEM simulations, which include the influence of the native water layer surrounding the tip when it touches the surface, both for an ideal tip and a tip with asperity. Figure 10(b) presents the FEM results for varying contact areas in the ideal case and the relationship between tip asperity and surface conductivity. A key observation across all cases is a capacitance plateau around 10 S/m, independent of the contact area. At higher surface conductivities (100 S/m), the system shows area dependence, with larger radii exhibiting larger capacitance values, except for the ideal tip in contact, Fig. 10(b). For this case, the opposite behavior is observed, with larger capacitances for smaller contact radius, which can be understood from the fact that a larger contact area shunts the capacitive component. The simple model by Jones59 [see inset of Fig. 10(d)] captures the tip with metallic asperity results.

From the qualitative analysis of sMIM modeling results, several key conclusions emerge:

  1. Impact of surface water and meniscus formation: The presence of surface water and the formation of a water meniscus can significantly change spatial resolution, regardless of the tip shape or whether the tip is in contact with the surface.60 Optimal resolution can be achieved with an ideal tip operating in attractive mode connected to a flat surface via a water meniscus.31,44

  2. Effect of tip asperities and debris: Tip asperities and other debris collected during scanning can affect both resolution and contrast; for dielectric particles, there is an associated decrease in the system sensitivity, whereas metallic tip asperities can produce high sensitivity and high resolution.13 

  3. Substrate properties: The nature of the substrate—whether it is insulating or metallic—affects both resolution and contrast in sMIM imaging, with insulating substrates with low permittivity leading to lower resolution due to the dispersion of the electric field.

  4. Optimal range for imaging: For 3 GHz, the range of maximum contrast happens around 1 S/m surface conductivity. Changing the frequency allows for the possibility of investigating samples and materials with different conductivities.

In scanning microwave impedance microscopy (sMIM), the experiment involves the vector measurement of reflected microwaves from a tip hovering over a surface, which may either touch the surface in the repulsive mode or remain tethered to it by a water meniscus in the attractive mode. The signal of interest, reflected microwaves, is typically described by the S-parameter in Eq. (6), representing the reflection coefficient.

A typical sMIM setup uses a heterodyne detection scheme to sample the reflected microwave signal. Figure 11 shows a simplified setup, where a microwave source is connected to the tip, and the incident and reflected signals are mixed to extract the DC component of the reflection.

FIG. 11.

Simplified schematics of the detection scheme. A microwave source is connected to a tip, and the incident and reflected signals at and from the tip are mixed, and the resulting signal is low passed to retain the DC component. The impedance matching network has been omitted for simplicity. Also not shown is a quadrature reference signal, A sin(ωt + θ + π/2), which is used to generate the out of phase, or imaginary component.

FIG. 11.

Simplified schematics of the detection scheme. A microwave source is connected to a tip, and the incident and reflected signals at and from the tip are mixed, and the resulting signal is low passed to retain the DC component. The impedance matching network has been omitted for simplicity. Also not shown is a quadrature reference signal, A sin(ωt + θ + π/2), which is used to generate the out of phase, or imaginary component.

Close modal
The key voltages associated with the experiment are given as
(11a)
(11b)
(12)
By mixing the reflected and reference signals, the real (in-phase) and imaginary (quadrature) components of the reflection can be isolated:
(13a)
(13b)

Here, A and B are the constants proportional to the incident and reflected signal strengths, ϕ is the phase difference between them, θ is an arbitrary offset, and η is the mixer conversion factor. Ψ is ABη  sin(ωt + ϕ + θ). The reference signals are used to compute the scalar product of the reflected signal, which allows the real and imaginary components to be extracted.

From Eqs. (6)(10), the reflection coefficient Γ is sensitive to impedance mismatches between the tip and the transmission line. An impedance matching network is often used to remove the reactive portion of the reflected signal (i.e., eliminate the capacitive component), mitigate cabling and connection losses, and ensure that the parameters B and ϕ are such that the tip impedance matches the source output impedance.

By setting the correct reference phase θ, the in-phase and quadrature components of the reflected signal can be computed as
(14)
and
(15)

This signal processing method allows for the precise extraction of surface properties such as the relative capacitance and conductivity. The phase relationships are illustrated by a phasor diagram (Fig. 12), depicting the reference and reflected signal vectors.

FIG. 12.

Phasor diagram.

A more detailed description of circuitry with shielded cantilevers to minimize parasitic capacitance has been discussed in depth by Lai et al.26 in their foundational work on the technique. The quantitative understanding of Scanning Microwave Impedance Microscopy (sMIM) signals has been significantly advanced through recent circuit-level analyses.61,62 The measured signal, ΔVMIM, reflecting changes in tip–sample interactions, is determined by a set of well-defined circuit parameters:
(16)
where G represents the system voltage gain, ΔY is the variation in tip–sample admittance, Y0 is the system admittance, and η = Vprobe/Vin is the voltage enhancement factor.

This relationship reveals the dependence of the MIM signal on circuit parameters. Crucially, the signal scales quadratically with the voltage enhancement factor (η2) under conditions of limited incident power, and linearly with η when constrained by maximum probe voltage. These scaling laws provide a framework for optimizing system sensitivity and bandwidth in sMIM applications.

By explicitly defining the quantitative relationship between circuit parameters and the MIM signal, this approach highlights the role of G as the primary tunable parameter. Optimizing G involves balancing amplification against noise introduced by each stage in the signal chain.

Shan et al.61,62 demonstrated that achieving maximum sensitivity often requires enhancing η through resonant impedance matching or low-loss circuit components. However, the use of broadband designs may offer advantages for spectroscopic applications by maintaining a more consistent response across frequencies.

This quantitative framework replaces qualitative approximations with precise relationships, enabling a systematic approach to improving sMIM performance. The implications extend to both narrow-band imaging and broadband spectroscopic applications, where sensitivity-bandwidth trade-offs are critical.

1. Impedance matching protocol for sMIM measurements in AFM systems

A critical aspect of performing sMIM is addressing the significant impedance mismatch between the tip–substrate load and the microwave source. Typically, the tip–substrate load exhibits extremely small capacitance with negligible resistance, often in the GΩ range, while the microwave generator and transmission lines typically operate at a much lower impedance, standardized at 50 Ω. Thus, the use of an impedance matching network is required to facilitate maximum power transfer from the microwave source to the probe, which directly impacts the effectiveness of sMIM measurements.63 

Recent developments have highlighted critical nuances in optimizing sMIM systems, particularly the relationship between impedance matching quality and sMIM responsivity.64 While impedance matching directly impacts responsivity, they are not synonymous; a distinction crucial for understanding and optimizing sMIM performance.28 This insight underscores that responsivity, a measure of the sMIM system’s sensitivity to changes in sample properties, is influenced by but not solely dependent on impedance matching. Factors such as probe characteristics, sample interactions, and microwave field distribution also play significant roles.

a. Impedance matching network design.

In some commercial sMIM systems such as the PrimeNano Inc., the impedance matching network is carefully configured to integrate seamlessly with the AFM probe holder and platform, without affecting its primary functions, such as topography measurements. For this particular instrument, impedance matching protocols allow the adjustment of network parameters to optimize microwave coupling. Typically, S11 measurements, which reflect the return loss, are monitored during this process. The goal is to minimize the reflection coefficient (S11), which corresponds to maximizing the microwave energy delivered to the probe. A well-optimized impedance matching network can achieve coupling efficiency exceeding 60 dB,65,66 as shown in network analyzer measurements in Fig. 13.

FIG. 13.

(a) Configuration of a transmission line near the probe. (b) S11 signal after impedance matching.

FIG. 13.

(a) Configuration of a transmission line near the probe. (b) S11 signal after impedance matching.

Close modal
b. Tip–holder contacts and their impact on microwave coupling.

An essential factor influencing successful microwave coupling is the integrity of the electrical contacts between the tip and the transmission line. In the system configuration, the sample probe is inserted into the probe holder and secured by a spring-loaded clip, which ensures both mechanical stability and electrical connectivity. Poor contact at this interface can significantly hinder impedance matching, resulting in suboptimal microwave transmission to the tip. The tip mounting process is thus critical and is monitored using S11 measurements, with adjustments made to maximize the microwave coupling65 (Figs. 14 and 15).

FIG. 14.

sMIM images of a calibration sample consisting of aluminum pillars deposited on a silicon oxide substrate (a) before phase calibration and (b) after phase calibration where channel 1 is identified as the capacitance channel.

FIG. 14.

sMIM images of a calibration sample consisting of aluminum pillars deposited on a silicon oxide substrate (a) before phase calibration and (b) after phase calibration where channel 1 is identified as the capacitance channel.

Close modal
FIG. 15.

AFM force curves presented side by side with simultaneously collected output of the two sMIM channels before and after phase calibration. The data shown was collected from a glass microscope cover slip. Whereas prior to calibration, both sMIM channels in (a) contain components associated with the capacitance contribution (b), the contribution of the capacitance signal to the conductance signal is completely eliminated after calibration as evidenced by the flat response of channel 1 during approach. Channel 1 is thus the channel measuring conductance and channel 2 is the capacitance.

FIG. 15.

AFM force curves presented side by side with simultaneously collected output of the two sMIM channels before and after phase calibration. The data shown was collected from a glass microscope cover slip. Whereas prior to calibration, both sMIM channels in (a) contain components associated with the capacitance contribution (b), the contribution of the capacitance signal to the conductance signal is completely eliminated after calibration as evidenced by the flat response of channel 1 during approach. Channel 1 is thus the channel measuring conductance and channel 2 is the capacitance.

Close modal

The operating frequency of the sMIM measurements is determined by the frequency at which the S11 response shows a minimum, indicating the point of maximum power transfer. This frequency can vary across different probes and systems, ranging up to a few GHz, depending on the specific probe characteristics and the system configuration.63 

c. Importance of fine-tuning the impedance matching.

The impedance matching protocol not only ensures the proper transfer of power but also enhances the sensitivity and accuracy of subsequent sMIM measurements. By achieving a minimized S11 signal, the AFM system can operate at its peak microwave efficiency, allowing for precise surface property analysis without interfering with other functionalities, such as topography mapping. This dual functionality of the AFM platform—optimized for both traditional AFM applications and microwave impedance mapping—marks a significant advancement in multifunctional AFM systems.65,66 However, it must be noted that during the impedance matching protocol, one can no longer measure the absolute values of capacitance and conductance.

Recent advancements emphasize that the relationship between impedance matching and responsivity is pivotal for sMIM optimization.64 Achieving high-quality impedance matching improves microwave energy delivery and enhances responsivity. However, fine-tuning sMIM responsivity also requires careful consideration of other system parameters, reinforcing that impedance matching and responsivity, though interrelated, represent distinct aspects of sMIM performance optimization.28 

2. Phase calibration in sMIM systems

Phase calibration is a crucial step in sMIM systems to ensure accurate separation of the real and imaginary components of the reflected microwave signals (S11), which correspond to surface impedance and capacitance, respectively. After mounting the probe and ensuring proper matching with the transmission line, calibration strategies can be implemented to fine-tune the system for accurate measurements.

One commonly employed strategy utilizes a calibration sample with known surface impedance, such as metal pillars on a dielectric substrate. For instance, Huber et al.6 demonstrated a technique using aluminum pillars deposited on silicon oxide. In their experiment, images were captured before and after phase calibration, illustrating the contrast between capacitance and surface conductivity. Before calibration, contrast is visible in both real and imaginary components (S11), which may include unwanted contributions from capacitive and non-capacitive sources. After calibration, the contrast in the real part (conductance) disappears, leaving only the capacitance channel active, indicating successful separation of the signal components.

A second strategy, the z-scan, also known as a force curve experiment, offers a more rigorous approach to phase calibration. Force curves measure the deflection of the AFM probe as it approaches and retracts from the sample, reflecting interactions between the tip and the surface. Simultaneous monitoring of the sMIM channels during these measurements allows for precise calibration by observing the behavior of both capacitance and conductance signals as the probe moves relative to the surface, as pioneered by Bryant and Gunn.19 Calibration is achieved when the capacitance signal (imaginary part of S11) changes predictably with tip height, while the conductance signal (real part of S11) remains constant or shows variation within the noise. This method is particularly effective because it accounts for stray capacitances and other artifacts that might interfere with accurate signal interpretation.63 

3. Quantitative analysis and calibration strategies

To convert the voltage outputs from sMIM to absolute values of capacitance and conductance, calibration techniques are essential. One straightforward method involves the use of samples with known capacitances for comparison. This allows for the calculation of a proportionality constant (α) that relates the measured signal to capacitance.6 The equation for this method is
(17)
where ΔS11 represents the difference in the reflection coefficient between a metallic pad and a dielectric region on the sample. This method effectively subtracts stray capacitance by comparing signals from different areas of the same sample.

Another calibration approach is the Short-Open-Load (SOL) method, adapted from network analyzer theory for sMIM applications.67 The SOL method uses three reference points (metallic, dielectric, and open) to correct the measured reflection coefficient (S11,m) to the actual value (S11,a), enabling accurate assessment of the tip’s impedance. This technique provides high precision but requires well-defined areas with distinct permittivities on the same substrate, or multiple substrates, which can introduce errors related to geometry.

Gramse et al.63 proposed a further refinement of the SOL method by incorporating z-scans over both metallic and dielectric regions. This approach solves issues related to different materials and minimizes the influence of sample topography. This strategy, subsequently employed in studies of bacteria’s electrical properties,32 offers a robust calibration procedure that eliminates topographical cross-talk and enhances the accuracy of sMIM measurements. However, the greatest challenge arises during calibration when impedance matching occurs, as this effectively eliminates parasitic capacitances and resistances. Several methodologies have been proposed to address this. Gramse offered a partial solution by significantly reducing ΔC.63 More recent work, such as that by Ma and his research group, has introduced a more comprehensive solution that enables absolute measurements28 rather than just relative ones.

4. Probes and their influence on measurement quality

One key important factor is that of reducing parasitic capacitances arising from the tip. Lai et al.26 initially developed a cantilever-based approach where through microfabrication, tips could be placed at the end of a shielded waveguide, which greatly reduces parasitics and allows for better signal performance. The disadvantages of this approach consist of special tip holders and tips, with the latter being quite sensitive to handling conditions, with the central conductor becoming severed depending on the operating conditions. One attempt to utilize a bare wire was employed by Cui et al.68 in a setup consisting of an electrochemically etched tungsten tip attached to a Quartz tuning fork. The tip’s high aspect ratio allowed for operation with performance quite superior to shielded probes.

Recent advancements have highlighted the significant potential of commercially available, gold-coated silicon cantilever probes in scanning microwave impedance microscopy (sMIM). As pointed out by Shan et al.,28 these probes can produce high-quality MIM images without requiring extensive modifications or complex fabrication processes. This result is particularly important as it reduces both the technical and economic barriers to sMIM adoption, making the technique more accessible for diverse applications in research and industry.

The use of conventional probes simplifies operational protocols while maintaining high spatial resolution and measurement stability.6,28 These advantages are critical for expanding sMIM’s applicability, particularly in cost-sensitive and multidisciplinary environments. Moreover, their compatibility with existing setups minimizes the need for specialized equipment, further expanding access to advanced nanoscale imaging techniques.

Scanning microwave impedance microscopy (sMIM) has rapidly advanced into a powerful tool for nanoscale characterization, with applications across various domains. One of its most prominent applications is in material science, where sMIM has been used extensively to probe the electrical properties of advanced materials such as graphene, transition metal dichalcogenides, and complex oxides. It allows researchers to investigate properties like conductivity, capacitance, and permittivity with high spatial resolution, providing critical insights into phenomena like quantum properties, charge carrier distribution, grain boundaries, and domain structures.

In semiconductor technology, sMIM is applied for failure analysis and the characterization of dopant profiles, crucial for device optimization. It can non-invasively image subsurface dopant layers in silicon with exquisite precision and dopant sensitivity.69 Also noteworthy is the careful analysis of tip band bending effects, which can be assessed by sMIM70 and must be taken into account for the characterization of semiconductor nanoscale systems.

Another emerging application of sMIM is in biological sciences, where it can be used to characterize biological tissues and single cells. This opens the door to investigating biological systems’ electrical and dielectric properties, offering potential insights into cellular processes, disease diagnosis, and treatment development.

From a future perspective, several key areas stand out for further development; however, here we will focus our efforts on two types:

  1. Broadening operational environments: Future advancements in sMIM could enable its effective operation under extreme conditions, such as cryogenic or elevated temperatures and liquid environments. This would not only enhance its applications in materials science but also open up new avenues in biological research. For instance, sMIM could be used to probe live cells, tissues, and biomolecules in their native environments, enabling the study of dynamic biological processes, membrane potentials, protein interactions, and even cellular differentiation with nanoscale precision.32 Operation in high frequency has also enabled the field of nanoelectrochemistry, achieving sub-atto-Ampere sensitivity in local cyclic voltammetry measurements at the solid–liquid interface and permitting Redox reactions to be spatially resolved down to 120 molecules.33,71–77

  2. Spectroscopy techniques: Scanning microwave impedance microscopy (sMIM) presents a significant opportunity for developing novel spectroscopy techniques,10 though literature exploring this potential remains limited. Currently, most studies focus on sMIM’s ability to map electronic properties at the nanoscale, but its application in spectroscopy—particularly in high-frequency regimes—remains underexplored. The potential for sMIM to perform absolute measurements and detect subtle variations in local conductivity and permittivity makes it a powerful tool for spectroscopic analysis. Continued advancements in this field could unlock new capabilities for investigating materials at unprecedented resolutions and frequencies.

Scanning microwave impedance microscopy (sMIM) has emerged as a promising technique for investigating nanoscale electronic properties. By leveraging the sensitivity of microwave frequencies to both the real and imaginary components of reflected signals, sMIM offers unprecedented spatial resolution in materials science, biology, and other fields. Despite challenges related to tip wear, water meniscus formation, and the complexities of field distribution, advances in finite element modeling and experimental calibration techniques have significantly improved its accuracy and applicability.

Continued research on probe design, environmental effects, and the refinement of signal processing methods will likely extend the capabilities of sMIM even further. However, it is the integration of sMIM with other microscopy techniques, such as atomic force microscopy, that promises to be a game-changer. This integration holds the potential to significantly enhance both the qualitative and quantitative analysis of materials, opening new avenues for non-destructive, high-resolution characterization at the nanoscale. These developments have far-reaching implications for nanotechnology and fundamental research in physics and materials science.

The authors acknowledge financial support from CNPq, FINEP, FAPEMIG, INCT IQNano, and CAPES.

The authors have no conflicts to disclose.

Diego Tami: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal). Douglas A. A. Ohlberg: Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal). Cássio Gonçalves do Rego: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal). Gilberto Medeiros-Ribeiro: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal). Jhonattan C. Ramirez: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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