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

Noninvasive and depth‐sensitive measurements of dielectric properties are becoming of great interest in advanced and complex nanostructured architectures. Here, a straightforward parallel approach applicable in peak force Kelvin probe force microscopy for a 3D measurement of dielectric constants at the nanoscale is demonstrated. The proposed approach features a simultaneous measurement of the mechanical, electrical, and depth‐dependent dielectric properties applied to embedded nanostructures. The findings provide initial elements for further development of experimental dielectric nano‐tomography methods for characterizing buried and embedded systems and dielectric interfaces.


Introduction
The stacking of nanostructures into functional architectures is currently the building-block approach for advanced technologies DOI: 10.1002/admi.202300503ranging from nano-biotechnology, molecular electronics, and 2D nanomaterials to semiconducting and piezoelectric devices for electronics and energy conversion applications.Atomic force microscopy (AFM) is well established in materials science as a unique characterization tool, offering tremendous possibilities to probe and image surfaces in real space at the nanoscale.Meanwhile, the last decade has witnessed a growing interest in extending the AFM measurements beyond the sample's surface to probe subsurface properties in complex multiphase nanostructures, by exploring a wide range of AFM configurations and modes. [1]ethods reported in the literature have covered local mechanical properties of the underlying systems such as biological cells, [2,3] interfacial properties related to induced mismatching stresses or local charges or dipoles formation, [4][5][6] charge carriers and doping properties, [7] embedded structures of different chemistries such as polymers or thin films, [8][9][10][11] carbon nanotubes (CNTs) and nanowires (NWs), [12][13][14][15] 2D materials, [16] nanoparticles (NPs), [17,18] and embedded air cavities [19] or confined water structures. [20]5][26][27][28][29][30] In parallel, the quantification of the dielectric constant of multiphase nanomaterials has recently gained increasing interest owing to its high importance in 3D heterostructure interconnects and embedded inclusions of nanostructures (nanotubes, nanowires, nanoparticles, and 2D materials). [15,17,31,32]Several works have been reported on the measurement of the dielectric constant using AFMbased methods, which fall into two main categories.The first methods typically operate in the contact mode, incorporating microwave signals.They include the so-called scanning microwave microscopy (SMM) [33,34] and scanning microwave impedance microscopy (sMIM). [35]Despite their importance for the measurement of dielectric constants in the radio-frequency range, these methods still suffer from the drawbacks associated with contact mode operation.In particular, the wear of the tip during scanning greatly affects the stability and repeatability of the measurements, hence limiting their application for soft fragile samples.It is worth noting that these limitations could be overcome in some particular cases such as the fast force mapping mode for peak force sMIM. [36]The second methods are based on the use of electric force microscopy (EFM), which operates in a noncontact mode to measure the dielectric constants. [37,38]Most of the reported EFM-based methods measure the amplitude of the cantilever oscillations' influenced by the electrostatic signal, which is proportional to the first-order derivative of the capacitance between the tip and the sample. [37,39,40][43][44][45] However, all reported methods operate without an applied Kelvin controller feedback loop, which makes the measured signals susceptible to uncompensated parasitic electrostatic contributions due to the presence of charges and differences in work functions.
In this work, we present an alternative approach bringing solutions to overcome the above limitations using the peak force Kelvin probe force microscopy (PF KPFM) method.Our approach combines peak force quantitative nanomechanical mapping (PF QNM) measurements with frequency modulated (sidebands) Kelvin probe force microscopy (FM-KPFM) in lift mode.In addition to the simultaneous acquisition of the mechanical (i.e., adhesion, deformation, and Young's modulus) and surface potential properties, the proposed approach allows a parallel real-time measurement of the second-order derivative of the tipsample capacitance without the need for additional instrumentation.

Background
The measurement in the PF KPFM mode is based on a double scan scheme, as shown in Figure 1.During the first scan, the surface topography and mechanical properties are measured in PF QNM mode. [46]The second scan is performed in the lift mode, i.e., by applying a lift height separation where the cantilever is mechanically actuated using a dither piezo element to oscillate at its resonance frequency (typically the fundamental) while a simultaneous electrical modulation signal is applied directly to the cantilever at much lower frequencies, typically ≈2-5 kHz. [47]n 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.This is reflected in the variations of the amplitude (A 1 ) and phase ( 1 ) of the cantilever's mechanical oscillations, measured at the output of the first lock-in amplifier (LIA1) during the second scan, as shown in Figure 1.In the limit of the linear approximation of the harmonic oscillator model for the vibrating cantilever during the second scan, the phase variations [51] at the output of LIA1 are given by: where, k and Q correspond to the spring constant and quality factor of the cantilever oscillations, respectively.F e is the electrostatic force that builds between the conductive AFM probe and the sample's surface, given by: C z where V dc is a variable bias voltage supplied by the Kelvin controller, V cpd is the contact potential difference equal to the difference in the work functions between the tip and the sample, and V ac sin ( e t) is the modulated electrical drive signal applied to the cantilever at the frequency f e = 2 e .Using Equation (2) in Equation ( 1) yields three spectral components of the phase variations at the output of the LIA1, as follows.
A closed Kelvin controller feedback loop is used during the PF KPFM measurement (Figure 1) to determine the work function difference by applying V dc = V cpd during the second scan.It follows that the expression in Equation ( 4) is nullified, whereas the remaining components become.
It is interesting to note that in this particular configuration, both the static and the 2 e modulated phase variation signals have the same magnitude, which is directly dependent of the second-order derivative of the tip-sample capacitance (  2 C z 2 ).Therefore, by monitoring the phase at the output of LIA1 (Figure 1), the variations in  2 C z 2 of the sample could be mapped.The importance of measuring this signal stems from its direct dependence on the local variations in the dielectric constant of the sample.Most of the reported measurements of the  2 C z 2 channel in KPFM have been based on the use of an additional third lock-in amplifier in the measurement setup. [47,52,53]However, to our best knowledge, the use of the static component (Δ 1,dc ) of the mechanical phase variations for this purpose has not been previously reported.As such, our approach presents two main advantages over existing methods.First, it does not require the use of an additional lock-in amplifier to detect the 2 e signal, which reduces costs, noise sources, and shortage on the bandwidth of the measurement chain.Second, it features the implementation of the simultaneous closed Kelvin controller feedback loop, which compensates for the electrostatic contributions.This is particularly important for measuring artefact-free variations in z 2 depending on the variations of the sample's dielectric constant.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 ( C 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).

MultiLevel Imaging
Our approach to map the phase signal change at the output of LIA1 in the PF KPFM mode provides simultaneous ability to measure both the top-surface and the subsurface properties of the sample.As proof-of-concept, Figures 1a,b show the surface topography and adhesion maps measured during the first scan on a GaN cross-section sample with p-doped stripes regions.The maps of the surface potential and Δ 1,dc , shown in Figures 1c,d, were measured during the second scan in the PF KPFM mode.Flake-like structures were clearly visible on top of the GaN surface (Figure 1a) with a distinct adhesion contrast (Figure 1b), indicating a difference in chemical termination at the surface of these flakes compared to the underlying substrate.The surface potential map (Figure 1c) showed no correlation with the surface topography and adhesion.The observed contrast revealed the variation in surface potential induced by the p-doping of the GaN sample in a stripes-like configuration.The lower surface potential at the dark stripes on the map in Figure 1c is the result of a higher work function of the sample in the doped regions due to the shift of the corresponding Fermi level.It should be noted that both adhesion and surface potential are surface-sensitive properties characterizing the quality of the sample's top surface.However, in the particular case of a doped semiconducting sample, GaN in this case, the implanted dopants alter the electronic configuration in the depth of the sample.This electronic alteration induces changes in the polarization-dependent response of the sample to an external electric field, which is translated into a local variation of the dielectric constant within the sample.Indeed, both stripes and flake-like features (Figure 1d) were detected in the static phase channel Δ 1,dc , which is directly related to the variations in the capacitance second-order derivative  2 C z 2 , as shown in Equation (6).To interpret the observed contrast, it is interesting to recall that the measurement was performed in the lift mode, with a continuously compensated contact potential difference in FM-KPFM.Therefore, the potential-dependent electrostatic contribution to the signal is not expected to leak into the phase channel at the output of LIA1.As a result, the capacitance gradientdependent contrast in Figure 1d is solely dependent on the local changes in the dielectric constant and polarization properties of the sample.Hence, our approach demonstrates the possibility of probing multi-depth levels of the samples' properties, providing new means for AFM-based dielectric constant nanotomography.This is mainly because the capacitance-gradient variation encompasses that of the dielectric constant of the entire dielectric medium between the tip and the back electrode.Therefore, in the case of different dielectrics within the bulk of a sample and at the top of its surface, a contrast is expected to build up in the Δ 1,dc channel, as is the case in Figure 1d.The simultaneous correlation with the surface potential and adhesion maps in Figure 1b,c allows distinguishing between the different depth levels of all contrast features.Several works have been reported in the literature demonstrating depth-sensitive AFM-based methods using the EFM method. [10,12,13,15]Virtually all the reported methods operate in the dynamic intermittent contact mode, in either single or double scan schemes, when probing electrostatic forces on embedded structures.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 main difference in our proposed method is the correlation with nano-mechanical measurements sensitive to top-surface structures.This approach helps discriminating the source of contrast in the images of capacitance variations depending either on embedded dielectric structures or those on top of the sample surface.

Validation of the Approach
A first look at the Δ 1,dc map in Figure 1d might raise the question whether the observed contrast is not due to crosstalk between the surface potential and the topography channels.To demonstrate the validity of this approach with respect to a plausible application of Equation ( 6), we used a commercially available test sample (PrimeNano Inc., Santa Clara, USA) formed by buried SiO 2 square structures beneath a flat polished Si 3 N 4 surface (see Experimental Section), as shown in Figure 2. Initially, two 20 × 20 μm 2 regions were scanned in contact mode while applying ±3 V bias voltages to implement charges in the Si 3 N 4 film.Then, a larger scan area (i.e., 50 × 50 μm 2 ) was imaged using our approach in PF KPFM mode.
The surface topography in Figure 2a shows a flat surface (i.e., z-range < 2 nm) with edge traces of the buried structures resulting from the fabrication process.The image in Figure 2b clearly shows the signature of the two charged regions as measured on the surface potential map obtained from the Kelvin controller closed loop.Interestingly, the contrast in the Δ 1,dc map in Figure 2c is solely related to the signature of the buried SiO 2 structures and shows no crosstalk with the surface potential channel.This result highlights the robustness and the validity of the approach, demonstrating the measurement of the z 2 variations with no electrostatically induced artefacts.Therefore, the contrast on the Δ 1,dc map directly reflects the variations in the local dielectric constant of the subsurface structures.Our method reproduces similar results shown with the sMIM method on the same sample, without the need to use additional instrumentation. [56]It is worth noting that a very careful observation of the contrast in Figure 2b reveals a vague, blurred signature of the buried structures on the surface potential channel.This results from imperfect Kelvin controller feedback loop settings, which did not totally nullify the sidebands in FM-KPFM.

Quantification of the Subsurface Dielectric Constant
The quantification of the dielectric constants of embedded structures remains an important aspect for a metrological application of this method.Pioneering works paved the way for the determination of the dielectric constant of various dielectric structures, including oxides, biological systems and, more recently, confined water. [37,39,57]However, most of the reported results describe measurements of top-surface structures.Nevertheless, the arguments detailed above for the results in Figure 2 clearly show that subsurface structures alter the measured capacitance signal used to determine the dielectric constant.Thus, for initially unknown structures, simple top-surface dielectric measurements are susceptible to fundamental artifacts.Recently, a numerical analysis approach was used to demonstrate the dielectric nanotomographic capabilities of the EFM method. [58]In the following, we describe an approach to quantify the dielectric constant of subsurface structures using the test sample shown in Figure 2 and applying data clustering methods for image processing.
Our approach consists of recording the phase signal at the output of the first LIA1 at different lift heights while the Kelvin controller feedback loop is closed in the FM-KPFM measurement.This signal corresponds to the variations in the static phase dc , as explained in Equation ( 6).The contrast observed in the images of the phase variation maps reveals the signature of the SiO 2 structures buried below the flat homogeneous Si 3 N 4 surface, as shown in Figure 3a.The principle of the proposed approach is illustrated in Figure 3b.First, a stack of images is created using the recorded phase signal maps at different lift heights.Then, local spectra are extracted at each pixel of the map scan by collecting the phase values from the images at each height.Finally, the phase spectra are converted into the variation of the second-order derivative of the tip-sample capacitance versus separation distance, as shown in Figure 3d, given by: where, V ac is the amplitude of the modulation voltage applied in KPFM.
As the topography of the sample was initially measured during the first scan in peak force tapping (PFT) mode, the zero position of the sample's surface is determined at the point of contact in the force curves applied in PFT.It follows that the lift height applied during the second scan in this case corresponds to the actual separation distance (z) between the tip and the sample.It is worth noting that this is in contrast to the previously reported methods where the determination of the tip-sample distance involves taking into account the amplitude of the cantilever's oscillation in tapping mode used during the first scan. [37,57]o mitigate the long-range capacitive couplings due to stray fields at large separation distances, we have considered the image at the highest distance, i.e., z = 170 nm as a reference.This was done by subtracting the corresponding phase image from all images recorded at different heights before extracting the capacitive gradient variation versus distance. [37,40]Two representative spectra were obtained at specific locations above the embedded SiO 2 structures and above the Si 3 N 4 matrix, respectively, as shown in Figures 3b,d.The two spectra exhibited a very small shift, which is related to the ratio of the dielectric constants of the two different materials (S3, Supporting Information).
To determine the sample dielectric constant, an analytical model of the tip-sample capacitance was used to fit the data. [37,59]nterestingly, the measurement of the phase shifts in the FM-KPFM mode is almost solely dependent on the contribution of the spherical tip apex to the detected electrical interaction force.This is mainly due to the sensitivity of the measured signal to the second derivative of the tip-sample capacitance.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 where, R = R(1 − sin), R is the radius of the spherical tip apex,  is the half-angle of the cone opening,  eq r is the equivalent relative dielectric constant of the entire dielectric film under the tip, and  = (ln [tan /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 3d.The model shows excellent agreement with the measured capacitance gradients up to a separation distance of z = 150 nm.
However, the fitting procedure requires knowledge of the tip radius value, R. In previous works, the tip radius has generally been determined from capacitance C (or dC/dz) curves versus tip-sample distance performed on metallic substrates, which were fitted with analytical expressions of the spherical tip apex capacitance with or without adding a contribution from the truncated cone. [40]These approaches yielded an estimated value of the tip radius R, which was subsequently used to fit experimental capacitance-gradient curves performed over dielectric films to extract the local dielectric constant values.Here, we have taken a slightly different approach, using the argument of a locally "electrically effective" tip radius.In fact, an exact determination of the tip apex radius for electrical measurements in AFM is a virtually unattainable task.This is mainly because the "electrically effective" radius as felt during the electrical interaction with the sample may show small discrepancies depending on the local geometrical nature of the interaction involving both the tip and the underlying surface.This argument would remain valid even if the actual physical dimension of the radius remained constant.
For this purpose, we have assumed that the value of the tip cone half-angle opening , which is more easily determined via scanning electron microscopy, is constant ( = 20°).This assumption is experimentally valid since the cone opening is not prone to damage during a properly controlled AFM measurement.Therefore, we used both R and  eq r as fitting parameters to fit the d 2 C/dz 2 versus z curves obtained from the images stack, which yielded an excellent agreement between the experimental data and the analytical model of the second-order derivative of the tip-apex capacitance.We found a mean value of the tip radius R = 130 ± 5 nm, which indicates a reasonably low standard deviation ≈3.8% consistent with the expected local discrepancies of the "electrically effective" radius (Figure S2, Supporting Information).
To generate a map of the dielectric constant variation of the sample's structures, the capacitance gradient spectra were fitted at each pixel of the scanned image area.Therefore, the map in Figure 4a corresponds to the equivalent dielectric constant of the entire dielectric film, regardless of its internal composition.The values of  eq r obtained in this case were determined using the capacitive model involving the probe (spherical apex and truncated cone), the airgap between the tip and sample, and the entire dielectric film of total thickness d = 280 nm, as indicated in Equation (9).At this stage, the application of this commonly known capacitive model yielded the dielectric constant of the entire dielectric film, independently of its internal structure.Nevertheless, our goal is to derive the dielectric constants of the different structures (i.e., Si 3 N 4 and SiO 2 ) forming the internal composition of the dielectric film of constant  eq r .For this, additional data treatment was required to automatically identify the zones on the AFM images where the dielectric film contains embedded SiO 2 structures relative to the zones forming the main matrix (i.e., Si 3 N 4 ).We applied the clustering paradigm, a machine learning technique that involves the specific grouping of data points, [62] using the k-medoids method [63] to partition the set of 256 × 256 pixels in the images into k groups or clusters (here, k = 2).The two clusters in our case correspond to the different contrasts in the images with respect to the squared zones above the SiO 2 embedded structures and to the dielectric matrix outside the squared zones, as shown on the clusters map in Figure 4c.By projecting the  eq r map (Figure 4a) onto the cluster map (Figure 4c), we obtained an averaged value for the equivalent dielectric constant for each cluster, i.e.,  eq r (squares) = 5.51 ± 0.20 and  eq r (matrix) = 5.74 ± 0.20.Based on the structure of the model sample used here (Figure 2), the relative dielectric constant over the matrix zone (outside the squares) corresponds to that of Si 3 N 4 .However, in the zones with embedded SiO 2 structures, the problem becomes equivalent to that of a dielectric film ( eq r ) formed by the stacking of two dielectrics ( Si 3 N 4 and  SiO 2 ) in series with known thicknesses (i.e., d 1 and d 2 , as depicted in Figure 2).We recall that considering the total dielectric film as formed by a stack of two different dielectrics is completely decoupled from the capacitive model in Equation (9), which was initially used to determine  eq r .To quantify the dielectric constant of the embedded SiO 2 structures, the equivalent dielectric constant of the two stacked dielectrics in series at the specific squared zones of the sample surface is given by: where,  eq r (squares) corresponds to the value of the dielectric constant determined from the map in Figure 4a at the locations of the squared zones.Therefore, the dielectric constant ( SiO 2 ) of the subsurface structures can be expressed as a function of the known dimensional parameters and the measured values of  eq r (squares) and  Si 3 N 4 r , as follows.
As a result, a quantitative map of the actual dielectric constant distribution of the sample was obtained, as shown in Figure 4e, where the dark contrast of the squares corresponds to the dielectric constant of the subsurface SiO 2 structures.Figure 4f shows the histogram of the dielectric constant values extracted from the map in Figure 4e and fitted with two distinct Gaussian distributions.We obtained  Si 3 N 4 = 5.72 ± 0.18 and  SiO 2 = 4.95 ± 0.37 using the central values of the peaks and the associated standard deviations.
To assess the accuracy of the values obtained, a comprehensive uncertainty budget was established, defining the various sources of uncertainty in the measurements and their respective weight of contribution to the result.

Uncertainty Analysis
To determine the combined uncertainty on the measured dielectric constants, it is important to estimate the uncertainties associated with all sources of error arising from the various experimental parameters involved in the measurement chain.In our case, the final determination of the dielectric constant of both dielectric structures explicitly depends on the experimental measurement of the phase variation signal, the analytical model expression involving the tip radius, the spring constant of the cantilever, the quality factor, and the thicknesses of the dielectric films.In addition, the image processing used to quantify the local capacitance gradient spectra and to estimate the statistical values of the dielectric constants plays a key role in the final uncertainty, depending on the pixel resolution and the local scattering of the values (Table 1).
The type A uncertainty evaluation method consists in deducing a probability density from the observed distribution of data.It is related to the pixel resolution and the histogram extracted from the final map of the dielectric constant distribution.Here, the type A uncertainty was calculated from the standard deviation obtained from each Gaussian distribution in Figure 4f.The relative uncertainties for the spring constant and the quality factor were determined using the thermal-tune method, as detailed in Experimental Section.They exhibited the highest contributions (i.e., 9% for k and 12% for Q) to the uncertainty on the values of the measured dielectric constants.This highlights the fundamental importance of a well-characterized cantilever's metrology for electrical AFM measurements.Such work was not performed in the present study; but is planned for a future dedicated work.As the tip radius was estimated from the fit of the capacitance gradient spectra at each pixel (Figure S2, Supporting Information), the relative uncertainty in this case is determined from the standard deviation of the values' scattering over the entire scanned area.Finally, the uncertainties of the dielectric structures' thicknesses were estimated based on the sample's manufacturer data.The combined relative uncertainty on the final values of the dielectric constants was determined using the well-known root-sumsquare (RSS) method, i.e., the root of the sum of the squared uncertainties from the different sources.This gives an uncertainty of 17% on the value of  Si 3 N 4 and 18% on the value of  SiO 2 .Thus, the dielectric constants of the two structures with respect to uncertainties are: These values lie within the expected dielectric constants for Si 3 N 4 and SiO 2 , as reported in the literature. [64,65]t is worth noting that the current study concerns the quantification of the dielectric constant using the 1D analytical model (i.e., Equation 9) applied to the stack of buried dielectrics with well-defined geometries and having small differences in dielectric constants.Our experimental results are in agreement with the modeling predictions performed by Fabregas and Gomila in this case. [58]However, for thick dielectric films with respect to the tip radius, our approach is no longer valid and numerical calculations [66] need to be considered to accurately quantify the dielectric constants.Besides, in the case of large differences in dielectric constants between layers, finite-elements-based 3D modeling approach is more appropriate to describe the fringing electric field lines that would strongly contribute to the obtained values.

Conclusion
In this study, we have demonstrated a straightforward approach for parallel imaging of the sample topography, mechanical properties, surface potential, and dielectric properties using the PF KPFM method.We have introduced the idea of simultaneously extracting the static mechanical phase variations signal at the output of the first lock-in amplifier during the second (lift) scan while maintaining the KPFM feedback loop.We showed that this particular combination of signals allows depth-sensitive discrimination between the various sources of contrast associated with top-surface and subsurface properties.We also demonstrated the ability to quantify the dielectric constants for subsurface structures using a test sample made of embedded SiO 2 pads in a Si 3 N 4 matrix.We were able to map the local variations in the dielectric constants with a direct discrimination of the subsurface properties.Combined with the top-surface mechanical properties and surface potential measurements, the approach proposed in this work opens new avenues toward a quantifiable AFM-based dielectric nanotomography.Our proposed method could be easily implemented in a fast force volume mode of operation, which would allow capturing the maps of the phase variations on the fly at different separation distances between the tip and the sample.The analysis of our quantification uncertainties revealed a predominant role of the uncertainty in the cantilever's spring constant.While the quantification of the spring constant receives particular attention in the AFM community when dealing with nanomechanical properties, our findings show that it is largely overlooked in electrical AFM applications.It is worth noting that our approach is based on the use of a relatively simple analytical model for the capacitance between the probe and the sample, which offers an easy means of implementation in experimental measurements.However, using 3D-based modeling of the capacitance would offer a higher accuracy in the quantification of the dielectric constants at the cost of complex analysis and higher computational resources.The dielectric constant measurements in our paper are demonstrated on a fully dielectric model sample with a known geometry.Further improvements of the method would be required for quantifying the dielectric constants of complex and unknown sample geometries.In these cases, the imaging approach in our paper remains valid and enables a qualitative mapping of dielectric-dependent properties.

Experimental Section
Experiments were carried out in ambient air at room temperature on a Dimension Icon AFM with a Nanoscope V controller from Bruker Nano-Surfaces (Santa Barbara, USA).The so-called peak force Kelvin probe force microscopy (PF KPFM) mode was used for the measurements, which is based on the combination of the peak force quantitative nanomechanical mapping (PF QNM) and the frequency modulation Kelvin probe force microscopy (FM-KPFM) methods, using a two consecutive scans approach.A metal-coated (Pt-Ir) silicon probe (SCM PIT, Bruker AFM probes) was used.The deflection sensitivity of the photodiode was determined from the averaged value of the slopes evaluated on ten consecutive approachesretract curves applied on a standard sapphire sample (Bruker, Santa Barbara, USA).The spring constant (k = 5 ± 0.4 N m -1 ) and quality factor (Q = 205 ± 25) of the probe were determined from the average of ten consecutive thermal tune fits using a software-implemented thermal-tune method. [46]The expanded uncertainty values for the measurement of k and Q in this case were determined at 9% and 12%, respectively.A gallium nitride GaN thin film prepared in cross section with p-type dopants implanted in a stripe-like configuration was used to demonstrate the multilevel depth sensitivity of the PF KPFM method.A test sample from Prime-Nano Inc. (Santa Clara, USA) was used to demonstrate the validity of the measurement approach and the measurement of the dielectric constant of subsurface structures.The test sample consisted of patterned SiO 2 structures embedded under a Si 3 N 4 film. [56]The nitride surface was polished to eliminate residual topographic features.A blanket SiO 2 film was deposited, patterned, and etched to produce 90 nm thick squared features (4 × 4 μm 2 ).A thick nitride film was deposited above the SiO 2 structures, and its surface was mirror polished.After the sample preparation, the nitride film thickness is of ≈280 nm, leaving 190 nm of nitride on top of the 90 nm thick oxide pads.The oxide and nitride film thicknesses were confirmed by optical interferometry according to the manufacturer data.The treatment of images in this work was performed using the Gwyddion software. [67]

Figure 1 .
Figure 1.Principle of experimental measurement in PF KPFM consisting of two consecutive scans.The first scan (left) is performed in PF QNM mode and the second scan (right) is performed in FM-KPFM at a lift height.a) The surface topography and b) the adhesion map of a doped GaN cross-section sample were acquired during the first scan.c) The surface potential and d) the phase variations at the output of LIA1 were acquired during the second scan.

Figure 2 .
Figure 2. Side view of the test sample structure (top panel) used for charging test experiments.a) Surface topography showing a flat surface (z < 2 nm).b) Surface potential map showing two charged zones created by applying ±3 V in contact mode scans over small areas.c) Simultaneously measured phase signal at the output of LIA1 showing the signature of the embedded structures with no crosstalk with the surface potential signal.

Figure 3 .
Figure 3. Overview of the measurement approach.a) 3D illustration of the measurement geometry.b) Image stacking approach of the phase signal at the output of the LIA1 at different lift heights.c) The electrostatic configuration of the measurement including the equivalent relative dielectric constant of the sample and the capacitance formed by the spherical tip apex and the sample.d) Variations of the second derivative of the capacitance as a function of the tip-sample separation derived from the stack of phase images at two different locations on the sample.The circles correspond to the experimental data and the solid lines represent the fit to the tip-apex capacitance model at the end of a truncated cone.

Figure 4 .
Figure 4. a) Map of the equivalent (apparent) dielectric constant corresponding to the total thickness of the dielectric sample shown in (b).c) Map of clusters obtained using the k-medoids method.d) Series capacitance model used to extract the map (e) of the actual dielectric constants of the Si 3 N 4 matrix and the embedded SiO 2 structures.f) Histogram distribution extracted from the dielectric constant map.

Table 1 .
List of the various uncertainties related to the different sources of errors associated with the estimation of all experimental parameters involved in the measurement chain to determine the final dielectric constants of the dielectric structures.