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[Nobuyuki Ishida](https://orcid.org/0000-0003-0161-0583), [Takaaki Mano](https://orcid.org/0000-0002-6955-260X)

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[Quantitative theoretical analysis of the electrostatic force between a metallic tip and semiconductor surface in Kelvin probe force microscopy](https://mdr.nims.go.jp/datasets/4161da21-8b1e-438a-a15a-d13ae45307ca)

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Quantitative theoretical analysis of the electrostaticforce between a metallic tip and semiconductorsurface in Kelvin probe force microscopyNobuyuki IshidaNational Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047,JapanE-mail: ishida.nobuyuki@nims.go.jpTakaaki ManoNational Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047,JapanOctober 2023Abstract. Theoretical analysis of the electrostatic force between a metallic tipand semiconductor surface in Kelvin probe force microscopy (KPFM) measurementshas been challenging due to the complexity introduced by tip-induced band bending(TIBB). In this study, we present a method for numerically computing the electrostaticforces in a fully three-dimensional (3D) configuration. Our calculations on a systemcomposed of a metallic tip and GaAs(110) surface revealed deviations from parabolicbehavior in the bias dependence of the electrostatic force, which is consistent withpreviously reported experimental results. In addition, we show that the tip radiiestimated from curve fitting of the theory to experimental data provide reasonablevalues, consistent with the shapes of tip apex observed using scanning electronmicroscopy. The 3D simulation, which accounted for the influence of TIBB, enablesa detailed analysis of the physics involved in KPFM measurements of semiconductorsamples, thereby contributing to the development of more accurate measurement andanalytical methods.21. BackgroundKelvin probe force microscopy (KPFM), a derivative of atomic force microscopy (AFM),is a pivotal analytical technique for measuring electrostatic potential distribution at thenanoscale [1–3]. Since its invention, KPFM has been widely used in characterizingvarious electronic [4–10] and ionic devices [11–16]. This method measures the contactpotential difference (CPD) by detecting the bias voltage at which the electrostaticforce between the probe tip and sample is minimized. The electrostatic force (Felec)is generally explained using a parallel-plate capacitor model, where the force isproportional to the square of the bias voltage (U), displaying parabolic behavior inFelec(U) characteristics. This behavior arises from the fact that for metal samples,the tip-sample capacitance is solely determined by the geometric positioning of thetip and sample, irrespective of the bias voltage. Consequently, the bias voltage ofminimal electrostatic force can be determined by fitting a parabolic curve to the Felec(U)characteristics [17–19].However, the situation differs for semiconductor samples. In this case, the lines ofelectric force (or electric field) generated between the tip and the sample by the appliedbias penetrate the semiconductor surface and enter the interior of the semiconductor, aphenomenon known as tip-induced band bending (TIBB) [20, 21]. The TIBB inducesa change in the tip-sample capacitance depending on the bias voltage, leading to adeviation from the parabolic behavior in the Felec(U) characteristics [10, 22]. Themagnitude and length scales of the TIBB generally depend not only on the bias voltagebut also on other parameters, including the tip radius, tip-sample separation, and dopingdensity of the sample [21, 23]. The complex nature of the TIBB complicates theoreticalanalysis of the electrostatic force. Although several studies have theoretically examinedthe bias dependence of the electrostatic force for semiconductor samples, they have beenlimited to one-dimensional approximations [24–26]. In addition, a direct quantitativecomparison between Felec(U) spectra from experiments and theoretical calculations, atype of line-shape analysis, has yet to be conducted.In this study, we developed a method for numerically computing the electrostaticforce between a metallic tip and semiconductor surface in a fully three-dimensional (3D)model while considering the effect of the TIBB. To achieve this, we utilized the Poissonsolver (SEMITIP) developed by Feenstra [21, 23, 27–30], which allowed us to calculatethe electrostatic potential distribution in both the vacuum and semiconductor regions,induced by an applied potential from a metallic probe tip near the semiconductor surface.The electrostatic forces acting on the tip were calculated using the electric field andcharge density on the tip surface as derived from the potential distribution providedby the SEMITIP. Due to the bias-dependent change in the magnitude of the TIBB,our calculation of the electrostatic force on a GaAs(110) surface revealed non-parabolicbehavior in the Felec(U) curve, which was consistent with our previously reported results[10]. We also achieved good quantitative agreement between the simulated spectra andthe experiments. In addition, we demonstrated that the tip radii used in the experiments3can be estimated from the curve fittings. Direct comparisons of numerical computationsand experiments enable a detailed analysis and rigorous understanding of the physicsinvolved in KPFM measurements of semiconductor samples. Given that many of thesamples analyzed by KPFM are semiconductors, our method provides significant benefitsfor achieving more precise electrostatic potential measurements and for advancing thedevelopment of new measurement and analytical methods.2. Methods2.1. Computation of electrostatic force using SEMITIPThe SEMITIP was originally developed to analyze experimental data obtained fromscanning tunneling microscopy (STM) and spectroscopy (STS) [21, 23]. It can computethe electrostatic potential and resulting tunneling current produced by a metallic probetip near the semiconductor surface. We extended the application range of the SEMITIPby developing a method to calculate the electrostatic forces acting between the tip andsample in AFM/KPFM measurements.The electrostatic force per unit area (felec) acting on the surface of a conductorcan be derived from the relationship felec =12σE, where σ is the surface charge densityand E is the electric field on the conductor surface [31]. On a conductor surface, theelectric field is always directed perpendicular to the surface and the magnitude can becomputed as E = E · n̂ = −dV/dn, where n̂ is the unit vector outward normal to thesurface, V represents the electrostatic potential, and dV/dn is the rate of change in thesurface-normal direction. In addition, the charge density on the conductor surface canbe calculated using the relationship σ = ε0E, where ε0 is the vacuum dielectric constant.Based on these relationships, we calculate the electrostatic force acting on thetip surface using the electrostatic potential distribution near the tip surface. We usedSEMITIP Ver.4, where calculations were performed using a cylindrical symmetric system[23, 32]. The radial direction is represented by r and the direction along the cylindricalaxis is represented by z. The script in the ”semitip_v4.f” file was slightly modified tooutput the coordinates (r and z values) of the grid points in the vacuum region andthe electrostatic potential at each grid point. The details of the script modifications arepresented in Supplementary Fig. S1.In the SEMITIP, the electrostatic potential (Vi,j) in the vacuum regions wascalculated using modified prolate spherical coordinates, denoted by ξi (i = 1, 2, . . . ,m)and ηj (j = 1, 2, . . . , n). Vi,n (i = 1, 2, . . . ,m) corresponds to the electrostatic potentialat the tip surface. To calculate the electrostatic force, we first calculated the electricfield on the tip surface along the η direction (Eetai ) (η direction is the direction traveledwhen i is fixed and j is changed) as follows:Eetai =Vi,n−1 − Vi∆l, (1)where ∆l is the distance between (ξi, ηn−1) and (ξi, ηn). The η direction was not alwaysperpendicular to the tip surface. Thus, to obtain an electric field perpendicular to the4tip surface (E = E · n̂), Eetai was multiplied by 1/ cos θi, where θi is the angle betweenthe surface normal and the lines connecting the two grid points (ξi, ηn−1) and (ξi, ηn).Subsequently, the surface charge density (σ) on the tip surface was calculated using therelationship: σ = ε0E. Finally, the electrostatic force per unit area (felec) acting on thetip surface was calculated based on the relationship felec =12σE.2.2. SPM measurementsAll the measurements were performed at 78 K under ultrahigh vacuum (UHV) conditions(< 1 × 10−8 Pa) using a low-temperature scanning probe microscopy (SPM) system(Unisoku USM-1400). The qPlus sensors with electrochemically etched tungsten (W)tips (P-100WS, Unisoku) were used as scanning probes. The resonance frequency of thesensor ranged from 24 to 31 kHz. The surface oxide layers on the W tips were removedby Ar+-ion sputtering (1.5 kV) for 15 min. The forces acting between the tip and samplewere acquired in frequency modulation (FM) mode [33] with an oscillation amplitude of580 pm. The tip-sample separation was regulated using the STM mode, i.e, an averagedtunneling current was employed as the feedback signal. A bias voltage was applied tothe sample with respect to the tip. The sample grown on an n-type GaAs(001) waferwith an n-type GaAs layer on top [10, 34] was cut into pieces of sizes of approximately 8mm × 3 mm. The sample was cleaved at room temperature to obtain a clean GaAs(110)surface and was immediately transferred to the low-temperature SPM head.3. Results and Discussion3.1. Electrostatic force distribution on tip surfaceUsing the method described in Section 2.1, we computed the electrostatic force on asystem composed of a metallic tip and n-type GaAs(110) surface (the doping densitywas set to 5 × 1017 cm−3). The main input parameters of the SEMITIP are the biasvoltage (U), tip radius (Rtip), tip-sample separation (s), and CPD. We used parametervalues of −2.0 V, 15 nm, 1.53 nm, and −0.850 V for U , Rtip, s, and CPD, respectively,which are typical values assumed in actual experiments. The other parameter valuesare shown in Supplementary Table I. Note that the definition of CPD in the SEMITIPdiffers slightly from ours; thus, the actual input for CPD had the opposite sign (+0.850V). In addition, we included the surface states originating from the Ga dangling bondswithin the conduction band with a Gaussian-type energy distribution [29]. The spatialdensity was set to 4.4 × 1014 cm−2, which corresponds to the spatial density of theGa dangling bond. The energy position of the Gaussian distribution ESS (defined asthe energy difference between the centroid of the Gaussian distribution and the valenceband maximum) and the full width at half maximum (FWHM) ∆ESS were introducedas input parameters. ESS and ∆ESS were set to 1.936 eV and 0.25 eV, respectively.Figure 1 shows the z component of the electrostatic force per unit area (f zelec =felec · ẑ), presented as absolute values and plotted against the radial distance. Note5Figure 1. z component of the electrostatic force per unit area (fzelec) acting on thetip surface plotted as a function of the radial distance. The forces are expressed asabsolute values.that, in general, in AFM/KPFM measurements, only the z component of the atomicforce is detectable. For clarity, the displayed range of r on the tip surface extended from0 to 30 nm, although the computation covered a total range of approximately 2000 nmon the tip surface. A plot for the full range of r is presented in Supplementary Fig. S2.The distribution of the electrostatic force revealed a peak at the center of the tip apex,steeply diminishing along the radial direction and asymptotically approaching zero. Ata distance of the tip radius, the force was reduced by approximately 94%. This resultdemonstrated that the electrostatic forces acting on the tip surface were predominantlyconcentrated within an area approximately equivalent to the tip radius.The z component of the total electrostatic force (F zelec) acting on the tip can becomputed by integrating f zelecdS along the radial direction, where dS is the area element,expressed as 2πrdr. In this study, the direction of the force was configured such thata negative electrostatic force represented an attractive force. F zelec calculated using thedata in Fig. 1 was −550 pN, which is consistent with the range of electrostatic forcestypically observed experimentally [35].3.2. Bias dependence of electrostatic forceBefore discussing the bias dependence of the total electrostatic force acting on thetip, we explain the method for converting the calculated electrostatic force into thefrequency shift measured in the actual experiments. In this study, we focused onKPFM measurements based on the FM mode of AFM [33]. In FM-AFM, the atomicforces between the tip and sample (Fts) are indirectly measured through the shift inthe resonance frequency of the oscillator ∆ν (here, we use ν to express the frequencybecause the letter f is used for force). ∆ν is approximately proportional to the forcegradient dFts/dz when the oscillation amplitude of the tip is small [36, 37]. By contrast,for a large oscillation amplitude in which the force gradient shows a substantial distance6dependence during the oscillation cycle, it is expressed by [36, 37]:∆ν = − ν0kA2〈Ftsq(t)〉 (2)where ν0 is the resonance frequency of the oscillator, A is the oscillation amplitude ofthe tip, and k is the spring constant of the oscillator. q(t) represents the deflection ofthe tip during the oscillation cycle, given by q(t) = A cos(2πν0t), where t is the time.The brackets represent the average over one oscillation cycle.To obtain the term given in the brackets, we computed the total electrostatic forces(F zelec) at 10 tip positions in a single oscillation cycle. For the terms preceding thebrackets, that is, − ν0kA2 , the resonance frequency and oscillation amplitude were set to25.813 kHz and 580 pm, respectively, based on the values used in the experiments. Weused a k value of 1852 N/m, theoretically estimated from the dimensions of the oscillator(the qPlus sensor) used in the experiments.The bias dependence of ∆ν, computed as described above, is depicted by the solidcircles in Fig. 2, revealing a parabolic-like behavior. The bias dependence of F zelec (beforeconversion to ∆ν) is presented in Supplementary Fig. S3, showing a curve shape similarto that of ∆ν signals. The blue dashed line in Fig. 2 represents the fitting of a quadraticfunction (2nd-order polynomial) to the simulated ∆ν signals. The fitting curve failed tofollow the curvature of the simulation, indicating deviations from the parabolic behaviorof the ∆ν(U) characteristics. This non-parabolic behavior can be attributed to thebias-dependent changes in the magnitude and length of the TIBB, leading to bias-dependent alterations in the tip-sample capacitance. The bias dependence of magnitudeof the TIBB (φsurf) measured relative to the potential energy at a point far inside thesemiconductor is shown in Supplementary Fig. S4.Due to the fitting errors, the CPD value obtained from the fitting (−0.657 V)deviated by approximately 0.2 V from the input CPD value (−0.850 V). By contrast,increasing the polynomial order for fitting significantly reduced the fitting errors, asillustrated by the orange dashed line in Fig. 2. Consequently, the CPD value derivedfrom the fitting (−0.852 V) closely matched the input CPD value. These results wereconsistent with our previously reported findings [10]. However, assigning physicalsignificance to an increase in polynomial order can be challenging. The success ofthe fitting is primarily attributed to the mathematical property whereby higher-orderpolynomials can follow the curvature of arbitrary curve shapes more accurately.3.2.1. Effect of Fermi level pinning The aforementioned calculations were performedon a GaAs(110) surface. The GaAs(110) surface is distinctive because no surface statesexist within the bandgap, and this results in a flat band from the bulk to the surfacewithout Fermi-level pinning [38]. Because of this characteristic, the magnitude of theTIBB varies significantly depending on the bias voltage, as shown in SupplementaryFig. S4. However, in many samples observed with KPFM, Fermi-level pinning occursbecause of surface states within the bandgap induced by surface reconstruction,oxidation, contaminations, and so on. In these cases, the bias dependence of the TIBB7Simulation2nd-order poly. fit9th-order poly. fitFigure 2. Bias dependence of ∆ν signals computed using the GaAs(110) surface.Curve fittings of 2nd- and 9th-order polynomials to the simulated spectrum arerepresented by blue and orange dashed lines, respectively.is expected to be highly suppressed. To incorporate the effect of Fermi-level pinning intoour simulation, we introduced virtual surface states at the center of the bandgap of theGaAs(110) (0.756 eV above the top of the valence band), as depicted in SupplementaryFig. S5 and calculated electrostatic forces.In Fig. 3(b), we show the ∆ν(U) spectrum (solid circles) computed whenconsidering the Fermi-level pinning effect. The blue dashed line represents the fittingcurve using a 2nd-order polynomial, which closely follows the simulation withoutsubstantial deviation. This indicates that the simulated curve was nearly parabolic. Thiswas because on this surface, the magnitude of the TIBB varied minimally with the biasvoltage due to Fermi-level pinning, resulting in a nearly constant tip-sample capacitance.Consequently, the bias voltage at which the electrostatic force was minimized could beaccurately determined by conventional parabolic curve fitting. However, the CPD valuedid not correspond to the difference in the work function between the tip (Wtip) andthe sample (Wsample) but rather the value expressed as ((Wsample + φ0surf) − Wtip)/e[25], where φ0surf is the magnitude of the intrinsic band bending of the surface (surfacepotential), and e is elementary charge. This result indicated that the CPD measured ona semiconductor surface was not necessarily the work function difference between thetip and sample but a value reflecting the surface potential.3.2.2. Insight into standard KPFM measurement Till here, we have focused onthe method known as Kelvin probe force spectroscopy (KPFS), where the ∆ν(U)characteristic is obtained, and each curve is fitted with a polynomial function todetermine the bias voltage at which the electrostatic force is minimized. However, thereis another widely used method for KPFM, which involves applying bias modulation toeither the tip or the sample and detecting the modulated force signals using a lock-in technique. In this method, the modulated force signal is minimized during KPFMmeasurements by adjusting the applied DC bias via a feedback circuit.When the ∆ν(U) characteristics deviate from the ideal parabolic behavior, the8Simulation2nd-order poly. fitFigure 3. Bias dependence of ∆ν signals computed using the GaAs(110) surfacewith virtual surface states at the center of the bandgap. Curve fittings of 2nd-orderpolynomials to the simulated spectrum are represented by blue dashed line. Thepositions of the input CPD value (−0.850 V) and the bias voltage corresponding tothe minimal electrostatic force (−0.09 V) are indicated by vertical dashed lines.modulation amplitude of the force signals detected by the lock-in technique doesnot respond linearly to the bias voltage. The lack of linearity may disturb accuratemeasurement of CPD [25] because the generally used feed back circuit assumes a linearresponse of the input signal.One simple method to reduce the nonlinear effect is to minimize the amplitude of themodulation voltage as much as possible. However, this solution is often impractical, asan amplitude of approximately 1 V is typically required to ensure sufficient sensitivity forforce detection. An alternative approach is to acquire ∆ν(U) characteristics at severalpoints on the sample before conducting the KPFM measurements and evaluate whetherthe curves deviate from parabolic behavior. If the ∆ν(U) characteristics exhibit typicalparabolic behavior, the CPD obtained through the feedback circuit can be assumed tobe accurate. This scenario may occur when the surface band is strongly pinned, asexplained in Section 3.2.1. On the other hand, if there is a deviation from parabolicbehavior, switching to the KPFS method would be effective, although the measurementtime would become significantly longer.3.2.3. Tip radius dependence In Fig. 4, we show the simulated ∆ν(U) spectra undertip radii of 5, 10, 30, 60, and 100 nm. The magnitude of the change in ∆ν variessignificantly depending on the tip radius. Similar situations are often encountered inactual experiments following accidental changes in the tip condition as a result of thetip crashing onto the surface. These changes are most likely induced by an increase inthe tip radius resulting from a crash. Although the tip radius might appear to affectonly the magnitude of the electrostatic force as a scaling factor, it also influences thecurvature of the ∆ν(U) spectrum across the entire bias range. This is because the tipradius affects the manner in which the TIBB changes depending on the bias voltage. Infact, a comparison of the normalized ∆ν(U) spectra between tip radii of 5 and 100 nm95 nm10 nm30 nm100 nm60 nmFigure 4. Simulated ∆ν(U) spectra with varying tip radii of 5, 10, 30, 60, and 100nm.showed a discrepancy between the two curves, as depicted in Supplementary Fig. S6.Next, we compare the simulated results with experimental data. For that, weestimated the radii of the W tips used in the experiments by fitting the simulated∆ν(U) spectra to the experimental data. The experimental spectra were obtained onthe n-type GaAs(110) surfaces with a doping density of 5× 1017 cm−3. The tip-sampleseparations for the bias spectroscopy were regulated using the STM mode. The same setpoint (a bias voltage of −2.5 V and an averaged tunneling current of 50 pA) was usedfor all the experiments. The radius of each tip was also evaluated after the experimentby observing its apex with scanning electron microscopy (SEM) and fitting a hyperboliccurve, as used in the SEMITIP simulation, to the tip contour in the SEM image (seeSupplementary Fig. S7). We performed the evaluations for nine W tips. According tothe vendor, the typical tip radius of the as-provided W tips is less than 20 nm. Toprepare W tips with larger radii, we heated the W tips with an electron beam (1 kV,3-5 mA) in the UHV before attaching them to the qPlus sensors.Figure 5 shows a typical result of the curve fitting after parameter optimization.The simulated curve showed good quantitative agreement with the experimental data,accurately reproducing the non-parabolic behavior of the ∆ν(U) spectra. Duringthe curve fittings, the CPD values were treated as fixed parameters, as they couldbe directly obtained from the experiments. The other main parameters were s andRtip. Additionally, the term preceding the brackets in Eq. 2, − ν0kA2 , was treated asa fitting parameter due to the uncertainty of k in the experiments. The uncertaintyof s was expected to be less than 0.4 nm because the tip-sample separation wasregulated using STM mode. To further narrow down the parameter range, we performedsimultaneous fitting of simulated tunneling spectroscopy and force spectroscopy data tothe experiment for one W tip, as explained in Supplementary Fig. S8. After that, thesame value of s was used for fitting the simulations to the ∆ν(U) spectra obtained withother W tips because the set point for the STM feedback was the same. While therestill might be an uncertainty of ±0.1 nm in s, within this small range, the change in the10ExperimentSimulationFigure 5. Experimentally obtained ∆ν(U) curve on the GaAs(110) surface (solidline). The oscillation amplitude of the qPlus sensor was 580 pm. Solid circles showthe simulated ∆ν(U) curve using parameter values of 15 nm, 1.53 nm, and 0.892 V forRtip, s, and CPD, respectively.shape of the ∆ν(U) spectra was small. The parameter that largely affected the shapeof the ∆ν(U) spectra was the value of Rtip. In our study, deviations from the optimaltip radius (±5 nm in R) prevented accurate fitting of the curvature on either side of theinflection point, as illustrated in Supplementary Fig. S9.In Fig. 6, we plot the tip radii, with values estimated from the simulations on thehorizontal axis and the values evaluated from SEM observations on the vertical axis. Thetip radii evaluated from SEM observations approximately scaled with those estimatedfrom the curve fittings. In most cases, the differences between the two evaluations werewithin ±5 nm. These results suggest that fitting the simulated ∆ν(U) spectra to theexperimental data is effective for estimating the radius of the tip used in experiments.The tip radius can also be estimated from the simulation of the tunneling spectroscopydata using SEMITIP [18]. However, since the curve shape of ∆ν(U) spectra is moresensitive than that of tunneling spectra, electrostatic force simulation can significantlynarrow down the parameter range.3.2.4. Tip-sample separation dependence We also investigated the tip-sampleseparation dependence of the simulated ∆ν(U) spectra and compared them with theexperiments. In the experiments, ∆ν(U) spectra were obtained by lifting the tip fromthe tip-sample separation regulated by STM feedback (a bias voltage of −2.5 V and anaveraged tunneling current of 50 pA). The lift heights were 0.0, 0.1, 0.3, 0.5, 1.0, and3.0 nm. In the simulations, the experimental data with a lift height of 0.0 nm was firstcurve-fitted with the simulation by parameter optimization. Subsequently, we computed∆ν(U) spectra by only changing the tip-sample separation according to the lift heights.The tip radius estimated from the parameter optimization was 12 nm, which was withinthe typical range of radius for W tips used in our study.The ∆ν(U) spectra from the experiments and simulations are displayed in Fig. 7 assolid lines and solid circles, respectively. In the experiments, the atomic force between11Figure 6. Plots of tip radii evaluated using two methods: curve fitting of theoreticalmodels and SEM observations. The horizontal axis represents the tip radii estimatedfrom the curve fittings, while the vertical axis shows the radii obtained from SEMobservations.the tip and sample (or ∆ν) did not reach zero, even at the bias voltage correspondingto the CPD due to the contribution from the van der Waals force. Therefore, we addedoffsets to the ∆ν signals in the simulation for the comparisons.For the data with a lift height of 0.0 nm, a fairly good agreement between theoryand experiment was achieved after parameter optimization. Furthermore, the spectrasimulated with increased tip-sample separations also quantitatively agreed well withthe experimental values without further parameter optimization. These results showthat the physical model used in the SEMITIP and the calculation of electrostatic forceare plausible for reproducing the experiments. However, when a W tip having a largertip radius (105 nm) was used, slight deviations were observed in fitting the simulatedspectra to the experiments with increased tip heights, as displayed in SupplementaryFig. S10. These discrepancies could not be mitigated by changing several parametersincluding the tip-sample separation regulated by STM feedback and the tip radius. Sofar, we have not been able to clarify the reason behind the discrepancies. In the future,the model of SEMITIP may need to be improved to increase calculation accuracy forlarge tip radii.4. SummaryWe developed a method for calculating the electrostatic forces between a metallic tipand semiconductor surface, which has been challenging in previous studies due to thecomplexity introduced by the TIBB. We showed that our calculations were consistentwith the expected properties of the electrostatic force, including (i) the concentration ofthe force around the tip apex, (ii) a parabolic-like behavior in the bias dependence, (iii) aminimum electrostatic force occurring at the bias voltage corresponding to the CPD, andREFERENCES 123.0 nm1.0 nm0.5 nm0.3 nm0.1 nm0.0 nmFigure 7. Tip-sample separation dependence of the ∆ν(U) spectra from theexperiments (solid lines) and simulations (solid circles). The offsets from the tip-sample separation regulated by the STM feedback were 0.0, 0.1, 0.3, 0.5, 1.0, and 3.0nm.(iv) a magnitude of the electrostatic force ranging from pN to nN. Simulations conductedon a GaAs(110) surface exhibited a ∆ν(U) spectra that deviated from parabolic behaviordue to a bias-dependent change in the TIBB, leading to a bias-dependent variation in thetip-sample capacitance. Curve fitting of the simulated ∆ν(U) spectra to experimentaldata showed good quantitative agreement. In addition, we demonstrated that thetip radii used in the experiments could be estimated from the curve fitting with anaccuracy of approximately 10 nm. The SEMITIP software used for the electrostatic forcecomputation is capable of handling inhomogeneous samples, including p-n junctions andhetero-interfaces of semiconductors. Thus, our method can be used for future theoreticalanalysis of physics involved in the KPFM measurements, including the manner in whichthe tip radius and oscillation amplitude affect the spatial resolution of KPFM and theextent to which the strength of Fermi-level pinning affects CPD measurements.AcknowledgmentsWe would like to thank R. M. Feenstra for the fruitful discussion concerning thecalculation of the electrostatic force using the SEMITIP. This work was partiallysupported by JSPS KAKENHI Grant Numbers JP17K06366, JP21H01818, andJP24K01367.ReferencesReferences[1] Nonnenmacher M, O’Boyle M P and Wickramasinghe H K 1991 Appl. Phys. Lett.58 2921[2] Melitz W, Shen J, Kummel A C and Lee S 2011 Surf. Sci. Rep. 66 1REFERENCES 13[3] Sadewasser S and Glatzel T (eds) 2012 Kelvin Probe Force Microscopy: Measuringand Compensating Electrostatic Forces (Springer)[4] Shikler R, Meoded T, Fried N and Rosenwaks Y 1999 Appl. Phys. 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