%\cref{fig:dC_vs_d2C_Freq_Sweep} -- Figure 6
%\cref{eq:calculation_d2C_from_amplitude_signal} -- eq. (8)
%\cref{eq:calculation_dC_from_amplitude} -- eq. (9)
%\cref{fig:F14H20_SCM_Pictures}  -- Figure 7
%\cref{fig:capacitors_overview_3D} -- Figure 2

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%\usepackage{dblcoltabularx}


\usepackage[per-mode = fraction,output-decimal-marker={.},separate-uncertainty=true,exponent-product = \cdot]{siunitx} %SI-Einheiten english!
\sisetup{separate-uncertainty} %Ausrichtung an \pm in Tabelle ohne Matheumgebung https://tex.stackexchange.com/questions/305395/table-alignment-around-pm-sign-how-to-have-good-alignment-without-pm-where-a
	\DeclareSIUnit\Molar{M}
	\DeclareSIUnit\U{U}
\usepackage[version=3]{mhchem} %Summenformeln angeben
\usepackage{pdfpages}
%\usepackage{acronym} %Abkürzungsverzeichnis % alt: \usepackage[nohyperlinks, printonlyused]{acronym}
\usepackage[printonlyused]{acronym}
\usepackage{comment} %Schreiben von Kommentaren



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%% Place any additional macros here.  Please use \newcommand* where
%% possible, and avoid layout-changing macros (which are not used
%% when typesetting).
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%\newcommand*{\CHCL}{\chem{CH_2Cl_2}}
%\newcommand*{\HNMR}{\chem{^{1}H~NMR}\xspace}
%\newcommand*{\CNMR}{\chem{^{13}C~NMR}\xspace}

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%% Beginning of the article
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\begin{document}

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%% Meta-data block
%% ---------------
%% The title of the article is given with the usual \title command.
%% The title of an article should be clear, concise and comprehensible to all
%% readers with the purpose of quickly identifying the focus of the reported
%% work. It should be brief and contain the most important keywords for search
%% engine optimization. The use of capitals should be restricted to the first
%% word and proper nouns. As far as possible abbreviations should be avoided.
%%
%% If you write a file for Supporting Information using manuscript=suppinfo,
%% you should give an additional title using the macro \sititle or the
%% optional argument of \title
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%% Each author needs to be given with a separate \author command.
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%% For corresponding authors please use \author* and give the email
%% address as a second mandatory argument.
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%% The affiliation of authors is given after the authors; the
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%% argument of \author and \author* to give the number of that
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%% end.
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\title{Nanoscale capacitance spectroscopy based on multi-frequency electrostatic force microscopy} 
%%%\sititle{} % when using manuscript=suppinfo
\author{Pascal N. Rohrbeck}%{rohrbeckp@mpip-mainz.mpg.de}
\affiliation[1]{Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany}
\affiliation{Department of Chemistry, University of Mainz, Duesbergweg 10-14, 55128 Mainz, Germany}
%\author[1,3]{Pravash Bista}
\author[1,3]{Lukas D. Cavar}
\author[1,2]{Franjo Weber}
\author[1]{Peter G. Reichel}
\author[1,3]{Mara Niebling}
\author*[1]{Stefan A. L. Weber}{stefan.weber@mpip-mainz.mpg.de; Stefan.Weber@ipv.uni-stuttgart.de}
%\affiliation{Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany}
\affiliation{Department of Physics, University of Mainz, Staudingerweg 7, 55128 Mainz, Germany}
\affiliation{Institute for Photovoltaics, University of Stuttgart, Pfaffenwaldring 47, 70569 Stuttgart, Germany}
\maketitle

\section{Abbreviations}

\begin{acronym}[Placeholderrrrrrrr]
\acro{AC}{alternating current}
\acro{AFM}{Atomic Force Microscopy}
\acro{AM}{amplitude modulation}
\acro{AM-AFM}{Amplitude Modulation Atomic Force Microscopy}
\acro{AM-KPFM}{Amplitude Modulation Kelvin Probe Force Microscopy}
%\acro{BACl}{\textit{n}-Butylammoniumchlorid (\ce{C4H12ClN})}
%\acro{BAPI}{Butylammoniumbleiiodid (\ce{BA2PbI4})}
%\acro{BCP}{Bathocuproin (2,9-Dimethyl-4,7-diphenyl-1,10-phenanthrolin)}
\acro{BW}{bandwidth}
\acro{CPD}{contact potential difference}
%\acro{CsFA}{Cäsiumformamidinium}
%\acro{CsFAPI}{Cäsiumformamidiniumbleiiodid (\ce{Cs_xFA_{1-x}PbI3})}
%\acro{CsI}{Cäsiumiodid}
\acro{DC}{direct current}
%\acro{DMF}{\textit{N},\textit{N}-Dimethylformamid  (\ce{HCON(CH3)2})}
%\acro{DMSO}{Dimethylsulfoxid(\ce{(CH3)2SO})}
\acro{H-SCM}[H-SCM]{heterodyne Scanning Capacitance Microscopy}
\acro{DS}{Dielectric spectroscopy}
\acro{DOS}{density-of-states}
%\acro{Dwell-Time}[\glqq\textit{Dwell-Time}\grqq]{Verweilzeit}
\acro{EDLC}{Electrical double-layer capacitors}
\acro{EFM}{Electrostatic Force Microscopy}
\acro{EIS}{Electrochemical Impedance Spectroscopy}
%\acro{ETL}{Electron Transport Layer}
\acro{EQE}{external quantum efficiency}
\acroplural{EQE}[EQEs]{external quantum efficiencies}
%\acro{FA}{Formamidinium}
%\acro{FAI}{Formamidiniumiodid}
%\acro{FAPI}{Formamidiniumbleiiodid (\ce{FAPbI3})}
%\acro{FFFR-Modus}{schneller freier Kraftwiederherstellungsmodus(\glqq \textit{Fast Free Force Recovery-Mode}\grqq{})}
%\acro{FF-TR-EFM}[FF-tr-EFM]{schnelle und freie zeitaufgelöste elektrostatische Kraftmikroskopie (\glqq \textit{Fast Free Time-Resolved Electrostatic Force Microscopy}\grqq{})}
\acro{FET}{field-effect transistor}
\acro{FFM}{fast force mapping}
\acro{FIB}{focused ion beam}
\acro{FM}{frequency modulation}
\acro{FM-AFM}{Frequency Modulation Atomic Force Microscopy}
\acro{FM-KPFM}{Frequency Modulation Kelvin Probe Force Microscopy}
%\acro{FTO}{Fluorine doped Tin(IV) Oxide; F$@$\ce{SnO2})}
\acro{F14H20}{Perfluoroalkyl-Alkane \ce{F(CF2)14(CH2)20H}}
\acro{GB}{Grain boundary}
\acroplural{GB}[GBs]{Grain boundaries}
\acro{H-KPFM}{Heterodyne Kelvin Probe Force Microscopy}
%\acro{HTL}{Lochtransportschicht (\glqq \textit{Hole Transport Layer}\grqq{})}
%\acro{I-Gain}{Integralverstärkungsfaktor}
\acro{invols}[InvOLS]{inverse optical lever sensitivity}
%\acro{IPA}{Isopropanol}
%\acro{I-Regler}{Integralregler}
\acro{IS}{impedance spectroscopy}
%\acro{ITO}{Indiumzinnoxid (\glqq \textit{Indium Tin Oxide}\grqq{}; \ce{In2O3 $\cdot$ (SnO2)_{x}})}
\acro{KPFM}{Kelvin Probe Force Microscopy}
%\acro{LASER}{\glqq Light Amplification by stimulated Emission of Radiation\grqq}
\acro{LED}{light-emitting diode}
\acro{LIA}{lock-in amplifier}
%\acro{MA}{Methylammonium}
%\acro{MACl}{Methylammoniumchlorid (\ce{CH3NH3+ + Cl-})}
%\acro{MAI}{Methylammoniumiodid (\ce{CH3NH3+ + I-})}
%\acro{MAPBr}{Methylammoniumbleibromid (\ce{MAPbBr3})}
%\acro{MAPI}{Methylammoniumbleiiodid (\ce{MAPbI3})}
%\acro{OACl}{\textit{n}-Octylammoniumchlorid (\ce{C8H20ClN})}
%\acro{O3}[O$_3$]{Ozon}
%\acro{PbI$_2$}{Bleiiodid}
%\acro{PCBM}{[6,6]-Phenyl \ce{C61}-Buttersäuremethylester}
%\acro{PEA}[\ce{PEA+}]{Phenylethylammonium (\ce{Ph-C2H4-NH3+})}
%\acro{PEACl}{Phenylethylammoniumchlorid (\ce{Ph-C2H4-NH3+ + Cl-})}
%\acro{PEAI}{Phenylethylammoniumiodid (\ce{Ph-C2H4-NH3+ + I-})}
\acro{PeLED}{perovskite light emitting diodes}
%\acro{PFN-Br}{Poly(9,9-bis(3$^\prime$-(\textit{N},\textit{N}-dimethyl)-\textit{N}-ethylammonium-propyl-2,7-fluorene)-alt-2,7-(9,9-dioctylfluorene))dibromide (\ce{(C56H80N2Br2)n})} %CAS:  889672-99-5 
%\acro{P-Gain}{Proportionalverstärkungsfaktor}
%\acro{PI-Regler}{Proportional- und Integralregler}
\acro{PLL}{phase-locked loop}
%\acro{ppm}{Anteile pro Million Teilchen (\glqq\textit{Parts Per Million}\grqq{})}
%\acro{P-Regler}{Proprotionalregler}
\acro{PS}{polystyrene}
\acro{PSC}{Perovskite solar cell}
\acro{PS-LDPE}[PS/LDPE]{Polystyrene low density polyethylene}
\acro{PS-PVAc}[PS/PVAc]{polystyrene / poly(vinyl acetate)}
\acro{PV}{Photovoltaik}
\acro{PVAc}{poly(vinyl acetate)}
\acro{QC}{quantum capacitance}
%\acro{Retrace}[\glqq\textit{Retrace}\grqq]{Rücklauf}
%\acro{Sample-Rate}[\glqq \textit{sampling rate}\grqq]{Datenerfassungsrate}
\acro{SCM}{Scanning capacitance microscopy}
\acro{SEM}{Scanning Electron Microscopy}
\acro{SNR}{Signal-to-Noise Ratio}
\acro{STM}{Scanning Tunneling Microscopy}
%\acro{SQ}{Shockley-Queisser}
\acro{TR}[tr]{time-resolved}
%\acro{Trace}[\glqq\textit{Trace}\grqq]{Hinlauf}
\acro{TR-EFM}[tr-EFM]{time-resolved Electrostatic Force Microscopy}
\acro{TR-KPFM}[tr-KPFM]{time-resolved Kelvin Probe Force Microscopy}
%\acro{UV}{Ultra violet}
%\acro{VE}{Vollentsalzt}
%\acro{V$_{OC}$}[$V_\text{OC}$]{Open Circuit Voltage}
%\acro{WW}{Wechselwirkung}
%\acro{2D}{Zweidimensional}
%\acro{3D}{Dreidimensional}
%\acro{IR}{Infrarot}
\end{acronym}

%\end{Abbreviations}

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%% The document should begin with an abstract, if appropriate. If one
%% is given and should not be, a warning is issued.
%% The abstract should summarize the context and purpose of the study,
%% the main findings and provide a brief summary and potential
%% implications. Abbreviations should be used sparingly in the abstract.
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%% standard characters are allowed.
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%\begin{abstract}

%\end{abstract}

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%% Keywords can be given with the \keywords command. Any number of
%% keywords can be given, but a number of at least five keywords is
%% recommended. The arguments are to be sorted alphabetically.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%alphabetische Reihenfolge!!!!! %vor der Abgabe erledigen !!!
%\keywords{atomic force microscopy; dielectric constant; dielectric spectroscopy; Kelvin probe force microscopy;   scanning capacitance microscopy; multi frequency AFM; polystyrene; nanoparticles; frequency modulation heterodyne; quantitative force spectroscopy}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The main text starts right here. For each required and optional
%% section of the chosen document type a special command is defined.
%%
%% It is strongly recommended to use BibTeX for managing references.
%% Citations and citation lists can be given with the \cite command.
%% Please note, that not all references have been added to the
%% example document.
%%
%% For references in floats \cite is locally redefined to
%% adds the reference to the end of the list of references.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% When referencing objects, LaTeX offers a label--ref mechanism.
%% The beilstein class extends this approach with the \cref command,
%% that adds the corresponding type of the object as well.
%%\cref{fig:AlkylidenationApproach}
%%
%% Tables, figures and schemes must have a single column or double
%% column width. To make life easier, some commands are defined.
%%
%% Captions (legends) will always be added at the correct place no
%% matter where you put in the source code.
%% Please note: labels always have to come /after/ the \caption.
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\begin{comment}
\begin{figure}
\centering
\caption{Contributions of the respective components to the second numeric derivative of the capacitance. The blue line marks the apex, the green line the cone, the red line the lever and the black line marks the entire system of the three components in parallel. It is noticeable that the contribution of the tip surpasses the cone-contribution for distances smaller than $\approx$ \SI{20}{\nano\meter}, while the contribution of the lever remains small.}
%\includegraphics[width=\textwidth,keepaspectratio]{Bilder/second_derivative.pdf}
%\label{fig:second_derivation_capacitance_signal_distance}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}
\end{comment}

\begin{comment}
\begin{scheme}
%\includegraphics[width=8.2cm,keepaspectratio]{scheme2}
\caption{Reaction of substituted silylketenes with ester-stabilised phosphoranes.}
\label{scheme:2}
\end{scheme}
\end{comment}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The Supporting Information is an essential part of many articles.
%% They are given inside the ``suppinfo'' environment with the
%% \sifile command which gets the following mandatory arguments:
%% #1: File name
%% #2: File type
%% #3: Descriptive File title
%% A long description can be given using the optional argument.
%%
%% You can label each entry and reference it in the text.
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\begin{suppinfo}
\begin{comment}
Supporting information features copies of \HNMR spectra of silylated diazoketones \CN{2} and silylketenes \CN{1}, plus \chem{{}^1H} and \CNMR spectra of allenylsilanes \CN{6}, \CN{7}, and \CN{14}--\CN{19}.
\sifile{S1.pdf}{PDF}{Experimental part}\label{si:1}
\sifile{S2.pdf}{PDF}{NMR spectra of compounds \CN{1}, \CN{2}, \CN{6} and \CN{7}}
\sifile{S3.pdf}{PDF}{NMR spectra of compounds \CN{14--19}}
\end{comment}

\begin{comment}
Supporting information features copies of \HNMR spectra of silylated diazoketones \CN{2} and silylketenes \CN{1}, plus \chem{{}^1H} and \CNMR spectra of allenylsilanes \CN{6}, \CN{7}, and \CN{14}--\CN{19}.
\sifile{S1.pdf}{PDF}{Experimental part}\label{si:1}
\sifile{S2.pdf}{PDF}{NMR spectra of compounds \CN{1}, \CN{2}, \CN{6} and \CN{7}}
\sifile{S3.pdf}{PDF}{NMR spectra of compounds \CN{14--19}}
\end{comment}

Supporting information features a comparison of the working principles of \ac{H-KPFM} and \ac{H-SCM}, all the raw and normalized  data of the \ac{H-SCM} frequency spectroscopy, the full  comparison of the \ac{H-SCM}, SF-\ac{EFM}, and \ac{H-KPFM} images on the \ac{F14H20} structures, and finally a comparison of the model data and the measured data on the microcapacitors.

\begin{figure}%↓
\centering
\caption{Schematic comparison of the excitation and detection frequencies in \ac{H-KPFM} and \ac{H-SCM}. The lower part shows the transfer function of the cantilever, where the amplitude is plotted vs the logarithmic angular frequency. The upper part shows the excitation frequencies   ($\downarrow$) and the detection frequencies ($\uparrow$) of the applied frequencies. The red arrow corresponds to topography- and the blue arrow to the electrical signal. Representation of Fig. S1 was inspired by \cite{Garrett2016,Axt2018a}.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/H-KPFM_and_H-SCM_scheme_working_principle_two.pdf}
%\label{fig:H_KPFM_H_SCM_scheme_working_principle}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%d2C
%d2C vs f overview

\begin{figure}
\centering
\caption{Non-normalized data of the comparison of the $C^{\prime\prime}$ frequency sweep shown in Figure 6 on the four spots while in \ac{H-SCM} (see eq. (8)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_d2C_vs_Frequency.pdf}
%\label{fig:si:Freq_Sweep_d2C_overview}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%d2C vs f zoom
\begin{figure}
\centering
\caption{Zoom of the non-normalized data from the comparison of the $C^{\prime\prime}$ frequency sweep shown in Figure 6 on the four spots while in \ac{H-SCM} (see eq. (8)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_d2C_vs_Frequency_zoom.pdf}
%\label{fig:si:Freq_Sweep_d2C_zoom}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%d2C vs f normed 
\begin{figure}
\centering
\caption{Normalized data of the  $C^{\prime\prime}$ frequency sweep shown in Figure 6 on the three spots while in \ac{H-SCM} (see eq. (8)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_d2C_vs_Frequency_normed.pdf}
%\label{fig:si:Freq_Sweep_d2C_normed}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%d2C vs f normed zoom
\begin{figure}
\centering
\caption{Zoomed and normalized data of the  $C^{\prime\prime}$ frequency sweep shown in Figure 6 on the three spots while in \ac{H-SCM} (see eq. (8)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_d2C_vs_Frequency_normed_zoom.pdf}
%\label{fig:si:Freq_Sweep_d2C_normed_zoom}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%phase d2C vs f
\begin{figure}
\centering
\caption{Non-normalized data of the phase signal $\varphi$ spectra of the comparison from the $C^{\prime\prime}$ frequency sweep shown in Figure 6 on the four spots while in \ac{H-SCM} (see eq. (8)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_elec_Phase_d2C_vs_Frequency.pdf}
%\label{fig:si:Freq_Sweep_d2C_Phase}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%phase d2C vs f zoom
\begin{figure}
\centering
\caption{Zoom of the non-normalized data of the phase signal $\varphi$ spectra of the comparison from the $C^{\prime\prime}$ frequency sweep shown in Figure 6 on the four spots while in \ac{H-SCM} (see eq. (8)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_elec_Phase_d2C_vs_Frequency_zoom.pdf}
%\label{fig:si:Freq_Sweep_d2C_Phase_zoom}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%dC

%dC vs f overview
\begin{figure}
\centering
\caption{Non-normalized data of the comparison of the $C^{\prime}$ frequency sweep shown in Figure 6 on the four spots while in SF-\ac{EFM} mode (see eq. (9)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_dC_vs_Frequency.pdf}
%\label{fig:si:Freq_Sweep_dC_overview}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%dC vs f zoom
\begin{figure}
\centering
\caption{Zoom of the non-normalized data of the comparison of the $C^{\prime}$ frequency sweep shown in Figure 6 on the four spots while in SF-\ac{EFM} mode (see eq. (9)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_dC_vs_Frequency_zoom.pdf}
%\label{fig:si:Freq_Sweep_dC_overview_zoom}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%dC vs f normed 
\begin{figure}
\centering
\caption{Normalized data of the  $C^{\prime}$ frequency sweep shown in Figure 6 on the three spots while in SF-\ac{EFM} mode (see eq. (9)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_dC_vs_Frequency_normed.pdf}
%\label{fig:si:Freq_Sweep_dC_normed}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%dC vs f normed zoom
\begin{figure}
\centering
\caption{Zoomed and normalized data of the  $C^{\prime}$ frequency sweep shown in Figure 6 on the three spots while in SF-\ac{EFM} mode (see eq. (9)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_dC_vs_Frequency_normed_zoom.pdf}
%\label{fig:si:Freq_Sweep_dC_normed_zoom}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%phase dC vs f
\begin{figure}
\centering
\caption{Non-normalized data of the phase signal $\varphi$ spectra of the comparison of the $C^{\prime}$ frequency sweep shown in Figure 6 on the four spots while in SF-\ac{EFM} mode (see eq. (9)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_elec_Phase_dC_vs_Frequency.pdf}
%\label{fig:si:Freq_Sweep_dC_Phase}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

%phase dC vs f zoom
\begin{figure}
\centering
\caption{Zoom of the non-normalized data of the phase signal $\varphi$ spectra of the comparison of the $C^{\prime}$ frequency sweep shown in Figure 6 on the four spots while in SF-\ac{EFM} mode (see eq. (9)). This was conducted with the $\mu$masch's  HQ:NSC18/Pt cantilever.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Freq_Sweep_Comparison_elec_Phase_dC_vs_Frequency_zoom.pdf}
%\label{fig:si:Freq_Sweep_dC_Phase_zoom}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}


\begin{figure}
\centering
\caption{Full version of the \ac{H-SCM} pictures given in Figure 7. \ac{H-SCM} pictures made on \ac{F14H20} with (a) the topography, (b) the $C^{\prime\prime} (\partial^2 C)$ picture at $1.59$ and \SI{1,98}{\mega\hertz}, (c) electric phase $\varphi_\text{el}$ of the $C^{\prime\prime} (\partial^2 C)$ signal at $1.59$ and \SI{1,98}{\mega\hertz}, (d) the $C^{\prime} (\partial C)$ picture at \SI{235,579}{\kilo\hertz}, (e) electric phase $\varphi_\text{el}$ of the $C^{\prime} (\partial C)$ signal at \SI{235,579}{\kilo\hertz}, (f) the \acs{CPD} picture, (g) the $C^{\prime\prime} (\partial^2 C)$ picture at $15.88$ and \SI{16,28}{\mega\hertz}, (h) electric phase $\varphi_\text{el}$ of the $C^{\prime\prime} (\partial^2 C)$ signal at $15.88$ and \SI{16,28}{\mega\hertz}, (i) the picture of the mechanical amplitude at the resonance frequency of \SI{74,580}{\kilo\hertz}, and (j) the picture of the mechanical phase at the resonance frequency of \SI{74,580}{\kilo\hertz}. This was conducted with the $\mu$masch's  HQ:NSC18/Pt Cantilever.}
\includegraphics[width=16.4cm,keepaspectratio]{Pictures/Results/Zeichnung_F14H20_Comparison_dC_d2C_SI.pdf}
%\label{fig:si:F14H20_SCM_Pictures}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}\SI{74,58039559423923}{\kilo\hertz}
\end{figure}


\begin{figure}
\centering
\caption{A comparison of the measured $C^{\prime\prime}$ values on various capacitors, as shown in Figure 2, is presented. The measurements, performed using the NuNano SPARK 70 Pt cantilever (solid lines), are contrasted with the theoretical contributions of the respective components to the first numerical derivative $C^{\prime}$ of the capacitance (dotted lines) as a function of the tip-to-sample distance, 
$z$ For the theoretical calculations, the properties of the  NuNano SPARK 70 Pt cantilever  ($w = \SI{30}{\micro\meter}$,  $l = \SI{225}{\micro\meter}$, $\alpha = \SI{11}{\deg}$, $h = \SI{12}{\micro\meter}$, $\theta = \SI{25}{\deg}$, $r = \SI{18}{\nano\meter}$, $\delta = \SI{3,7e-7}{}$)  with an mechanical amplitude of $A_\text{m} = \SI{10}{\nano\meter}$,  an excitation voltage of    $V_\text{AC} = \SI{2}{\volt}$, and a total amount of calculated points of  100,000, was used for these.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Comparison_Model_&_Data_d2C_vs_ZSensor_data_model_2_500.pdf}
%\label{fig:si:Data_and_Model_Peter_d2C}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}

\begin{figure}
\centering
\caption{A comparison of the measured $C^{\prime}$ values on various capacitors, as shown in Figure 2, is presented. The measurements, performed using the NuNano SPARK 70 Pt cantilever (solid lines), are contrasted with the theoretical contributions of the respective components to the first numerical derivative $C^{\prime}$ of the capacitance (dotted lines) as a function of the tip-to-sample distance, 
$z$. For the theoretical calculations, the properties of the  NuNano SPARK 70 Pt cantilever  ($w = \SI{30}{\micro\meter}$,  $l = \SI{225}{\micro\meter}$, $\alpha = \SI{11}{\deg}$, $h = \SI{12}{\micro\meter}$, $\theta = \SI{25}{\deg}$, $r = \SI{18}{\nano\meter}$, $\delta = \SI{3,7e-7}{}$)  with an mechanical amplitude of $A_\text{m} = \SI{10}{\nano\meter}$,  an excitation voltage of    $V_\text{AC} = \SI{2}{\volt}$, and a total amount of calculated points of  100,000, was used for these.}
\includegraphics[width=8.2cm,keepaspectratio]{Pictures/Results/Comparison_Model_&_Data_dC_vs_ZSensor_data_model_2_500.pdf}
%\label{fig:si:Data_and_Model_Peter_dC}
%\includegraphics[width=8.2cm,keepaspectratio]{figure1}
%% \sglcolfigure{figure1}
\end{figure}


\end{suppinfo}


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