Insights into Kelvin probe force microscopy data of insulator-supported molecules

We present a detailed analysis and understanding of Kelvin probe force microscopy (KPFM) data for a system of point charges in a vacuum-dielectric tip-sample system. Explicit formulae describing the KPFM signal $\Delta V$ are derived for the two KPFM operation modes, namely amplitude modulation and frequency modulation (FM). The formulae allow for a physical interpretation of the resulting KPFM signal, reveal contributing parameters, and especially disclose an additive behavior. We numerically evaluate these equations and show exemplary KPFM slice data for a single point charge. The theoretical analysis is complemented by two-dimensional FM-KPFM maps obtained experimentally on 2,5-dihydrobenzoic acid on calcite $(10.4)$. The molecules assemble in two coexisting phases understood as protonated and deprotonated molecules. The two-dimensional maps reveal a convex shape of the KPFM signal $\Delta V$ across the islands and a clear difference between the two phases is revealed. Furthermore, we apply the theoretically suggested difference strategy to extract the molecular component from the measured total KPFM signal.

Figure 1

(a) Electrostatic potential distribution for a fixed tip-sample distance of $d=5$ nm. (b) Vertical line profiles of ${\Phi }^{\left(0\right)}$ at $x=0$ for selected tip-sample distances $d$. (c) Electrostatic potential ${\Phi }^{\left(0\right)}$ at $(x,z)=(0,0.2\mathrm{nm})$ and capacitance $C^{\left(0\right)}$, both with respect to the tip-sample distance $d$.

Figure 2

(a) Setup of the tip-sample system considered in the KPFM model, consisting of the void capacitor of tip, dielectric, and sample back contact. Charges (also in the form of dipoles or multipoles) can be present within this void capacitor. (b) Energy diagram of the void tip-sample capacitor, illustrating the contact potential difference $\Delta \varphi$ between the sample back contact and the tip.

Figure 3

KPFM signal for all four KPFM conditions, namely (a) AM-KPFM for small $A_0$, (b) AM-KPFM for $A_0=2\mathrm{nm}$, (c) FM-KPFM for small $A_0$, and (d) FM-KPFM for $A_0=2\mathrm{nm}$. Data are calculated for a single point charge $2\AA$ above a dielectric sample of thickness 1 mm with $\epsilon =8$. The same color scale is chosen for each data pair of the AM and FM modes. (e) Comparison of vertical line profiles extracted at the center of each data slice. (f) Top: Horizontal line profiles extracted at the lowest slice position ($5\AA$ above the dielectric-vacuum boundary). Bottom: Profiles normalized to each maximum.

Figure 4

2,5-DHBA on calcite $(10.4)$. (a) Topography image presenting coexisting protonated (prot, striped appearance) and deprotonated islands (depr) on the calcite substrate (bare calcite surface regions indicated by arrows). (b) Corresponding KPFM image data. Both island types appear darker (more negative) than the bare calcite areas with the deprotonated islands showing the largest negative contrast.

Figure 5

(a) $Z\text{−}X$ KPFM data slice on islands of protonated and deprotonated 2,5-DHBA on calcite $(10.4)$. (b) Topography data indicating the position of the slice data, and (c) curves extracted from the protonated (dark and light blue) and deprotonated (red) regimes as marked in (a).

Figure 6

$Z\text{−}X$ KPFM data acquired on a pristine calcite $(10.4)$ surface. (a) Slice data showing small lateral variations. (b) Representation of all distance-dependent curves (in gray) and the averaged curve (in blue).

Figure 7

Charges (also in the form of dipoles or multipoles) are abstracted to four regimes within the void capacitor: charges $Q_{\mathrm{tip},i}$ at the tip, $Q_{\mathrm{mol},i}$ in adsorbed molecules, and $Q_{\mathrm{bulk},i}$ in the dielectric substrate as well as dipoles $p_{\mathrm{mol},i}$ of the adsorbed molecular layer.

Figure 8

(a) $Z\text{−}X$ KPFM data acquired on the molecule-covered calcite $(10.4)$ surface across protonated and deprotonated molecular islands. (b) Data with the calcite background subtracted, following Eqs. (22) and (23). In both panels, lateral slice data across different islands are acquired using the same tip. Vertical line profiles are extracted from these data at the indicated positions and the reference data for subtraction are included in (a), right panel.

Figure 9

Representation of all measurement channels in a KPFM slice experiment: frequency shift ($\Delta f$), KPFM signal ($\Delta V$), KPFM error signal ($\delta V$), amplitude ($A$), and dissipation ($\Gamma$) raw data.