Alignment method for the accurate and precise quantification of tip-surface forces

We introduce a procedure to determine accurate and precise force-distance curves in dynamic force measurements utilizing a sharp tip. While single force curves are prone to systematic, often unnoticed errors, we present their self-consistent retrieval by an alignment procedure using repetitive measurements with the force probe oscillating at varied amplitude. By processing model data, we show that the procedure provides the valid force curve, the actual oscillation amplitude, and fully compensates thermal drift. The benefit of the method is demonstrated by application to experimental data.

Figure 1

Stages of model data processing with the FCA method. (a) Simulated frequency shift $\mathrm{\Delta }{f}^i[z_{\text{p}}+\mathrm{\Delta }z\left(t^i\right)]$ data for six ascending and six descending voltage amplitudes $V_A^i$ (see inset) including thermal drift $\mathrm{\Delta }z\left(t\right)$ (see inset). $z_{\text{p}}$ denotes the piezo position. (b) Curves $F^i\left(z_{\text{tip}}\right)$ of the normal force as a function of the tip position $z_{\text{tip}}$ derived from the $\mathrm{\Delta }{f}^i[z_{\text{p}}+\mathrm{\Delta }z\left(t^i\right)]$ data using the starting value $S_{\gamma }=480\mathrm{nm}/\mathrm{V}$, zero drift, and the Sader-Jarvis force inversion algorithm [48]. $z_{\text{tip}}$ is defined as the minimum tip-surface distance attained in the lower turning point of the oscillatory tip motion (see Fig. 4 in Appendix pp1). The dash-dotted line is the model force curve used for generating the artificial frequency shift data. Black circles are optimization nodes used for the FCA correction. (c) Simplified FCA alignment (without considering thermal drift) yielding $S_{\text{simple}}=389\mathrm{nm}/\mathrm{V}$. (d) Accurate force curves resulting from applying the full FCA procedure yielding $S_{\text{FCA}}=415\mathrm{nm}/\mathrm{V}$. The inset demonstrates the excellent match between the $\mathrm{\Delta }z\left(t^i\right)$ corrections resulting from FCA and the artificial $\mathrm{\Delta }z\left(t\right)$ input. (e) Residual error ${\epsilon }_{\text{RMS}}$ as a function of $S$ together with the positions of the approximations $S_{\gamma }$ and $S_{\text{simple}}$ as well as the accurate value $S_{\text{FCA}}$.

Figure 2

Experimental force measurements on a ${\mathrm{CeO}}_2\left(111\right)$ surface decorated with gold clusters. Results compiled in (a)–(c) represent measurements taken with the bare silicon tip while panel (d) displays the final results of measurements performed with a Pt/Ir coated silicon tip. The steps of force inversion and full FCA optimisation were carried out in exactly the same manner as for the model data analysis shown in Fig. 1. Measurements were performed for positions at the center of gold clusters (on-cluster) as well as on atomically flat terraces far from step edges and other disturbances (off-cluster) and are coded with blue/green and black/orange colors, respectively. On-cluster and off-cluster positions are exemplary shown as red dots in the inset of (a), which is a NC-AFM topography image of the gold decorated ${\mathrm{CeO}}_2\left(111\right)$ surface taken with the bare silicon tip. The green and orange curves shown in (c) and (d) are averaged curves of the force curve data plotted as gray points. The left insets in (c) and (d) display the drift polynomial $\mathrm{\Delta }z\left(t\right)$ yielded by the FCA optimisation (solid line) and the deviations from the mean curve found for the six optimisation nodes (circles). The right insets in (c) and (d) display ${\epsilon }_{\text{RMS}}\left(S\right)$, similar to Fig. 1, for on- and off-cluster measurements.

Figure 3

(a) Interaction force $F$ used for the simulations. A combined Morse (green line) and van der Waals (blue line) interaction [69] is adjusted to fit an experimentally measured force-distance curve (circles). (b) Frequency shift $\mathrm{\Delta }f$ and (c) normalized frequency shift $\gamma$ calculated for different amplitudes from the force in (a). Experimental data included as circles.

Figure 4

Axis system describing the vertical $z$ positions. The notation follows Ref. [44].

Figure 5

Determination of the sensitivity factor $S$ utilizing the $\gamma$-method for a silicon probe over a ceria surface. $S_{\gamma ,z}$ follows from the slope of a linear regression and $S_{\gamma }$ from a correction for the inclination $\alpha$ between the cantilever oscillation and the surface normal ($\alpha \approx 12.5^{\circ }$ for the herein used system).

Figure 6

Simulation and experimental results for determining $S$ using the $\gamma$-method. Experimental data are represented by circles while solid lines are the result of simulations. The value $S_{\gamma }=480\mathrm{nm}/\mathrm{V}$ marked by “Fig. 1” is used as the starting value $S_{\gamma }$ for the FCA simulations in Fig. 1.

Figure 7

Effect of the cantilever inclination towards the sample surface. The lateral movement of the tip $\mathrm{\Delta }{x}^{\mathrm{n}}$ from the measurement position above an Au cluster is a function of the oscillation amplitude $A_{\mathrm{z}}^{\mathrm{n}}$.

Figure 8

Data acquisition protocol for FCA.

Figure 9

Schematic construction of the struct example_data.mat in matlab.

Figure 10

Schematic overview of the output data of FCA.

Figure 11

Flowchart for FCA analysis procedure.

Figure 12

Error analysis for the FCA method. The width of the ${\epsilon }_{\text{RMS}}\left(S\right)$ curves $25\%$ above the minimum ${\epsilon }_{\text{RMS}}$ value is marked by horizontal lines. Analysis for (a) silicon and (b) Pt/Ir-coated tip.

Figure 13

Results of the FCA method utilising the matrix force inversion algorithm [47] for (a) a simulated data set (identical model force as in the main paper, see also Fig. 1) and experimental measurements performed with (b) a silicon tip [off-cluster, see also Fig. 2], and (c) a Pt/Ir-coated tip [off-cluster, see also Fig. 2].