Time-averaged cantilever deflection in dynamic force spectroscopy

We theoretically and experimentally study a hitherto neglected effect in dynamic force spectroscopy. The time-averaged deflection of the cantilever and the frequency shift of its second flexural mode were measured for 11 oscillation amplitudes between 12.8 and 0.51 nm above a maximum protrusion of the atomically resolved topography on KBr(001). A small but measurable time-averaged deflection was observed and the magnitude increased with decreasing amplitude. Interaction force curves essentially coincident over the whole attractive range were obtained from the measured frequency shifts with a rms noise linearly decreasing with amplitude. The correction of the tip-sample distance with the measured time-averaged deflection changes the strength of the interaction force. The deflection calculated from the extracted interaction force agrees with the direct measurement and is approximately proportional to the frequency shift except at the smallest amplitudes.

Figure 1

(Color online) (a) Schematic drawing of a cantilever oscillating above the sample surface in the attractive range. [(b)–(d)] Calculated two-dimensional maps of (b) the interaction force, (c) the frequency shift, and (d) the time-averaged deflection. Assumed parameters: $f_{1\text{st}}=150\text{kHz}$, $A_{1\text{st}}=1\text{nm}$, $k_0=k_{1\text{st}}=30\text{N}/\text{m}$, $R_{\text{tip}}=5\text{nm}$, and $\lambda =79\text{pm}$.

Figure 2

(Color online) (a) A series of frequency shift curves vs nominal distance measured using 11 different amplitudes of the second flexural mode on a KBr(001) surface. The inset image was obtained with $\Delta f_{2\text{nd}}=-3.6\text{Hz}$ and $A_{2\text{nd}}=12.8\text{nm}$. (b) Simultaneously recorded time-averaged deflection curves. Cantilever parameters; $f_{2\text{nd}}=1039369\text{Hz}$, $k_0=33\text{N}/\text{m}$, $k_{2\text{nd}}=1442\text{N}/\text{m}$, and $Q_{2\text{nd}}=11727$. (c) Time-averaged deflection curves estimated from the measured frequency shift using Eq. (4).

Figure 3

(Color online) Essentially unique conservative interaction force extracted from frequency shifts measured with 11 different amplitudes plotted against the corrected tip-sample distance $Z_{\text{c}}$. The colors, one for each amplitude, are the same as in Fig. 2a. The $Z_{\text{c}}$ scale is adjusted so that all curves match at $Z_{\text{c}}=0.44\text{nm}$ and that $F\approx 0$ at $Z_{\text{c}}=0$.

Figure 4

(Color online) (a) Long-range force curves, each offset by 0.4 nN for clarity. (b) Rms force noise in the same range, decreasing linearly with the amplitude. The colors, one for each amplitude, are the same as in Fig. 2a.

Figure 5

(Color online) (a) Comparison of the extracted and time-averaged forces measured with the smallest amplitude of 0.51 nm. $F_{\text{D}}\left(Z_{\text{c}}\right)$ and $F_{\text{D}}\left(Z_{\text{c}}^{\prime }\right)$ denote the extracted force curves with and without correction for the time-averaged deflection. $F_{\text{s-cal}}$ and $F_{\text{s-exp}}$ denote the time-averaged force calculated from the extracted force using Eq. (1) and from the measured time-averaged deflection, namely, $k_0\Delta Z_0$. (b) Magnification of $F_{\text{D}}\left(Z_{\text{c}}\right)$ and $F_{\text{D}}\left(Z_{\text{c}}^{\prime }\right)$ in the most attractive force region.