Reconstruction of tip-surface interactions with multimodal intermodulation atomic force microscopy

We propose a theoretical framework for reconstructing tip-surface interactions using the intermodulation technique when more than one eigenmode is required to describe the cantilever motion. Two particular cases of bimodal motion are studied numerically: one bending and one torsional mode, and two bending modes. We demonstrate the possibility of accurate reconstruction of a two-dimensional conservative force field for the former case, while dissipative forces are studied for the latter.

Figure 1

Appearance of new frequencies in the spectrum of the tip motion in ImAFM due to the nonlinearities in the tip-surface interaction. The dashed line shows the characteristic range of the tip-surface force.

Figure 2

Schematic representation of the two-dimensional cantilever and detection principle in AFM.

Figure 3

Simulated spectrum of the engaged cantilever motion with two flexural modes. Two peaks are clearly seen above noise level near the resonant frequencies ${\omega }_1$ and ${\omega }_2$ which consist of the intermodulation products depicted in insets. The spectrum is obtained by integrating Eq. (16) with the tip-surface force [Eq. (22)] and subsequent addition of white noise.

Figure 4

Total transfer function $\chi$ (black solid line) for the cantilever with two flexural modes with transfer functions ${\chi }_1$ and ${\chi }_2$ (green and blue dotted lines, respectively).

Figure 5

(a) Response spectra for powers of a single dynamic variable with two resonances and (b) two separate variables, each with one resonance. Two frequency windows near resonances ${\omega }_1$ and ${\omega }_2$ are highlighted with light green and blue colors, respectively. Red dashed line means that the corresponding maximum response lies outside the frequency bands. Tables on the right indicate how dynamical variable combinations contribute to the corresponding narrow spectral bands: almost no contribution (red), maximum contribution into the first band (light green), second band (light blue), or both bands.

Figure 6

(a) The actual vdW-DMT force with position dependent damping used in the simulation shown for the region of the free tip motion with bimodal drive of fixed total amplitude ($A_1+A_2=25$ nm). White and black dashed lines circumscribe the regions of phase space for single mode drive of the first and second modes, respectively, with amplitude 25 nm (difference between maximum velocities is about an order of magnitude). (b) Reconstructed force shown in the region of the engaged tip motion. Cross sections (a)–(c) are depicted in Figs. 7a, 7b, 7c to highlight agreement between the actual force used in simulation and the reconstructed force.

Figure 7

Cross sections of the reconstructed tip-surface force (black) with comparison to the actual force used in simulation (red) for the cantilever driven at two flexural modes ($z_1$ and $z_2$). (a) ${\tilde{F}}_z(z,\dot{z}=0)$; (b) ${\tilde{F}}_z(z,\dot{z}=0.2{\dot{z}}_{\max })-{\tilde{F}}_z(z,\dot{z}=0)$; and (c) ${\tilde{F}}_z(z=0.75z_{\min },\dot{z})$. The linear fit for nonlinear damping nicely recovers the overall trend and the conservative part shows excellent agreement with the actual force.

Figure 8

Absolute error of the reconstruction of (a) conservative part of the tip-surface force [Fig. 7a] and (b) dissipative part [Fig. 7b] versus number of spectral components taken into account in the first frequency band (blue circles) and in the second (green squares).

Figure 9

Components $F_z$ and $F_y$ of actual two-dimensional conservative force used in the simulation shown for the region of the free tip motion [(a) and (b)] and reconstructed components shown in the region of the engaged tip motion [(c) and (d)]. Cross-sections (a)–(c) are illustrated in Figs. 10a, 10b, 10c to highlight agreement between the actual force used in simulation and the reconstructed force.

Figure 10

Cross sections of reconstructed tip-surface force (black) with comparison to the actual force used in simulation (red) for the cantilever driven at one flexural and one torsional mode ($z$ and $y$). (a) ${\tilde{F}}_z\left(z\right)$; (b) ${\tilde{F}}_y(z,y=0.2{\dot{z}}_{\max })$; and (c) ${\tilde{F}}_y(z=0.75z_{\min },y)$. The reconstructed force is in good agreement with the actual force near the surface. Perfect agreement is reachable if we assume model for $F_z$ independent of $y$.