Theory of Multifrequency Atomic Force Microscopy

We develop a theory that explains the origin of the high force sensitivity observed in multifrequency force microscopy experiments. The ability of the microscope to extract complementary information on the surface properties is increased by the simultaneous excitation of several flexural cantilever modes. The force sensitivity in multifrequency operation is about 0.2 pN. The analytical model identifies the virial and the energy dissipated by the tip-surface forces as the parameters responsible for the material contrast. The agreement obtained among the theory, experiments and numerical simulations validates the model.

Figure 1

(a) Cantilever geometries of the first (left) and second flexural modes (right). (b) First mode amplitude curve for monomodal (open dots) and bimodal-frequency excitations (dark dots). (c) Second mode amplitude curve for monomodal (open dots) and bimodal (dark dots) excitations. (d) First mode and (e) second mode phase shifts. $A_{01}=10\mathrm{nm}$, $A_{02}=1\mathrm{nm}$; $H=9.03\times {10}^{-20}\mathrm{J}$. See cantilever description in the main text for the other parameters. The same applies for Figs. 2, 3, 4.

Figure 2

(a) Time dependence of the tip deflection under bimodal-frequency excitation in the absence of tip-surface forces. (b) FFT of the wave form shown in (a). (c) Time dependence of the tip deflection under bimodal-frequency excitation for a tip that interacts with a surface, $A_1=4.4\mathrm{nm}$. (d) FFT of the signal shown in (c) $A_{01}=6\mathrm{nm}$, $A_{02}=1\mathrm{nm}$, $H=9.03\times {10}^{-20}\mathrm{J}$.

Figure 3

Dependence of the second mode phase shift with respect to the amplitude of the first mode as given by numerical simulations, analytical model and experiments. (a) Comparison between the numerical solution and the analytical expression [Eq. (7)]; $H=1\times {10}^{-20}\mathrm{J}$. (b) Comparison between theory and experimental measurements on mica $H=9.03\times {10}^{-20}\mathrm{J}$ (adapted from Ref. 27) and results given by the analytical equation.

Figure 4

Dependence of the virial for two different materials, open dots $H=4.7\times {10}^{-20}\mathrm{J}$, dark dots $H=9.03\times {10}^{-20}\mathrm{J}$. (a) First mode virial with respect to the first mode amplitude. (b) First mode virial with respect to the second mode amplitude. (c) Second mode virial with respect to the first mode amplitude. (d) Second mode virial with respect to the second mode amplitude.