2.3 MOVIMIENTOS OCULARES E IDENTIFICACIÓN DEL CONTENIDO
2.3.1 LOS MOVIMIENTOS OCULARES Y LA LECTURA
To confirm the validity of the model equations 3.18 and 3.19, the force dependence of
Ztip, contained in the couplinggin equation 3.19 was studied by recording force-distance
curves at different tip positions. As the local normalized resonator amplitude n(⃗x0)
varies for these positions, so do the frequency shift and induced damping for a given force. However, the linear theory predicts that all curves can be scaled to coincide with each other. Thus an entire dataset of frequency shift and damping for multiple positions can be fitted with a single set of parameters, wheren(⃗x0)enters the fit routine as the scaling
factor for the respective curve, and the real and imaginary part of the tip response function
ˆ
ηconstitute the only two global fitting parameters. Figure 3.9 shows the outcome of such a measurement and the fit of our model for a resonator with an unperturbed resonance frequency of f0 = 6.65MHz and damping rate of γ0 =173 s−1. Here the force-distance
curves were measured for three different tip positions in the clamping region, indicated by color. The rescaled data show very good quantitative agreement with the model. The detailed shape of the curves depend on the contact model used and good agreement with the employed DMT-model is found. As described in section 3.2.6, the coupling constant
g that enters the frequency shift and damping is given by equation 3.22. The presented data are taken from the retraction part of the force distance curves up to the point where the snap out of contact appears. Very similar results can be obtained with the approach curves, but with a restricted accessible range of forces.
To enable simultaneous fitting of all curves the following procedure is employed: First, the data are aligned sequentially along the force axis. Subsequently, the fitting function is defined as a piecewise function along this axis by shifting the zero-point of the model to the respective minimal force value of each data curve. This way, datasets for multiple measurement points can be adjusted with a single fit. After the nonlinear regression has been executed, the data can be rescaled by the fittedn(⃗x0), and the force
axes are restored to the original, actual forces.
The system parameters (obtained idependently, i.e. not fitted) used in this fit routine are as follows:
a b -20 -15 -10 -5 0 5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 fre qu en cy sh ift (a .u .) force (nN) damping (a.u.) -20 -15 -10 -5 0 5 0 1 2 3 4 da m pi ng (a .u .) force (nN) -20 -15 -10 -5 0 5 0 1 2 3 4 fre qu en cy sh ift (a .u .) force (nN) -20 -15 -10 -5 0 5 force (nN)
Figure 3.9: Force dependence of frequency shift and damping. Force-distance curves are measured at three points in the clamping region (red squares, green dots, blue triangles) with increasing distance to the resonator. Since a linear response theory is valid, all curves can be scaled to coincide with each other. Thus the model can be fitted to the whole dataset, i.e. all frequency shift and damping curves, with one single set of parameters. For clarity, only the retraction curves are shown. In all panels red squares represent the data for the point of highest interaction (closest to the clamp), green dots for an intermediate and blue
triangles for the point of lowest interaction. (a) Force dependence of the frequency shift of the fundamental
out-of-plane mode with a free resonance frequency of 6.65 MHz. The solid black line is a fit of the model. The inset shows the data and fitted model of the third harmonic mode with a free resonance frequency of
20.0 MHz, which was measured simultaneously. (b) Force dependence of the tip induced damping for the
same modes, including fit as solid line.
Effective mass m 2.07⋅10−15kg
Adhesion force (measured) Fad 20 nN
Young’s modulus silicon nitride ESiN 250 GPa
Young’s modulus silicon ESi 200 GPa
Poisson’s ratio silicon nitride νSiN 0.27
Poisson’s ratio silicon νSi 0.22
Tip radius (measured in SEM) R 80 nm Cantilever force constant kca 0.05 nN/nm
Furthermore, the cantilever spring constant of the silicon contact mode cantilever was calibrated using the procedure described in 2.1.2 and the Young’s moduli and Poisson’s ratios are literature values [Sha10]. Finally, the fit parameters for the fundamental out-of- plane mode and its third harmonic are the following:
Mode Re[ηˆ] Im[ηˆ] n,1 n,2 n,3
Fund.,n=0 -510 (N/m)−1 -15 (N/m)−1 2.1⋅10−4 3.2⋅10−4 3.7⋅10−4
3rd harm.,n=2 -518 (N/m)−1 -19 (N/m)−1 4.0⋅10−4 6.5⋅10−4 6.8⋅10−4
For the three curves of the fundamental resonance presented in Fig. 3.9, the maximal values of the frequency shift are: δf0,1 = 220 Hz, δf0,2 = 405 Hz, and δf0,3 = 799 Hz. In
the case of the third harmonic mode, the shifts are δf2,1 = 725 Hz, δf2,2 = 1484 Hz, and
δf2,3 =2600 Hz. The values for the n,i are of the order of 10−4 as predicted byComsol
simulations (c.f. section 3.4). As expected from eq. (3.20), the values of the local nor- malized resonator amplitudes n,i increase for the curves with a higher frequency shift.
Moreover, the real and imaginary part of the tip response function do not change for the two modes, signifying that the assumption of a quasi-continuum of cantilever states is jus- tified. This in turn means that the presented method of mechanical impedance mismatch imaging is not restricted to certain frequencies coinciding with cantilever resonances. Thus the procedure is in principle applicable to all kinds of nanomechanical systems.