Self-Driving Capacitive Cantilevers for High-Frequency Atomic Force Microscopy

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A major trend in atomic force microscopy (AFM) and other cantilever-based microelectromechanical systems (MEMS) is to use cantilevers with higher resonance frequencies f 0 to allow for faster measurements and greater sensitivity.High-frequency measurements have been useful for ultrafast topographic scanning, 1 monitoring real-time dynamics of biological processes, 2,3 mechanical measurements through contact resonance, 4 and biodetection. 5,6nventional dynamic AFM uses cantilevers with f 0 ~ 200 kHz driven by a piezoelectric element on the holder. 7The utility of conventional piezoelectric drive is diminished at high frequencies f 0 ≥ 1 MHz because unwanted mechanical modes in the cantilever holder play a greater role. 8chniques have been developed to drive the cantilever directly in order to improve its coupling with the actuator.Laser illumination has been used to drive cantilevers through the photothermal effect, 9 and electrodes on the sample have been used to electrostatically actuate cantilevers. 10tuators have also been fabricated onto cantilevers to drive them directly, such as piezoelectrics, 11 Schottky diodes 12 and transistors. 13Capacitors with air 14,15 and dielectric 16 gaps integrated with AFM cantilevers have previously been used to drive and sense deflections of cantilevers with low (≤ 60 kHz for air gaps and ~ 2 kHz for dielectric gaps) resonance frequencies.
In this paper, we demonstrate that capacitive driving is a simple and robust method for actuating cantilevers at high frequencies.We begin by describing how a driving force is created by a thin film capacitor on the surface of a cantilever.To compare their performances at different frequencies, capacitors are fabricated on conventional and high-frequency AFM cantilevers.Our measurements show that capacitively-driven cantilevers display improved drive fidelity over mechanical driving.High-frequency cantilevers show a dramatic improvement with capacitive driving because mechanical driving by a piezoelectric element is complicated by many spurious modes.We find that the magnitude of the vibration amplitude is in good agreement with a simple model of capacitive drive.
A thin-film capacitor fabricated on the surface of an AFM cantilever can be used to mechanically actuate the cantilever by applying a voltage V across the capacitor. 16A rectangular cantilever of length L, width W, and thickness T is shown in Fig. 1(a).A capacitor with capacitance C is formed by the conducting cantilever and a metal layer deposited on its entire bottom surface separated by an insulator of thickness D and relative permittivity ε.When the cantilever is deflected a distance z at its tip, the bending creates mechanical strain in the capacitor 17 which directly changes its dimensions and changes ε through electrostriction, 18 as shown in Fig. 1(b).Conversely, when a voltage V is applied across C, the dependence of C on z leads to an effective force F on the tip, which to first order in strain is: where ε 0 is the permittivity of free space, and ( ) is the reduction in ε due to electrostriction, which depends on the Poisson's ratio ν, the shear electrostriction constant a 1 , and the bulk electrostriction constant a 2 . 18We calculate Δε = 3.3 for the material used below -low stress silicon nitride grown with plasma-enhanced chemical vapor deposition (PECVD) for which ε = 5.5 19 and ν = 0.21. 20The constant β depends on the cantilever shape, β = 3/8 for a rectangular cantilever and β = 1/4 for a triangular cantilever with base width W. The cantilever spring constant k also depends on the geometry of the cantilever. 21ectrostatic actuation by a capacitor is useful for driving cantilevers near their mechanical resonance.The vibration amplitude A of a cantilever in response to the capacitive force F from Eq. ( 1), when driven by an ac voltage near the cantilever's resonant frequency f 0 is: where Q is the quality factor of the resonance.If the cantilever is capacitively driven with a voltage V = V DC + V AC sin(2πft), the quadratic dependence of F on V from Eq. ( 1) indicates that there will be force contributions at dc, f, and 2f.A robust driving force can be obtained by driving at half the resonant frequency with f = f 0 /2.
Capacitive cantilevers are made using a fabrication protocol similar to our procedure for making coaxial AFM probes. 22,23Conventional tapping mode (Arrow-NCPt, NanoAndMore AG) and high-frequency (Arrow-UHF, NanoAndMore AG) conducting AFM cantilevers are chemically cleaned and mounted on a metal carrier.The cantilevers are coated with 25 nm of Ti using electron-beam evaporation (EE) for adhesion.PECVD is used to deposit a ~ 50 nm thick insulating layer of low-stress SiN x on the cantilever.An additional ~ 2 µm insulating layer is deposited on the cantilever holder to make it mechanically and electrically robust.Finally, 30 nm of Ti and 50 nm of Au are deposited on the cantilevers by EE.AFM experiments are performed in a commercial system (MFP-3D, Asylum Research) with a custom tip holder that makes separate electrical contact to the top and bottom of the probe chip.A glass substrate is used to calibrate cantilever deflection, but all measurements are taken with the cantilever ~1 cm from the substrate.
Figure 2 shows a comparison of AFM cantilever response to capacitive and mechanical driving for a conventional tapping mode cantilever (f 0 = 243 kHz) and a high-frequency cantilever (f 0 = 1.5 MHz).The left column in Fig. 2 shows data taken with the tapping mode cantilever shown in Fig. 2 9) from ref. [24]) added in quadrature to white noise, 25 with results summarized in Table I.Figures 2(c) and 2(g) show the vibration amplitude A observed when mechanically driving the cantilevers with the piezoelectric element.
The resonance peak for the tapping mode cantilever shows only a slight deviation from the fit, while A of the high-frequency cantilever is complicated by many spurious features that make it impossible to locate the true resonance peak.This deviation is due to the activation of unwanted modes of vibration between the piezoelectric drive source and the cantilever, which produces spurious modes through intermodulation distortion.
Figures 2(d) and 2(h) show high-quality capacitive driving of the tapping mode and highfrequency cantilevers, respectively.In both cases, the response is clean and fits Eq. ( 2) very well.
The high level of agreement between A and the fit indicates that the vibrational mode is being driven with high fidelity.Capacitive driving is able to effectively avoid the spurious modes encountered with piezoelectric drive because it is directly attached to the cantilever.For the tapping mode cantilever, the fit indicates that f 0 = 243.2kHz and Q = 533 ± 1 with drive voltages V , where α exp is the experimentally determined capacitive driving coefficient.Values of α exp for two cantilevers are compared with the theoretically predicted capacitive driving coefficient α theory in Table 1; α theory is found by combining Eq. ( 1) with Eq. ( 2) using f = f 0 /2 to find, Values of α theory are calculated for the rectangular tapping mode and triangular high-frequency cantilevers using ε = 5.5 and Δε = 3.3 and the parameters given in Table I.Both cantilevers are in good agreement with the theory.The high-frequency cantilever is less ideal, possibly due to the thickness of the cantilever being similar to the thickness of the capacitor, resulting in deviations from behavior as an ideal beam.
Capacitive driving can in principle be used at high frequencies (above 1 GHz) and for driving higher vibrational or torsional modes.As sensitivity and bandwidth increase with cantilever resonance frequency, 6,26 it is important to consider the maximum frequency at which capacitive driving technique remains practical.While electrical operation at substantially higher frequencies requires impedance matched electrodes for optimal performance, the high frequency limit can be estimated by the RC roll-off frequency f RC = (2πRC) -1 of the drive capacitor, where R ~ 1 Ω is the resistance of the capacitor plate.Using the dimensions of the high-frequency cantilever shown in Fig. 2(e), we find C ~ 1.5 pF, which gives f RC >> 1 GHz, so operation at GHz frequencies is attainable.Capacitive driving can also be used to drive higher order vibrational or torsional modes of the cantilever.The efficiency with which the driving force couples to a given mode is related to the positional overlap between the driving force and the deflection of the mode. 27In our case, the capacitor covers the entire cantilever, thus the driving force is uniformly distributed.In the first resonance mode, the entire cantilever is moving in one direction, so a uniform force drives the first mode efficiently.For higher modes or torsional modes, where the sign of the deflection varies along the cantilever, it is ideal to pattern the capacitor only in regions of deflection of a single sign, thus maximizing dot product between the driving force and displacement profile of the desired mode.
(a), while the right column shows data taken with the high-frequency cantilever shown in Fig. 2(e).The principle mechanical resonance is found by observing the thermal noise spectra n A , shown in Figs.2(b) and 2(f).The resonances appear cleanly in n A and are fit to the thermal vibration spectrum of the peak (Eq.( and V AC = 1 V.For the high-frequency cantilever, the fit gives f 0 = 1.526MHz and Q = 187 ± 3 with V DC = 5 V and V AC = 5 V.Both cantilevers show excellent agreement between the resonances measured in the capacitive drive response and n A .The vibration amplitude A found for capacitive drive is in good agreement with our model.Figures3(a) and 3(b) show A vs. V AC for a tapping mode and a high-frequency cantilever, respectively, both driven at f = f 0 /2.Both curves are well-fit by A = α exp 2 AC

FIG. 1 .FIG. 2 .FIG. 3 .
FIG. 1.(a) Schematic of a cantilever with a thin-film capacitor on its bottom face.(b) The application of a voltage across the capacitor causes a deflection that changes the dimensions and dielectric properties of the capacitor.

TABLE I .
Properties of the conventional (1) and high-frequency (2) cantilevers used in this study.f 0 and k are found through fitting of n A from Figs. 2(b) and 2(f), typical cantilever geometries are measured in a SEM, and D is estimated from ellipsometry of a test wafer.