Abstract
An important issue in the emerging field of multifrequency atomic force microscopy (MFAFM) is the accurate and fast demodulation of the cantilevertip deflection signal. As this signal consists of multiple frequency components and noise processes, a lockin amplifier is typically employed for its narrowband response. However, this demodulator suffers inherent bandwidth limitations as highfrequency mixing products must be filtered out and several must be operated in parallel. Many MFAFM methods require amplitude and phase demodulation at multiple frequencies of interest, enabling both zaxis feedback and phase contrast imaging to be achieved. This article proposes a modelbased multifrequency Lyapunov filter implemented on a fieldprogrammable gate array (FPGA) for highspeed MFAFM demodulation. System descriptions and simulations are verified by experimental results demonstrating high tracking bandwidths, strong offmode rejection and minor sensitivity to crosscoupling effects. Additionally, a fivefrequency system operating at 3.5 MHz is implemented for higher harmonic amplitude and phase imaging up to 1 MHz.
Introduction
Atomic force microscopy (AFM) [1] has been integral in the field of nanoscale engineering since its invention in 1986 by Binnig et al. By sensing microcantilever tip–sample interactions [2], atomic scale resolution imaging is achieved, which far exceeds the optical diffraction limit. An image generated by constantforce topography AFM depends entirely on its feedback control loop. The composition of a sample is visualized in threedimensions by plotting the control signal against the lateral scan trajectories of the nanopositioner.
In staticmode AFM (contact mode), the control loop attempts to maintain a constant contact force [3]. Where as in dynamic modes, for example intermittentcontact constantamplitude AFM [4], the control loop acts to maintain a constant cantilever oscillation amplitude. This is achieved by feeding back the demodulated fundamental frequency present in the deflection signal. In intermittentcontact mode AFM [5], the tapping amplitude is chosen such that only gentle tip–sample interactions occur. This is particularly suitable for studying biological samples, allowing for biophysical processes to be studied [68].
Multifrequency AFM (MFAFM) methods allow for the study of tip–sample interactions occurring at multiple frequencies [9]. This extends imaging information beyond the topography to a range of nanomechanical properties including sample stiffness, elasticity and adhesiveness [10]. The acquisition of these observables requires tracking the amplitude and phase of additional frequencies of interest. These include higher harmonics of the fundamental frequency [11], higher flexural eigenmodes [12] and intermodulation products [13]. Higher harmonic methods have demonstrated the ability to image relatively large biological objects, such as cells [14,15], while bimodal AFM has successfully imaged properties of protein complexes [16]. Intermodulation AFM is a novel extension to the bimodal method that focuses on the mixing products of a slightly below and above resonance bimodal drive. It has been shown to achieve increased image contrast [17] and lead to further insights into nanomechanical properties [18]. Regardless of which particular MFAFM method is employed, they each require the demodulation of amplitude and phase to form observables for the characterization of nanomechanical properties.
Due to the large bandwidth requirements of tracking high frequencies in MFAFM, every component of the zaxis feedback loop detailed in Figure 1 needs to be optimized for speed. This includes the lateral and vertical nanopositioner for each axis (x, y and z), cantilever, vertical feedback controller and demodulator. In this article, the demodulator component is improved with respect to its key performance metrics: tracking bandwidth, sensitivity to other frequency components, and implementation complexity. Tracking bandwidth is defined as the point in a frequency response where the output of the demodulator drops by −3 dB with respect to the input modulating signal. Offmode rejection (OMR) is a term that describes the attenuation of unwanted frequencies present in the input signal, which lie outside the modeled carrier of interest.
It has been shown that conventional highspeed demodulation techniques are incompatible with MFAFM, due to the lack of sensitivity to multiple frequency components [19]. These include the peak detector [20], peakhold [21] and RMStoDC [22] conversion methods. A typical MFAFM demodulator employs multiple lockin amplifiers (LIA) in parallel, as each provides an accurate estimation of a particular frequency component. However, lowpass filters are employed to diminish mixing products, which severely limits the demodulator bandwidth [23].
Motivated by improving MFAFM demodulation capabilities, previous work by the authors includes a multifrequency Kalman filter [24]. It was shown to outperform a commercially available LIA in terms of both tracking bandwidth and noise performance. However, a major disadvantage of the Kalman filter is the large computational expense of each additional frequency modeled. This reduces its realizable performance through limitations of the sample rate. An estimator in the form of a Lyapunov filter [25] was demonstrated to perform similarly to the Kalman filter [26]. However, the Lyapunov filter complexity scales significantly better than the Kalman filter when multiple frequencies are modeled [27].
This article extends previous work by providing a thorough performance analysis of the multifrequency Lyapunov filter in terms of tracking bandwidth, offmode rejection and crosscoupling effects. In addition, MFAFM demodulation is demonstrated by performing higher harmonic imaging with amplitude and phase on both a stiff and compliant sample.
Lyapunov Filter
System modeling
A singlefrequency cantilever deflection signal is modeled as a sine wave with carrier frequency f_{c}, timevarying amplitude A(t) and phase (t) of the form
For readability, explicit dependencies on time of the amplitude A(t) and phase (t) are dropped from this point onward. By extension, a deflection signal consisting of multiple frequencies is modeled as
where i = 1, 2, …, n denotes the ith modeled frequency. Through the doubleangle trigonometric identity, this model is linearly parameterizable such that vector pairs within the state vector x = represent quadrature and inphase components of each particular modeled frequency. That is, each individual sine wave is represented by
Based on the parametrization of the signals in Equation 3, the timevarying amplitude and phase of a particular frequency is recovered by
Filter description
The Lyapunov filter [28] is implemented as a linear observer as shown in Figure 2. A key property of the filter is exponential convergence of the estimated states [29], with the tunable loop gain constant γ governing the speed of convergence. The filter is shown to have a negative feedback loop in which integral action regulates the error. By feeding back an estimate of the input signal obtained from the parameterized states in the form of Equation 3, an error signal is generated. Regulation of this error through feedback leads to the much desired suppression of the 2f_{c} mixing components.
The update law for the singlefrequency Lyapunov filter [28] can be extended to a multifrequency form, resulting in
where
and
In this form, represents the estimated output and the amplitude A_{i} and phase are available by applying Equation 4 to each quadrature and inphase pair of . A key requirement to ensure exponential convergence of to x, is to guarantee that c is persistently excited [29]. Convergence is shown for the singlefrequency filter in [28], and can easily be extended for the multifrequency case. Furthermore, exponential convergence of means that and also converge.
Results and Discussion
Hardware
The Lyapunov filter was implemented on a highspeed FPGA to achieve the necessary sample rate for accessing higher harmonics during imaging. A Xilinx Kintex7 KC705 evaluation board (model: XC7K325T) paired with a DCcoupled highspeed 4DSP input/output (I/O) card (model: FMC151) was utilized. The FPGA clock is synchronized with the highspeed I/O card at 250 MHz. The I/O card has a twochannel 14bit analogtodigital converter (ADC) and a twochannel 16bit digitaltoanalog converter (DAC), which sample at 250 MHz and 800 MHz, respectively.
Implementation
Figure 2 shows the block diagram of a singlefrequency Lyapunov filter (SFLYAP). Here, the digital components required for FPGA implementation can be seen: multipliers, adders, registers, sample rate control for feedback and a programmable direct digital synthesizer (DDS). The DDS generates the sine (inphase) and cosine (quadrature) signals required to model carrier frequencies. It may be tuned through control of its frequency word which is calculated by
where f_{out} is the desired output frequency, FW represents the binary word required to program the DDS, n is the length of FW and f_{clk} is the speed of the FPGA board.
A SFLYAP was successfully implemented at a sampling rate of f_{s} = 5 MHz. As stability is of priority, the chosen data representation is floating point in the standard IEEE 754 format. The integration method used is backward Euler, as this ensures stability when γ is large [30]. The output equation (Equation 4) is realized with the Xilinx Coordinate Rotation Digital Computer (CORDIC), set to a 16bit configuration such that the amplitude and phase are formatted for the I/O card. The carrier frequency f_{c}, γ and any necessary output gains for amplifying very small signals during imaging are tunable in realtime using the Xilinx Virtual Input Ouput (VIO) tool.
In Figure 3, the block diagram of the implemented multifrequency Lyapunov filter (MFLYAP) is shown. Here, it can be seen that an MFLYAP involves several SFLYAPs running in parallel. Channel crosscoupling occurs in the combined output feedback as dictated by the output equation (Equation 6). The Lyapunov filters timing constraints for a fivefrequency system result in a maximum sampling rate of f_{s} = 3.5 MHz. This is a large improvement over the multifrequency Kalman filter [24], which was 1.5 MHz for a threefrequency system. The Kalman filter equations [24] can be shown to have a complexity of , while that of the Lyapunov filter is for n modeled frequencies. This stark difference in complexity arises from the computations required for the Kalman gain and covariance matrix update.
Experimental setup
A LIA (Zürich Instruments HF2LI) was used inconjunction with a laboratory function generator (Agilent 33521A waveform generator) to experimentally verify the performance of the implemented Lyapunov filter. These investigations include a frequency response experiment to measure the tracking bandwidth and channel crosscoupling. Additionally, offmode rejection of channels in both highspeed and slow configurations was explored through a carrier sweep.
Tracking bandwidth
The tracking bandwidth of the Lyapunov filter was characterized through frequency responses from both a simulated and experimentally implemented system. For each frequency response, the modulating signal A(t) in Equation 1 was swept from DC to 1.5 MHz while the carrier frequency was held constant. The tracking bandwidth experiment examines the relationship between the −3 dB point and γ for a 1 MHz carrier frequency and γ values ranging between 5 × 10^{4} and 1 × 10^{7}. Figure 4 shows the results of (a) simulated and (b) experimental tracking frequency responses, where it can be seen that the two systems match closely. The similarity was achieved by maintaining a consistent sample rate and integration method for both simulation and experimental implementation. In Figure 4c, the simulated and experimental −3 dB points are shown as a function of the tunable loop constant γ. For both systems, the tracking bandwidth approaches the carrier frequency f_{c}.
Figure 5 demonstrates several cases in which the Lyapunov filter is achieving a high tracking bandwidth of f_{c}, the equivalent of single cycle tracking. This was achieved for the five carrier frequencies 100 kHz, 200 kHz, 500 kHz, 700 kHz and 1 MHz with γ values of 1.2 × 10^{6}, 2.2 × 10^{6}, 3.7 × 10^{6}, 4.4 × 10^{6} and 5.1 × 10^{6}, respectively.
Crosscoupling
The effect of channel crosscoupling on the tracking bandwidth was examined for both a simulated and experimentally implemented system. For simplicity, crosscoupling was demonstrated with a twofrequency MFLYAP wherein the modeled carrier frequencies are 100 kHz and 500 kHz for channels 1 and 2, respectively. Each channel is considered for two fixed tracking bandwidth settings, low (1 kHz) and high (50% of f_{c}), while the other channel is increased in speed.
Figure 6a–c shows that the tracking bandwidth of a channel will increase from its original setting as the other channel is tuned faster. Conversely, Figure 6d shows channel 2 slowing down as channel 1 is increased in speed. This is explained by the fact that channel 2 is set to a tracking bandwidth of 250 kHz (50% f_{c}), which is higher than the maximum obtainable speed of channel 1. Throughout this investigation, the simulation and experimental results agree. The results show that crosscoupling effects are more pronounced in lowspeed channels. They are, however, negligible if tracking bandwidths of channels remain below 10% of f_{c}.
Offmode rejection
The offmode rejection of the multifrequency Lyapunov filter was analyzed by performing a singletone sine sweep on the input signal and recording the demodulated amplitude magnitude of each channel. For each frequency response, the carrier frequency f_{c} in Equation 1 was swept from DC to 1.25 MHz with a constant amplitude A. This experiment used a fivefrequency MFLYAP with channels set to carriers of 100 kHz, 200 kHz, 500 kHz, 700 kHz and 1 MHz for both a simulated and experimentally implemented system.
Figure 7a–e shows offmode rejection for a fast (10% of f_{c}) tracking bandwidth setting. For each channel, a full recovery (0 dB) of the signal can be seen to occur at its modeled carrier frequency, as expected. There is strong offmode rejection occurring at the other modeled carrier frequencies, due to output feedback crosscoupling sharing state information between channels. Figure 7f–j demonstrates offmode rejection for a slow (1 kHz) tracking bandwidth setting. Here, the narrowband response is a direct result of the reduced γ_{i} values of each channel. This causes a less distinct, but still visible, modeled offmode rejection at the other carrier frequencies. It can be seen that the slower system achieves greater offmode rejection outside of the modeled frequencies than the fast system. Again, a similar performance between the simulated and experimental results can be observed. The less distinct offmode rejection in the experimental results compared to the simulations is due to a finite DC offset from the DAC. This precludes the direct measurement of signals smaller than this value.
AFM imaging
Imaging setup
The Lyapunov filter as a multifrequency AFM demodulator was validated through a series of imaging experiments where it is compared sidebyside to a lockin amplifier. To ensure a fair comparison, the demodulators were tuned to the same tracking bandwidth in both experiments. This is required as the noise performance has been shown to be a function of the tracking bandwidth [19]. The lockin amplifier is the stateoftheart multifrequency method due to its strong offmode rejection, however it can not achieve the same speed as the Lyapunov filter due to postmixing filtering [19]. As the highspeed superiority of the Lyapunov filter is well established, it is compared to the lockin amplifier in a lowspeed environment.
Using an NTMDT NTEGRA AFM, amplitude and phase higher harmonic imaging was performed with a NTMDT NSG01 and Bruker DMASP cantilever. These cantilevers were found to have fundamental resonance frequencies of 168.8 kHz and 46.1 kHz, respectively. The samples used are a zcalibration grating (NTMDT TGZ3) with periodic height features of approx. 500 nm and a blend of polystyrene (PS) and polyolefin elastomer (LDPE) available from Bruker (PSLDPE12M). Due to the different elastic moduli of the PS and LPDE regions, the sample is widely used for qualitative imaging the material contrast.
Imaging a TGZ3 calibration grating
Higherharmonic amplitude images with the first, second, third, sixth and seventh harmonics were obtained by the MFLYAP and multiple parallel LIAs. The frequency response of the NSG01 cantilever in free air and the power spectrum of its deflection signal during contact are shown in Figure 8. Here, the fundamental and second resonance frequencies can be seen in the cantilever frequency response. The deflection signal spectrum shows additional higher harmonics and minor intermodulation products are present. These are due to nonlinear atomic forces exciting the cantilever during contact.
Amplitude imaging results are shown in Figure 9. As the sixth and seventh harmonics are closely spaced to the second resonance frequency of the cantilever, they provide an increased signaltonoise ratio. The MFLYAP can be seen to perform comparably to the LIA when tuned to similar measurement bandwidth settings. When imaging with higher harmonics, the offmode rejection of each channel was tuned to suppress the large fundamental frequency.
Imaging a PS/LPDE calibration grating
Higherharmonic phase images were obtained for the first five harmonics of a Bruker DMASP cantilever. The frequency response of the cantilever in free air and the power spectrum of its deflection signal during contact are shown in Figure 10. Here, the fundamental resonance frequency and higher eigenmodes can be seen in the cantilever frequency response. As before, the deflection signal contains additional higher harmonics and intermodulation products due to the nonlinear atomic excitation. Note that the DMASP cantilever uses integrated piezoelectric actuation [31], which results in a clean frequency response when compared to the base excited NSG01 as seen in Figure 8.
The higherharmonic phase imaging results are shown in Figure 11. For both demodulators, we see a strong material contrast between the PS and LPDE regions. This was expected from the rich frequency content present in the deflection signal, as visible in Figure 10. We note that the images show particularly strong contrast for the second harmonic, which is due to its proximity to the second mode of the cantilever. This fact is also visible in the increased noise floor around that frequency in the deflection signal. Similarly to the amplitude imaging, the large fundamental frequency contribution required tuning higher harmonics for increased offmode rejection. For this reason, we tuned the first harmonic demodulators to 1 kHz bandwidth (LIA LPF 1 kHz, LYAP γ = 20 × 10^{3} and the higher harmonics to 200 Hz (LIA LPF 200 Hz, LYAP γ = 2 × 10^{3}).
Conclusion
This article describes a multifrequency Lyapunov filter for highspeed demodulation in MFAFM. The performance and flexibility of the proposed Lyapunov filter is demonstrated through simulations and experiments. The filter may reach tracking bandwidths up to the modeled carrier frequency, the equivalent of singlecycle tracking. Additionally, the offmode rejection of the system was found to be controlled by its bandwidth as dictated by the tunable loop constant γ. The relationship between γ and the bandwidth was shown to be linear, up to the modeled carrier frequency. Channel crosscoupling, which occurs due to output feedback, was found to cause distinct rejection of other modeled frequencies during the offmode rejection experiments. An investigation into this crosscoupling revealed it has negligible effect on the tracking bandwidth of the system.
The multifrequency Lyapunov filter as a flexible, highspeed demodulator was verified through higher harmonic MFAFM imaging for both amplitude and phase. This demonstrates the filters ability to be used as a demodulator in various MFAFM techniques involving higher harmonic, higher eigenmode or intermodulation frequency components. In the presented AFM images, the proposed filter performed comparably to a stateoftheart lockin amplifier setup. In comparison to the Kalman filter, the Lyapunov filter is similar in terms of speed, offmode rejection and operation. However, it was found to be significantly easier to implement, which is a priority when considering an extension to multiple frequencies.
References

Binnig, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930–933. doi:10.1103/PhysRevLett.56.930
Return to citation in text: [1] 
Rabe, U.; Janser, K.; Arnold, W. Rev. Sci. Instrum. 1996, 67, 3281–3293. doi:10.1063/1.1147409
Return to citation in text: [1] 
Abramovitch, D. Y.; Andersson, S. B.; Pao, L. Y.; Schitter, G. A tutorial on the mechanisms, dynamics and control of atomic force microscopes. 2007 American Control Conference; 2007; pp 3488–3502.
Return to citation in text: [1] 
García, R. Amplitude modulation atomic force microscopy; John Wiley & Sons: New York, NY, U.S.A., 2011.
Return to citation in text: [1] 
Zhong, Q.; Inniss, D.; Kjoller, K.; Elings, V. B. Surf. Sci. 1993, 290, 688–692. doi:10.1016/00396028(93)905825
Return to citation in text: [1] 
Möller, C.; Allen, M.; Elings, V.; Engel, A.; Müller, D. J. Biophys. J. 1999, 77, 1150–1158. doi:10.1016/S00063495(99)769663
Return to citation in text: [1] 
Kodera, N.; Yamamoto, D.; Ishikawa, R.; Ando, T. Nature 2010, 468, 72. doi:10.1038/nature09450
Return to citation in text: [1] 
Chiaruttini, N.; RedondoMorata, L.; Colom, A.; Humbert, F.; Lenz, M.; Scheuring, S.; Roux, A. Cell 2015, 163, 866–879. doi:10.1016/j.cell.2015.10.017
Return to citation in text: [1] 
García, R.; Herruzo, E. T. Nat. Nanotechnol. 2012, 7, 217–226. doi:10.1038/nnano.2012.38
Return to citation in text: [1] 
Garcia, R.; Proksch, R. Eur. Polym. J. 2013, 49, 1897. doi:10.1016/j.eurpolymj.2013.03.037
Return to citation in text: [1] 
Stark, R. W.; Heckl, W. M. Rev. Sci. Instrum. 2003, 74, 5111–5114. doi:10.1063/1.1626008
Return to citation in text: [1] 
Martínez, N. F.; Lozano, J. R.; Herruzo, E. T.; Garcia, F.; Richter, C.; Sulzbach, T.; Garcia, R. Nanotechnology 2008, 19, 384011. doi:10.1088/09574484/19/38/384011
Return to citation in text: [1] 
Platz, D.; Tholén, E. A.; Pesen, D.; Haviland, D. B. Appl. Phys. Lett. 2008, 92, 153106. doi:10.1063/1.2909569
Return to citation in text: [1] 
Raman, A.; Trigueros, S.; Cartagena, A.; Stevenson, A. P. Z.; Susilo, M.; Nauman, E.; Contera, S. A. Nat. Nanotechnol. 2011, 6, 809. doi:10.1038/nnano.2011.186
Return to citation in text: [1] 
CartagenaRivera, A. X.; Wang, W.H.; Geahlen, R. L.; Raman, A. Sci. Rep. 2015, 5, 11692. doi:10.1038/srep11692
Return to citation in text: [1] 
Herruzo, E. T.; Perrino, A. P.; Garcia, R. Nat. Commun. 2014, 5, 3126. doi:10.1038/ncomms4126
Return to citation in text: [1] 
Forchheimer, D.; Forchheimer, R.; Haviland, D. B. Nat. Commun. 2015, 6, 6270. doi:10.1038/ncomms7270
Return to citation in text: [1] 
Thorén, P.A.; de Wijn, A. S.; Borgani, R.; Forchheimer, D.; Haviland, D. B. Nat. Commun. 2016, 7, 13836. doi:10.1038/ncomms13836
Return to citation in text: [1] 
Ruppert, M. G.; Harcombe, D. M.; Ragazzon, M. R. P.; Moheimani, S. O. R.; Fleming, A. J. Beilstein J. Nanotechnol. 2017, 8, 1407–1426. doi:10.3762/bjnano.8.142
Return to citation in text: [1] [2] [3] 
Ando, T. Nanotechnology 2012, 23, 062001. doi:10.1088/09574484/23/6/062001
Return to citation in text: [1] 
Ando, T.; Kodera, N.; Takai, E.; Maruyama, D.; Saito, K.; Toda, A. Proc. Natl. Acad. Sci. U. S. A. 2001, 98, 12468. doi:10.1073/pnas.211400898
Return to citation in text: [1] 
Kitchin, C.; Counts, L. RMS to DC Conversion Application Guide; Analog Devices: Cambridge, MA. U.S.A., 1986.
Return to citation in text: [1] 
Ruppert, M. G.; Karvinen, K. S.; Wiggins, S. L.; Moheimani, S. O. R. IEEE Trans. Control Syst. Technol. 2016, 24, 276–284. doi:10.1109/TCST.2015.2435654
Return to citation in text: [1] 
Ruppert, M. G.; Harcombe, D. M.; Moheimani, S. O. R. IEEE/ASME Trans. Mechatronics 2016, 21, 2705–2715. doi:10.1109/TMECH.2016.2574640
Return to citation in text: [1] [2] [3] 
Ragazzon, M. R. P.; Gravdahl, J. T.; Fleming, A. J. On Amplitude Estimation for HighSpeed Atomic Force Microscopy. 2016 American Control Conference (ACC); 2016; pp 2635–2642.
Return to citation in text: [1] 
Ruppert, M. G.; Harcombe, D. M.; Ragazzon, M. R. P.; Moheimani, S. O. R.; Fleming, A. J. Frequency domain analysis of robust demodulators for highspeed atomic force microscopy. 2017 American Control Conference (ACC); 2017; pp 1562–1567.
Return to citation in text: [1] 
Harcombe, D. M.; Ruppert, M. G.; Ragazzon, M. R. P.; Fleming, A. J. Higherharmonic AFM imaging with a highbandwidth multifrequency Lyapunov filter. 2017 IEEE International Conference on Advanced Intelligent Mechatronics (AIM); 2017; pp 725–730.
Return to citation in text: [1] 
Ragazzon, M. R. P.; Ruppert, M. G.; Harcombe, D. M.; Fleming, A. J.; Gravdahl, J. T. IEEE Trans. Control Syst. Technol. 2017, PP, 1–8. doi:10.1109/TCST.2017.2692721
Return to citation in text: [1] [2] [3] 
Ioannou, P. A.; Sun, J. Robust Adaptive Controls; Dover Publications, Inc.: Mineola, NY, U.S.A., 2012.
Return to citation in text: [1] [2] 
Cellier, F. E.; Ernesto, K. Continuous System Simulation; Springer: Berlin, Germany, 2006.
Return to citation in text: [1] 
Ruppert, M. G.; Moheimani, S. O. R. Beilstein J. Nanotechnol. 2016, 7, 284. doi:10.3762/bjnano.7.26
Return to citation in text: [1]
29.  Ioannou, P. A.; Sun, J. Robust Adaptive Controls; Dover Publications, Inc.: Mineola, NY, U.S.A., 2012. 
28.  Ragazzon, M. R. P.; Ruppert, M. G.; Harcombe, D. M.; Fleming, A. J.; Gravdahl, J. T. IEEE Trans. Control Syst. Technol. 2017, PP, 1–8. doi:10.1109/TCST.2017.2692721 
30.  Cellier, F. E.; Ernesto, K. Continuous System Simulation; Springer: Berlin, Germany, 2006. 
1.  Binnig, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930–933. doi:10.1103/PhysRevLett.56.930 
5.  Zhong, Q.; Inniss, D.; Kjoller, K.; Elings, V. B. Surf. Sci. 1993, 290, 688–692. doi:10.1016/00396028(93)905825 
18.  Thorén, P.A.; de Wijn, A. S.; Borgani, R.; Forchheimer, D.; Haviland, D. B. Nat. Commun. 2016, 7, 13836. doi:10.1038/ncomms13836 
4.  García, R. Amplitude modulation atomic force microscopy; John Wiley & Sons: New York, NY, U.S.A., 2011. 
19.  Ruppert, M. G.; Harcombe, D. M.; Ragazzon, M. R. P.; Moheimani, S. O. R.; Fleming, A. J. Beilstein J. Nanotechnol. 2017, 8, 1407–1426. doi:10.3762/bjnano.8.142 
3.  Abramovitch, D. Y.; Andersson, S. B.; Pao, L. Y.; Schitter, G. A tutorial on the mechanisms, dynamics and control of atomic force microscopes. 2007 American Control Conference; 2007; pp 3488–3502. 
16.  Herruzo, E. T.; Perrino, A. P.; Garcia, R. Nat. Commun. 2014, 5, 3126. doi:10.1038/ncomms4126 
31.  Ruppert, M. G.; Moheimani, S. O. R. Beilstein J. Nanotechnol. 2016, 7, 284. doi:10.3762/bjnano.7.26 
2.  Rabe, U.; Janser, K.; Arnold, W. Rev. Sci. Instrum. 1996, 67, 3281–3293. doi:10.1063/1.1147409 
17.  Forchheimer, D.; Forchheimer, R.; Haviland, D. B. Nat. Commun. 2015, 6, 6270. doi:10.1038/ncomms7270 
11.  Stark, R. W.; Heckl, W. M. Rev. Sci. Instrum. 2003, 74, 5111–5114. doi:10.1063/1.1626008 
13.  Platz, D.; Tholén, E. A.; Pesen, D.; Haviland, D. B. Appl. Phys. Lett. 2008, 92, 153106. doi:10.1063/1.2909569 
19.  Ruppert, M. G.; Harcombe, D. M.; Ragazzon, M. R. P.; Moheimani, S. O. R.; Fleming, A. J. Beilstein J. Nanotechnol. 2017, 8, 1407–1426. doi:10.3762/bjnano.8.142 
10.  Garcia, R.; Proksch, R. Eur. Polym. J. 2013, 49, 1897. doi:10.1016/j.eurpolymj.2013.03.037 
14.  Raman, A.; Trigueros, S.; Cartagena, A.; Stevenson, A. P. Z.; Susilo, M.; Nauman, E.; Contera, S. A. Nat. Nanotechnol. 2011, 6, 809. doi:10.1038/nnano.2011.186 
15.  CartagenaRivera, A. X.; Wang, W.H.; Geahlen, R. L.; Raman, A. Sci. Rep. 2015, 5, 11692. doi:10.1038/srep11692 
19.  Ruppert, M. G.; Harcombe, D. M.; Ragazzon, M. R. P.; Moheimani, S. O. R.; Fleming, A. J. Beilstein J. Nanotechnol. 2017, 8, 1407–1426. doi:10.3762/bjnano.8.142 
9.  García, R.; Herruzo, E. T. Nat. Nanotechnol. 2012, 7, 217–226. doi:10.1038/nnano.2012.38 
24.  Ruppert, M. G.; Harcombe, D. M.; Moheimani, S. O. R. IEEE/ASME Trans. Mechatronics 2016, 21, 2705–2715. doi:10.1109/TMECH.2016.2574640 
6.  Möller, C.; Allen, M.; Elings, V.; Engel, A.; Müller, D. J. Biophys. J. 1999, 77, 1150–1158. doi:10.1016/S00063495(99)769663 
7.  Kodera, N.; Yamamoto, D.; Ishikawa, R.; Ando, T. Nature 2010, 468, 72. doi:10.1038/nature09450 
8.  Chiaruttini, N.; RedondoMorata, L.; Colom, A.; Humbert, F.; Lenz, M.; Scheuring, S.; Roux, A. Cell 2015, 163, 866–879. doi:10.1016/j.cell.2015.10.017 
12.  Martínez, N. F.; Lozano, J. R.; Herruzo, E. T.; Garcia, F.; Richter, C.; Sulzbach, T.; Garcia, R. Nanotechnology 2008, 19, 384011. doi:10.1088/09574484/19/38/384011 
24.  Ruppert, M. G.; Harcombe, D. M.; Moheimani, S. O. R. IEEE/ASME Trans. Mechatronics 2016, 21, 2705–2715. doi:10.1109/TMECH.2016.2574640 
22.  Kitchin, C.; Counts, L. RMS to DC Conversion Application Guide; Analog Devices: Cambridge, MA. U.S.A., 1986. 
21.  Ando, T.; Kodera, N.; Takai, E.; Maruyama, D.; Saito, K.; Toda, A. Proc. Natl. Acad. Sci. U. S. A. 2001, 98, 12468. doi:10.1073/pnas.211400898 
29.  Ioannou, P. A.; Sun, J. Robust Adaptive Controls; Dover Publications, Inc.: Mineola, NY, U.S.A., 2012. 
28.  Ragazzon, M. R. P.; Ruppert, M. G.; Harcombe, D. M.; Fleming, A. J.; Gravdahl, J. T. IEEE Trans. Control Syst. Technol. 2017, PP, 1–8. doi:10.1109/TCST.2017.2692721 
27.  Harcombe, D. M.; Ruppert, M. G.; Ragazzon, M. R. P.; Fleming, A. J. Higherharmonic AFM imaging with a highbandwidth multifrequency Lyapunov filter. 2017 IEEE International Conference on Advanced Intelligent Mechatronics (AIM); 2017; pp 725–730. 
28.  Ragazzon, M. R. P.; Ruppert, M. G.; Harcombe, D. M.; Fleming, A. J.; Gravdahl, J. T. IEEE Trans. Control Syst. Technol. 2017, PP, 1–8. doi:10.1109/TCST.2017.2692721 
25.  Ragazzon, M. R. P.; Gravdahl, J. T.; Fleming, A. J. On Amplitude Estimation for HighSpeed Atomic Force Microscopy. 2016 American Control Conference (ACC); 2016; pp 2635–2642. 
26.  Ruppert, M. G.; Harcombe, D. M.; Ragazzon, M. R. P.; Moheimani, S. O. R.; Fleming, A. J. Frequency domain analysis of robust demodulators for highspeed atomic force microscopy. 2017 American Control Conference (ACC); 2017; pp 1562–1567. 
23.  Ruppert, M. G.; Karvinen, K. S.; Wiggins, S. L.; Moheimani, S. O. R. IEEE Trans. Control Syst. Technol. 2016, 24, 276–284. doi:10.1109/TCST.2015.2435654 
24.  Ruppert, M. G.; Harcombe, D. M.; Moheimani, S. O. R. IEEE/ASME Trans. Mechatronics 2016, 21, 2705–2715. doi:10.1109/TMECH.2016.2574640 
© 2018 Harcombe et al.; licensee BeilsteinInstitut.
This is an Open Access article under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The license is subject to the Beilstein Journal of Nanotechnology terms and conditions: (https://www.beilsteinjournals.org/bjnano)