We investigate a mathematical model of tapping mode atomic force microscopy (AFM), which includes surface interaction via both van der Waals and meniscus forces. We also take particular care to include a realistic representation of the integral control inherent to the real microscope. Varying driving amplitude, amplitude setpoint and driving frequency independently shows that the model can capture the qualitative features observed in AFM experiments on a flat sample and a calibration grid. In particular, the model predicts the onset of an instability, even on a flat sample, in which a large-amplitude beating-type motion is observed. Experimental results confirm this onset and also confirm the qualitative features of the dynamics suggested by the simulations. The simulations also suggest the mechanism behind the beating effect; that the control loop over-compensates for sufficiently high gains. The mathematical model is also used to offer recommendations on the effective use of AFMs in order to avoid unwanted artefacts.
Atomic force microscopes (AFMs) are remarkable mechanical devices that are capable of resolving features on a surface to nanometre precision (e.g. McMaster et al. 1994; Alsteens et al. 2009). While conventional microscopes use lenses to magnify the surface optically, an AFM works by ‘feeling’ the topography of the surface using a sharp tip at the end of a small cantilever moving over the area of interest (Hansma et al. 1994). As the tip moves over the surface, the position of the tip is measured, producing a three-dimensional image with high resolution in the topography height. As the cantilever has a low stiffness, it is possible to image delicate biological samples without damaging them. Another advantage of AFMs, over electron-based imaging methods, is that samples submerged in fluid may be imaged, as the sample does not have to be made from or covered in a conducting material. Along with imaging, AFMs can also be used to measure small forces, for example those required to unravel a strand of DNA (Rief et al. 1999).
There are many modes of operation of AFMs; see Alessandrini & Facci (2005) for a comprehensive review. The simplest is known as contact mode, where the tip is in constant contact with the surface. Although still used for many imaging tasks, such a mode of operation is not advisable if the sample is considered too delicate (Zhong et al. 1993; Höper et al. 1995). Instead, a form of intermittent contact called tapping mode (Zhong et al. 1993) is often used. When an AFM is used in the tapping mode, the cantilever is driven using a piezoelectric device at an oscillation frequency close to its resonant frequency. Away from the surface, the tip oscillates freely. When the tip is brought close to the individual atoms on the sample surface, it is affected by both long-range attractive and short-range repulsive van der Waals forces (Ciraci et al. 1992; Dankowicz et al. 2007). Under standard humidity conditions, a layer of water, typically nanometres in thickness, forms over the surface and the tip (Beaglehole & Christenson 1992); as the tip approaches this layer, a meniscus is formed (Zitzler et al. 2002; Hashemi et al. 2008). Both these effects change the amplitude of oscillation of the cantilever, which in turn is carefully measured by the AFM apparatus and is used to build up an image of the surface.
(a) Imaging with an atomic force microscope in tapping mode
Figure 1 depicts the apparatus used to create tapping mode AFM images, such as those presented in figure 2. Images are put together using multiple sweeps along the fast scan direction (the y axis in figure 1); the tip is incrementally stepped in the slow scan direction (the x axis in figure 1) at the completion of each sweep. The fast scan motion has a typical frequency of 2 Hz and the slow scan can take over a minute to complete one period. A tube made from piezoelectric material moves the tip in both the fast and slow scan directions as well as the surface displacement (z) direction, in order to try to keep the amplitude of oscillation constant during the scan. The piezoelectric tube changes its dimensions as the voltage placed across it varies. The displacement of the tip in the z direction is measured by reflecting a focused laser off the end of the cantilever onto a position-sensitive photodiode. As the tip taps across the surface, a feedback control loop attempts to keep the amplitude of oscillation of the tip constant by altering the distance between the sample and the tip. Feedback loops and error analysis algorithms are used to deduce the required change in extension of the piezoelectric tube to produce the required tip-sample distance in order to keep the recorded amplitude constant. Computer software records the movement of the tip in the z-direction owing to the control and plots it against the position of the tip in the x and y directions to create a final image like the one in figure 2.
Tapping mode AFMs are now well-established experimental measurement devices with numerous celebrated successes (e.g. Round & Miles 2004; Yang et al. 2007). Yet, there remain relatively few modelling studies, especially those that aim to correctly capture features observed in commercial AFM devices with feedback control; although see Melcher et al. (2008) for a description of general purpose software Veda for simulations of the dynamics of an AFM cantilever. Cantilevers and tips are expensive to produce, making simulation an attractive alternative to trial and error experiments. Also, by accurately capturing the behaviour of an AFM in simulations, multiple parameters may be altered and the results recorded automatically, saving many hours of equipment time and diminishing the risk of damaging the piezoelectric tube by crashing the tip into the surface.
The purpose of this paper is not to produce a simulation tool per se, but to examine a low-order mathematical model of a tapping mode AFM that contains all the important physical effects, including a combination of nonlinear tip–sample interaction and the effects of the feedback loop control laws. Such a low-order model is amenable to performing parameter sweeps, to produce so-called brute-force bifurcation diagrams, and to investigate potential instabilities. Nevertheless, as can be seen in figure 2, when imaging with an AFM the image is rarely perfect. In order to understand the possible systematic sources of such artefacts, to distinguish them from thermal noise, it is imperative that the fully nonlinear properties of the AFM be modelled as closely as possible.
Previous research has derived models for tapping mode AFMs; see Song & Bhushan (2008) for a review. In particular, studies have included nonlinear surface interaction forces and meniscus effects (Zitzler et al. 2002; Dankowicz et al. 2007; Song & Bhushan 2008), which are highly nonlinear and can lead to chaotic motion if the desired amplitude of oscillation (also known as the setpoint) is chosen incorrectly (Hu & Raman 2006). Such complex motion can be caused by the so-called grazing bifurcations (Jamitzky et al. 2006; Dankowicz et al. 2007; Yagasaki 2007; Hashemi et al. 2008), as the amplitude setpoint is decreased. Theoretically, such bifurcations are shown to underlie a functional bistability between two distinct but nearby equilibrium amplitudes for a given set of parameters.
However, such phenomena have typically been found in open-loop systems. In this paper, we shall specifically investigate how the value of gain used in a simple model of the control loop affects the stability of the AFM system. We will also study the effects of changing amplitude setpoint in a closed-loop system. At each stage of the investigation, we take experimental results to support the model.
The rest of the paper is outlined as follows. Section 2 briefly describes the experimental apparatus that we have used to compare with the predictions of our mathematical model. Section 3 then carefully explains the steps taken in developing the model, including how we implement the control law in an attempt to mirror how it operates in practise. In §4 we produce brute-force bifurcation diagrams through numerical simulation of the model applied to the imaging of a flat surface. A form of instability is found for sufficiently high gains. We also show how experimental bifurcation diagrams contain the same qualitative behaviour. An attempt is made to explain the physical mechanisms that underlie the observed phenomenon. Section 5 contains a detailed comparison between the simulations and experiments when applied to the calibration grid in figure 2, by variation of several key parameters. Finally, §6 draws conclusions and makes practical recommendations for effective AFM operation in order to avoid the effects described here.
2. Experimental methodology
The mathematical model we shall develop is based on a specific experimental set-up. All experimental data reported here were obtained using a Digital Instruments Dimension 3100 AFM, Model D31005-1, with the Digital Instruments piezoelectric tube and photodetector head model DMLS, and a Nano World Arrow-NCR-W, S/N 67209l697 Arrow silicon SPM sensor cantilever and tip (of length 160 μm, width 45 μm and thickness 4.6 μm, with resonance approx. 285 kHz and force constant 42 N m−1). Many of the parameters of the apparatus can be accurately altered via a data acquisition computer running NanoScope V. 5.3 software. Measurements are collected using the AFM in tapping mode, imaging a three-dimensional calibration grid (Digital Instruments 498-000-026). This grid has a 10 μm pitch with holes of depth 200 nm. An AFM image of a 26×26 μm area of the grid can be seen in figure 2.
Using a signal access module (SAM), it was possible to intercept, measure and record the raw data from the movement of the laser over the photodiode before the signal is fed into the controller. An oscilloscope (Tektronix TDS2012D) with a sampling frequency of 100 MHz was used to capture the data. The oscillation of the cantilever was sampled every 2×10−8 s, which corresponds to about 175 samples per period of the cantilever’s oscillation at 285 kHz. The difference between the maximum and minimum amplitude values within 1000 samples (roughly six periods of oscillation) is recorded as the peak-to-peak amplitude. The advantage of this method is that the control loop can remain active while the raw data are collected, allowing the viewing of both the image and the raw data. Hence aspects of the control loop can be inferred.
3. Model development
(a) Free oscillation
When the cantilever is being driven away from the surface, the position of the tip of the probe with time can be faithfully modelled as a driven damped simple harmonic oscillator (Rabe et al. 1996; Turner et al. 1997) oscillating in the z direction only. Applying a periodic piezoelectric force of amplitude γ and frequency ω, the equation of motion for the freely oscillating cantilever can thus be written in the form 3.1where m is the effective mass of the cantilever together with its tip, c the damping coefficient and k its linear stiffness. We can rewrite the effective mass and damping of the cantilever and tip in terms of its natural frequency ω0 and a quality factor, Q via 3.2The equation of motion (3.1) then takes the form 3.3The values of these modal parameters can be found in table 1.
Although many harmonics may be present on the cantilever (Stark et al. 2004), the control loop in our experimental system considers motion in the first transverse mode only, as there is only a single laser reflected off the end of the cantilever. Thus, a model that ignores higher cantilever harmonics is broadly justified. Any features in the images that may be caused indirectly by the excitation of higher modes will not form part of this paper. See Melcher et al. (2008) for multi-modal simulations of AFMs.
Closer to the surface, we must include an additional term on the right-hand side of equation (3.1) that captures the forces the tip experiences owing to sample–tip interactions. These forces are owing to two separate effects, van der Waals forces and capillary action. We shall now treat each in turn.
(b) van der Waals forces
Figure 3 defines the distance d=z+u(t)−ζ(y) between the sample and the cantilever tip. As d tends to zero, the tip experiences a force due to atomic interactions, the so called van der Waals force. The tip is not atomically sharp, and even if it were, it would not interact with a single molecule at a time. Therefore, we model the tip as a spherical surface of radius R coming in contact with a locally flat sample surface. The contact force is modelled as a combination of surface adhesion and the restoring forces generated by the elastic spherical probe tip pushed against a flat surface. We use an approximation of the contact region of the van der Waals force called the Derjaguin, Muller and Toporov (DMT) model (Derjaguin et al. 1975) to avoid the singularity as d tends to zero; at tip sample distances, d≤a, where a is the intermolecular distance, the adhesion and attractive part of the DMT contact force is assumed to be held fixed at the value of the van der Waals force at the distance d=a. Thus, we approximate the contact force F as 3.4where Vt and Vs are the Poisson’s ratios, and Et and Es are the elastic moduli of the tip and sample, respectively.
(c) Capillary action
It has been established that under standard atmospheric conditions, a thin water layer covers both the surface of the tip of the AFM and the sample (Zitzler et al. 2002). This layer acts to attract the tip while the tip’s fluid layer is joined with the surface’s fluid layer by means of a fluid meniscus bridge. The attractive force G is proportional to the volume of water contained in the meniscus (Zitzler et al. 2002; Hashemi et al. 2008). Thus, we assume that the total meniscus force as a function of the distance from the sample is 3.5where h is the thickness of the fluid layer on the sample (assumed uniform and to be equal to that on the tip) and se is the surface energy of the fluid. Also the distance don of the tip from the surface at which the meniscus forms is not equal to the distance doff at which the meniscus breaks (Hashemi et al. 2008). The meniscus is formed when the tip is travelling down towards the surface () and d<don and is broken only when the tip is moving away from the surface () and d>doff. This inequality leads to a hysteresis in the overall force that the tip experiences.
The equilibrium displacement of the tip above the surface is a function of the surface topography, ζ(y), measured relative to the height of initial contact with the surface, and the movement in the z direction owing to the control loop u(t) (figure 3), so 3.6
The control system in the experimental AFM used in this study (Digital Instruments dimension controller, model D3100HP-2) is effectively a black box system; however, it is known that a form of proportional, integral and derivative (PID) controller is used. As mentioned above, the role of the control in a tapping mode AFM is to maintain a constant amplitude of oscillation by shifting the origin of the z oscillation relative to the surface. The rest of this section will describe how a control law was implemented in the model, which appears to mimic that of the experimental apparatus.
Preliminary simulations and experiments with the AFM using different values of PID gains indicated that for the relatively large features on the calibration grid (or lack of features on a stationary flat silicon surface), the integral gain had by far the greatest influence on the results. Furthermore, a faithful fit to the experimental results could be obtained by setting the proportional and derivative gains to zero and allowing the integral gain to vary.
In order for the AFM to function properly, ze must always be less than half the peak-to-peak amplitude γe of the oscillating cantilever; else the tip will be oscillating away from the surface and not imaging. This is achieved by selecting a peak-to-peak amplitude setpoint, sa=γe(1−ra), in which .
As the cantilever travels across the surface, the simulated control measures the peak-to-peak amplitude γm of the oscillating cantilever, by recording the maximum and minimum z-values in a small time interval Δt, where Δt≫2π/ω, so that many periods of motion are contained in this window (table 1). The controller compares γm with the setpoint amplitude sa over a period of time τ.
We allow the cantilever to oscillate for 2000 time-sampling intervals Δt away from the surface, then measure the equilibrium peak-to-peak amplitude reached; we call this amplitude γe. We then reduce γe by an amount raγe to give the peak-to-peak amplitude setpoint sa. As sa decreases, the control will try and keep the tip oscillating closer to the surface and vice versa. Thus 3.7We approximate this integral with a summation (which also mimics the behaviour of the control scheme in the experimental apparatus): 3.8Effectively, this sum calculates a mean error over the N previous samples. This error is then multiplied by a gain value gint to give the final control input u(t).
We found that the size of N plays a significant role in the quality of the image produced by the simulated AFM: too small, and the tip either crashes into the surface or is lifted fully away; too large, and any features on the surface are smeared out. The value of τ is unknown in the actual AFM, our value of N was chosen so that the simulated calibrated images closely match those from the experiments (§5).
We attempted weighting the amplitudes of previous sampling intervals to give more recent amplitudes a larger effect on the control u(t) in either an exponential or linear fashion. The results were, however, discouraging. In fact, the most acceptable fit between the simulation and the experimental data was obtained using equation (3.8) with N=40.
(e) Parameter fitting
In order to fit parameters accurately, only a single parameter was varied in each physical or simulated run. The baseline values used are given in table 1. The physical parameter values have been chosen owing to their success in the existing simulations in the literature or from the equipment manufacturer in the case of the cantilever properties. In the case of the control parameters, their values were obtained by careful comparison with the experimental data, which we explain as follows.
Experimentally, the amplitude setpoint is given as a voltage, although physically it affects the equilibrium distance between the tip and the surface (sa in the simulations) through the microscope control. The voltage is passed to the piezoelectric device that controls the movement of the sample in the z axis, and is proportional to the extension of this device, although the constant of proportionality is not known.
In §4 we find that, upon increasing the gain gint, an instability sets in at a specific integral gain value, in both the simulation and the experimental results. We use the onset of this instability to calibrate the integral gain value gint against the experimental gain value G. Our findings suggest that the relationship between them does not follow a linear scale. Instead, the relationship 3.9provides a satisfactory fit for most values of experimental gain used.
(f) Running simulations
The model was solved using the Matlab differential equation solver ode45, using a relative error tolerance of 1×10−3, which implies an absolute tolerance of 𝒪(10−10). Matlab’s event handling was used to locate the boundaries d=don/off and d=a, so that the correct forces G(d) and F(d) are applied.
At each sampling interval, t=nΔt, the value of ζ(y) progresses to the next value by stepping in y(t). Over a flat sample surface, this remains constant but to simulate the effect of the calibration grid, a square waveform is fed with a time period of 600Δt and a peak-to-peak amplitude the same as the depth of the troughs in the physical calibration grid (200 nm). The value of u(t) calculated for a given sampling point t=nΔt is then used in the following sampling point t=(n+1)Δt to calculate the value of ze using the value of ζ(y) at the new time. A Matlab event file is then used to update the new value of d given by equation (3.6). The delay of one sampling time-step in the control is justified by the time taken for the amplifiers in the real AFM to extend or retract the piezoelectric tube. Nevertheless, this time is small when compared with the controller’s lock-in time, and so the effect on the overall dynamics of the system is negligible.
4. Nonlinear effects for a flat sample
(a) Variation of setpoint and gain
Figure 4 shows the results of running quasi-static simulations for a range of setpoint amplitudes ra, for different integral gain values gint. For each value of sa, we plot the peak-to-peak amplitudes of the oscillation of the tip calculated for the final 200 sample intervals Δt, after allowing transients to decay over 2000Δt. As the setpoint is reduced (by increasing ra), note the various different regions of either periodic motion (represented by a single amplitude value) or non-periodic motion (a sample of different amplitude values). These effects are more pronounced when the gain is largest. For gains lower than the lowest values used in figure 5, there is no evidence at all of non-periodic motion.
The effects seen in figure 4 can also be seen in similar simulated bifurcation diagrams, where the setpoint sa is held fixed and gain is varied quasi-statically. The results are shown in figure 5a–c for the three separate gain values indicated by the numbers 1–3 in figure 4c. The left-hand panels of figure 5a–c show the corresponding experimental data. Gaps in the experimental plots are owing to data having not been collected for the particular gain and setpoint combination. Nevertheless, qualitative and quantitative agreement can be seen between the experiments and the simulations. In particular, in each case an instability sets in at a specific value as the gain is increased, and the amplitude variation of the non-periodic response becomes more pronounced as the gain is further increased.
Figure 5d–f reproduces the phase portraits at the correspondingly labelled points in figure 5b. The plots display the simulation data from the final 200 sampling intervals after 2000 intervals of transient behaviour. Note that the phase portrait (in figure 5d) of stable operation corresponds to a pure periodic orbit, whereas those in the instability regions (figure 5e,f) correspond to non-periodic motion. It is hard to say from the phase portraits whether this motion is truly chaotic. Rather, the shape of the phase portraits seems to be more consistent with the complicated quasi-periodic motion in which several independent frequencies are present.
(b) Cause of the instability
One possible explanation of the instability seen in figures 5 and 6 might be some kind of dithering between the two distinct stable equilibria in the amplitude of open-loop tapping mode AFMs, as reported by Dankowicz et al. (2007). According to their results, those two different setpoints are best characterized by their very different values of phase difference between the driving and the response of the cantilever. Hence we replot in figure 6 the bifurcation diagram shown in figure 4c, plotting now the phase difference on the y-axis.
Note from the figure that there is a clear correlation between the amplitude and the phase, as the amplitude setpoint is reduced. The regions of instability have corresponding large ranges of phase difference. Note that we do not see any evidence of passage close to two distinct equilibrium amplitudes. Rather, the regions of instability visit a wide range of different phase differences.
Another feature of the bistability found by Dankowicz et al. (2007) is hysteresis upon decrease and subsequent increase in the setpoint amplitude. To test for this, we ran simulations similar to those in figure 4c by first quasi-statically decreasing then increasing sa to check for hysteresis in the locations of the regions of instability. We found limited evidence of hysteresis in the points of onset of the instability. The forward and backward sweeps did indicate the presence of multiple attractors at some parameter values inside the instability regions.
To further test the accuracy of our model, equivalent simulations were carried out using Veda (Melcher et al. 2008) without feedback control, but with other parameters set as in table 1. These simulations showed results that are remarkably similar to those using our model, but showed none of the instabilities we have found here for sufficiently high gains. This gives some confidence in our assertion that the instability seen in the experiments is indeed owing to the action of the feedback control.
To gain further insight into the cause of the instability, it is useful to consider the dynamics at unstable parameter values in more detail, in both the experiments and the simulations. Figure 7 shows the z movement of the cantilever against time, from experimental data (figure 7a–d) and the corresponding numerical simulation (figure 7e–h). In experiments, we keep the surface stationary in the x and the y directions and record the actual z displacement by intercepting the signals sent to the photodiode via the SAM box, and sampling them at 100 MHz. The experimental results are thus only proportional to the displacement in the z direction.
Note the close qualitative agreement between experiment and numerics. For high enough gains both theory and experiments show a repeating significant feature of growing amplitude followed by arch-like slow excursions as the control pushes the tip into the surface. For low gains, the arches are not present; instead it appears that an envelope wave exists with a ‘beating’ frequency of roughly 500 Hz (from the experimental data using a driving frequency of 285 kHz).
Beating motion has previously been observed in tapping mode AFMs, owing to the tip hitting the surface and bouncing off too high (Stark & Heckl 2000) when the impulse from the surface coincides constructively with the oscillation piezoelectric device in the head of the AFM. This still might be playing a part in the formation of the structures but our bifurcation studies make it clear that the structures are most strongly dependent on the value of the integral gain, rather than any nonlinear detuning causing constructive interference.
We are left with the inevitable conclusion that the instability is not primarily due to the non-smooth nature of the cantilever dynamics, but is a delay-induced instability (e.g. Stepan 1989; Gopalsamy 1992) caused by choosing too great a value for the integral gain. Qualitatively, the extreme form of the instability when the arch-like features are present comes about because, if the gain is increased, the resulting control movement for a given error will increase. In the situation where the cantilever is oscillating at an amplitude lower than its amplitude setpoint, the control will therefore move the oscillation midpoint up away from the surface. If the gain is too high, the tip may become free of the surface leading to a large gain in the peak-to-peak z amplitude. Owing to the integral nature of the control, it is not until a number of sample intervals later that the control once again pushes the cantilever towards the surface causing an arch-like feature. There is a period of 𝒪(NΔt) before the cantilever makes contact with the surface again, whereupon the high gain again causes an over-compensation, which results eventually in another lift-off event.
5. Measurement of test surfaces
We now investigate the effect of altering the driving amplitude (γ), driving frequency (ω) and setpoint (sa) on the image of the calibration grid. In simulating the calibration grid, ζ(y) is no longer kept constant with time; instead, ζ(y) is a square wave with a peak-to-peak amplitude the same as the depth of the calibration grid troughs (200 nm).
(a) Variation of driving amplitude
With all other parameters remaining constant (as given in table 2), the amplitude of oscillation was varied. A selection of experimental results with increasing amplitude can be seen in figure 8a–c. Below 100 mV the tip fails to establish contact with the surface.
As the amplitude of oscillation increases, two features in the height trace can be seen to vary. First, as the tip moves over the grid into a trough, the decay rate of the recorded height increases, which decreases the ‘shadow’ effect. Second, the amount of detail that the AFM picks up from the surface changes with amplitude. At low amplitudes, the tip seems to ‘bounce’ over the surface (Stark & Heckl 2000); at intermediate amplitudes, the AFM works as it was designed to and produces good images. If the amplitude is increased further then the tip seems to have trouble imaging the higher parts of the image. However, the troughs of the grid are resolved well, with the exception of a dip being formed as the tip drops off the edge of the grid and hits the bottom of the trough.
A ‘shadow’ is caused when ze is larger than the half peak-to-peak amplitude of oscillation of the cantilever for more than one time step. Until the control loop can lower the value of ze enough for ze to be less than the half peak-to-peak amplitude, no information is collected from the surface topography as the tip is oscillating free from the surface, and so is not being affected by G(d) or F(d). As seen in figure 8, the region of shadow can be minimized using a larger driving amplitude. In doing so, the amplitude increases at a quicker rate when the tip passes over a falling edge to meet the surface below.
To simulate the above amplitude experiment, the driving amplitude was increased from 5×10−9 to 1×10−7, while all other parameters were held constant (as given in table 1). We show the results in figure 8d–f. Figure 8a shows that when the drive amplitude leads to a half oscillation amplitude of the tip which is less than the setpoint, the AFM is not capable of imaging the surface. Put simply, the tip is not making contact with the surface but is just oscillating above it. When the driving amplitude is just large enough for the tip to make contact with the surface, a situation similar to that shown in figure 8b can be seen where the AFM image has a ‘shadow’ after any sharp drop in the topography height. The shadow is caused by the tip not being in contact with the surface. The oscillation amplitude of the tip needs to increase sufficiently to make contact with the surface, while the control moves the surface up to meet the tip. The effect of this is that the fall of the input step function is smeared into a slope. When the driving amplitude is increased further, as shown in figure 8c, the shadow is all but removed owing to the reduced time required for the tip’s oscillation amplitude to become large enough for the tip to make contact with the surface again after falling off the top of a step.
(b) Variation of driving frequency
In order for the AFM to function properly, the piezoelectric device in the head of the microscope must drive the cantilever at a frequency at or just below its resonant frequency. Figure 9 shows the results of altering the driving frequency about the resonant frequency of the tip used (285 kHz experimentally and the scaled 100 Hz in the simulation), which correspond to traces in figure 9b,g. The quality of the trace degrades rapidly either side of the corrected resonant frequency. Experimentally, the AFM control loop can image when the driving frequency is reduced to 0.2 per cent below the resonant frequency and increased to 0.07 per cent above. Even when the driving frequency is at the boundary of the usable frequencies, the rising edge can be recognized, but flat surfaces are not recorded well in the tip traces. Neither are the falling edge nor the trough of the grid recorded well at these extreme frequencies. At the correct driving frequency (figure 10b,g), the plots give a very good indication of the position of any change in topography.
The experimental and simulation results (figure 9a–e and 9f–j, respectively) taken as the driving frequency was altered can be understood by realizing that moving the driving frequency away from the natural frequency has the effect of reducing the driving oscillation amplitude. When the resulting amplitude falls below the setpoint, then the AFM is no longer able to image. In particular, the same key features are seen when the frequency is either too low or too high. However, in terms of quantitative comparison, the results in figure 9 are somewhat less convincing than the corresponding results in figures 7, 8 and 10. This could be owing to the possibility that higher modes of vibration becoming increasingly important as the difference between ω and ω0 increases. It is also believed that as ω moves away from ω0, some of the physical parameters of the system would change. These changes were seen as being beyond the scope of this paper and so not included in the model. A more complicated spatially extended approach to modelling the cantilever would be required to look into this further (e.g. Song & Bhushan 2008).
(c) Variation of the setpoint
We now proceed to investigate the effect of changes in setpoint; the results are shown in figure 10. For the experimental results, figure 10a–f, the parameters are held constant at the values stated in table 2, while the driving frequency is altered. The driving amplitude is also increased from 119 to 200 mV so that structures can be seen in more detail near the edges of the range of frequencies used. At a driving amplitude of 200 mV, the tip makes contact with the surface at a setpoint of 0.9614 V (figure 10a), corresponding to the AFM beginning to record just the tops of the calibration grid. As the setpoint is decreased further more detail is picked up from the surface. When the tip starts to get very close to the surface (figure 10f), the amplitude traces pick out the changes in topography very clearly. This could be owing to the tip having only the opportunity to increase its amplitude of oscillation while the control loop and piezoelectric tube are temporarily increasing the setpoint during a change in surface topography. It is also very clear that the cross section becomes much less noisy as we lower the setpoint. This supports the results found in §4, especially those in figures 4 and 7, where the amplitude variance owing to the beating effect generally decreases once the setpoint is below half the value at which the tip starts to make contact with the surface (ra<0.5).
Figure 10g–l shows the results of numerical simulations where the amplitude setpoint, sa, is changed; sa varies between 102 and 60 per cent of half the equilibrium peak-to-peak amplitude (1.96×10−7 m), while all other parameters remain as given in table 1. Even at setpoints greater than half the equilibrium peak-to-peak amplitude, the tip is still weakly able to pick up the surface (figure 10g). This is because of the control loop over-compensating for a correction and bringing the cantilever closer to the surface than sa by a few microns and so the tip makes contact with the surface. The simulated results agree well with those collected experimentally. In both simulation and experimental results, the AFM does not image the surface correctly until the setpoint is reduced to 90 per cent of the half peak-to-peak amplitude of the freely oscillating cantilever. Another common feature is the decrease in the noise with a decrease in setpoint. Presumably this noise is because of the instability we have investigated in §4. While it is not always possible to use lower amplitude setpoints for fear of damaging a delicate sample surface (the lower the setpoint the more force the tip will exert on the surface), it is clear from the results above that a setpoint below 50 per cent of the half peak-to-peak amplitude of the freely oscillating cantilever leads to much better image quality.
In this paper we have investigated a model of an AFM in tapping mode. Specifically, a damped, driven oscillating mass model was extended using surface interactions (van der Waals and fluid meniscus forces) to simulate the cantilever in an AFM in tapping mode. A simple model of a control with integral gain, as present in the experimental apparatus within a commercial black box controller, was included in the model. The predictions of the mathematical model correctly describe the experimental results from the AFM, under variation of the key parameters that may be varied in an experiment: amplitude setpoint, driving amplitude and frequency.
The key finding of our study has been that the choice of too large a value for the gain parameter in the control loop can cause an instability. For very high gains, the instability can lead to the arch-like structures in the oscillation amplitude seen when imaging a flat surface in figure 8c,d. In reality, the AFM is not likely to be used with such high gains, but typical operating conditions could easily be high enough to cause the envelope waves seen in figure 8a,b to be produced, degrading the final image as a result.
It would be interesting in future work to probe further the cause of this instability. The presence of beating-type behaviour is strongly suggestive of a delay-induced oscillatory bifurcation (see Stepan 1989; Gopalsamy 1992). A precise mathematical analysis of such an instability is beyond the scope of this paper, and is likely to be challenging, because the underlying state is a large-amplitude non-smooth periodic orbit. It would be interesting to see though whether such a form of instability can be captured in a simplified ‘toy’ model of the combined cantilever/controller dynamics. We note similarities with other work particularly (Dombovari et al. 2008) with respect to the delay and loss of contact.
We have also not probed in detail whether the dynamics are chaotic or not inside the regions of instability. Clearly, the dynamics have a lot of structure, which suggests that the dominant part of the motion is a periodic modulation of the periodic tapping motion. Nevertheless, the fine structure we observe in the brute force bifurcation diagrams might also be consistent with a (weak) chaotic modulation of the underlying quasi-periodic motion. The mechanism seems very different to that which appears to underlie the previous observation of chaotic dynamics in an AFM (Hu & Raman 2006). In a sense, whether or not chaos is present in the motion we observe is not as serious an issue in practice as how to avoid the instability in the first place.
One motivation of this work has been to help understand how cleaner AFM images may be produced in tapping mode. Specifically, we have provided a theoretical framework for understanding how appropriate machine parameters should be selected. The results for the flat sample show the importance of selecting values for the amplitude setpoint and integral gain within the regions of stability in the bifurcation diagrams shown in figures 5 and 6. For example, the results show that stability can always be achieved by lowering the amplitude setpoint, or by lowering the integral gain. Nevertheless, there is a trade off; too low a value of the setpoint can cause the cantilever to crash into the surface, and too small a value of gain will stop the AFM from functioning effectively, as the transient effects will be too large.
The results we obtained from experiments and simulations on a calibration grid suggest further optimal conditions for imaging such a surface. The best results were obtained from a peak-to-peak amplitude setpoint of about 60 per cent that of free oscillation, with as large a driving amplitude as possible and as slow a scan speed. However, it is also important to put these conclusions in context. Although these parameters may provide the best images for a hard surface, the force at which the tip hits the surface, and any damage caused by these forces, has not been taken into account. For softer surfaces another force curve such as the so-called Johnson, Kendall and Roberts (JKR) approximation (Johnson et al. 1971) may yield better results than the DMT model, although the models of the control loop and fluid layer would still apply, as would the general simulation methodology. The results from this investigation, however, demonstrate that models of the AFM, which include the control loop and fluid layer allow the tip dynamics to be recreated, understood and explained.
O.P. gratefully acknowledges the financial support of the Engineering and Physical Sciences Research Council (grant no. EP/E032249/1). The authors also acknowledge helpful conversations with Arvind Raman and Harry Dankowicz.
- Received August 26, 2010.
- Accepted November 15, 2010.
- This journal is © 2010 The Royal Society