Anewcontrol strategy for high-speed atomic force microscopy

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING Nanotechnology 15 (2004) Anewcontrol strategy for high-speed atomic force microscopy NANOTECHNOLOGY PII: S (04) GSchitter 1,FAllgöwer 2 and A Stemmer 1,3 1 Nanotechnology Group, Swiss Federal Institute of Technology, ETH Zentrum CLA, CH-8092 Zurich, Switzerland 2 Institute for Systems Theory in Engineering, University of Stuttgart, Pfaffenwaldring 9, D Stuttgart, Germany stemmer@nano.mavt.ethz.ch Received 25 June 2003, in final form 22 September 2003 Published 14 November 2003 Online at stacks.iop.org/nano/15/108 (DOI: / /15/1/021) Abstract An advanced controller consisting of a feedback and feedforward part is presented to improve the performance of an atomic force microscope (AFM) enabling topography measurements at higher scan rates with a reduced measurement error. The tip sample interaction force is held constant by an H -controller while the scanner is simultaneously tracked to the topography of the last recorded scan line by a model-based feedforward controller. The designed controller is implemented on a commercial AFM system to comparethe performance to a standard proportional integral controlled AFM. The new controller reduces the measurement error and enables imaging at higher speedsandatsmaller tip sample interaction forces. 1. Introduction Atomic force microscopes (AFM) are used to trace the topography of a nanoscale specimen by a sharp tip supported on amicro-mechanical cantilever [1]. The spatially resolved topography is measured by scanning the sample laterally under the probing tip by means of a piezoelectric tube scanner [2]. Due to the sample s topography the cantilever gets deflected; this is monitored by an optical lever and a segmented photo diode [3, 4]. In some AFM designs the probing tip is scanned [5] instead of the sample. However, the issues for controlling the AFM are the same as in the scanning sample design, dealt with in this paper. A detailed description of the components and the function of the AFM can be found in [1, 6 8]. In contact mode the tip permanently touches the specimen. In the so-called constant force mode the cantilever deflection, and thus the tip sample interaction force, is held constant in a closed-loop operation. Changes in the deflection signal of the cantilever due to the sample s topography get compensated in the feedback-loop by varying the position of the sample along the Z-direction, which is along the axis of symmetry of the piezoelectric tube. In commercial realizations the closed-loop AFM system (figure 1), consisting of the scanner s voltage 3 Author to whom any correspondence should be addressed. Figure 1. Block diagramofa standardpi-controlledclosed-loop AFM system. The dashed box includes the components of the AFM in the Z-direction. amplifier, the scanner, the cantilever, the photodiode, and the controller, is operated by a proportional integral (PI) element. The setpoint in the control loop (figure 1) predetermines the nominal value of the tip sample interaction force. By recording the feedback-generated voltage applied to the Z-axis of the scanner piezo simultaneously with the lateral position of the sample, spatially resolved images of the sample surface can be drawn. Although the AFM is capable of imaging the topography of a specimen at high resolution, it only does so at relatively low speeds. The low scanning speed is mostly due to the limited bandwidth of the PI-controlled AFM given by the main dynamics of the piezo scanner. Depending on the settings of the PI controller faster imaging results either in an /04/ $ IOP Publishing Ltd Printed in the UK 108

2 High speed AFM increased cantilever deflection around the setpoint value or in oscillations of the piezoelectric tube. Both situations result in varying imaging forces, which might cause damage to tip or sample. Additionally, in the latter case the oscillations of the closed-loop AFM system would distort the topography image. First efforts to improve the performance of scanning probe microscopes in the Z-direction have been made by introducing high frequency piezo segments to the probe [9 11] or scanner [12, 13] or by implementing a modern model-based feedback controller [14]. However, multiple actuators in the vertical direction necessitate controlling the AFM in a nested feedback loop [15], which requires a lot of tuning knowledge from the AFM user when adjusting two competing PI loops. The control system presented in this paper does not contain competing feedback loops and still may be applied to multiple actuator systems. Here we present a new approach to considerably reduce the measurement error of the AFM. One can take advantage of the fact that two adjacent scan lines normally are quite similar. Thus we delay the topography of the last recorded scan line by one period of the scanning motion and a feedforward controller tracks the Z-direction of the piezoelectric tube scanner to this delayed topography signal. A simpler solution that neglects the dynamics of the piezo scanner was implemented earlier on a commercial AFM system [16]. Data from the previous scan line was used to make a one-pixel prediction and this information was added to the control action of the PI controller [17]. We present the design and implementation of a twodegree-of-freedom (2DOF) controller, which consists of a feedforward and a feedback part that takes into account the dynamics of the piezo scanner. For the design of the new controller a mathematical model of the AFM system s dynamic behaviour is required which is obtained from a black-box identification using sub-space methods [18]. The feedback operation is performed similarly to conventional AFM systems but at an increased bandwidth by utilizing modern model-based control methods. To this end an H -feedback controller is proposed for accelerating the scanning process by considering the high frequency dynamics of the AFM system. Simultaneously, a model-based feedforward controller, which is also designed in the H - framework, tracks the Z-direction of the piezoelectric tube scanner to the topography of the previous scan line. Thus, the feedback controller only has to compensate the difference between the previous andtheactual scan line. We demonstrate that this use of prior knowledge reduces the cantilever deflection and improves the performance of the AFM system compared to a well-tuned PI-controlled AFM. 2. Identification of the scanner dynamics To obtain a mathematical model of the AFM system we performed a system identification [19] similar to the procedure described in [14]. For hard samples and small amplitudes the model can be reduced to the dynamic behaviour of the piezoelectric tube scanner and a gain factor modelling the voltage amplifier and the deflection detection system [14] Experimental setup Tests were performed on a NanoScope-IIIa MultiMode-AFM (Veeco, USA) to identify a J -class piezoelectric tube scanner. For measurement of the input and output data an external digital signal processor (DSP) 4 was employed whichwasalsoused for the implementation of the controller presented in this paper. The input to the AFM system was given by the voltage amplifier of the piezo scanner in the Z-direction and the system output was the Z-signal of the segmented photo diode (see figure 1). Mica was used as a specimen because of its hard surface and the ability to avoid surface contamination by cleaving the sample just before the experiment. The scanner s extension and retraction in the Z-direction were detected by the usual AFM measurement setup using a contact lever of type NP-S, Silicon Nitride Probes, 200 wide (Veeco, USA). Aband-limited white noise signal was applied to the system input to get the input and output data for system identification. Due totheconstant power spectrum of the white noise signal, generated by the white noise block of Matlab Simulink [20] running on the DSP, the tube scanner was excited atallfrequencies within thebandwidth ofinterest. The excitation signal and the response of the AFM system to the applied excitation were recorded simultaneously by the external DSP running at a sampling time of 16.5 µswhichwas also the sampling time of the controller presented in this paper. The noise power of the excitation signal was chosen to get a maximum signal amplitude of the system s response between 2.5 and 10 V System identification A linear model can be calculated from the measured input and output data using a standard algorithm based on subspace methods [18], that is implemented in the Identification Toolbox of Matlab [20]. The piezo dynamics along the Z-axis can be modelled linearly as shown also in [21] and [14]. The identified model includes the dynamic behaviour of the piezoelectric tube scanner as well as the gain factor modelling the voltage amplifier and the optical deflection detection system. The model order is set to 5 because an increased model order does not reduce the modelling error substantially but compromises the sampling rate of the DSP at the implementation of the new controller by increasing the order of the controller as well [22]. Figure 2 displays acomparison between the simulated output of the identified model (solid curve) and the measured output of the AFM system (dashed curve) using validation data that have been measured in a separate experiment. The two curves fit very well showing only marginal, occasional separations. The identified model consists of two stable resonances, a low pass, a pair of complex zeros in the left half of the Laplace plane, and a pair of conjugate complex zeros in the right half of the Laplace plane. In figure 3 the corresponding bode diagram is plotted, clearlyshowingthe fundamental resonance of the piezoelectric tube scanner at about 8.5 khz and the anti-resonance at about 12 khz given by the pair of zeros in the left half of the Laplace plane. 4 DSP-System processor board DS1005, 16-bit A/D-board DS2001, 16-bit D/A-board DS2102 dspace: Germany. 109

3 GSchitter et al Figure 2. Verification of the system identification of the scanner model in the Z-direction. Output of the identified model: solid curve; measured signal: dashed curve. Figure 4. Structure of the extended mathematical model for the design of (a) the feedforward and (b) the feedback controller. Figure 3. Bode diagram of the identified model showing amplitude and phase of the piezoelectric tube scanner s transfer function in the Z-direction. Duetothe piezo dynamics the commercial, PI-controlled AFM is limited to a bandwidth below 1.5 khz, because the PI controller is not able to handle the high order dynamics of such oscillatory systems [23, 14]. To increase the control performance of the AFM a more sophisticated controller has to be designed and implemented which is the focus of the next section. 3. Controller design The design of our new controller is split into two parts, a model-based feedforward controller and a feedback controller. Both controllers can be designed separately but are combined to make a 2DOF controller (see, e.g. [22]) for operating the AFM in a feedback loop while the sample simultaneously is tracked to the previous scan line by the feedforward controller. Due to this feedforward operation the cantilever deflection is reduced, most notably when topographical steps or ramps occur perpendicular to the fast scanning direction. The new feedback controller is calculated utilizing modern model-based control methods to improve the performance of the closed-loop operated AFM system [14]. For the design of the feedback a linear H -controller [22] was chosen, because of the possibility to specify requirements to the closed-loop operated system, such as fast sequential control, no steady-state deviation, suppression of disturbances, rejection of measurement noise and robustness against model uncertainties. The feedforward controller is designed as an H -filter. The requirements for the feedforward and the feedback system have to be formulated as mathematical design specifications (weights), by which the mathematical model obtained from the system identification is extended (figure 4). The outputs of the extended mathematical model v u, v y, z e, z u, and z y in figure 4 do not exist in the real AFM and are used only during the design of the new controllers. These mathematical outputs are introduced to rate the performance of the control loop when calculating the model-based feedforward and feedback controller, respectively. The achieved performance of the feedforward or feedback operated system is evaluated by simulated step responses. During the design of the controllers the corresponding mathematical weights W are adjusted iteratively, such that therespective system output y 1 and y 2 tracks the guidance step in r 1 and r 2 as fast as possible without oscillations of the sample position. The feedforward and feedback controllers, which are also part of the extended model, are designed by minimizing the H -norm of the weighted model [22] as usual for H -control. Amoredetailed description of the design of the new controller can be found in [24]. In the weighted model for the design of the feedforward controller (figure 4(a)) the guidance signal r 1 represents the previous scan line. The position of the sample is given by the scanner output y 1. The output of the extended model v y is used to rate the tracking performance of the feedforward operated system by setting the tracking weight W y1 to low pass characteristics. The output v u penalizes the input to the piezoelectric tube scanner u 1 and is used to dampthe fundamental resonance of the scanner by the appropriate choice of the weight W u1. Figure 4(b) shows the weighted model for the design of the feedback controller. The setpoint r 2 determines the desired cantilever deflection. W u2 is again chosen to damp the resonance of the piezoelectric tube scanner by penalizing 110

4 High speed AFM Figure 5. Block diagram of the 2-degree-of-freedom-controlled AFM systeminthez-direction and the simulation of the scanner for data acquisition. The blocks within the dashed rectangle represent the new controller implemented on the DSP system. Figure 6. Flow chart of signals and tuning parameters of the model-based controlled AFM system. the scanner input u 2. The weight W y2 for the achieved cantilever deflection y 2 is set to high pass characteristics to suppress measurement noise and achieve robustness against uncertainties of the identified model beyond the control bandwidth. The weight W e2 assesses the control error e 2 and is set to low pass characteristics. By increasing the bandwidth of W e2 the feedback controller is designed for fast sequential control and suppression of disturbances, which in the eyes of acontrol engineer are given by the cantilever deflection due to the sample topography (see figure 1). Figure 5 shows the structure of our new 2DOF controller combining the feedforward and the feedback controller. The blocks labelled gain FF and gain FB are two additional tuning parameters. The block gain FF is used to define the rate of the feedforward trajectory applied to the tube scanner, which can be chosen between zero and one. A gain of zero means the AFM system is purely operated by the feedback controller. Setting the gain to one denotes the last recorded scan line isapplied to the piezo scanner at 100%, which reduces the control error nearly to zero for structures that do not change too much from one scan line to the next. For samples having avery unstructured surface, for example dust particles that appear just in one scan line, the feedforward gain can be reduced. The block gain FB is used to adapt changes in the gain of the optical deflection detection system, which can occur using another cantilever or different alignment of the laser beam on the cantilever. Aside from these two gain factors the 2DOF controller needs no further tuning because it is designed for the complete dynamics of the AFM system, in contrast to the PI-controlled AFM. For acquisition of the topographical data the position of the sample is simulated by the model of the piezo scanner [14]. To compensate the scanner dynamics at the increased control bandwidth, the control signal still oscillates while the position of the sample, and thus the cantilever deflection, already settled to the desired setpoint value. Due to these persistent oscillations, the voltage applied to the piezo in the Z-direction no longer represents the sample topography. The simulated position is additionally delayed by T, which is one period of the scanning motion minus the reaction time of the open-loop system in the Z-direction, to generate the trajectory applied to the feedforward controller. The derived Figure 7. Silicon calibration grid imaged at 10 Hz line-scan rate by the PI-controlled AFM ((a), (c), (e), and (g)) and the 2 DOF-controlled system ((b), (d), (f), and (h)). Topography ((a) and (b)), topographical cross section ((c) and (d)) marked in image (a) and (b), cantilever deflection ((e) and (f)), and error cross section ((g) and (h)) marked in panel (e) and (f). All images are recorded from right to left and measure 6 6 µm 2. mathematical models of the scanner, the feedforward and the feedback controller are implemented in their state-space representation, which are sets of first order matrix difference equations, on the DSP (see footnote 4) according to figure 5. Figure 6 shows a flow chartof the signals and tuning parameters between the AFM electronics, the AFM, the DSP and the host computer, which is used to control the DSP and to adjust the feedforward and the feedback gain. The topographical data generated by our new controller are 111

5 GSchitter et al Figure 8. Control error of the PI-controlled AFM ((a), (c), (e), and (g)) and the 2DOF-controlled system ((b), (d), (f), and (h)) in response to 26 nm topographical steps recorded at a line-scan rate of 10 Hz ((a) and (b)), 15 Hz ((c) and (d)), 20 Hz ((e) and (f)), and 30 Hz ((g) and (h)). All line scans are recorded from right to left and measure 6 µm inwidth. recorded by an A/D-converter of the commercial AFM system to use the AFM s imaging system for data acquisition. To avoid lateral oscillations of the piezoelectric tube due to the scanning motion an additional model-based open-loop controller is implemented on thedspto filterthe scanning voltages applied to the piezo tube, such as is described in [25]. 4. Experimental results All experiments shown in figures 7 9 were performed on a NanoScope-IIIa MultiMode-AFM. The PI controller of the commercial AFM system was tuned well [16] for each measurement to compare the control performance between the standard PI-controlled and the new 2DOF-controlled AFM system. The images presented in figures 7 and 8 were recorded using a cantilever of type NSC12/Si3N4/50, type E (MikroMasch, Estonia) with a nominal force constant of 0.3 Nm 1.This cantilever is of a different type from the one used for the identification experiment and demonstrates that the cantilever dynamics can be neglected in contact-mode AFM. The experiments shown in figure 9 were imaged with a contact lever of type NP-S, Silicon Nitrite Probes, 200 wide (Veeco, USA) with a nominal force constant of 0.12 N m 1. Figure 7 shows a comparison of the standard PI-controlled AFM system with the 2DOF-controlled AFM recorded at 10 Hz line scan rate. The test specimen is a silicon calibration grid (TGZ01, MikroMasch, Estonia) showing 26 nm deepetched moats with a pitch of 3 µm. By comparing images 7(e) with 7(f) and 7(g) with 7(h) the reduction of the cantilever deflection by operating the AFM system with our new 2DOF controller can be seen. A reduction of the maximum deflection reduces the variation of the tip sample interaction force and results in more reliable topographical data. Furthermore, it reduces the maximum tip sample interaction force and prevents damage to the tip and/or sample. In figure 8 the control error of single line scans of the specimen shown in figure 7 recorded by the PI-controlled AFM and the 2DOFcontrolled system is shown for selected line scan rates. The control error is always smaller in caseof the 2DOF-controlled AFM compared with the PI-controlled system. Single line scans of a silicon calibration grating (TGZ03, MikroMasch, Estonia) showing 530 nm deep-etched moats with a pitch of 3 µm areshowninfigure 9 to compare the control performance of the PI-controlled and the 2DOFcontrolled AFM. The overshoot at topographical steps is an imaging artefact stemming from theclosed-loop operation. Areduction of this overshoot can be seen when the AFM is operated by our new controller. Since the measured topography is always a dilation of the sample topography and the probing tip [26, 27], topographical steps that are much higher than the tip radius occur as a ramp in the topography signal according to the profile of the tip geometry. A closed-loop system with just one integrator (single pole at s = 0), such as the PI-controlled AFM, is not able to track a ramp exactly, and follows the guidance signal with a delay resulting in a constant control error [28]. In the case of the PI-controlled AFM this causes a deflection signal with an offset to the desired setpoint during the topographical 112

6 High speed AFM Figure 9. Single line scans recorded on a 530 nm calibration grating by the PI-controlled AFM ((a) (f)) and the 2DOF-controlled system ((g) (m)). Panels (a) (c) and (d) (f) show the topography and the control error of the PI-controlled AFM at 10 Hz ((a) and (d)), 20 Hz ((b) and (e)), and 30 Hz ((c) and (f)) line scan rate. The topography and error signal of the 2DOF-controlled AFM is shown in panels (g) (i) and (k) (m) recorded at 10 Hz ((g) and (k)), 20 Hz ((h) and (l)) and 30 Hz ((i) and (m)), respectively. All line scans are recorded from right to left and measure 15 µm inwidth. ramp, which can be seen in figures 9(d) (f). The constant delay during the tracking of the ramp signal also causes a shift of the recorded topography with respect to the real topography. The new 2DOF controller applies the topographical ramp to the piezoelectric tube scanner through the feedforward path, enabling the feedback controller to track the desired imaging force with much lower variations and to circumvent the shift between the real and recorded topography. The setpoint was adjusted to 0.0 V for all scans measured with the 2DOF-controlled AFM (figures 9(g) (m)). In order to avoid loss of tip sample contact in the case of the PIcontrolled AFM (figures 9(a) (f)), the deflection setpoint had to be increased to 0.2 V for a line scan rate of 10 Hz, to 1.0 V for 20 Hz, and to 1.4 V for 30 Hz line scan rate. Increasing the setpoint by 1.4 V results in an increase of the tip loading force of about 12 nn, which is calculated from the deflection sensitivity, the setpoint and the typical value of this lever type s spring constant. Due to the reduced cantilever deflection when controlling the AFM by our new 2DOF controller the system can be operated closer to the minimum force value without loss of tip sample contact resulting in smaller imaging forces. 5. Conclusion A 2DOF controller was designed and implemented to reduce the measurement error and improve imaging speed of an AFM. The dynamic behaviour of the AFM system was identified to obtain a mathematical model for the design of the model-based feedforward and feedback controllers and for simulation of the piezoelectric tube scanner to get the actual topographical trajectory used for data acquisition. Additionally this signal is delayed by one period of the trace retrace cycle of the scanning motion to generate the target trajectory of the next scan line, which is applied to the feedforward controller. This use of prior knowledge improves the performance of the AFM considerably. The new 2DOF controller enables fast scanning at significantly reduced measurement errors. Further tuning of the controller by the user is not necessary because the 2DOF controller is designed for the complete dynamics of the AFM system. Due to the noticeable reduction of the maximum cantilever deflection imaging can be accomplished at smaller tip sample interactionforces without lossofcontact, preventing damage to tip or sample. Although the controller presented in this paper is designed for contact-mode AFM, we expect that a feedforward compensation also reduces the control error in tapping mode. For systems with multiple actuators in the vertical direction (e.g. [15]) a model-based controller would remove the necessity to tune two competing PI loops, because this controller would generate the control action for both actuators. Implementing a 2DOF controller as proposed here would increase the performance of such an AFM even further. Acknowledgments The authors thank Dr Robert Stark of the Nanotechnology Group of the Swiss Federal Institute of Technology for his help and fruitful discussions. 113

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