Control of Scanning Quantum Dot Microscopy
Michael Maiworm, Christian Wagner, Taner Esat, Philipp Leinen, Ruslan Temirov, F. Stefan Tautz, Rolf Findeisen
CControl of Scanning Quantum Dot Microscopy
Michael Maiworm , Christian Wagner , Taner Esat , Philipp Leinen , Ruslan Temirov ,F. Stefan Tautz , Rolf Findeisen Abstract — Scanning quantum dot microscopy is a recentlydeveloped high-resolution microscopy technique that is basedon atomic force microscopy and is capable of imaging theelectrostatic potential of nanostructures like molecules or singleatoms. Recently, it could be shown that it not only yieldsqualitatively but also quantitatively cutting edge images evenon an atomic level. In this paper we present how control is akey enabling element to this. The developed control approachconsists of a two-degree-of-freedom control framework thatcomprises a feedforward and a feedback part. For the latterwe design two tailored feedback controllers. The feedforwardpart generates a reference for the current scanned line basedon the previously scanned one. We discuss in detail variousaspects of the presented control approach and its implicationsfor scanning quantum dot microscopy. We evaluate the influenceof the feedforward part and compare the two proposed feedbackcontrollers. The proposed control algorithms speed up scanningquantum dot microscopy by more than a magnitude and enableto scan large sample areas.
I. I
NTRODUCTION
An important aspect in nanotechnology is the determina-tion of characteristics of the fundamental building blocks ofmatter, namely atoms and molecules. One of these charac-teristics are the electrostatic properties that govern in manycases the functionality of nanoscale objects and systems. Thisis particularly important for new materials and devices as-sociated with nanoscale electronics, such as semiconductors.The investigation of electrostatics at the nanoscale becomestherefore more and more important and is a vivid field ofongoing research [1], [2], [3], [4], [5].Scanning quantum dot microscopy (SQDM), introducedin [6], [7] allows to measure the electrostatic potentialsof nanostructures with sub-nanometer resolution at largeimaging distances. It generates 2D images of the electro-static potential (Fig. 1). It furthermore allows to separatelyimage the electrostatic potential and the surface topogra-phy. Recently it was shown in [5] that SQDM does notonly generate qualitative but also quantitative images ofthe electrostatic potential of nanostructures. The paper alsodemonstrated large-scale imaging (Fig. 1), resolving bothsmall (single atoms and molecules) and large structures (anisland composed of several hundreds of molecules) in thesame image, whereas in the previous publications of SQDM[6], [7] only isolated atoms and molecules were imagedindependently of each other. Thereby, SQDM became a Otto-von-Guericke-Universit¨at Magdeburg, Laboratory forSystems Theory and Automatic Control, Germany, { rolf.findeisen,michael.maiworm } @ovgu.de. Peter Gruenberg Institute (PGI-3), JuelichResearch Center, Germany, [email protected]. Fig. 1. Top: Scanning tunneling microscope image of a sample presentedin [5]. The image size is 600 ×
600 ˚A. Bottom: 2D electrostatic potentialimage of the same sample at height z = 20 ˚ A . The image was divided into200 ×
200 pixels. mature widely applicable microscopy technique for the areaof nanotechnology.This paper is the control engineering counterpart of [5],wherein the controller was only briefly mentioned. Wepresent in detail the tailored two-degree-of-freedom controlalgorithm that is a key enabling element that turned SQDMinto a well applicable microscopy technique. The controlleris a variant of the one presented in [8], which included aGaussian process as a feedforward signal generator. How-ever, that approach is not yet ready for the deployment a r X i v : . [ ee ss . S Y ] M a r ig. 2. Schematic of Scanning Quantum Dot Microscopy (left). The tip ofa frequency modulated non-contact AFM is decorated with a quantum dot(QD) and a bias voltage source is connected between the microscope tip andthe sample. Depending on the electrostatic potential of the nanostructure onthe sample surface and the bias voltage V b , a single electron ( e − ) tunnelsback and forth between the AFM tip and the quantum dot. The tip togetherwith the quantum dot is moved in a raster scanning pattern (right). in the experiment because further research, in particularregarding the computational issues for the online learningof the Gaussian process, has to be conducted. Henceforth,the objective of this work is to present the controller versionthat was actually used in [5]. Opposed to the initial methodof generating SQDM images based on spectroscopy grids(see [6]), the controller now allows to continuously scanthe sample, which yields order-of-magnitude faster imagegeneration and eliminates the need for spectroscopy. Thisputs it in line with other microscopy techniques like scanningtunneling microscopy [9] and atomic force microscopy [10].Furthermore, the controller now also allows to scan im-ages with highly varying electrostatic potentials. Previously,using spectroscopy grids one had to set up the spectrumrange, which increases with the electrostatic potential, beforestarting the grid. This was a huge setback in speed if theelectrostatic potential and with that the spectrum range waslarge because the entire voltage range had to be scanned.After this introduction, the paper is structured as follows.Sec. II provides the fundamentals of scanning quantum dotmicroscopy. The 2DOF control framework is presented inSec. III and thoroughly analyzed in simulations in Sec. IV.Further experimental results are presented in Sec. V and thepaper is concluded in Sec. VI.II. S CANNING Q UANTUM D OT M ICROSCOPY
This section explains the working principle of scanningquantum dot microscopy and the associated image generationprocess. The process is analyzed and a model derived thatwill be used in Sec. IV-B for simulations.
A. Working Principle
SQDM is able to measure electric surface potentials andallows to distinguish between topographical and electrostaticeffects. It utilizes a frequency modulated non-contact atomicforce microscope (NC-AFM) [10], operated in ultra-highvacuum and at a temperature of 5 K. The atomically sharptip (Fig. 2) is mounted to a tuning fork (qPlus sensor [11])that oscillates with a frequency f = f + ∆ f of around30 kHz, where f is the free resonance frequency and ∆ f Fig. 3. The spectrum ∆ f ( V b ) describes how the tuning fork oscillationfrequency changes with the bias voltage V b at one particular position inspace. The overall parabolic shape of the curve is a consequence of thetip-sample capacitance (see [18]). The SQDM specific dips result from thecharging events of the quantum dot. The voltage values where the two dipsreach their minimum are indicated by V − and V + . is the frequency change that is caused by a vertical forcegradient acting on the tip. Typically, this force is the resultof the tip-sample interaction.The AFM tip is decorated ([12], [13], [14], [15], [16])with a quantum dot (QD) , a nano-sized object whoseenergy levels can take only discrete values. Changes of theelectrostatic potential Φ s of the surface can change the QD’scharge state via gating as an electron tunnels from the tipinto the QD. This leads to an abrupt change in the tip-sample force. These tip-sample force changes are detected bythe NC-AFM, effectively transducing the information aboutthe electrostatic potential of, e.g. a nanostructure, into themeasurable quantity ∆ f . Monitoring the charging events ofthe QD while scanning the sample is the basic workingprinciple of SQDM.To detect the charging events, a bias voltage source V b is connected to the sample while the tip is grounded. Theassociated electrostatic potential Φ b that is generated by V b is superimposed on the intrinsic electrostatic surface potentialof the sample Φ s . Accordingly, a change in V b then leads toa change of the effective electric potential at the QD. If thisreaches a threshold value, a change in the QD’s charge stateis triggered.Charging of the QD leads to a change in the tip-sampleforce, whose gradient is proportional to ∆ f for small ampli-tudes of the AFM tip oscillation (see [17]). The changes ofthe tip-sample force generated by the charging events appearin the so-called spectrum ∆ f ( V b ) as features that we denoteas dips (Fig. 3). The ∆ f ( V b ) spectrum is the superpositionof a parabola ([18]) and two dips, one at negative V b valuesand one at positive V b values. The dips separate V b intervalswith different charge states of the QD.We denote the voltage values at which the dips reach theirminimum with V − and V + , or short V ∓ . These values Currently a PTCDA (Perylenetetracarboxylic dianhydride) moleculeserves as the QD. haracterize the dips’ positions within the spectrum. Bymeans of Φ ∗ ( p ) = V − · ∆ V ( p )∆ V − V − ( p ) , (1)the effective surface potential Φ ∗ at the position of the tip p = ( x, y, z ) can be calculated, where ∆ V = V + − V − and V − , ∆ V are reference points. The actual surface potential Φ s can be recovered from Φ ∗ through deconvolution in post-processing [19]. In the rest of this paper we will deal with Φ ∗ unless otherwise indicated. For more details see [19].The V ∓ ( p ) maps together with Φ ∗ ( p ) of the sample shownin Fig. 1 are depicted in Fig. 4. Fig. 4. From left to right: the V − map, the V + map, and the resultingelectrostatic potential image Φ ∗ of Fig. 1. B. Original Image Generation Process
The image generation process that had been previouslyused in [6], [7] is as follows. The sample (Fig. 1) isdiscretized in pixels (Fig. 2) and the tip with the QD ismoved from pixel to pixel. At the first pixel, a completespectrum (like Fig. 3) is measured and the positions of thedips V ∓ are determined. It has to be assumed that the intervalin which V ∓ ( p ) will change while scanning the sampleis approximately known a priori. For the following pixels,the bias voltage V b is swept accordingly within these twointervals (e.g. a voltage range of 0.2 V instead of 6 V forthe complete spectrum). This results in the measurement oflocal dip spectra. After obtaining the local dip spectra for allpixels, the V ∓ ( p ) values are determined for each pixel andused in (1) to generate Φ ∗ .The main limitation of this image generation process isthe required large total measurement time. For instance,measuring the local dip spectra takes about 3 s for eachdip and pixel for a certain V b interval size. Hence, thedetermination of the complete V ∓ maps in Fig. 4 wouldrequire 66.7 h. This severely limits the applicability ofSQDM. In particular, • the microscope is blocked for several hours for thegeneration of one image, • the longer the measurement time, the higher the proba-bility of failures, and • effects like drift increase and deteriorate image quality.Furthermore, obtaining Φ ∗ images from a grid of spectralimits SQDM in several ways: The data shown in Fig. 4 was generated by the proposed control approachin this paper, which results in significantly smaller measurement times. • Generation of fast and rough images for a first impres-sion is not possible. • Generation of images like the one presented in Fig. 1and Fig. 4 become practically impossible. • The measurement times of the local spectra increaseif the aforementioned voltage intervals, wherein thedips move, increase. This can occur, for instance, forother substrate-sample combinations with stronger elec-trostatic variations and thus, reduce the applicabilityeven further.
C. Simulation Model
The controllers, as outlined in Section III, are going tobe evaluated in simulations in Section IV. This requires aSQDM simulation model, which we derive in the following.
Fig. 5. SQDM block diagram.
Fig. 5 illustrates the SQDM system in a block diagram. Inthe experiment, the current bias voltage V b and tip position p lead to an oscillation frequency f of the tuning fork differentfrom the free resonance frequency f . Hence, V b and p arethe inputs to the system and the frequency change ∆ f is theoutput. The bias voltage V b is the control input to the systemand can be chosen freely, whereas the tip position p changesaccording to the raster scanning pattern as shown in Fig. 2.As V b changes, the frequency changes almost instantaneouslyaccording to Fig. 3. Therefore, the first block in Fig. 5 can beconsidered stationary. The relatively small frequency change ∆ f (only up to a few Hz, compared to 30 kHz of the freeresonance frequency f ) is determined from the oscillationsignal using a phase-locked loop (PLL), which is modeled asa first order system with bandwidth ω PLL and whose outputis corrupted by white Gaussian noise.To simulate SQDM and the image generation process, thespectrum ∆ f ( V b ) has to be available as an analytic function.It consists of a parabola and the two dips, which we modelas Gaussian curves . The ansatz for the spectrum is ∆ f ( V b ) = ∆ f para ( V b ) + ∆ f − ( V b ) + ∆ f + ( V b ) (2)with the parabola function ∆ f para ( V b ) = p V b + p V b + p and the Gaussian curves ∆ f − ( V b ) = d − · exp (cid:32) − (cid:18) V b − V − w − (cid:19) (cid:33) (3) ∆ f + ( V b ) = d + · exp (cid:18) − g (cid:18) V b − V + w + (cid:19)(cid:19) (4) Note that in the actual experiment, the shape of the dips is somewherebetween a Gaussian curve and a half-circle, depending on the chosen tiposcillation amplitude, the value of V ∓ , and the width of the electronic levelof the QD. See also [20]. or the dips, where d ∓ , V ∓ , w ∓ are the respective depth,position, and width of the dips. The function g ( · ) in (4) isthe polynomial g ( x ) = a x + a x + a x . The parametersin (2), (3), and (4) are then fitted using experimental data.The resulting fit in Fig. 3 and Tab. I shows that the proposedansatz is well suited to model the experimentally acquired ∆ f spectrum. TABLE I ∆ f S PECTRUM F IT P ARAMETERS c = − . a = 0 . d − = − . d + = − . c = 0 . a = − . V − = − . V + = 4 . c = − . a = 1 . w − = 0 . w + = 0 . To simulate the changing tip position, the experimentalreference data of Fig. 4 for V ∓ for every pixel is fed to (3)and (4). Thus, at each new pixel in the simulation, the dipsare shifted according to the experimental reference.III. C ONTROLLER D EVELOPMENT
The effective electrostatic potential of the sample at theposition of the tip Φ ∗ ( p ) changes in the three dimensionalspace. However, since SQDM is based on AFM, only 2Dimages can be generated. Therefore, we consider in thefollowing Φ ∗ ( x, y ) that, according to (1), depends in turnon the quantities V ∓ ( x, y ) , i.e., Φ ∗ (cid:0) V ∓ ( x, y ) (cid:1) . The V ∓ values are a priori unknown, cannot be measured directly, andchange with the tip position p = ( x, y ) . Therefore, SQDMwith the changing V ∓ values can be regarded as a parametervarying system and the objective of this paper is to design acontrol framework that automatically determines and tracksthe unknown parameters while scanning the sample.As outlined, V ∓ cannot be measured directly and a modelof the respective dynamics is unavailable because it depends,besides the scan speed, on the electrostatic potential, whichitself is unknown. Thus, a potential control approach has toadapt V b indirectly, based on a quantity that can be measured,in this case the frequency change ∆ f . Hence, we are lookingfor a control law V b ( t ) = κ (∆ f ) .In the following we present a two-degree-of-freedom(2DOF) control approach, consisting of a feedback and afeedforward (FF) part (Fig. 6). We furthermore developtwo different feedback controllers, leading to two differentversions of the 2DOF controller. The first feedback controlleris an extremum seeking controller (ESC) and the secondcontroller is called slope tracking controller (STC). Thecentral idea to both controllers is, instead of measuring thedip spectrum (time consuming, contains a lot of unnecessarydata), to track one specific reference point in each dip.A major difference between the ESC and the STC is thisreference point. While the ESC tracks directly V ∓ , the STCtracks a point on the dip’s slope (thus the name).The individual 2DOF parts will be discussed in more detailin the remainder of this section. Fig. 6. Block diagram of the closed loop: The bias voltage V b is the sumof the feedback and feedforward output. The feedforward part is influencedby the current tip position p . A. Extremum Seeking Control
The fact that V ∓ characterize the minima of the localconvex dips ∆ f ∓ satisfying V − ( x, y ) = arg min V b ∆ f − ( x, y, V b ) (5) V + ( x, y ) = arg min V b ∆ f + ( x, y, V b ) , (6)can be exploited for the determination of V ∓ . Since ∆ f is measured online by the phase-locked loop, the minimaof ∆ f ∓ can be determined continuously by appropriatelyadapting the input V b in (5) and (6) during the scanningprocess.This can be achieved by employing methods of extremumseeking control [21], [22]. As the name indicates, thesemethods are designed to find the extremum, i.e., a minimumor a maximum, of the output of a given system. Thecore principle includes a seeking element that continuouslysamples the output signal at the current operating point toobtain some kind of direction or gradient information. If ameasure of the gradient is detected, the current operatingpoint is changed accordingly. If the optimum is reached,the gradient is zero and the operating point is not changedanymore. Thus, ESC approaches are closely related to opti-mization and are therefore also called real-time optimization methods [21]. In principle, an ESC can be realized with manydifferent optimization methods, though they have to satisfya variety of additional properties, such as, for instance, lowcomputational load or the ability to deal with constraints.Works on extremum seeking date back to as early as 1922[23] or 1951 [24]. Though many other works were publishedin the second half of the 21th century (see [25]), it wasn’tuntil the paper of [26] in the year 2000 that provided thefirst general stability analysis of ESC. Since then, interest hassparked again and found its realizations in applications suchas anti-lock-breaking systems [27], [28], maximum-power-point-tracking [29], [30], source seeking [28], [31], beamcontrol in particle accelerators [32], or the control of plasmain a Tokamak reactor [33]. Theoretic extensions for discretetime systems [34], for multivariable systems [35], systemswith partial model information [25], [36], and further resultson stability [37], [32] have been presented. Generalizationsof the ESC scheme such as a unifying framework [38] andwith other optimization approaches, such as newton likeextremum seeking [39], [40], stochastic [41], [42] and aon-gradient approach [43] have been developed too. Forextensive lists of further publications see [44], [22], [32].In this work we employ an adapted version of the approachof [26] as shown in Fig. 7. In this setup we deal with a localconvex function h ( u ) for which there exists a minimum at u = u ∗ . At this minimum we have naturally d h d u (cid:12)(cid:12) u ∗ = 0 . Tosearch for u ∗ , a dither signal d ( t ) = a d sin( ω d t ) is used toperturb the current u . The resulting signal y ( t ) = h ( u ( t )) is passed through a high pass filter ss + ω H to eliminate anyconstant offsets. The filtered signal ξ ( t ) is then multipliedby the phase shifted dither signal and low pass filtered with ω L s + ω L . This leads to ξ ( t ) and it can be shown that lim t →∞ ξ ( t ) = a d h d u (cid:12)(cid:12)(cid:12) ˆ u at the current operating point ˆ u . Choosing K := − / a d , oneobtains lim t →∞ ξ ( t ) = − d h d u (cid:12)(cid:12)(cid:12) ˆ u i.e., ξ ( t ) tends to the negative gradient of h (ˆ u ) . Hence, ξ ( t ) can be used subsequently to implement a gradient descentapproach, realized by the integration of ξ ( t ) , turning ˆ u intoan estimation of the minimizer u ∗ . Fig. 7. Block diagram of the extremum seeking control approach.
Applied to SQDM we have h ( u ) = ∆ f ( V b ) as a local convexfunction and the ESC computes the derivative ∆ f (cid:48) = d∆ f d V b by modulating the dither signal d ( t ) onto the V b signal. Since ∆ f (cid:48) = 0 characterizes the dips’ minima, it also characterizesexactly the value of V ∓ . Thus, if a potential controllerregulates ∆ f (cid:48) to zero , it automatically yields V b = V ∓ .Therefore, ∆ f (cid:48) = 0 is used as the reference and the gradientdescent is achieved by using an integral controller V b,C ( t ) = K ESC (cid:90) t e d ( τ ) d τ that minimizes the error e d ( t ) = ∆ f (cid:48) ref − ∆ f (cid:48) ( t ) with ∆ f (cid:48) ref =0 and K ESC > . The voltage applied to the AFM cantileveris V b,mod ( t ) = V b,C ( t ) + d ( t ) + V b,FF ( t ) , where V b,FF is the feedforward signal computed as detailedin Section III-C. Note that the signal V b = V b,C + V b,FF The phase of ξ ( t ) is shifted by the high pass filter by φ . In order to bein phase, the dither signal that is multiplied with ξ ( t ) is shifted also by φ . Note that only one dip can be tracked at a time. without the dither signal is used for image generation. Thecorresponding block diagram is depicted in Fig. 8.
Fig. 8. Block diagram of the closed loop with the ESC.
The ESC parameters that need to be chosen are the dithersignal amplitude a d and frequency ω d , the low and high passfilter cut-off frequencies ω L and ω H , and the control gain K ESC . In the following we present some general guidelinesfor choosing these parameters based on the characteristics ofthe respective dip and the phase-locked loop.
1) Dither Signal:
The interval in which the applied biasvoltage varies locally is [ V b − a d , V b + a d ] . The gradientof the dip d∆ f d V b is then approximated within this interval,i.e., an average gradient around the current bias voltage V b is computed. A natural upper limit is therefore the dipwidth w ∓ itself. On the other hand, the smaller a d , the moreaccurate the gradient at V b . However, the ∆ f measurementsfor the gradient computation are corrupted by noise, whichposes a lower limit on a d . Thus, we get a d ≤ w ∓ , where a d should be chosen as small as possible.Regarding the choice of the dither signal frequency ω d :the higher the frequency, the faster the gradient estimateconverges but also the higher the variance of the estimate.Additionally, if ω d is much larger than the system’s band-width, the dither signal is damped and shifted significantlyand in consequence the gradient computation deteriorates.Hence, an upper bound depends on the maximum bandwidthof the system dynamics. In SQDM, the phase-locked loop isthe limiting element with its bandwidth ω PLL . We have found ω PLL ≤ ω d ≤ ω PLL to work well, where larger values decrease convergence timebut increase variance.
2) Filter Cut-Off Frequencies:
The objective of the highpass filter is to remove the constant offset of the ∆ f signal.This can be sufficiently fast achieved for ω H ≥ . ω d . The objective of the low pass filter is to smoothen thesignal ξ . The smaller ω L the stronger the smoothing. Thestronger the smoothing, the less oscillatory the gradientapproximation becomes but also the slower the convergence.ence, choosing ω L is a trade-off between convergence speedand variance, similar to ω d . We have found that . ω d ≤ ω L ≤ . ω d works well. Accordingly, ω L can be chosen within thisinterval, depending on the objective of fast convergence (fastscanning) or small variance.
3) Phase Shift:
The phase-locked loop and the high passfilter introduce an additional phase shift φ = arg (cid:0) G PLL (j ω d ) G HP (j ω d ) (cid:1) w.r.t. the dither signal, where G PLL (j ω d ) = ω d + ω PLL is thePLL transfer function and G HP (j ω d ) = j ω d j ω d + ω H the high passfilter transfer function. This can be accounted for by addingthe same phase shift φ to the dither signal that is multipliedwith the high pass outcoming signal (see Fig. 7).
4) Control Gain:
To facilitate gain tuning, we define K ESC via K ESC = k | G PLL (j ω d ) G HP (j ω d ) | , (7)where k > is the new tunable ESC gain. This redefinitionautomatically compensates the amplitude change of ξ ( t ) introduced by the PLL and the high pass filter. This way thePLL and the high pass filter can be changed without influ-encing the amplitude. Note that the low pass filter G LP (j ω d ) is not included in (7) because then one loses the possibilityof adjusting the convergence of ξ ( t ) independently of thatof ˆ u ( t ) .The larger k , the faster the convergence to the minimumbut also the more oscillatory the estimated minimizer ˆ u ( t ) = V b ( t ) . Hence, the larger k , the faster we can scan thesample but at the cost of less accurate tracking. In particular,oscillations due to high k values eventually appear in thefinal image as noise. Furthermore, if the oscillations in V b become too large, the dip might be lost. For instance, whenthe negative dip is tracked the oscillations might cause V b to leave the dip towards the left side of the dip. There thegradient descent then leads the controller to further decreasethe V b value down the parabola, making it impossible torecover the dip. B. Slope Tracking Control
Another possibility to control SQDM is by tracking apoint on the dips’ slope (Fig. 9), instead of tracking thedips’ minima. The resulting ∆ f value is then set as thereference value ∆ f ref for the whole sample and the deviations e f ( t ) = ∆ f ref − ∆ f ( t ) are used directly as an error to theintegral controller V b,C ( t ) = K STC (cid:90) t e f ( τ ) d τ (8)that adapts V b accordingly with K STC < . The resultingvoltage applied to the AFM cantilever is V b ( t ) = V b,C ( t ) + V b,FF ( t ) Fig. 9. Measured spectrum of the negative dip with controller referencepoints for ESC and STC. The dashed black line indicates that the ∆ f ref point of the STC has three crossings with the blue spectrum. Hence, threeequilibria of the closed-loop system exist for the STC. The one on the dip’sinner slope (on the right hand side for the negative dip) is the point wewant to stabilize.Fig. 10. Block diagram of the closed loop with the STC. with V b,FF computed as detailed in Section III-C. The result-ing block diagram is depicted in Fig. 10.The STC parameters that need to be chosen are the exactposition of the reference point ∆ f ref and the control gain K STC .
1) Reference Point:
The choice of the STC reference point ∆ f ref is very important. In particular because most of the ∆ f values of a dip appear three times in the local spectrumaround the respective dip (see Fig. 9). We choose ∆ f ref tolie on the inner slope of the dips (i.e., on the right slope ofthe negative dip and on the left slope of the positive dip) anddetail the reasons in the following.First note, that the inner slope is always larger than theouter slope (compare Fig. 3 and Fig. 9), which suggestsbetter control performance on the inner slope. Second, if thereference point is located on the inner slope, according to (8)the controller is able to drive V b to ∆ f ref even if V b has leftthe dips towards the vertex of the parabola (the sign of e f doesn’t change). This would not be the case if the referencepoint was chosen to lie on the outer slope and V b had left thedips towards the part of the parabola where values decreaseindefinitely. In that case the controller would drive the biasvoltage to even larger absolute values until the corresponding f ref value on the parabola is reached (crossing of the dashedblack line and the blue parabola on the left side in Fig. 9).In general, the STC won’t be able to recover the dip in thatcase.Regarding the exact position, ∆ f ref should lie relativelyfar away from the dip minimum at V ∓ because this isanother critical point for the STC. If V b moves over thispoint (e.g. V b < V − for the negative dip) the control error e f becomes smaller instead of larger, which automaticallyhas a deteriorating effect on the control performance andincreases the probability that the dip is left towards the partof the parabola where values decrease indefinitely.
2) Controller Gain:
Regarding the STC gain K STC , thesame holds as for the ESC gain K ESC . The larger K STC thefaster the convergence to ∆ f ref but also the more oscillatory.Thus, the larger K STC the faster the sample can be scannedbut at the cost of less accurate tracking.The STC does not require the computation of the derivativeand is therefore faster. However, it introduces a systematicerror due to the difference between V ∓ and the V b value atthe STC reference ∆ f ref (approximately
20 mV in Fig. 9)that we denote by e STC . Additionally, this error is notconstant and changes while scanning because when the dipchanges its position it slides the parabola up- or downwards.For further clarification imagine that the negative dip movesvertically upwards (no horizontal movement). In that case, V − does not change but V b (∆ f ref ) changes to more negativevalues, moving closer to V − . Hence, e STC decreases as thedip moves vertically upwards and decreases as the dip movesdownwards. This effect also occurs when the dip movesalong the parabola because there is always a vertical motioncomponent. In particular, the effect is larger for the positivedip because it is typically located at steeper parts of theparabola where the vertical motion is more pronounced.A comparison of the advantages and drawbacks of the ESCand STC is provided in Tab. II.
TABLE IIC
OMPARISON OF FEEDBACK APPROACHES
Advantages DrawbacksESC robust slow, many parametersSTC fast, few parameters delicate, systematic error
C. Feedforward x , y , and z direction. The z -piezo is controlled accordingto the topography feedback signal (e.g. tunneling currentin scanning tunneling microscopy or force in contact modeatomic force microscopy [45], [46]), whereas the piezos in x Since the positive dip is wider, this error is larger for the positive dip. and y direction are controlled such that the tip follows spe-cific reference trajectories in the ( x, y ) -plane that implementthe raster scanning pattern. For the main scanning direction,this is usually a triangular signal [47]. The controllers areoften based on models of the respective piezo stages. Thefeedforward part of the 2DOF controllers is therefore alsousually model based and techniques like H ∞ , l -optimal,model inversion, or iterative learning control are employed.For good overviews on this topic see [47], [48], [49], [50],[51], [52].As already detailed throughout this section, the objectiveis to track the dips’ minima (with the ESC) or a point on thedips’ inner slope (with the STC). This objective is taken careof by the respective feedback controller. However, as alreadymentioned in Section III-B, it is possible (and has occurred)that the dips change their position faster than the respectivecontroller can adapt the bias voltage, which eventually leadsto the controller “losing the dip”. A combination of a rapidchange of the electrostatic potential, a high scan speed, andthe controller dynamics is usually the cause. In that case,the scanning process has to be aborted and restarted fromthe beginning.This risk can be substantially reduced by the generationof an appropriate feedforward signal, such that the initialvalue for each controller at each tip position stays alwayswithin the dips’ interval. In that way, the feedforward hasnot only the potential for increased performance, and withthat eventually higher scan speeds, but is even essential forcorrect operation of SQDM.Since an a priori model of the electrostatic potential is notavailable, a natural choice is an approach that is based on thepreviously scanned line. Fig. 11 shows a block diagram of thefeedforward signal generator for SQDM, which comprisesthe following elements and features. Fig. 11. Feedforward block diagram. p = ( x, y ) denotes the current tipposition. Current V b values are stored in a buffer, delayed by one line andoutputted through a mean filter at the next line as the feedforward signal V b,FF . • Buffer:
The V b ( x, y ) values of the currently scannedline y are stored in a buffer alongside the indexing x values within the line. Hence, x is the fast scandirection. At the same time, already stored V b ( x, y − values of the previously scanned line y − are usedas a basis for the feedforward signal V b,FF ( x, y ) of thecurrent line. • Filter:
The measured and buffered V b values are cor-rupted by noise and small ripples caused by the ESC(if this is used for control), which have a deterioratingeffect on the control performance, especially in regionswhere the electrostatic potential is relatively flat. There-fore, while scanning the current line y , the previouslyeasured V b ( x, y − values are smoothed to generatethe feedforward signal V b,FF ( x, y ) using the mean filter V b,FF ( x, y ) = 1 n n (cid:88) i =1 V b (cid:16) x − n i, y − (cid:17) with filter window length n . • Scan speed adaptation:
The previous line y − isindexed using the measured x -position values. In thecurrent line scan, the active x -position is determined andused for picking the right reference value. This allowsfor varying scan speed within a line.This approach to generate the feedforward signal V b,FF issimple, straightforward to implement, yet practically power-ful. By adding V b,FF to V b , the controller has only to correctthe difference between the current and the previous line. Thisresults in a decreased control error and therewith a decreasedprobability that the “dips are lost”. Accordingly, the scanspeed can be increased while maintaining the same imagequality. IV. S IMULATIONS
In this section, we simulate the SQDM process togetherwith the ESC and the STC and evaluate their perfor-mances qualitatively and quantitatively. To this end, weuse the simulation model of Section II-C implemented inMATLAB/Simulink. We use a fixed-step ode3 (Bogacki-Shampine) solver with a sampling time of T s = 5 ms .Experimentally acquired spectra (Fig. 12) and the V ∓ mapsof Fig. 4 are used as reference data. The cutoff frequencyof the phase-locked loop was determined by fitting a stepresponse and was computed as ω PLL = 10 s − . The normallydistributed white noise has a standard deviation of σ n =0 .
03 Hz . The parameters of the two controllers are listed inTab. III.
Fig. 12. Experimentally acquired negative dip (top) and positive dip(bottom) with model fit.
A. Influence of the Dip Parameters
In Sec. III we have established a connection, in the form ofconstraints, between some of the controllers’ parameters andthe parameters of the dips and the phase-locked loop. Here
TABLE IIIC
ONTROLLER P ARAMETERS ( IF NOT GIVEN OTHERWISE )Negative Dip Positive DipESC ESC a d = 1 mV a d = 1 mV ω d = 4 ω PLL ω d = 4 ω PLL ω L = 0 . ω d ω L = 0 . ω d ω H = 3 ω d ω H = 3 ω d k = − · − k = − · − STC STC ∆ f ref = 1 · w − ∆ f ref = − . · w + K STC = 0 . K STC = − . we furthermore investigate how the controllers’ performanceis influenced by the dips’ depth d ∓ and width w ∓ .As can be seen in Fig. 13 and Fig. 14, the deeper andnarrower, i.e., the sharper the negative dip, the better becausethe faster the convergence. The same holds for the positivedip, as well as using the STC because sharper dips havesteeper slopes, which improve tracking. We omit the corre-sponding plots for the sake of brevity. Hence practitioners ofSQDM should aim for sharp dips. This can be achieved, forinstance, with small oscillation amplitudes of the microscopecantilever [20]. However, at the same time such amplitudesincrease the noise, which ultimately imposes a lower boundon the oscillation amplitude. Thus, the selection of theoscillation amplitude is a trade-off. Fig. 13. ESC simulations of the negative dip with gain k = 10 − . Thesimulations were performed for different dip depths m · d − with the scalingfactors m as indicated in the legend. At t = 10 s the current bias voltage V b was shifted towards the right slope of the dip such that the ESC had to regainthe minimum at V − . All lines converge to the same signal because eachsimulation used the same noise realization for the sake of better comparison.Similar results are obtained for the positive dip. B. Simulation Results
We now compare the ESC and STC with and without feed-forward using the experimentally acquired reference data. Westart with an exemplary time evolution of the involved signalsand illustrate the influence of the feedforward. Afterwards weturn our attention to the whole image generation and discussthe final images qualitatively and quantitatively. Using thesame measures, we also quantify the influence of the scanspeed on the image quality. ig. 14. ESC simulations of the negative dip with gain k = 10 − . Thesimulations were performed for different dip widths m · w − with the scalingfactors m as indicated in the legend. At t = 10 s the current bias voltage V b was shifted towards the right slope of the dip such that the ESC had to regainthe minimum at V − . All lines converge to the same signal because eachsimulation used the same noise realization for the sake of better comparison.Similar results are obtained for the positive dip.
1) Time Evolution:
In Fig. 15 and Fig. 16 exemplary timeevolutions of the V b signal for the ESC and STC withoutand with feedforward are shown. Both simulations wereperformed with a scanning time of T scan = 2 h for therespective dip map. To mimic the experiment, each line isscanned back and forth. This results in approximately
18 s foreach single line scan, which in turn equals a scanning speedof approximately 33.3 ˚A/s with the given area of 600 ×
600 ˚A.In case of the negative dip (Fig. 15), at the beginningof the depicted evolution all four controller instances areable to adequately track the dip, though the STC with aconstant error due to the fact that it tracks a point on the dips’slope. As the variations in the electric potential increase,both controllers without the feedforward are unable to fullyfollow the reference. The ESC is unable to follow the V − variations (beginning around t = 160 s ) and the STC losesthe reference around t = 340 s . In case of the positive dip(Fig. 16), the STC without feedforward again loses the dip(around t = 565 s ) and the ESC tracking results in largererrors for the shown plot as compared to its version withfeedforward. Indeed at later times, which are not depicted forreasons of clarity, the ESC without feedforward presents thesame behavior as in Fig. 15, where it is unable to follow the V ∓ variations. Hence, we can conclude at this point that withfeedforward the probability of both controllers successfullytracking the dips is significantly higher and the resultingtracking error is reduced.
2) Image Plots:
Now we turn to the simulation of thecomplete scans of the negative and positive dip maps V ∓ ( x, y ) and the resulting electrostatic potential Φ ∗ ( x, y ) computed by (1). For the sake of brevity we provide onlythe final Φ ∗ ( x, y ) images.The two resulting potential images of the ESC+FF(Fig. 17) and STC+FF (Fig. 18) look almost identical to thereference in Fig. 1 and 4. Hence, the dips are successfullytracked over the whole sample. The respective error images(Fig. 19 and Fig. 20) reveal that the errors are in the range of − to . . As the total Φ ∗ variation is . ,the relative errors w.r.t. this variation are − . to . .Furthermore, it can be seen that the STC version of the2DOF controller performs slightly better with smaller errorson average and with less noise. On the other hand, above themolecular island (right part in Fig. 20) it can be observedthat the average STC error is shifted roughly about 1.5 mVtowards more negative values. This is due to the change ofthe systematic STC error e STC as discussed in Section III-B.2. Nevertheless, the STC performs slightly better than theESC, as is also confirmed in Tab. IV, where we quantifythe image quality using the image mean-square error (MSE)and the image peak signal-to-noise ratio (PSNR) per pixelrespectively .The total scan time for each of the results in Fig. 17 andFig. 18 was 4 h. This is approximately 17 times faster thanthe original image generation process based on spectroscopygrids that would have taken 66.7 h (see Sec. II-B). TABLE IVMSE
AND
PSNR
PER PIXEL FOR F IG . 17 AND F IG . 18.MSE PSNR [ mV ] [ dB ] ESC 0.427 63.7STC 0.346 64.6
3) Influence of scan time:
Finally we investigate how theresulting image quality is influenced by the scan time T scan .To this end we repeat the simulation with scan times from 2 hto 6 h. This equals scan speeds of 33.3 ˚A/s to 11.1 ˚A/s. Afterevery simulation we compute the MSE and PSNR of theresulting electrostatic potential image and plot it over T scan .The results are depicted Fig. 21. The slower we scan, theslower the V ∓ variations and thus, the feedback controllersare better able to follow the references, which is reflected inlower MSE and larger PSNR values. Remark:
Of course, if the variability in the electrostaticpotential is large enough, there exists always a scan timethat is too small such that good tracking is not possible, evenwith the feedforward. Nevertheless, we could show in thissection that the probability to successfully track the referenceis significantly higher if the proposed feedforward signal isemployed and that reasonable scan times can be reached forlarge samples. V. E
XPERIMENTAL R ESULTS
In this section, we present information on the implemen-tation of the 2DOF controller in the experimental setup,discuss the SQDM operation procedure, and present furtherexperimentally generated images of nanoscale samples withthe presented control approach of this work. These measures are often used in imaging science as objective qualitymeasures. See, for instance, [53], [54].ig. 15. The top figure shows the reference evolution ( V − ) together with the results of the different controllers. The bottom figure shows the error.Fig. 16. The top figure shows the reference evolution ( V + ) together with the results of the different controllers. The bottom figure shows the error. A. Implementation
The 2DOF controller is built in MATLAB/Simulink andautomatically converted into C-code and then loaded to andexecuted on a ds1104 controller board by dSPACE. Thecontroller board is mounted into a standard desktop PC andconnected via analog-digital and digital-analog convertersto a Createc non-contact atomic force/scanning tunnelingmicroscope (STM/NC-AFM) that operates at and underultra high vacuum. The microscope is equipped with aqPlus sensor ([55]) tuning fork with resonance frequency f = 31 . , stiffness κ = 1800 Nm − . An amplitude of A = 0 . ˚ A was used in the measurements. B. Image Generation Procedure
The procedure used to acquire a new image is as follows:1) Move to first pixel and measure the local spectrum ofthe respective dip ∆ f ∓ ( V b ) .2) Adjust V b manually to the reference point (the selectionof the reference point depends on whether the STC orthe ESC is used, see Fig. 9).3) Start the 2DOF controller. ig. 17. Resulting image with the 2DOF controller employing the ESC.Fig. 18. Resulting image with the 2DOF controller employing the STC.
4) Start the raster scanning protocol.5) Enable the feedforward after one or more lines havebeen scanned.6) Increase scanning speed desired.7) Finish the scan for the current dip ∆ f ∓ ( V b ) .8) Goto 1) and repeat for the other dip ∆ f ± ( V b ) . C. Images
Here we show a so far unpublished result obtained with thedescribed control approach. The 2DOF controller with theSTC has been employed together with the above describedprocedure to investigate the electrostatic potential of thesample presented in Fig. 22. The resulting SQDM imageis shown in Fig. 23. The sample shows three features ona Ag(111) surface, namely a PTCDA molecule (top left), aPTCDA-Ag complex (top right), and a single Ag adatom Fig. 19. Resulting image error with the 2DOF controller employing theESC.Fig. 20. Resulting image error with the 2DOF controller employing theSTC. (bottom). VI. C
ONCLUSION
In this paper we have presented a two-degree-of-freedomcontrol approach for scanning quantum dot microscopy. Theapproach consists of a feedback part and a feedforward signalgenerator, where the latter is based on the previous linescan. For the feedback part we have presented two differentcontrollers, namely an extremum seeking control approachthat directly tracks the dip minimum and a controller thattracks a reference point on the dip slope. We have discussedthe individual working principles, respective advantages anddrawbacks, and provided guidelines for controller parame-terization tailored to scanning quantum dot microscopy.In simulations we could show how the utilization of the ig. 21. Influence of increasing scan times on the mean square error (MSE)and the peak signal-to-noise ratio (PSNR) of the resulting electrostaticpotential image.Fig. 22. Scanning tunneling microscope image of a sample with threedistinct features. Measured with a voltage of 20 mv and a tunneling currentof 50 pA. feedforward decreases the probability of losing the dips anddecreases the resulting image error at the same time. Thisleads to a several times faster image generation processand enables to scan larger images in reasonable time thanbefore, which was also verified in experiments. In addition,the presented control approach now allows to continuouslyscan a sample, which puts scanning quantum dot microscopyin line with other scanning probe microscopy techniques likescanning tunneling or atomic force microscopy.Furthermore, we could also show that the sharpness of thedips plays a central role in the control performance, whichled to the recommendation that one should aim for sharperdips if possible.The next steps are going to include an automatic gainadaptation for varying scan speeds and the improvementof the feedforward by using previous lines to generate anaccurate prediction of the current line using a Gaussianprocess and implement it in the experimental setup [8].R
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