Dynamic behavior of amplitude detection Kelvin force microscopy in ultrahigh vacuum

ARTICLE IN PRESS Ultramicroscopy 110 (2010) 162–169

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Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic

Dynamic behavior of amplitude detection Kelvin force microscopy in ultrahigh vacuum H. Diesinger , D. Deresmes, J.-P. Nys, T. Me´lin Institut d’Electronique, Microe´lectronique et Nanotechnologie (IEMN), CNRS UMR 8520, B.P. 60069, Avenue Poincare´, 59652 Villeneuve d’Ascq, France

a r t i c l e in f o

a b s t r a c t

Article history: Received 20 April 2009 Received in revised form 15 September 2009 Accepted 27 October 2009

The acquisition rate of all scanning probe imaging techniques with feedback control is limited by the dynamic response of the control loops. Performance criteria are the control loop bandwidth and the output signal noise power spectral density. Depending on the acceptable noise level, it may be necessary to reduce the sampling frequency below the bandwidth of the control loop. In this work, the frequency response of a vacuum Kelvin force microscope with amplitude detection (AM-KFM) using a digital signal processing (DSP) controller is characterized and optimized. Then, the main noise source and its impact on the output signal is identified. A discussion follows on how the system design can be optimized with respect to output noise. Furthermore, the interaction between Kelvin and distance control loop is studied, confirming the beneficial effect of KFM on topography artefact reduction in the frequency domain. The experimental procedure described here can be generalized to other systems and allows to locate the performance limitations. & 2009 Elsevier B.V. All rights reserved.

Keywords: Non-contact atomic force microscopy (AFM) Kelvin force microscopy

1. Introduction Like most scanning probe imaging techniques, Kelvin force microscopy (KFM) is based on a closed feedback control loop. The objective of the control loop is the suppression of the electric field between probe and sample. The DC voltage VKelvin applied to the probe and required to cancel the field corresponds to the contact potential difference (CPD) and is imaged as function of position on the sample of the applied bias if an operating device is imaged. The misleading designation of DC component is due to the fact that typically it is a slowly varying signal compared to a superimposed AC signal that is required by the control loop to detect and cancel the mismatch CPD  VKelvin . However as soon as a sample with areas of different work functions or a biased circuit is imaged, the ‘‘DC’’ voltage is subject to time evolution, and in order to acquire an image in reasonable time or to study a biased circuit, the dynamic behavior becomes a crucial property of the setup. The KFM studied in this work uses amplitude detection of the electrostatically excited second cantilever resonance (AM-KFM) as first proposed by Kikukawa [1]. The first resonance of the probe is used for topography in non-contact frequency modulation atomic force microscopy (nc-FM-AFM). The idea of

 Corresponding author.

E-mail address: [email protected] (H. Diesinger). 0304-3991/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2009.10.016

using two resonances of a cantilever simultaneously dates back to the work of Henning et al. who used amplitude detection at both frequencies for both AFM and KFM imaging [2] in ambient conditions. Shortly thereafter, Kikukawa combined amplitude detection on the second cantilever resonance for surface potential imaging with nc-FM-AFM on the first cantilever resonance proposed by Albrecht [3] to adapt the setup to vacuum conditions. The objective of this work is the detailed study and optimization of the dynamic behavior. Therefore, the frequency response of individual components of the Kelvin loop is first measured and interpreted. In a second step, the optimization of the closed loop response allows to obtain a bandwidth of 200 Hz. Finally, the closed loop response is computed from the previously determined open loop response of the loop components. Then the noise of the output signal is studied: a power spectral density of noise of the closed loop Kelvin signal is measured and computed, showing that the shape of the noise spectrum can be precisely modeled by nothing more than the location of the main noise source and the response of the loop components. For a reasonable CPD resolution, it is found necessary to limit the bandwidth below the cutoff frequency of the control loop to reduce the integrated noise, showing that the cutoff frequency is not the only factor limiting the acquisition rate. The influence of loop design and tip parameters on the performances is discussed. Finally, the interaction between the distance control loop and the Kelvin control loop is addressed.

ARTICLE IN PRESS H. Diesinger et al. / Ultramicroscopy 110 (2010) 162–169

2. Experimental conditions

electrostatic force:

The KFM is based on an Omicron ultrahigh vacuum variable temperature atomic force microscope (UHV-VT-AFM) and uses two resonance frequencies of the cantilever. It is driven by a Nanonis scanning probe microscopy (SPM) microscopy controller entirely based on digital signal processing (DSP). The probe used in these experiments is a platinum–iridium coated Nanosensors Point Probe Plus EFM tip with a spring constant between 0.5 and 9.5 N/m, having its first and second resonance frequencies at fres1 ¼ 67:6 kHz and fres2 ¼ 420:3 kHz and respective Q-factors of Qres1 ¼ 24 500 and Qres2 ¼ 13 300. The sample is a gold coated silicon substrate (Omicron test sample). The setup is shown in Fig. 1. The AFM consists of two blocks, a phase locked loop (PLL) that measures the frequency shift of the first resonance frequency of the probe, and a distance controller. The PLL controller, consisting of a voltage controlled oscillator (VCO), phase comparator and error amplifier with individually adjustable proportional and integral gains (PI controller), mechanically excites the tip near its fundamental resonance frequency f0 . The shift Df relative to this frequency is function of the Van-der-Waals tip–sample interaction. The distance controller receives a Df signal from the PLL, compares it to a frequency setpoint and amplifies the error signal by another PI amplifier acting on the z piezo element. All KFM measurements are performed while distance control is enabled. It is detailed later that the simultaneous operation of AFM and KFM is beneficial for both control loops. The CPD is measured by the Kelvin controller. It is based on detection of the signal at the second resonance fres2 of the probe at about 6.25 times the first resonance frequency fres1 . An external lock-in amplifier (Signal Recovery, model 7280) therefore superposes a small signal AC voltage VAC on the probe and detects the real and imaginary projections of the tip oscillation at this frequency. The error signal of the surface potential is the imaginary projection of the tip deflection at fres2 caused by the

F¼

In

L o c k − in

163

1 dC ðVAC cosð2pfres2 tÞ  CPDÞ2 2 dz

only the mixed term at fres2 , Fres2 ¼ 

dC VAC CPD cosð2pfres2 tÞ dz

3. Frequency response characterization In the following, it is shown how the dynamic response of a control loop is measured by a lock-in amplifier. Then, the Kelvin loop is decomposed into two blocks with a forward and a feedback gain A and F respectively. These are studied individually while the loop is opened (open switch in Fig. 1). Next, the loop is closed again and the missing feedback PI parameters are determined empirically to obtain an optimized closed loop response. Finally, from the known open loop responses of both blocks, including the previously obtained PI parameters, the closed loop response is computed and compared to the measured spectrum.

PI A mp lifier

A −D Conv erter

Vkelvin

op en closed

Osc

VPD

Kelvin Controller

Osc

+ +

PI A mp l

M ag Out L o c k − in

ϕ Out

"Δf"

fres1= 67 .6 kHz

Dith er

Σ

=

PI A mp l

Laser Diode Photodetector In

ð2Þ

is retained since the static term and the one at 2fres2 are far from any resonance frequency and cause negligible tip deflection. The tip is excited to oscillate at its resonance by Fres2 , its excursion is detected optically and demodulated with respect to fres2 by the lock-in amplifier. The demodulated oscillation then undergoes analog to digital (A-D) conversion and is amplified by the PI controller which then adjusts the tip voltage. The CPD and the feedback voltage VKelvin are often called ‘‘DC’’ voltages in contrast to the high frequency AC bias used for tip excitation. However the CPD is varying as the sample is scanned. If the control loop is closed and properly configured, it should ideally track the CPD, corresponding to a closed loop gain near unity.

Y

fres2= 420 .3 k Hz V AC

ð1Þ

PLL Controller

Mag i nput Outp ut

CPD

VCO

f input

Vpert

PI A mp l Z piezo

f S etp oin t Distance Controller

Fig. 1. (Color online). Setup of AM-KFM. Distance control and topography imaging is achieved by the PLL controller and distance controller (blue middle block and green lower block, respectively). CPD tracking is performed by the Kelvin control loop (red upper block), consisting of a lock-in amplifier, A-D converter, PI controller, and signal adder. The lock-in amplifier and signal adder of the Kelvin loop are stand-alone units while the PI controller preceded by a sample-and-hold AD converter input is part of the digital SPM controller. The Kelvin loop can be opened by a switch.

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ΔY (static) or YN (noi se PS D)

3.1. Measurement of the closed loop response To characterize the dynamic behavior of the Kelvin loop, a varying CPD is simulated by adding a perturbation to the surface potential. Therefore a voltage Vpert cosð2pfpert tÞ is added to the sample as shown in Fig. 1, with fpert ranging from zero to 300 Hz. The response of the Kelvin controller to the perturbation signal is monitored using a lock-in amplifier. Scanning is stopped so that the CPD itself is indeed a real DC component and has no effect on the result of this dynamic study. An important feature of the DSP controller is an integrated lock-in module that can be configured for measuring the transfer function between any two signals. It is used for all dynamic response measurements in this work. An example of a closed loop response is shown in Fig. 2. The Bode diagram of the KFM signal allows to extract the following information:

Vpert V

+ −

kelvin

op en

=

Block A Forward gain

Y

closed

Block F Feedb ack gain Fig. 3. (Color online). Schematic diagram of the AC study of the KFM feedback loop. Only the AC signal component with fpert time dependence is considered (CPD ¼ 0 in Fig. 1). The transfer function of block A is defined as the ratio of the output signal y of the lock-in amplifier to the Vpert  VKelvin . Block A consequently contains the electrostatic excitation via tip–sample interaction, tip transfer function, the optical detection, and the demodulation by the 7280 lock-in amplifier. Block F contains the A-D conversion at the analog input of the controller and the PI amplifier. A noise source is included as voltage generator at the lock-in output.

 the static gain at low frequency which should be close to unity for ideal tracking;

 the harmonic overshoot for control loops of order Z 2;  the bandwidth determined by the cutoff frequency fc at 3 dB;  the order of the system determined from the phase rotation Df (red dotted arrow) or from the slope of the magnitude. 3.2. Decomposition into a generalized feedback scheme, open loop responses In Fig. 3, the setup of Fig. 1 is described by a general closed loop scheme consisting of blocks A (forward gain) and F (feedback gain). The description is adapted to the dynamic response and noise study. The real DC component CPD is neglected, and instead one is interested in the response of the output signal VKelvin at the frequency fpert of the perturbation signal Vpert to which it responds. The decomposition of the components of Fig. 3 into blocks A and F is arbitrary, since A and F are connected in sequence and the Kelvin output signal is the output of F. Only the product of the gains A and F will appear in the expression of the closed loop gain. Here, the delimitation between blocks A and F is chosen behind the lock-in output Y. Consequently, the open loop gain A is defined as the transfer function between Vpert and the lock-in output signal Y in Fig. 1. Among the components of Fig. 1, it contains the lock-in amplifier that generates VAC

Fig. 4. (Color online). Open loop response of block A, transfer function between a gap voltage modulation and the output voltage of the Kelvin lock-in amplifier, measured with AC excitation voltage VAC ¼ 200 mV.

with its oscillator, the electrostatic tip-sample interaction, the tip transfer function (deflection divided by force), the photodetector and the demodulation by the lock-in amplifier. Block F contains the A-D conversion and the PI amplifier. To measure the dynamic response of block A alone, the loop is opened by the switch, a perturbation voltage Vpert cosð2pfpert tÞ is added to the sample, and the output Y is monitored with a lock-in amplifier using fres2 as reference. To treat the simulated surface potential variation analytically, Vpert cosð2pfpert tÞ has to be inserted into Eq. (1) instead of the CPD to obtain the electrostatic tip excitation force. Eq. (2) describing the fres2 force component becomes Fres2 ¼ 

Fig. 2. (Color online). Typical closed loop transfer function illustrating the information that can be obtained: under-unity gain, harmonic overshoot, cutoff frequency, and phase shift.

dC VAC Vpert cosð2pfpert tÞ cosð2pfres2 tÞ dz

ð3Þ

It is known that the Q-factor of the cantilever introduces a decay time constant and consequently a lowpass behavior of block A with a cutoff frequency fc ¼ fres =ð2Q Þ. Therefore, the deflection of the tip resulting from the modulated excitation force is also an oscillation at fres2 and modulated at fpert , but the modulation function is subject to lowpass filtering. After demodulation of the tip oscillation by the Kelvin lock-in amplifier, the Y

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signal is YðtÞ ¼ Vpert VAC

dC Q HPD ðfres2 ÞHlockin dz k

1 expð2pif pert tÞ 2iQf pert 1þ fres2

ð4Þ

HPD is the deflection to voltage conversion gain of the photodetector, and Hlockin the gain (fullscale output divided by input range) of the Kelvin lock-in amplifier. The gain of the lock-in amplifier used to demodulate the oscillation is independent of fpert since its output bandwidth is chosen much bigger than fpert . The gain of block A is consequently Aðfpert Þ ¼

Y dC Q ðfpert Þ ¼  VAC HPD ðfres2 ÞHlockin Vpert dz k

1 2iQf pert 1þ fres2

ð5Þ

For a given cantilever, the higher order flexural modes n lead to higher lowpass cutoff frequencies fc;n ¼ fres;n =ð2Qres;n Þ since the Q-factor of these modes is smaller and simultaneously their resonance frequency is higher. Therefore it may be interesting to use the higher flexural modes particularly in vacuum where the Q factors are important [4].

165

the transfer function between the lock-in output Y and VKelvin . Block F contains the A-D conversion HAD ðfpert Þ at the analog input of the DSP controller and the gain HPI ðfpert Þ of the PI controller: Fðfpert Þ ¼

VKelvin ðfpert Þ ¼ HAD ðfpert ÞHPI ðfpert Þ Y

To determine HAD ðfpert Þ of the SPM controller, the controller is characterized by its own lock-in module. A sinusoidal signal is generated on one of the analog outputs, which is connected to the analog input of the controller so that the signal undergoes the A-D conversion. The frequency fpert of the test signal is swept from 5 Hz to 5 kHz. The result is shown in Fig. 5. The static gain is 1.12 instead of unity. It is found that the phase rotates without limitation as the frequency is increased above 100 Hz, and the amplitude decreases with a slope that is increasing even on a logarithmic scale. Curve fitting of Fig. 5 (not shown) proves that the behavior is perfectly described by a sample-and-hold circuit operating at 10 kHz and that introduces an additional signal processing delay of 0.6 ms. It is also preceded by a 5 kHz second order anti-aliasing lowpass filter. The amplitude of the fitting transfer function is 3

2 6 6 HAD Fðfpert Þ ¼ 0:872  6 6 4

7   7 1 1 7  sinc pfpert pffiffiffi p ffiffiffi þ 27 2 5 kHz fpert ð10 kHz  fpert Þ 5 2if pert 2ið10 kHz  fpert Þ 1þ  1þ  5 kHz 5 kHz ð5 kHzÞ2 ð5 kHzÞ2

Fig. 4 shows the measured gain of block A, i.e., the transfer function between the applied perturbation voltage Vpert and the Y output voltage of the Kelvin lock-in amplifier (imaginary projection of the tip oscillation with respect to fres2 ) as function of the frequency fpert . A first order lowpass with a steady state gain jA0 j ¼ Y=Vpert ¼ 0:5 and a cutoff frequency of 16 Hz is found to fit the observed transfer function with good agreement (fitting function not shown): Aðfpert Þ ¼

Y ðfpert Þ ¼ A0 Vpert

ð7Þ

1 1 ¼ 0:5 2iQf pert if pert 1þ 1þ fres 16 Hz

The pre-factor of the fitting function is 0.872 whereas the static gain of Fig. 5 is 1.12 because some aliasing takes place despite the anti-aliasing filter. The first term in the square brackets is a second order low-pass filter with Butterworth characteristic and cutoff at 5 kHz, the second term is the contribution of the first aliasing introduced by sampling at 10 kHz. At zero frequency, the sum of both terms is 1:12=0:872. In good approximation, the contribution of further aliasing is neglected. The phase of the transfer function is

ð6Þ

The observations are in agreement with Eq. (5). fc ¼ fres2 =ð2Q Þ ¼ 16 Hz is associated with the high quality factor Q ¼ 13 300 at the second resonance of the tip in vacuum. With the preceding definition of block A, block F is consequently defined as the remaining part of the loop, representing

ð8Þ

fF ðfpert Þ ¼  360  fpert  0:6 ms þ

180

p

 arctan

  Im½  Re½ 

ð9Þ

where [ ] denotes the square brackets of Eq. (8) and represents the phase shift caused by the sample and hold treatment. The first term of Eq. (9) is due to an additional signal delay of 0.6 ms. The 3 dB cutoff frequency of the amplitude is about 2 kHz. The PI controller by definition has the response: ! 1 1 ¼ P 1þ HPI ðfpert Þ ¼ P þI 2pif pert 2pif pert t

ð10Þ

In the present setup, the integral amplification is chosen via a time constant t instead of the parameter I. 3.3. Empirical optimization of the feedback PI parameters

Fig. 5. (Color online). Response of the A-D input stage of the Nanonis controller.

Since the Kelvin control loop contains the first order lowpass of block A, the complicated A-D conversion gain and the PI controller, we do not consider determining the feedback parameters analytically. For a second order system consisting of two lowpasses, an analytical treatment would be straightforward, but we do not know about a comparable case in the literature that gives analytical expressions for the optimized feedback

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parameters of a closed loop similar to the present case. Consequently the PI parameters are determined empirically by changing P and t until the best response according to Fig. 2 is obtained. The objective of the empirical parameter sweep is a flat response with maximum cuttoff frequency. The resulting closed loop transfer function is shown in Fig. 6, black curve, for P ¼ 11 and t ¼ 60 ms.

3.4. Computation of the closed loop gain Even though it seems difficult to derive expressions for the PI parameters of the optimized loop, it is nevertheless possible to compute the gain of a closed loop for a particular set of PI parameters and with the known responses of all other components. Generally for a feedback loop consisting of blocks A and F according to Fig. 3, when the switch is closed and the noise source is neglected, the output voltage is VKelvin ¼ ðVpert  VKelvin Þ  AF

ð11Þ

By resolving first for VKelvin and dividing by Vpert , the closed loop gain is HCL ðfpert Þ ¼

Aðfpert ÞFðfpert Þ VKelvin ¼ Vpert 1þ Aðfpert ÞFðfpert Þ

ð12Þ

The closed loop gain HCL is computed from the complex expressions A and F including the empirically determined values of P and t. The result is plotted in Fig. 6 (red dotted curve). It is in good agreement with the measured gain and validates the preceding description of the components of the loop. The cutoff frequency is fc ¼ 200 Hz. It is far above the cutoff frequency of the open loop gain A. It is common to have a closed loop cutoff frequency above the lowest open loop response cutoff frequency. This is most often the case in AFM distance control loops where the same Q-factor induced lowpass behavior occurs. The limitation of fc to 200 Hz can be attributed to the transfer function of the A-D conversion: if the control loop was a second order system consisting only of the Q-factor induced lowpass and the error amplifier, its cutoff frequency could be set to an arbitrary value, irrespective of the cutoff frequency of block A which is here as low as 16 Hz. Newer systems are less limited since due to the availability of more performant acquisition cards, analog signals may be sampled at a much higher rate, or several lock-in amplifiers may be controlled digitally, making sampling of analog inputs obsolete.

4. Sampling rate limitation due to noise The sampling rate is limited not only by the cutoff frequency of the Kelvin closed loop transfer function but also by the noise level that one is willing to accept after sampling the signal into image pixels. Depending on the acceptable noise in the Kelvin signal, which is the integral of the noise power spectral density (PSD) over the bandwidth, the output bandwidth and sampling rate may have to be reduced below the closed loop cutoff frequency, or system design and tip parameters might be adapted to reduce noise.

4.1. Origin of noise and its propagation through the control loop In the following, the origin of noise, its propagation in the closed loop and the influence of loop components on the noise PSD of the closed loop Kelvin signal are discussed. Therefore, the noise is measured in open loop at the photodetector output and at the lock-in output, and in closed loop in the Kelvin output signal. The measured closed loop Kelvin noise PSD is compared to a noise spectrum computed from the known open loop transfer functions A and F and assuming that the previously measured noise PSD at the output of block A is the main noise source. In open loop, when measuring a PSD of the tip deflection in the vicinity of the second resonance without electrostatic tip excitation, a constant background of tip deflection is obtained in the range fres2 7500 Hz. For thermal tip excitation, the force PSD would be constant, and the deflection PSD would have the shape of the resonance curve. However, in our microscope, thermal excitation of the second resonance frequency could never be observed because the resonance peak is buried in a constant background deflection PSD. Consequently, the background is photodetector and preamplifier noise rather than thermally excited tip pdeflection. A constant deflection noise PSD of ffiffiffiffiffiffiffi DN ¼ 13 mV= Hz has been measured with the built-in spectrum analyzer. With a photodetector conversion rate of 100 nm/V, the noise can be calibrated to tip deflection in terms ofpdisplacement rather ffiffiffiffiffiffiffi is a typical than output voltage, yielding 1:3  1012 m= Hz. This p ffiffiffiffiffiffi ffi value even though noise PSDs lower than 1014 m= Hz have been reported in the literature [5]. At the output of block A, after demodulating the photodetector output with the Kelvin lock-in amplifier using fres2 as reference, a nearly constant noise PSD YN is found at frequencies between zero and 500 Hz with a value

mV YN ¼ 350 pffiffiffiffiffiffiffi  Hlockin  DN : Hz

ð13Þ

The white noise DN at the output of the photodetector in the vicinity of fres2 7 500 Hz becomes, after demodulation with fres2 as reference, white noise YN from zero to 500 Hz. The gain Hlockin ¼ ðfullscale output=input rangeÞ ¼ 2:5 V=100 mV applies between the noise PSDs at the input and output of the lock-in amplifier which is approximately verified here. This noise PSD YN has been directly measured with the integrated Nanonis spectrum analyzer and is shown in Fig. 7. In the following, it is assumed that this output noise of block A is the main noise source of the Kelvin loop. It can be represented as a voltage noise generator that adds noise to the output of block A. To model its effect on the Kelvin signal, an error source DY is included in Fig. 3 and the reasoning of Eqs. (11)–(12) is repeated. Eq. (11) then becomes Fig. 6. (Color online). Closed Kelvin loop transfer function with P ¼ 11 and t ¼ 60 ms, black curve: measured with Vpert ¼ 200 mV; red dotted curve: modeled.

½ðCPD  VKelvin Þ  A þ DY  F ¼ VKelvin

ð14Þ

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If resolved for VKelvin , it gives VKelvin ¼ CPD

AF F þ DY 1 þ AF 1 þ AF

ð15Þ

To study the effect of noise, pffiffiffiffiffiffiffi Eq. (15) is rewritten in terms of noise PSD YN in units ½V= Hz: the PSD of noise in the Kelvin signal, VKelvin;N ðfpert Þ, as function of the PSD of noise of block A, YN becomes VKelvin;N ðfpert Þ ¼ YN

Fðfpert Þ 1 þAðfpert ÞFðfpert Þ

ð16Þ

With this expression, the noise PSD of the closed loop Kelvin signal can be computed from the constant noise PSD YN and the known transfer functions of the blocks A and F, similar to the computation of the closed loop gain. For the case where jAFjb 1, the Kelvin noise PSD VKelvin;N ðfpert Þ can be approached: VKelvin;N ðfpert Þ 

YN jAðfpert Þj

ð17Þ

Fig. 8 shows the Kelvin closed loop PSD of noise VKelvin;N , as measured by the Nanonis integrated spectrum analyzer, and the noise PSD computed precisely according to Eq. (16) and with the approximation of Eq. (17) for jAFj b1. The good agreement between the measured and the precisely computed PSD supports the previous assumption that the main noise source is the white noise (i.e., constant PSD from zero to

167

fpert ¼ 500 Hz) at the output of block A, originating from the deflection noise PSD that is nearly constant in the vicinity of fres2 7 500 Hz. The approximation of Eq. (17) is indeed valid in the range where jAFjb 1, which coincides with the range where the closed loop gain has near unity gain according to Eq. (12) and Fig. 6. Beyond the cutoff frequency of the closed loop Kelvin response, the noise PSD decreases again. However this decrease cannot be exploited to limit the total output noise since at frequencies above the cutoff frequency, the gain is considerably under-unity, and therefore the simplified approach of Eq. (17) is sufficient for the subsequent discussion. The total noise is the square root of the squared Kelvin noise PSD of Eq. (17) integrated over the bandwidth B: "Z  #0:5 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B YN 2 S¼ df pert ð18Þ /VKelvin jAðfpert Þj 0 If the full bandwidth from zero to the cutoff frequency 200 Hz of the closed loop response is exploited pffiffiffiffiffiffiffi and a constant (frequency independent) value YN ¼ 350 mV= Hz is used for the bandwidth, and if the numerical expression of Eq. (6) is used for the gain A, the noise integrates to 62 mV. It is obvious from the slope of the PSD of noise (Fig. 8) that this integral can be reduced considerably by reducing the output bandwidth and sampling rate, implying lower scan speeds. 4.2. Effect of system design Concerning the improvement of system design to decrease the Kelvin noise PSD, according to Eq. (17) the gain of block A has to be increased or the white noise at the output of block A has to be reduced. Inserting the expression of A of Eq. (5) into Eq. (17), one obtains VKelvin;N ðfpert Þ 

    YN ðfpert Þ  YN ðfpert Þ k i2fpert Q dC  ¼ 2VAC  1þ jAðfpert Þj HPD ðfres2 ÞHlockin Q fres2 dz 

     DN ðfres2 Þ k i2fpert Q dC  2VAC    1þ HPD ðfres2 Þ Q fres2 dz 

Fig. 7. Noise PSD of the output signal of block A.

ð19Þ

The dominant noise source is expected to be the photodetector or its preamplifier. The most simple means of increasing the gain of A, e.g. by increasing HPD with another amplifier or increasing the gain Hlockin , would therefore amplify the noise as well. The objective is to increase the conversion gain of the photodetector in the stages preceding the noise source. The ratio of the intrinsic detector noise PSD divided by the conversion gain of the photodetector corresponds exactly to the noise expressed in pffiffiffiffiffiffi ffi terms of m= Hz. The minimization of this value is manufacturer dependent but the user can for example change the light source of the photodetector or assure that the highest possible light intensity is used which in our measurements might not have been the case. It is obvious that a higher electrostatic excitation bias VAC increases A and reduces the Kelvin noise. Here we have an AC excitation voltage of 200 mV. Increasing the AC bias is however limited by surface potential errors due to asymmetric band bending on semiconductor surfaces [6] and undesirable effects on the distance control loop (tip retraction, snap to contact). Consequently, fixing the AC excitation voltage to a certain value is the base of fair comparisons between different implementations. 4.3. Effect of the probe parameters

Fig. 8. (Color online) Kelvin PSD of noise VKelvin;N : measured (black), computed according to Eq. (16) (red dashed), and according to Eq. (17) (green dotted).

Another possibility is to focus on the probe parameters. Since the Kelvin PSD is frequency dependent, we first calculate the mean square deviation of the Kelvin voltage by integrating the square of the PSD (Eq. (19)) over the bandwidth B, before

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considering how this value can be reduced by the probe parameters. ! ! Z 2 4fpert Q2 k2 B k2 4 B3 Q 2 /ðDVKelvin Þ2 SðBÞp 2 df 1þ ¼ B þ pert 2 2 3 fres2 Q 0 Q2 fres2 pffiffiffiffiffiffiffiffiffiffi If fres2 is replaced by k=m, this yields  2  k 4 /ðDVKelvin Þ2 SðBÞpB þ B2 km 2 3 Q

ð20Þ

ð21Þ

This shows that depending on the bandwidth the first or second term of the sum is dominant. In the first case it would be favorable to use a softer cantilever and increase the Q-factor, whereas in the second case it should be soft and have little mass irrespective of its Q-factor. However these indications are only valid as long as these parameters can be changed independently which is certainly not the case in a wide range. These parameters might influence each other and even have an influence on the gain A by modifying the pre-factor dC=dz or by imposing a reduction of VAC to avoid snap-to-contact if the cantilever is made very soft, or they might have an effect on the lateral resolution. However the minimization of Eq. (21) can be performed by probe developers who are aware of the mutual dependencies of the probe parameters. Detailed cantilever design optimization is therefore beyond the scope of this work. The important aspect is that the result is slightly different from the conclusions that one would draw if thermal cantilever excitation was considered as the main noise source. Thermal cantilever excitation appears as a frequency independent force PSD: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4kkB T ð22Þ FN;th ¼ ores2 Q This force PSD can be directly converted into Kelvin noise PSD by dividing by the frequency independent coefficient 1=2VAC dC=dz that converts the CPD into electrostatic excitation force. Integrating the square of the Kelvin noise PSD over the bandwidth gives the mean square deviation of VKelvin : ! pffiffiffiffiffiffiffi  2 kmkB T 4kkB T 1 dC 2 /ðDVKelvin Þ SðBÞ ¼ B pB VAC ð23Þ ores2 Q 2 dz Q The result (second expression) shows that noise optimization in the case of dominating thermal cantilever noise requires softer and lighter cantilevers with higher Q-factors, irrespective of the bandwidth, since the bandwidth appears only to the power 1 in contrast to the detector noise treatment. In both cases, dominant thermal noise or dominant detector noise, the mutual dependencies between cantilever parameters k, m and Q need to be known if Kelvin noise is to be reduced by modifying the cantilever.

5. Interference between the distance and the kelvin control loop In AM-KFM, there are two points of interactions where the distance control loop and the Kelvin control loop might interfere: the first is the electrostatic excitation of the cantilever, the second is the influence of an uncompensated tip–sample electric field that might cause the distance controller to withdraw the tip and introduce topography artefacts. 5.1. Parasitic mechanical tip excitation at the second resonance frequency by capacitive crosstalk Concerning the first point, it has been found that AM-KFM setups using a self-oscillating loop to mechanically excite the first

resonance of the tip have a major drawback: if a capacitive crosstalk couples some of the electrostatic excitation signal at fres2 to the photodiode, the self-oscillating loop feeds back this parasitic signal to the piezo dither. Therefore the excitation at fres2 is partly electrostatic and partly mechanical, leading to wrong KFM measurements. Either the microscope cabling has to be modified [6] or an external active crosstalk compensation as we suggested earlier [7] can be used. The advantages of tip excitation by a PLL controller versus a setup in which the tip is part of a self-oscillating loop have been shown in the past [8], and in the particular case of AM-KFM on the second cantilever resonance, resolve completely the problem of capacitive crosstalk. By using a PLL controller, the problem is avoided since the mechanical excitation is driven by a clean sinewave of the VCO and no longer part of the self-oscillating loop subject to injection of the electrostatic excitation signal due to stray capacitance. Hence, excitation at the second resonance is caused purely by electrostatic tip–sample interaction and therefore proportional to VKelvin  CPD.

5.2. Surface potential induced topography artefacts: benefit of Kelvin control on topograpy imaging A major criteria of KFM implementations is the interaction between potential and topography imaging. On one hand, the application of the electrostatic excitation signal may itself introduce artefacts. It is reminded that the application of an AC voltage at fres2 to the tip introduces additional force gradients at f ¼ 0, f ¼ fres2 , and f ¼ 2fres2 . The term at zero frequency cannot be avoided but kept small enough to cause a negligible tip retraction by keeping the excitation voltage low ðVpert ¼ 200 mVÞ and the topography feedback setpoint hard enough. The terms containing the AC electrostatic tip excitation frequency certainly cannot produce topography artefacts since they are several orders above the cutoff frequency of the distance controller. On the other hand, varying surface potential is a problem in bare AFM setups since it creates topography artefacts. This might be reduced by simultaneous KFM imaging since the Kelvin controller applies a voltage to the tip to track surface potential variations. Here in the context of a study of dynamic behavior, we are interested in this beneficial effect of simultaneous KFM. Therefore the effect of the perturbation voltage Vpert on tip retraction is studied by applying a perturbation voltage to the sample and measuring the transfer function between Vpert and the z signal with disabled and enabled Kelvin control. Since the electrostatic force gradient to which the probe is exposed is proportional to the square of the perturbation voltage, the distance control loop is expected to retract the tip at twice the frequency of the applied perturbation, and consequently the tip retraction is demodulated at 2fpert . Fig. 9 shows the cross transfer function between the perturbation of 1 V amplitude and the z signal at 2fpert , without Kelvin controller (upper curve) and with Kelvin control enabled (lower curve). With enabled Kelvin controller, the response of the distance controller to the perturbation voltage is efficiently reduced. The surface potential induced artefact on topography imaging is reduced by a factor up to 100 at low frequency by enabling the Kelvin control loop. Such a beneficial effect has been mentioned in the previous works [9] but demonstrated in the time domain only. It is noteworthy that without Kelvin control, the cross transfer does not have a plateau at low frequencies. As explanation, we emphasize that the transfer function of the AFM distance controller Dz=Dz had been optimized before the perturbation was applied and had a plateau and a cutoff frequency of about 100 Hz. However, when the large perturbation voltage of 1 V amplitude is applied, the tip retracts from the

ARTICLE IN PRESS H. Diesinger et al. / Ultramicroscopy 110 (2010) 162–169

Fig. 9. (Color online). Comparison of the cross transfer between a perturbation of the surface potential and the response of the distance control with and without Kelvin controller enabled (red dotted and black solid curve respectively).

surface, and its increased mean distance reduces the gain of the distance controller. As a result, the system does not have the ‘‘ideal’’ behavior any more (ideal in the sense of an optimized control loop) but has two distinct poles, the first of which is below 3 Hz so that a decreasing response is seen from the beginning of the scale. Without Kelvin control, the cross transfer function is huge at low frequency and scanning a surface with varying CPD would introduce undesirable artefacts, whereas with enabled Kelvin control they would be efficiently suppressed.

6. Conclusions The dynamic behavior of an AM-KFM was studied by measuring its response in closed loop, and agreement was found with the computed response based on transfer functions of individual components acquired in open loop. The optimization of the feedback parameters has allowed to obtain a cutoff frequency of 200 Hz. This particular setup is limited by the 10 kHz A-D conversion between an external lock-in amplifier and the digital SPM controller. Ideally, if the Kelvin loop was a second order system consisting only of the lowpass imposed by the probe response and a PI amplifier, the cutoff frequency of the closed loop could be set to an arbitrary value. The cutoff frequency of the closed loop however does not present a bottleneck in this setup: if the full bandwidth was used, the mean deviation of the Kelvin signal would be 62 mV

169

due to noise propagation. It is likely that one would deliberately choose a lower bandwidth and sampling frequency to reduce noise. In the present case the main noise source is white noise at the photodetector output. Its propagation through the control loop has been modeled and leads to a shape of the noise power spectral density of the Kelvin voltage which is increasing above the frequency reciprocal to the cantilever decay time constant. This is in contrast to the noise introduced by thermal cantilever excitation which is negligible here but which would convert to white noise in the Kelvin signal. Some indications are given on noise optimization: first, the photodetector should have a maximum signal to noise ratio which is equivalent toffi pffiffiffiffiffiffi minimizing its noise in terms of deflection expressed in m= Hz. Second, it is shown how noise can be reduced by the probe parameters. Furthermore, the study demonstrates that the Kelvin control loop efficiently suppresses topography artefacts introduced by varying surface potential. The study is of general interest to the nearfield microscopy community: the method of decomposing a closed control loop, measuring the gain of individual components in open loop configuration and modeling the closed loop gain as well as noise propagation, is universal and can be applied to most scanning probe microscopes and other control loop applications.

Acknowledgments This work was achieved with the financial support of the European Union, the French Government and the Regional Council. Further financial aid from the French Agence Nationale de Recherche (Grants JC-05-46152 and ANR-06-NANO-070-05) is gratefully acknowledged. We thank Dr. Romain Stomp from Nanonis GmbH for fruitful discussions.

References [1] [2] [3] [4] [5] [6]

A. Kikukawa, S. Hosaka, R. Imura, Appl. Phys. Lett. 66 (1995) 3510. A.K. Henning, T. Hochwitz, J. Appl. Phys. 77 (1995) 1888. T.R. Albrecht, P. Gruetter, D. Horne, D. Rugar, J. Appl. Phys. 69 (1991) 668. P. Girard, M. Ramonda, R. Arinero, Rev. Sci. Instrum. 77 (2006) 96105. T. Fukuma, Rev. Sci. Instrum. 80 (2009) 23707. C. Sommerhalter, T.W. Matthes, T. Glatzel, A. Jager-Waldau, M.C. Lux-Steiner, Appl. Phys. Lett. 75 (1999) 286. [7] H. Diesinger, D. Deresmes, J.-P. Nys, T. Melin, Ultramicroscopy 108 (2008) 773. [8] B.I. Kim, Rev. Sci. Instrum. 75 (2004) 5035. [9] D. Ziegler, J. Rychen, N. Naujoks, A. Stemmer, Nanotechnology 18 (2007) 5.