APPLIED PHYSICS LETTERS 88, 043102 共2006兲

Signal amplitude and sensitivity of the Kelvin probe force microscopy T. Ouisse,a兲 F. Martins, M. Stark, and S. Huant Laboratoire de Spectrométrie Physique, Université Joseph Fourier, Grenoble 1 and CNRS (UMR C5588), 140, rue de la physique, B.P. 87, 38042, Saint-Martin d’Hères Cedex, France

J. Chevrier CNRS/LEPES, 25, avenue des Martyrs, 38042 Grenoble cedex 9, and European Synchrotron Radiation Facility (ESRF), BP220 38043 Grenoble Cedex, France

共Received 15 February 2005; accepted 1 December 2005; published online 24 January 2006兲 When the tip-sample distance is small, Kelvin probe force microscopy is affected by parametric amplification. This is due to the fact that the electric force has two components; the higher one having a frequency exactly twice as high as the lower. The oscillation amplitude may substantially depart from what is usually expected. Those phenomena are analytically modeled and experimentally shown, and the optimal parameter values which must be used for voltage detection are established. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2168251兴 Kelvin probe force microscopy 共KPFM兲, a useful variant of atomic force microscopy 共AFM兲, allows one to investigate surface electrostatic properties.1–6 This noncontact and dynamic AFM technique combines the ability to detect small potential variations and to investigate materials at a local and even atomic scale.3,4 Probing the nanometer range requires one to work with AFM tips maintained as close as possible to the sample. We show that in such a regime, parametric amplification phenomena emerge and may even prevail. They make the optimization of KPFM more complex than is actually believed. Hence, such effects must be carefully assessed for optimizing the measurement conditions or evaluating the ultimate KPFM sensitivity. The aim of this letter is two fold. First, we provide a comprehensive analytical framework for describing parametric effects in a KPFM configuration, which we validate through experiment. Secondly, we derive the best measurement conditions to enhance the measurement sensitivity. The experiment principle is schematized in Fig. 1. The average distance between the tip and sample z0 is usually maintained constant via a feedback control of the cantilever oscillation imposed by a piezoelectric bimorph. The electric force, Fel, results from the application of a sinusoidal electric signal V applied between the metallized tip and the substrate at a frequency ␻ = ␻0 + ⌬␻ close to the natural cantilever resonance ␻0. Although it is not ignored that in addition to the ␻ component, there is also a 2␻ force component 共Fel is proportional to V2, see, e.g., Ref. 7兲, its effects are usually neglected. However, close to the sample, this 2␻ component acts as a pump and induces parametric amplification of the ␻ component. Depending on the operating point, the oscillation amplitude and sensitivity may substantially depart from the expected ones. In AFM, or for microactuation, nonlinear and parametric phenomena are not unknown.8–11 In Ref. 11 we described them in a general case. Here, we focus on KPFM and determine the best parameter values from analytical modeling. The electric force is Fel = 共V2 / 2兲 ⳵ C / ⳵z,1 where C is the tip/sample capacitance and V = V0 + V P sin ␻t. V0 = V00 + ⌬⌽ is the sum of the offset voltage V00 and the work function a兲

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difference ⌬⌽ between the tip and sample. In the conventional modeling, the oscillation amplitude A due to Fel is proportional to both V0 and V P, and ⌬⌽ is assessed by finding the V00 value which cancels the oscillation. The sensitivity can be defined as the minimum voltage Vmin for which A exceeds the noise amplitude AN at ␻,2 A共Vmin兲 ⬎ AN, and if Vmin is small, this reduces to Vmin = AN / 共⳵A / ⳵V0兲. Hence, the best sensitivity is obtained by maximizing ⳵A / ⳵V0, with V0 close to zero, or by minimizing the whole quantity AN / 共⳵A / ⳵V0兲, if AN also depends on the parameters to be tuned. In a point-mass approximation and with a first-order development of Fel, the cantilever equation of motion is11

冉

冊

1 ⳵ Fel d 2z ␻0 dz F p共t兲 Fel共0,t兲 2 共t兲 z + = + , 共1兲 2 + ␻0 − dt m ⳵z Q dt m m where z共t兲 , k , m, and Q are the cantilever displacement, the spring constant, the mass, and the quality factor, respectively. In Eq. 共1兲, the periodic part of the electric force gradient is usually neglected. However, when the tip approaches the sample close enough, this assumption is no longer justified, and this periodic term makes the system parametric.11 Hence, the 2␻ electric component of the force gradient may induce a parametric amplification of the forced oscillation at ␻ 共degenerate parametric mode.12兲 Keeping the time varying terms

FIG. 1. Scheme of the KPFM experimental apparatus in the point-mass approximation.

0003-6951/2006/88共4兲/043102/3/$23.00 88, 043102-1 © 2006 American Institute of Physics Downloaded 24 Jan 2006 to 193.48.255.141. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

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in the force gradient and introducing normal variables, it can be shown that the oscillation amplitude at ␻ = ␻0 + ⌬␻ is given by11 A=

− 2Q␥V0V P k ⫻

冑1 + Q2关␣共V20 + V2P/4兲/k + 2⌬␻/␻0兴2 1 − Q2关共␣V2P/4k兲2 + 共␣共V20 + V2P/2兲/k + 2⌬␻/␻0兲2兴

, 共2兲

if the piezoelectric excitation and electric frequencies are incommensurate, with 2␣ = ⳵2C / ⳵z2 and 2␥ = ⳵C / ⳵z. From Eq. 共2兲, the flat-band voltage condition is still obtained by nullifying the oscillation with V0. For small V0 , V P, and ␻ = ␻0, Eq. 共2兲 reduces to the conventional amplitude Aconv = −2Q␥V0V P / k.2 However, A can also substantially differ from Aconv. In particular, for small z0 and large V P, Eq. 共2兲 gives A ⬇ −共8V0 ⳵ C / ⳵z兲 / 共3V P⳵2C / ⳵z2兲, which means that the amplitude can decrease with either V P or when the tip gets closer to the sample 共for a plane capacitor A is then proportional to z0兲. This is in complete contrast with the conventional model. Besides, from Eq. 共2兲, A becomes infinite if V P ⬎ V PC = 共4k / Q␣兲1/2 and if ⌬␻ is comprised in the interval given by ⌬␻⬁± = 关− ␣共V20 + V2P/2兲 ± 冑␣2V4P/16 − k2/Q2兴/2m␻0 . 共3兲 Of course, the actual motion remains finite and is limited by the higher-order terms neglected in Eq. 共1兲. But outside the instability domain defined by V P ⬎ V PC and Eq. 共3兲, Eq. 共2兲 is a good approximation of the real amplitude.11 The instability domain is not suited for KPFM, because spontaneous oscillations can set in easily. But getting close to it enhances the parametric effect and substantially modifies the oscillation amplitude. This is usually the case when the tip gets close to the sample, as is required for a local detection. Outside the instability domain, one can face two different situations: Either V P ⬍ V PC and ␻0 + ⌬␻⬁− ⬍ ␻ ⬍ ␻0 + ⌬␻⬁+ , or ␻ is located outside the frequency interval defined by Eq. 共3兲. In the latter case, due to the parametric effect, ⳵A / ⳵V0 does not grow indefinitely with V P, but is maximized for a value which can be straightforwardly derived from Eq. 共2兲, and is given by a fourth-degree polynomial equation. Although its analytical expression is complicated in the general case, for V0 close to zero and ␻ = ␻0, it reduces to V POPT = 2共−1 + 2 / 冑3兲1/4共k / Q␣兲1/2. This is illustrated by Fig. 2, which shows the oscillation amplitude calculated from Eq. 共2兲 in the V0 , V P plane, for a realistic parameter set and ␻ = ␻0. Both A and ⳵A / ⳵V0 are higher on the line V P = V POPT. The parametric effect induces a profound departure from the conventional analysis, for which the amplitude is proportional to both V P and V0, and the isoamplitude lines are simple hyperbolae. Hence, in contrast to a widespread belief, the sensitivity does not continuously improve with increasing. V P, as suggested by the common 共and incorrect at high V P兲 amplitude formula Aconv = −2Q␥V0V P / k. Indeed, without a careful parameter choice, the real amplitude can be either better or worse than usually expected. For realistic systems, V PC is in the range of a few volts. One can also choose to work in the potentially unstable frequency domain, below ␻0. Then, as long as V P is smaller than V PC, the frequency ␻ M which leads to a maximum amplitude A M is given as a root of a third-degree polynomial equation, derived by maximizing

FIG. 2. Oscillation amplitude in the V0 , V P plane for a cantilever + tip system and a metal plane, obtained from the parametric Eq. 共2兲; capacitance modeled 共as in Ref. 13兲. Parameters: Cone length 15 ␮m, apex radius 100 nm, cone angle 5°, cantilever area 170⫻ 30 ␮m2, tip-sample distance 40 nm, k = 2.5 N / m , Q = 200, and m = 10−11 kg. Isoamplitude lines from 1 to 25 nm, with a 1 nm spacing.

Eq. 共2兲. This complicate root is not given here, but a good approximation 共to a few percent兲 is obtained by minimizing the denominator in Eq. 共2兲. Then, ␻ M = ␻0 − ␣共V2P / 2 + V20兲 / 2m␻0 and the amplitude is A M = 8␥V0V P共V4P + V4PC兲1/2 / ␣共V4PC − V4P兲. This case is illustrated by Fig. 3, which depicts the variation of the amplitude with frequency for the parametric and conventional models, respectively. From the numerical solving of the differential equation using the full electric force expression Fel = 共V2 / 2兲 ⳵ C / ⳵z 共points in Fig. 3兲, it is clear that only the parametric approach correctly captures the cantilever motion. From the above analysis, we can now suggest optimized KPFM measurement conditions. If the noise limit is not imposed by thermomechanical noise, the fluctuations are not affected by parametric amplification, and the best sensitivity is achieved with the parameters which maximize ⳵A / ⳵V0. Hence, two KPFM measurement strategies can be considered. First, one can choose a conservative approach which avoids the instability domain, with a given tip-sample distance and ␻ close to ␻0. Then, one must apply the optimum pump voltage V POPT = 1.254共k / Q␣兲1/2, which only depends on the capacitance coupling between the probe and the

FIG. 3. Resonance curves estimated either from a numerical integration of the full differential movement equation 共points兲, the analytical parametric model 共solid lines兲, or the conventional amplitude formula 共dashed lines兲 for a tip close to the sample; plane capacitor model. Q = 300, k = 2 N / m , f 0 = 53.3 kHz, and z0 = 55 nm, capacitor area 0.1⫻ 0.1 ␮m2. Downloaded 24 Jan 2006 to 193.48.255.141. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

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FIG. 4. Experimental variation and fit to the resonance amplitude vs pump voltage, under ambient conditions. V0 = −80 mV, extracted parameters: k = 7.3 N / m , Q = 208, f 0 = 54.09 kHz; capacitance modeled 共as in Ref. 13兲 with tip and cantilever dimensions as given by the tip supplier, fitted apex radius r = 0.34 ␮m.

sample and the cantilever properties. A second strategy is to work at frequencies below ␻0 and belonging to the potentially unstable range defined by Eq. 共3兲. Then, V P must be kept below V PC, and it is preferable to work at ␻ = ␻ M . Such an approach is more difficult to manage, for one must avoid entering the instability domain, but the closer to it 共i.e., the closest is V P to V PC兲, the better the sensitivity. And, this sensitivity can be made much larger than the conventional one. However, if thermomechanical fluctuations prevail, the noise amplitude at ␻ can be represented as Ath = Ath1 sin ␻t + Ath2 cos ␻t, where Ath1 and Ath2 are random amplitudes which vary slowly with respect to ␻.9 Ath1 has the same phase as the electric signal, and is amplified with the same parametric gain G. Hence, the signal/noise ratio is not expected to change with the parametric effect, and remains the same as in a conventional analysis.2 The easiest way to experimentally show the parametric effect is to stabilize the tip close to the sample, and to measure resonance curves while varying V P. Typical results are shown in Fig. 4, obtained with an AFM setup already described in Ref. 11, and a degenerate Si substrate covered with Au. Figure 4 represents the resonance amplitude A M versus V P, for z0 = 55 nm i.e., the maximum in the experimental amplitude versus frequency curves. The variation is strongly nonlinear. Following Ref. 13, we modeled the tip-

sample with a sphere + cone geometry, in parallel with a plane capacitor corresponding to the whole cantilever. The cantilever dimensions were that given by the tip supplier. k , Q, and ␻0 were experimentally extracted. The apex size was adjusted 共typically a few tens of nanometers兲. The analytical expression 共2兲 gives an excellent fit to the data. Here, the offset from the flat-band condition is kept rather small 共V0 ⬇ −80 mV兲. Nevertheless, parametric amplification results into premature tapping around V P = 5.7 V. Without the parametric effect, tapping would have been expected at V P ⬇ 17.5 V. For distances larger than a few ␮m, the amplitude is again linear with V P. Although such points are usually not detailed in the literature, this might explain the usual practice of operating with relatively small V P values, which here is justified by the concern for avoiding premature tapping due to parametric amplification. Using small V P values reduces the sensitivity, but allows one to avoid abrupt variations in amplitude. In contrast, close to the sample, working with ␻ ⬍ ␻0 avoids the amplitude vanishing which may result from the parametric effect. From the above analysis, it is clear that for optimizing KPFM measurements close to the sample, parametric effects must be taken into account; either for avoiding or using them. 1

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