Bias Potential for Tip--Plane Systems in Kelvin Probe ... - IOPscience

Japanese Journal of Applied Physics 49 (2010) 025201

REGULAR PAPER

Bias Potential for Tip–Plane Systems in Kelvin Probe Force Microscopy Imaging of Non-uniform Surface Potential Distributions Krzysztof Sajewicz, Franciszek Krok, and Jerzy Konior Marian Smoluchowski Institute of Physics, Jagiellonian University, 30-059 Krako´w, ul. Reymonta 4, Poland Received August 1, 2009; accepted November 20, 2009; published online February 22, 2010 The bias potential, Vbias , is the key quantity for the Kelvin probe force microscopy (KPFM) measurements and interpretation. Using an efficient method for electrostatic force determination, Vbias has been calculated for tip–plane systems, with realistic tip geometry and for non-uniform potential distributions on the plane. The considered potential distributions on the plane include a potential step, a quadratic potential island, and two quadratic potential islands with varying separation. Vbias has been evaluated along three different schemes, i.e., from the minimization of electrostatic force, from the force gradient, and from the integral formula. We have studied Vbias as a function of tip–surface distance, island size, vibration amplitude, and tip sharpness radius (the so called nanotip). We have found that there are substantial differences between the gradient and integral schemes for Vbias evaluation. We have determined that the nanotip presence favors an accurate potential mapping, particularly for small potential islands. The implications of the obtained results for KPFM method are also discussed. # 2010 The Japan Society of Applied Physics DOI: 10.1143/JJAP.49.025201

1.

Introduction

Since the invention by Binning et al. in 1986,1) the atomic force microscopy (AFM) has been used as a powerful technique for measurements of surface properties at a nanometer scale for a variety of materials.2) Among many different schemes developed on the basis of AFM, the noncontact AFM (NC-AFM) allows for measuring not only the topography, but also for obtaining the atomic contrast for insulators,3,4) imaging of p–n junction,5) grains on semiconductor surfaces,6) or even single molecules.7,8) In NCAFM, a vibrating cantilever with a sharp tip at its end changes slightly the vibration frequency due to the interaction with the surface. From the measured changes of vibration characteristics, tiny details of the tip–surface interaction may be deduced.2) Kelvin probe force microscopy (KPFM) is a version of NC-AFM, in which a bias voltage Vbias is applied to the sample in order to minimize the effect of the electrostatic tip–surface interaction. By this, in addition to topography, information about the surface potential distribution may be obtained.9,10) Moreover, two different sub-methods have been developed in KPFM, the so called frequency modulation and amplitude modulation modes of KPFM (FM-KPFM and AM-KPFM, respectively).11) For the simplest case, when the tip and sample surfaces have uniform potential distributions, Vbias is strictly related to the work function difference W ¼ Wtip Wsurface between the tip and the sample. In such a case, this difference defines the so called contact potential difference VCPD ¼ W=e, where e is the electron charge, and in this situation in KPFM, Vbias ¼ VCPD . Even for such a simple system, there are many important practical problems, related to the KPFM measurements, i.e., tip geometry effect, tip– plane distance and cantilever size and shape influence, and so on.12,13) For the simplest nonuniform surface potential distribution, the sample is considered as divided into two regions with ðiÞ two different values of the work function (Wsurface , with i ¼ 1 or 2), which, for example, may be due to chemical composition changes or because of different doping levels. These regions define two values of the contact potential

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ðiÞ ðiÞ ¼ W ðiÞ =e (W ðiÞ ¼ Wtip Wsurface ). Howdifference, VCPD ever, in KPFM, if only the tip is placed far away from the ðiÞ border line, the relation Vbias ¼ VCPD is true. When the tip crosses the boundary line between two sample regions, Vbias ð1Þ ð2Þ changes its value continously from VCPD to VCPD . Since properties of this system are directly related to many KPFM measurements, it has been extensively studied by different theoretical methods.14–18) As variations of this system, there have been some studies of systems with a plane with potential stripe,16) with a single island of quadratic or circular shape.16,19) In the most general case, the electrostatic potential on the sample surface corresponds to N (N > 2) potential islands. Experimentally, KPFM measurements are performed in the same way as for a uniform sample surface, but the interpretation of the results is not so straightforward. In ðiÞ principle, there are N different values VCPD (i ¼ 1; 2; . . . ; N), and being a complex functional of all those values, the measured bias potential is does not correspond directly to ðiÞ any VCPD , since the potential island shape and size also determine the measured properties. Though very important, this is a very little studied problem. Therefore, to get a better understanding of this system and in order to establish a ðiÞ relation between the corresponing VCPD ’s of a tip–sample system and measured Vbias , further studies and new models are necessary. In this paper we introduce and study such a model, for which Vbias may be effectively evaluated. Calculation of Vbias may proceed through three different paths. In the simplest case, the value of the electrostatic force is required to be minimum and from this condition the values of Vbias is deduced.15) This approach is related to AM-KPFM. For actual dynamic FM-KPFM experiments, the condition for Vbias is that the tip frequency shift f gets minimum. In the so called gradient approximation, f is deduced from the electrostatic force gradient. Consequently, Vbias is deduced from this frequency shift.16,18,19) In the general case of large vibration amplitude, f should be calculated from the integral formula that involves the electrostatic interaction over the whole oscillation amplitude and Vbias results from this integral minimization.20) Here, we focus on the forward problem, trying to develop and explore efficient methods for electrostatic force evalua-

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tion. In the so called inverse problem, the aim is to restore the actual surface potential distribution from the measured two-dimensional KPFM image. Although a full solution of this extremely important problem is still lacking, some important results have been already obtained.21–23) While studying the forward problem, we would like to stress that any solution (partial or full) to the inverse problem would require an efficient force determination method. With this in mind, we are convinced that the method presented in this work might help in getting progress in the inverse problem solution. There is another important problem, related to the electrostatic tip–surface interaction, which should be mentioned here. Bocquet et al.24,25) studied an analytical model of the electrostatic interaction on ionic systems and they found a possibility of the atomic contrast on the local contact potential difference observed. Their findings are based on the existence of the short-range electrostatic forces, resulting from the Madelung potential arising at the surface of an ionic crystal. However, in this paper we do not consider the phenomena related to the short-range electrostatic forces, restricting ourselves to plane potential distributions varying in nanometer rather than atomic scale. We have performed an extensive investigation of the bias potential for nonuniform potential distribution on the sample surface. The computation uses an efficient method for calculating the electrostatic force in tip–plane systems, recently published.26) The paper is organized as follows. In §2, the basic formalism for the bias potential calculation is introduced. §3 presents the calculated values of the bias potential for different plane potential distribution and for different tip–surface geometries. Finally, in §4, the presented results are summarized and discussed. 2.

Bias Potential Evaluation

In the KPFM, the metallic tip with potential V0 is placed at a certain height above the surface (x–y plane) and vibrates with the resonant frequency f0 . In the plane, given is a potential distribution V1 ðx; yÞ and we assume V1 ðx; yÞ in the form of one or more islands, each having a fixed value of the electrostatic potential Vi , where i ¼ 1; 2; . . . ; N; the rest of the plane has the potential value equal to zero. With this form of V1 ðx; yÞ, we can think of the considered system as ðiÞ having N values of VCPD ¼ Vi V0 , as defined in the previous chapter. In such a system, the electrostatic energy Wel is given by the formula27) N 1X Wel ¼ Cij Vi Vj ; ð1Þ 2 i; j¼0 where Cij is a relative capacitance of a pair of potential islands i and j. The component Fz of the electrostatic force acting on the tip is given by the expression N @Wel 1X @Cij Fz ¼ ð2Þ ¼ Vi Vj ; 2 i; j¼0 @z @z where z is the tip–surface distance. From eq. (2) it follows that for fixed potentials Vi (i ¼ 1; 2; . . . ; N), Fz is a quadratic function of the tip potential V0 Fz ¼ þ V0 þ V02 ;

ð3Þ

where , , and are geometry dependent parameters. Generally, for the defined problem with realistic tip–plane

geometry, numerical calculation of electrostatic tip–plane forces is not a trivial task. The main difficulty is related to different length scales associated with the tip and the plane. In our study we have used an efficient method that was proposed for solving this problem.26) The essential point of the proposed scheme is that the plane potential distribution is being integrated out analytically through the Green function formalism. As a result, only the tip part has to be solved numerically and that is a significant numerical advantage over the algorithms where both the tip and the surface have to be included into the numerical part of the evaluation. With the electrostatic force given by eq. (3), the bias st potential Vbias is defined as such a value of the tip potential V0 that minimizes the absolute value of Fz .9) From eq. (3) we obtain (force) Vbias ¼

: 2

ð4Þ

Experimentally, this approach can be to some extend related to the AM-KPFM. In the AM mode, the cantilever is excited with an ac bias of frequency !, corresponding to the second resonance. The amplitude induced by the electrostatic forces is measured directly and a dc bias voltage is used to nullify the cantilever oscillations at !.11) In the FM-KPFM experiment, the tip is mounted at the end of a cantilever with spring coefficient k and vibrates with an amplitude A. When compared to elastic forces, the tip– sample interaction is rather weak, so the tip changes its frequency by the amount of f f0 . If the force gradient may be considered as constant over the whole amplitude of oscillations, the relative frequency shift is given by the gradient approximation formula28) f 1 @Fz ðzÞ ¼ ; ð5Þ f0 2k @z z¼dmin where dmin is the minimum tip–surface distance. After differentiating eq. (3), the force gradient is still a quadratic function of V0 and this allows for writing an analogous formula for the bias potential (grad) ¼ Vbias

0 ; 2 0

ð6Þ

where x0 ¼ @x=@zjz¼dmin , with x ¼ or x ¼ . The potential (grad) Vbias given by eq. (6) is considered as the bias potential for the gradient approximation. In the general case of FM-KPFM, the value of oscillations amplitude A is so big that the force gradient during the oscillation cycle cannot be considered as being constant. Then, the improved formula is derived by the classical perturbation theory, leading to the following expression20) Z1 f 1 u du ¼ Fz ½dmin þ Að1 þ uÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi : ð7Þ f0 kA 1 1 u2 It is interesting to note that for small values of A, eq. (7) takes exactly the form of eq. (5), as it should be. From eq. (3) used together with eq. (7), it follows that the calculated value of f = f0 is again a quadratic function of V0 , but with modified coefficients 1 , 1 , and 1 , which result from corresponding integration of Fz in eq. (7). Therefore, for the general nonlinear case, the bias potential may be written as

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(int) Vbias ¼

1 : 21

K. Sajewicz et al.

ð8Þ

It is one of the main tasks in this work to study the difference between the gradient [eq. (5)] and full nonliner integral approach [eq. (7)]. In the experiment, both these approaches are related to FM-KPFM mode, where the frequency shift amplitude f is minimized at a certain ac sample bias. In general, the FM-KPFM is expected to provide higher lateral resolution of the measured CPD compare to the AM-KPFM, as the force gradient is much more confined to the tip front end than the force.11) In many papers Vbias , as defined above, is denoted just as contact potential difference, i.e., CPD. Here, we reserve the term CPD for its usual meaning, that is as defined for any pair of materials with uniform surfaces. The role of these approaches, as applied to the conventional nc-AFM, has been investigated thoroughly and is now rather well understood. We would like to stress that the main novelty of our study is to investigate the difference between the gradient and integral approach, as applied to KPFM. 3.

Fig. 1. Geometry of the tip–plane system studied in this work, with details of the tip apex geometry shown in the inset. The potential distribution on the plane is marked as V1 ðx; y Þ and the tip surface has potential V0 .

Numerical Results and Discussion

Before the bias potential may be calculated, geometry of the tip–plane system has to be specified. In this paper, we follow the previous studies12,14,26) and the geometry of the tip has been chosen to model shape and size of the real tips, typically used in experiments. Figure 1 presents the details of the tip–sample geometry. Length of the whole tip Ltip ¼ 10 mm, the cone angle 2 ¼ 20 , and radius of the lower semisphere rtip ¼ 10 nm. To study the effect of a sharp tip termination, the so called nanotip, following29,30) we consider two values of rnano , namely, 0 and 0.5 nm. We verified by explicit computation that for considered systems, cantilever presence has little influence for the calculated quantities, typically the error is less the 1%. Therefore, for simplicity, we have neglected the cantilever presence. For this problem we also refer the reader to published works.12,13,31) To investigate the effect of the plane potential distribution V1 ðx; yÞ, we start from the step function 0; for x < 0, ð9Þ V1 ðx; yÞ ¼ 1 V; for x > 0. The results for the bias potential are presented in Fig. 2. For smaller tip–surface distance dmin ¼ 1 nm, essentially there is no difference between the ‘‘force’’ and ‘‘integral’’ case. However, for a larger value of dmin ¼ 5 nm, both approaches differ immediately around xc ¼ 0, leading to different values of the resolution. The presented results might be compared and found consistent with those presented by Shen et al.16) (theoretical calculations) and by Zerweck et al.19) (theory and experiment) for the AM and FM mode of KPFM. Generally, in the experiment FMKPFM yields a better resolution than AM-KPFM, as it is based on the force gradient rather than on the force alone. Our results suggest that this is indeed the case if only the tip–surface distance is not very small. For very small tip– surface distances, when the geometrical details of the tip do not play a significant role, both approaches yield similar accuracy.16)

Fig. 2. Vbias for the step function, as a function of the tip apex (force) coordinate xc . Vbias data are from the ‘‘force’’ formula, [eq. (4)], and (int) Vbias data are from the integral formula with A ¼ 40 nm [eq. (8)].

As the second model system, we have considered quadratic potential islands with 1 V potential and of different size a, with the plane potential distribution V1 ðx; yÞ given by 1 V; for jxj < a=2 and jyj < a=2, ð10Þ V1 ðx; yÞ ¼ 0; elsewhere. (force) Figure 3 presents the calculated values of Vbias , as function of the tip apex coordinate xc , scanned along the symmetry axis of the island, i.e., for yc ¼ 0; to study the effect of island size, two values a ¼ a1 ¼ 5 nm and a ¼ a2 ¼ 10 nm have been considered. The best conditions for accurate mapping of the island are if the island size a is big and the tip–plane distance dmin is small. In this case the calculated bias potential corresponds only to about 50 – 60% of the actual surface potential. For smaller islands and larger tip–plane distances, the mapping of the actual plane potential distribution is worse with the accuracy of mapping of the true surface potential on the level of 5% (over the center of the island). This effect is also visible in Fig. 4, (force) where within the same approach, the dependence of Vbias on the island size a is shown. The subsequent curves were obtained for the tip hold at different values of height dmin

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(force) Fig. 3. Bias potential Vbias for quadratic islands, computed from the ‘‘force’’ formula, [eq. (4)], as a function of the tip apex coordinate xc , scanned along the symmetry axis of the island, i.e., for yc ¼ 0. The data are for selected values of island size a (a1 and a2 ) and tip–plane distance dmin . The island size is marked with thin vertical dotted lines. There is no nanotip at the tip apex.

(force) Fig. 4. Bias potential Vbias for quadratic islands, computed from the ‘‘force’’ formula, [eq. (4)], as a function of the island size a for different values of the tip–sample distance dmin . The subsequent curves were obtained for the tip hold above the center of the islands (xc ¼ yc ¼ 0) and there is no nanotip at the tip apex.

above the center of the islands (xc ¼ yc ¼ 0). For small values of the island size a, the calculated bias potential is much smaller than the actual value of the island potential. Summarizing, the results of Figs. 3 and 4 suggest that better mapping is possible for larger island size a and/or for smaller tip–plane distance dmin . The calculated results for the dynamical cases (gradient and integral) are presented in Figs. 5 and 6. The results are for the fixed tip–surface distance dmin ¼ 1 nm and for two values of the island size, namely, a ¼ 5 nm (Fig. 5) and a ¼ 10 nm (Fig. 6). Furthermore, to investigate the tip sharp end effect, a nonzero value of the tip apex radius, rnano ¼ 0:5 nm has been introduced and compared with the case rnano ¼ 0. The linear dynamic case may be considered as a limiting case of the nonlinear approach, with a very small vibration amplitude, i.e., A ! 0. To study the effect of finite A, two values of A have been considered, namely A ¼ 2 and 20 nm. It is interesting to observe that for considered values of A, the best mapping is obtained for small values of A, i.e.,

Fig. 5. Bias potential for a quadratic island as a function of the tip apex coordinate xc , as calculated along the island symmetry axis (yc ¼ 0), for the square island with a ¼ 5 nm and for the tip–plane distance dmin ¼ 1 nm. The data for the gradient approach [eq. (6)], correspond to the oscillation amplitude A ¼ 0, while for the integral approach [eq. (8)], A 6 ¼ 0. Additionally, the results are given for two values of the nanotip radius rnano . The inset shows the bias potential as a function of the oscillation amplitude A, as evaluated for xc ¼ yc ¼ 0.

Fig. 6.

The same as in Fig. 5 but for the island size a ¼ 10 nm.

mapping for A ¼ 2 nm is generally much better than that of A ¼ 20 nm. This effect is also demonstrated in the insets of Figs. 5 and 6, where Vbias is presented as a function of A and this conclusion is true for both considered values of rnano . The important role of the oscillation amplitude is particularly visible for a smaller potential island, as comparison between Figs. 5 and 6 demonstrates. Therefore, the important physical conclusion is that it is the big value of the oscillation amplitude — if taken into account properly — that makes the mapping procedure less accurate. The presented results clearly indicate that the commonly used gradient approximation may lead to substantial errors in interpreting the experimental data and this is the important conclusion from the presented results. As for the nanotip role, similarly to the ‘‘force’’ case, adding a nanotip makes the mapping more accurate, what int means that with rnano 6¼ 0, the calculated values of Vbias get closer to the actual values of the ‘‘scanned’’ plane potential V1 ðx; yÞ. This is apparent from the presented results in Figs. 5 and 6 and the presented results are consistent with the calculations made by the other methods.18,19) Further-

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(int) Fig. 7. Bias potential Vbias for the dynamic case together with the experimental results for KBr islands on the InSb(001),32,33) as a function of the island size a. The theoretical results were calculated for fixed values of the vibration amplitude A ¼ 40 nm, the tip–surface relative coordinates dmin ¼ 1 nm, and xc ¼ yc ¼ 0, for three given values of the tip apex radius rtip . There is no nanotip at the tip apex. The curves between the symbols are guide to eye only.

more, as the insets of Figs. 5 and 6 demonstrate, including the nanotip is equally important for both small and large oscillation amplitudes A. However, comparing Fig. 5 with Fig. 6 we may observe that the role of nanotip is less decisive, while scanning larger potential islands. The last but not least is that, as in the ‘‘force’’ case, even with the nanotip present, the calculated bias potential is between 0.2 and 0.8 V, which is clearly below the actual value of the potential (i.e., 1 V). The general nonlinear method for the bias potential evaluation has been used for a direct comparison with the experimental values, as obtained in32,33) for KBr islands on InSb(001) surface. The results are plotted in Fig. 7. In the calculations, we used the experimental value of amplitude of oscillation A ¼ 40 nm and fixed the tip–surface distance at dmin ¼ 1 nm. The only varying paramter was the tip radius, for which we have chosen three values, namely, rtip ¼ 5, 10, and 20 nm. The best agreement between the calculation and experiment is reached if a value of rtip ¼ 20 nm is selected and this value of rtip ¼ 20 has been also found in the experiment.33) The results of Fig. 7 suggest that there might be a possibility of determining the tip geometry parameters from the bias potential behavior, when measurement and calculation are performed for a series of island size.30) As the final step of our study, we have investigated a system that comprises two quadratic island of equal size, with varying separation. To out knowledge, such a system — though very important from the experimental point of view — has not been theoretically studied so far. The results for two islands of a ¼ 5 nm are presented in Fig. 8. There are at least two features of the presented data, which should be commented. First, the important question related to the presented data is: when can the two islands be detected as separate entities? For the presented data, even for quite small island–island separation, two maxima and one minimum of the bias potential are clearly visible, which is in a good agreement with experimental data, where two KBr islands, separated by about 4 nm from each other, could be distinguished.34) Approximately, that would correspond to

(int) Fig. 8. Bias potential Vbias for the ‘‘integral’’ case [calculated from eq. (8)], as a function of the tip apex coordinate xc , as calculated along the system symmetry axis (yc ¼ 0), for double island configuration, as presented in the inset. The calculations were performed for the oscillation amplitude A ¼ 40 nm, the tip–surface distance dmin ¼ 1 nm and without nanotip (rnano ¼ 0 nm). The value a=d ¼ 1:0 corresponds to one island of 2a a size.

the value of d=a ¼ 2:0 in Fig. 8 (edge–edge distance being equal to 5 nm). The presented data show that in this case two int are well separated by the minimum maxima in the Vbias between them, thus resembling two island presence. The calculated data in Fig. 8 also suggest that the two islands might be distinguished even for the d=a ¼ 1:5 case. However, for the d=a ¼ 1:25 value, the local minimum in (int) Vbias in xc ¼ 0 gets too shallow, since the maximum– minimum difference of Vbias is only around 5% of maximum (int) value of Vbias . Such a small maximum–minimum difference might not be sufficient to get the islands resolved experimentally. Therefore, the presented results suggest that there would be the smallest inter-island separation, below which the considered islands could not be distinguished. The second question is even more subtle and it concerns the task of quantative determining the islands’ boundaries (and sizes) from the behavior of Vbias . This is an extremely complex problem that forms a part of the inverse problem, which we do not attempt to investigate in this work. However, even the presented forward problem results might help to estimate the islands size, for example, by approximating each island contribution to Vbias by a gaussian curve. Obviously, the quantitative results could be obtained only with the full deconvolution procedure, with geometrical factors (tip size and shape, its distance to the surface) properly included. 4.

Conclusions

In this paper, the bias potential Vbias , resulting from the electrostatic tip–surface interaction, has been calculated for realistic tip–surface geometry and for different, non-uniform potential distribution on the plane. Three different schemes have been studied as leading to different bias potential values, namely, the ‘‘force’’, the ‘‘gradient’’, and the ‘‘integral’’ method. The effect of tip geometry has been investigated and it has been shown that a sharp tip end makes the bias potential get closer to the actual surface potential distribution. In the case of oscillations, results of presented calculations show that there is a better mapping

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with Vbias for small amplitudes of the oscillations and/or with a nanotip added. Also, the commonly used gradient approxiamtion might give lead to appreciable differences with the correct integral formula. The obtained results for two island system are consistent with the experimental data. We have calculated the bias potential for given tip geometry and plane potential distribution and combining the experimental data with the calculation, there might be a possibility to establish geometrical features of the tip. Another possibility would be finding of potential island boundaries. There are several effects not studied in this work, which we leave for subsequent investigations. These include non-quadratic potential islands and islands with different values of the potential. The present formalism is suitable for dealing with all those cases. Acknowledgments

This work was supported by the European Commission within the Project under FP7 Coordination and Support Action, ‘‘Nano-scale ICT Devices and Systems Coordination Action—NanoICT’’, Grant Number 216165.

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