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Dynamic AFM using the FM technique with constant excitation amplitude
Applied Surface Science 188 (2002) 355–362
Dynamic AFM using the FM technique with constant excitation amplitude B. Gotsmann*, H. Fuchs Physikalisches Institut, Universita¨t Mu¨nster, D-48149 Mu¨nster, Germany Received 3 September 2001; accepted 4 October 2001
Abstract Dynamic atomic force microscopy using the frequency modulation technique is investigated for the case that the excitation amplitude is kept constant. This mode of operation has very unique properties. A computer simulation is used to investigate the distance dependence of the measurement signals frequency and amplitude using various conservative and non-conservative interaction forces. It is shown how the two measurement channels are interlinked and influenced by both conservative and dissipative interactions. Further, discontinuous frequency shift versus distance curves as reported in the literature were not obtained. # 2002 Elsevier Science B.V. All rights reserved. Keywords: AFM; Frequency modulation; Constant excitation; Dynamic force spectroscopy
1. Introduction Dynamic atomic force microscopy (AFM) is a powerful tool for the characterization of various surfaces under different conditions. There are numerous modes of operation some of which are particularly useful for special applications. Among these modes of operation the frequency modulation (FM) technique is in particular useful for the application in vacuum [1] and leads to atomic resolution images [2–4]. In the FM technique the driving-signal of the cantilever excitation (performed by a small piezo plate, e.g.) is generated through a feedback loop. The detector, sensing the cantilever movement, produces an ac-signal which is amplified with a certain gain-factor, * Corresponding author. Present address: IBM Research, Zurich Research Laboratory, 8803 Ru¨schlikon, Switzerland. Tel.: þ41-1-724-8223; fax: þ41-1-724-8529. E-mail address: [email protected] (B. Gotsmann).
phase shifted and then used as an excitation signal. The frequency f of the vibrating lever then varies in response to the tip–sample interaction forces. The force induced shift of the resonance frequency Df in the self-excited loop is used as a control signal to keep the tip–sample distance constant while scanning. In order to maintain a stable vibration, however, it is essential to control the gain factor of the vibrating system. There are two ways which are used to determine an appropriate gain factor. On one hand, an automatic gain control (AGC) unit can be used to keep the cantilever vibration amplitude at a constant value [1]. On the other hand, the excitation amplitude can be held constant [5] (either by an AGC unit or by a delimiter). In order to distinguish between the two, we denote the first by CA mode and the latter by CE mode. This paper is denoted to the analysis of the measurement using CE mode by means of a computer simulation. Differences between CA and CE mode become obvious showing that the CE mode has its own unique properties.
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discussed below. It might become important for future applications.
2. Theoretical procedure Fig. 1. Schematic of a dynamic AFM explaining the variables: z (the deflection of the lever), d (the lever support distance) and ~z ¼ z þ d the total tip–sample distance.
The most striking difference of the CE mode with respect to the CA mode is that the vibration amplitude is not fixed. This means that the amplitude decays as the vibrating cantilever/tip is approached towards the surface and comes into repulsive contact. Consequently, when the cantilever holder and henceforth the position of the undeflected lever d (see Fig. 1) is moved towards the surface the closest tip–sample distance may not penetrate the repulsive interaction regime in a similar manner. In an extreme case this minimal tip–surface distance ~zmin d A can be constant (i.e. @AðdÞ=@d ¼ 1 or ~zmin ¼ constant). Sugawara et al. [6] and Ueyama et al. [5] suggest that degradation of the tip can be reduced in this way. The energy loss which is associated with a repulsive contact may lead to a corresponding decrease of amplitude. Further, they observed that the frequency shift as a function of displacement Df(d) is a monotonous function of d in the CE mode. In contrary, in the CA mode Df(d) runs through a minimum. This restricts the regimes of stable imaging in the CA mode. This argumentation does not seem to reveal a principle failure of the CA–FM mode, because it has been shown that this mode can be used well for high resolution imaging [2–4]. Still, there might be surfaces which can be more easily imaged using the CE mode. However, if one is interested in probing forces in the repulsive regime, then the CE mode cannot be used, because the decay of amplitude prevents a strong penetration into the repulsive regime. In the following analysis, we will describe the theoretical treatment of the problem and look at the solutions Df(d) and A(d) (DFS curves) for different model force interactions. One of the most interesting observations (which has only been reported for the CE mode yet) are non-continuous Df(d)-curves on chemically reactive surface areas, probably indicating a sensitivity to chemical binding [6–10]. This phenomenon shall be
2.1. Analytical equations The basic problem of modeling the CE technique is that for both, amplitude A(d) and frequency shift Df(d) expressions have to be found. The problem can be approached analytically using the assumptions of a harmonic movement of the cantilever/tip. For this two coupled equations can be found in the literature. It is known that the frequency shift can be expressed to a good approximation by the following equation [11,20]: Z 2p f0 Df ðdÞ ¼ Fts ðd þ A cosðxÞÞcosðxÞ dx (1) 2pkl A 0 Here, f0 denotes the resonance frequency of the free lever, kl the spring constant, A the amplitude and Fts the tip–sample force. This equation assumes a selfexcited feedback loop with optimum phase lag [26]. The amplitude A has a strong influence on the value of this integral. The amplitude can be determined mainly via dissipative forces using a second expression which can easily be derived [12]: we can expect that the power inserted into the vibrating lever (averaged over one period of oscillation) Z 1=f in ¼ f P Pin ðtÞ dt ¼ pfAexc A sin f (2) 0
is always equal to the power loss into both intrinsic damping of the lever 2 2 intr ¼ pf kl A P f0 Q
(3)
and the power dissipated into the tip–sample contact Z 1=f ts ¼ f P Fts ð~zðtÞ; z_ ðtÞ; . . .Þ_zðtÞ dt (4) 0
Hence, we find pfAexc A sinðfÞ ¼
pf 2 kl A2 f0 Q Z 1=f þf Fts ð~zðtÞ; z_ ðtÞ; . . .Þ_zðtÞ dt (5) 0
B. Gotsmann, H. Fuchs / Applied Surface Science 188 (2002) 355–362
Here, Q is the quality factor of the free lever, and f the phase shift between excitation and vibration usually fixed at p/2 in experiments. The excitation amplitude is denoted by Aexc in units of force. The deflection of the lever is z so that the tip–sample distance becomes ~z ¼ d þ z with d being the position of the non-deflected tip. Consequently, conservative and dissipative forces are interlinked in their influence on the two measurement channels Df and A (Eqs. (1) and (5)). This complicates the analysis. In contrary, in the CA mode the two channels are independent to a good degree [13]. We believe that more theoretical analysis is needed in order to make the CE mode a quantitative tool, and in order to interpret specific measurements such as non-continuous Df(d)-curves. Rather than solving these two coupled equations (Eqs. (1) and (5)), we chose to tackle the problem from a different angle by solving the equation of motion directly using a computer simulation. The equation of motion will be described below. We show solutions for the functions A(d) and Df(d) which are known as dynamic force spectroscopy curves. 2.2. Equation of motion Treating the system as a one-dimensional movement of the tip and the free lever as a harmonic oscillator yields the force-based equation of motion meffzðtÞ þ bl z_ ðtÞ þ kl zðtÞ Fts ðd þ zðtÞÞ ¼ Fexc ðtÞ (6) where z denotes the deflection of the cantilever, meff its effective mass, bl the damping coefficient of the cantilever motion and kl the spring constant. The tip– sample interaction force is denoted by Fts as a function of the absolute tip–sample separation ~z ¼ d þ z (see Fig. 1). Fexc describes the excitation through an actuator-piezo element. In our simulation Fexc is held constant through normalization of the gain factor by the actual vibration amplitude Fexc ðtÞ ¼
A0exc zðt tphase Þ AðtÞ
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tphase is usually set to 1/4 of the actual frequency f ¼ f0 þ Df . The differential equation was solved with the Verlet algorithm [14]. All input parameters can be determined experimentally. We chose parameters which are of typical order of magnitude for non-contact AFM applications: f ¼ 300 kHz, A ¼ 10 nm (i.e. 20 nmpp), kl ¼ 40 N/m, Q ¼ 20 000. It needs to be emphasized, however, that a drastic change of any of the parameters, like for example going to soft cantilevers (kl < 1 N/m), may have a strong influence on the general characteristics of the dynamic solution of Eq. (6). 2.3. Model forces In order to give a broad picture of the special features of the CE mode, we used various different forms of the tip–sample interaction forces in order to consider very different cases. We would like to point out that the exact form of such forces does not play a major role here, as we are interested in rather general conclusions. The exact form of the interaction depends on a lot of experimental parameters like the material of tip and sample, the shape of the tip and the bias voltage. Thus, the force curves may vary strongly in decay length and magnitude. We chose model forces which produce right orders of magnitude in strength and reasonable decay lengths and distance dependencies. 2.3.1. Conservative force We used the model by Muller et al. [15] (MYD model) as depicted in Fig. 2a. This model produces complete force curves including attractive and repulsive regime and the transition regime between the two. Typically, contact is established at the minimum of such a force curve. It is based on continuum theory and uses van der Waals attraction as well as elastic deformation of the solid. Other, more adequate force interactions use short range forces which account for more specific adhesion forces like electronic interaction [16–19]. For our purposes the MYD model is sufficient.
(7)
The actual vibration amplitude (peak-to-peak value) is denoted by A(t), and A0exc is the set point value of the excitation and tphase the phase shift. The inverse of
2.3.2. Damping force It is known that in dynamic AFM energy is dissipated into the tip–sample interaction [21]. A realistic dynamic tip–sample contact cannot be purely conservative.
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The characteristic features of such a damping force are defined via the damping coefficient gð~zÞ, which depends only on the tip–sample distance. A suitable gð~zÞ could have the form ~z (9) g ¼ g0 exp exp z0 This function reflects the strong increase of dissipation with increasing strength of tip–sample interaction. It includes a part decaying into the non-contact region. Note that gð~zÞ may be an effective quantity and Eq. (8) an effective damping force. True damping mechanisms are still under debate and this form may serve as a starting point. In order to match experimental orders of magnitude [13,28] we chose the constants g0 exp ¼ 108 N s/m and z0 ¼ 0:5 nm. For a solution of the integral (4), see, e.g. [13,22]. 2.3.3. Hysteretic force Another possible mechanism for the energy loss is the hysteresis between approach and retraction [23–26]. Here, we chose a hysteretic force which stems from the MYD model for an extremely soft surface. A detailed description can be found in [24]. Again, we would like to point out that the specific form of the curve is not of importance here. Other ‘atomistic theories’ also predict hysteresis of similar form and order of magnitude (e.g. [27]). Hence, this might serve as a general model for hysteresis effects of different origins.
3. Results and discussion 3.1. Conservative force
Fig. 2. Model forces used in the simulations as a function of the absolute tip–sample distance. (a) Conservative force from the MYD model with parameters for a silicon tip and silicon surface (for details see [13]). The point of first repulsive contact at the apex of the tip (with overall force still attractive), i.e. the contact point is taken as the zero point of the distance axis. (b) Damping coefficient: exponential decay. (c) Hysteretic force from the MYD model (for details see [24]).
In order to introduce damping effects into the equation of motion, one may choose a damping force of the form Ftsdiss ð~z; z_ Þ ¼ gð~zÞ_z
(8)
The results of a simulation with the pure conservative force from Fig. 2a is shown in Fig. 3. The resulting Df(d)-curve and A(d)-curve are shown in Fig. 3a and b, respectively. Most striking is that the Df(d)-curve is almost identical to the case of CA–FM–AFM [13]. The resulting Df(d)-curve appears slightly stretched around zero. The marginal difference stems mainly from a slight deviation of the amplitude from its starting value. From the trajectory z(t) the forces acting at the turning point of the oscillation were determined as shown in Fig. 3c. Repulsive interaction between the
B. Gotsmann, H. Fuchs / Applied Surface Science 188 (2002) 355–362
Fig. 3. Computer simulation of a DFS curve using the pure conservative force from Fig. 2a.
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Fig. 4. Computer simulation of a DFS curve using the conservative force from Fig. 2a and a damping force as in Eq. (8) (Fig. 2b).
apex of the tip and the sample occurs for the first time at the minimum of the force curve. This corresponds to a point slightly before the minimum (on the right hand side of the minimum, near the turning point) of Df(d). The average force given by the static deflection of the lever is depicted in Fig. 3d. This turns repulsive after repulsion at the turning point of the oscillation has been established. The amplitude decay in the repulsive regime is small compared to experimental findings which can, therefore, only be explained using energy dissipation (see below). The small increase of the amplitude in the attractive regime, however, has been observed experimentally [8,9]. 3.2. Conservative force þ damping force The next example includes both a conservative force as in Fig. 2a and a damping force of the form of Eq. (8). From the results displayed in Figs. 4 and 5 it can be seen that the DFS curves are changed crucially by the presence of the damping force. The amplitude drops and the range of the trajectory of the oscillation changes accordingly. This influences the frequency
Fig. 5. Computer simulation of a DFS curve using the conservative force from Fig. 2a and a damping force as in Eq. (8) (Fig. 2b).
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shift. The amount of energy dissipated at the contact point is of the order of 1014 W, which is rather little compared to experimental data obtained with the CA– FM technique [13,28]. With two respects experimental data [5,6,8,9] using the CE mode are different. First, a long range of drop of amplitude does not seem to take place in experiment. The transition regime between the part of the curve where A is almost constant to the part where the amplitude drops linearly with a slope close to 1 is of the order of a few Angstroms only. Secondly, the dissipation in the repulsive regime is stronger in the experiments (i.e. @AðdÞ=@d ! 1 in the simulation, and @AðdÞ=@d 1 in the experiments). Hence, we used a different model of the dissipation in another simulation which ignores long range effects and starts out stronger at the beginning of the repulsive regime gð~zÞ ¼ g0~z
(10) 4
2
with g0 ¼ 3:33 10 N s/m if ~z < 0, and g0 ¼ 0 else. As is shown in Fig. 5 the simulation results reproduce more the expected behavior. (The small increase of amplitude in the attractive regime (at d A0 > 0 where no damping force acts) is equivalent to the case of a pure conservative force discussed above.) It seems, that long range dissipation effects as observed in [13,18,28] for the CA mode are not present in [5,6,8,9] for the CE mode. Hopefully, more experimental data for different tip and sample materials can explain this apparent contradiction.
Fig. 6. Computer simulation of a DFS curve using the noncontinuous force from Fig. 2c (i).
a discontinuity, but changes the slope when the turning point passes the discontinuity of the force curve. The ˚ , but it remains frequency changes rapidly within 1 A continuous. The general behavior is not surprising with regard to Eq. (1) (see [24]). Again it becomes clear that for an interpretation of experimental data it is not sufficient to regard conservative forces only.
3.3. Discontinuous conservative force 3.4. Hysteretic conservative force With regard to experimental data which showed discontinuous Df(d)-curves, it has been speculated that these are originated in force curves which show a sudden increase of attractive forces after a certain distance is reached [6–8]. For a model calculation it is straightforward to choose just the curve: (i) of the hysteretic force in Fig. 2c for both approach and retraction. The resulting force is conservative on the whole range except at the point of discontinuity. The condition of a ‘sudden onset of stronger attractive forces’ is clearly given for such a force interaction. The results of a simulation are shown in Fig. 6. As there is no energy dissipation into the tip–sample contact the amplitude remains almost constant (as in Fig. 3). The frequency shift curve does not show
The situation is different when in addition to the discontinuity a hysteresis as in Fig. 7 is introduced. Then a dissipative channel is opened. Consequently, we find a decay of the amplitude. Firstly, the simulated curves are identical to the ones in Fig. 6 in the regime where the hysteresis is not yet reached (Region I in Fig. 7). Then, starting from the point where the tip runs through the hysteresis for the first time, the amplitude decays linearly with a slope of @AðdÞ=@d ¼ 1 (Region II). Finally, the corresponding energy dissipation rises almost linearly until the dissipation corresponds to PðdÞ ¼ ½f0 þ Df ðdÞ DEhys (Region III). Each time the tip runs through the hysteresis an energy of DEhys ¼ 0:660 eV is dissipated. On the other hand,
B. Gotsmann, H. Fuchs / Applied Surface Science 188 (2002) 355–362
Fig. 7. Computer simulation of a DFS curve using the hysteretic force from Fig. 2c (i) and (ii).
the amount of energy which can be dissipated is given by the actual value of the amplitude. Hence a stable solution of the analytic coupled Eqs. (1) and (5) (i.e. an oscillation with a constant amplitude for a given distance d) cannot strictly be found in the specified Region II. The simulation shows, that the mean value of the amplitude has a constant value for a given d. However, individual oscillation cycles vary by some pico meters, so that only a certain fraction of cycles n ¼ nðdÞ undergoes the hysteresis. As a result the energy dissipation can be expressed as PðdÞ ¼ n DEhys ½f0 þ Df ðdÞ . As a direct consequence the frequency shift drops also almost linearly in that region. Unlike the case of CA–FM–AFM the Df(d)-curve remains continuous. Region I becomes clear that this is a non-linear problem which can be solved with the numeric solution of Eq. (6). 3.5. Interpretation of discontinuous Df(d)-curves None of the various tip–sample interaction forces lead to non-continuous Df(d)-curves as were reported
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in the experiments [6–10]. The question arises how to interpret these experimental findings. A straightforward explanation could be that tip or sample might be irreversibly changed after contact between tip and sample was established. However, this explanation can be ruled out by regarding the reproducibility of experimental data [7]. A closer look at the experimental data, however, ˚ in shows that the resolution of the d-axis (about 1 A [8]) is not sufficient to rule out a behavior as seen in Fig. 6 in the non-continuous case. There the frequency ˚ by about 60 Hz. Apparently, drops within less than 1 A there is no need to assume hysteresis in order to explain the behavior of the experimental Df(d)-curves, but maybe it can be used to explain the amplitude decay after the ‘discontinuity’ of Df(d) in experiments (i.e. left-hand side). In order to resolve such a specific bend of Df(d) as in Fig. 6 in the experiment a very defined and stable experimental setup is needed. It seems that the required force interaction needs to change rapidly (not necessarily in a discontinuous form as was used in our example of Fig. 6) after a certain distance is reached, but has associated an energy loss mechanism, possibly a hysteresis.
4. Summary and conclusions We have presented some examples of different tip– sample interactions and showed how they are probed with CE mode AFM. We have seen that although analytic approximate solutions can be applied, a full solution of the equation of motion is capable of solving the problem of two-linked spectroscopic channels (conservative and dissipative forces being probed with frequency and amplitude). It became clear that for a full understanding of the curves both conservative and disspative interactions have to be considered. In the case of hysteretic forces a stable solution with a constant amplitude cannot be found. A hysteresis has a completely different effect on DFS-curves as in the CA mode. Although, it is not so straightforward, it should be possible to learn a lot about dissipative forces from the experimental data. It might be useful to summarize some of the most important differences between this and other modes of operating a dynamic AFM.
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In the CE mode an observed amplitude decay is mainly a sign of energy dissipation. This is mainly due to the phase shift between excitation and vibration is kept constant by the electronics in the CE mode. Hence, a detuning off resonance (as in tapping mode) does not occur. In contrast, in tapping mode an amplitude decay can also be expected for pure conservative forces due to a detuning off resonance [29]. As speculated in previous papers [5], the CE mode leads to a relative protection of tip. However, for high resolution a careful handling is essential anyway. Published high quality data obtained with the CA– FM technique suggest that this point is not essential. Further, in an accidental instability the maximum amount of energy stored in the oscillator of about 104 eV is enough to damage the tip. For force-spectroscopy applications it might be interesting to note that the regime of strong repulsive forces cannot be probed using the CE mode. This can either be a disadvantage or an advantage depending on the experimental goal. On the whole the interpretation of DFS is rather more difficult than in the CA mode, due to mixing of spectroscopic channels. References [1] T.R. Albrecht et al., J. Appl. Phys. 69 (1991) 668. [2] F.J. Giessibl, Science 267 (1995) 68. [3] Proceedings of the First International Workshop on Noncontact AFM, Appl. Surf. Sci. 140 (1999) 243–456. [4] Proceedings of the Second International Workshop on Noncontact AFM, Appl. Surf. Sci. 157 (2000) 207–428. [5] H. Ueyama, Y. Sugawara, S. Morita, Appl. Phys. A 66 (1998) S295.
[6] Y. Sugawara, H. Ueyama, T. Uchihashi, M. Ohta, Y. Yanase, T. Shigematsu, M. Suzuki, S. Morita, Mater. Res. Symp. Proc. 442 (1997) 15. [7] S. Morita, Y. Sugawara, K. Yokoyama, T. Unchihashi, Nanotechnology 11 (2000) 120. [8] T. Uchihashi, Y. Sugawara, T. Tsukamoto, M. Ohta, S. Morita, Phys. Rev. B 56 (1997) 9834. [9] Y. Sugawara, S. Orisaka, S. Morita, Appl. Surf. Sci. 157 (2000) 239. [10] T. Uchihashi, Y. Sugawara, T. Tsukamoto, T. Minobe, S. Orisaka, T. Okada, S. Morita, Appl. Surf. Sci. 140 (1999) 304. [11] F.J. Giessibl, Phys. Rev. B 56 (1997) 16010. [12] B. Anczykowski, B. Gotsmann, H. Fuchs, J. Cleveland, Appl. Surf. Sci. 140 (1999) 376. [13] B. Gotsmann et al., Phys. Rev. B 60 (1999) 11051. [14] L. Verlet, Phys. Rev. 159 (1967) 98. [15] V.M. Muller, V.S. Yushenko, B.V. Derjaguin, J. Coll. Interf. Sci. 77 (1980) 91. [16] H. Ho¨ lscher, A. Schwarz, W. Allers, U.D. Schwarz, R. Wiesendanger, Phys. Rev. B 61 (2000) 12678. [17] M. Guggisberg et al., Phys. Rev. B 61 (2000) 11151. [18] B. Gotsmann, H. Fuchs, Phys. Rev. Lett. 86 (2001) 2597. [19] R. Perez, M.C. Payne, I. Stich, K. Terakura, Phys. Rev. Lett. 78 (1997) 678. [20] A.I. Livshits, A.L. Shluger, A.L. Rohl, A.S. Foster, Phys. Rev. B 59 (1999) 2436. [21] J. Cleveland, B. Anczykowski, A.E. Schmid, V.B. Elings, Appl. Phys. Lett. 72 (1998) 2613. [22] U. Du¨ rig, Appl. Phys. A 72 (2001) S43. [23] U. Du¨ rig, Appl. Phys. Lett. 75 (1999) 433. [24] B. Gotsmann, H. Fuchs, Appl. Phys. A 72 (2001) S55. [25] N. Sasaki, M. Tsukada, Jpn. J. Appl. Phys. 39 (2000) L1334. [26] H. Ho¨ lscher, B. Gotsmann, W. Allers, U.D. Schwarz, H. Fuchs, R. Wiesendanger, Phys. Rev. B 64 (2001) 075402. [27] S.H. Ke, T. Uda, R. Perez, I. Stich, K. Terakura, Phys. Rev. B 60 (1999) 11631. [28] C. Loppacher, R. Bennewitz, O. Pfeiffer, M. Guggisberg, M. Bammerlin, S. Scha¨ r, V. Barwich, A. Baratoff, E. Meyer, Phys. Rev. B 62 (2000) 13674. [29] B. Anczykowski, D. Kru¨ ger, H. Fuchs, Phys. Rev. B 53 (1996) 15485.
Dynamic AFM using the FM technique with constant excitation amplitude B. Gotsmann*, H. Fuchs Physikalisches Institut, Universita¨t Mu¨nster, D-48149 Mu¨nster, Germany Received 3 September 2001; accepted 4 October 2001
Abstract Dynamic atomic force microscopy using the frequency modulation technique is investigated for the case that the excitation amplitude is kept constant. This mode of operation has very unique properties. A computer simulation is used to investigate the distance dependence of the measurement signals frequency and amplitude using various conservative and non-conservative interaction forces. It is shown how the two measurement channels are interlinked and influenced by both conservative and dissipative interactions. Further, discontinuous frequency shift versus distance curves as reported in the literature were not obtained. # 2002 Elsevier Science B.V. All rights reserved. Keywords: AFM; Frequency modulation; Constant excitation; Dynamic force spectroscopy
1. Introduction Dynamic atomic force microscopy (AFM) is a powerful tool for the characterization of various surfaces under different conditions. There are numerous modes of operation some of which are particularly useful for special applications. Among these modes of operation the frequency modulation (FM) technique is in particular useful for the application in vacuum [1] and leads to atomic resolution images [2–4]. In the FM technique the driving-signal of the cantilever excitation (performed by a small piezo plate, e.g.) is generated through a feedback loop. The detector, sensing the cantilever movement, produces an ac-signal which is amplified with a certain gain-factor, * Corresponding author. Present address: IBM Research, Zurich Research Laboratory, 8803 Ru¨schlikon, Switzerland. Tel.: þ41-1-724-8223; fax: þ41-1-724-8529. E-mail address: [email protected] (B. Gotsmann).
phase shifted and then used as an excitation signal. The frequency f of the vibrating lever then varies in response to the tip–sample interaction forces. The force induced shift of the resonance frequency Df in the self-excited loop is used as a control signal to keep the tip–sample distance constant while scanning. In order to maintain a stable vibration, however, it is essential to control the gain factor of the vibrating system. There are two ways which are used to determine an appropriate gain factor. On one hand, an automatic gain control (AGC) unit can be used to keep the cantilever vibration amplitude at a constant value [1]. On the other hand, the excitation amplitude can be held constant [5] (either by an AGC unit or by a delimiter). In order to distinguish between the two, we denote the first by CA mode and the latter by CE mode. This paper is denoted to the analysis of the measurement using CE mode by means of a computer simulation. Differences between CA and CE mode become obvious showing that the CE mode has its own unique properties.
0169-4332/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 1 ) 0 0 9 5 0 - 3
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discussed below. It might become important for future applications.
2. Theoretical procedure Fig. 1. Schematic of a dynamic AFM explaining the variables: z (the deflection of the lever), d (the lever support distance) and ~z ¼ z þ d the total tip–sample distance.
The most striking difference of the CE mode with respect to the CA mode is that the vibration amplitude is not fixed. This means that the amplitude decays as the vibrating cantilever/tip is approached towards the surface and comes into repulsive contact. Consequently, when the cantilever holder and henceforth the position of the undeflected lever d (see Fig. 1) is moved towards the surface the closest tip–sample distance may not penetrate the repulsive interaction regime in a similar manner. In an extreme case this minimal tip–surface distance ~zmin d A can be constant (i.e. @AðdÞ=@d ¼ 1 or ~zmin ¼ constant). Sugawara et al. [6] and Ueyama et al. [5] suggest that degradation of the tip can be reduced in this way. The energy loss which is associated with a repulsive contact may lead to a corresponding decrease of amplitude. Further, they observed that the frequency shift as a function of displacement Df(d) is a monotonous function of d in the CE mode. In contrary, in the CA mode Df(d) runs through a minimum. This restricts the regimes of stable imaging in the CA mode. This argumentation does not seem to reveal a principle failure of the CA–FM mode, because it has been shown that this mode can be used well for high resolution imaging [2–4]. Still, there might be surfaces which can be more easily imaged using the CE mode. However, if one is interested in probing forces in the repulsive regime, then the CE mode cannot be used, because the decay of amplitude prevents a strong penetration into the repulsive regime. In the following analysis, we will describe the theoretical treatment of the problem and look at the solutions Df(d) and A(d) (DFS curves) for different model force interactions. One of the most interesting observations (which has only been reported for the CE mode yet) are non-continuous Df(d)-curves on chemically reactive surface areas, probably indicating a sensitivity to chemical binding [6–10]. This phenomenon shall be
2.1. Analytical equations The basic problem of modeling the CE technique is that for both, amplitude A(d) and frequency shift Df(d) expressions have to be found. The problem can be approached analytically using the assumptions of a harmonic movement of the cantilever/tip. For this two coupled equations can be found in the literature. It is known that the frequency shift can be expressed to a good approximation by the following equation [11,20]: Z 2p f0 Df ðdÞ ¼ Fts ðd þ A cosðxÞÞcosðxÞ dx (1) 2pkl A 0 Here, f0 denotes the resonance frequency of the free lever, kl the spring constant, A the amplitude and Fts the tip–sample force. This equation assumes a selfexcited feedback loop with optimum phase lag [26]. The amplitude A has a strong influence on the value of this integral. The amplitude can be determined mainly via dissipative forces using a second expression which can easily be derived [12]: we can expect that the power inserted into the vibrating lever (averaged over one period of oscillation) Z 1=f in ¼ f P Pin ðtÞ dt ¼ pfAexc A sin f (2) 0
is always equal to the power loss into both intrinsic damping of the lever 2 2 intr ¼ pf kl A P f0 Q
(3)
and the power dissipated into the tip–sample contact Z 1=f ts ¼ f P Fts ð~zðtÞ; z_ ðtÞ; . . .Þ_zðtÞ dt (4) 0
Hence, we find pfAexc A sinðfÞ ¼
pf 2 kl A2 f0 Q Z 1=f þf Fts ð~zðtÞ; z_ ðtÞ; . . .Þ_zðtÞ dt (5) 0
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Here, Q is the quality factor of the free lever, and f the phase shift between excitation and vibration usually fixed at p/2 in experiments. The excitation amplitude is denoted by Aexc in units of force. The deflection of the lever is z so that the tip–sample distance becomes ~z ¼ d þ z with d being the position of the non-deflected tip. Consequently, conservative and dissipative forces are interlinked in their influence on the two measurement channels Df and A (Eqs. (1) and (5)). This complicates the analysis. In contrary, in the CA mode the two channels are independent to a good degree [13]. We believe that more theoretical analysis is needed in order to make the CE mode a quantitative tool, and in order to interpret specific measurements such as non-continuous Df(d)-curves. Rather than solving these two coupled equations (Eqs. (1) and (5)), we chose to tackle the problem from a different angle by solving the equation of motion directly using a computer simulation. The equation of motion will be described below. We show solutions for the functions A(d) and Df(d) which are known as dynamic force spectroscopy curves. 2.2. Equation of motion Treating the system as a one-dimensional movement of the tip and the free lever as a harmonic oscillator yields the force-based equation of motion meffzðtÞ þ bl z_ ðtÞ þ kl zðtÞ Fts ðd þ zðtÞÞ ¼ Fexc ðtÞ (6) where z denotes the deflection of the cantilever, meff its effective mass, bl the damping coefficient of the cantilever motion and kl the spring constant. The tip– sample interaction force is denoted by Fts as a function of the absolute tip–sample separation ~z ¼ d þ z (see Fig. 1). Fexc describes the excitation through an actuator-piezo element. In our simulation Fexc is held constant through normalization of the gain factor by the actual vibration amplitude Fexc ðtÞ ¼
A0exc zðt tphase Þ AðtÞ
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tphase is usually set to 1/4 of the actual frequency f ¼ f0 þ Df . The differential equation was solved with the Verlet algorithm [14]. All input parameters can be determined experimentally. We chose parameters which are of typical order of magnitude for non-contact AFM applications: f ¼ 300 kHz, A ¼ 10 nm (i.e. 20 nmpp), kl ¼ 40 N/m, Q ¼ 20 000. It needs to be emphasized, however, that a drastic change of any of the parameters, like for example going to soft cantilevers (kl < 1 N/m), may have a strong influence on the general characteristics of the dynamic solution of Eq. (6). 2.3. Model forces In order to give a broad picture of the special features of the CE mode, we used various different forms of the tip–sample interaction forces in order to consider very different cases. We would like to point out that the exact form of such forces does not play a major role here, as we are interested in rather general conclusions. The exact form of the interaction depends on a lot of experimental parameters like the material of tip and sample, the shape of the tip and the bias voltage. Thus, the force curves may vary strongly in decay length and magnitude. We chose model forces which produce right orders of magnitude in strength and reasonable decay lengths and distance dependencies. 2.3.1. Conservative force We used the model by Muller et al. [15] (MYD model) as depicted in Fig. 2a. This model produces complete force curves including attractive and repulsive regime and the transition regime between the two. Typically, contact is established at the minimum of such a force curve. It is based on continuum theory and uses van der Waals attraction as well as elastic deformation of the solid. Other, more adequate force interactions use short range forces which account for more specific adhesion forces like electronic interaction [16–19]. For our purposes the MYD model is sufficient.
(7)
The actual vibration amplitude (peak-to-peak value) is denoted by A(t), and A0exc is the set point value of the excitation and tphase the phase shift. The inverse of
2.3.2. Damping force It is known that in dynamic AFM energy is dissipated into the tip–sample interaction [21]. A realistic dynamic tip–sample contact cannot be purely conservative.
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The characteristic features of such a damping force are defined via the damping coefficient gð~zÞ, which depends only on the tip–sample distance. A suitable gð~zÞ could have the form ~z (9) g ¼ g0 exp exp z0 This function reflects the strong increase of dissipation with increasing strength of tip–sample interaction. It includes a part decaying into the non-contact region. Note that gð~zÞ may be an effective quantity and Eq. (8) an effective damping force. True damping mechanisms are still under debate and this form may serve as a starting point. In order to match experimental orders of magnitude [13,28] we chose the constants g0 exp ¼ 108 N s/m and z0 ¼ 0:5 nm. For a solution of the integral (4), see, e.g. [13,22]. 2.3.3. Hysteretic force Another possible mechanism for the energy loss is the hysteresis between approach and retraction [23–26]. Here, we chose a hysteretic force which stems from the MYD model for an extremely soft surface. A detailed description can be found in [24]. Again, we would like to point out that the specific form of the curve is not of importance here. Other ‘atomistic theories’ also predict hysteresis of similar form and order of magnitude (e.g. [27]). Hence, this might serve as a general model for hysteresis effects of different origins.
3. Results and discussion 3.1. Conservative force
Fig. 2. Model forces used in the simulations as a function of the absolute tip–sample distance. (a) Conservative force from the MYD model with parameters for a silicon tip and silicon surface (for details see [13]). The point of first repulsive contact at the apex of the tip (with overall force still attractive), i.e. the contact point is taken as the zero point of the distance axis. (b) Damping coefficient: exponential decay. (c) Hysteretic force from the MYD model (for details see [24]).
In order to introduce damping effects into the equation of motion, one may choose a damping force of the form Ftsdiss ð~z; z_ Þ ¼ gð~zÞ_z
(8)
The results of a simulation with the pure conservative force from Fig. 2a is shown in Fig. 3. The resulting Df(d)-curve and A(d)-curve are shown in Fig. 3a and b, respectively. Most striking is that the Df(d)-curve is almost identical to the case of CA–FM–AFM [13]. The resulting Df(d)-curve appears slightly stretched around zero. The marginal difference stems mainly from a slight deviation of the amplitude from its starting value. From the trajectory z(t) the forces acting at the turning point of the oscillation were determined as shown in Fig. 3c. Repulsive interaction between the
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Fig. 3. Computer simulation of a DFS curve using the pure conservative force from Fig. 2a.
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Fig. 4. Computer simulation of a DFS curve using the conservative force from Fig. 2a and a damping force as in Eq. (8) (Fig. 2b).
apex of the tip and the sample occurs for the first time at the minimum of the force curve. This corresponds to a point slightly before the minimum (on the right hand side of the minimum, near the turning point) of Df(d). The average force given by the static deflection of the lever is depicted in Fig. 3d. This turns repulsive after repulsion at the turning point of the oscillation has been established. The amplitude decay in the repulsive regime is small compared to experimental findings which can, therefore, only be explained using energy dissipation (see below). The small increase of the amplitude in the attractive regime, however, has been observed experimentally [8,9]. 3.2. Conservative force þ damping force The next example includes both a conservative force as in Fig. 2a and a damping force of the form of Eq. (8). From the results displayed in Figs. 4 and 5 it can be seen that the DFS curves are changed crucially by the presence of the damping force. The amplitude drops and the range of the trajectory of the oscillation changes accordingly. This influences the frequency
Fig. 5. Computer simulation of a DFS curve using the conservative force from Fig. 2a and a damping force as in Eq. (8) (Fig. 2b).
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shift. The amount of energy dissipated at the contact point is of the order of 1014 W, which is rather little compared to experimental data obtained with the CA– FM technique [13,28]. With two respects experimental data [5,6,8,9] using the CE mode are different. First, a long range of drop of amplitude does not seem to take place in experiment. The transition regime between the part of the curve where A is almost constant to the part where the amplitude drops linearly with a slope close to 1 is of the order of a few Angstroms only. Secondly, the dissipation in the repulsive regime is stronger in the experiments (i.e. @AðdÞ=@d ! 1 in the simulation, and @AðdÞ=@d 1 in the experiments). Hence, we used a different model of the dissipation in another simulation which ignores long range effects and starts out stronger at the beginning of the repulsive regime gð~zÞ ¼ g0~z
(10) 4
2
with g0 ¼ 3:33 10 N s/m if ~z < 0, and g0 ¼ 0 else. As is shown in Fig. 5 the simulation results reproduce more the expected behavior. (The small increase of amplitude in the attractive regime (at d A0 > 0 where no damping force acts) is equivalent to the case of a pure conservative force discussed above.) It seems, that long range dissipation effects as observed in [13,18,28] for the CA mode are not present in [5,6,8,9] for the CE mode. Hopefully, more experimental data for different tip and sample materials can explain this apparent contradiction.
Fig. 6. Computer simulation of a DFS curve using the noncontinuous force from Fig. 2c (i).
a discontinuity, but changes the slope when the turning point passes the discontinuity of the force curve. The ˚ , but it remains frequency changes rapidly within 1 A continuous. The general behavior is not surprising with regard to Eq. (1) (see [24]). Again it becomes clear that for an interpretation of experimental data it is not sufficient to regard conservative forces only.
3.3. Discontinuous conservative force 3.4. Hysteretic conservative force With regard to experimental data which showed discontinuous Df(d)-curves, it has been speculated that these are originated in force curves which show a sudden increase of attractive forces after a certain distance is reached [6–8]. For a model calculation it is straightforward to choose just the curve: (i) of the hysteretic force in Fig. 2c for both approach and retraction. The resulting force is conservative on the whole range except at the point of discontinuity. The condition of a ‘sudden onset of stronger attractive forces’ is clearly given for such a force interaction. The results of a simulation are shown in Fig. 6. As there is no energy dissipation into the tip–sample contact the amplitude remains almost constant (as in Fig. 3). The frequency shift curve does not show
The situation is different when in addition to the discontinuity a hysteresis as in Fig. 7 is introduced. Then a dissipative channel is opened. Consequently, we find a decay of the amplitude. Firstly, the simulated curves are identical to the ones in Fig. 6 in the regime where the hysteresis is not yet reached (Region I in Fig. 7). Then, starting from the point where the tip runs through the hysteresis for the first time, the amplitude decays linearly with a slope of @AðdÞ=@d ¼ 1 (Region II). Finally, the corresponding energy dissipation rises almost linearly until the dissipation corresponds to PðdÞ ¼ ½f0 þ Df ðdÞ DEhys (Region III). Each time the tip runs through the hysteresis an energy of DEhys ¼ 0:660 eV is dissipated. On the other hand,
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Fig. 7. Computer simulation of a DFS curve using the hysteretic force from Fig. 2c (i) and (ii).
the amount of energy which can be dissipated is given by the actual value of the amplitude. Hence a stable solution of the analytic coupled Eqs. (1) and (5) (i.e. an oscillation with a constant amplitude for a given distance d) cannot strictly be found in the specified Region II. The simulation shows, that the mean value of the amplitude has a constant value for a given d. However, individual oscillation cycles vary by some pico meters, so that only a certain fraction of cycles n ¼ nðdÞ undergoes the hysteresis. As a result the energy dissipation can be expressed as PðdÞ ¼ n DEhys ½f0 þ Df ðdÞ . As a direct consequence the frequency shift drops also almost linearly in that region. Unlike the case of CA–FM–AFM the Df(d)-curve remains continuous. Region I becomes clear that this is a non-linear problem which can be solved with the numeric solution of Eq. (6). 3.5. Interpretation of discontinuous Df(d)-curves None of the various tip–sample interaction forces lead to non-continuous Df(d)-curves as were reported
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in the experiments [6–10]. The question arises how to interpret these experimental findings. A straightforward explanation could be that tip or sample might be irreversibly changed after contact between tip and sample was established. However, this explanation can be ruled out by regarding the reproducibility of experimental data [7]. A closer look at the experimental data, however, ˚ in shows that the resolution of the d-axis (about 1 A [8]) is not sufficient to rule out a behavior as seen in Fig. 6 in the non-continuous case. There the frequency ˚ by about 60 Hz. Apparently, drops within less than 1 A there is no need to assume hysteresis in order to explain the behavior of the experimental Df(d)-curves, but maybe it can be used to explain the amplitude decay after the ‘discontinuity’ of Df(d) in experiments (i.e. left-hand side). In order to resolve such a specific bend of Df(d) as in Fig. 6 in the experiment a very defined and stable experimental setup is needed. It seems that the required force interaction needs to change rapidly (not necessarily in a discontinuous form as was used in our example of Fig. 6) after a certain distance is reached, but has associated an energy loss mechanism, possibly a hysteresis.
4. Summary and conclusions We have presented some examples of different tip– sample interactions and showed how they are probed with CE mode AFM. We have seen that although analytic approximate solutions can be applied, a full solution of the equation of motion is capable of solving the problem of two-linked spectroscopic channels (conservative and dissipative forces being probed with frequency and amplitude). It became clear that for a full understanding of the curves both conservative and disspative interactions have to be considered. In the case of hysteretic forces a stable solution with a constant amplitude cannot be found. A hysteresis has a completely different effect on DFS-curves as in the CA mode. Although, it is not so straightforward, it should be possible to learn a lot about dissipative forces from the experimental data. It might be useful to summarize some of the most important differences between this and other modes of operating a dynamic AFM.
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In the CE mode an observed amplitude decay is mainly a sign of energy dissipation. This is mainly due to the phase shift between excitation and vibration is kept constant by the electronics in the CE mode. Hence, a detuning off resonance (as in tapping mode) does not occur. In contrast, in tapping mode an amplitude decay can also be expected for pure conservative forces due to a detuning off resonance [29]. As speculated in previous papers [5], the CE mode leads to a relative protection of tip. However, for high resolution a careful handling is essential anyway. Published high quality data obtained with the CA– FM technique suggest that this point is not essential. Further, in an accidental instability the maximum amount of energy stored in the oscillator of about 104 eV is enough to damage the tip. For force-spectroscopy applications it might be interesting to note that the regime of strong repulsive forces cannot be probed using the CE mode. This can either be a disadvantage or an advantage depending on the experimental goal. On the whole the interpretation of DFS is rather more difficult than in the CA mode, due to mixing of spectroscopic channels. References [1] T.R. Albrecht et al., J. Appl. Phys. 69 (1991) 668. [2] F.J. Giessibl, Science 267 (1995) 68. [3] Proceedings of the First International Workshop on Noncontact AFM, Appl. Surf. Sci. 140 (1999) 243–456. [4] Proceedings of the Second International Workshop on Noncontact AFM, Appl. Surf. Sci. 157 (2000) 207–428. [5] H. Ueyama, Y. Sugawara, S. Morita, Appl. Phys. A 66 (1998) S295.
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