Frequency Analysis and Identification in Atomic Force Microscopy

Transcription

1 Frequency Analysis and Identification in Atomic Force Microscopy Michele Basso, Donatello Materassi July 14, 26 1 Introduction In the last few years nanotechnologies have had a considerable development due to the realization of devices capable of obtaining images with a nanometric resolution. The Atomic Force Microscope (AFM) is one of the most diffused instruments of this kind [Meyer et. al. (23)]. The concept of force microscopy is based on the measurement of forces between a sharp tip and a sample surface. The tip is mounted on the end of a centilever which acts as a force sensor. Depending on the operating mode, either the static deflection of the cantilever or the change in its dynamic properties due to the tip-sample interaction can be exploited. In this chapter, we limit the study to the so-called tapping or dynamic, whose schematic is depicted in Figure 1: the cantilever is periodically forced by a piezo placed under its support inducing a periodic oscillation that is naturally influenced by the interaction forces between the cantilever tip and the sample to analyze. The topography can be inferred by slowly moving the cantilever along the sample surface by means of a piezoactuator and by measuring the amplitude of the cantilever deflection through an optical lever method. A feedback controller driving the piezo input voltage is employed to reject variations of the separation between the sample and the tip center of oscillation due to the sample topography. To this aim, the separation-amplitude curve, that is the plot of the oscillation amplitude as a function of the above separation, is a crucial information in order to retrieve the tip-sample distance from the amplitude signal along the scanning lines. Moreover, topography is not the only information one might be interested in. In many biological applications, other sample features and characteristics are even more important than imaging, and are strictly related to the identification of the tip-sample interaction forces. Unfortunately, these forces have a highly nonlinear behavior depending on a number of factors, such as different materials properties, different surface forces, operating conditions, etc., which affect the separation-amplitude relation. These phenomena make the important problem of identifying different surface forces quite challenging. 1

2 Proportional, integral, and derivative feedback control Piezo Voltage Laser Photodetector Amplitude / phase Error signal High-frequency signal generator Z Piezo T - B Diff. amp. Set point Sample Figure 1: Schematic of a tapping-mode AFM In practice, the standard methods to infer some of the sample characteristics from specific experiments are one of the following: 1. directly retrieving static force-distance curves; 2. computing force-distance curves from separation-amplitude ones. While the first approach has many limitations, specially when dealing with soft biological samples, the latter is generally based on the identification of a tipsample interaction force model for the AFM. Many such models are available in literature (see, e.g., [Cappella and Dietler (1999)]), although in most cases the separation-amplitude curve is generally obtained by computationally intensive numerical simulations. In this respect, only a few models are capable to provide interesting analytical results. For instance, in [Salapaka et. al. (2)] and [Materassi et. al. (23a)] an analytical expression is obtained for two simplified classes of models. In this work we analytically evaluate the separation-amplitude curve for a large class of models comprising some of most common interaction potential functions studied in the literature, such as the classical Lennard-Jones potential [Cappella and Dietler (1999)]. The proposed approach relies on the use of harmonic balance techniques, providing an useful insight of many nonlinear known phenomena with respect to fully numerical approaches. Other peculiar features of the proposed class of models are: i) can easily account for energy losses by incorporating the hysteresis function approach developed in [Materassi et. al. (23a)]; ii) is suited for nonlinear frequency-domain identification techniques such as those proposed in [Sebastian et. al. (21)] and [Basso et. al. (1997)]. The paper is organized as follows. In Section 2 the considered AFM model class is described; in Section 3 the Harmonic Balance (HB) technique is exploited 2

3 for the analysis of the AFM tip oscillations, whereas in Section 4 identification methods and some preliminary experimental results are provided. Finally, in Section 5 main results are briefly discussed. 2 Modeling AFM cantilevers operating in dynamic mode are almost always modeled as a feedback interconnection of a SISO linear system L and a nonlinear static function h as depicted in Figure 2. Models of this kind are well-known in γ(t) + L y(t) h( ) δ(t) + l Figure 2: A feedback interconnection of a linear system and a nonlinear static function control literature and are usually referred to as Lur e models [Khalil (1996)]. Using a symbolic notation, we can write the system equation in the following form ( ) d L y(t) = h[δ(t)] + γ(t) (1) dt where L(s) is the transfer function of the system L, y(t) is the measured output (that is the cantilever tip position), l is the separation, δ(t) = y(t)+l represents the tip-sample distance, and γ(t) is the external periodic forcing γ(t) = Γcos ωt. From a physical point of view, the transfer function L(s) describes the free cantilever dynamics only, while the nonlinear function h represents the interaction between the tip and the sample. AFM cantilevers have their lengths in the range of 1 3µm. When operating in dynamic mode, common values for the tip oscillation amplitudes are of the order of 1 1nm, that is less than 1 3 times the whole cantilever length. Thus, it seems reasonable to assume a linear behaviour for an AFM cantilever, which, practically, acts as simple elastic beam. The identification of L(s) can be achieved indipendently from h in many manners. The most popular techniques perform this identification task measuring the system response to the thermal noise or a simple frequency sweep 3

4 excitation when the sample is absent [Gibson et. al. (25)]. In Figure 3 the result of an identification based on the thermal response is shown. 2.5 x Power Spectral Density [mv2] Frequency [KHz] Figure 3: The system thermal response (solid line) is very well fitted by a fourth order model (dashed line) Conversely, modeling the tip-sample interaction force h is still a challenging task. The main difficulty lies in the choice of a suitable class of functions to describe the force potential. In fact, microscopic bodies are subject to reciprocal forces that are strongly repulsive at short distances and weakly attractive at long ones. In addition to such a highly nonlinear behaviour, in dynamic mode it has also some relevance to take into account dissipations and energy losses onto the sample surface. Force curves are traditionally obtained by moving the sample relative to the cantilever in a quasi-steady state manner when the cantilever is not forced. The sample is moved slow enough that it can be assumed the cantilever tip settles to the equilibrium point corresponding to the given separation l and previous initial conditions. Then, the measured data can be fitted using a proper class of functions. However, such static force curves do not characterize the tip-sample interaction behavior when the cantilever is oscillating. In fact, they cannot describe the damping and lossy processes of the sample, as those dissipation phenomena typically depend on the cantilever velocity and are absent when the velocity is zero. Instead, it is reasonable to think that the interaction force could show a distinct behaviour when then tip is continuosly moving, as it happens in the dynamic mode. This motivates the development of tools to perform a dynamic identification in order to model the forces acting on the cantilever, using functions depending on the tip velocity ẏ = δ, too. 4

5 We consider the following class of functions N K n h n (δ) if δ < h(δ, δ) = N K+ n h n (δ) if δ (2) where h n (δ) are a base of suitable non-negative functions and the dependence on δ occurs in h considering only its sign. From a physical perspective, we can consider (2) as a simple hysteresys law made of two different positional forces: the first one acts when the tip and the sample are approaching and the second one when they are getting further. In order to actually make the interaction described by h(δ, δ) dissipative, we should also impose some additional constraints on the parameters K + n and K n. For example, the condition K n K + n is sufficient since it makes every base element dissipative by itself (although it is not a necessary one). In our analysis, we will consider two special cases of this hysteretic interaction since they allow to obtain analytically or, at least, computationally efficient results. 2.1 Piecewise interaction force The first class of potential functions we treat contains the functions h( ) in the form (2) where N = 2 and h n (δ) = { if δ δ n 1 if δ < (3) In Figure 4 the shape of such a kind of functions is depicted. The variable δ in this model represents the distance between the tip and the region where the sample begins the interaction. 2.2 Lennard Jones-like interaction force The adoption of the Lennard-Jones potential h(δ) = K n δ n + K m δ m n,m N;K n,k m R (4) is a quite common choice when fitting statically measured curves. As analogy, we consider a possible generalization of the Lennard-Jones Potential in the form (2) where h n (y) = 1 δ n. (5) The variable δ now represents the tip distance from the sample region where the interaction force becomes infinitely large. As it is obvious, apart an additive constant the two δ s in the proposed classes of models have identical meaning and roles. 5

6 K1 K 2 (y + l) h K 1 + K+ 2 (y + l) l y K 1 K + 1 Figure 4: 3 Frequency Analysis via Harmonic Balance Considering physical values for the mass and the elastic constant of the cantilever, the linear part of the Lur e system given by L(s) in (1) typically shows a marked filtering effect beyond the first resonance peak due to a very small damping. This suggests to exploit the harmonic balance (HB) method to analyze the system periodic behavior [Khalil (1996)]. Limiting to the first order harmonic, we can assume y(t) Re[A+Be iωt ] and approximate the corresponding output of the nonlinear hysteresis block h ( y(t) + l,ẏ(t) ) Re[N A + N 1 Be iωt ], where 8 >< N = N (A, B, ω) := 1 Z 1 T h`y(t), ẏ(t) dt A T >: N 1 = N 1(A, B, ω) := 1 B 2 T Z T h`y(t), ẏ(t) e iωt dt are the constant and harmonic gains of the nonlinear block usually denoted as the describing functions of the nonlinearity. For the general class of hysteretic force models introduced, we obtain N = 1 2πA = 1 2πA NX Z K n + h n(l + A + B cos τ)dτ + π NX Z +π Σ n h n[b(q + cos τ)]dτ Z +π (6) «Kn h n(l + A + B cos τ)dτ = where Σ n := K n + K + n 6

7 h(y,y) y l SAMPLE Figure 5: and Similarly, we find for N 1 N 1 = 1 NX Z K n + h n[b(q + cos τ)]e iτ dτ + πb π = 1 πb q := l + A B. (7) Z +π NX Z +π Z +π Σ n h n[b(q + cos τ)] cos τdτ i n «Kn h n[b(q + cos τ)]e iτ dτ = «h n[b(q + cos τ)] sin τdτ (8) where n := K n + Kn. The periodic solutions of system (1) with a sinusoidal forcing γ(t) = Re[Γe i(ωt+φ) ] can be computed through the classical describing function method which originates the following equation to be solved in A, B, φ or, equivalently, A + Be iωt = L()N A + L(iω)[ N 1 B + Γe iφ ]e iωt t (9) { [1 + L()N (A,B)]A = [1 + L(iωt)N 1 (A,B)] B = L(iωt)Γe iφ. (1) Finally, we can easily decouple the variable φ from (1) [1 + L()N (A,B)]A = 1 + L(iωt)N 1 (A,B) B = L(iωt) Γ. φ = arg [ L(iω) 1 + N 1 (A,B) ] (11) 7

8 3.1 Piecewise interaction model analysis For the piecewise-linear potential described in Section 2.1 we obtain { N = 1 A [Σ 1R 1 (q) + Σ 2 R 2 (q)b] N 1 = 1 B [Σ 1S 1 (q) + i 1 T 1 (q)] + [Σ 2 S 2 (q) + i 2 T 2 (q)] (12) where R 1 (q) := acos(q) 2π 1 q 2 S 1 (q) := π T 1 (q) := 1 q π Finally, by the substitutions R 2 (q) := qacos(q) 1 q 2 2π S 2 (q) := acos(q) q 1 q 2 2π (1 q)2 T 2 (q) := 2π χ(q) := Σ 1 R 1 (q) Φ(q) := Σ 1 S 1 (q) + i 1 T 1 (q) Ω(q) := Σ 2 R 2 (q) Ψ(q) := Σ 2 S 2 (q) + i 2 T 2 (q) we can obtain a formally handy expression for the describing functions N = 1 [χ(q) + Ω(q)B] A N 1 = Φ(q) B + Ψ(q). In this model, the variable q can represent the penetration of the tip in the sample. In fact, assuming as exact the first harmonic approximation, we have that for q > 1 the tip does not get in contact with the sample; for q = 1 the tip grazes the sample and for q < 1 we have an effective interaction. The case for q < 1 is meaningless in this model since it would indicate that the tip completely oscillates into the sample; so we will not need to analyze it deeply. From (1), it is also possible to formally write B as a function of q only. In fact Γ B = L(iω) 1 (13) + N 1 implies L(iω) 1 B + BN 1 2 = L(iω) 1 B + Φ + ΨB 2 = Γ 2. (14) The substitutions ˆΦ := Φ and ˆΨ := Ψ + L(iω) 1 yield (ˆΦ + ˆΨB)(ˆΦ + ˆΨ B) = Γ 2 ˆΨ 2 B 2 + 2Re[ˆΦˆΨ ]B + ˆΦ 2 Γ 2 = (15) which is a simple second order algebraic equation whose roots are B = Θ(q) := Re[ˆΦˆΨ ] ± Re[ˆΦˆΨ ] 2 ˆΨ 2 ( ˆΦ 2 Γ 2 ). (16) ˆΨ 2 8

9 Substituting in (11) and reminding that l = qb A, we can finally write A(q) = L()[χ(q) + Ψ(q)Θ(q)] B(q) = Θ(q) φ(q) = arg [ L(iω) 1 + N 1 (A,Θ(q)) ] l(q) = qθ(q) + L()[χ(q) + Ψ(q)Θ(q)] (17) The variable q, for its definition (7), depends on A, B and l, therefore equations (17) are only implicit relations, formally masked as explicit ones. System (17) can not be solved in closed form since it involves transcendental equations. However, it is possible to obtain its solution through a conceptually easy method. Since the variable l is a known parameter of the model, it is possible by the last of (17), to determine the corresponding values of q and then A, B and φ by exploiting the remaining equations. In other words, we have transformed the problem of solving the whole system (1) into the easier problem of solving a single real equation in the unknown q. Experimentally, the separation-amplitude curve is obtained slowly moving the sample towards the cantilever and measuring both the amplitude of the first harmonic and the separation. Although it is not possible to derive an explicit analytical form for B = B(l), we can give a parametric form for it. By using the q-explicit equations in (17) we can consider the parametric curve { l = l(q) q R. (18) B = B(q) In Figure 6 we can observe the comparison beetween simulated results and the predicted ones using the analytical expression (18). The two curves show a good agreement. In addition, we note that there is a range of values of l for which three different periodic solutions exist, two stable and one unstable. This critical behavior, exhibited for some values of the parameters, generates jumps depending on the scan direction (l increasing or decreasing). Both experiments and simulations confirm this phenomenon [Kyhle et. al. (1997)]. The proposed approach has allowed, in addition, to detect the unstable periodic solutions which are difficult to obtain by simulation. A similar behavior can be observed in the phase-separation diagram and can be explained exactly in the same way. In addition, even in this simple model, we can observe that a relation of quasilinearity exists between amplitude and separation [Salapaka et. al. (2)]. 3.2 Lennard Jones-like hysteretic model analysis For the generic hysteretic interaction force of the class (5), we can again evaluate the describing functions N and N 1 of the nonlinearity h: { N N 1 = N = N Σ n AB n R n (q) 1 B n+1 [Σ n S n (q) + i n T n (q)] 9

10 Theoretical Curve Simulation [approaching] Simulation [departing] First harmonic amplitude (nm) Separation (nm) Figure 6: Comparison beetween the separation amplitude curve obtained analytically and by a simulation. where the functions R n (q) := 1 2π S n (q) := 1 π T n (q) := 1 π π π π 1 (q + cos τ) n dτ cos τ (q + cos τ) n dτ sin τ (q + cos τ) n dτ. can be analitically evaluated for any given n and q > 1. Imposing harmonic balance we get { N A = L() Σ n Rn(q) [ Γe iφ N B n Σ ns n(q)+i nt n(q) B n ] L(iω) = B. The second equation of (19) can be expressed in the form L(iω)Γe iφ = L(iω) N (19) Σ n S n (q) + i n T n (q) B n + B (2) We can remove φ by multiplying each term by its conjugate. Finally, multiplying by B 2N the equation can be easily rewritten as a (N + 2)-degree polynomial in the variable B whose coefficients depend on the variable q only p(b) = 2N+2 C n (q)b n =. (21) 1

11 It can be easily shown that C 2N+2 = 1/ L(iω) 2, C 2N+1 = and C 2N = Γ 2. For sufficiently large q (that is when the interaction is neglegible) we have that C k =, k < 2N, such that p(b) = ( L(iω) 2 B 2 Γ 2 )B 2N = (22) One evident root is B = Γ L(iω) with the meaning that, if the cantilever tip is far away from the sample, a periodic solution is detected and its first harmonic amplitude is about l = Γ L(iω), the free oscillation amplitude. For every q > 1, the polynomial equation (21) can be efficiently solved in B keeping only solutions with a physical meaning (the real and positive ones). Then, the constant component of the periodic solution A can be easily evaluated exploiting the first of (19). The phase φ can also be similarly obtained as a function of the parameter q { } N φ(q) = arg L 1 Σ n S n (q) + i n T n (q) (iω) + B n+1. (23) Finally, the parameter l is given by the original relation l(q) = qb(q) A(q). (24) The final result is that the variables A, B, φ and l are all expressed with respect to the parameter q. The separation-amplitude diagram can be therefore obtained considering the pair (l(q),b(q)) which describes a curve in a parametric form. A similar procedure can be used to obtain the relation between any two variables. The proposed analysis technique is computationally efficient for this class of models such that it easily allows to investigate the influence of model parameters in the system behavior. 4 Identification of the Tip-Sample Force Model In this section we develop a method to identify the parameters defining the proposed interaction models directly from experimental data. 4.1 Model identification method As previously discussed, the transfer function L(s) can be indipendently estimated, so it assumed known. A, B, φ are quantities which can be immediately evaluated by the measured signal y(t). Moreover l is a known quantity (apart an additive offset which can be easily estimated). Thus, the variable q = A+l B can be assumed known, too. The class of functions (2) chosen to model the interaction has the useful property to be linear in its parameters Kn and K n +. This allows to develop a simple 11

12 identification procedure which makes again use of HB techniques. Let us define The first order harmonic balance equation leads to a set of 2N linear equations in the unknown variables Σ m and n { Γcos(φ) Im[L 1 (iω)]b = N Σ n Sn(A,B) B n+1 Γsin(φ) Re[L 1 (iω)]b = N n Tn(A,B) B n+1 (25) Assuming to have M > 2N experimental measurements (A 1,B 1,φ 1,l 1 )...(A M,B M,φ M,l M ) and adopting a more compact notation, we can write two indipendent matrix equations P S Σ = Q S P D = Q D (26) where Σ := are the unknown vectors and P S [m,n] := Sn(Am,Bm) Bm n+1 P D [m,n] := Tn(Am,Bm) Bm n+1 Σ 1. Σ N := 1. N (27) Q S [m] := Γcos(φ) Im[L 1 (iω)]b m Q D [m] := Γsin(φ) Im[L 1 (28) (iω)]b m are constant matrices. Σ and can be evaluated from (26) using a least-square method. The presence of linear constraints on the unknown parameters (for example to force the identified force to be dissipative) can be easily handled, too. In fact, the problem just reduces to a contrained quadratic optimization. 4.2 Experimental results A simplified interaction model compared to (5) has been considered h(y,ẏ) = { K na δ na K + na δ na + K nr δ nr + K+ nr δ nr with n r > n a (29) The choice of this simplified model is motivated by simplicity and by an analogy with the standard Lennard-Jones Potential. The identification technique described in the previous section has been employed on a set of experimental data using different values for n a and n r. The transfer function L(s) has been previously identified using a thermal-response based approach. The results obtained for the two cases n a = 2,n r = 3 and n a = 6,n r = 13 are reported in Figure 7 and Figure 8, respectively. It can be noted that the quality of the identification improves increasing the values of the two esponents n a and n r. 12

13 3 identified model experimental data 25 2 Amplitude [nm] Separation [nm] (a) Normalized Force distance δ [nm] (b) Figure 7: An experimental separation-amplitude curve (a) fitted using a Lennard-Jones like model (n a = 2, n r = 3) for the interaction force (b) 13

14 3 identified model experimental data 25 Amplitude [nm] Separation [nm] x (a) Normalized Force distance δ [nm] (b) Figure 8: An experimental separation-amplitude curve (a) fitted using a Lennard-Jones like model (n a = 6, n r = 13) for the interaction force (b) 14

15 5 Conclusions In this chapter we have proposed a class of models for tip-sample interacion dynamics in atomic force microscopy via impact dynamics. The use of a hysteresis law can be well combined with harmonic balance techniques for the analysis of oscillatory behaviors, providing interesting analytical results. For instance, the presence of jump phenomena discovered in many experiments is well-predicted and explained. The suggested method is based on a first order harmonic approximation and gives good quantitative results since the linear part L(s) of the considered Lur e system has a strong filtering effect. In such a situation, the HB technique has advantages over standard numerical approaches since it requires a computational effort much smaller than the one required by a pure simulation of the same model. References [Basso et. al. (1997)] Basso M., Genesio R., Tesi A., Torrini G., On describing systems with periodic behaviour in terms of simple nonlinear models, Conference Control of Oscillations and Chaos, [Cappella and Dietler (1999)] Cappella B., Dietler G., Force distance by atomic force microscopy, Surface Science Reports, 34, 1-14, [Gibson et. al. (25)] Gibson C., Smith D., Roberts C., Calibration of silicon atomic force microscope cantilevers, Nanotechnology, Vol. 16, , 25. [Khalil (1996)] Khalil H., Nonlinear Systems, Second Edition, Prentice-Hall, Upper Saddle River, [Kyhle et. al. (1997)] Kyhle A., Sorensen A., J. Bohr, Role of attractive forces in tapping tip force microscopy, Journal of Applied Physics, 81, , [Materassi et. al. (23a)] Materassi D., Basso M., Genesio R., A Model for Impact Dynamics and its Application to Frequency Analysis of Tapping- Mode Atomic Force Microscopes, Proc. of IEEE Int. Conference on Decision and Control, Maui, 23. [Meyer et. al. (23)] Meyer E., Hug H., Bennewitz R., Scanning Probe Microscopy: The Lab on a Tip, Springer, Berlin, 23. [Salapaka et. al. (2)] Salapaka M., Chen D., Cleveland J., Linearity of amplitude and phase in tapping-mode atomic force microscopy, Physical Review B, 61, , 2. [Sebastian et. al. (21)] Sebastian A., Salapaka M., Chen D., Cleveland J., Harmonic and power balance tools for tapping-mode AFM, Journal of Applied Physics, Vol. 89, ,