WO2003094174A1

WO2003094174A1 - Continuous temporal evolution method and implementation thereof in order to optimise the frequency response of a force microscope

Info

Publication number: WO2003094174A1
Application number: PCT/ES2003/000191
Authority: WO
Inventors: Tomás R. RODRÍGUEZ FRUTOS; Ricardo García García
Original assignee: Consejo Superior De Investigaciones Científicas
Priority date: 2002-05-03
Filing date: 2003-04-30
Publication date: 2003-11-13
Also published as: AU2003224180A1; ES2194608B1; ES2194608A1

Abstract

The invention relates to an oscillating mode force microscope (AFM) which has been transformed into one of the most powerful techniques used to characterise surfaces on a nanometric scale. However, the use thereof in liquid media is greatly conditioned by the decline in the quality factor (Q) of the microlever of the AFM. The aforementioned decline, which is the result of an increase in the hydrodynamic friction with the fluid, produces effects such as a reduction in topography sensitivity, slower microscope operation and the presence of greater forces on the sample. As a result, different modes of effectively increasing factor Q have been developed. Nonetheless, all of the analyses of earlier methods are based on the hypothesis of a stationary response.

Description

TITLE

THE METHOD OF CONTINUOUS TEMPORARY EVOLUTION AND ITS IMPLEMENTATION TO OPTIMIZE THE FREQUENCY RESPONSE OF A FORCE MICROSCOPE

SECTOR

Microscopy

BACKGROUND In 1998 Anczykowski et al. (1) suggested that an effective increase in the microbar lever quality factor of a oscillating mode microscope could facilitate the operation of the microscope in the so-called attractive mode in the presence of long-range electrostatic forces. They proposed that this would increase the stability of the microscope. For this they designed an electronic circuit that contained a phase shifter, a signal amplifier and an adder (scheme 1). All this was intended to offset the instantaneous signal, amplify it and add it to the excitation. The same authors made a theoretical analysis of the system based on the equation

• • • mz = -k „z -a - _~ z + E„ coscot) ^c eff 0 with a _e r = a + G, in this way by adjusting the gain it is possible to modify the effective quality factor Q _ej f = m ü / a _e ff. Implicitly in the previous analysis it was assumed that the movement of the micro lever could be described by a solution of the type z (t) ∞A sinωt.

Which, as will be seen later, deeply limits the description of the system.

SHEET DΕ REPLACEMENT RULE 26 To be

function generator adder

Scheme 1

In 2000 Tamayo and collaborators (2) applied the previous idea for the use of a microscope of forces in liquid media. As in the previous case, these authors designed an electronic circuit (scheme 2) to perform the increase in the quality factor, and also performed an analysis of the behavior of the system based on the equation,

mz = -k _c z - γ z + F cos (ωt) + GA. cos (ωt - φ + π / 2) + E. mt

The previous authors solved the aforementioned equation using an iterative method for the choice of Ai and y. Like Ancykowski and collaborators, Miles's group in Bristol assumed a stationary response and dominated by the first harmonic z (t) ∞A sinωt.

Our development is based on both theoretical and experimental work carried out over eight years in the Forces and Tunnel laboratory of IMM for which it is responsible (RG). Numerous original results on the dependence of the amplitude of the oscillation, phase and resonance frequency of the micro lever with the intensity and non-linear nature of the forces existing in the microscope have been published in internationally renowned journals (3) and presented as Invited contributions at numerous international conferences. On the other hand, the work most directly related to this development proposed here, which contemplates the continuous temporal evolution of the micro lever with special emphasis on the relevance of the transitory terms, will remain unpublished until the resolution of this request (4).

AFM head

Rilípnentnción's tie

Electronics + control unit

Scheme 2

The current proposal presents the following improvements with respect to the previous proposals.

(to). The method of continuous temporal evolution allows, by solving the system equation from the initial moment, greater stability in the response in frequencies to high gains. In addition, it also allows greater speed in the response of the system as the

SUBSTITUTE RULE 26 feedback is possible during transients, which does not make it necessary for them to disappear to assume the stationary solution. All of this would allow the development of a digital program that would increase the effective quality factor of the system from 5 to 1000. Which, as far as we know, is not possible with programs based on stationary feedback as its transitory diverges for effective Q factors greater than 200. All this would result in a more precise and faster operation of the modules responsible for modifying the quality factor (scheme 3).

(b). By being able to simulate the frequency curve throughout its range, that is, from frequencies that are much lower than that of resonance at much higher frequencies, the microscope operation can be selected at any frequency regardless of the gain. On the other hand, it would make possible the on-demand choice of the dynamic response of the system before conducting the experiment. This would mean, on the one hand, a greater reproducibility when establishing the experimental conditions, on the other hand, a greater flexibility by being able to excite the system at frequencies other than that of resonance.

(one). B. Ancykowski, J.P. Cleveland, D. Krüger, N. Elings and H. Fuchs, Appl. Phys. A66,

S885-S889 (1998).

(two). J. Tamayo, A..D.L. Humphris, and M.J. Miles, Appl. Phys. Lett. 77, 582 (2000); A.D.L. Humphris et al. Langmuir 16, 7891 (2000); J. Tamayo et al. Biophys J. 81, 526

(2001).

(3). R. García et al. Appl. Phys. A66, S309 (1998); R. García and A. San Paulo, Phys. Rev.

B 60, 4961 (1999); J. Tamayo and R. García, Langmuir 12, 4430 (1996); R. García et al.

Surf. Anal interface 27, 312 (1999); R. García and A. San Paulo, Ultramicroscopy 82, 79 (2000); A. San Paulo and R. García, Biophys. J. 78, 1599 (2000); R. García and A. San

Paulo, Phys. Rev. B 61, R13381 (2000); A. San Paulo and R. García, Phys. Rev. B 64,

193411 (2001); A. San Paulo and R. García, Surface Sci. 471, 71 (2001); T.R. Rodríguez and R. García, Appl. Phys. Lett. 81 (2002)

(4). T.R. Rodríguez and R. García (pending publication).

GLA 26 Acquisition and signal processing

espe

To be

Function generator

Scheme 3

DESCRIPTION OF THE INVENTION

As mentioned earlier, this proposal proposes to program the frequency response of a force microscope based on the continuous evolution method. To better understand the scope of this proposal, we will review the developments used in references (1) and (2) of the previous section to understand the limitations associated with these approaches.

1. Quality factor control through stationary approaches

The equation of the starting movement is

x + x

This equation is identical to that used by Ancykowski and in practice is the one used by the works referenced in (2).

Where x is the deflection, Q _nat is the unmodified quality factor, coo is the natural frequency of the system, G is the gain and f is the excitation force. This equation has as a general solution the sum of a particular solution and the homogeneous solution.

1.1. Particular solution.

We assume: x _p (t) = _4e ' ^{(ω (_φ)} => substituting in the equation

ω ₀ ² -ω ² e _! l

Now we consider that G is a constant. The new quality factor will depend on the frequency in this way:

β '(ω) GQ „

1 - preserve,

The new free amplitude also depends on the frequency through the quality factor: / β '(ω)

A (ω) = mω „

To set G and f, we set the quality factor and the free amplitude in the resonant frequency that will be two of our input parameters:

Define: Q '(ω _Q ) ≡ Q'; A ' ₀ (ω ₀ ) ≡ A ^< ₀

With these considerations we have that the amplitude curve versus the frequency of the particular solution is as follows:

Figure 1 shows this curve for different values of Q 'and Q _mt . We see that for a system with a natural Q of 5 to which we apply the corresponding gain to have a Q 'of 200, the amplitude curve is identical to that of a system with a natural Q of 200.

A priori this would be the ideal case of Q-control operation. However, in any real system before reaching the steady state there are some transitory terms that have not been considered in this analysis and, as we will see below, dramatically modify the behavior of the system. To do this we also have to solve the homogeneous equation. 1.2. Homogeneous solution

I guess: x _h (t) = e ^kl - ^ replacing in the equation

The general solution will be: x (t) = x _h (i) + x _p (t) = e ^~ ° "(R cosfut + C senτσt) + _4 (ω) cos (ωt -φ (ω))

With: ω „α = -

2fl 2ω

R, C constants that depend on the initial conditions

We have to study in which case the system stops decaying exponentially as a function of Q '. This will happen when α <0, that is: ω ₀

<0: 1 <. -β nat ^ 2β „2ω QQ _n ω

ω

1- Qn • UNSTABLE OR rx Q

This can be seen in Figure 2. These curves are calculated by solving the equation of motion numerically by varying the excitation frequency. We measure the amplitude after 4000 periods. We see that for the case with Q _mt = 5 and Q '= 200 the curve becomes unstable at ω / ωo = 0.975. Figure 3 shows the divergence of the signal over time when we excite with a frequency less than 0.975o_o. Both the instability and the divergence observed in the previous figures show the limitations of the frequency response programs of the microscope of forces based on stationary or pseudo-stationary feedback. The previous theoretical description corresponds to that made by most of the articles that refer to the subject and also predict instabilities of the type mentioned here (1-2). All its argument lies in the following, the extra excitation is due to a feedback with the output signal out of phase 90 ° and if the movement of the micro lever is sinusoidal, said excitation is proportional to the speed and, from there, it reaches the equation of motion we pointed out at the beginning. However, this is not correct since for that signal to be the velocity we have to be in a steady state all the time, which is incorrect because of ignoring the transient terms that are excited whenever a disturbance or excitation is introduced to the oscillating system.

2. Control of the Q-factor through feedback based on the method of continuous temporal evolution

The starting equation contains an outdated term a quarter of a period as this is the term that increases the quality factor,

x + cosωt -Gx (t- ~)

If the solution were stationary (x (t) ~ cos (ωt-φ)), then the feedback term will be identical to the previous case with velocity. However, the feedback includes both the stationary and the transient component and therefore the description must be made including both. 2.1. Particular solution.

For greater agility in the solution, we use complex variables. This is how it is obtained,

x + - —x + G> ² x = - cosωt -Gx (t - ^ -): zínat m (assuming x (t) = Ae ' ^(a) )

β 'and Q have the same meaning as in the previous case. Therefore, Figure 1 is also valid for this type of description. The contribution of the transients of the system is obtained from solving the homogeneous equation.

2.2. Homogeneous solution:

x + - - x + ωlx + Gx {t — ξ) = Q ^> (assuming x _h (t) = e ^kt )

* ¿Nat k ', 2 ^l +, - ^ω 0 -k 7, +, ω „2 ² + G ~ e - ^~ í * - = 0 fí nat

We see that there is a transcendent equation for complex unknowns k. Therefore, the

2 x (t) = ∑B ^¡ '+ _4 (ω) cos (ωt -φ) particular solution will be the following:

Where B, they are constants that depend on the initial conditions and k¡ are the complex roots of the previous transcendent equation. In this case we cannot decide with certainty if the general solution is going to be stable or not due to the difficulty of finding the roots k¡. Therefore, we will simply draw the amplitude curves against the frequency of Figure 2 but solve the equation with feedback based on the method of continuous temporal evolution. The simulations have been carried out as follows,

1st. We have as input the natural quality factor Q _nat , the resonance frequency fo and the constant of the micro lever k _c .

2nd. We also choose Q 'and _4 ₀ ' as input so that in the particular solution (which is supposed to be stationary) there is that quality factor and that free amplitude. This sets us / and G.

3rd. The integration is done in a total time equal to 4000 periods. The feedback is "turned on" after 50 periods from the initial moment to initially feed back a steady state. The feedback is then left for the remaining time.

A typical example of time swing simulation can be seen in Figure 4. Figure 4.b shows in detail where the feedback is turned on.

4th. The amplitude versus frequency curves are obtained from the curves of Figure 4 by varying the excitation frequency and measuring the amplitude of the oscillation in the last period.

To make a comparison with the stationary method the same curves are calculated as in figure 2 (figure 5). Here we draw on each graph three curves, the resonance curve for the feedback system, the least squares adjustment of this curve to a Lorentzian and the resonance curve for an unrefeed system with a Q _nat equal to the Q 'of the feedback. The following results are observed,

ON RULE 26 1. The resonance curves are more stable in this case than in the stationary case because divergences are no longer observed at low frequencies.

2. As we change the gain and strength to obtain a higher Q 'and with a constant AO, we see that we obtain both a free amplitude and a lower quality factor than expected if we had a stationary described only by the particular solution .

3. The system's resonance frequency shifts to higher frequencies as we increase the gain. In any case, the displacement is less than 0.1 kHz but at high Q 'can cause a significant drop in amplitude at the natural resonance frequency of the system / Q. This can be seen in Figure 6. The oscillation at frequency jo is represented in a system with an O, „ _α = 1000 and in another system with a Q _mt = 5 and Q '= 1000 (Q _e ff = 664) and we see that the amplitude in the latter is much smaller due to the combined effect of the decrease in free amplitude and the displacement of the resonance.

conclusion

The theoretical explanation given in the articles we referred to above is not correct (1-2) since it is not equivalent to consider the 90 ° offset deflection signal as the velocity at all times. Precisely the description that considers feedback in deflection at an earlier time gives stable resonance curves, although these curves do not correspond to the particular solution of the equation.

Case of a micro lever in the vicinity of a sample

Next, the behavior of the system is studied when the micro lever is in the vicinity of a surface. For this we use the equation of motion with feedback based on the method of continuous temporal evolution described above. In addition, we include the tip-sample interaction given by the DMT model for short-range forces and van der Waals forces for long-range forces. We perform amplitude curves versus distance over time, varying the

STITUTION RULE 26 Tip-sample distance z _c with a round-trip ramp. The choice of the gain G and the exciter force / is carried out as described in the case without sample.

Simulation details: We choose the parameters (A ', k, Q', G, f _sxc = f). In the initial moment the cantilever is 3 microns. The first 100 periods allow the system to evolve without any feedback until a stationary is reached. After that time the feedback is switched on (still at 3 microns) and 400 periods are allowed to pass. Again a stationary is reached with the amplitude slightly less than_4 or because the general solution of the movement is not only the particular solution (see the case without sample). Subsequently the tip is approached to a distance equal to (A '+5) ran. From there the distance is reduced to 0 nm to increase again to reach (A'Q + 5) nm. Each curve is simulated to be performed at approximately 4 Hz.

With respect to the interactions we have to say that the repulsive force will be the same for all cases, with a sample of reduced elastic modulus of 1.51 GPa. In the case of the van der Waals force and the adhesion force we will have a Hamaker constant of 6.4 10 ^"20 J for QnaXlO while for β„ _β. <10 we will have a 6.4 10 ^"21 J constant, that is, an order of magnitude less. The radius of the tip in all cases is 20 nm. The resonant frequency, force constant and AO in all cases are 350 kHz, 40 N / m and 20 nm respectively.

Results

The magnitudes we represent are deflection, average force, maximum force and minimum distance between the tip and the sample. We want to see the differences that exist when we feedback or not a system with a low Q, _ιai . We see this in figure 8. In red the deflection, the average force, the maximum force and the minimum distance for a system with a Q _nat = 5 without feedback are represented and in black the same system but feedback so that a Q '= 50. Notable differences are observed. First, the deflection has a greater slope and is more symmetrical in the case of feedback. Second, both the average force and the maximum force are much greater in the

TION RULE 26 case without feedback with a low Q _mt . The minimum distance (deformation if it is less than zero) is much less in the feedback system.

In addition, with this system we can reduce the bistability range for systems with a high quality factor increasing it further. This can be seen in Figure 8, where the natural system favors the attractive operating regime.

DESCRIPTION OF THE FIGURES

Figure 1. Amplitude of the particular solution as a function of frequency for two cases without feedback with natural quality factors equal to 5 (blue) and 200 (black dots). In red a case fed from 5 to 200.

Figure 2. Amplitude of the complete solution as a function of the frequency for the cases described in Figure 1. Note the drastic influence of the homogeneous solution in the case fed back for low frequencies (b). Figure 3. Divergent temporal response of the micro lever in the feedback system at a slightly lower excitation frequency than that of resonance.

Figure 4. Temporary response of the micropalnca for a feedback system according to the method of continuous temporal evolution. Figure 4.b shows in detail where the feedback is turned on. Figure 6. Differences ( _c) a system with a high natural Q (1000) and another with a low natural Q (5) fed back _> using the present method until its particular solution has a high Q '(1000).

Figure 7. Deflection curve (a), average force (b), maximum force (c) and minimum distance (d) as a function of the tip-surface distance for a case with a low natural Q without feedback (red) and feedback according to the method of continuous temporal evolution (black).

Figure 8. Deflection curve (a), average force (b), maximum force (c) and minimum distance (d) as a function of point-to-surface distance for a natural system that has bistability and no feedback system.

TION RULE 26

Claims

(to). The method of temporal evolution continues to resolve the frequency response of the micro lever of a force microscope. This method considers that the response of the micro lever is affected both by the terms from solving the particular equation and the homogeneous system. And this makes it possible to establish that the most optimal feedback to effectively increase the gain of the G system is to add to the instantaneous deflection the existing signal a quarter of an earlier period.

(b). Implementation of the method mentioned in (a) in a computer program to control the dynamic response of the microscope of a force microscope. Given the characteristics of the method of continuous temporal evolution, this implementation would allow to avoid the existing divergences at frequencies lower than the resonance presented by programs developed with stationary solutions when G> γω.

(c). The method described in (a) provides higher Q quality factors for G values lower than those obtained by stationary methods because the feedback is always out of date T / 4 with respect to the instantaneous signal.

(d). Program the response in frequencies of the movement of the micro lever. This includes: (1) the program for acquiring the frequency response of the unmodified micro lever. (2) The adjustment of the experimental data to a Lorenzian curve. From this adjustment, the free amplitude, the resonant frequency oo and the natural quality factor Q _{nat are obtained} . With these values and the program described in (b) the user can choose the frequency response of the desired microscope.

STITUTION RULE 26