On an Overview of Nonlinear and Chaotic Behavior and Their Controls of an Atomic Force Microscopy ( AFM ) Vibrating Problem

© 2012 Balthazar et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. On an Overview of Nonlinear and Chaotic Behavior and Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem


Introduction
It is known that in Brazil and in the whole world, the development of mathematical modeling techniques for the study of problems involving new technologies and the design of nonlinear structures is an emerging area in Engineering Science. To background to the general problems of non-linear dynamics, see the comprehensive monographs: (Awrejcewicz, 1991) and (Awrejcewicz and Lamarque, 2003).
This chapter addresses this issue, involving the dynamical behavior of Electrical Mechanical Systems (EMS), that are systems responsible for electromechanical energy conversion. It is common knowledge that an electromechanical system can be classified in three different groups, according to scales dimensions: i) Macro-Systems >1mm, ii) Micro-Systems <1mm and >0.1μm, and iii) Nano-Systems <0.1 μm and >0.1nm.
The Macro-Systems are characterized by being visible without using microscope and they are present in our Day to Day activities. The Micro-Systems are generally constructed on silicon chips, and are employed in manufacturing integrated electronic circuits. The Nano-Systems can only be seen and manipulated with the use of the Atomic Force Microscope.
In this research field -MEMS and NEMS -, the researches are based on the fact that, in opposition to traditional macro-scale problems, the atomic forces effects and surface phenomena are preponderant over the forces of inertia and gravity. It is also known that the manufacture of these new products have been based on the binomial consuming time and an enormous amount of financial resources. Thus, it becomes necessary some investment in an interface, which addresses the design and manufacturing that, gives designers of

General aspects of the Atomic Force Microscope (AFM)
In Binnig et al. (1986) the atomic force microscope ("AFM") was presented. The AFM has been described as one of the most efficient tools for obtaining high-resolution images of the samples, by exploiting its surface, both in air and in liquid media. At the core of the "AFM", there is a tip, mounted on the end of a micro cantilever, while the tip vibrates to scan the sample, its vibration is detected by a laser system, which emit signals to a photo detector that generate images of the object under examination at high resolution. This movement may vary depending on the need and on the type of material being analyzed. These variations include techniques such as static contact mode and dynamic techniques, such as non-contact mode and intermittent or tapping mode. The principle of operation of the "AFM" is to the measure the deflection of the microcantilever on whose free end the probe is mounted. The deflections (analyzed during the scan) are caused by forces acting between the tip and sample. These forces act on medium to large distances -typically ≥ 100 Å -such as attractive Van der Waals forces, magnetic forces and Coulomb forces (Eisenschitz, 1930). It can be said that when the tip approaches to the sample, it is first attracted by the surface, due to a wide range of attractive forces in the region. This attraction increases when the probe is very close to the sample. However, when the atoms of the tip and sample become very close, the orbital electrons begin to repel. The forces are canceled when the distance between the atoms is of the order of several angstroms (the distance of the order characteristic of a chemical bond). When the forces become positive, it can be said that the atoms of the tip and the sample are in contact and ultimately the repulsive forces dominate.
In Figures: 1,2 and 3, the main characteristics and the operation of an AFM can be observed. Figure 3; display the basic configuration of an AFM. The micro cantilever is V-shaped or rectangular with a sharp tip. This tip is usually of pyramidal or conical shape ( Figure 2).    Typically, probes are made predominantly of silicon nitride (Si3N4); its upper surface is coated with a thin reflective surface, generally gold (Au) or aluminum (Al). The probe is brought to inside and out of contact with the sample surface, by using a piezo-crystal (Hilal and Bowen, 2009). In the illustration of a typical micro-manipulator, shown in Figure 3, it is possible to observe that it consists of a movable stage mounted under an optical microscope. The movement of the base can be controlled by an electronic control console. Large deflections are required to achieve high sensitivity. Therefore, the spring should be very "soft" (slightly stiff).

Operation modes in AFM
The probe mounted on the AFM performs the scan on the sample in a raster fashion. The movement of the microcantilever over the sample is carried out by the piezoelectric scanner, which comprises piezoelectric material that expands and contracts according to the applied voltage. There are several modes of operation for scanning and mapping surface. These modes include non-contact, contact and intermittent contact modes. These three modes of operation differ from each other, basically, by the tip and sample distance.
The Lennard-Jones potential describes the relationship of the tip and sample interaction forces as depending on the tip and sample surface distance, considering the potential energy of a pair of particles, and is given by: 12 6 () 4 Ur rr where  and  are constants depending on the sample properties,  is approximately equal to the diameter of the particles involved. Deriving potential function (U) in relation to the distance (r) gives an expression for the force (F) versus distance (r) (Equation (1a)). This force is represented in Figure 4.
The region above the r-axis corresponds to the region where the repulsive forces dominate (contact region). The region below the r-axis corresponds to the region where attractive forces dominate (non-contact region). Also in red, it can be seen the distance region that the tapping-mode technique is applied.

Mathematical modeling of the Atomic Force Microscope (AFM)
The mathematical models governing the dynamics of the AFM micro-beams, usually result from the discretization of the classical beam equations, based on its mode of vibration, leading to one or more degrees of freedom. Several models of this kind are described in the literature, e.g., Wang (Paulo & Garcia, 2002). Different AFM techniques provide a number of possibilities for topographical images of the samples, generating a wide range of information. In this chapter models for intermittent contact mode are discussed.

Mathematical modeling of AFM: with inclusion of the cubic (spring) term
The physical model of the AFM tip-sample interaction can be considered as shown in Figure  5 The energy of the system scaled by the mass of the cantilever is given by (,' ,) EXX Z : Replacing 1 XX  and 2 XX   , then, from equation (3) results: The dynamic AFM system in Figure 5 is obtained replacing (2) and (3) into (4): where b Z the distance from the equilibrium position. The molecular diameter is 6 where Ah is the Hamaker constant and R is the sphere radius. Considering only attractive Van der Waals force, and that the cantilever is being excited by mfcos(wt), where w is the natural frequency, the system equations can be written as: is the damping force. Considering the relations: and wt   , the system (6) may be rewritten in the following dimensionless form: On an Overview of Nonlinear and Chaotic Behavior and Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 51

AFM Mathematical modeling: intermittent mode and hydrodynamic damping
The microcantilever schematic diagram of the AFM operating in intermittent mode can be seen in Figure 6. The base of the microcantilever is excited by a piezoelectric actuator generating a displacement cos( ) fw t . According to (Zhang et al., 2009), considering only the first vibration mode, the (AFM) can be modeled as a spring-mass-damper, as shown in Figure 7. The "tip" is considered as being a of radius R and Z0 is the distance from the equilibrium position of the cantilever to the sample. The position of the cantilever is given by x, measured from the equilibrium position. According to (Rutzel et al., 2003) the tipsample interaction can be modelled as a sphere-flat surface interaction, given by: Uxz is the Lennard-Jones potential(LJ), Hamaker constant for the attractive and repulsive potential, respectively, with 1  and 2  the densities of the interacting components, and 1 c and 2 c are constants from interaction. It should be noted that, when the "cantilever" is close to the sample, attractive van der Waals  force must be considered. These forces can be represented as the sum of the attractive and repulsive forces (Rutzel et al., 2003), expressed by: In the intermittent mode (TM-AFM) the probe touches the surface of the sample at the point of maximum amplitude of oscillation. During the scanning the microcantilever is driven to oscillate according to the force cos f wt , resulting that the tip-sample contact generates the force F . The contact between the tip and sample is delicate, and this mode of operation is suitable for fragile samples. Then, considering the Lagrangian LTV   and the Euler- are the kinetic energy, the gravitational potential energy, and nonconservative forces, respectively, the equation of motion for the microcantilever tip displacement x is given by: with: where eff  is an effective coefficient of viscosity, B the width of the cantilever and L is the length of the cantilever (Zhang et al. (2009)). Defining: and considering the following relations in (11) Writing equation (14) 2009)). The displacement can be seen in Figure 8a and the phase portrait can be seen in Figure 8b.

Chaos in mathematical model with cubic spring
According to  the chaotic behavior of AFM depends on the damping of the excitation and on the distance between the tip and the sample, suggesting that a feedback control of the states can be used to eliminate the possibility of chaotic behavior. According to that, considering the system (7) in nondimensional form:

Chaos with hydrodynamic damping in TM-AFM
The elastic constant of the cantilever c k must be less than the effective elastic constant of the interatomic coupling at k of the sample.
, can be seen in Figure 11b, indicating that the system has a chaotic attractor.

Scanner position control
A laser beam focus on the top of the microcantilever and the reflection is detected by a photodiode. The light is converted into an electrical signal, and stored in the computer as a reference. An oscillation of the microcantilever deflects the laser beam on the photodiode, allowing the system to compute the microcantilever motion. The error signals are then forwarded, and the piezoelectric scanner moves vertically to scan the sample, as shown in Figure 3. The control techniques are diverse: PID or PD, sliding mode, LQR, or other.

Feedback control for the model with hydrodynamic damping
Considering the model (17) and defining a periodic orbit as a function of () xt  . The desired regime is given by: Since u  control the system in the desired trajectory, and () xt  is a solution of (19), without the term control u F , then 0 u   , resulting: The feedforward control u  is given by: Replacing (19) in (18) and defining the deviation from the desired trajectory as: 11 22 xx e xx The system (18) can be represented as follows: the control u is obtained. In Figure 12 the tip displacement without and with control are shown. 11 where: Uuu   , u is the feedback control, u  is the feedforward control, given by : Replacing (29) in (28) and defining the deviation from the desired trajectory as: where x  is the desired orbit, and rewriting the system in deviations, results: where P is a symmetric matrix, solution of the reduced Riccati equation: Defining the desired trajectory as: The matrices 01 0.14668 0 In Figure 13 it can observed the tip displacement with and without control.

AFM mathematical modeling with Phase-Locked Loops (PLLS)
As mentioned above, the Atomic Force Microscopy started in 1986 when the Atomic Force Microscope (AFM) was invented by Binnig in 1986. Since then many results have been obtained by simple contact measurements. However, the AFM cannot generate truly atomic resolution images, by simple contact measurement, in a stable operation. Besides, since 1995, using noncontact techniques, it was possible to obtain atomic resolution images, with stable operation, under attractive regime at room temperature (Giessible, 1995; Morita et. al., 2009). Noncontact AFM operates in static and dynamic modes. In the static mode the tip-sample interaction forces are translated into measured microcantilever deflections, and the image is a map z(x,y,Fts) with Fts constant. In the dynamic mode the microcantilever is deliberately vibrated. The Amplitude Modulated AFM and the Frequency Modulated AFM are the most important techniques. In both AM-AFM and FM-AFM the amplitude and frequency of the microcantilever are kept constant by two control loops. The AGC (Automatic Gain Control) and the ADC (Automatic Distance Control). The AGC controls the amplitude of oscillation and the ADC controls the frequency by adjusting the distance between tip and sample. The oscillatory behavior of the microcantilever is illustrated in Figure 14. The FM-AFM block is shown in Figure 15. In the FM-AFM the control signal of the AGC loop is used to generate the dissipation images and the ADC control signal is used to generate the topographic images. The FM-AFM improved image resolution and for surface studies in vacuum is the preferred AFM technique (Morita et. al., 2009;Bhushan, 2004). From Figure 15 it can be seen that the PLL generates the feedback signal for both control loops, therefore the PLL performance is vital to the FM-AFM. The PLL is a closed loop control system that synchronizes a local oscillator to a sinusoidal input. The PLLs are composed of a phase detector (usually a multiplier circuit), of a low-pass filter and of a VCO (Bueno et. al., 2010;Bueno et al., 2011), as it can be seen in Figure 16, and additionally, shows the PM and AM outputs used in the AFM system. The AGC loop also depends on the amplitude detector output, shown in Figure 17. The amplitude detector is composed of diode followed by a first-order low-pass filter. The circuit holds the output A(t) for a while, allowing the AGC to determine the control signal.

Mathematical model of the FM-AFM
The mathematical model of the FM-AFM considers the microcantiler dynamics, the tipsample interaction, the amplitude detector circuit and the PLL. The microcantilever is assumed to be a second order system with natural frequency c  and damping coefficient  .
Concerning to tip-sample interaction, there are short, medium and long range forces. Since the FM-AFM operates in long-range distance, the predominant force is the Van Der Waals Equation 45 establishes a design criterion that assure the local stability of the PLL, i.e., for small phase and frequency steps the PLL synchronizes to the microcantilever oscillation.
Additionally, from the design parameters  and n  the loop gain G has a superior bound, and can be determined in order to satisfy the requirements of performance and stability.
On an Overview of Nonlinear and Chaotic Behavior and Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 63 Despite the good transient response and high frequency noise rejection -such as the double frequency jiter -, provided by the all pole filter the steady state response may need improvement. The PLL must demodulate an FSK (Frequency Shift Keying) signal, that actually is a frequency step. In order to track a frequency shift the loop filter must have at least a pure integration, i.e., the PLL must be at least a type 2 system (Bueno et al., 2010 andBueno et al., 2011;Bueno, 2009, Ogata, 1993. According to that, and considering the loop  This shows that the PLL design must provide strong damping to noise and to the double frequency jitter. Besides, if the PLL is not at least of type 2 the PLL presents steady state error in the FSK demodulation, impairing the AFM imaging process. The PLL perfomance was analysed and presented under the FM-AFM perspective. Besides, a PLL design method was shown and illustrated by simulation, making clear the PLL performance importantance in the AFM control system. Figure 18. PLL response to a FSK signal.

Conclusions
This chapter deals with emergent problems in the Engineering Science research, presenting study and research related to NEMS systems, specially microcantilevers with many modes of vibration, for which the tip-sample interaction forces are highly nonlinear, impairing the stability of the latent image, while the others modes of vibration can be explored in order to improve the AFM performance.
In the context of this work, the following specific problem have been approached: The understanding of the relations of the properties and the structure of the nanoscopic and molecular materials, through the atomic force microscopy, using microcantilevers, giving subsidiary information to next generation of microscopy instrumentation.