**7. 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.

**Figure 13.** (a): Tip displacement without control (b): Tip Displacement with control

Considering the system (31) in the following way:

*a* 

1 2

*y y*

2 *<sup>y</sup> <sup>y</sup> <sup>y</sup>* ;

solving the following equation:

Defining the desired trajectory as:

The matrices 0 1

a map z(x,y,Fts) with Fts constant.

*A*

where: <sup>1</sup>

;

3 3

*gx gx* () () *b b a y x ax*

> 0.14668 0 *<sup>A</sup>*

*R* 0.1 , results after using the matlab(R) to obtain *u*:

 

0 1 *<sup>B</sup>*

0

where *P* is a symmetric matrix, solution of the reduced Riccati equation:

, <sup>0</sup> 1 *<sup>B</sup>*

In Figure 13 it can observed the tip displacement with and without control.

**7. 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

2 1 1 21 2 2

3 3 2 11 2 1 1 21 2 2

 

and

*zy x zx*

11 1

*y ay a y x ax u*

*zy x zx*

11 1

*y Ay g x g x Bu* () () (32)

<sup>1</sup> *<sup>T</sup> u R B Py* (33)

*x t* 2cos( ) (35)

, and defining Q and R as 250 0

1 2 *u yy* 49.8535 17.3120 (36)

<sup>1</sup> 0 *T T PA A P PBR B P Q* (34)

(31)

. The control *u* is obtained by

0 20 *<sup>Q</sup>* 

,

*b b*

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.

**Figure 14.** Microcantilever oscillatory behavior. Source: (Bueno et al., 2011).

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.

On an Overview of Nonlinear and Chaotic Behavior and

*c* and damping coefficient

and amplitude ( ) *<sup>o</sup> rtv* ,

.

*<sup>d</sup> RC* 

Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 61

 *t t* 

The mathematical model of the FM-AFM considers the microcantiler dynamics, the tipsample interaction, the amplitude detector circuit and the PLL. The microcantilever is

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

7.1. Besides, the microcatilever is excited by an external forcing signal with a previously

where *r t*( ) is the AGC signal, and *<sup>o</sup> v* is constant. The tip-sample interaction forces cause modulations both in the amplitude and in the frequency of oscillation of the AFM microcantilever. The modulations are detected by the PLL and used by the AGC and by the ADC, in order to control the microcantilever, drivin it to oscillate according to

The mathematical model of the amplitude detector is given by equation 38, where <sup>1</sup>

mathematical model in equation 39 follows Bueno et al., 2010 and 2011. Considering the

phase detector gain, *<sup>o</sup> k* is the VCO gain, *<sup>o</sup> v* is the VCO output amplitude and *Ac* is the nominal microcantilever amplitude of oscillation, Figure 14. Equations 40 and 41 represent

, where AH is the Hamaker constant and d(t) is the tip-sample distance, Figure

*c c* . The microcantilever mathetamical model is given by equation

. The PLL model can be seen in many works in the literature. The

2

*dt zt*

6 () () *H*

*A*

*G k kvA moo c* , where *mk* is the

and the gain <sup>1</sup>

(37)

<sup>2</sup> () () ()

 

**7.1. Mathematical model of the FM-AFM** 

 <sup>2</sup> 6 () () *AH dt zt*

37.

*z t A t sen t t* 

and ( ), ( ) 0 0, ( ) 0 *<sup>d</sup> zt zt*

*z t* 

filter transfer function <sup>0</sup>

*z*

assumed to be a second order system with natural frequency

determined amplitude. The signal is a sinusoid with phase ( ) *<sup>c</sup>*

**Figure 17.** Amplitude detector. Source (Bueno et al., 2011)

*f t*

2

2

 

*s s* 

 

1 0

the AGC and ADC, respectively. Equations 37 to 41 are the model of the FM-AFM.

*c o co*

*z z t z t r t v sen t t*

**Figure 15.** Block Diagram of the FM-AFM control system. Source: (Bueno et al., 2011).

**Figure 16.** PLL block diagram
