**2. Analysis**

In contact mode, the AFM cantilever moves down by a small amplitude (1-5 nm) when the cantilever tip processes a sample surface. Therefore, the linear model can be used to describe the tip-sample interaction. The atomic force microscope cantilever, shown in Fig.1, is a small elastic beam with a length *L* , thickness *H* , width *b* , and a tip with a width of *w* and length *h* . *x* is the coordinate along the cantilever and *vxt* ( ,) is the vertical deflection in the *x*direction, as shown in Fig.2. One end of the cantilever, at 0 *x* , is clamped, while the other end, from *L w* to *L* , has a tip.

When the sampling is in progress, the tip makes contact with the specimen, resulting in a vertical reaction force, ( ) *F t <sup>y</sup>* and a horizontal reaction force, ( ) *F t <sup>x</sup>* , both of which functions of time *t* . Assuming that the reaction forces act on the tip end, the product of the horizontal force and the tip length can form a bending stress on the bottom surface of the cantilever. The sampling system can be modeled as a flexural vibration motion of the cantilever. The motion is a function of mode shape and natural frequency, and its transverse displacement depends on time and the spatial coordinate *x* [7 and 8]. When the sampling forces are

the analysis. However, for AFM-based cantilever direct mechanical nanomachining, the indentation and sampling of solid materials (e.g. polymer silicon and some metal surfaces) are performed. The effects of transverse shear deformation and rotary inertia in the vibration analysis should be taken into account for cantilevers whose cross-sectional dimensions are comparable to the lengths. Neglecting the effects of transverse shear deformation and rotary inertia in the vibration analysis may result in less accurate results. Hsu et al. (Hsu, et al., 2007) studied the modal frequencies of flexural vibration for an AFM cantilever using the Timoshenko beam theory and obtained a closed-form expression for the frequencies of vibration modes. However, the solution of the vibration response obtained using the modal superposition method for AFM cantilever modeled as a Timoshenko beam, and the response of flexural vibration of a rectangular AFM cantilever which has large shear

In this chapter, the response of flexural vibration of a rectangular AFM cantilever subjected to a sampling force is studied analytically by using the Timoshenko beam theory and the modal superposition method. Firstly, the governing equations of the Timoshenko beam model with coupled differential equations expressed in terms of the flexural displacement and the bending angle are uncoupled to produce the fourth order equation. Then, the sampling forces which are applied to the end region of the AFM cantilever by means of the tip, are transformed into an axial force, distributed transversal stress and bending stress. Finally, the response of the flexural vibration of a rectangular AFM cantilever subjected to a sampling force is solved using the modal superposition method. Moreover, a validity comparison for AFM cantilever modeling between the Timoshenko beam model and the Bernoulli-Euler beam model was conducted using the ratios of the Young's modulus to the shear modulus. From the results, the Bernoulli-Euler beam model is not suitable for AFM cantilever modeling, except when the ratios of the Young's modulus to the shear modulus are less than 1000. The Timoshenko beam model is a better choice for simulatimg the flexural vibration responses of AFM cantilever,

In contact mode, the AFM cantilever moves down by a small amplitude (1-5 nm) when the cantilever tip processes a sample surface. Therefore, the linear model can be used to describe the tip-sample interaction. The atomic force microscope cantilever, shown in Fig.1, is a small elastic beam with a length *L* , thickness *H* , width *b* , and a tip with a width of *w* and length *h* . *x* is the coordinate along the cantilever and *vxt* ( ,) is the vertical deflection in the *x*direction, as shown in Fig.2. One end of the cantilever, at 0 *x* , is clamped, while the other

When the sampling is in progress, the tip makes contact with the specimen, resulting in a vertical reaction force, ( ) *F t <sup>y</sup>* and a horizontal reaction force, ( ) *F t <sup>x</sup>* , both of which functions of time *t* . Assuming that the reaction forces act on the tip end, the product of the horizontal force and the tip length can form a bending stress on the bottom surface of the cantilever. The sampling system can be modeled as a flexural vibration motion of the cantilever. The motion is a function of mode shape and natural frequency, and its transverse displacement depends on time and the spatial coordinate *x* [7 and 8]. When the sampling forces are

deformation effects, are absent from the literature.

especially for very small shear modulus.

end, from *L w* to *L* , has a tip.

**2. Analysis** 

Fig. 1. Schematic diagram of an AFM tip-cantilever assembly processing a sample surface.

applied, the loads transmitted from the tip holder act on the end of the AFM cantilever, and can be modeled as the three parts shown in Fig.2, termed axial force *N t*( ) , transverse excitation ( ,) *<sup>l</sup> p x t* , and bending excitation ( ,) *<sup>b</sup> p x t* .

Assuming that the transverse excitation is uniformly distributed on the bottom surface of the AFM cantilever, then it can be written as:

$$p\_I(\mathbf{x}, t) = F\_y(t)\mu(\mathbf{x} - L + w) / \, w \,\, \tag{1}$$

where *ux L w* ( ) is the unit step function.

 The relationship between ( ) *F t <sup>x</sup>* and ( ) *F t <sup>y</sup>* can be expressed as 2 cos *F F <sup>x</sup> <sup>y</sup>* for a cone shape cantilever tip, where is the half-conic angle. The bending excitations, which result

Vibration Responses of Atomic Force Microscope Cantilevers 61

where *x* is the distance along the center of the cantilever, *vxt* ( ,) is the transverse

density, *K* is the shear factor ( *K =* 5/6 for rectangular cross-section), and *A* is the rectangular

Equations (5) and (6) may be uncoupled to produce a fourth order equation in *vxt* ( ,) . Considering the axial force effect, the classical uncoupled Timoshenko-beam partial

4 2

4 2 2 ( ,) ( ,) ( ,) *<sup>t</sup> v x t EI v x t <sup>I</sup> <sup>I</sup> p xt KG <sup>t</sup> KG x t*

The mode-superposition analysis of a distributed-parameter system is equivalent to that of a discrete-coordinate system once the mode shapes and frequencies have been determined because in both cases, the amplitudes of the modal-response components are used as generalized coordinates in defining the response of the structure. In principle, an infinite number of these coordinates are available for a distributed-parameter system, since it has an infinite number of modes of vibration. Practically, however, only those modal components which provide significant contributions to the response need be considered (Ray & Joseph, 1993 ; William, 1998). The essential operation of the modesuperposition analysis is the transformation from the geometric displacement coordinates to the modal-amplitude or normal coordinates. For a one-dimensional system, this

> 1 1 ( ,) ( ) () ( ,) *nn n n n vxt xY t q xt*

*x* is the *n* -th mode shape of the AFM cantilever. In order to find the natural

where ( ,) *nq xt* is the response contribution of the n-th mode, ( ) *Y t <sup>n</sup>* is the normal coordinate,

2 22 2 2 2 , ,, . *x AL I EI b rs L EI AL KAGL*

( ) ( ) ( ) cosh sinh cos sin ( ) ( ) *n n n n <sup>n</sup> R R RR R C b b b <sup>b</sup>*

frequencies and mode shapes, the following non-dimensional variables are defined:

4

 

is the non-dimensional length along the beam, and

 

*x* can be given by (White, et al., 1995):

 

 

*vxt vxt* ( ,) ( ,) *v xt*, *EI <sup>A</sup> N t x t x x*

4 4

 

Young's modulus, *G* is shear modulus, *I* is the area moment of inertia,

4 2

( ,) *x t* is the rotation of the neutral axis during bending, *E* is

(7)

(8)

(9)

 

1 3 1 3 2 4 2 4

*R RR R RR*

 

is the radian frequency. Then,

  (10)

is the volume

displacement, *t* is time,

cross-sectional area of the cantilever.

where () () *Nt F t <sup>x</sup>* is the axial force.

transformation is expressed as:

and ( ) *<sup>n</sup>* 

Here 

( ) *<sup>n</sup>* 

> 

differential equations can be written as (White, et al., 1995):

Fig. 2. Schematic diagram of excitations acting on the AFM cantilever

from the horizontal sampling force ( ) *F t <sup>x</sup>* , act on the bottom surface of the AFM cantilever within the region from *L w* to *L* . They can be written as:

$$p\_b(\mathbf{x}, t) = \left[12h(2L - w - 2\mathbf{x})\cos\theta \;/\,\pi w^3\right] \mathbf{F}\_y \mu(\mathbf{x} - L + w) \tag{2}$$

By summing the above two excitations, the total transverse excitation ( ,) *<sup>t</sup> p x t* can be expressed as (Horng, 2009):

$$p\_t(\mathbf{x}, t) = \mathbf{C}(\mathbf{x}) F\_y(t) \mu(\mathbf{x} - L + w) \tag{3}$$

$$\text{where } \mathcal{C}(\mathbf{x}) = \frac{1 + 12h(2L - w)\cos\theta \,/\, \pi w^2}{w} - \frac{24h\cos\theta}{\pi w^3} \mathbf{x} \tag{4}$$

Vibration behaviors of an AFM cantilever are examined using the Timoshenko beam theory. The effects of the rotary inertia and shear deformation are taken into account during contact with the sample. The governing equations of the Timoshenko beam model are two coupled differential equations expressed in terms of the flexural displacement and the angle of rotation due to bending. When the beam support is constrained to be fixed and all other external influences are set to zero, we obtain the classical coupled Timoshenko-beam partial differential equations (Hsu, et al., 2007):

$$
\rho A \frac{\partial^2 v}{\partial t^2} - K A G(\frac{\partial^2 v}{\partial \mathbf{x}^2} - \frac{\partial \boldsymbol{\mu}}{\partial \mathbf{x}}) = 0 \tag{5}
$$

$$EI\frac{\partial^2 \nu}{\partial \mathbf{x}^2} + \text{KAG}(\frac{\partial \nu}{\partial \mathbf{x}} - \boldsymbol{\nu}) - \rho I \frac{\partial^2 \nu}{\partial t^2} = 0 \tag{6}$$

*z*

from the horizontal sampling force ( ) *F t <sup>x</sup>* , act on the bottom surface of the AFM cantilever

*L*

*x*

<sup>3</sup> ( , ) 12 (2 2 )cos / ( ) *b y p xt h L w x w Fux L w* 

By summing the above two excitations, the total transverse excitation ( ,) *<sup>t</sup> p x t* can be

( ,) ( ) ()( ) *t y p xt CxF tux L w* (3)

Vibration behaviors of an AFM cantilever are examined using the Timoshenko beam theory. The effects of the rotary inertia and shear deformation are taken into account during contact with the sample. The governing equations of the Timoshenko beam model are two coupled differential equations expressed in terms of the flexural displacement and the angle of rotation due to bending. When the beam support is constrained to be fixed and all other external influences are set to zero, we obtain the classical coupled Timoshenko-beam partial

> 2 2 2 2 ( )0 *v v A KAG t x x*

2 2 2 2 () 0 *<sup>v</sup> EI KAG <sup>I</sup> x t x*

 

1 12 (2 )cos / 24 cos ( ) *hLw w h C x <sup>x</sup>*

 

(2)

*y*, ( ,) *vxt*

2

*w w*

> 

3

(5)

(6)

(4)

 

Fig. 2. Schematic diagram of excitations acting on the AFM cantilever

( ,) *<sup>b</sup> p x t*

within the region from *L w* to *L* . They can be written as:

( ,) *<sup>l</sup> <sup>p</sup> x t <sup>w</sup>*

expressed as (Horng, 2009):

*N t*( )

where

differential equations (Hsu, et al., 2007):

where *x* is the distance along the center of the cantilever, *vxt* ( ,) is the transverse displacement, *t* is time, ( ,) *x t* is the rotation of the neutral axis during bending, *E* is Young's modulus, *G* is shear modulus, *I* is the area moment of inertia, is the volume density, *K* is the shear factor ( *K =* 5/6 for rectangular cross-section), and *A* is the rectangular cross-sectional area of the cantilever.

Equations (5) and (6) may be uncoupled to produce a fourth order equation in *vxt* ( ,) . Considering the axial force effect, the classical uncoupled Timoshenko-beam partial differential equations can be written as (White, et al., 1995):

$$
\rho I I \frac{\hat{\sigma}^4 v(\mathbf{x}, t)}{\hat{\sigma} \mathbf{x}^4} + \rho A \frac{\hat{\sigma}^2 v(\mathbf{x}, t)}{\hat{\sigma} t^2} - \frac{\hat{\sigma}}{\hat{\sigma} \mathbf{x}} \left[ N(t) \frac{\partial v(\mathbf{x}, t)}{\hat{\sigma} \mathbf{x}} \right] + 
$$

$$
+ \rho I \frac{\rho}{K \mathbf{G}} \frac{\hat{\sigma}^4 v(\mathbf{x}, t)}{\hat{\sigma} t^4} - \left( \rho I + \frac{\rho EI}{K \mathbf{G}} \right) \frac{\hat{\sigma}^4 v(\mathbf{x}, t)}{\hat{\sigma} \mathbf{x}^2 \hat{\sigma} t^2} = p\_t(\mathbf{x}, t) \tag{7}
$$

where () () *Nt F t <sup>x</sup>* is the axial force.

The mode-superposition analysis of a distributed-parameter system is equivalent to that of a discrete-coordinate system once the mode shapes and frequencies have been determined because in both cases, the amplitudes of the modal-response components are used as generalized coordinates in defining the response of the structure. In principle, an infinite number of these coordinates are available for a distributed-parameter system, since it has an infinite number of modes of vibration. Practically, however, only those modal components which provide significant contributions to the response need be considered (Ray & Joseph, 1993 ; William, 1998). The essential operation of the modesuperposition analysis is the transformation from the geometric displacement coordinates to the modal-amplitude or normal coordinates. For a one-dimensional system, this transformation is expressed as:

$$w(\mathbf{x},t) = \sum\_{n=1}^{\infty} \varphi\_n(\mathbf{x}) Y\_n(t) = \sum\_{n=1}^{\infty} q\_n(\mathbf{x},t) \tag{8}$$

where ( ,) *nq xt* is the response contribution of the n-th mode, ( ) *Y t <sup>n</sup>* is the normal coordinate, and ( ) *<sup>n</sup> x* is the *n* -th mode shape of the AFM cantilever. In order to find the natural frequencies and mode shapes, the following non-dimensional variables are defined:

$$\text{s} \ \text{\$ } \text{s} = \frac{\text{x}}{\text{L}}, \text{ } \text{s}^2 = \frac{\rho A L^4}{EI} \text{o}^2, \text{ } r^2 = \frac{I}{AL^2}, \text{ } s^2 = \frac{EI}{\text{KAGL}^2}. \tag{9}$$

Here is the non-dimensional length along the beam, and is the radian frequency. Then, ( ) *<sup>n</sup> x* can be given by (White, et al., 1995):

$$\varphi\_n(\xi) = \mathbb{C}\left[\cosh b\_n a \xi - \frac{(R\_1 - R\_3)}{(R\_2 - R R\_4)} \sinh b\_n a \xi - \cos b\_n \beta \xi + \frac{R(R\_1 - R\_3)}{(R\_2 - R R\_4)} \sin b\_n \beta \xi\right] \tag{10}$$

$$\begin{array}{l}\text{where}\\\text{where}\end{array}\tag{1}\begin{cases}a\\\beta\end{cases}=\left(1\Big/\sqrt{2}\right)\left\{\mp(r^2+s^2)+\left[(r^2-s^2)^2+4\Big/b\_n\right]^2\right\}^{1/2}\tag{11}$$

$$R = \frac{\left(a^2 + s^2\right)}{a} \frac{\beta}{\left(\beta^2 - s^2\right)}\tag{12}$$

Vibration Responses of Atomic Force Microscope Cantilevers 63

<sup>0</sup> ( ) () *L*

( ) ( ) () *<sup>L</sup> <sup>n</sup> n n EI d x T I x dx KG dx*

( ) () ( )

<sup>0</sup> () ( ) ( ,) *L Pt x n nt* 

() () *P t cF t n ny* (29)

1 12 (2 )cos / 24 cos ( ) ( ) *<sup>L</sup>*

Then, the Normal-Coordinate Response Equation, which is exactly the same equation

( ) () [ () ] ( )/ ( ) ( )/ *n n <sup>n</sup> <sup>x</sup> <sup>n</sup> <sup>n</sup> nn n n n <sup>y</sup> <sup>n</sup>*

*dY t dY t GF t M <sup>M</sup> T S Y t cF t S*

Assuming a zero initial condition, with *v x*( ,0) 0 , *v x* ( ,0) 0 , *v x*( ,0) 0 and *v x*( ,0) 0 , and providing that the sampling force ( ) *F t <sup>y</sup>* is a series of harmonics, ( ) *F t <sup>y</sup>* can be written as:

> 1 ( ) sin( ) *m y i i i Ft F t*

The main goal of this study is to analyze the flexural vibration responses in nanoscale processing using atomic force microscopy modeled as a Timoshenko beam. To demonstrate the validity of the analytical solution, numerical computations were performed. The

2300 / *km m* , 125 *L m*

The modulus-ratio *REG* , defined as the ratio of *E* to *G* (i.e. / *REG E G* ), is introduced to define the values of shear modulus *G* and to describe the effects of shear deformation. In this

*hLw w h c x x u x L w dx w w* 

 

> 2 2

*dx* 

2

 

*Mn n* 

> > 0

0 3

4 22

, <sup>3</sup>

Kutta method is introduced to solve the above fourth-order system.

geometric and material parameters considered were as follows:

.

and 3 *m* , <sup>9</sup> 1000 /(2 1) 10 *F i <sup>i</sup>*

*dt dt S*

*L <sup>n</sup> n n d x G t x dx*

Using Eq. (3) and Eq. (4), Eq. (28) can be rewritten as

considered for the discrete-parameter case, can be solved.

4 2

*n n*

When the *j-th* excitation frequency

**3. Results and discussion** 

*E Gpa* 170 , *m km m* 0.2898 /

, 2 30 

where

2

2 0 2

*A x dx* (25)

(26)

(27)

*<sup>p</sup> x t dx* (28)

(32)

, 4.2 *H m*

*<sup>n</sup>* , the Runge-

 , 5 *h m* 

 

(31)

*<sup>j</sup>* is equal to the n-th natural frequency

 , 30 *b m* 

(30)

*n*

$$R\_1 = (b\_n \,/\ a) \sinh b\_n a \tag{13}$$

$$R\_2 = (b\_n / a) \cosh b\_n a \tag{14}$$

$$R\_3 = (b\_n \; / \; \beta) \sin b\_n \beta \tag{15}$$

$$R\_4 = -(b\_n \,/\, a) \cos b\_n \beta \tag{16}$$

and *nb* are the non-dimensional natural frequencies, which can be obtained using the characteristic equation

$$\left(\frac{\alpha^2 + s^2}{\alpha}\right) (R\_3 R\_4' - R\_3' R\_4 + R\_4 R\_1' - R\_1 R\_4') + \left(\frac{\beta^2 - s^2}{\beta}\right) (R\_2 R\_3' - R\_2' R\_3 + R\_1 R\_2' - R\_2 R\_1') = 0 \quad \text{(17)}$$

$$\text{where}\\
\qquad\qquad R'\_1 = \left[\left(a^2 + s^2\right) / \, a\right] \flat\_n a \cosh \vartheta\_n a\tag{18}$$

$$R\_2' = [(a^2 + s^2) / a] b\_n a \sin b\_n a \tag{19}$$

$$R\_3' = -[\left(\beta^2 - s^2\right) / \beta] b\_n \beta \cos b\_n \beta \tag{20}$$

$$R\_4' = -[\left(\beta^2 - S^2\right) / \beta] b\_n \beta \sin b\_n \beta \tag{21}$$

Equation (8) simply states that any physically permissible displacement pattern can be modeled by superposing appropriate amplitudes of the vibration mode shapes for the structure. Substituting Eq. (8) into Eq. (7) and using orthogonally conditions gives

$$S\_n \frac{d^4 Y\_n(t)}{dt^4} + (M\_n + T\_n) \frac{d^2 Y\_n(t)}{dt^2} + [-G\_n F\_x(t) + \alpha\_n^2 M\_n] Y\_n(t) = P\_n(t) \tag{22}$$

where *<sup>n</sup>* is the *n* -th mode natural frequency of the AFM cantilever obtained using:

$$a\_n = b\_n \sqrt{\frac{EI}{\rho AL^4}}\tag{23}$$

*Sn* , *Mn* ,*Tn* and *<sup>n</sup> p* are the generalized constants of the *n* -th mode, given by

$$S\_n = (\rho I \frac{\mathcal{P}}{\mathcal{K}\mathcal{G}}) \int\_0^L \rho\_n(\mathbf{x})^2 d\mathbf{x} \tag{24}$$

$$M\_n = (\rho A) \int\_0^L \rho\_n(\mathbf{x})^2 d\mathbf{x} \tag{25}$$

$$T\_n = (\rho I + \frac{\rho EI}{KG}) \int\_0^L \rho\_n(\mathbf{x}) \frac{d^2 \rho\_n(\mathbf{x})}{d\mathbf{x}^2} d\mathbf{x} \tag{26}$$

$$G\_n(t) = \int\_0^L \left[\varphi\_n(\mathbf{x}) \frac{d^2 \varphi\_n(\mathbf{x})}{d\mathbf{x}^2} \right] d\mathbf{x} \tag{27}$$

$$P\_n(t) = \int\_0^L \varphi\_n(\mathbf{x}) p\_t(\mathbf{x}, t) d\mathbf{x} \tag{28}$$

Using Eq. (3) and Eq. (4), Eq. (28) can be rewritten as

$$P\_n(t) = c\_n F\_y(t) \tag{29}$$

where

62 Atomic Force Microscopy – Imaging, Measuring and Manipulating Surfaces at the Atomic Scale

2 2

( )

<sup>1</sup> *Rb b* ( / )sinh *n n* 

<sup>2</sup> *Rb b* ( / )cosh *n n* 

<sup>3</sup> *Rb b* ( / )sin *n n* 

<sup>4</sup> ( / )cos *Rb b n n* 

and *nb* are the non-dimensional natural frequencies, which can be obtained using the

34 34 41 14 23 23 12 21 ( ) ( ) <sup>0</sup> *s s RR RR RR RR RR RR RR RR*

 

 

<sup>1</sup> *R sb b* [( ) / ] cosh

2 2 <sup>2</sup> *R sbb* [( ) / ] sin

2 2 <sup>3</sup> [( ) / ] cos *R sbb*

2 2 <sup>4</sup> *R Sbb* [( ) / ] sin

Equation (8) simply states that any physically permissible displacement pattern can be modeled by superposing appropriate amplitudes of the vibration mode shapes for the

> ( ) ( ) ( ) [ () ] () () *n n <sup>n</sup> n n nx n n n n dY t dY t S MT GF t M Y t P t*

*<sup>n</sup>* is the *n* -th mode natural frequency of the AFM cantilever obtained using:

*n n* 4 *EI <sup>b</sup> AL*

<sup>0</sup> ( ) () *L S I x dx n n KG* 

 

2

structure. Substituting Eq. (8) into Eq. (7) and using orthogonally conditions gives

*Sn* , *Mn* ,*Tn* and *<sup>n</sup> p* are the generalized constants of the *n* -th mode, given by

2 2

*s*

 *n n* (18)

 *n n* (19)

 *n n* (20)

 *n n* (21)

2

(23)

(24)

( )

 

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(22)

where 1 2 2 2 2 22 2 1 2 (1 2 ) ( ) [( ) 4 ] *<sup>n</sup> rs rs b*

2 2 2 2

4 2

4 2

*dt dt*

where 2 2

*<sup>s</sup> <sup>R</sup>*

characteristic equation

where 

$$c\_n = \int\_0^L \rho\_n(\mathbf{x}) \left( \frac{1 + 12h(2L - w)\cos\theta \,/\,\pi w^2}{w} - \frac{24h\cos\theta}{\pi w^3} \mathbf{x} \right) \mu(\mathbf{x} - L + w) d\mathbf{x} \tag{30}$$

Then, the Normal-Coordinate Response Equation, which is exactly the same equation considered for the discrete-parameter case, can be solved.

$$\frac{d^4Y\_n(t)}{dt^4} + \left(M\_n + T\_n\right) / S\_n\\\frac{d^2Y\_n(t)}{dt^2} + \frac{\left[-G\_nF\_x(t) + o\_n^2M\_n\right]}{S\_n}Y\_n(t) = c\_nF\_y(t) / S\_n\tag{31}$$

Assuming a zero initial condition, with *v x*( ,0) 0 , *v x* ( ,0) 0 , *v x*( ,0) 0 and *v x*( ,0) 0 , and providing that the sampling force ( ) *F t <sup>y</sup>* is a series of harmonics, ( ) *F t <sup>y</sup>* can be written as:

$$F\_y(t) = \sum\_{i=1}^{m} F\_i \sin(o\_i t) \tag{32}$$

When the *j-th* excitation frequency *<sup>j</sup>* is equal to the n-th natural frequency*<sup>n</sup>* , the Runge-Kutta method is introduced to solve the above fourth-order system.

#### **3. Results and discussion**

The main goal of this study is to analyze the flexural vibration responses in nanoscale processing using atomic force microscopy modeled as a Timoshenko beam. To demonstrate the validity of the analytical solution, numerical computations were performed. The geometric and material parameters considered were as follows:

*E Gpa* 170 , *m km m* 0.2898 / , <sup>3</sup> 2300 / *km m* , 125 *L m* , 30 *b m* , 4.2 *H m* , 5 *h m* , 2 30 and 3 *m* , <sup>9</sup> 1000 /(2 1) 10 *F i <sup>i</sup>* .

The modulus-ratio *REG* , defined as the ratio of *E* to *G* (i.e. / *REG E G* ), is introduced to define the values of shear modulus *G* and to describe the effects of shear deformation. In this study, the flexural vibration responses at the end of the AFM cantilever were obtained using the contribution of the first five vibration modes. A non-dimensional response was used to normalize the static response as given in <sup>3</sup> *F L EI* <sup>1</sup> /(3 ) , and (2 1) *i n i r* , and set as the simulated values of the excitation frequency of the vertical sampling force. Thus, ( ) *F t <sup>y</sup>* is taken as:

$$F\_y(t) = 1000 \left( \sin r o\_n t + \frac{1}{3} \sin 3 \times r o\_n t + \frac{1}{5} \sin 5 \times r o\_n t \right) \tag{33}$$

Vibration Responses of Atomic Force Microscope Cantilevers 65

shape of the 1-st mode shape of the 2-nd mode shape of the 3-rd mode shape of the 4-th mode shape of the 5-th mode

The first five vibration Modes of AFM

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 0.1 0.3 0.5 0.7 0.9 Dimensionless distance from the fixed end

Fig. 5. The effect of various small modulus-ratios ( *REG* ) on the response of the end point

.

for the excitation frequency far away from the first natural frequency, i.e. <sup>1</sup> 0.1

Fig. 4. The shape of the first five vibration modes for an AFM cantilever.






0


1

0.5

2

1.5

3

2.5

where *r* is the frequency ratio that can used to describe the deviation between the excitation frequency and modal frequencies. The modal frequency *<sup>n</sup>* and modal shapes( ) *x* of the first five vibration modes for an AFM cantilever are shown in Fig.3 and Fig.4, respectively.

Fig. 3. Natural frequencies of the first five vibration modes for an AFM cantilever.

In order to investigate the effects of transverse shear deformation, the response histories at the end point of the AFM cantilever between different small and large modulus-ratios, with respect to excitation frequencies far away from(*r* 0.1) and close to (*r* 0.9) the first natural

frequency, are shown in Fig.5 to Fig.8. Figure 5 and Fig.6 indicate that the responses are similar for the various modulus-ratios when they are less than 1000. This means that if the effects of transverse shear deformation are small enough to be negligible, the Timoshenko beam model can be reduced to the Bernoulli-Euler beam model. Figure 6 also reveals that the resonance effect occurs when the AFM cantilever has small modulus-ratios and the excitation frequencies are close to the modal frequencies.

Figure 7 and Fig.8 show the response histories at the end point of the AFM cantilever between different large modulus-ratios for excitation frequencies far away from ( *r* 0.1) and close to (*r* 0.9) the first modal frequency, respectively. The results are quite different

study, the flexural vibration responses at the end of the AFM cantilever were obtained using the contribution of the first five vibration modes. A non-dimensional response was used to

simulated values of the excitation frequency of the vertical sampling force. Thus, ( ) *F t <sup>y</sup>* is

 1 1 ( ) 1000 sin sin 3 sin 5 3 5 *F t r t r t r t nN y nnn*

where *r* is the frequency ratio that can used to describe the deviation between the excitation

first five vibration modes for an AFM cantilever are shown in Fig.3 and Fig.4, respectively.

The first five natural frequency of AFM 1-st mode 2-ed mode 3-rd mode 4-th mode 5-th mode

0 2 4 6 810 Time (e-6 sec)

In order to investigate the effects of transverse shear deformation, the response histories at the end point of the AFM cantilever between different small and large modulus-ratios, with respect to excitation frequencies far away from(*r* 0.1) and close to (*r* 0.9) the first natural frequency, are shown in Fig.5 to Fig.8. Figure 5 and Fig.6 indicate that the responses are similar for the various modulus-ratios when they are less than 1000. This means that if the effects of transverse shear deformation are small enough to be negligible, the Timoshenko beam model can be reduced to the Bernoulli-Euler beam model. Figure 6 also reveals that the resonance effect occurs when the AFM cantilever has small modulus-ratios and the

Figure 7 and Fig.8 show the response histories at the end point of the AFM cantilever between different large modulus-ratios for excitation frequencies far away from ( *r* 0.1) and close to (*r* 0.9) the first modal frequency, respectively. The results are quite different

Fig. 3. Natural frequencies of the first five vibration modes for an AFM cantilever.

(33)

*i r*

*<sup>n</sup>* and modal shapes

, and set as the

( ) *x* of the

normalize the static response as given in <sup>3</sup> *F L EI* <sup>1</sup> /(3 ) , and (2 1) *i n*

frequency and modal frequencies. The modal frequency

0E+000

4E+007

2E+007

excitation frequencies are close to the modal frequencies.

6E+007

1E+008

1E+008

8E+007

Hz

1E+008

2E+008

taken as:

Fig. 4. The shape of the first five vibration modes for an AFM cantilever.

Fig. 5. The effect of various small modulus-ratios ( *REG* ) on the response of the end point for the excitation frequency far away from the first natural frequency, i.e. <sup>1</sup> 0.1.

Vibration Responses of Atomic Force Microscope Cantilevers 67

r=0.1

REG=1000 REG=2000 REG=3000 REG=5000 REG=10000 REG=15000 REG=20000

0 2 4 6 810 Time (e-6 Sec)

0 2 4 6 810 Time(e-6 Sec)

Fig. 8. The effect of various large modulus-ratio ( *REG* ) on the response of the end point for

the excitation frequency which is close to the first natural frequency, i.e. <sup>1</sup> 0.9

> .

Fig. 7. The effect of various large modulus-ratio ( *REG* ) on the response of the end point for

the excitation frequency far away from the first natural frequency, i.e. <sup>1</sup> 0.1

r=0.9 REG=1000 REG=2000 REG=3000 REG=5000 REG=10000 REG=15000 REG=20000





0

Non-dimensional response at the free end of AFM

4

8

12

0

0.5

Non-dimensional response at the free end of AFM

1

1.5

2

from those of Fig.5 and Fig.6. Figure 7 indicates that the magnitude of the transversal response increases and its oscillating frequency decreases when the modulus-ratio increases. This is because that large shear deformation increases the transversal response, which slows down the oscillating frequency when the excitation frequency is far away from the natural frequency. However, Fig. 8 tells us that the magnitude of the transversal response decreases and its oscillating frequency becomes small when the modulus-ratio increases. The reason for this is that the effects of resonance were counteracted by the transverse shear deformation, resulting in the small transversal response when the AFM cantilever has the sufficiently large modulus-ratios and the excitation frequencies of AFM cantilever are close to the modal frequencies. Consequently, Fig.7 and Fig.8 imply that when a sufficiently small shear modulus is used in AFM cantilever, the effect of transverse shear deformation has a significant effect on the transversal response and the Timoshenko beam model is the proper choice for simulating AFM cantilever dynamic behavior.

Figures 9 shows the effects of various tip holder widths, *w* , on the response of the end point. The widths are normalized by the length of the AFM cantilever. Fig.9 shows that the response at the free end decreases as the width of the tip increases. Therefore, an AFM cantilever with a large tip width is suggested to reduce the response at the end of the AFM cantilever. Figure 10 shows the response histories at the end point of the AFM tip for various tip lengths *h* . From the simulation results shown in Fig.10, the responses are relatively small when the tip length is large. This is due to a large tip length producing large bending effects. Therefore, an AFM tip with large tip length is suggested to reduce the response at the end of the AFM cantilever.

Fig. 6. The effect of various small modulus-ratios ( *REG* ) on the response of the end point for the excitation frequency close to the first natural frequency, i.e. <sup>1</sup> 0.9.

from those of Fig.5 and Fig.6. Figure 7 indicates that the magnitude of the transversal response increases and its oscillating frequency decreases when the modulus-ratio increases. This is because that large shear deformation increases the transversal response, which slows down the oscillating frequency when the excitation frequency is far away from the natural frequency. However, Fig. 8 tells us that the magnitude of the transversal response decreases and its oscillating frequency becomes small when the modulus-ratio increases. The reason for this is that the effects of resonance were counteracted by the transverse shear deformation, resulting in the small transversal response when the AFM cantilever has the sufficiently large modulus-ratios and the excitation frequencies of AFM cantilever are close to the modal frequencies. Consequently, Fig.7 and Fig.8 imply that when a sufficiently small shear modulus is used in AFM cantilever, the effect of transverse shear deformation has a significant effect on the transversal response and the Timoshenko beam model is the proper

Figures 9 shows the effects of various tip holder widths, *w* , on the response of the end point. The widths are normalized by the length of the AFM cantilever. Fig.9 shows that the response at the free end decreases as the width of the tip increases. Therefore, an AFM cantilever with a large tip width is suggested to reduce the response at the end of the AFM cantilever. Figure 10 shows the response histories at the end point of the AFM tip for various tip lengths *h* . From the simulation results shown in Fig.10, the responses are relatively small when the tip length is large. This is due to a large tip length producing large bending effects. Therefore, an AFM tip with large tip length is suggested to reduce the

r=0.9

REG=2 REG=20 REG=100 REG=200 REG=400 REG=800 REG=1000

0 2 4 6 810 Time (e-6 Sec)

> .

Fig. 6. The effect of various small modulus-ratios ( *REG* ) on the response of the end point

for the excitation frequency close to the first natural frequency, i.e. <sup>1</sup> 0.9

choice for simulating AFM cantilever dynamic behavior.

response at the end of the AFM cantilever.



0

4

Non-dimensional response at the free end of AFM

8

12

Fig. 7. The effect of various large modulus-ratio ( *REG* ) on the response of the end point for the excitation frequency far away from the first natural frequency, i.e. <sup>1</sup> 0.1

Fig. 8. The effect of various large modulus-ratio ( *REG* ) on the response of the end point for the excitation frequency which is close to the first natural frequency, i.e. <sup>1</sup> 0.9.

Vibration Responses of Atomic Force Microscope Cantilevers 69

The modal superposition method and the Timoshenko beam theory were applied to determine the flexural vibration responses at the end of the AFM cantilever during AFMbased nanoprocessing process. As expected, the Bernoulli-Euler beam model for AFM cantilever applies to the small effects of transverse shear deformation, but not for modulusratios greater than 1000. When modulus-ratios are greater than 1000, the Timoshenko beam model is the proper choice for simulating the flexural vibration responses of AFM cantilever. Moreover, the oscillating frequency of transversal response decreases due to the transverse shear deformation and the magnitudes of the transversal response depend on the deviation between the excitation frequencies and the modal frequencies. In conclusion, one can reduce the response at the end of AFM cantilever by decreasing the shear modulus when the frequencies of processing are far away from the modal frequencies, and by increasing the shear modulus when the frequencies of processing are close to the modal frequencies. Furthermore, an AFM cantilever with a large tip width and length is suitable for reducing

This work was supported by the National Science Council, Taiwan, Republic of China,

Chang, W.J. & Chu, S.S. (2003). Analytical solution of flexural vibration response on taped atomic force microscope cantilevers, Phys. Letter A. Vol. 309, pp. 133-137. Fang, T.H. & Chang, W.J. (2003). Effects of AFM-based nanomachining process on aluminum surface, J. Phys. Chem. Solids, *J. Phys. Chem. Solids*, Vol. 64 913-918. Girard, P.; Ramonda, M. & R. Arinero, (2006). Dynamic atomic force microscopy operation based on high flexure modes of the cantilever, *Rev. Sci. Instrum.* Vol. 77, 096105. Horng, T.L. (2009). Analytical Solution of Flexural Vibration Responses on Nanoscale

Horng, T.L. (2009). Analyses of Vibration Responses on Nanoscale Processing in a Liquid Using Tapping-Mode Atomic Force Microscopy, *Appl. Surf. Sci.* Vol. 256 311-317. Hsu, J.C.; Lee, H.L.& Chang, W.J. (2007). Flexural Vibration Frequency of Atomic Force

Ilic, B.; Krylov, S.; Bellan L.M. & H.G. Craighead, (2007). Dynamic characterization of

Kageshima, M.; Jensenius, H.; Dienwiebel, M.; Nakayama, Y.; Tokumoto, H. ; Jarvis, S.P. &

Kobayashi, K.; Yamada, H. & Matsushige, K. (2002). Dynamic force microscopy using FM detection in various environments. Appl. Surf. Sci. Vol.188, pp. 430-434.

Processing Using Atomic Force Microscopy, *J. Mater. Pro. Tech.*, Vol. 209, pp. 2940-

Microscope Cantilevers Using the Timoshenko Beam Model, *Nanotechnology.* Vol.

nanoelectromechanical oscillators by atomic force microscopy, J. *Appl. Phys.* Vol.

Oosterkamp, T.H. (2002). Noncontact atomic force microscopy in liquid environment with quartz tuning fork and carbon nanotube probe. *Appl. Surf. Sci.*

**4. Conclusions** 

the response at the end of the AFM cantilever.

**5. Acknowledgements** 

**6. References** 

2945.

18, 285503.

101, 044308

Vol. 188, pp.440-444.

under grant NSC 99-2221-E-168-021.

Fig. 9. Response histories at the end point for various tip widths with 10000 *REG* .

Fig. 10. Response histories at the end point for various tip lengths with 10000 *REG* .

#### **4. Conclusions**

68 Atomic Force Microscopy – Imaging, Measuring and Manipulating Surfaces at the Atomic Scale

Fig. 9. Response histories at the end point for various tip widths with 10000 *REG* .

Response of various tip legths

0.5L 0.1L 0.04L 0.01L 0.001L

with REG=10000

0 2 4 6 810 Time ( e-6 sec)

Fig. 10. Response histories at the end point for various tip lengths with 10000 *REG* .



0

Non-dimensional response at the free end of AFM

2

4

The modal superposition method and the Timoshenko beam theory were applied to determine the flexural vibration responses at the end of the AFM cantilever during AFMbased nanoprocessing process. As expected, the Bernoulli-Euler beam model for AFM cantilever applies to the small effects of transverse shear deformation, but not for modulusratios greater than 1000. When modulus-ratios are greater than 1000, the Timoshenko beam model is the proper choice for simulating the flexural vibration responses of AFM cantilever. Moreover, the oscillating frequency of transversal response decreases due to the transverse shear deformation and the magnitudes of the transversal response depend on the deviation between the excitation frequencies and the modal frequencies. In conclusion, one can reduce the response at the end of AFM cantilever by decreasing the shear modulus when the frequencies of processing are far away from the modal frequencies, and by increasing the shear modulus when the frequencies of processing are close to the modal frequencies. Furthermore, an AFM cantilever with a large tip width and length is suitable for reducing the response at the end of the AFM cantilever.

#### **5. Acknowledgements**

This work was supported by the National Science Council, Taiwan, Republic of China, under grant NSC 99-2221-E-168-021.

#### **6. References**


Lin, S.M. (2005). Exact Solution of the frequency shift in dynamic force microscopy, *Appl. Surf. Sci.* Vol. 250, pp. 228-237.

**0**

**5**

*Italy*

**Wavelet Transforms in Dynamic**

*Interdisciplinary Laboratories for Advanced Materials Physics (i-LAMP) and*

*Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, I-25121 Brescia*

Since their invention scanning tunneling microscopy (STM, (Binnig et al., 1982)) and atomic force microscopy (AFM, (Binnig et al., 1986)) have emerged as powerful and versatile techniques for atomic and nanometer-scale imaging. In this review we will focus on AFM, whose methods have found applications for imaging, metrology and manipulation at the nanometer level of a wide variety of surfaces, including biological ones (Braga & Ricci, 2004; Garcia, 2010; Jandt, 2001; Jena & Hörber, 2002; Kopniczky, 2003; Morita et al., 2009; 2002; Yacoot & Koenders, 2008). Today AFM is regarded as an essential tool for nanotechnology

AFM relies on detecting the interaction force between the sample surface and the apex of a sharp tip protruding from a cantilever, measuring the cantilever elastic deformation (usually its bending or twisting) caused by the interaction forces. Fig. 1a shows a schematic interaction force dependence on tip-sample distance in vacuum (Hölscher et al., 1999). As the distance between the cantilever and the sample surface is reduced by means of a piezoelectric actuator, the tip first experiences an attractive (typically van der Waals) force, that increases to a maximum value. During further approach, the attractive force is reduced until a repulsive force regime is reached. Therefore the AFM is a sensitive force gauge on the nanometer and atomic scale (Butt et al., 2005; Cappella & Dietler, 1999; Garcia & Perez, 2002; Giessibl, 2003;

The use of AFM in such tip-sample force measurements is commonly referred to as *force spectroscopy*. The simplest technique used for quantitative force measurements involves directly monitoring the static deflection of the cantilever as the tip moves towards the surface (approach curve) and then away (retraction curve), providing a deflection versus distance plot. To obtain a force-distance curve, the cantilever deflection is converted to tip-sample interaction force using Hooke's law (Butt et al., 2005; Cappella & Dietler, 1999), after calibration of the cantilever spring constant (Hutter & Bechhoefer, 1993; Sader, 1999). A typical force curve at room temperature and in air is shown in Fig. 1b (Butt et al., 2005; Cappella & Dietler, 1999). During the approach to the surface, an attractive long-range force on the probe bends the cantilever toward the surface. Then the tip suddenly jumps into contact with the surface due to the large gradient of the attractive force near the sample surface

**1. Introduction**

Mironov, 2004).

and a basic tool for material science in general.

**Atomic Force Spectroscopy**

Giovanna Malegori and Gabriele Ferrini

