Surface electrostatic attraction between the atoms in the tip of the probe and those in the surface is simulated with flexural and longitudinal contact stiffness

_______________________________

*Corresponding author: [email protected]

ANALYTICAL SOLUTION OF INTERACTIVE DAMPING AND

LONGITUDINAL AND NORMAL CONTACT STIFFNESS ON SENSITIVITY OF VIBRATION MODES OF RECTANGULAR AFM CANTILEVERS

J. YAZDANI^a, S.YAZDANI^b,*, S. BIGHAM^b

aSchool of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran

bSchool of Mechanical Engineering, University of Tehran, Tehran, Iran

The interactive damping sensitivity and the resonant frequency of normal vibration and longitudinal vibration of an atomic force microscope (AFM) rectangular cantilever have been analyzed. Surface electrostatic attraction between the atoms in the tip of the probe and those in the surface is simulated with flexural and longitudinal contact stiffness.

Theoretical investigation of normal and longitudinal interaction individually and both, have been presented as normal and longitudinal sensitivity. Also using the sensitivity equations the effects of material property and geometrical parameters can be specified.

(Received November 24, 2008; accepted December 5, 2008)

Keywords: Atomic force microscope; Sensitivity; Resonant frequency; Nano

1. Introduction

The vibration of the tiny oscillators can be measured by tapping with an atomic force microscope (AFM). An AFM uses a tiny probe that moves slowly just above a surface.

Electrostatic attraction or repulsion between the atoms in the tip of the probe and those in the surface causes the probe to move up and down, creating an image of the surface so detailed that individual atoms show up as bumps. Alternatively, the AFM can be used in tapping mode, literally bouncing off the surface. To measure the vibration of a Nano-Mechanical oscillator, the AFM probe moves along the length of the oscillating rod. The result is a complex bouncing interaction between the probe and the oscillator imagine shaking one end of a spring and watching the vibrations at the other end from which the frequency of vibration of the oscillator can be determined mathematically. To obtain atomic resolution, the AFM cantilever should not be too soft, and at the same time, it should have a high resonant frequency, in order to minimize sensitivity to vibrational noise from the building and to have a large imaging bandwidth.

Furthermore, the resonant frequency of the cantilever can influence the imaging rate in the operating process. More information exists in the literature [1-5]. Dynamic responses of the AFM cantilever have been investigated by [6-8]. Some of researchers [9–14] have been studied the vibration response of an AFM cantilever for convenience without considering the interactive damping but Turner et al. [15] and Rabe et al. [16], have been shown the effect of damping on the vibration response of an AFM cantilever is very important. Chang et al. [17] have been investigated effect of interactive damping on sensitivity of vibration modes of rectangular AFM cantilevers. They found that the sensitivity of flexural mode 1 clearly decreases with increasing normal interactive damping coefficient and the higher damping coefficient can influence the larger range of βn value. They also presented two equations for normal and torsional sensitivity. Effect of

tip length and normal and lateral contact stiffness on the flexural vibration responses of atomic force microscope cantilevers have been investigated by Wu et al. [18]. They didn't consider the damping effects in their study. More recently Chang et al.[19] studied the sensitivity of the first four flexural modes of an AFM cantilever with a sidewall probe. Coupled lateral bending–

torsional vibration sensitivity of atomic force microscope cantilever have been studied by Haw- Long Lee and Win-Jin Chang [20]. Their results showed that each mode has a different resonant frequency to variations in contact stiffness and each frequency increased until it eventually reached a constant value at very high contact stiffness. The normal and lateral interactive forces between the cantilever tip and the sample surface can be modeled by a set combination of a damper spring parallel to a dashpot in the normal direction and a similar combination in the lateral direction.

Many researchers have studied the vibration response of an AFM cantilever, but there is no any investigation to consider the effects of normal and longitudinal contact stiffness both, by a set combination of a damper spring parallel simulation. At the current study analytical solution of interactive damping and normal and longitudinal contact stiffness on sensitivity of vibration modes of rectangular AFM cantilevers is presented.

2. Analysis

The schematic of the problem is shown in Fig.1. The cantilever has a length L, thickness b, width a, and tip length h. The tip interacts with the sample by a linear spring kn and dashpots Cn

for normal interaction and for longitudinal Kl and Cl for longitudinal interaction. It is assumed that the atomic force microscope cantilever here is a rectangular elastic beam and the dashpots are assumed to create a linear viscous type of damping.

2.1. Flexural vibration

When the AFM cantilever tip interacts with the sample by a normal spring Kn, normal dashpot Cn, and longitudinal spring kl, longitudinal dashpots Cl the cantilever will vibrate flexural.

The linear differential equation of motion for the free vibration of the cantilever beam is [21]

) 0 , ) (

, (

2 2 4

4

=

∂ + ∂

∂

∂

t y x A y

x t x

EI y ρ

(1)

Where E is the modulus of elasticity, I is the area moment of inertia, ρ is the volume density and there are related to material property and A is the uniform cross-sectional area of the cantilever.

Fig. 1. Schematic of a rectangular AFM cantilever in contact with a sample.

The corresponding boundary conditions are

Y(0,t) =0 (2)

) 0 , 0

( =

∂

∂ x y t

(3)

) ) , ( (

) , ) (

,

(

2 2

2 2

x t l y h t

x c t L h y

x k t L

EI y

_{l l}

∂

∂

∂

− ∂

∂

− ∂

∂ =

∂

(4)

t t y L c t l y x k

t L

EI y =

_n

+

_n

∂ ∂

∂

∂ ( , )

) , ) (

, (

3 3

(5)

The boundary condition of the probe at

x =

0 is assumed fixed end; then the boundary conditions given by Eq. (2), (3) correspond to conditions of zero displacement and zero slope. The boundary conditions given by Eq. (4), (5) correspond to zero moment at x=L and the force is balanced between the beam and a combination of the linear tip-sample stiffness and dashpot.

A general solution of equations. (1) to (5) is

e

iwt

x k a

x k a

x k a x k a t x

y ( , ) = (

₁

cos +

₂

sin +

₃

cosh +

₄

sinh )

(6)

where aj, j=1-4 , are constants determined from the boundary conditions, w is the angular frequency, k is the flexural wave number.

A k EI

w ρ

=

2 (7)

By setting Eq. (6) into Eq. (1)

2 0

4

− Aw =

k

EI ρ

(8)

Frequency can be shown as function of wave number

A EI f L

π ρ γ

2 2

=

2 (9)

From the quations, (2), (3), (6) can be found:

3 0

1

+ a =

a

,

a

₂

+ a

₄

=

0 (10)

From the equations (4), (6):

+ + +

− +

− (cos cosh ) ( ) (sin sinh )]

[

^{2 2}

1

k EI kL L k c wi h k kL kL

a

_{l l}

0 )]

cosh (cos

) (

) sinh (sin

[

^{2 2}

2

− k EI kL + kL + k + c wi h k kL − kL =

a

_{l l} (11)

From the equations (5), (6):

[ ⁻

³

^{(sinh − sin ) + ( + )(cosh − cos )} ] ⁺

1

EIk kL kl k c wi kL kL

a

_{n n}

[

³

^{(cosh cos ) ( )(sinh sin )} ] ⁰

2

− EIk kL + kL + k + c wi kL − kL =

a

_{n n} (12)

By introducing

) sinh (sin

) (

) cosh

(cos

²

2

1

EI k k L k L k c wi h k k L k L

X = − + −

_l

+

_l

+

) cosh (cos

) (

) sinh

(sin

²

2

2

EIk k L k L k c wi h k kL kL

X = − + +

_l

+

_l

−

cos ) )(cosh

( ) sin (sinh

3

1

EIk k L k L k c wi k L k L

Y = − − +

_n

+

_n

−

(13)

) sin )(sinh

( ) cosh

3

(cos

2

EI k k L k L k c wi kL kL

Y = − + +

_n

+

_n

−

And with

0

2 1

2

1

=

Y Y

X

X

(14)

Then the characteristics equation can be found:

) , , ( k_n k_l

C γ = ⁸

0

1

∑

⁼

=

= j

j

η

j (15)

Now suppose that

ε

₁

= [ E

²

I

²

k

⁵

+ ( k

n

+ c

n

w i )( k

l

+ c

l

wi ) k h

²

]

ε

₂

= [

2

EI k

⁴

h

²

( k

_l

+ c

_l

wi ) ]

( )]

) (

[

^{4 2 2}

3

= −

2

EI k h k

_l

+ c

_l

wi +

2

E I k k

_n

+ c

_n

wi

ε

(16)

)]

( )

(

[

^{4 2 2}

4

=

2

E I k h k

_l

+ c

_l

wi +

2

E I k k

_n

+ c

_n

wi

ε

] )

)(

(

[

^{2 2 2 5}

5

=

2

k

_n

+ c

_n

wi k

_l

+ c

_l

wi h k −

2

E I k

ε

And by introducing

η

1 =

ε

₁

sinh

²

γ

η

2 =

− ε

₁

sin

²

γ

η3 =

− ε

₂

sin

²

γ cos

²

γ

(17)

η

4 =

− ε

₃

sinh γ cos γ

η

5 ⁼

− ε

₄

cosh γ sin γ

η

₆ ⁼

− ε

₁

cos

²

γ

η

₇ ⁼

ε

₅

cosh γ cos γ η

8 ⁼

− ε

₁

cosh

²

γ

where γ=K×L is the normalized longitudinal wave number, βl=(Kl×L³/EI) is the normal stiffness ratio, βn=(Kn×L³/EI) is the normal stiffness ratio between the normal contact stiffness and that of the cantilever.

If we negligible Cn, Kn

0 ] cosh sin cos sinh cos )[sin (

) cosh cos 1 ( ) , (

2 2

2

1 = + + +

γ γ

+

γ γ

+

γ γ

=

ρ γ γ β

γ γ

β

γ

AE

L i c L

C _l h ^{l l (18)}

The longitudinal and normal sensitivity of the cantilever can be calculated from the frequency, which can be measured. The sensitivity of the mode of the cantilever changes significantly for small variations of stiffness as the cantilever crosses the sample. Differentiation of Eq. (18) with respect to βl yields

γ β β

γ

∂ ∂

∂ ∂

−

=

1 1

c c d

d

_l

l

(19)

The relationship between longitudinal frequency fl and contact stiffness βl can be expressed as

l l l

l

f

f β γ

β ^∂ γ ^{∂ ∂}

= ∂

∂

∂

(20)

Then the following equation can be obtained

)]

cosh sin cos sinh cos

2 (sin 21 [

2 2

2 γ γ γ γ γ γ γ

π ρ

β ⁼ L ^{EIA −} _L^{h + +}

d df

l l

) cosh sin

sinh (cos )

cosh cos

1

{( + γ γ + γ γ γ − γ γ

×

) cosh sin

cosh sinh

cos

2

(sin

2 2

γ γ

γ γ

γ ρ γ

γ + +

+ L

h AE

i L c

_l

1 2

2 2

2 2

)}

cosh cos

2 sin

)(cos

( + − +

⁻

+ γ γ γ γ

ρ β γ

AE iL c L

h

_l

l (21)

Equation (18) and (21) are complex types due to the complex representation of damping. The absolute value of the complex quantity is used in the calculation. Eq. (21) can be expressed in normalized form as

A EI L

df d

S

_l ^l

π ρ β

2 2

1

=

(22)

By Setting Kl and Cl and equal to zero with neglected Kn and Cn, From the Eq. (15), the characteristics equation can be found:

) cosh sin cos )(sinh (

) cosh cos 1 ( ) , (

2 3

2

γ γ γ γ

ρ β γ γ γ γ

β

γ = + − + −

AEI L i

C

_{n n}

c

^{n (23)}

The normal sensitivity of the cantilever can be calculated from the frequency, which can be measured. The sensitivity of the mode of the cantilever changes significantly for small variations of stiffness as the cantilever crosses the sample. Differentiation of Eq. (23) with respect to βn

yields

γ β β

γ

∂ ∂

∂ ∂

−

=

2 2

c c d

d

n

n

(24)

The relationship between normal frequency fn and contact stiffness βl can be expressed as

n n n

n

f

f β γ

β γ ∂ ∂

∂ ∂

∂ =

∂

(25)

Then the following equation is obtained

[2 (sinh cos sin cosh )]

21

2 ρ γ γ γ γ γ

β ⁼ π EIA ⁻

d L df

n n

) sin cosh cos

(sinh )

cosh cos

1 ( 3

{ γ

²

+ γ γ + γ

³

γ γ − γ γ

×

(26)

1 2

)}

sin cosh cos

(sinh 2 1

sin sinh ) (

2 + − − ⁻

+ γ γ γ ρ γ γ γ γ

ρ β γ

L AEI EI ic

A iL c

n n

n

That equation (26) is same as presented equation, for this case in [17]. The absolute value of the complex quantity is used in the calculation. Eq. (26) can be expressed in normalized form as

A EI L df d

S_n ⁿ

π ρ β

2 2

1

= (27)

With considering normal and longitudinal effects both, characteristics equation can be found as

∑

⁼

=

=

=

⁸

1

3

( , , )

^j

0

j j l

C γ β

n

β φ

(28)

Where

γ γ

ρ β γ

ρ β γ

γ

φ

₁

[

⁵

(

²

)(

²

)

²

] sinh

²

L h AEI iL c AEI

iL

L +

_n

+ c

ⁿ _l

+

^l

=

.

γ γ

ρ β γ

ρ β γ

γ

φ

₂

[

⁵

(

²

)(

²

)

²

] sin

²

L h AEI iL c AEI

iL

L −

_n

+ c

ⁿ _l

+

^l

−

=

.

γ γ

ρ β γ

φ 2 γ ( )] sin cos

[

^{4 2 2}

3

AEI

iL c L

h

_l

l

+

−

=

.

γ ρ γ

β γ ρ γ

β γ

φ 2 γ ( ) 2 ( )] sinh cos

[

2 2

2 2 4

4

AEI

iL L c

AEI iL c L

h

_n

n l

l

+ + +

−

=

. (29)

γ ρ γ

β γ ρ γ

β γ

φ 2 γ ( ) 2 ( )] sin cosh

[

2 2

2 2 4

5

AEI

iL L c

AEI iL c L

h

_n

n l

l

+ − +

−

=

.

γ γ

ρ β γ

ρ β γ

γ

φ

²

2 2 2

5

6

[ ( )( ) ] cos

L h AEI iL c AEI

iL

L −

_n

+ c

ⁿ _l

+

^l

−

=

.

γ γ γ

ρ β γ

ρ β γ

γ

φ [ 2 2 ( )( ) ] cos cosh

2 2 2

5

7

L

h AEI iL c AEI

iL

L +

_n

+ c

ⁿ _l

+

^l

−

=

γ γ

ρ β γ

ρ β γ

γ

φ

₈

[

⁵

(

²

)(

²

)

²

] cosh

²

L h AEI iL c AEI

iL

L −

_n

+ c

ⁿ _l

+

^l

−

=

For determine normal and longitudinal sensitivity the set of following equation are obtained

γ β βγ

∂ ∂

∂ ∂

−

=

3 3

c c d

d _l

l

(30)

l l l

l

f

f β γ

β γ ^∂

∂ ∂

= ∂

∂

∂

(31)

γ β β

γ

∂ ∂

∂ ∂

−

=

3 3

c c d

d _n

n

(32)

n n n

n

f

f β γ

β γ ∂ ∂

∂ ∂

∂ =

∂

(33)

By introducing new parameters Sl and Sn as function of βl and βn for the first the effects of normal and longitudinal both, is considered. In the previous works parameter Sl isn't introduced and Sn is function of βl only.

EIA L df d

S ^l

l n

l l

π ρ β β

β

2 2

) 1 ,

( = (34)

EIA L df d

S ⁿ

n l

n n

π ρ β β

β

2 2

) 1 ,

( = (35)

The sensitivity parameter is also a functional of material property such as the modulus of elasticity and the volume density and geometrical property such as the area moment of inertia and cross-sectional area of the cantilever. The above equation denotes also the effects of this property.

3. Conclusion

The interactive damping and normal and longitudinal contact stiffness on sensitivity of vibration modes of rectangular AFM cantilevers has been investigated. The effect of damping on the vibration response of an AFM cantilever is significant and cannot be disregarded. Theoretical investigation of normal and longitudinal interaction individually and both, have been presented.

Using the sensitivity equations the effects of material property and geometrical parameters can be specified.

References

[1] G. Binnig, H. Rohrer, C. Gerber, E. Weible, Phys. Rev. Lett. 49, 57 (1982).

[2] G. Binnig, C.F. Quat, C. Gerber, Phys. Rev. Lett. 56, 930 (1986).

[3] K. Holmberg, A. Matthews, Coatings Tribology: Properties, Techniques and Applications in Surface Engineering, Elsevier, New York, 1994.

[4] B. Bhushan, Handbook of Micro/Nanotribology, second ed., CRC, Boca Raton, FL, 1999.

[5] B. Bhushan, Nanotribology and Nanomechanics: An Introduction, Springer, Heidelberg, Germany, 2005.

[6] O. Nakabeppu, M. Chandrachood, Y. Wu, J. Lai, A. Majumdar, Appl. Phys. Lett.

66(6), 694 (1995).

[7] U. Rabe, K. Janser, W. Arnold, Rev. Sci. Instrum. 67(9), 3281 (1996).

[8] T. Drobek, R.W. Stark, M. Graber, W.M. Heckl, New J. Phys. 1, 15.1 (1999).

[9] S. Hirsekom, U. Rabe, W. Arnold, Nanotechnology 8, 57 (1997).

[10] K. Yamanaka, A. Noguchi, T. Tsuji, T. Koike, T. Goto, Surf. Interface Anal. 27, 600 (1999).

[11] G.G. Yaralioglu, F.L. Degertekin, K.B. Crozier, C.F. Quate, J. Appl. Phys.

87(10), 7491 (2000).

[12] R.F. Fung, S.C. Huang, ASME J. Vibr. Acoust. 123, 502 (2001).

[13] R. Levy, M. Maaloum, Nanotechnology 13, 33 (2002).

[14] H. Kawakatsu, S. Kawai, D. Saya, M. Nagashio, D. Kobayashi, H. Toshiyoshi, Rev. Sci.

Instrum. 73(6), 2317 (2002).

[13] J. A. Turner, S. Hirsekorn, U. Rabe, W. Arnold, J. Appl. Phys. 82(3), 967 (1997).

[14] U. Rabe, J.A. Turner, W. Arnold, Appl. Phys. A 66, S277 (1998).

[15] J.A. Turner, S. Hirsekorn, U. Rabe, W. Arnold, J. Appl. Phys. 82(3), 967 (1997).

[16] U. Rabe, J.A. Turner, W. Arnold, Appl. Phys. A 66, S277 (1998).

[17] Win-Jin Chang, Te-Hua Fang, Huann-Ming Chou, Physics Letters A 312,158 (2003).

[18] Tser-Son Wu, Win-Jin Chang, Jung-Chang Hsu, Microelectronic Engineering 71, 15 (2004).

[19] Win-Jin Chang, Haw-Long Lee, Terry Yuan-Fang Chen, Ultramicroscopy 108, 619 (2008).

[20] Haw-Long Lee, Win-Jin Chang, Ultramicroscopy 108, 707 (2008).

[21] W. J.Weaver, S.P. Timoshenko, D.H. Young, Vibration Problems in Engineering, 5th Edition, Wiley, New York, 1990.