Sensitivity of V-shaped atomic force microscope cantilevers
based on a modified couple stress theory
Haw-Long Lee and Win-Jin Chang *
Department of Mechanical Engineering
Kun Shan University, Tainan 71003, Taiwan
*
Corresponding author: e-mail: changwj@mail.ksu.edu.tw
Tel: +886(6)2050883; Fax: +886(6)2050509
1
Abstract
A relationship based on a modified couple stress theory is developed to investigate the flexural
sensitivity of a V-shaped cantilever of an atomic force microscope (AFM) taking into account the
normal interaction force between the cantilever tip and the sample surface. An approximate solution
to the flexural vibration problem is obtained using the Rayleigh  Ritz method. The results show
that the sensitivity of the V-shaped AFM cantilever using the modified couple stress theory is
smaller than that using the classical beam theory for the lower contact stiffness. However, when the
contact stiffness becomes higher, the situation is reversed. Furthermore, as the ratio of cantilever
thickness to internal material length scale parameter decreases, the sensitivity of the cantilever
decreases.
Keywords: Atomic force microscope; V-shaped cantilever; Flexural sensitivity; A modified couple
stress theory.
PACS: 07.79.Lh; 62.25.Jk; 68.37.Ps
2
1. Introduction
The atomic force microscope (AFM) has been widely used as a tool for imaging the surface
topographies of conductors and insulators on an atomic scale [1  4]. It is well known that a
cantilever with a sharp tip at the free end plays an important role in AFM measurements. AFM
cantilever induces a dynamic interaction force between the tip and the surface when a tip scans
across a sample surface. The dynamic behavior is complicated and precise analysis is difficult, but
it is related with the resolution of the surface image. Therefore, the dynamic vibration analysis is
significant enough to warrant further investigation.
Rectangular and V-shaped cantilevers are commercial available and are most commonly used
in AFM. A Rectangular cantilever can easily reflect the optical signal in measurements. However, it
is also prone to lateral twisting once the cantilever jumps into and out of contact with the specimen.
On the other hand, a V-shaped cantilever can reduce lateral twisting, but its vibration response is
difficult to analyze due to the complicated geometries [5]. In the last few years, there is a growing
interest in studying the vibration frequency and sensitivity of AFM cantilever for both
configurations using different methods, including analytical and numerical methods [6-12].
Recently, Kahrobaiyan et al. [13] studied the resonant frequency and sensitivity of AFM
rectangular cantilevers using a modified couple stress theory. The theory introduces additional
material parameters used to modify the classical continuum model applying to micro-scale
structures [14-16]. In order to compare the vibration response of an AFM cantilever obtaining using
3
the two theories, in this article we analyze the sensitivity of a V-shaped cantilever. In addition, the
sensitivities of the V-shaped and rectangular cantilevers are compared based on the modified couple
stress theory.
2. Analysis
A schematic diagram of a V-shaped AFM probe cantilevered at one end as shown in Fig. 1.
The cantilever has a length L, thickness h, width B and legs width b with length L1 . A tip with mass
M is attached at the free end of the cantilever. The tip interacts with the sample that is modeled by a
spring constant k. X is the coordinate along the center of the cantilever; Y(X,t) is the vertical
deflection in x-direction at time t. The linear differential equation of motion for free vibration of the
V- shaped cantilever is [13,17]
2
2
 2Y ( X , t )
2  Y ( X , t)
[(
EI
(
X
)

GA
(
X
)
l
)
]


A
(
X
)
0,
X 2
X 2
t 2
(1)
where E and G are the elastic modulus and the shear modulus, respectively; l is the material length
scale parameter which indicates the size-dependent behavior of the microcantilever based on the
modified coupled stress theory; ρis the volume density; A(X) and I(X) are the cross-sectional area
and the area moment of inertia of the cantilever, respectively, and they are a function of X.
The corresponding boundary conditions are
Y (0, t )  0
(2)
Y (0, t )
0,
X
(3)
4
 2Y ( L, t )
 0,
X 2
(4)

 2Y ( L, t )
 2Y ( L, t )
,
[( EI  GAl 2 )
]

kY
(
L
,
t
)

m
X
X 2
t 2
(5)
The cantilever is assumed to be fixed at X = 0. The boundary conditions given by Eqs. (2) and (3)
correspond to the conditions of zero displacement and zero slope at X = 0. In addition, Eq. (4)
represents zero moment at X = L . Eq. (5) is the combination of the normal tip-sample interaction
force and the inertia force of tip mass. Since a linear model is used to describe the tip-sample
interaction force, the cantilever is restricted to small displacements.
Eqs. (1)-(5) define the flexural vibration problem. The harmonic solution to the problem
can be expressed as
Y ( X , t )  W ( X )eit
(6)
where  is the angular frequency.
The dimensionless variables are defined as
x
EI
A
X
W
, y  ,  ( x) 
,  ( x) 
,
L
L
EI 0
 A0

m
,
 A0 L
2 
 2  A0 L4
EI 0
,


 A0l 2
EI 0

kL3
,
EI 0
12
E h / l 
2
(7)
where EI 0 and  A0 are the bending stiffness and mass density at X = 0, respectively;  ,  ,  ,
and  denote the dimensionless contact stiffness, tip mass, natural frequency, and the effective
flexural rigidity based on the modified coupled stress theory, respectively. Meanwhile, h/l is the
ratio of cantilever thickness to internal material length scale parameter.
Applying the above harmonic solution and dimensionless parameters, we can obtain a
5
dimensionless fourth-order ordinary differential equation and four corresponding boundary
conditions. Since the parameters  ( x ) and  ( x) are dependent on the position x along the cantilever,
a numerical method for solving the boundary value problem is proposed. In this article, the
Rayleigh-Ritz method is used to calculate the natural frequencies. we set
r
y ( x )   ai ui ( x).
(8)
i 1
where ai are constants, and ui ( x) is the admissible function which requires to satisfy the
geometric boundary conditions, but need not satisfy the natural boundary conditions. In this study,
ui ( x)  x i 1 ,
i  1, 2,3,...,10 is chosen, then applying the Rayleigh quotient, the following
eigenvalue problem can be obtained:
Ka   2 Ma
(9)
where a is the eigenvector of expansion coefficients and
1
 d 2ui ( x)   d 2u j ( x) 
K i j   ( ( x)   ( x)) 
dx   ui (1)u j (1),

2
2
0
 dx   dx 
1
M i j    ( x)ui ( x)u j ( x)dx   ui (1)u j (1),
(10)
(11)
0
Due to the fact that the mass matrix M does not depend explicitly on  , and assuming the
eigenvector to changes in stiffness is negligible. After the derivative of the eigenvalue problem, Eq.
(9), is taken with respect to  , then we can get the following equation
K

a  2
Ma


Finally, the normalization condition
(12)
aT Ma  1 is introduced, the dimensionless flexural
6
sensitivity, S n , can then be expressed as
Sn 

1 T K

a
a
 2

(13)
When the contact stiffness is given, the natural frequencies and corresponding mode shapes can be
calculated from Eq. (9) and then the flexural sensitivity for each vibrational mode can be obtained
from Eq.(13).
While the parameter  ( x )  1 and  ( x)  1 , it imply the probe is a rectangle shape which
flexural frequency and sensitivity are also analytical. From Eqs.(1) to (7), the dimensionless
fourth-order ordinary differential equation and boundary conditions were simplified as
d 4 y ( x)
2

y ( x)  0
dx 4
(1   )
(14)
and
y (0)  0 ,
dy (0)
d 2 y (1)
d 3 y(1)    2
 0,

0
,

y(1)
dx
dx 2
dx3
1 
(15)
The dimensionless flexural frequency equation and sensitivity can be analytical derived as

C ( ,  )   3 (1  cosh  cos  )  (
  4 )(cosh  sin   sinh  cos  )
1 
(16)
and
2
S na 

3

(sinh  cos   cosh  sin  )
(1  cosh  cos  )  (1  4 )(sinh  cos   cosh  sin  )  2(

 (1   )
3
  ) sinh  sin 
(18)
where
4
2
1 
(19)
7
3. Results and discussion
In order to calculate the flexural sensitivity of a V-shaped AFM cantilever, a ten-term
polynomial expansion is used. The geometric and material parameters of the V-shaped cantilever
are as listed in Table 1, which are chosen to be similar to typical AFM cantilevers.[17] In addition,
the tip mass used in the analysis is 2 1013 kg [18]. The ratio of cantilever thickness to internal
material length scale parameter, h/l, is assumed to be 4. The dimensionless flexural sensitivities of a
V-shaped AFM cantilever based on modified couple stress theory and classical beam theory are
depicts in Fig. 2. Unlike the classical beam theory, the modified couple stress theory includes the
effect of small-scale structure and that leads to a stiffer cantilever. Therefore, when the contact
stiffness is small, it can be seen that the sensitivity of the V-shaped cantilever using the modified
couple stress theory is smaller than that using the classical beam theory. When the contact stiffness
becomes large, the situation is reversed. This indicates that a stiffer cantilever is better for scanning
a harder surface. In addition, a softer cantilever is easier to bent and more sensitive for imaging of
soft biological samples.
The dimensionless flexural sensitivities of the cantilever for the V-shaped and the rectangular
cross section with h / l  4 based on modified couple stress theory are illustrated in Fig. 3. When the
contact stiffness is low, the sensitivity of the V-shaped cantilever is larger than that of the
rectangular cross section cantilever. However, the situation is reversed when the contact stiffness
becomes large. It implies that a V-shaped cantilever is suitable for scanning a softer surface, while a
8
rectangular cross section cantilever is better for scanning a stiffer surface.
The effect of the ratio of microcantilever thickness to internal material length scale parameter
h/l on the sensitivity is shown in Fig.4. The material length scale parameter h/l represents the size
effect based on modified couple stress theory. It is noted that for the case of h / l   , this implies
that the classical beam theory is used in the analysis. Hence, it can be seen that the effect of h/l on
the sensitivity becomes significant as the thickness of cantilever is close to the internal material
length scale parameter for the lower contact stiffness. In addition, the sensitivity decreases as the
value of h/l decrease. This indicates the thickness-dependent behavior of an AFM cantilever.
4. Conclusions
The sensitivity of the flexural vibration modes for a V-shaped atomic force microscope
cantilever has been analyzed using the Rayleigh-Ritz method. According to the analysis, the
sensitivity of the V-shaped cantilever using the modified couple stress theory was smaller than that
using the classical beam theory. The V-shaped cantilever was suitable for scanning a softer surface,
while it was a better choice for scanning a stiffer surface using a rectangular cross-section cantilever.
In addition, the sensitivity of a V-shaped cantilever decreased as the ratio of cantilever thickness to
internal material length scale parameter decreased.
Acknowledgment
9
The authors wish to thank the National Science Council of the Republic of China in Taiwan for
providing financial support for this study under Projects NSC 99-2221-E-168-019 and NSC
99-2221-E-168-031.
References
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[2]B. Bhushan, Handbook of Micro/Nanotribology, 2nd edn., CRC, Boca Raton, FL, 1999.
[3]Trends in Nanotechnology Research, edited by Eugene V. Dirote, Nova Science Publishers, New
York, 2004.
[4]Nanophysics, Nanoclusters and Nanodevices, edited by Kimberly S. Gehar, Nova Science
Publishers, New York, 2006.
[5]S. K. Jericho, and M. H. Jericho, Rev. Sci. Instrum. 73 (2002) 2483.
[6]W.J. Chang, Nanotechnology 13 (2002) 510.
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10
[11]T. Ogawa, S. Kurachi, M. Kageshima, Y. Naitoh,Y. J. Li, and Y. Sugawara, Ultramicroscopy
110 (2010) 612
[12]Q. Huang and Q. Hou, Appl. Mech. Mater. 44-47 (2011) 489.
[13]M.H. Kahrobaiyan, M. Asghari, M. Rahaeifard, and M.T. Ahmadian, Int. J. Eng. Sci. 48 (2010)
1985.
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Table 1
11
The geometric and material parameters of a V-shaped AFM cantilever.
Figure captions
Fig. 1. Schematic diagram of a V-shaped AFM cantilever in contact with a sample. The interaction
with the sample surface is modeled by a normal spring.
Fig. 2. Dimensionless flexural sensitivities of a V-shaped AFM cantilever based on modified couple
stress theory and classical beam theory.
Fig. 3. Dimensionless flexural sensitivities of V-shaped and rectangular AFM cantilevers
12
with h / l  4 based on modified couple stress theory.
13
Fig. 1. Schematic diagram of a V-shaped AFM cantilever in contact with a sample. The interaction
with the sample surface is modeled by a normal spring.
14
100
mode 1
mode 2
Dimensionless sensitivity, Sn
10-1
mode 3
mode 4
mode 5
10-2
Classical beam theory
Modified couple stress theory
-3
10
-4
10
-2
10
10
-1
10
0
1
10
10
Dimensionless contact stiffness, 
2
3
10
Fig. 2. Dimensionless flexural sensitivities of a V-shaped AFM cantilever based on modified couple
stress theory and classical beam theory.
15
100
mode 1
mode 2
Dimensionless sensitivity, Sn
mode 3
-1
10
mode 4
mode 5
10-2
V- shape
Rectangular
-3
10
-4
10
10-2
10-1
100
101
102
Dimensionless contact stiffness, 
103
Fig. 3. Dimensionless flexural sensitivities of V-shaped and rectangular AFM cantilevers
with h / l  4 based on modified couple stress theory.
16
Dimensionless sensitivity of mode 1, Sn
0.7
0.6
h/l=4
h/l=8
h/l=
0.5
0.4
0.3
0.2
0.1
0
10-2
10-1
100
101
102
Dimensionless contact stiffness, 
103
Fig. 4. Dimensionless flexural sensitivity of mode 1 for an AFM V-shaped cantilever at various
ratios of microcantilever thickness to internal material length scale parameter h/l.
17
Dimensionless sensitivity of mode 2, Sn
0.2
0.15
0.1
h/l=4
h/l=8
h/l=
0.05
0
10-2
10-1
100
101
102
Dimensionless contact stiffness, 
Fig 5
18
103
Dimensionless sensitivity of mode 3, Sn
0.1
0.08
0.06
h/l=4
h/l=8
h/l=
0.04
0.02
0
10-2
10-1
100
101
102
Dimensionless contact stiffness, 
Fig 6
19
103
0.6
n
Dimensio
less s
1,
f mode
o
y
it
iv
it
s
en
Sn
0.6
0.5
0.5
0.4
0.4
0.3
7
0.3
10
8
6
6
L1 / b
5
4
4
Fig 7
2
β=1
20
h/l
0.2
2,
f mode
o
y
it
iv
it
s
n
nless se
io
s
n
e
im
D
Sn
0.2
0.18
0.18
0.16
0.16
0.14
0.14
0.12
0.12
0.1
7
0.1
10
8
6
6
L1 / b
5
4
4
2
Fig 8
21
h/l
f mode 3, S n
o
y
it
iv
it
s
n
e
s
s
Dimensionles
0.1
0.1
0.09
0.09
0.08
0.08
0.07
0.07
0.06
0.06
0.05
7
0.05
10
8
6
6
L1 / b
5
4
4
2
Fig 9
22
h/l
23