ARTICLE IN PRESS
Ultramicroscopy 107 (2007) 245–253
www.elsevier.com/locate/ultramic
Energy dissipation and dynamic response of an amplitude-modulation
atomic-force microscopy subjected to a tip-sample viscous force
Shueei Muh Lin
Department of Mechanical Engineering, Kun Shan University, Tainan, Taiwan 710-03, Republic of China
Received 10 February 2006; received in revised form 2 August 2006; accepted 2 August 2006
Abstract
In a common environment of atomic force microscopy (AFM), a damping force occurs between a tip and a sample. The influence of
damping on the dynamic response of a cantilever must be significant. Moreover, accurate theory is very helpful for the interpretation of a
sample’s topography and properties. In this study, the effects of damping and nonlinear interatomic tip–sample forces on the dynamic
response of an amplitude-formulation AFM are investigated. The damping force is simulated by using the conventional Kelvin–Voigt
damping model. The interatomic tip–sample force is the attractive van der Waals force. For consistance with real measurement of a
cantilever, the mathematical equations of the beam theory of an AM-AFM are built and its analytical solution is derived. Moreover, an
AFM system is also simplified into a mass–spring-damper model. Its exact solution is simple and intuitive. Several relations among the
damping ratio, the response ratio, the frequency shift, the energy dissipation and the Q-factor are revealed. It is found that the resonant
frequencies and the phase angles determined by the two models are almost same. Significant differences in the resonant quality factors
and the response ratios determined by using the two models are also found. Finally, the influences of the variations of several parameters
on the error of measuring a sample’s topography are investigated.
r 2006 Elsevier B.V. All rights reserved.
Keywords: AFM; AM; Energy dissipation
1. Introduction
Atomic force microscopy (AFM) has been widely
developed as a powerful technique for obtaining atomicscale images and the material surface properties [1,2]. For
example, AFM is used to scan DNA, proteins and
polymers in air or liquids [2]. When a soft sample such as
DNA, protein and polymer is scanned, there exists a
damping force between a cantilever tip and a sample [3–16].
For studying the morphologies and nanostructures of
samples, the energy dissipation, the frequency shift and the
phase angle of an AFM subjected to a damping force must
be investigated. Moreover, an accurate analysis can
improve greatly the studies of surface image, interaction
energies and interaction forces.
In general, dynamic behavior of a AFM is simulated by
using the beam theory [10,12,13,17–21] and the effective
Tel.: +866 62050496; fax: +866 62050509.
E-mail address: sm.lin@msa.hinet.net.
0304-3991/$ - see front matter r 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.ultramic.2006.08.001
spring–mass-damper model [4–11,14–16,22]. The elementary beam theory is commonly known as the Euler–Bernoulli beam theory. A large slender ratio and a small
deformation are the assumptions of the Euler–Bernoulli
theory. Generally, the Euler–Bernoulli beam theory is
suitable for the AFM probe. It is well known that if the
beam theory is used to simulate a vibrating motion of a
AFM, the equation of motion is a partial differential
equation. Moreover, in a real measurement, the tip–sample
interacting force is nonlinear. Obviously, the mathematical
problem involved of the two conditions is very difficult to
solve so that some approximated methods, such as (1) the
force gradient method [10,12,13] and (2) the mode superposition method [19,20], are proposed. Firstly, if the
tip–sample interacting force is described by using the force
gradient method, the nonlinear tip–sample force is replaced
by a linearized one. Therefore, the solution of the simplified
problem can be derived by using a conventional method.
However, the force gradient method has been verified to
result in inaccurate results [17]. Secondly, it is well known
ARTICLE IN PRESS
246
S.M. Lin / Ultramicroscopy 107 (2007) 245–253
that the mode superposition method is suitable only for a
linear system with a proportional damping. In other words,
the mode superposition method cannot be used to
investigate arbitrary tip–sample damping force.
Alternatively, the cantilever is usually approximated by
an effective spring–mass-damper model. The equation of
motion of an spring–mass-damper model is an ordinary
differential equation which is easily solved. However,
because an effective spring–mass-damper model has one
degree freedom, only the first mode can be commonly
derived. It should be noted that according to the
fundamental natural frequency, an effective spring constant or an effective mass of an effective spring–massdamper model is derived. Therefore, the model can result in
accurate results only for some special conditions. Rodriguez and Garcia [20] found that simulation based on a
mass–spring-damper model was suitable to describe a
cantilever tip motion with relatively high Q factor. So far,
due to the complexity of the beam theory subjected to the
van der Waals and a viscous forces no analytical solution
of the system has been proposed.
In this study, an analytical solution of the dynamic
response of an AM-AFM in the Bernolli–Euler beam
theory is derived. Because the mass–spring-damper model
is simple and helpful for interpreting the morphologies and
nanostructures of a sample, the exact solution of the
mass–spring-damper system is also derived here. Moreover, the assessment of the two models is made. The effects
of several parameters on the energy dissipation, the
frequency shift and the response ratio are investigated.
viscous and a nonlinear interatomic van der Waals forces
between the tip and the sample are considered. Nonuniform cross-section of the beam is considered. The material
of beam is homogenous. In terms of the following
dimensionless quantities,
2. Damped beams
at x ¼ 1
In this study, a cantilever is excitated harmonically by a
piezoelectric shaker at the root end, as shown in Fig. 1. A
3
bðxÞ ¼ IðxÞ
Ið0Þ ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
c̄t ¼ ct L EIð0ÞrAð0Þ
AH RL
cv ¼ EIð0ÞL
3 ;
0
D̄ ¼
f v ¼ cv2
m ¼ rðxÞAðxÞ
rð0ÞAð0Þ ;
mt
m̄t ¼ rð0ÞAð0ÞL
;
qffiffiffiffiffiffiffiffiffi
o ¼ OL2 rAð0Þ
EIð0Þ
x ¼ Lx ;
6D̄
D
L0
wðx; tÞ ¼ WLðx;tÞ
;
0
qffiffiffiffiffiffiffiffiffi
EIð0Þ
t ¼ Lt2 rAð0Þ
(1)
the dimensionless governing differential equation of the
system is [17,23]
q2
q2 w
q2 w
bðxÞ
¼ 0.
þ
mðxÞ
qt2
qx2
qx2
(2)
The associated boundary conditions are:
at x ¼ 0
w ¼ A0 cos ot.
qw
¼ 0,
qx
q2 w
¼ 0,
qx2
Fig. 1. Geometry and coordinate system of a microprobe.
ð3Þ
(4)
ð5Þ
ARTICLE IN PRESS
S.M. Lin / Ultramicroscopy 107 (2007) 245–253
q
q2 w
q2 w
qw
bðxÞ 2 m̄t 2 c̄t
¼ f v ðtÞ,
qx
qt
qt
qx
2
2 d
d W̄ c
d
d W̄ s
b
b
cos ot þ
sin ot
dx
dx
dx2
dx2
þ m̄t o2 W̄ c cos ot þ W̄ s sin ot
þ c̄t o W̄ c sin ot þ W̄ s cos ot ¼ f~v ,
(6)
where A is the cross-sectional area of the beam, AH the
Hamaker constant, A0 the amplitude of excitation, ct the
interacting viscous coefficient, D the tip–surface distance, E
the Young’s modulus, fv the dimensionless van der Waals
force [9], I the area moment of inertia, L the length of the
beam, and L0 the characteristic length. A small value of L0
is introduced to avoid numerical transaction error. mt is the
tip mass, R the tip radius, t the time variable, W the flexural
displacement, x the coordinate along the beam, r the mass
density per unit volume and O the frequency of excitation.
3. Solution method
3.1. Characteristic governing equations and boundary
conditions
ð16Þ
where the dimensionless
2 van der Waals force is
f~v ¼ cv =6 D̄0 W ð1; tÞ .
On multiplying Eq. (16) by cos ot and integrating it
from 0 to the period T ¼ 2p/o, Eq. (16) becomes
b
_
d3 W̄ c
þ m̄t o2 W̄ c þ c̄t oW̄ s ¼ f vc ,
3
dx
_
2
(17)
2
2
where f vc ¼ cv W̄ c ð1Þ=3ðD̄0 W̄ c ð1Þ W̄ s ð1ÞÞ3=2 .
Similarly, on multiplying Eq. (16) by sin ot and
integrating it from 0 to the period T ¼ 2p/o, Eq. (16)
becomes
b
_
d3 W̄ s
þ m̄t o2 W̄ s c̄t oW̄ c ¼ f vc ,
3
dx
_
2
(18)
2
2
where f vc ¼ cv W̄ s ð1Þ=3ðD̄0 W̄ c ð1Þ W̄ s ð1ÞÞ3=2 .
The solution of the system is assumed as
W ðx; tÞ ¼ W̄ c ðxÞ cos ot þ W̄ s ðxÞ sin ot
3.2. Characteristic equations
¼ W̄ ðxÞ cosðot yÞ,
ð7Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
where W̄ ðxÞ ¼ W̄ c þ W̄ s , tan y ¼ W̄ s ðxÞ=W̄ c ðxÞ and y
is the phase angle. Substituting the solution into the
governing equation and the boundary conditions and
taking ‘cos ot’ and ‘sin ot’ apart, coupled differential
equations can be obtained:
d2
d2 W̄ c
bðxÞ
(8)
o2 mðxÞW̄ c ¼ 0,
dx2
dx2
d2
d2 W̄ s
bðxÞ
o2 mðxÞW̄ s ¼ 0.
dx2
dx2
247
(9)
At x ¼ 0:
W̄ c ð0Þ ¼ A0 ,
(10)
W̄ s ð0Þ ¼ 0,
(11)
dW̄ c ð0Þ
¼ 0,
dx
(12)
dW̄ s ð0Þ
¼ 0.
dx
(13)
At x ¼ 1:
d2 W̄ c ð1Þ
¼ 0,
dx2
(14)
d2 W̄ s ð1Þ
¼ 0,
dx2
(15)
The general solutions of the characteristic differential
Eqs. (8) and (9) can be expressed as, respectively,
W̄ c ðxÞ ¼ C c1 V c1 ðxÞ þ C c2 V c2 ðxÞ þ C c3 V c3 ðxÞ þ C c4 V c4 ðxÞ,
(19)
W̄ s ðxÞ ¼ C s1 V s1 ðxÞ þ C s2 V s2 ðxÞ þ C s3 V s3 ðxÞ þ C s4 V s4 ðxÞ,
(20)
where Vci and Vsi, i ¼ 1, 2, 3, 4, are the linearly
independent fundamental solutions of Eqs. (8) and (9),
respectively. They are assumed to satisfy the following
normalization conditions at the origin of the coordinated
system:
3
3
2
2
V c1 V c2 V c3 V c4
V s1 V s2 V s3 V s4
7
7
6 0
6 0
6 V c1 V 0c2 V 0c3 V 0c4 7
6 V s1 V 0s2 V 0s3 V 0s4 7
7
7
6
6
¼ 6 00
7
7
6 00
6 V c1 V 00c2 V 00c3 V 00c4 7
6 V s1 V 00s2 V 00s3 V 00s4 7
5
5
4
4
000
000
000
000
000
000
000
V 000
V
V
V
V
V
V
V
c1
c2
c3
c4 x¼0
s1
s2
s3
s4 x¼0
3
2
1 0 0 0
7
6
60 1 0 07
7
6
ð21Þ
¼6
7,
60 0 1 07
5
4
0
0
0
1
where primes indicate differentiation with respect to the
dimensionless spatial variable x. It should be noted that
because the parameters of Eq. (8) are the same as those of
Eq. (9), their normalized fundamental solutions are same,
i.e., V ci ðxÞ ¼ V si ðxÞ. The exact fundamental solutions can
be easily derived by using the method by Lin [24].
Substituting Eq. (19) into the boundary conditions (10),
(12) and (14), one finds that the coefficients are Cc1 ¼ A0,
ARTICLE IN PRESS
S.M. Lin / Ultramicroscopy 107 (2007) 245–253
248
Cc2 ¼ 0 and C c4 ¼ A0 V 00c1 ð1Þ þ C c3 V 00c3 ð1Þ =V 00c4 ð1Þ. Similarly, substituting Eq. (20) into the boundary conditions
(11), (13) and (15), the coefficients are Cs1 ¼ Cs2 ¼ 0 and
C s4 ¼ C s3 V 00s3 ð1Þ=V 00s4 ð1Þ. Subsituting these back into
Eqs. (19) and (20), the general solutions can be expressed as
A0 V 00c1 ð1Þ
V c4 ðxÞ
W̄ s ðxÞ ¼ A0 V c1 ðxÞ V 00c4 ð1Þ
V 00c3 ð1Þ
þ C c3 V c3 ðxÞ 00
V c4 ðxÞ ,
V c4 ð1Þ
W̄ s ðxÞ ¼ C s3
V 00 ð1Þ
V
V s3 ðxÞ s3
ð
x
Þ
.
s4
V 00s4 ð1Þ
ð22Þ
(23)
1
E~ k ¼
2
Z
1
0
h
i
2
2
m W̄ c ðxÞ þ W̄ s ðxÞ dx
i
1 h 2
2
ð28Þ
þ m̄t W̄ c ð1Þ þ W̄ s ð1Þ .
2
The energy lost per cycle due to the tip–sample viscous
force is
I qwðL; tÞ
E loss ¼
ct
dwðL; tÞ:
(29)
qt
Substituting the solution (7) into Eq. (29), the energy lost
2
per cycle is E loss ¼ apc̄t oW̄ ð1Þ. Substituting Eqs. (27)–(29)
back into Eq. (24), the effective Q factor is
2E~ s þ o2 E~ k
Q¼
.
(30)
2
c̄t oW̄ ð1Þ
Given the tip amplitude W̄ ð1Þ and the dimensionless
frequency of excitation o and substituting Eqs. (22) and (23)
into Eqs. (17) and (18), the coefficients Cc3 and Cs3 and the
amplitude of excitation A0 at the root can be easily determined
by using the numerical method proposed by Lin [24].
4. Effective mass–spring-damper model
3.3. Relation between energy dissipation and Q factor
4.1. Relations among several parameters
In general, a quality factor is used to express the intrinsic
property of a dynamic system. The Q factor is expressed in
terms of the ratio of total energy stored in a system to the
energy dissipation per cycle [25] as follows:
For simplicity, the beam system is usually simulated by
an effective mass–spring-damper model. The equation of
motion can be expressed as [16]
Q ¼ 2p
E total
,
E loss
(24)
where Etotal is the total energy and E loss is the energy lost
per cycle. It is obvious that the phase angle is a function of
position variable x. It means that when the tip is at the top
dead position, i.e., the velocity of the tip is zero, the
velocity at the other position of beam is not zero.
According to this fact, the total energy is considered to
be an average value of a cycle as follows:
Z
1 T
E total ¼
ðE s ðtÞ þ E k ðtÞÞ dt;
(25)
T 0
where Es and Ek are the strain and kinetic energies,
respectively,
Z
1 L
E s ðtÞ ¼
EIðq2 w=qx2 Þ2 dx,
2 0
2
Z
1 L
qw
1
qwðL; tÞ 2
rA
dx þ mt
.
ð26Þ
E k ðtÞ ¼
2 0
qt
2
qt
Substituting the solution (7) into Eqs. (25) and (26), the
total energy is expressed as
1
E total ¼ a E~ s þ o2 E~ k ,
(27)
2
where
Eð0ÞIð0Þ 2
Lc ,
L3
Z
1 1 h 00 2 00 2 i
b W c ðxÞ þ W s ðxÞ dx,
E~ s ¼
4 0
a¼
m€z þ ct z_ þ kzðtÞ ¼ F ts þ gkzðt t0 Þ,
(31)
where z represents the tip displacement at time t, and k, m,
and ct are the spring constant, the effective mass, and the
tip–sample viscous coefficient of cantilever, respectively. Fts
is the tip–sample force and g the gain factor which is the
amplification factor of the displacement signal in the
closed-loop control system. gz(tt0) is the root displacement. 1/g respresents the response ratio of the tip
amplitude to the root one. If the second term was replaced
by 2pf 1 =Qres where Qres is the Q factor at the resonant
frequency, the equation of motion (31) becomes the same
as that given by Hölscher et al. [16]. The relation between
Qres and the resonant response ratio is discussed later.
Moreover, it is well known that the energy dissipation ratio
changes with the frequency of excitation. Therefore, the Q
factor changes with the frequency of excitation [18]. A
detailed investigation is given in Section 4.3. The solution
of Eq. (31) is assumed to be
zðtÞ ¼ A cosð2pftÞ.
(32)
Substituting Eq. (32) into Eq. (31), multiplying it by
cos(2pft) and integrating it from 0 to the period T ¼ 2p/O,
one obtains
Z 2p
1
g cos 2pft0 ¼ s2 þ 1 F ts cos w dw,
(33)
pkA 0
where s ¼ f/f0. Similarly, substituting Eq. (32) into Eq.
(31), multiplying by sin(2pft) and integrating it from 0 to
the period T ¼ 2p/O, one obtains
Z 2p
2pf 0 ct
1
s
g sin 2pft0 ¼ F ts sin w dw.
(34)
pkA 0
k
ARTICLE IN PRESS
S.M. Lin / Ultramicroscopy 107 (2007) 245–253
Taking square of (33) and (34) and summing these, one
obtains
2
Z 2p
1
2
2
g ¼ s þ 1 F ts cos w dw
pkA 0
2
Z 2p
2pf 0 ct
1
s
þ F ts sin w dw .
ð35Þ
pkA 0
k
where sres ¼ fres/f0. It should be noted that Fts is an
arbitrary force. If the van derR Waals force is considered,
2p
2
F ts ¼ AH R=½6ðD0 W
R 2pð1ÞÞ , 0 F ts cos w dw ¼ 2pAAH R=
2 3=2
2
½6ðD0 A Þ and 0 F ts sin w dw ¼ 0. Dividing Eq. (34)
by Eq. (33), the corresponding phase angle f is obtained:
tan f ¼
s2
2pf 0 ct s=k
h
i,
1 þ AH R= 3kðD20 A2 Þ3=2
(38)
where g is the damping ratio pf0ct/k. It is observed from
Eq. (38) that increasing the damping ratio and van der
Waals force increases the resonant frequency shift. In the
other way, given the frequency shift, the tip amplitude, and
the tip–sample distance, the damping ratio can be easily
determined via the formula (38). If g ¼ 0, and
AH R=½3kðD20 A2 Þ3=2 51, the resonant frequency shift
(38) becomes
f 0 AH R
6kðD20 AÞ3=2
.
(39)
It is the same as that derived by the conventional
perturbation method [17]. Substituting Eq. (38) back into
Eq. (35), the resonant response ratio is obtained:
1
1
¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .
AH R
gres 2g 1 g2 3kðD2 A2 Þ3=2
Δf = -5.06 Hz
6
4
Δf = -10.7 Hz
2
Δf = -51.8 Hz
0
0
0.001
0.002
Δγ
0.003
0.004
Fig. 2. Error of measured step height due to the resonant frequency shift
Df and the variation of damping at different surface positions Dg.
(A ¼ 5 nm, b̄ ¼ 45 mm, h̄ ¼ 3:5 mm, L ¼ 200 mm, Lc ¼ 10 nm, b ¼ m ¼ 1,
mt ¼ 0.067 1013 kg, R ¼ 20 nm, E ¼ 70.3 109 Pa, r ¼ 2.5 103 kg/
m3, AH ¼ 1019 J, f1 ¼ 74941.4 Hz).
(37)
where f ¼ 2pft0. The resonant frequency shift of an AFM
subjected to the van der Waals force is derived from
Eq. (36):
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2v
3
#
u"
u
A
R
H
15 ,
Df ¼ f res f 0 ¼ f 0 4t 1 2g2 3kðD20 A2 Þ3=2
Df ¼ 8
error (nm)
In the amplitude modulation, the amplitude of tip A is
constant. At resonance the amplitude of the exciting root
gA is minimum. Letting the gradient of the square of gain g
to be zero, the follwing relation among the resonant
frequency fres, the viscous coefficient, the tip–sample force,
and amplitude is derived:
Z 2p
2p2 c2t f 20
1
s3res þ 1 þ
þ
F
cos
w
dw
sres
ts
pkA 0
k2
Z 2p
ct f
20
F ts sin w dw ¼ 0,
ð36Þ
k A 0
249
(40)
0
The viscous damping ratio g can be calculated via Eq. (40)
by measuring the resonant response ratio 1/gres. If Fts ¼ 0,
the resonant response ratio
(40)ffi becomes the well-known
pffiffiffiffiffiffiffiffiffiffiffiffi
formula, 1=gres;0 ¼ 1=2g 1 g2 [23]. If the damping ratio
is very small, the resonant response ratio is equal to the
resonant quality factor Qres,0. Therefore, the resonant
response ratio usually represents the energy dissipation
ratio per cycle of a system.
Moreover, the resonant
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
frequency becomes f res ¼ f 0 1 2g2 .
4.2. Error of measuring a sample’s topography
The effects of several parameters on the error of
measuring a sample’s topography are investiged here.
Eq. (38) can be rewritten as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
#2=3
u"
u
AH R
t
D0 ¼
(41)
þ A2 .
3k 1 2g2 s2res
Consider AFM scanning a sample’s step height, as shown
in Fig. 1. The step height between two surface points ‘a’
and ‘b’ is to be measured. For a perfect measurement,
whenever any point of a sample’s surface is measured, the
amplitude, the resonant frequency and the damping ratio
must be kept constant. If these conditions are to be kept,
the height of the piezoelectric scanning table mus be
adjusted so that the tip–sample distances are same while
measuring the two surface points ‘a’ and ‘b’, D0,b ¼ D0,a.
The adjusted step height Dhmea is the real height Dhab
between the two points. Unfortunately, there must exist the
tolerances of the tip amplitude, the resonant frequency or
the damping difference over a sample’s surface in the
ARTICLE IN PRESS
S.M. Lin / Ultramicroscopy 107 (2007) 245–253
250
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2
32=3
u
u
u
AH R
5 þ A2a .
t4 3k 1 2g2a s2res;a
proceeding of measurement such that D0,b6¼D0,a and
|Dhmea6¼Dhab|. The corresponding error of the measured
step height is |DhmeaDhab| ¼ |D0,bD0,a|. According to
Eq. (41), one can determine the error of a measured step
height due to the tolerences of amplitude, resonant
frequency, and damping difference. Assuming that
Ab ¼ Aa+DA, sres,b ¼ sres,a+Ds, and gb ¼ ga+Dg where
{DA,Ds,Dg} are the tolerences. Substituting these into
Eq. (41) and the relation error ¼ |D0,bD0,a|, the error of
a measured step height between the two points a and b is
expressed as
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
v
2
32=3
u
u
u
AH R
2
error ¼ t4 2 2 5 þ ðAa þ DAÞ
3k 1 2 ga þ Dg sres;a þ Ds
It is observed from Eq. (42) that increasing the tip radius R
and the damping ratio g increases the error. Increasing the
effective spring constant k decreses the error. Besides, the
influence of the variation of the damping ratio exsiting at
different surface positions on an error of measuring
topography is investigated here. For simplicity, the
variations of the resonant frequency and the amplitude
are neglected. The relation among frequency shift, the
variation of damping ratio and the error of measuring a
sample’s topography is shown in Fig. 2. It shows that the
variation of damping ratio increases the error, especially
120
0.04
: ct = 105.5 nN-s/m
: ct = 10.55 nN-s/m
: discrete model
: distributed model
0.03
: ct = 105.5 nN-s/m
: ct = 10.55 nN-s/m
: discrete model
: distributed model
80
40
θ
A0 (nm)
ð42Þ
0.02
0
-40
0.01
-80
0
73000
(a)
-120
73500
74000
74500
75000
75500
73000
73500
74000
(b)
f (Hz)
74500
75000
75500
f (Hz)
1200
Q-factor
800
: ct = 105.5 nN-s/m
: ct = 10.55 nN-s/m
: discrete model
: distributed model
400
0
73000
(c)
73500
74000
74500
75000
75500
f (Hz)
Fig. 3. With a constant tip amplitude A, the influence of the frequency of excitation O and the viscous coefficient ct on the amplitude of excitation A0, the
phase angle f, and the Q factor of a uniform beam. (The tip–sample distance D0 ¼ 4 nm, the tip amplitude A ¼ 1 nm, the width of cross-section of beam
b̄ ¼ 45 mm, the thickness of cross-section of beam h̄ ¼ 3:5 mm, L ¼ 200 mm, Lc ¼ 10 nm, b ¼ m ¼ 1, mt ¼ 3.18 1013 kg, R ¼ 50 nm, E ¼ 70.3 109 Pa,
r ¼ 2.5 103 kg/m3, AH ¼ 1019 J, the first natural frequency without damping and the ver der Waals force, f1 ¼ 74355.8 Hz).
ARTICLE IN PRESS
S.M. Lin / Ultramicroscopy 107 (2007) 245–253
for a smaller frequency shift. It should be known that
increasing the tip–sample distance decreases the resonant
frequency shift. In other words, decreasing the tip–sample
distance decrease the error due to the variation of damping
ratio.
4.3. Relation between energy dissipation and Q factor
The Q factor of the discrete mass–spring-damper model
is derived here. The corresponding total energy and the lost
energy lost per cycle are, respectively,
Z 1 T 1 2 1
m_z þ kðzðtÞ gzðt t0 ÞÞ2 dt
Ē total ¼
T 0 2
2
2
A
mO2 þ k 1 þ g2 2g cos Ot0 ,
¼
ð43Þ
4
I
DE loss ¼
ct z_ dz ¼ 2p2 fct A2 .
(44)
Substutiting these back into Eq. (24), the Q factor is
expressed as
Q¼
1 þ s2 þ g2 2g cos 2pft0
.
4gs
(45)
Further, substituting Eqs. (38) and (40) into Eq. (45), the
resonant Q factor is
AH R
2
4
2
1 g 2g 1=2 þ 2g 3kðD2 A2 Þ3=2
0
Q̄res ¼
.
(46)
1=2
HR
2g 1 2g2 3kðDA2 A
2 3=2
Þ
0
If the van der Waals parameter AH R=½3kðD20 A2 Þ3=2 and
g are far less than one, Q̄res 1=2g 1=gres which is the
same as that of a conventional system without the van der
Waals force [23]. Substituting it into Eq. (44), one obtains
that E loss;res pkA2 s=Q̄E res . Further, if fresEf0, then
E loss;res pkA2 =Q̄res [26]. For example, a cantilever beam
is made of SiO2. The following parametes are taken: the
width of cross-section of beam b̄ ¼ 45 mm, the thickness of
cross-section of beam h̄ ¼ 45 mm, the length L ¼ 200 mm,
the tip mass mt ¼ 3.18 1013 kg, the radius of tip
R ¼ 50 nm, the Young’s modulus E ¼ 70.3 109 Pa, the
density r ¼ 2.5 103 kg/m3, the Hamaker constant
AH ¼ 1019 Joule, the first natural frequency without
damping and the ver der Waals force, fI ¼ 74933.28 Hz,
the tip amplitude A ¼ 2 nm, the tip–sample distance
D0 ¼ 4 nm, the van der Waals parameter AH R=½3kðD20 A2 Þ3=2 ¼ 0:000946. In general, the Q factor of an AFM is
about between 50 and 10000 and the corresponding
damping ratio g is between 0.00004 and 0.01. Obviously,
the damping ratio g and van der Waals parameter of an
AFM is far less than one. Further, it is observed from
Eq. (46) that the influence of the van der Waals force on the
resonant Q factor is very small.
251
5. Numerical results and discussion
The comparison of the numerical results determined by
using the effective mass–spring-damper and beam models
are made. Fig. 3 shows the influence of the frequency of
excitation O and the viscous coefficient ct on the amplitude
of excitation A0, the phase angle f, and the Q factor of an
uniform beam with constant tip amplitude A. It shows that
the root amplitude at the resonance is minimum. The
resonant frequencies and the phase angles determined by
using the two models are almost same. Rodriguez and
Garcia [20] found that simulations based on a mass–springdamper model is suitable to describe a cantilever tip motion
with relatively high quality factor, Q102–103. Fig. 3 also
shows that the resonant Q factor is maximum. When the
viscous damping coefficient is large enough, the Q factors
of the two models are less different. But if the viscous
damping coefficient is small, there exits a peak of the Q
factor curve of the beam model at the resonance. It is well
known that the resonant Q factor and the resonant
response ratio of a system with small damping ratio are
the same. When the frequency of excitation approaches the
resonant frequency, the response ratio is maximum. In
other words, there is a peak of the curve of the response
ratio and the frequency of excitation at the resonance. If
the Q factor were the response ratio, there would be a peak
of the curve of the Q factor and the frequency of excitation
at the resonance.
Fig. 4 shows the influence of the damping ratio g on the
resonant Q factors and the response ratios of the two
models. In the mass–spring-damper model, if g 1,
100000
10000
1000
100
10
: w(L)/A0
: Qres , Eq. (21)
: Qres , Eq. (33)
: 1/gres , Eq. (30)
: 1/2γ
1
0.1
1E-005
0.0001
0.001
0.01
0.1
1
γ
Fig. 4. Comparison of the Q factor and the response ratio g, (D0 ¼ 5 nm,
A ¼ 2 nm, b̄ ¼ 45 mm, h̄ ¼ 3:5 mm, L ¼ 200 mm, Lc ¼ 10 nm, b ¼ m ¼ 1,
mt ¼ 3.18 1013 kg, R ¼ 50 nm, E ¼ 70.3 109 Pa, r ¼ 2.5 103 kg/m3,
AH ¼ 1019 J, f1 ¼ 74355.8 Hz).
ARTICLE IN PRESS
S.M. Lin / Ultramicroscopy 107 (2007) 245–253
252
120
4
80
: ct = 10.55 nN-s/m
: ct = 105.5 nN-s/m
: ct = 10.55 nN-s/m
: ct = 105.5 nN-s/m
40
θ
A0 (nm)
3
0
2
-40
1
-80
0
-120
0
200
400
f (kHz)
(a)
600
800
0
200
(b)
400
f (kHz)
600
800
10000000
1000000
100000
Q-factor
10000
1000
100
: ct = 10.55 nN-s/m
: ct = 105.5 nN-s/m
10
1
0.1
0
200
(c)
400
600
800
f (kHz)
Fig. 5. Influence of the frequency of excitation O and the viscous damping constant c̄t on the amplitude of excitation A0, the phase angle and the Q factor
of a uniform beam. (D0 ¼ 5 nm, A ¼ 3 nm, b̄ ¼ 45 mm, h̄ ¼ 3:5 mm, L ¼ 200 mm, Lc ¼ 10 nm, b ¼ m ¼ 1, mt ¼ 3.18 1013 kg, R ¼ 50 nm,
E ¼ 70.3 109 Pa, r ¼ 2.5 103 kg/m3, AH ¼ 1019 J, f1 ¼ 74355.8 Hz, f2 ¼ 466030 Hz).
Q̄res 1=2g 1=gres . In the beam model, the resonant
quality factor Qres is different from the response ratio
wðLÞ=A0 by 20%. Moreover, the resonant quality factors of
the two models are different by 21%.
Commonly, using the mass–spring-damper model one
can derive only the first mode. However, the beam model
can be used to determine the responses of the higher
modes. Fig. 5 shows the behavior of the first two modes.
With constant tip amplitude, Fig. 5(a) shows that at
resonance the root amplitudes of the two modes are locally
minimum. Fig. 5(b) shows that at resonance the phase
angle of the two modes change suddenly. Fig. 5(c) shows
that the effects of the frequency of excitation and the
damping coefficient on the Q factor are significant. There
exits a peak of the Q factor curve of the beam model at the
resonance.
6. Conclusions
In this study, analytical solutions of several modes of an
AM-AFM subjected to a tip–sample viscous force are
presented. Moreover, an exact solution of an effective
mass–spring-damper model is also derived. Although only
the attractive force is discussed here, the proposed methods
ARTICLE IN PRESS
S.M. Lin / Ultramicroscopy 107 (2007) 245–253
can be easily used to investigate an AFM system subjected
to a viscous and arbitrary tip–sample forces. Based on an
effective mass–spring-damper model, simple and complete
relations among the resonant response ratio, the damping
ratio and the van der Waals force are revealed. These
relations are intuitive and very useful for interpreting the
surface properties of a sample. In order to realize the
accuracy of an mass–spring-damper model, the numerical
results of an AM-AFM are determined by using the beam
and mass–spring-damper models. It is found that the
resonant frequencies and the phase angles determined by
using the two models are almost same. Moreover,
increasing the damping ratio decreases slightly the resonant
frequency. The effects of the frequency of excitation and
the damping coefficient on the Q factor are significant.
However, with a constant damping coefficient, the
resonant Q factor of the mass–spring-damper model will
be underestimated by about 20%.
Because the relation among the resonant frequency, the
damping ratio and the dimensionless van der Waals force
has been proved to be accurate, several trends are obtained
via the relation as follows:
(1) Increasing the tip radius and the damping ratio
increases significantly the error of measuring a sample’s
topography.
(2) Increasing the effective spring constant decreases the
error of measuring a sample’s topography.
(3) The variation of damping ratio increases the error of
measuring a sample’s topography, especially for smaller
frequency shift.
In addition, it is well known that decreasing the
tip–sample distance increases the resonant frequency shift.
According to these facts, it is further found that decreasing
the tip–sample distance will decrease the error of measuring
a sample’s topography due to the variation of damping.
Acknowledgment
The support of the National Science Council of Taiwan,
ROC, is gratefully acknowledged (Grant number: Nsc952212-E168-007).
References
[1] F.J. Giessibl, Advances in atomic force microscopy, Rev. Mod. Phys.
75 (2003) 949–983.
[2] R. Garcia, R. Perez, Dynamic atomic force microscopy methods,
Surf. Sci. Rep. 47 (2002) 197–301.
[3] J.N. Israelachvili, Intermolecular and Surface Forces, Academic
Press, New York, 1985.
[4] G.Y. Chen, R.J. Warmack, Harmonic response of near-contact
scanning force microscopy, J. Appl. Phys. 78 (3) (1995) 1465–1469.
253
[5] J. Tamayo, R. Garcia, Effects of elastic and inelastic interactions on
phase contrast images in tapping-mode scanning force microscopy,
Appl. Phys. Lett. 71 (16) (1997) 2394–2396.
[6] L. Nony, R. Boisgard, J.P. Aime, Nonlinear dynamic properties of an
oscillating tip-cantilever system in the tapping mode, J. Chem. Phys.
111 (4) (1999) 1615–1627.
[7] D.P. Behrend, F. Oulevey, D. Gourdon, E. Dupas, A.J. Kulik, G.
Gremaud, N.A. Burnham, Harmonic response of near-contact
scanning force microscopy, Appl. Phys. A 66 (1998) s219–s221.
[8] H. Bielefeldt, F.J. Giessibl, A simplified but intuitive analytical model
for intermittent-contact-mode force microscopy based on Hertzian
mechanics, Surf. Sci. 440 (1999) L863–L867.
[9] L. Wang, The role of damping inphase imaging in tapping mode
atomic force microscopy, Surf. Sci. 429 (1999) 178–185.
[10] L. Delineau, R. Brandsch, G. Bar, M.H. Whagbo, Harmonic
responses of a cantilever interacting with elastomers in tapping mode
atomic force microscopy, Surf. Sci. 448 (2000) L179–L187.
[11] M.V. Salapaka, D.J. Chen, J.P. Cleveland, Linearity of amplitude
and phase in tapping-mode atomic force microscopy, Phys. Rev. B 61
(2) (2000) 1106–1115.
[12] P.J. James, M. Antognozzi, J. Tamayo, T.J. McMaster, J.M.
Newton, M.J. Miles, Inyterpretation of contrast in tapping mode
AFM and shear force microscopy: a study of nafion, Langmuir 17
(2001) 349–360.
[13] M. Antognozzi, D. Binger, A.D.L. Humphris, P.J. James, M.J. Miles,
Modeling of cylindrically tapered cantilevers for transverse dynamc
force microscopy (TDFM), Ultramicroscopy 86 (2001) 223–232.
[14] S.H. Ke, T. Uda, K. Terakura, Frequency shift and energy
dissipation in non-contact atomic-force microscopy, Appl. Surf. Sci.
157 (2000) 361–366.
[15] B. Gotsmann, C. Seidel, B. Anczykowski, H. Fuchs, Conservative
and dissipative tip–sample interaction forces probed with dynamic
AFM, Phys. Rev. B 50 (15) (1999) 11051–11061.
[16] H. Hölscher, B. Gotmann, W. Allers, U.D. Schwarz, H. Fuchs, R.
Wiesendanger, Measurment of conservative and dissipative tip–sample interaction forces with a dynamic force microscope using the
frequency modulation technique, Phys. Rev. B 64 (2001) 075402.
[17] S.M. Lin, Exact solution of the frequency shift in dynamic force
microscopy, Appl. Surf. Sci. 250 (2005) 228–237.
[18] S.M. Lin, Energy dissipation and frequency shift of a damped
dynamic force microscopy, Ultramicroscopy 106 (2006) 516–524.
[19] R.W. Stark, W.M. Heckl, Higher harmonics imaging in tappingmode atomic-force microscopy, Rev. Sci. Instrum. (2003) 5111–5114.
[20] T.R. Rodriguez, R. Garcia, Tip motion in amplitude modulation
(tapping-mode) atomic-force microscopy: comparison between continuous and point-mass models, Appl. Phys. Lett. 80 (2002)
1646–1648.
[21] T.R. Rodriguez, R. Garcia, Compositional mapping of surfaces in
atomic force microscopy by excitation of the second normal mode of
the microcantilever, Issue Series Title: Appl. Phys. Lett. 84 (2004)
449–451.
[22] S. Crittenden, A. Raman, R. Reifenberger, Probing attractive forces
at the nanoscale using higher-harmonic dynamic force microscopy,
Phys. Rev. B 72 (2005) 235422.
[23] L. Meirovitch, Analytical Methods in Vibrations, Collier-Macmillan
Inc., London, 1967.
[24] S.M. Lin, Dynamic analysis of rotating nonuniform Timoshenko
beams with an elastically restrained root, ASME J. Appl. Mech. 66
(3) (1999) 742–749.
[25] A.D. Nashif, D.I.G. Jones, J.P. Henderson, Vibration Damping,
Wiley, New York, 1985.
[26] S. Morita, R. Wiesendanger, E. Meyer, Springer, Berlin, 2002,
pp. 395–431.