PHYSICAL REVIEW B, VOLUME 64, 085409

Dynamics of the cantilever in noncontact dynamic force microscopy:
The steady-state approximation and beyond
M. Gauthier,1 N. Sasaki,2,3 and M. Tsukada1
1

Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan2
Department of Materials Engineering, Graduate School of Engineering, University of Tokyo,
Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan
3
Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Corporation (JST),
2-1-6 Sengen, Tsukuba, Ibaraki 305-0047, Japan
共Received 9 March 2001; published 3 August 2001兲
A detailed analysis of models that have been used for describing the dynamics of the cantilever in largeamplitude noncontact dynamic force microscopy is presented. For each model, attention is especially paid to
the nonlinear nature of the problem and its implication for experiments. In particular, in the steady-state
approximation where the equations of motion for an externally and self-driven oscillator are rigorously identical, the theory predicts the existence of unstable solutions whether the interaction force is dissipative or not.
The need to go beyond the steady-state approximation by including the finite-time response of the system to achange of tip-surface separation or interaction during the scanning is also discussed.
DOI: 10.1103/PhysRevB.64.085409

PACS number共s兲: 68.37.Ef, 07.79.Lh, 87.64.Dz

I. INTRODUCTION

Noncontact dynamic force microscopy 共DFM兲 共Refs.
1– 4兲 is a powerful tool for investigating various surfaces at
an atomic scale. In experiments, the shift of the resonant
frequency of the cantilever, which serves as a frequency determining element, is used for regulating the tip-surface
separation. It is also possible to obtain additional atomicscale information by simultaneously measuring the dampingof the lever oscillation due to the surface5–10 关these experiments have been performed with phase lock loop 共PLL兲 techniques兴. The damping can straightforwardly be interpreted as
dissipation if the oscillator behaves linearly. However, ingeneral, the cantilever is operated with large amplitude compared to the range of the tip-surface nonlinear potential, and
it is yet not clear to what extent the problem remains linear.
In interpreting surface images, it is therefore important tounderstand the dynamics of the cantilever before asserting
what the apparatus is actually measuring. Most of the analytical works on the dynamics of the lever have focused on
describing the steady-state solutions of the cantilever equation of motion.11–15 In this approximation, quantities such as
the amplitude, driving frequency, or excitation amplitude aretreated as parameters. Unfortunately, the stability of the solutions, which is understood,12,14 has only been briefly described and the implication of unstable solutions for real experiments has not been emphasized enough. Moreover,stability has not been discussed when including dissipation
to the surface. Consequently, the validity of the steady-state
approximation has never been seriously challenged. Not included in this approximation is the effect of the time dependence of the quantities mentioned above in actual experiments and the fact that time-averaged observables, such as
the peak-to-peak oscillation amplitude,16 are involved in theelectronics.
In this paper, we analyze in detail the models that havebeen used for describing the dynamics of the lever. Attention
0163-1829/2001/64共8兲/085409共9兲/$20.00

is paid to the nonlinear features of the solutions. For instance, in the steady-state approximation where the equationsof motion for an externally driven and self-driven oscillator
are rigorously identical, the existence of unstable solutions is
an unavoidable consequence of the nonlinearity of the interaction. This is also true when including the effects of dissipation to the surface on the lever of the dynamics. The failure to systematically observe mechanical instabilities and toreproduce the order of magnitude of the atomic scale corrugation of the damping signal 共assuming it results from dissipation兲 based on first-principles theory and the fact thatquantities such as the driving frequency and excitation amplitude are dynamically determined indicate the need to go
beyond the steady-state approximation by including thefinite-time response of the system to a change of separation
or interaction.
II. STEADY-STATE APPROXIMATION
A. Weakly nonlinear integrable Hamiltonian

The theoretical description of DFM operated with large
amplitude was begun with the work of Giessibl.17 The
Hamiltonian isH⫽mẋ 2 /2⫹kx 2 /2⫹U i 共 x⫹L;X;Y 兲 ,共1兲

where L is the mean tip-surface separation, k the cantilevereffective spring constant, and U i the nonlinear potential. X
and Y are the lateral coordinates of the tip. The equation of
motion can be written as a set of first-order differential equations referred to as a flow:

冉冊 冉
ẏ

ẋ

⫽

⫺ ␻ 20 x⫹F i 共 x⫹L 兲 /m
y

冊

,

共2兲

with ␻ 20 ⫽(2 ␲ f 0 ) 2 ⫽k/m. The flow is two dimensional andautonomous 共no explicit time dependence兲. Because the nonlinear potential 兩 U i 兩 is much smaller than kx 2 /2, the system
is only weakly nonlinear. One can therefore use first-order

64 085409-1©2001 The American Physical Society

M. GAUTHIER, N. SASAKI, AND M. TSUKADAPHYSICAL REVIEW B 64 085409

perturbation theory17 or Fourier analysis for obtaining an expression for the cantilever resonant frequency shift,f0⌬ f NIH⫽⫺ r 共 A,L 兲 ,2

共3兲

where r(A,L) is some unitless average of the interaction
force over one oscillation period,r 共 A,L 兲 ⬅

1
kA ␲

冕

2␲

0

F i 共 Acos␪ ⫹L 兲 cos␪ d ␪ .

共4兲

The dependence of the frequency shift, Eq. 共3兲, on the oscillation amplitude A and its order of magnitude agree wellwith various experiments 共see, e.g., Ref. 8兲. Note that within
this model, the atomic-scale damping is not related to the
dynamics of the cantilever. Following the work of Giessibl,
Eq. 共1兲 has also been used as a starting point by a number ofauthors.18 –24
B. Inclusion of an intrinsic dissipative and forcing term

The treatment can easily be extended by including intrinsic dissipation and forcing. The equation of motion can have
many variations depending on the specific model used for thedriving term. Here two models are discussed: the so-called
externally driven oscillatorẍ⫹␻0ẋ⫹ ␻ 20 x⫽F i 共 x⫹L 兲 /m⫹A d␻ 20 cos共 ␻ t 兲
Q0

共5兲

and the self-driven oscillator
mẍ⫹

␻ 0m
ẋ⫹kx⫽F i 共 x⫹L 兲 ⫹Gkx 共 t⫺t 0 兲 .
Q0共6兲

A d , G, t 0 , and Q 0 are the excitation amplitude, gain factor,
time delay, and intrinsic quality factor, respectively. Equation
共5兲 describes PLL electronics. The driving term is exactly
sinusoidal because it is the voltage-controlled oscillator
共VCO兲 output signal.25 Equation 共6兲 describes an experimental setup with an automatic gain control 共AGC兲 where thetime-delayed cantilever signal is used to excite thecantilever.1,16 Here A d and ␻ 关Eq. 共5兲兴, or G 共Ref. 16兲 and t 0
关Eq. 共6兲兴 can be complicated functions of time for describing
real electronics. Within the steady-state approximation, however, they are treated as parameters because one seeks
steady-state solutions of the form x(t)⫽Acos(␻t⫺␾) for Eq.
共5兲 or x(t)⫽Acos(␻t) for Eq. 共6兲. The main difference withthe previous case is that these equations are no longer autonomous 关in Eq. 共6兲, x(t⫺t 0 ) cannot be expressed in termsof x(t)兴. Equivalently, Eq. 共5兲 can be written as a threedimensional autonomous flow,

冉冊冉
ẏ

ẋ

␰˙

⫽

␻0
⫺
y⫺ ␻ 20 x⫹F i 共 x⫹L 兲 /m⫹A d␻ 20 cos共 ␻␰ 兲
Q0
y
1

冊

,

共7兲

and similarly for Eq. 共6兲. Here 兵 L,A, ␾ ,A d , ␻ ,X,Y 其 and
兵 L,A,t 0 ,G, ␻ ,X,Y 其 span a seven-dimensional parameter
space for an externally and self-driven oscillator, respectively. It is easy to demonstrate that the equation of motion
for an externally driven and self-driven oscillator are rigorously identical within this approximation. One simply needs
to insert the ansatz for x(t) into Eq. 共6兲, make the change ofvariable t⫺t 0 →t, and the assignments ␾ ⫽2 ␲ ⫺ ␻ t 0 共assuming ␲ ⭐ ␻ t 0 ⭐2 ␲ ) and G⫽A d /A. Physically, these are
equivalent because, for a steady-state, the causality relationship has the property that there is no distinction between thelever responding to a periodic external excitation or the leverexciting itself with a time delay. The equivalence of thesemodels in the steady state approximation has motivatedmany authors11–15,26 to use the equation of motion of an externally driven oscillator for successfully studying DFM.
Note the striking resemblance between Eq. 共5兲 and the wellstudied Duffing equation27 with a small anharmonic perturbation.
The effective potential energy U i (x⫹L)⫹kx 2 /2 is obviously asymmetric around x⫽0. Although this may naively
be thought to imply an anharmonic cantilever motion, this is
not the case since, as pointed out above, the magnitude of F i
is much smaller than 兩 kx 兩 .24 In the tapping mode, the cantilever motion is still sinusoidal for soft enough surfaces eventhough the magnitude of the interaction is usually significantly larger.28 –30 An explicit evaluation of the importance ofanharmonicity in DFM based on Fourier analysis is presented in the Appendix. While Fourier analysis can be used
for obtaining the solutions of the equation of motion, it does
not provide information on their stability. One must haverecourse to the theory of nonlinear differential equations27
for this purpose. The fixed points, corresponding to thesteady-state solutions, can be determined by using the time
averaging and perturbation approach or Poincaré map. The
idea is to reduce the flow, Eq. 共7兲, to an averaged twodimensional autonomous one 关the sign of ␾ is determined by
the form of the ansatz x⫽Acos(␻t⫺␾)兴:

冉 冊 冉

冊

⫺ ␻␻ 0 A/Q 0 ⫹A d␻ 20 sin ␾
1⫽.
2 ␻ 共 ␻ 2 ⫺ ␻ 20 兲 A⫹A d␻ 20 cos ␾ ⫹A ␻ 20 r 共 A,L 兲
˙
A␾
共8兲
Ȧ

˙ to zero.
The fixed points are obtained by fixing Ȧ and A ␾This leads to the following equations of the steady-state approximation:

␻ 20 ⫺ ␻ 2
Adcos ␾ ⫽
⫺r 共 A,L 兲
A
␻ 20

共9兲

1
Ad
sin␾ ⫽
.
A
Q0

共10兲

and

Equations 共9兲 and 共10兲 共Refs. 12 and 14兲 define a hypersurface H in the seven-dimensional parameter space. In thesteady-state approximation, this hypersurface describes all
possible solutions of the equation of motion Eq. 共5兲 or Eq.

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PHYSICAL REVIEW B 64 085409

共6兲. The local stability of a given fixed point is determined bylinearizing the flow in its vicinity and finding the eigenvalues. The secular equation is (␭ represents the eigenvalues兲
␭ 2⫹

冉

冊

␻␻ 0 A d␻ 20 sin ␾⫹␭⫹ 兵 ␻ 2o 关 1⫺g 共 A,L 兲兴 ⫺ ␻ 2 其
Q0
A⫻␻␻ 0 A d 2
Ad 2
␻ 0 cos ␾ ⫹␻ sin ␾A
Q0 A 0⫽0,

共11兲

where g(A,L), a quantity similar to r(A,L), is defined as
g 共 A,L 兲 ⬅

1
k␲

冕

2 ␲ dF 共 A cos ␪ ⫹L 兲
i

dx

0

cos2 ␪ d ␪ .

共12兲
FIG. 1. Tip-surface interaction force used for the calculations. A
Morse potential is assumed with parameters described in the text.

A steady-state solution is unstable if the condition

冉

S⬅ 1⫺

␻2

⫺g 共 A,L 兲
2

␻0

冊

Ad
1
cos ␾ ⫹ 2 ⬍0
A
Q0

共13兲

is satisfied 共i.e., ␭⬎0), or else it is stable. In the following, S
is referred to as the stability function. Equations 共9兲, 共10兲,and 共13兲 provide all the information about the dynamics ofthe lever in the steady-state approximation for a conservative
force.
A solution of particular interest is the one having a phase␲ /2. According to Eqs. 共13兲 and 共9兲, it is stable and corresponds to a frequency shift equal to ⌬ f NIH . 12,14,15 Since thisstate has the maximum amplitude for a given A d , it is naturally referred to as the resonant state. Therefore the introduction of intrinsic dissipation and driving does not alter theconclusion that ⌬ f ⫽⌬ f NIH and the damping is unrelated tothe dynamics of the cantilever.
It should be clearly understood that the hypersurface H
defined by Eqs. 共9兲 and 共10兲 is an object whose properties are
independent of the way it is sampled. In order to emphasize
this point and demonstrate the relevance of unstable solutions for real experiments, the hypersurface H is projected
onto four constant parameter hypersurfaces: 兵 L,A d ,X,Y 其 ,
兵 ␾ ,A d ,X,Y 其 , 兵 A, ␾ ,X,Y 其 , and 兵 A,L,X,Y 其 . For the sake ofillustration, the following constants are defined: L *⫽103.5 Å , A * ⫽100 Å , A d* ⫽A * /Q 0 and U i (X * ;Y * ) is
assumed to be a Morse potential,
U i 共 x;X * ;Y * 兲 ⫽U 0

再冋冉1⫺exp ⫺2b

x⫺R c
Rc

冊 册冎

,

共14兲

with U 0 ⫽2.273 eV, b⫽1.497, and R c ⫽2.357 Å .31 The
force F i is plotted in Fig. 1. The cantilever is characterizedby k⫽40 N/m, Q 0 ⫽104 , and f 0 ⫽150 kHz.
1. Constant parameter hypersurface ˆL * ,A d* ,X * ,Y * ‰The intersection of the hypersurface H with this constantparameter hypersurface is shown in Figs. 2共a兲 and 2共b兲. It
corresponds to a phase variation experiment with constantexcitation. The shape of the curves are qualitatively very
similar to that of the Duffing’s problem.27 The curves are
extremely distorted 共multivalued兲 around the solution(A * , ␲ /2). The emergence of such distorted structures is anunavoidable consequence of large-amplitude operation evenwhen the system is only weakly nonlinear 共as is the casehere兲. All solutions are stable except those corresponding to
the lower part of the needle edge which are indicated by thedotted line. Solutions are unstable when the stability function
S defined in Eq. 共13兲 is smaller than zero, as shown in Fig.
2共c兲.
2. Constant parameter hypersurface ˆ ␾ * ,A d* ,X * ,Y * ‰When the phase ␾ * is fixed to ␲ /2, all solutions have anamplitude A * and are stable because S⫽1/Q 20 ⬎0. However,unstable solutions exist for phases ␾ * ⫽90°. Even for asmall diminution of the phase—for example, when ␾ *⫽89.5°—unstable solutions emerge in the near-noncontact
region as shown in Figs. 3共a兲 and 3共b兲. In Figs. 3共c兲 and 3共d兲,
the phase is slightly larger than 90°—that is,␾ * ⫽90.5°—and the situation is reversed; some solutions in
the contact region are unstable while all solutions in the noncontact region are stable.
Since DFM experiments are usually conducted with constant amplitude, the following constant parameter hypersurfaces are more directly of interest.
3. Constant parameter hypersurface ˆA * , ␾ * ,X * ,Y * ‰This projection corresponds to the normal operation for
DFM. When ␾ * ⫽90°, all solutions are stable independentlyof the separation L, as shown in Figs. 4共a兲 and 4共b兲. The
situation is drastically different for phases ␾ * ⫽90°. For example, Figs. 4共c兲 and 4共d兲 show that for a phase ␾ * ⫽80°,no stable solution is possible for some separation in the nearnoncontact region. It is not difficult to see that if these unstable solutions were to be projected onto the constant parameter hypersurface similar to Fig. 2, they would be on thelower part of the needle edge. For a phase ␾ * ⫽100° 关Fig.
4共e兲 and 4共f兲兴, some solutions in the contact region become
unstable and all others are stable.
4. Constant parameter hypersurface ˆA * ,L * ,X * ,Y * ‰This projection, shown in Fig. 5, corresponds to a phasevariation experiment at constant amplitude. The nonlinear

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M. GAUTHIER, N. SASAKI, AND M. TSUKADAPHYSICAL REVIEW B 64 085409

As mentioned above, the expression for the frequency
shift is identical to that found prior to the introduction of
intrinsic dissipation and forcing. Even if the solution with␾ ⫽ ␲ /2 is stable, it has been shown that many other solutions are not. Since the results presented in Figs. 4 and 5 aredirectly comparable with DFM experiments, the existence ofinstability can and should be verified experimentally. It isclear that the steady-state approximation is not satisfactory ifthese instabilities cannot be observed. As will be explainedin the following section, instability should exist even whenthe effect of dissipation to the surface on the dynamics of thecantilever is included.
C. Inclusion of dissipation to the surface

So far, we have ignored the possibility that energy dissipates to the surface. For a dissipative interaction force
F i (x;ẋ), Eqs. 共9兲, 共10兲, and 共13兲 generalize to

and

冉

S⬅ 1⫺

␻2
␻ 20

␻ 20 ⫺ ␻ 2
Adcos ␾ ⫽⫺r ⫹ 共 A,L, ␻ 兲 ,
A
␻ 20

共15兲

1
Ad
⫹r ⫺ 共 A,L, ␻ 兲 ,sin ␾ ⫽
A
Q0

共16兲

⫺g ⫹ 共 A,L, ␻ 兲

冉

冊

冉

Ad
1
cos ␾ ⫹⫹g ⫺ 共 A,L, ␻ 兲
A
Q0

冊

共17兲

冊

1
⫻
⫹r ⫺ 共 A,L, ␻ 兲 ⬍0,
Q0

respectively. The average functions r and g have been generalized tor ⫾ 共 A,L, ␻ 兲 ⬅

1
kA ␲
⫻

nature is even more apparent; for phases smaller than ␲ /2,
there is no stable solution. Phase variation experiments atconstant amplitude have been presented using PLL techniques and no instability was reported.32

g ⫾ 共 A,L, ␻ 兲 ⬅

2␲

0

F i 共 A cos ␪ ⫹L;⫺A ␻ sin ␪ 兲

再 冎
cos ␪
sin ␪

and
FIG. 2. Steady-state approximation: intersection of the hypersurface defined by Eqs. 共9兲 and 共10兲 with the constant parameter hypersurface 兵 L * ,A d* ,X * ,Y * 其 . This corresponds to a phase variation
experiment at constant excitation amplitude. 共a兲, 共b兲, and 共c兲 show
the amplitude, phase, and stability function, respectively. 共a兲, 共b兲
Extremely distorted structures around the solution (A *⫽100 Å , ␲ /2) can be seen and unstable solutions exist because 共c兲
S is not always positive. The general features are similar to thesolutions of Duffing’s equation with a small anharmonic
perturbation.

冕

1
k␲
⫻

d␪

冕 冉 ⳵⳵
2␲

0

再 冎
cos ␪
sin ␪

Fi
⳵Fi
␻ sin ␪cos ␪ ⫺
x
⳵ ẋ
d␪.

共18兲

冊
共19兲

Equations 共15兲 and 共16兲 are equivalent to Eqs. 共12兲 and共13兲 in Ref. 33. We immediately examine the nature of thefrequency dependence of the function r ⫹ . A dissipative term
which is a separable function of x and ẋ does not affect r ⫹ .
This is the case for the atomic-scale stochastic dissipation34关 ⬃ ␥ (x)ẋ兴 and dissipative currents for small-amplitudeoperation.35 This is also the case for a piecewise continuousfunction of x in phase with ẋ like phenomenological meta-

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PHYSICAL REVIEW B 64 085409

FIG. 3. Steady-state approximation: intersection of the hypersurface defined by Eqs. 共9兲 and 共10兲 with the constant parameter hypersurface 兵 ␾ * ,A d* ,X * ,Y * 其 . 共a兲 and 共c兲 关共b兲 and 共d兲兴 show the amplitude 关stability function兴 at 89.5° and 90.5°, respectively. For a phase 90°, allsolutions have an amplitude A * ⫽100 Å and are stable since S⫽1/Q 20 ⬎0 independently of the separation. 共a兲, 共b兲 However, even for asmall diminution of the phase—for example, ␾ * ⫽89.5°—some solutions in the near-noncontact region become unstable since S⬍0. 共c兲, 共d兲
Similarly, for a slight augmentation of the phase ␾ * ⫽90.5°, there are unstable solutions in the contact region.

stable stick-slip processes.18,24,33 However, Loppacher et al.8
have used a different model for the dissipative current; theyused a retarded dissipative force of the form F⫽F̃ cos (␻t⫹⌬␸) where F̃ and ⌬ ␸ are functions of ␻ . This leads to aterm F̃ cos ⌬␸/kA that contributes to r ⫹ . Moreover, any adhesion hysteresis 共i.e., stick-slip processes兲 which are notexactly in phase with the velocity but have some phase⌬ ␤ ( ␻ ) could also account for the frequency dependence ofr ⫹ . It is illuminating to rewrite Eqs. 共15兲 and 共16兲 astan ␾ ⫽ 关 1/Q 0 ⫹r ⫺ 共 A,L, ␻ 兲兴冒冉

␻ 20 ⫺ ␻ 2
␻ 20⫺r ⫹ 共 A,L, ␻ 兲

冊

共20兲

FIG. 4. Steady-state approximation: intersection of the hypersurface defined by Eqs. 共9兲 and 共10兲 with the constant parameter hypersurface 兵 A * , ␾ * ,X * ,Y * 其 . This corresponds to the normal DFM operation. 共a兲, 共c兲, and 共e兲 关共b兲, 共d兲, and 共f兲兴 show frequency shift profile关stability function S兴 at 90°, 80°, and 100°, respectively. 共a兲 For a phase 90°, all solutions are stable because 共b兲 S is always larger than zero.
The situation is drastically different when the phase differs from 90°. 共c兲, 共d兲 When ␾ * ⫽80°, some solutions become unstable in thenear-noncontact region; 共e兲, 共f兲 for ␾ * ⫽100°, some of the solutions in the contact region are unstable.
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M. GAUTHIER, N. SASAKI, AND M. TSUKADAPHYSICAL REVIEW B 64 085409

According to Eq. 共20兲, the dependence of r ⫹ on the frequency implies that dissipation affects the frequency of thesolution corresponding to ␲ /2:⌬ f ⫽⫺

f0 ⫹r 共 A,L, f 0 ⫹⌬ f 兲 ⫽⌬ f NIH .
2

共22兲

On the other hand, Eqs. 共20兲 and 共21兲 imply that this solution
would not necessarily have the maximum amplitude becauseof the frequency dependence of r ⫺ . For constant amplitude,
the minimum excitation amplitude A d occurs at the frequency ␻ M given by冉

2␻ 20 ⫺ ␻ M

␻ 20
⫽

冉

⫺r ⫹ 共 A,L, ␻ M兲

冊冉

2␻M

⫹
2

␻0

冊 冏

冏冊

⳵r⫹
⳵␻

␻M

1
⳵r⫺
⫹r ⫺ 共 A,L, ␻ M兲
.
Q0
⳵␻ ␻

共23兲

M

The minimum of the excitation curve coincides with thephase ␲ /2 only if ⳵ r ⫺ / ⳵␻ ⫽0; otherwise, there is a phaseshift ␾ M :tan ␾ M⫽

FIG. 5. Steady-state approximation: intersection of the hypersurface defined by Eqs. 共9兲 and 共10兲 with the constant parameter hypersurface 兵 A * ,L * ,X * ,Y * 其 . This corresponds to a phase variation
experiment at constant amplitude. 共a兲, 共b兲, and 共c兲 show the excitation curve, phase, and stability function S, respectively. For ␾⬍ ␲ /2, all solutions are unstable since S⬍0.

and

Ad
⫽
A

冑冉

␻ 20 ⫺ ␻ 2
␻ 20⫺r ⫹ 共 A,L, ␻ 兲冊2⫹ 关 1/Q 0 ⫹r ⫺ 共 A,L, ␻ 兲兴 2 .
共21兲

冉

2␻M

⫹
2

␻0

冏 冊冒 冏

⳵r⫹
⳵␻ ␻

M

⳵r⫺
.
⳵␻ ␻

共24兲

M

It is emphasized that these frequency dependences are linked
to slow dissipation processes but not to the atomic-scale stochastic dissipation34 because the time scale of the motion of
surface atoms is much smaller than that of the cantilever. It is
worth noting that if the frequency dependence of the slow
dissipation processes is roughly linear around ␻ M , then Eq.
共21兲 can be rewritten as A d( ␻ M⫹ ␦ ␻ )/A⬇ 冑K 1 ( ␦ ␻ ) 2 ⫹K 2 ,where ␦ ␻ represents any departure from ␻ M such that ␦ ␻Ⰶ ␻ M , and K 1 , K 2 are constant. In other words, as long as
the slow dissipation mechanisms are not very sensitive functions of the frequency, then the excitation curve should beapproximately symmetrical around ␻ M . In any case, thesefrequency dependences need to be more cautiously studied
and are beyond the scope of this work.
Apart from these possible small frequency dependences,the physical origin of r ⫹ and r ⫺ is very different 共this is also
true for g ⫹ and g ⫺ ). Here r ⫹ is essentially related to theaverage force over an oscillation period; r ⫺ depends on thedissipation mechanism involved. It is thus unrealistic to assume that the average force and dissipation mechanism could
ensure that the projection similar to that shown in Fig. 2 isalways singled valued 关i.e., the stability function S in Eq.
共17兲 always larger than zero兴 for any separation L. In otherwords, unstable solutions should exist even when some dissipation to the surface takes place.

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PHYSICAL REVIEW B 64 085409

III. FULLY NONAUTONOMOUS TREATMENT

In real DFM experiments, A, ␾ , L, ␻ , X, Y, and A d are not parameters but 共time-dependent兲 variables. The fully nonautonomous three-dimensional flow is

冉冊冉
ẏ

ẋ

⫽

⫺

␻0
y⫺ ␻ 20 x⫹F i „x⫹L 共 ␰ 兲 ;X 共 ␰ 兲 ;Y 共 ␰ 兲 …/m⫹A d共 ␰ 兲 ␻ 20 cos关 ␻ 共 ␰ 兲 ␰ 兴
Q0
y

␰˙

1

a similar generalization holds for Eq. 共6兲. It is urgent toclarify whether or not this extra nonautonomicity can alter
the dynamics of the lever in DFM. The inclusion of the response of the system to a change of L, X, or Y could add
extra structure to the global stable manifold of the periodic
cycles. A step in this direction is found in Ref. 36 where it
was argued that nonautonomous nonlinear effects (L was avariable兲 could contribute to the atomic-scale corrugation of
the damping signal for PLL electronics 共see, for example,
Ref. 6兲 even though the cantilever motion is sinusoidal to ahigh degree of accuracy. Note that the simulations presented
in Ref. 16 which aim at understanding a different type ofelectronics setup, do not show any departure from thesteady-state approximation 共the phase was always ␲ /2兲.
There are a number of reasons to believe that the fully
nonautonomous equation of motion needs to be considered.
First, instabilities predicted in the steady-state approximation
have not been observed experimentally. Second, the hypersurface H is extremely distorted around solutions with ␲ /2,
which can become unstable even for a small perturbation. On
the other hand, it is well known that three-dimensional flows
can lead to a practically infinite set of new phenomena 共e.g.,chaotic behavior兲.27 This may not the case in the steady-state
approximation because 兵 L,A, ␾ ,A d , ␻ ,X,Y 其 are treated asparameters, which means that the system has in fact been
assumed to respond instantaneously to any change in theseparation or interaction. In real experiments, however, theresponse of the system is not instantaneous. For example,
PLL electronics was reported to take ⬃1 ms to adjust its
phase to an input signal phase step.25 Also the electronics
depends on time-averaged observables. For instance, the excitation or gain factor depends on the peak-to-peak oscillation amplitude which is a time-averaged quantity. Moreover,the sensitivity to a sudden change may be reduced because
the feedback circuit depends on the history of such timeaveraged observables 关see, for example, Eq. 共2兲 in Ref. 16兴.
Consider, for example, the case where the tip is initiallylocated directly above one surface atom and a small change
in X occurs 共during the scanning兲. The situation is equivalent
to a sudden small change in the interaction force 共diminution兲. If the lever is oscillating at one of the fixed points with␾ ⫽ ␲ /2 in the hypersurface H and if the response of theelectronics is not instantaneous, then there is a chance thatfor some time the lever would be oscillating at a frequency
below that of the new resonance, which may lead to substantially different dynamics.

冊

;

共25兲

Thus, these considerations in addition to the discussion
presented in Ref. 36 for PLL electronics and the failure toobserve instability experimentally may indicate that a proper
treatment of the fully nonautonomous flow is essential forunderstanding the dynamics of the lever operated with large
amplitude. We emphasize that the equations for an externallydriven and self-driven oscillator are potentially different only
within a fully nonautonomous treatment. In fact, there is no
reason to believe that nonautonomous nonlinear effects, ifany, should be the same since the equivalence between thetwo models only holds within the steady-state approximation. This may explain why different electronics setup canlead to different results.
IV. SUMMARY

In summary, in the steady-state approximation where theequations of motion for an externally and self-driven oscillator are rigorously identical, Eqs. 共9兲, 共10兲, and 共13兲 关Eqs.
共15兲, 共16兲, and 共17兲兴 provide all information about the solutions for a conservative 关dissipative兴 force. It is explained
why instability is an unavoidable consequence of the nonlinear nature of the interaction for large-amplitude operationeven when the interaction force is dissipative and how this
could be verified experimentally. Because the solutions with␾ ⫽ ␲ /2 are located in a very highly nonlinear region of thehypersurface and because in this approximation the responseof the system is in fact assumed to be instantaneous, it isimportant to consider the dynamical problem in a fully nonautonomous way and to include the finite-time response of
the system. We are currently working on extending our previous works in this direction. In parallel, effort should also
be put into understanding atomic-scale dissipation mechanisms in DFM based on first-principles theory. An importantand original proposal was recently made by Kantorovich fortreating the stick-slip process as a nonequilibrium process.37
In doing so he may be able to treat atomic-scale
stochastic34,38 and metastable stick-slip processes33 within
the same theoretical framework.
ACKNOWLEDGMENTS

This work was supported in part by a Grant-in-Aid on
Priority Areas from the Ministry of Education, Science andCulture, Japan. M.G. acknowledges support from the Ministry of Education of Japan.

085409-7

M. GAUTHIER, N. SASAKI, AND M. TSUKADAPHYSICAL REVIEW B 64 085409

APPENDIX

The cantilever motion can be rigorously described by introducing the quantity ␦ x representing the cantilever motion
departure from purely sinusoidal,x⫽x 0 ⫹ ␦ x⫽A cos共 ␻ t⫺ ␾ 兲 ⫹ ␦ x,

共A1兲

⬁
⬁
⫺
␦ x⬅ ␦ 0 /2⫹ 兺 n⫽2
␦⫹n cos n(␻t⫺␾)⫹兺n⫽2␦n sin n(␻twhere⫺␾). By Fourier expanding the interaction force F i , the following can be obtained:tan ␾ ⫽␻ ⬘ /Q 0 ⫹ ␦ r 2
共 1⫺ ␻ ⬘ 兲 ⫺ 共 r 1 ⫹ ␦ r 1 兲
2

,

共A2兲

A ⬘ 2d ⫽ 关共 1⫺ ␻ ⬘ 2 兲 ⫺ 共 r 1 ⫹ ␦ r 1 兲兴 2 ⫹ 关共 ␻ ⬘ /Q 0 兲 ⫹ 共 ␦ r 2 兲兴 2 ,共A3兲

FIG. 6. Anharmonic effect. The quantity Ã provides informationon the importance of anharmonic effects in DFM. In the noncontact
limit, Ã can be calculated with the procedure explained above. Ã
differs from zero by at most ⬃0.015 Å at the separation L *⫽103.5 Å .

冉where ␻ ⬘ ⬅ ␻ / ␻ 0 and A d⬘ ⬅A d /A. Here r n is given by
1
r n⬅
k A␲

冕

2␲

0

F i 共 x 0 ⫹L 兲 cos n 共 ␪ ⫺ ␾ 兲 d ␪ ,␦ r 1⬇

␦ 0⬘

⫹
⫹
g 10

兺 g 1l⫹ ␦ ⬘ ⫹l ,

2

冊

M

⫹ ⬘
␦ 0⬘ ⫺ 兺 g 0l
␦ l⫹ ⬇r 0 ,
l⫽2

共A4兲
共 1⫺n

with ␪ ⬅ ␻ t. Here ␦ r 1 , ␦ r 2 represent corrections from firstharmonic motion. In the noncontact limit, ␦ r 1 , ␦ r 2 can be
written
⬁

1⫺

⫹
g 00

⫺
⫹
兲 ␦ ⬘⫹n ⫹n/Q 0 ␦ ⬘ n ⫺g n0

␦ 0⬘
2

M

⫺

兺 g ⫹nl ␦ ⬘ ⫹l ⬇r n ,

l⫽2
M

共 1⫺n

⬁

⫺
␦ r 2 ⬇ 兺 g 1l
␦ ⬘⫺
l ,

2

2

⫹
兲 ␦ ⬘⫺n ⫺n/Q 0 ␦ ⬘n ⫺兺 g ⫺nl ␦ ⬘ ⫺l ⬇0.

l⫽2

共A7兲

⫾
The coefficients ␦ ⬘ ⫾n ⬅ ␦ n /A can be determined by solving
the following set of 2 M ⫺1 linear equations, M representing
the order of the highest harmonic included in the calculation:

Roughly speaking, Eqs. 共A5兲 and 共A7兲 are accurate as long⫹
兩 /2Ⰶ1. A useful quantity for measuras 兩 ␻ ⬘ ⫺1 兩 Ⰶ1 and 兩 g 00
ing the deviation from purely sinusoidal is Ã⬅ 冑x 2 ⫹(ẋ/ ␻ ) 2 ⫺A. This is plotted in Fig. 6 for the force
described in the text at L * ⫽103.5 Å . The quantity Ã departs from zero by at most ⬃0.015 Å . Apart from the introduction of the driving term, these considerations are similar to those discussed by Dürig in Ref. 24. Here weemphasize the following. In the noncontact region, Eq. 共A7兲provides a very simple way to calculate accurately the anharmonic terms; one simply has to solve a set of linear equations.

1

8

2

l⫽2

l⫽2

共A5兲
⫾
defined aswith g nl

g⫾
nl ⬅

1
k␲

冕

2 ␲ dF

0

dx

i

冏

1
兵 cos关共 l⫺n 兲共 ␪ ⫺ ␾ 兲兴2x0⫾cos关共 l⫹n 兲共 ␪ ⫺ ␾ 兲兴 其 d ␪ .

共A6兲

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