DOI 10.1007/s11071-015-2014-4 O R I G I NA L PA P E R
Control of bistability in non-contact mode atomic force microscopy using modulated time delay
Ilham Kirrou · Mohamed Belhaq
Received: 26 July 2014 / Accepted: 6 March 2015
© Springer Science+Business Media Dordrecht 2015
Abstract The control of bistability in non-contact mode atomic force microscopy subjected to a harmonic base excitation and in the presence of time-delayed feedback is investigated in this paper. We consider the case where the time-delayed feedback gain is modu- lated with a frequency higher than the natural frequency of the system and the frequency of the base displace- ment. We assume that the tip–sample interaction force is described by Lennard–Jones potential, and we con- sider a lumped single degree of freedom model describ- ing weakly nonlinear dynamics of the microcantilever in the range of attracting force. Analytical investigation is performed to obtain approximation of the frequency response of the microcantilever as well as the regions of bistability in different parameter planes. The influence of the time-delayed feedback gain on the bistability behavior as well as the vibration amplitude of the tip motion is studied near primary resonance for different parameter variation.
Keywords Non-contact mode AFM·Bistability· Time delay control·Time-periodic gain·Perturbation analysis
I. Kirrou (B)·M. Belhaq
Laboratory of Mechanics, University Hassan II-Casablanca, Casablanca, Morocco e-mail: ilhamkirrou@gmail.com
1 Introduction
Atomic force microscopy (AFM) [1] has been amply used to provide hight-resolution surface topography at the atomic and molecular level. The dynamics of AFM basically depend on the interaction of a microcan- tilever with surface forces through a surface–tip inter- action potential. This interaction property which influ- ences the frequency response near the resonance can be exploited for material characterization and quanti- tative measurements of surface mechanical properties.
One of the performance modes of AFM operation is the non-contact mode for which the probe operates in the attractive force region while remains separated from the sample without affecting it. In most situations, such interactions cause the tip to oscillate at a low ampli- tude with low (attractive) force offering advantages for characterization of soft samples [2]. While short range repulsive interactions produce hardening nonlin- ear response [3,4] leading the tip oscillation to switch from a purely non-contact to tapping mode, attractive forces, on the other hand, induce softening behavior [5].
In the present study, only the case of attractive forces in the limit of purely non-contact operating mode is considered.
The dynamics of a microcantilever tip–sample inter- action in non-contact mode AFM have been studied by many authors [6–10]. In [11], the Gakerkin method was applied to truncate an AFM system subjected to tip–
surface interaction and the periodic and chaotic behav- iors were studied numerically. In recent years, various
studies have been published in the field of AFM non- linear dynamics; see for instance [10,12,13].
Because the microcantilever tip is operating under a highly nonlinear attractive force in non-contact mode AFM, bistable behavior may occur in the proximity of the sample [14,15]. This bistability, causing hystere- sis and jump phenomena in the system, produces in general discontinuous transition of imaging character- istics when scanning soft matter [16]. Thus, it is of importance to develop strategies for controlling such a bistability regime in the system.
In this paper, we deal with the control of bistabil- ity in non-contact mode AFM using the time-delayed feedback [17]. This method was performed numeri- cally to control the dynamics in AFM [18], to sup- press chaotic motions in tapping mode AFM [19,21]
or to control quality factor [20]. In an experimental work [22], time-delayed feedback introduced in veloc- ity was used to control bistability and chaotic response in AFM using a magnetic excitation instead of typical piezoelectric excitation. Time-delayed feedback was also implemented experimentally to control bistability in capacitive MEMS resonator [23,24].
The intent of this paper is to study analytically the bistability behavior in non-contact mode AFM under time-delayed feedback with a modulated gain.
We assume that the tip–sample interaction force is described by Lennard–Jones potential, and we consider a lumped single degree of freedom model describing weakly nonlinear responses of the system [6] in the range where only attractive force is dominant (pure non-contact mode operation). Following the experi- mental investigation [22], we consider that the time- delayed feedback controller is introduced in velocity such that the control input is given by the difference signal between the current output and the retarded one [17]. A focus is placed on examining the influence of the gain modulation on the dynamics of the tip motion near the primary resonance. One assumes that the fre- quency of the modulation is larger than the natural fre- quency and the frequency of the excitation. We begin in Sect.2by describing the model under investigation and apply the method of direct partition of motion to obtain the main equation governing the slow dynamic of the system. Section3applies the method of multiple scales on the slow dynamic to derive the modulation equations of the slow dynamic near the primary reso- nance. The influence of variation in system parameters on the bistability regime is reported in Sect. 4. Sec-
tion5discusses the response charts in the context of bistability problem. Conclusions follow in Sect.6.
2 Equations of motion and slow dynamic
Considering the first mode of vibration in the AFM dynamics, the vertical motion of the microcantilever tip can be modeled by a lumped single degree of freedom system, as shown in Fig. 1. If one assumes that the microcantilever is under a time-varying delay control and subjected to a harmonic base excitation d(t) = F sin1t , the equation of tip motion can be written in the form [25,26]
mx¨+c0(x˙− ˙d)+k0(x−d)
= fL J +G(t)[ ˙x(t−τd∗)− ˙x(t)] (1) where x is the cantilever tip displacement relative to the fixed base frame, m, c0and k0denote, respectively, the cantilever tip mass, damping and spring stiffness coefficients, while fL J denotes Lennard–Jones attrac- tion/repulsion force (AFM tip–sample surface interac- tion force) for lumped-parameter model defined explic- itly by [11,26]
fL J = A2R
6(z0−x)2 − A1R
180(z0−x)8 (2) where z0 is the distance from the fixed base frame coordinate to the sample in the absence of any inter- action between the tip and the sample, A1and A2are
Fig. 1 Equivalent lumped-parameter model of the AFM micro- cantilever
the Hamaker constants for the repulsive and attractive potentials, respectively, R represents the cantilever tip radius, and the time-varying gain G(t)is given by G(t)=G∗0+G∗1cos2t (3) where G∗0and G∗1denote the amplitudes of unmodu- lated and modulated gains, respectively,2is the fre- quency of the modulation, andτd∗ is the time delay.
The displacement x is defined by considering the static problem as x=zs+X and the quantity z∗=z0−zs
as the equilibrium gap between the tip and the sample, where zsis the static position and X is the displacement from the static position.
In order to simplify the system identification, it is useful to normalize Eq. (1) using the following dimen- sionless variables: u = zX∗, τ = ω0t , τd = ω0τd∗, ω20 = km0,ω = ω10, = ω20, c = mcω00, G0 = mGω∗00, G1= mGω∗10 and f = zF∗. Expanding the Lennard–Jones force around the equilibrium position using the Taylor series expansion up to third order, the dimensionless equation of motion takes the form
¨
u+ω21u+cu˙+β2u2+β3u3= f sinωτ
+cωf cosωτ+(G0+G1cosτ)[ ˙u(τ−τd)− ˙u(τ)]
(4) whereω21 =1+β1,β1 =8α−2β,β2=36α−3β, β3 = 120α − 4β with α = 180mA1ωR2
0z∗9 and β =
A2R
6mω20z∗3. Note that the third-order Taylor expansion in the Lennard–Jones force can be considered as a good approximation for the non-contact mode opera- tion in the range where only the attractive force effect is dominant (as in the case under investigation). Figure2 shows the comparison between the original Lennard- Jones force and its third-order Taylor approximation.
A good match is depicted in the range of non-contact mode operation. The jump-to-contact phenomena that is a limit of non-contact operation is ignored in the present study.
Equation (4) includes a slow dynamic due to the base harmonic motion assumed to be slow and a dynamic produced by the frequency of the time-delayed feed- back modulation supposed larger than the natural and the external frequencies,ω1 andω, respectively.
To analyze the effect of the modulated time-delayed feedback on the frequency response, it is convenient to extract the slow dynamic of (4) using the method of direct partition of motion [27,28] by introducing two
Fig. 2 Comparison between the original LJ force and the third- order Taylor approximation
different timescales: a fast time T0=τand slow time T1=τ. Next, we split up the solution u(τ)into a slow part z(T1)and a fast partεφ(T0,T1)as
u(τ)=z(T1)+εφ(T0,T1) (5) and
u(τ−τd)=z(T1−τd)+εφd(T0−τd,T1−τd) (6) where z describes the slow main motions at timescale of oscillations, εφ stands for an overlay of the fast motions, and ε indicates that εφ is small compared with z. Sinceis considered as a large parameter, we chooseε≡−1, for convenience. The fast partφand its derivatives are assumed to be 2π−periodic functions of fast time T0with zero mean value with respect to this time so that<u(τ) >=z(T1)and<u(τ −τd) >=
z(T1−τd) where <>≡ 21π2π
0 ()d T0 defines time- averaging operator over one period of the fast excita- tion with the slow time T1fixed. Introducing Dij ≡∂∂jTji
yieldsddτ =D0+D1,ddτ22 =2D20+2D0D1+D12, and substituting Eqs. (5) and (6) into Eq. (4) gives ε−1D02φ+2D0D1φ+εD12φ+D12z+c D1z
+c(D0φ+εD1φ)+ω21(z+εφ)+β2(z+εφ)2 +β3(z+εφ)3−G0[D1z(T1−τd)−D1z(T1)]
−G0(D0+εD1)[φ(T0−τd,T1−τd)
−φ(T0,T1)] =G1[D1z(T1−τd)−D1z(T1)]cos T0
+G1(D0+εD1)[φ(T0−τd,T1−τd)
−φ(T0,T1)]cos T0+ f sinωτ+cωf cosωτ (7)
Averaging (7) leads to
D21z+c D1z+ω21z+β2z2+β3z3
−G0(D1z(T1−τd)−D1z)+(β2+3β3z)ε2φ2
−G1[(D0φd+εD1φd)cos T0
− (D0φ+εD1φ)cos T0] = f sinωτ+cωf cosωτ (8) Subtracting (8) from (7), an approximate expression for εφis obtained by considering only the dominant terms of orderε−1as
D20φ= G1
[D1z(T1−τd)−D1z(T1)]cos T0 (9) The stationary solution to the first order forφis then written as
εφ(T0,T1)= −G1
2[D1z(T1−τd)−D1z(T1)]cos T0
(10) Insertingφfrom (10) into (8), using that<cos2T0>=
1/2 and neglecting terms of orders greater than three in z,we find the approximate equation for slow motions
D21z+c D1z+ω21z+β2z2+β3z3
−G0[D1z(T1−τd)−D1z(T1)] +(λ1+λ2z)[D21z +D12z(T1−τd)−2D1z(T1)D1z(T1−τd)]
+λ3[D1z(T1−2τd)−D1z(T1−τd)]
−λ4D12z(T1−τd)+λ5D12z(T1−2τd)
= f sinωτ+cωf cosωτ (11) where λ1 = β22G421, λ2 = 3β23G421, λ3 = G221 sinτd, λ4=2G212 +2G212 cosτdandλ5=2G212cosτd.
3 Modulation equations
In this section, we approximate the frequency-response of the slow dynamic near the primary resonance using the multiple scales method [29,30]. Introducing a book- keeping parameterε and scaling as c = 2c, β3 = 2β3, G0 = 2G0, λi = 2λi(i = 1, . . . ,5) and f = 2f (the other parameters being of order ), Eq. (11) can be rearranged in the form
¨
z+ω21z= −β2z2−2{cz˙+β3z3
−G0[˙z(τ−τd)− ˙z]
+(λ1+λ2z)[˙z2+ ˙z2(τ−τd)−2˙zz(τ˙ −τd)]
+λ3[˙z(τ−2τd)− ˙z(τ−τd)] −λ4¨z(τ−τd) +λ z(τ¨ −2τ )− f sinωτ−cωf cosωτ} (12)
Steady-state solution are expanded as
z(T0,T1,T2)=z0(T0,T1,T2)+z1(T0,T1,T2) +2z2(T0,T1,T2)+0(3) (13) where T0=τ, T1=τand T2=2τ. In terms of the variables Ti (i =0,1,2), the time derivatives become
d
dτ = D0+D1+2D2+O(3)and ddτ22 = D02+ 2D01+2D21+22D02+O(3), where Di = ∂∂T
i.
Substituting (13) into (12) and equating the terms with the same order ofyield
D20z0+ω21z0=0 (14) D20z1+ω21z1= −2D0D1z0−β2z20 (15) D20z2+ω21z2= −2D0D1z1−(D12+2D0D2)z0
−c D0z0−2β2z0z1−β3z30−G0[D0z0(T0−τd)
−D0z0] −(λ1+λ2z0)[D12z0+D21z0(T0−τd)
−2D1z0D1z0(T0−τd)] −λ3[D1z0(T0−2τd)
−D1z0(T0−τd)] +λ4D21z0(T0−τd)
−λ5D12z0(T0−2τd)+ f sinωτ+cωf cosωτ (16) The solution of Eq. (14) can be written in the form z0(T0,T1,T2)=A(T1,T2)eiω1T0+cc (17) Eliminating the terms in Eq. (15) that produce secular terms in z1yields D1A=0 or A=A(T2). Hence, the solution of Eq. (15) becomes
z1(T0,T2)=β2A2
3ω21 e2iω1T0 −2β2AA¯
ω21 +cc (18) where A(T2)is a complex amplitude and cc stands for the complex conjugate of the preceding terms. The con- dition for primary resonance is written as
ω=ω1+σ (19)
in whichσ is a detuning parameter representing the deviation from natural frequency. Substituting (17), (19), (18) into (16) and removing secular terms, we obtain
10β22
3ω21 −3β3+γ3
A2A¯+
γ1−G0ω1sinω1τd
A
+i
−2ω1D2A+γ4A2A¯−(cω1+G0ω1
−G0ω1cosω1τd−γ2)A +
i f 2+cωf
2
eiσT2=0 (20) where
γ1=λ3ω1(sin 2ω1τd−cosω1τd)+λ4ω21cosω1τd
−λ5ω21cos 2ω1τd (21)
γ2=λ3ω1(cos 2ω1τd−cosω1τd)−λ4ω21sinω1τd
+λ5ω21sin 2ω1τd (22)
γ3=3λ2ω12−λ2ω21(cos 2ω1τd+2 cosω1τd) (23) γ4=λ2ω21(sin 2ω1τd−2 sinω1τd) (24) Equation (20) can be solved for the complex amplitude by introducing the polar form A = 21aeiθ. Substitut- ing into (20) and separating real and imaginary parts, we obtain the modulation equations of amplitude and phase as
⎧⎪
⎪⎨
⎪⎪
⎩ ω1
da dt =γ4
8
a3+1 2
−cω1−G0ω1+G0ω1cosω1τd+γ2
a+
cωf
2 sinϕ− f 2 cosϕ
ω1adϕ dt =
5β22 12ω21 −3β3
8 +γ3
8
a3+
σω1+1
2G0ω1sinω1τd+γ1
2
a+ cωf
2 cosϕ+ f 2 sinϕ
(25)
in whichϕ = σT2−θ. Equilibria of this slow flow, corresponding to periodic solutions of Eq. (12), are determined by setting dadt = ddtϕ = 0. This leads to the amplitude–frequency response equation
A J3+B J2+C J+D=0 (26) where
A= 3
4β3−5β22 6ω21−γ3
4 2
+
−γ4
4 2
B =
3
4β3−5β22 6ω21−γ3
4
×(−2σ ω1−G0ω1sinω1τd−γ1)
+(cω1+G0ω1−G0ω1cosω1τd−γ2)γ4
4 C =(cω1+G0ω1−G0ω1cosω1τd−γ2)2
+(−2σ ω1−G0ω1sinω1τd−γ1)2 D= −f2−(cωf)2
J =a2
It is worth noticing that the slow flow (25) may also exhibit periodic solutions that correspond to quasiperi- odic responses of the slow dynamic (12) [32,33]. Such a quasiperiodic regime, which was reported experimen- tally in the AFM dynamic under time-delayed feedback [22], can be investigated analytically applying the dou- ble perturbation method as in [31–33]. This problem is not considered in the current study.
In what follows, the analysis of the dynamic of the tip–sample system is investigated for a representative case: the interaction of soft monocrystalline silicon
microcantilever with the (111) reactive face of a flat silicon sample. The properties of the cantilever and interaction properties of the sample are taken from [11] and given by k = 0.11 N m−1, R = 150 nm, ω0 = 74129.12 rad s−1, Q = 100, A1 = 1.3596× 10−70J m6, A2 = 1.865×10−19 J and f = 0.008.
Note that the chosen physical values of the gain leading to an effective control of the bistability are of the same order of magnitude than the viscous damping coeffi- cient which is of order 10−6.
4 Controlling bistability and vibration amplitude Next, we examine the effect of the modulated time- delayed feedback on the frequency response and on the bistability behavior considering different cases of time delay.
4.1 Case without time delay
In the absence of the feedback gains(G0=0 and G1= 0), Fig. 3 shows the amplitude–frequency response, as given by (26), for two different values of the gap z∗ [z∗ = 10 nm (Fig.3a) and z∗ = 6 nm (Fig.3b)].
The solid lines correspond to stable branches, while the dashed line corresponds to the unstable one. The plots depict that as the vibrating tip approaches the sam- ple (Fig.3b), the attraction force increases leading the softening characteristic to increase and the frequency response to shift toward lower frequencies, as expected.
Due to the dependence between the tip–sample dis- tance, the interaction force and the frequency shift, the interaction force or the optimal tip–sample distance, for which the microcantilever can operate in stable regime, can be estimated.
The effect of the amplitude of the excitation f on the frequency response is shown in Fig.4a, while the effect of damping is reported in Fig.4b. Figure4a indi- cates that a small increase in the excitation amplitude leads to an increase in the vibration amplitude of the microcantilever, while Fig.4b shows that an increase in the damping decreases the vibration amplitude of
0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 0
0.2 0.4 0.6 0.8 1
ω
a
(a)
0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1
0 0.2 0.4 0.6 0.8 1
ω
a
(b)
Fig. 3 Frequency response for c=0.01, f =0.008, G0=0, G1=0 and for different values of the gap z∗; a z∗=10 nm, b z∗=6 nm
0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ω
a
(a)
f=0.008
f=0.004
0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ω
a
(b)
c=0.02 c=0.01
Fig. 4 Frequency response for z∗=8 nm, G0=0, G1=0; a c=0.01, b f =0.008
the system. It can be concluded that an appropriate choice of the amplitude of the excitation and damp- ing can optimize the nonlinear dynamics effect on the microcantilever vibration in order to better deduce a high quality measurement of desired quantities. Such effects were also reported in [10,13].
4.2 Case with unmodulated time delay
In the remaining of the study, we fix the parameters c = 0.01, f = 0.008, = 4 and z∗ = 7 nm.
Next, we analyze the effect of the unmodulated feed- back gain (G0 =0,G1 =0) on the dynamic of non- contact mode AFM near the primary resonance. Fig- ure 5 illustrates the frequency response for different values of G . For validation, analytical approxima-
tion [solid (dashed) lines for stable (unstable)] solu- tions are compared to results obtained by numerical simulations (circles) using dde23 algorithm [34,35].
It can be seen that increasing the feedback gain G0
decreases the vibration amplitude of the microcan- tilever and suppresses bistability. This indicates that in the case where the physical parameters of the system in non-contact mode AFM are fixed during imaging operation, time-delayed feedback control may be used to suppress bistability leading the microcantilever to operate in a certain safe region where only one sta- ble equilibrium exists. It is worthy to point out that the suppression of the bistable behavior (by increas- ing G0) is accompanied by a decrease in the vibration amplitude. The shift in the frequency response obtained in Fig. 5a–c is depicted in Fig. 5d superimposing Fig.5a–c.
0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ω
a
(a)
0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ω
a
(b)
0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ω
a
(c)
0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ω
a
G0=0.3
G0=0 G0=0.4
(d)
Fig. 5 Frequency response forτd = 0.5; a G0 =0, b G0 =0.3, c G0 =0.4, d global plots of Fig.5a–c. Solid lines analytical approximation; circle numerical simulation
Figure6a shows the frequency response versusωfor a given time delayτdand for increasing negative values of the unmodulated gain G0, while Fig.6b illustrates the frequency response for a given value of G0 and for increasing values of time delayτd. One observes a decrease in the amplitude and a shift toward higher fre- quencies for increasing negative G0(Fig.6a) and only a decrease in the amplitude for increasingτd(Fig.6b).
These results indicate that the feedback gain G0influ- ences not only the bistability and the amplitude of the tip vibration but also the frequency shift and thereby it can be used to estimate the interaction force and the optimal tip–sample distance ensuring the stability.
4.3 Case with modulated time delay
Here we explore the effect of the amplitude modulation of time delay(G1= 0)on the frequency response of the non-contact mode AFM. Figure7shows the influ- ence of G1on the frequency response in the absence
of the unmodulated gain, G0 =0 and for a givenτd. It can be seen that in the case when only the modu- lated gain G1 is introduced (Fig. 7b), the amplitude of the response decreases and the bistability is elimi- nated. Nevertheless, the value of G1needed to achieve the suppression of bistability is higher than the case where only the unmodulated gain G0is acted.
In Figs.7and8, we can appreciate the effect of G0
and G1on the amplitude vibration when acted sepa- rately or simultaneously. Figures7a and8a depict the influence of G0alone, Fig.7a, b shows the influence of G1alone, and Figs.7a and8b illustrate the combined effect of G0and G1. One can conclude that the com- bined effect of both components G0and G1is relevant to optimal control of bistability and vibration ampli- tude.
The variation in the amplitude of vibration versus time delay is shown in Fig.9for fixed G0and differ- ent increasing values of G1. Figure 9a–c shows that for small values of the modulated gain G1, the bista- bility behavior occurs in certain periodic intervals of
