Laurent Nony
email: laurent.nony@l2mp.fr.

Alexis Baratoff

Dominique Schaer

Oliver Pfeiffer

Adrian Wezel

Ernst Meyer

Dominique Schär

Adrian Wetzel

Noncontact atomic force microscopy simulator with phase-locked-loop controlled frequency detection and excitation

Keywords: numbers: 07.79.Lh, 07.50.Ek, 46.40.Ff virtual machine, non-contact AFM, dissipation, damping, apparent dissipation, Phase Locked Loop

I. INTRODUCTION

Since almost a decade, non-contact atomic force microscopy (nc-AFM) has proven capable of yielding images showing contrasts down to atomic scale on metals, semiconductors, as well as insulating ionic crystals, with or without metallic or adsorbate overlayers 1,[START_REF] Garcia | Dynamic atomic force microscopy methods[END_REF][START_REF] Giessibl | Advances in atomic force microscopy[END_REF] . Like other scanning force methods, the technique relies on a micro-fabricated tip grown at the end of a cantilever. However, unlike the widely used contact or the tapping modes, the cantilever deflection is neither static nor driven at constant frequency, but is driven at a frequency f 0 = ω 0 /2π equal to its fundamental bending resonance frequency, slightly shifted by the tip-sample interaction. A sufficiently large oscillation amplitude prevents snap into contact.

A quality factor exceeding 10 4 , readily achieved in UHV, together with frequency detection by demodulation provide unprecedented force sensitivity [START_REF] Albrecht | Frequency modulation detection using high-Q cantilevers for enhanced force microscope sensitivity[END_REF][START_REF] Dürig | Interaction force detection in scanning probe microscopy: Methods and applications[END_REF] . A phase-locked loop (PLL) is typically used for that purpose. Since f 0 varies with the tip-surface distance, it deviates from f 0 , the fundamental bending eigenfrequency of the free cantilever. Upon approaching the surface, the tip is first attracted, in particular by Van der Waals forces, which decrease f 0 .

The negative frequency shift, ∆f = f 0 -f 0 , varies rapidly with the minimum tip-distance d, usually as d -n with n ≥ 1.5, and then as exp(-d/λ) a few angströms above the surface, owing to short-range chemical and/or steric forces [START_REF] Guggisberg | Separation of interactions by non-contact force microscopy[END_REF] . When ∆f is used for distance control, contrasts down to the atomic scale can be achieved. Another specific feature of the nc-AFM technique is that the oscillation amplitude A is kept constant while approaching or scanning the surface at constant ∆f .

Controlling the phase of the excitation so as to maintain it on resonance and to make the frequency matching a preset f 0 -value, as well as the driving amplitude so as to keep the tip oscillation amplitude constant, respectively, requires dedicated electronic components. Amplitude control is usually achieved using a proportional integral controller (PIC), hereafter referred to as APIC, whereas phase and frequency control can be performed in two ways.

In both cases the AC deflection signal of the cantilever is filtered, then phase-shifted and multiplied by the APIC output and by a suitable gain. The most common method consists in using a band-pass filtered deflection signal [START_REF] Gauthier | Interplay between nonlinearity, scan speed, damping and electronics in frequency modulation atomic force microscopy[END_REF][START_REF] Couturier | Noncontact atomic force microscopy: stability criterion and dynamical responses of the shift of frequency and damping signal[END_REF] . This is referred to as the self-excitation mode. The second method, extensively analyzed hereafter, consists in using the PLL to generate the time-dependent phase of the excitation signal. The PLL output is driven by the AC deflection signal and phase-locked to it, provided that the PLL settings are properly adjusted. Then, the PLL continuously tracks the oscillator frequency f 0 with high precision.

Moreover, the phase lag introduced by the PLL itself can be compensated. For reasons of clarity, this mode will be referred to as the PLL-excitation mode. The choice of the PLL as the excitation source has initially been motivated to take benefit of the noise reduction due to PLLs [START_REF] Best | Phase Locked Loops: Design, Simulation and Applications[END_REF] . A further advantage is that the noise reduction does not only optimize the detection of the frequency shift, but also the excitation signal. In both modes, the phase shifter is adjusted so as the phase lag ϕ between the excitation and the tip oscillation equals -π/2 rad throughout an experiment. If all adjustments and controls were perfect, the oscillator would then always remain on resonance.

The nc-AFM technique therefore requires the simultaneous operation of three controllers : PLL, APIC and distance controller, which keeps constant a given ∆f while scanning the surface. Since the tip-surface interaction makes the dynamic of the oscillator non-linear, the combined action of those three controllers becomes complex. They can conditionally interplay [START_REF] Gauthier | Interplay between nonlinearity, scan speed, damping and electronics in frequency modulation atomic force microscopy[END_REF][START_REF] Couturier | Noncontact atomic force microscopy: stability criterion and dynamical responses of the shift of frequency and damping signal[END_REF] and therefore influence the dynamics of the system. Consider for instance the time the PLL spends to track f 0 is long compared to the time constant of the APIC. Then, the cantilever is no longer maintained at f 0 , but at a frequency slightly higher or lower. Consequently, the oscillation amplitude drops 10 and the APIC increases the excitation to correct the amplitude reduction. Such an apparent loss of energy, which can as well be interpreted as a damping increase of the cantilever, does not result of a dissipative process occurring between the tip and the surface, but is the consequence of the bad tracking of f 0 . So-called apparent dissipation (or apparent damping) remains under discussions in the nc-AFM community, which hinders the quantitative interpretation of the experimental proofs of dissipative phenomena on the atomic scale over a wide variety of samples [START_REF] Gotsmann | Conservative and dissipative tip-sample interaction forces probed with dynamic afm[END_REF][START_REF] Loppacher | Dynamic force microscopy of copper surfaces: atomic resolution and distance dependence of tip-sample interaction and tunneling current[END_REF][START_REF] Loppacher | Experimental Aspects of Dissipation Force Microscopy[END_REF][START_REF] Bennewitz | Atomically Resolved Steps and Kinks on NaCl islands on Cu(111): Experiment and Theory[END_REF][START_REF] Gotsmann | Dynamic Force Spectroscopy of Conservative and Dissipative Forces in an Al-Au(111) Tip-Sample System[END_REF][START_REF] Hoffmann | Energy dissipation in atomic force microscopy and atomic loss processes[END_REF][START_REF] Giessibl | Friction traced to the single atom[END_REF][START_REF] Hoffmann | Atomic resolution imaging and frequency versus distance measurements on NiO(001) using low-temperature scanning force microscopy[END_REF][START_REF] Pfeiffer | Distance dependence of force and dissipation in non-contact atomic force microscopy on Cu(100) and Al(111)[END_REF][START_REF] Hembacher | Local spectroscopy and atomic imaging of tunneling current, forces, and dissipation on graphite[END_REF] . Thus, addressing the problem of apparent dissipation turns out to be mandatory but requires to understand the complex interplay between controllers as well as to analyze the system time constants. Although several models of physical dissipation, connected or not to the conservative tip-surface interaction have been proposed [START_REF] Gauthier | Theory of noncontact dissipation force microscopy[END_REF][START_REF] Sasaki | Effect of Microscopic Nonconservative Process on Noncontact Atomic Force Microscopy[END_REF][START_REF] Gauthier | Damping mechanism in dynamic force microscopy[END_REF][START_REF] Dürig | Interaction sensing in dynamic force microscopy[END_REF][START_REF] Kantorovich | Stochastic friction force mechanism of energy dissipation in noncontact force microscopy[END_REF][START_REF] Boisgard | Analysis of mechanisms inducing damping in dynamic force microscopy: surface viscoelastic behavior and stochastic resonance process[END_REF][START_REF] Boisgard | Surface mechanical instabilities and dissipation under the action of an oscillating tip[END_REF][START_REF] Trevethan | Physical dissipation mechanisms in non-contact atomic force microscopy[END_REF][START_REF] Trevethan | Atomistic simulations of the adhesion mechanism of atomic scale dissipation in non-contact atomic force microscopy[END_REF]30 and reviewed 31 , the question of apparent dissipation in the self-excitation scheme has been addressed by two groups [START_REF] Gauthier | Interplay between nonlinearity, scan speed, damping and electronics in frequency modulation atomic force microscopy[END_REF][START_REF] Couturier | Noncontact atomic force microscopy: stability criterion and dynamical responses of the shift of frequency and damping signal[END_REF][START_REF] Couturier | Stability of an oscillating tip in noncontact atomic force microscopy: Theoretical and numerical investigations[END_REF] . M. Gauthier et al. [7] emphasize the interplay between the controllers and the conservative tip-sample interaction which, although weak, can significantly affect the damping. They put in evidence resonance effects which can conditionally occur in damping images upon scan speed and APIC gains. G. Couturier et al. [START_REF] Couturier | Noncontact atomic force microscopy: stability criterion and dynamical responses of the shift of frequency and damping signal[END_REF] address a similar problem numerically and analytically. They show that the self-excited oscillator can be conditionally stable within a narrow domain of K p and K i gains of the APIC, but that consequent damping variations can as well be generated upon conservative force steps which change the borders of the stability domain. The results mentioned above are valid for the self-excitation mode, but the question of apparent dissipation remains open regarding the PLL-excitation mode. However recently, J. Polesel-Maris and S. Gauthier [33] have proposed a virtual dynamic AFM based on the PLL-excitation scheme. Their work is targeted at images calculations including realistic force fields obtained from molecular dynamics calculations 34 . Their conclusions stress the contribution of the scanning speed and of the experimental noise to images distortion but do not address the potential contribution of the PLL upon operating conditions.

The goal of the present work is two-fold : 1-providing a detailed description of the PLL-excitation based electronics of a home-built AFM used in our laboratory; 2-assessing the contribution of the various controllers to the dissipation signal and in particular the contribution of the PLL. The paper is organized as follows. In section II, an overview of the chart of the microscope and of the attached electronics (cf. fig. 1), is given in terms of blocks, namely ; oscillator and optical detection (block 1), RMS-to-DC converter (block 2), amplitude controller (block 3), PLL (block 4), phase shifter (block 5) and tip-surface distance controller (block 6). In section III, the detailed description of the numerical scheme used to perform the calculations is given on the base of coupled integro-differential equations ruling each block. Section IV provides an estimate of the minimum detectable dissipation by the instrument with the goal to assess the relevance, compared to experimental results, of the apparent dissipation which can be conditionally generated numerically. Section V reports the results. In the first part, the simulation is validated by comparing a numerical ∆f vs. distance curve to the analytic expression of the ∆f due to Morse and Van der Waals interactions which does not take into account the finite response of the various controllers.

Then the dynamic properties of the numerical PLL and APIC upon gains are compared to those of the real components. Section V C gives some examples on how apparent dissipation can be produced upon working conditions of the PLL. Section V D finally shows scan lines computed while varying PLL gains, scan speed and APIC gains. A discussion and a conclusion end the article.

II. OVERVIEW

A. Description

The electronics consists of analog and digital (12 bits) circuits which are described by six interconnected main blocks operating at various sampling frequencies (f s ). The highest sampling frequency among the digital blocks is the PLL one, f s 1 = 20 MHz. The PLL electronics has initially been developped by Ch.Loppacher [35].

Block 1 represents the detected oscillating tip motion coupled to the sample surface. In the simulation, the block is described by an equivalent analog circuit. More generally, all the analog parts of the electronics are described in the simulation using a larger sampling frequency compared to f s 1 , namely f s 2 = 400 MHz. This is motivated by the ultra-high vacuum environment within which the microscope is placed, thus resulting in a high quality factor of the cantilever, typically Q = 30000 at room temperature. Besides, nc-AFM cantilevers have typical fundamental eigenfrequencies f 0 ≃ 150 kHz. The chosen sampling frequency should therefore insure a proper integration of the differential equations with an error weak enough. The signal of the oscillating cantilever motion goes into a band pass filter which cuts-off its low and high frequencies components. The bandwidth of the filter is typically 60 kHz, centered on the resonance frequency of the cantilever. Despite the filter has been implemented in the simulation, no noise has been considered, so far. The signal is then sent to other blocks depicting the interconnected parts of two boards, namely an analog/digital one, the "PLL board", and a fully digital one which integrates a Digital Signal Processor (DSP), the "DSP board". The boards share data via a "communication bus" operating at f s 3 = 10 kHz, the lowest frequency of the digital electronics. Block 2 stands for the lone analog part of the PLL board (f s = f s 2 ). It consists of a RMSto-DC converter. The block output is the rms value of the oscillations amplitude, A rms (t).

A rms (t) is provided to block 3, one of the two PICs implemented on the DSP (f s = f s 3 ).

When operating in the nc-AFM mode, the block output is the DC value of the driving amplitude which maintains constant the reference value of the oscillations amplitude, A set 0 . This is why it is referred to as the amplitude controller, APIC. For technical reasons due to the chips, the signal is saturated between 0 and 10 V.

The dashed line in fig. 1 depicts the border between analog and digital circuits in the PLL board. The digital PLL, block 4 (f s = f s 1 ), consists of three sub-blocks : a Phase Detector (PD), a Numerical Controlled Oscillator (NCO) and a filtering stage consisting of a decimation filter and a Finite Impulse Response (FIR) low pass filter in series. The PLL receives the signal of the oscillation divided by A rms (t) plus an external parameter : the "center frequency", f cent = ω cent /2π. f cent specifies the frequency to which the input signal has to be compared to for the demodulation frequency stage. This point is particularly addressed in section III D. The NCO generates the digital sin and cos waveforms of the time-dependent phase, ϕ nco (t)+ϕ pll (t), ideally identical to the one of the input signal. ϕ pll (t) is correlated to the error which is potentially produced while the frequency demodulation, upon operating conditions. The sin and cos waveforms are then sent to a digital phase shifter, block 5 (f s = f s 1 ) which shifts the incoming phase ϕ nco (t) + ϕ pll (t) by a constant amount, ϕ ps , set by the user. Since the cantilever is usually driven at f 0 , ϕ ps is adjusted to make that condition fulfilled 36 , namely :

ϕ nco (t) + ϕ pll (t) + ϕ ps = ω 0 t, (1)
Indeed, the PLL produces the phase locked to the input, that is

ϕ nco (t) + ϕ pll (t) ≃ ω 0 t-π/2.
If it optimally operates, ϕ pll (t) ≃ 0. ϕ ps has therefore to be set equal to +π/2 to maintain the excitation at the resonance frequency prior to starting the experiments. Consequently, ϕ = -π/2 rad. The block output, sin [ϕ nco (t) + ϕ pll (t) + ϕ ps ], is converted into an analog signal and then multiplied by the APIC output, thus generating the full AC excitation applied to the piezoelectric actuator to drive the cantilever.

Block 6 is the second PIC of the DSP (f s = f s 3 ). It controls the tip-surface distance to maintain constant either a given value of the frequency shift, or a given value of the driving amplitude while performing a scan line (switch 3 set to location "a" or "b", respectively in fig. 1). The output is the so-called "topography" signal. The block is referred to as the distance controller, DPIC.

Finally, a digital lock-in amplifier detects the phase lag, ϕ, between the excitation signal provided to the oscillator and the oscillating cantilever motion. The block mimics the photodiodes acquiring the signal of the motion of the oscillating cantilever. The equation describing its behavior is given by the differential equation of the harmonic oscillator :

z(t) + ω 0 Q ż(t) + ω 2 0 z(t) = ω 2 0 Ξ exc (t) + ω 2 0 F int (t) k c (2)
ω 0 = 2πf 0 , Q, k c stand for the angular resonance frequency, quality factor and cantilever stiffness of the free oscillator, respectively. z(t), Ξ exc (t) and F int (t) are the instantaneous location of the tip, excitation signal driving the cantilever and the interaction force acting between the tip and the surface, respectively. The equation is solved with a modified Verlet algorithm, so-called leapfrog algorithm [START_REF] Rapaport | The Art of Molecular Dynamics Simulation[END_REF] , using a time step ∆t s 2 = 1/f s 2 = 5 ns. In the followings, the time will be denoted by its discrete notation :

t → t i = i × ∆t s 2 .
The instantaneous value of the driving amplitude Ξ exc (t i ) (units : m) can be written as :

Ξ exc (t i ) = K 3 A exc (t i )z ps (t i ) (3)
K 3 (units : m.V -1 ) represents the linear transfer function of the piezoelectric actuator driving the cantilever. A exc (t i ) (units : V) is the APIC output (cf. section III C). It is proportional to the damping signal according to :

K 3 A exc (t i ) = Γ(t i )A 0 ω 0 , (4)
Γ(t i ) and A 0 (units : s -1 and m, respectively) being the damping signal and oscillations amplitude of the cantilever when driven at f 0 , respectively. When the cantilever is externally driven and if no interaction occurs, A exc (t i ) can be written as a function of A 0 and of the quality factor of the cantilever :

K 3 A exc,0 = A 0 Q (5)
Then the damping of the free cantilever equals :

Γ 0 = ω 0 Q (6)
In nc-AFM, the dissipation is commonly expressed in terms of dissipated energy per oscillation cycle, E d 0 . For a cantilever with a high quality factor oscillating with an amplitude A 0 :

E d 0 (A 0 ) = πk c A 2 0 Q = πk c A 2 Γ 0 ω 0 (7)
In UHV and at room temperature, Q = 30000. Besides, nc-AFM commercial cantilevers have typical stiffnesses 39 k c ≈ 40 N.m -1 . Considering A 0 = 10 nm, the intrinsic dissipated energy per cycle of the cantilever is then E d 0 ≃ 2.6 eV/cycle.

In equation 3, z ps (t i ) is the AC part of the excitation signal (cf. section III E). It is provided by the PS when the PLL is engaged. When the steady state is reached, e.g.

t i ≫ t steady ≃ 2Q/f 0 , the block output is :

K 1 z(t i ) = K 1 A(t i ) sin [ωt i + ϕ(t i )] (8)
K 1 (V.m -1 ) depicts the transfer function of the photodiodes which is assumed to be linear within the bandwidth (3 MHz in the real setup). If the damping is kept constant, the amplitude and the phase, A(t i ) and ϕ(t i ) respectively, are supposed to be constant as well. This is no longer true once the various controllers are engaged, therefore their time dependence is explicitly preserved.

In equation 2, the interaction force F int (r) = -∂ r V int (r) is derived from a conservative potential consisting of two components : a long-range part, depicted by a Van der Waals term defined between a sphere and a half-plane and a short-range part, prevailing at closer distances, depicted by a Morse potential :

V int (r) = - HR 6r -U 0 2e -r-rc λ -e -2(r-rc) λ (9)
H and R are the Hamaker constant of the tip-vacuum-surface interface and tip's radius, respectively. U 0 , r c and λ are the depth, equilibrium position and range of the Morse potential. The instantaneous tip-surface separation is r(

t i ) = D(t i ) -z(t i ), where D(t i ) is
the distance between the surface location and the cantilever position at rest. So far, neither elastic deformation of the sample and tip, nor dissipative interaction have been considered.

The signal K 1 z(t i ) then gets into the band pass filter (BPF), the central frequency of which, f c = ω c /2π, equals the resonance frequency of the cantilever, f 0 , with a bandwidth B W ≃ 60 kHz. The output, z bpf (t i ) (units : V), is ruled by :

zbpf (t) + 2πB W żbpf (t) + ω 2 c z bpf (t) = 2πB W K 1 ż(t) (10)
z bpf (t i ) is then provided to the RMS-to-DC converter of the PLL board.

B. Block 2: RMS-to-DC converter

The converter is the only analog part of the PLL board. The related differential equation is integrated at f s 2 . The chip (AD734) computes the square root of the squared value of the incoming signal, preliminary filtered by a first-order low pass filter, the cut-off frequency of which is f co = 400 Hz. The output is the amplitude (DC value) of the oscillation, A rms (t i ) (units : V) :

A rms (t i ) = V s (t i ), (11)
V s (t i ) being the output of the first-order low pass filter :

τ rms Vs (t) + V s (t) = z 2 bpf (t i ), (12)
with τ rms = 1/(2πf co ) ≃ 400 µs.

z bpf (t i ) is then divided by A rms (t i ) in order to normalize the amplitude of the waveform.

The signal thus normalized is sent to the ADC1 to be operated by the digital PLL.

C. Block 3: amplitude controller

The block represents a digital PI controller implemented in the DSP board. The controller receives the RMS-to-DC output signal via the communication bus. Since the bus operates at f s 3 = 10 kHz, the time step used to solve the related differential equation is ∆t s 3 = 1/f s 3 = 100 µs. Besides A rms , the controller receives three external parameters :

the proportional and integral gains, K ac p and K ac i respectively (units : dimensionless and s -1 , respectively), and the reference amplitude expected to be kept constant as soon as the controller is engaged (switch 1 set to location "b" in fig. 1), A set 0 (units : V). The block output is the DC value of the excitation, previously referred to as A exc (t i ) (cf. equ.3) :

A exc (t i ) =K ac p A set 0 -A rms (t i ) + i k=0 K ac i A set 0 -A rms (t k ) ∆t s 3 (13)
Engaging the APIC makes the nc-AFM mode effective. This requires the PLL-excitation mode (block 4, cf. section III D) to be already engaged. If operating at f 0 , then A rms /K 1 equals the resonance amplitude, A 0 . A exc is then minimal.

D. Block 4: PLL

Before starting this section, note that some of the elements detailed hereafter are adapted from the book by R.Best [9]. The digital PLL consists of three sub-units : a Phase Detector, a decimation filter and a FIR low pass filter in series and a NCO. The block operates at f s 1 = 20 MHz, with the related time step ∆t s 1 = 1/f s 1 = 50 ns. In the electronics, various FIR low pass filters have been implemented upon the desired sensitivity in the frequency detection, among which a 19 th order filter with a 3 kHz cut-off frequency and a 45 th order filter with a 500 Hz cut-off frequency. Both of them can be used in the simulation.

Phase detector

The PD is analogous to a multiplier regarding the two input signals : the BPF output divided by A rms and the cos waveform coming out of the NCO (cf. fig. 1). Their product is multiplied by a further gain, K d (units : V) converting the dimensionless signal into volts to be operated by the FIR low pass filter. The instantaneous block output is referred to as

K d z e (t i ) : K d z e (t i ) = K d z bpf (t i ) A rms (t i ) cos (ϕ nco (t i )) (14)

Filtering stage

Assume that ω 0 (t i ) and ω nco (t i ) are the instantaneous angular frequencies of the cantilever and of the signal generated by the NCO, respectively. K d z e (t i ) consists of a high frequency component : ω 0 (t i ) + ω nco (t i ) and a low frequency one :

δω(t i ) = ω 0 (t i ) - ω nco (t i ).
The FIR low pass filter cuts off the high frequency component and produces

u f (t i ) ∝ sin {δω(t i )t i } ∝ [δω(t i )] × t i ,
which can be referred to as an error signal of the PLL. Indeed, when the PLL optimally operates, ω nco (t i ) almost perfectly matches ω 0 (t i ).

The instantaneous value u f (t i ) can therefore be interpreted as a correction term in the PLL cycle.

Before being operated by the FIR low pass filter, the signal is processed by the decimation filter. The filter averages K d z e (t i ) over N ds PLL cycles upon the FIR low pass filter cut-off frequency. For instance, N ds = 400 for the 3 kHz low pass filter. The updating rate of the FIR low pass filter is therefore f s 1 /N ds = 50 kHz. The digital data are averaged over those N ds cycles. The average value is fed at the first entry of a buffer B consisting of N fir entries.

The entries of the buffer are then all shifted by one into the buffer. At a given moment in time, t i , u f (t i ) is given by the following algorithm :

         B k = N ds j=k-N ds K d ze(t j )
N ds shift of the buffer entries

u f (t i ) = i k=i-N fir c k × B(t k ) ( 15
)
N fir is the order of the FIR low pass filter (N fir ≪ N ds ) and c k is the k th coefficient of the FIR low pass filter. Once the buffer is transmitted, it is initialized and filled again. Finally,

u f (t i
) is multiplied by a further gain, K 0 , which depicts the linear conversion of the signal from volts to rad.s -1 (units: rad.V -1 .s -1 ) and provided to the NCO.

Numerical Controlled Oscillator

We first assume that the frequency tracker of the PLL is disengaged (switch 2 set to location "a" in fig. 1). Its role is carefully addressed in section III D 5. The NCO adds the instantaneous angular frequency K 0 u f (t i ) to an external input, the center angular frequency of the PLL, ω cent . ω cent is fixed equal to the angular resonance frequency of the free cantilever ω 0 , prior to starting the experiments. The signal is then integrated, which produces the related phase, ϕ nco (t i ), locked to the one of the cantilever :

ϕ nco (t i ) = i k=pll [ω cent + K 0 u f (t k )] ∆t s 1 , (16)
t pll being the moment when the PLL is engaged. Obviously, the PLL has to be engaged once the oscillator has reached its steady state and before the APIC.

Frequency demodulation

When the tip is located far from the surface, ω 0 = ω 0 . Once approached close enough from it, ω 0 starts decreasing. Meanwhile, the NCO produces

ω nco (t i ) = ω cent + K 0 u f (t i ),
as mentioned above. When the frequency tracker is disengaged, ω cent is kept constant and matches the resonance frequency of the free cantilever,

ω cent = ω 0 . Therefore K 0 u f (t i ) is
nothing but the instantaneous frequency shift (actually 2π × ∆f ) of the tip interacting with the surface. In other words :

ω cent + K 0 u f (t i ) -ω 0 = 2π∆f (t i ) ( 17
)
K 0 is a key parameter of the PLL. It sets its capability to get locked to the input signal and in turn it sets its stability. R.Best defines K 0 from the locking range ∆ω l of the PLL, e.g. the frequency gap with respect to the center frequency the PLL can detect remaining locked [START_REF] Best | Phase Locked Loops: Design, Simulation and Applications[END_REF] .

On the hardware level, the control signal u f (t) is limited to a range which is smaller than the supply voltages, usually ±5V. Assuming u fm and u f M be the minimum and maximum values allowed for u f , Best defines K 0 as :

K 0 = 3∆ω l u f M -u fm (18)
Therefore K 0 is related to the maximum frequency shift detectable per volt within the detection range of the low pass filter. Practically, the value of K 0 is not accessible a priori.

It's easier to set the locking range ∆ω l . For an oscillation at f 0 = 150 kHz, frequency shifts of about a few hundreds of hertz are typically expected 40 . We can therefore choose the 3 kHz FIR low pass filter to insure a proper detection of ∆f , which sets the locking range to ∆ω l = 2π × 6000 rad.s -1 . The maximal value of K 0 expected is then ≃ 11000 rad.V -1 .s -1 , which is an excellent estimate as detailed in section V.

Frequency tracker

The frequency tracker is a specific feature of our digital PLL. When engaged (switch 2 set to location "b" in fig. 1), the center frequency is continuously updated by the FIR low pass filter output :

ω cent (t i ) = ω cent (t i-1 ) + K 0 u f (t i ) (19)
The updating frequency is 2.5 kHz. The frequency tracker has been implemented in order to compensate the fact that the frequency demodulation was performed via the lone proportional gain K 0 . Thus, as mentioned before, K 0 u f (t i ) can be interpreted as the error signal produced in the frequency detection compared to ω cent . Consequently, this error is also integrated by the NCO, which leads to an additional phase lag added to ϕ nco at each PLL cycle and previously referred to as ϕ pll . ϕ pll per PLL cycle can approximately be estimated to :

∆ϕ pll = ϕ pll (t i+1 ) -ϕ pll (t i ) ≈ K 0 u f (t i ) × ∆t s 1 (20)
ϕ pll would be zero if no frequency shift occurred, which is the case in most of the applications using PLLs. But while approaching, ∆f decreases continuously, therefore so does ϕ pll . On the contrary, when the frequency tracker is engaged, ω cent is continuously updated. The error in the frequency detection drops to zero. More exactly, it is equal to the difference between two consecutive values of

ω cent : ǫ ≃ ω cent (t i ) -ω cent (t i-1
), but is necessarily small and so is ϕ pll .

To assess how sensitive to frequency changes the phase is, the following experiment is carried out. A 150 kHz sinusoidal waveform is generated by means of a function generator and sent to the real PLL. The frequency is then slowly detuned from -150 Hz up to +150 Hz.

The phase lag between input and output waveforms, ϕ pll , is recorded with a lock-in amplifier (Perkin Elmer 7280) and reported as a function of the detuning. The PLL center frequency is fixed to f cent = 150 kHz. The experiment is repeated the frequency tracker being engaged and disengaged. Two amplitudes of the PLL input waveform are used. In this experiment, the input waveform stands for the oscillatory motion of the cantilever and the tuning for the shift occurring when the tip is approached towards the surface upon attractive or repulsive forces. The results are reported in fig. 2. When the tracker is disengaged, the maximum detuning corresponds to a phase lag of ±80 degrees, which means that the cantilever would then be driven off resonance severely. On the opposite, when engaged, the phase lag reduces (inset) to ±0.05 degree.

This feature has no consequence when the PLL is only used as a frequency demodulator like in the self-excitation mode. On the opposite in the PLL-excitation scheme, this point is crucial since the PLL is produces the excitation signal. Therefore particular attention has to be paid on the way it is produced. If it is abnormally phase shifted, then the oscillation amplitude drops and consequently apparent dissipation is generated, as shown in section V.

E. Block 5: phase shifter

The PS receives the sin and cos waveforms generated by the NCO. A further input to the block is the phase lag, ϕ ps , fixed prior to starting the experiments to make the cantilever oscillating at f 0 . The PS digitally computes :

z ps (t i ) = sin [ϕ nco (t i ) + ϕ pll ] × cos (ϕ ps ) + cos [ϕ nco (t i ) + ϕ pll ] × sin (ϕ ps ) = sin [ϕ nco (t i ) + ϕ pll + ϕ ps ] (21)
When the system is being operated in the PLL-excitation mode, z ps (t) is converted into an analog signal by the DAC1 and multiplied by the APIC output.

F. Block 6: distance controller

The distance controller is the second digital PI controller implemented in the DSP operating at f s 3 . The block gets the setpoint value of the signal (∆f or damping) onto which the control of the tip-sample distance is performed and the proportional and integral gains, K dc p and K dc i , respectively. Here, let's assume that the reference signal is the frequency shift, as depicted in fig. 1. We have arbitrarily chosen not to describe the transfer function of the z-piezo drive. Therefore K dc p and K dc i have natural units (nm.Hz -1 and nm.Hz -1 .s -1 , respectively). The controller is described by :

D(t i ) =D(t dc ) + K dc p [∆f set -∆f (t i )] + i k>dc K dc i [∆f set -∆f (t k )] ∆t s 3 , (22)
D(t dc ) being the tip-surface distance when the DPIC is engaged.

G. Lock-in amplifier

The description of the lock-in amplifier implemented in the simulation does not depict the detailed operational mode of the real lock-in which is used to monitor the phase shift of the oscillator (Perkin Elmer 7280). Its purpose is to provide an easy way to estimate the phase shift between the excitation and the oscillation. The calculation of the phase is performed at 2.5 kHz. The buffer used to extract the phase therefore consists of n lock-in = f s 1 /2.5 kHz = 8000 samples. The numerical code used to describe it is :

tan(ϕ(t i )) = i k=i-n lock-in z bpf (t k ) × sin[ϕ nco (t k ) + ϕ pll + ϕ ps ] i k=i-n lock-in z bpf (t k ) × cos[ϕ nco (t k ) + ϕ pll + ϕ ps ] (23)
H. Code implementation

The numerical code has been implemented with LabView This is true for a free cantilever, but the contribution of the thermal noise is expected to be even more pronounced when the tip is close to the surface. Then, the fluctuations of the interaction force δF int have a strong influence on the nonlinear dynamics of the cantilever, in particular when the tip is at distances involving short-range forces where the nonlinearity is more pronounced.

The instrumental noise (cf. Ch.2 in refs. [1] and [33]), essentially due to electronic components, is not considered and we further assume that the electronic blocks (RMS-to-DC, PI controllers, PLL) operate perfectly. Doing so, δE d is under-estimated but the framework of this section is to provide a ground value to be compared to the values obtained with the simulation.

A. Connection between δE d and δF int

The fluctuation of the dissipated energy per cycle can be connected to the fluctuation of damping δΓ, via equ.7 :

δE d = πk c A 2 0 δΓ ω 0 (24)
Besides, because

A exc = A 0 /Q = A 0 Γ 0 /ω 0 = F exc /k c
on resonance and because the tipsample interaction force F int can be treated, to first order 41 , on the same level as F exc , a fluctuation of F int should produce, a fluctuation of damping :

δΓ ω 0 = δA exc A 0 = δF int k c A 0 , (25)
Consequently :

δE d = πA 0 δF int (26)

B. Estimate of δF int

For large oscillation amplitudes (that is larger than the minimum tip-surface distance, a few angströms), F int is connected to the so-called normalized frequency shift 42 ,

γ ≡ ∆f k c A 3/2
0 /f 0 , via the equation 43 (cf. also Ch.16 in ref. [1]) :

γ(r) ≃ 0.43 V int (r)F int (r), (27)
where V int (r) and F int (r) are the interaction potential and force, respectively, between the tip and the sample at a location r. The fluctuation in the relative frequency shift δ∆f /f 0 = δf /f 0 , that is the cantilever frequency noise, due to a fluctuation of F int is then given by :

δf f 0 ≃ 0.43 2k c A 3/2 0 V int (r) F int (r) δF int (28)
δf f 0 = 2k B T B π 3 k c A 2 0 f 0 Q (29)
Therefore, the dissipated energy due to thermal fluctuations of the cantilever close to the surface can be estimated to :

δE d ≃ 4.6 2k B T Bk c A 3 0 F int (r) πf 0 QV int (r) (30)
The measurement bandwidth B can be estimated out of the following considerations. As mentioned by F.J. Giessibl (cf. Ch.2 in ref. [1]), B is a function of the scan speed v s and the distance a 0 between the features which need to be resolved :

B = v s a 0 (31)
For UHV investigations, a 0 is of about one atomic lattice constant, that is a few angströms. corresponds to 5% of the intrinsic dissipated energy of the free cantilever. This corresponds to about 150 meV/cycle with typical conditions for UHV investigations carried out at room temperature (cf. equ.7 and discussion below). Besides, as mentioned before, this value is underestimated. A straightforward consequence is that the strength of apparent dissipation should overcome this limit to be relevant. With a moderate quality factor in the Van der Waals regime like in high vacuum for instance, the limit drops by almost a factor 3 (1.8%).

Thus, apparent dissipation effects might occur more easily under these conditions 46 . At low temperatures, in the short-range regime the ratio is 2.67%. However, this value is likely still too high because then, the thermal drift being drastically reduced, the measurement bandwidth can be lowered and apparent dissipation more likely to be measured. and relative damping, respectively are reported vs. the tip-surface separation. Phase and amplitude remain almost constant while approaching, within, however, deviations limited to 0.3% compared to -90 degrees and A set 0 = 7 nm, respectively. In the repulsive part of the potential, steep phase changes occur, but the amplitude does not dramatically drops, at least up to ∆f = +100 Hz. Consequently, the relative damping remains constant.

B. Numerical vs. real setups

PLL dynamics

The dynamic behaviors of real and simulated PLLs have then been compared. The experiment consists in locking the PLL onto a 150 kHz sinusoidal waveform according to the same procedure than in section III D 5. The 3 kHz FIR low pass filter is used. At a certain moment, a frequency step of +10 Hz is applied to the center frequency, resulting in a shift of -10 Hz (ω 0 + 2π∆f = ω cent ). The step response is recorded for various values of the so-called loop gain (real PLL) and various values of K 0 K d (simulation). The variations of ∆f vs. time are fitted with simple decaying exponential functions, the characteristic time of which stands for the locking time of the PLL. The results are reported in figs.4(a) and (b). A rather long locking time is noticed for low values of the gains whereas the PLLs lock faster when the gains become larger. For the latter case, the PLLs can operate up to the limit of the locking range as shown by the oscillations.

The locking time deduced from each fit is plotted as a function of the gains of both PLLs.

In order to make the curves comparable, the loop gain must be rescaled by an arbitrary constant which depends on the electronics. The best agreement between the curves was achieved with 91 × 10 3 (cf. fig. 5). A single master curve of the PLL dynamics can thus be extracted. The rather good agreement between the two curves provides evidence that the simulation reasonably describes the real component, at least within the locking range.

For values of the gains up to K 0 K d = 6000 rad.s -1 , the PLL is stable and able to track frequency changes within the locking range around 150 kHz. Above 6000 rad.s -1 , the PLL introduces overshoot in the output waveform while attempting to lock the input signal.

For higher gains, the PLL is not able to properly track the input signal, even though its frequency is within the locking range. The border is reached for K 0 > 10 4 rad.V -1 .s -1 , in good agreement with the value expected from R.Best's criterion (cf. discussion in section III D 4). The arrow in fig. 5 indicates the usual loop gain value which is chosen to perform the experiments using the 3 kHz low pass filter, corresponding to K 0 K d = 5000 rad.s -1 . The related locking time of the PLL is then ≃ 0.35 ms. For those values of gains, the locking range is about ±400 Hz.

APIC dynamics

In order to extract a typical time constant of the component, similar experiments have been carried out with the APICs. The cantilever remaining far from the surface, a step is applied to the setpoint amplitude A set 0 resulting in an abrupt change in A rms upon gains. The results are reported in fig. 6(a, real setup) and (b, simulated setup). The curves exhibit over-(no overshoot at all) under-(oscillating behavior) or critically damped (single overshoot) behaviors upon chosen gains. So as to extract the APIC response time, we focus at curves which exhibit a single time constant, that is curves for which a weak overcritically or a critically damped response is observed (cf. insets in figs. 6(a) and (b)). This is motivated by the controller response which is then the fastest, while preserving an overall stable behavior. The changes in A rms are fitted with decaying exponentials functions and the related characteristic time is extracted. The variation of the so-called response time of the controller (t resp ) vs. gains is reported in fig. 7. The restriction to curves exhibiting a single time constant is similar to restricting the analysis to a single gain of the controller 48 . Thus, a single master curve which describes the dynamics of both APICs can be extracted as well. In fig. 7, the K p gain of the real controller has been rescaled to make it matching K ac p (the best rescaling factor is 1/40000). t resp decreases as K ac p increases (being given a single K ac i per K ac p ). However, the controller is limited to an optimum t resp of about 2 ms as shown by the plateau reached for K ac p ≃ 10 -3 [49] (arrow in fig. 7). So far, the origin of the saturation remains unclear. Nevertheless, a brief analysis of the response function of the controller to a step wherein the contribution of the RMS-to-DC converter is neglected (cf. appendix, section B) emphasizes that the dynamic behavior can reasonably be predicted (triangles in fig. 7) up to 2 ms. The best agreement between the experimental results and the model is found when considering the weak overcritically damped regime, namely :

t resp ≃ 1 c + c 2 -ω 0 2 K 1 K 3 K ac i , (32)
with :

c = ω 0 4 1 Q + K 1 K 3 K ac p ( 33
)
The origin of the saturation might thus be attributed to the contribution of the RMS-to-DC converter.

As expected, the shortest APIC response time is approximately 6 times longer than the optimal PLL locking time, ≃ 0.35 ms. Thus the PLL should track frequency changes much faster than amplitude changes. Therefore, with PLL gains insuring a locking time much shorter than the APIC one, the two blocks can be considered as operating separately. Then, no amplitude changes which would be the consequence of a bad tracking of the resonance frequency can occur.

It might be objected that the experiments and the analysis, despite consistent, have been performed without considering the tip-sample interaction. Regarding the PLL, the way the dynamics is affected when the tip is close to the surface has not yet been investigated. But regarding the amplitude controller, Couturier et al. [8] have reported a theoretical analysis of the controller stability upon the gains and the strength of the non-linear interaction in the self-excitation scheme. The analysis stresses that the stability domain of the controller shrinks when the contribution of the non-linear interaction (pure Van der Waals) increases.

Thus, a couple (K ac p ;K ac i ) initially inside the stability domain might correspond to an unstable behavior of the controller close to the surface, thus introducing apparent dissipation [START_REF] Couturier | Noncontact atomic force microscopy: stability criterion and dynamical responses of the shift of frequency and damping signal[END_REF] .

Nevertheless, considering their parameters with a tip-surface distance ranging from infinity down to 0.8 Å (corresponding to ∆f -250 Hz), that is very close to the surface for operating in nc-AFM 50 , the stability domain weakly shrinks 51 . A similar analysis for the PLL-excitation scheme is still lacking and should be performed for quantitative comparison and discussion. But comparing their analysis to the tip-surface distances and frequency shifts which are being used in this work, we believe that the contribution of the non-linear interaction to the APIC dynamics remains weak and thus would not change drastically its coupling to the PLL. This point is strengthened by the results given in the following section (V C 1).

C. Apparent dissipation

Contribution of the PLL gains

Section V B 1 has proved that the PLL gains were controlling the PLL locking time.

Within the locking range, the higher K 0 K d , the faster the PLL. The test performed here is to compute approach curves for various values of K 0 K d . Except K 0 K d , the parameters are similar to those given in fig. 3. In particular, Q = 30000, K ac p = 10 -3 and K ac i = 10 -4 s -1 , corresponding to t resp = 2 ms. 4 sets of K 0 K d have been used, namely : 11000, 5000, 1000 and 100 rad.s -1 , corresponding to locking times of ≃ 0.2 ms, 0.35 ms, 1.8 ms and > 4 ms, respectively. Note that the two later values are almost similar or larger, respectively, than t resp . The 3 kHz FIR low pass filter has been used and the frequency tracker has been engaged. The results are reported in fig. 8.

With the four sets of data, no effect on the frequency shift is observed. For K 0 K d = 11000 and 5000 rad.s -1 , changes in phase, amplitude and damping are noticeably similar. The phase and the amplitude remain constant and subsequently, no damping occurs. On the opposite, for K 0 K d = 1000 and 100 rad.s -1 , that is for a PLL locking time of about or larger than the APIC one, the changes are more pronounced. With K 0 K d = 1000 rad.s -1 (set 3), the phase strongly varies along the repulsive part of the interaction, from -93 to -79 degrees.

Consequently, the amplitude starts dropping and the damping increases. This trend is more pronounced with K 0 K d = 100 rad.s -1 , for which the phase in the repulsive regime reaches -60 degrees, requiring the APIC to produce ≃ 14% more excitation. According to the discussion put forward in section IV, such an effect is expected to be detected if it would occur.

Therefore, if the PLL does not lock the incoming signal fast enough, that is for locking times of about or larger than 1 ms, corresponding approximately to t resp /2, an undesirable phase shifted signal is produced, resulting in an amplitude decrease and producing significant apparent dissipation.

It's peculiar to notice that, when the cantilever is driven out of resonance as shown with the phase changes with K 0 K d = 100 rad.s -1 , no abnormal frequency shift occurs.

As a matter of fact, close to the resonance, the phase changes of the free cantilever scale as, to first order :

ϕ u→1 = π/2 -2Q(u -1)
, where u = f /f 0 . Considering f 0 = 150 kHz and δϕ = ±30 degrees, then δf = f 0 δϕ/2Q = ±1.3 Hz, which is not visible in fig. 8(a).

Similar effects have been reported by H.Hölscher et al. [52]

. This effect should be more (less) pronounced with low (high) Q values (±8 Hz or ±0.08 Hz with Q = 5000 or 500000, respectively) and the apparent dissipation higher (lower).

Contribution of the frequency tracker

As mentioned in paragraph III D 5, the frequency tracker updates ω cent with the goal to prevent the phase due to the frequency shift, ϕ pll , be added to the NCO output. Figure 9 reports two approach curves computed upon the frequency tracker is engaged or not.

When it is disengaged, the phase continuously decreases along the attractive part of the interaction potential, as expected from equ.20. At a tip-sample separation corresponding to the minimum of the interaction potential, r = 2.35 Å, the phase reaches -120 degrees, meaning that the oscillator is then seriously driven out of resonance. Following the phase change, the amplitude continuously decreases and the damping strives to compensate the amplitude reduction, thus reaching 15% of the intrinsic damping of the free oscillator. Here again, such an effect should be measurable. When the tip is further approached towards the surface, the repulsive regime makes the frequency shift increasing and so does ϕ pll . The amplitude increases back to reach A set 0 and the damping is obviously reduced. An amplitude growth is not expected in the repulsive region of the interaction potential, but it is the consequence of the bad tracking of the resonance frequency. On the other hand, when the tracker is engaged, as already mentioned, the phase remains constant and no apparent dissipation occurs.

D. Scan lines

In addition to the analysis of the time constant of the various blocks, it is important to focus at variables changes when the tip is scanned along a surface. Two types of surfaces have been investigated : 1-a sinusoidally corrugated surface with a spatial wavelength of 6.6 Å and a corrugation of ±0.1 nm, consistently with the lattice constant of KBr, a sample regularly used in the group, and 2-a surface with two opposite steps with a step height of 3.3 Å. The steps are built out of arctan functions and spread out laterally over 5 Å. The upper terrace spreads out over 3 nm (cf. insets in fig. 12).

The results shown here have all been obtained by ∆f regulation. The scan lines have been initiated from the approach curve shown in fig. 3 with ∆f set = -60 Hz, corresponding to an initial tip-surface separation of about 5 Å, that is in the short-range regime (cf. fig. 3(a)). The gains of the distance controller have been chosen in order to insure a critically damped response of the controller to a step of -1 Hz when ∆f set is reached, namely K dc p = 2 × 10 -3 nm.Hz -1 and K dc i = 2 nm.Hz -1 .s -1 . For all of the following curves, the frequency tracker has been engaged. Three sets of parameters have been varied : the PLL gains, the scan speed and the APIC gains.

Contribution of the PLL gains

Paragraph V B 1 has proven how K 0 and K d gains were controlling the PLL locking time. Figure 10 shows scan lines computed for three values of K 0 K d , namely : 100, 1000 and 5000 rad.s -1 , corresponding to locking times > 4, 1.8 and 0.35 ms, respectively. The unchanged parameters are : scan speed = 7 nm.s -1 , K ac p = 10 -3 and K ac i = 10 -4 s -1 . The latter gains correspond to t resp ≃ 2 ms (arrow in fig. 7). Each signal, namely topography, ∆f , ϕ, A rms and relative damping, consists of 256 samples. The topography signal follows accurately the surface corrugation. No contribution due to the gains is revealed. ∆f is modulated around ∆f set = -60 Hz, with an amplitude ranging from ±2 to ±3 Hz upon K 0 K d . The accuracy of the distance control is then of about 97%. Note also that the nonlinear interaction makes the modulation asymmetric around ∆f set and the maxima are mismatched compared to the maxima of the surface. However, the mismatch does not depend on K 0 K d . If the PLL is slow, a rather important phase lag is observed, ranging from -65 to -110 degrees, resulting in small amplitude changes. Here also, the asymmetry around -90 degrees is manifest and it's interesting, despite expected, to notice the doubling in the periodicity of the amplitude fluctuation. The related relative apparent damping fluctuates accordingly, reaching about 12%Γ 0 , which should be experimentally detectable.

Contribution of the scan speed

Five scan speeds ranging from 1 to 20 nm.s -1 have been used, accordingly to typical experimental values for such a scan size. The unchanged parameters are : K 0 K d = 5000 rad.s -1 , K ac p = 10 -3 and K ac i = 10 -4 s -1 . The surface with opposite steps has been used and each signal consists of 256 samples. The results are given in fig. 11. At high speed, the topography channel starts being distorted as a consequence of a bad distance regulation as shown by the large ∆f variations. The phase varies accordingly, but within a narrower domain. Therefore neither relevant variations of amplitude nor of relative damping (±2% only) are revealed.

Contribution of the APIC gains

Similar experiments have been carried out by varying the APIC gains. Thirteen sets of K ac p and K ac i values have been used over 2 orders of magnitude for each gain. The unchanged parameters are : scan speed = 5 nm.s -1 and K 0 K d = 5000 rad.s -1 . The surface with opposite steps has been used. For those curves, each signal consists of 1024 samples. The sets of gains have been chosen such that the response time of the controller is varied from 2 to 20 ms, according to fig. 7. The results are reported in fig. 12. Here again, the topography signal accurately follows the corrugation. In particular, no unwanted overshoot is observed at the step edge despite ∆f varies significantly. A small phase variation is observed at the step edge. For the latter three channels, no dependence is observed upon the gains such that the curves match each other. The small phase variation induces a tiny amplitude change, barely visible in the inset of fig. 12(d), but the overall fluctuations are weak, which corresponds to relative damping fluctuations of ±3% (worse case, t resp ≈ 20 ms). Following discussion of section IV, this value is below the threshold limit of thermal noise, at least for experiments carried out in UHV and at room temperature. Upon gains, a small spatial shift is observed in the amplitude or in the relative damping signals up to a maximum value of 0.1 Å.

VI. DISCUSSION

The above analysis stresses five important results :

• The PLL dynamics plays a major role in the occurrence of relevant apparent damping if the locking time is about or larger than 1 ms, that is only twice faster than the APIC optimum response time. By "relevant damping", we mean, on the base of the discussion given in section IV, a damping which would be detectable experimentally upon operating conditions (cf. table I).

• The frequency tracker, the aim of which is to update the PLL center frequency to make it matching the actual resonance frequency, plays also a major role in the occurrence of apparent damping. It has to be mandatorily engaged when performing approachcurves, otherwise unwanted additional phase shift due to the PLL occurs and the cantilever is then driven off resonance.

• The PLL optimal locking time is about 0.35 ms that is 6 times shorter than the shortest APIC response time of the free cantilever. Therefore the resonance condition is expected to be always properly maintained. Consequently, when the PLL operates properly, no amplitude changes due to a bad tracking of the resonance frequency are expected to occur. If they would, this should rather be the consequence of the APIC and/or the DPIC dynamics.

• The APIC response time seems to be limited to t resp ≃ 2 ms due to the RMS-to-DC converter. There is a priori no fundamental restriction to the APIC response time, as shown by fig. 7. However, it must be stressed that if the APIC is made faster, the PLL should be made faster accordingly.

• A weak contribution of the APIC to apparent dissipation is observed. Although spatial shift and apparent dissipation can conditionally be generated, the overall strength of the effect remains weak and should hardly be measurable for UHV investigations at room temperature.

In order to compare our results to other works, the contribution of the noise (equ.30) to the dissipation has been estimated with the parameters given by Gauthier et al. [7].

The maximum of relative apparent damping they report is about 5% (cf. fig. 4 2.3% at a distance 53 s = 4 Å. Therefore, due to the use of a low amplitude [START_REF]The amplitude dependence in equation 30 implies that δE d ([END_REF] , the strength of the effect would not be balanced by the noise and could be detectable experimentally.

On the opposite, considering smaller scan speeds, the maximum of apparent dissipation decreases down to 2%. Such effects become then unlikely to be observed experimentally.

Finally, let's note that the contribution of the third controller, the DPIC, has not been assessed in this work. Nevertheless, as mentioned by H. Hug and A. Baratoff [31], in order to minimize feedback errors and resulting image distortions, the time constant of the distance controller must be shorter than the speed in the fast scanning direction / lateral extent of the smallest feature to be resolved and necessarily (much) larger than other time constants (RMS-to-DC, APIC, PLL). If those conditions are satisfied, then its contribution to the overall stability of the setup should be weak. This is what is readily seen in figs.10, 12 and, to a certain extend in fig. 11, with reasonable speeds. This is also confirmed by Couturier et al. who have recently found out that the distance controller had no effect on the stability diagram in the self-excitation scheme and that only the amplitude controller played a role [START_REF] Couturier | Contribution of the instrumentation to the damping signal in nc-AFM[END_REF][START_REF] Couturier | Damping and instability in non-contact atomic force microscopy: the contribution of the instrument[END_REF] .

To summarize, this analysis has emphasized that a maximum of about 15% of apparent dissipation, mainly due to the PLL and not to the APIC, could be generated (cf. fig. 10).

The contribution of the APIC, within the range of gains used, is systematically smaller. We finally infer that, with our setup (UHV and room temperature) and under typical experimen-tal conditions (scanning speed, PLL and APIC gains...) corresponding to E d 0 ≃ 2 eV/cycle, apparent dissipation in the range of a few eV per cycle, as frequently reported in the literature, is unlikely to occur. A striking example is provided by R. Hoffmann et al. [18] who put in evidence a dissipation of 3 eV/cycle on a NiO(001) sample, despite operating at 4 K [57]. This strongly suggests the interpretation of the experimental damping images in terms of "physical" dissipation.

VII. CONCLUSION

The realization and successful testing of a modular nc-AFM simulator implemented with LabView TM is reported. The design is based on a real electronics which includes a digital PLL, the output of which is used to detect the frequency shift but also to generate the time dependent phase of the excitation signal. Good agreement is obtained between the locking behavior of the real PLL and the PLL from the simulator. The optimum locking time of the PLL is found to be about 0.35 ms. The behavior of the amplitude controller is also found to correctly describe the real setup with an optimum response of 2 ms. The analysis of the time constants of the former two components provides evidence that the electronics tracks properly the cantilever dynamics if the PLL runs more than twice faster than the amplitude controller. When the system is operated with properly chosen parameters, frequency shift vs. distance curves successfully compare to an analytic expression which ignores the finite response of the electronics. No phase deviation resulting in apparent dissipation occur if the center frequency of the PLL tracks the resonance frequency shifted by the tip-sample interaction. An estimate of the minimum dissipation expected to be detected experimentally gives some insights on the relevance of apparent dissipation which can conditionally be generated numerically. This provides a framework to discuss the overall contribution of apparent dissipation during experiments. Computations of scan lines show that when the system is operated with experimentally relevant parameters, the contribution of the proportional and integral gains of the amplitude controller and the scan speed (up to 20 nm.s -1 ) do not lead to significant apparent dissipation. To give orders of magnitude, the worse situation (frequency tracker disengaged) leads to a maximum of 15% more dissipation than the intrinsic dissipation of the free cantilever. This is below the values which are experimentally reported. This strongly suggests that nc-AFM damping images mainly reflect physical channels of dissipation and not electronics artifacts. Besides Ref. [START_REF] Couturier | Damping and instability in non-contact atomic force microscopy: the contribution of the instrument[END_REF] , two publications dealing with the implementation and/or performance of "virtual force microscopes" have appeared recently. These simulation codes are analogous to ours but differ in detail. Kokavecz et al. [START_REF] Kokavecz | Mechler Novel amplitude and frequency demodulation algorithm for a virtual dynamic atomic force microscope[END_REF] proposed and tested a numerical scheme designed to produce response times of the whole simulated setup, as well as of separate blocks (amplitude, phase and distance controllers) as short as 0.1 ms. Trevethan et al. [START_REF] Trevethan | Modelling atomic scale manipulation with the non-contact atomic force microscope[END_REF] used the scheme described in Ref. [START_REF] Polesel-Maris | A virtual dynamic atomic force microscope for image calculations[END_REF] to compute fingerprint-like responses in the frequency shift, the minimum tip-sample distance and the damping signal caused by an atomic-scale configuration change at the surface of the sample. This change was first predicted from atomistic simulations and then induced upon approach under distance control down to a judiciously chosen frequency setpoint. A manual describing the combined atomistic simulation 30 and virtual force microscope codes is now available online 62 .

Fc = k c A ∆f f 0 - f + 2f 0 Q Fd kcA 2f 0 Q tan(ϕ) (39) Fd = k c Af 2f 0   K 3 A exc A 1 + ∆f f 0 (1 + 1/ tan(ϕ)) - 1 Q   (40)
Equation 39 illustrates that, if the phase is properly maintained at -π/2 rad, the frequency shift is essentially due to the conservative interaction, whatever the dissipative contribution.

Besides, equ.39 gives the well known expression of the frequency shift due to the conservative contribution [START_REF] Giessibl | Physical interpretation of frequency-modulation atomic force microscopy[END_REF] :

∆f f 0 = Fc k c A (41)
The expression of Fc related to the Morse potential given in equ.9 can be integrated analytically :

S Morse int = 2π ω 0 V Morse int (t)dt = - 2πU 0 ω 2Υ 0,1/λ -Υ 0,2/λ , (42) where :
Υ α,β = e -β(D-rc) × BesselI(α, βA)

BesselI(α, βA) is the modified Bessel function of first kind for the parameters α and βA. As shown by equ.37, the method not only requires to estimate the action related to S Morse int , but also its derivative with respect to the amplitude A :

∂ A S Morse int = - 4πU 0 ωλ Υ 1,1/λ -Υ 1,2/λ (44)
Regarding the Van der Waals contribution, it was demonstrated 58 :

∂ A S VdW int = - πHRA 3ω (D 2 -A 2 ) 3/2 (45)
The frequency shift due to both contributions can finally be deduced :

∆f f 0 = - 1 k c A HRA 6 (D 2 -A 2 ) 3/2 + 2U 0 λ Υ 1,1/λ -Υ 1,2/λ (46)
with :

   Kac p = K 1 K 3 K ac p Kac i = K 1 K 3 K ac i (52)
The proportional and integral gains are scaled by the transfer functions of the piezoelectric actuator and of the photodiodes, K 3 and K 1 , respectively.

Analogy

Equation 51 has two poles :

s 1,2 = -c ± c 2 -b Kac i , ( 53
)
where c is the parameter given in equation 33, which can also be written :

c = a + b Kac p 2 (54)
Equation 51 is thus almost analog to a standard 2 nd order system, the canonical form of which can be written :

G(s) = ω 2 n s 2 + 2ζω n s + ω 2 n , (55)
where ω n and ζ are the characteristic frequency and damping factor of the system, respectively. Thus :

ζ = c b Kac i (56)
Now, it's well known that the position of ζ with respect to 1 defines the overall behavior of

with ξ ′ = b Kac i -c 2 .

Summary

The transition from the over to the under critically damped regime is thus controlled by the parameter c, that is the proportional gain, and occurs when the condition c = b Kac i is fulfilled. In the critically damped regime, the relationship between the two gains is :

Kac i = ω 0 8 1 Q + Kac p 2 (62)
The time constant of the system can be extracted upon the regime and the time scale that are considered. For short time scales, the response time of the controller is typically 1/c in the critically damped regime and t resp = 1/(c + c 2 -b Kac i ) in the overcritically damped regime (cf. equ.59), which is given in equation 32. Figure 7 shows that the response times of the real system and of the simulated machine reasonably match t resp while the RMS-to-DC converter has a negligible influence on the system dynamics. not engaged, the phase continuously drifts during the approach due to the increase of the attractive interaction meaning that the oscillator is not driven on resonance. Subsequently, the amplitude drops and the APIC strives to keep it constant by increasing the excitation. 15% more excitation is thus produced, that is above thermal noise (cf. section IV).

  TM 6.1, supplied by National Instruments TM . It consists of a user interface where all the parameters are tunable at runtime, like during a real experiment. The couple of integro-differential equations 2, 3, 10, 11, 12, 13, 14, 15, 16, 21 and 22 are integrated at their respective sampling frequencies. The monitored signals are the oscillation amplitude A rms (equ.11), the frequency shift ∆f (equ.17), the phase ϕ (equ.23) and the relative damping Γ/Γ 0 -1=QK 3 A exc /A 0 -1, deduced from the APIC output (equ.4). The connection to the dissipated energy per cycle E d is given by equation 7, that is Γ/Γ 0 -1 = E d /E d 0 -1.IV. APPARENT DISSIPATION VS. MINIMUM DISSIPATIONAddressing the question of apparent dissipation requires to estimate the minimum dissipation which is detectable by the instrument upon operating conditions. Beyond the specificities of the PLL-or self-excitation modes, important parameters like quality factor Q, temperature and bandwidth of the measurement must be considered.We here focus on the minimum dissipated energy, δE d , due to thermal fluctuations of the cantilever when it oscillates close to a surface. Thermal driving forces are connected to the energy dissipation by the Q factor of the cantilever. The thermal kicks introduce fluctuations of amplitude and phase and therefore fluctuations of the energy dissipation.

  At room temperature, due to thermal drift, atomic scale images are usually recorded at scan speeds of about 6 lines (3 forwards plus 3 backwards) per second. Let's consider for instance a moderate resolution of 6 pixels per atomic period. Then, a line consisting of 256 pixels should be acquired with a bandwidth B = 6 × 256/6 = 256 Hz.

  fig.3(a)), r = 5 Å.

  of the numerical setup Frequency shift vs. distance curves obtained from the simulation have first been compared to the analytic expression of ∆f due to Van der Waals and Morse potentials (cf. appendix, section A). The results are shown in fig.3(a). The parameters chosen to perform the simulation are consistent with typical parameters used during experiments performed in UHV. The parameters of the interaction potential have been taken from ref.[47]. They are representative of the interaction between a silicon tip and a silicon(111) facet. An excellent agreement is observed between numerical and analytic curves along the attractive and repulsive parts of the interaction potential, thus validating the numerical scheme. The parameters used to perform the calculation are given in the caption. Let's also notice that the frequency tracker was engaged. In figs. 3(b), (c) and (d), the variations of ϕ, A rms

  in the above reference, curve ♯6), corresponding to a scanning speed of about 90 Å.s -1 . The spatial wavelength of their surface model being 8 Å, the bandwidth of the corresponding measurement is B ≃ 11 Hz (cf. equ.31). The non-linear interaction is depicted by a Rydberg function with U 0 = 4 × 10 -11 J, λ = 0.599 Å and r c = 3 Å. With A 0 = 1.5 nm, f 0 = 159154 Hz, k c = 26 N.m -1 , Q = 24000 (E d 0 = 4.8 × 10 -2 eV/cycle), we get δE d /E d 0 ≃

3 . 1 s ( 58 )• 60 )•

 315860 Undercritically damped regime⇒ ζ < 1 ⇔ c < b Kac i Critically damped regime ⇒ ζ = 1 ⇔ c = b Kac i Overcritically damped regime ⇒ ζ > 1 ⇔ c > b Kac i Analysis to a step responseTo assess how fast the controller reacts, we investigate the response of the controller to a step in amplitude upon it is in the over-, under-or critically damped regime. Let's assume a step of amplitude A s , the corresponding transfer function is G s (s) = A s /s and the transfer function of the closed loop system to which the step is applied is thereforeG cls (s) = G s (s) × G cl (s), that is : G cls (s) = A s b Kac p (s + c) 2 -c 2 + b Kac i + A s b Kac i (s + c) 2 -c 2 + b Kac i ×The time-dependent solutions g cls (t) = L -1 {G cls (s)} are :• Overcritically damped regime : c > b Kac i g cls (t) = A s 1 + ς -e -(c+ξ)t -ς + e -(c-ξ)t(59) with ς ± = (c ± ξ -b Kac p )/2ξ and ξ = c 2 -b Kac i . Critically damped regime : c = b Kac i g cls (t) = A s 1 -e -ct + b Kac p -c te -ct (Undercritically damped regime : c < b Kac i g cls (t) = A s 1 -e -ct × cos (ξ ′ t) + c -b Kac p ξ ′ sin (ξ ′ t)

FiguresFIG. 1 :

 1 Figures

FIG. 3 :FIG. 9 :

 39 FIG.3: Numerical approach curves. The parameters are A set 0 = 7 nm, f 0 = 150 kHz, k c = 30 N.m -1 , Q = 30000, therefore Γ 0 = 31.4 s -1 and E d 0 = 0.96 eV/cycle, K 1 = 0.1 V.nm -1 ,K 3 = 1/K 1 nm.V -1 , K d = 1 V, K 0 = 5000 rad.V -1 .s -1 , K ac p = 10 -3 , K ac i = 10 -4 s -1, approach speed 2 nm.s -1 . The parameters of the interaction potential have been taken from ref.[47] :H = 1.865 × 10 -19 J, R = 5 nm, U 0 = 3.641 × 10 -19 J, λ = 1.2 Å,and r c = 2.357 Å. Except in (a), the signals are monitored at 10 kHz. (a)-Comparison between the ∆f computed numerically (open circles) and the analytic expression of ∆f (thick grey line) due to Van der Waals and Morse interactions (equ.46). The two curves match accurately along the attractive and repulsive parts of the interaction potential. For clarity reasons, 10 times less samples are displayed compared to plots shown in (b), (c) and (d). The dotted line depicts the analytic ∆f due to a pure Van der Waals potential, thus showing where the short-and long-range interaction regimes are discernable. (b)-Phase lag, ϕ. The frequency tracker being engaged, the phase remains constant and equal to -90 degrees within deviations limited to 0.3%, thus maintaining the cantilever driven on resonance.(c)-Amplitude A rms (t)/K 1 . Since no phase variation occur, the amplitude remains constant as well throughout the approach. (d)-Relative damping.

  C. Estimate of δf /f 0 Y. Martin et al. [44], T.R. Albrecht et al. [4], H. Dürig et al. [45] and F.J. Giessibl (Ch.2 in ref.[1]) have calculated the thermal limit of the frequency noise in frequency-modulation technique over a measurement bandwidth B. It is given by :

Table I

 I gives some estimates of the relative dissipated energy due to thermal fluctuations of the cantilever δE d /E d 0 close to the surface in the short-range or pure Van der Waals

	Q	Interaction	E d 0	δE d	δE d /E d 0
		regime	(eV/cycle)	(eV/cycle)
	5000 (298 • K) VdW + short-range 7.69	0.177	2.3%
		VdW only	7.69	0.141	1.8%
	30000 (298 • K) VdW + short-range 1.28 7.25 × 10 -2 5.7%
		VdW only	1.28 5.78 × 10 -2 4.5%
	500000 (4 • K) VdW + short-range 0.077 2.06 × 10 -3 2.7%
		VdW only	0.077 1.64 × 10 -3 2.1%

regimes at various temperatures and for various quality factors. In UHV at room temperature, our experimental conditions, the minimum dissipated energy which is detectable

TABLE I :

 I Dissipated energy of the free cantilever E d 0 (equ.7

This value corresponds to the tip-surface separation shown in fig.3in the above reference.

Acknowledgments

The authors acknowledge the Swiss National Center of Competence in Research on Nanoscale Science and the National Science Foundation for financial support. They are grateful to R. Bennewitz (Mac Gill Univ., Canada), C. Loppacher (Dresden Univ., Germany), to colleagues of the electronics workshop of the University of Basel for discussions and advices, in particular Ch. Wehrle H.-R. Hidber and M. Steinacher, to S. Gauthier, J. Polesel-Maris and X. Bouju (CEMES, France) for stimulating discussions on noise, H. Hug, N. Pilet and T. Ashworth (Basel Univ.) for discussions on the frequency tracker and to G. Couturier for having provided a copy of his poster presented at the 2004 nc-AFM conference (Seattle, USA).

k = 40 N.m -1 in UHV (Q = 30000) and at room temperature. 41 To first order, F int = k ef f z(t), where k ef f = k c + dF int /dz(t).

42 F.J. Giessibl. Forces and frequency shifts in atomic-resolution dynamic-force microscopy. Phys.

Rev. B, 56 (24):16010-16015, 1997. 45 U. Dürig, H.R. Steinauer, and N. Blanc. Dynamic force microscopy by means of the phasecontrolled oscillator method. J. Appl. Phys., 82:3641-3651, 1997. 46 However, in the Van der Waals regime, the tip is relatively distant from the surface. No atomic feature is expected to be observed. The bandwidth can therefore be made bigger accordingly. 48 In the critically damped regime, it is known that K p and K i are not independent, but coupled (cf. appendix, section B). 49 The scaling of the gains K ac p and K ac i essentially depends on the transfer functions of the piezo actuator K 3 and of the photodiodes K 1 as detailed in appendix, section B (equation 52).

Therefore, K ac p can be chosen smaller than 1 considering K 1 K 3 = 1 (cf. caption of fig. 3). 50 On the base of the work by R. Pérez with the Si(111)5 × 5 (ref. [47]), S.-H. Ke (Ch.16 in ref. [1]), state that even at tip-surface distances of ≈ 5 Å, there is an onset of covalent chemical bonding between the dangling bonds on the tip and the surface. 51 In fig. 3 of ref. [8], the stability domain is an K ac i vs. K ac p chart that can qualitatively be described by a parabola. For a tip-surface distance ranging from infinity down to 0.8 Å, the (K ac p , K ac i )

gains corresponding to the maximum of the parabola range from (61, 5.1 × 10 4 ) down to only (55, 4.5 × 10 4 ). The analytical method gives analytical, tractable expressions of the frequency shift versus distance upon the force expression. Couple of years ago, F.J. Giessibl and M.Guggisberg have proposed approached expressions for the frequency induced by a Morse potential [START_REF] Guggisberg | Separation of interactions by non-contact force microscopy[END_REF]42 .

Here a complete expression is provided. The following calculation is based on a variational method extensively detailed in former articles [START_REF] Nony | Stability criterions of an oscillating tip-cantilever system in dynamic force microscopy[END_REF] . We start from equation 2 and postulate a solution of the differential equation where the amplitude and the phase are not constant, but are allowed to vary slowly within time, that is over durations much longer than the oscillating period, namely :

Equation 2 is then equivalent to :

with :

Fc and Fd are the first Fourier components of the conservative and dissipative forces experienced by the tip over one oscillation period T , respectively :

Setting Ȧ(t) = Ä(t) = φ(t) = φ(t) = 0 yields to the steady equations of the oscillator in amplitude and phase out of which the relevant variables, namely ∆f = f -f 0 and driving amplitude, related to the damping, can be extracted :

B. APIC response time

The problem consists in analyzing the dynamics of the oscillator the amplitude of which is controlled by a proportional-integral controller. For that purpose, all the components are assumed to be linear. No interaction occurs between the oscillator and the surface. The contributions of the RMS-to-DC and of the band pass filter are neglected. The system under investigation is given in figure 13.

Transfer function of the closed loop

We start from the set of equations 35 wherein the phase is assumed to be constant and

This assumption is argued by the PLL behavior which maintains the phase almost constant, even when the interaction occurs (cf. fig. 3(b)). The assumption implies : ω = ω 0 , φ(t) = φ(t) = 0 and obviously, we only focus at changes occurring in the resonance amplitude A(t) = A 0 (t). The set of equations 35 is then equivalent to :

Keeping β = γ yields to :

The transfer function of the block standing for the oscillator can thus be written :

where :

The transfer function of the APIC being G AP IC (s) = K ac p + K ac i /s, the transfer function of the closed loop G cl (s) = K 1 A 0 (s)/A ref 0 (s) can now be calculated : FIG. 12: Calculated cross-section of a surface with two opposite steps for various sets of the APIC gains, namely (K ac p = 10 -2 ; K ac i = 10 -3 s -1 ) = •, (10 -3 ; 10 -3 ) = ⋄, (10 -2 ; 10 -2 ) = ×, (10 -3 ; 10 -2 ) = +, (10 -2 ; 10 -4 ) =△, (10 -3 ; 10 -4 ) = •, (10 -4 ; 10 -4 ) = , (10 -4 ; 10 -3 ) = , (10 -3 ; 10 -5 ) = , (10 -2 ; 10 -5 ) = and (10 -4 ; 10 -5 ) = ⊙. The scan lines have been initiated

upon the same conditions than in fig. 10


Faridah Djellal
email: faridah.djellal@univ-lille1.fr

Faïz Gallouj
email: faiz.gallouj@univ-lille1.fr

Services and crisis: stop shooting at the ambulance!

Keywords: services, innovation, crisis

Paradoxically, despite the extent of their contribution to wealth and employment, services are regularly blamed (in both political discourses and certain theoretical works) to be directly or indirectly responsible for the economic crisis. This short note aims at refuting such a paradox, arguing that the service economy is an economy of knowledge, skills and innovation.

Introduction

In contemporary developed economies, services account for about three-quarters of wealth and jobs. Any work dedicated to these activities consistently starts with this statistical observation. One could add that one of the characteristics of the so-called emergent countries (for example, Brazil, China,…) is the acceleration of their tertiarization going hand in hand with their industrial success. Paradoxically, despite the extent of their contribution to wealth and employment, services are regularly blamed (in both political discourses and certain theoretical works) to be directly or indirectly responsible for the economic crisis. This short note aims at refuting such a paradox, arguing that the service economy is an economy of knowledge, skills and innovation.

The rise of the service economy and the establishment of the service theory

If the concept of service is ancient (it is particularly discussed by Adam Smith in the « Wealth of Nations », 1776; Jean-Baptiste [START_REF] Say | Traité d'économie politique[END_REF]Frédéric Bastiat, 1848), the concept of service sector was born in the 1930s and early 1940s in relation with national accountancy issues [START_REF] Clark | The Conditions of Economic Progress[END_REF][START_REF] Fisher | The Clash of Progress and Security[END_REF]. It was reinforced in the 1950s and 1960s, in the United States, by the statistical works of [START_REF] Kuznets | Quantitative aspects of the economic growth of nations. Industrial distribution of national product and labor forces[END_REF]especially Fuchs (1968).

To simplify, one can say that there has been a struggle between two major theses in order to explain the growth of the service sector: the post-industrial thesis, on the one hand, and the neo-industrial thesis, on the other hand. The former one, and especially Daniel [START_REF] Bell | The Coming of Post-Industrial Society, a Venture in Social Forecasting[END_REF], advocates an optimistic and idealized vision of the service society in which the tertiarization is explained by a demand law (Engel's law) and a productivity law (Fourastié's law). Postindustrial society allegedly constitutes a new stage in human progress, based on the production and consumption of services and the pre-eminence of a higher, white-collar tertiary sector. The neo-industrial thesis, whose figurehead is Jonathan Gershuny, is more critical regarding services. Thus for Gershuny (1978, see also Gershuny and[START_REF] Gershuny | The new service economy[END_REF], technology and social innovation sound the death knell of the "service society" and replace it with a "self-service" society, in which consumers reject market services in favour of domestic production based on a technological system (the DVD rather than the cinema, the microwave and the pizza rather than the restaurant). For other authors belonging to the neo-industrialist perspective, services are subordinated to the manufacturing industry (the only driving force), when they are not just parasitical. These debates, which culminated in the 1970s-1980s, quietened down little by little, as services became established in an irremediable way in the socioeconomic landscape, as illustrated by the macroeconomic indicators.

Services and the economic crisis: the logic of the scapegoat

The fact remains that according to the old logic of the scapegoat, at every downturn in the economy, these polemics are revived. Services are then more or less explicitly designated as responsible for the economic difficulties, and especially the weakening of the industrial basis (deindustrialization). Influential politicians and distinguished economists alike (who are sometimes the same persons) establish in their statements a questionable causal relationship between tertiarization and deindustrialization, as if the opening of a hotel or a restaurant, the creation of a consulting firm, or an association could cause the closing of a plant! Therefore, to limit oneself to these two recent French examples1 , Nicolas Sarkozy, while he was Minister of Economy declared « France cannot be only an economy of banks, insurance and services ». (France Info., April 16th, 2004). Similarly, Arnaud Montebourg, at the head of the Ministry of the "manufacturing recovery" declared in 2012 « « Aluminium, textile industry, wood… our project is to recover all the industries which have gone abroad. The idea of France which succeeds without plants, is finished (…). Goodbye the service economy, long life to the recovery by manufacturing, by hard material. Our country should not become a large ski area for rich men, a luxury hotel with spa ». (Le Parisien, 2012). This suspicion regarding services is also regularly reflected in the literature, as illustrated by the following more or less recent titles: "Too few producers" [START_REF] Bacon | Britain's Economic Problem : Too Few Producers[END_REF], "Manufacturing matters" [START_REF] Cohen | Manufacturing Matters[END_REF], "France without its plants" [START_REF] Artus | La France sans ses usines [France without its plants[END_REF], "Reindustrialisation, I write your name" (Levet and al., 2012).

If nobody can deny the economic crisis which characterizes most of the European countries, what is the mechanism which leads to (implicitly or explicitly) attribute this crisis to services and to shout death to these activities? In some way, Adam Smith, whose thought keeps influencing the visions of the contemporary economists (and of politicians alike), bears the responsibility for this stigmatization of services. In an analysis limited, it should be acknowledged, to the work of domestic servants, of artists and of the servants of the State, he defined services as « unproductive of any value » and as activities "perishing at the very instant of their production ». However it must be acknowledged that other well-known myths systematically re-emerge at each crisis pick: the myths of the low capital intensity of services (the absence of plants), of their low productivity, of their disability to innovate, their maladjustment to exchange and international trade, the myth of the « society of services as a society of servants » (according to the expression of the philosopher André Gorz [START_REF] Gorz | Métamorphoses du travail et quête de sens [The metamorphoses of work and the seek for sense[END_REF], …

The service society: a knowledge and innovation society

However, today more than in the past, these myths do not hold any more. It is mainly in services that ICTs, the emblematic technologies of our century are invasive. These ICTs facilitate the exchange and the international trade, productivity gains and innovation dynamics. Even if it is dual, the service society, as it is illustrated by statistics, is more an engineers' society than a society of servants: service organizations are the main employers of executives, engineers and managers. If they also recruit less skilled employees (« bad jobs » or « hamburger jobs »), is it always necessary to complain about it, especially in periods of economic difficulties? Nevertheless the services economy is also the economy of knowledge intensive services (engineering, consultancy), which are not only particularly innovative for themselves, but which constitute essential support for the innovation of other sectors (especially manufacturing sectors). As William Baumol (2002) pointed out in a very evocative paper (« Service as leaders and the leader of the services »), R-D is a service (one can add that this also holds for education). More generally, growing research works emphasise the overpowering rise of the capacity of innovation of services, or the recognition of a dynamics of innovation which was invisible to our analytical tools (characterized by a technologistic and industrialist bias). One could go a step further in the improvement of the image of services, by paradoxically considering that the service economy does not exist any more (or is fast disappearing), that service and goods are consequently inextricably linked, as it is expressed in a certain number of recent theories: economics of functionalities [START_REF] Stahel | The Functional Economy: Cultural and Organizational Change[END_REF], economics of experience (Pime and Gilmore, 1999), approaches in terms of characteristics [START_REF] Gallouj | Innovation in services[END_REF], service-dominant logic [START_REF] Lusch | Service-Dominant Logic: reactions, reflections and refinements[END_REF]…

Conclusion

All in all, in the search for the reasons for economic crisis, we must not choose the wrong target. If it is necessary to seriously tackle the problem of the deindustrialization of our economies, it is not by attacking services. Services are not the problem. They are conversely often a (part of the) solution. A defensive or therapeutic solution, in certain cases, in a general perspective of a social and solidarity-based economy. But also and above all an offensive solution, taking advantage from the capacity of innovation of services in general and from the driving effect of some of them (the knowledge intensive business services) over the national innovation dynamics and economic growth. Therefore let's stop shooting at the ambulance!

It would be easy to provide examples for other countries.


Camille Juin

Valérie Thiéry

Jean-Paul Cadoret

Laurent Picot

Towards the Clinical Use of Phytoplankton Carotenoid Pigments to Cure Cancer

come   L'archive ouverte pluridisciplinaire

Editorial

Towards the clinical use of phytoplankton carotenoid pigments to cure cancer.

Juin C 1 , Thiéry V 1 , Cadoret JP 2 and Picot L 1* .

1

Université de La Rochelle, UMRi CNRS 7266 LIENSs, 17042 La Rochelle, France. 2

Ifremer, Laboratoire BRM/PBA, Rue de l'Ile d'Yeu, 44311 Nantes, France.

*

Corresponding author : Dr Laurent PICOT, lpicot@univ-lr.fr

Beyond their major ecolophysiological functions, phytoplankton pigments exert biological and pharmacological activities in human cells that allows to consider their clinical use to cure various pathologies. Although much of our knowledge relating to their cell pharmacology and bioactivity has come from in vitro studies in cell culture models, recent in vivo studies have validated the potential of phytoplankton carotenoid pigments to limit inflammation and metabolic disorders, retinal diseases, degenerative diseases, tumor progression, and hepatotoxicity. Aside from these promising results, additional studies are now required to precise their pharmacokinetics, pharmacological targets, and clinical efficacy in humans. The availability of highly purified pigments at rational costs will be a milestone to set up clinical trials and develop new therapies using microalgae pigments. This short paper focuses on the great potential of phytoplankton carotenoid pigments to prevent and cure cancers.

Marine and freshwater microalgae have evolved a wide range of pigments that belong to the chlorophylls, carotenoids and phycobiliproteins families. Extensive research has proved that microalgae pigments exert significant biological and pharmacological activities in human cells. Beyond their well-known antioxidant activity, used as a commercial argument to sell algae-based cosmetics and nutraceutics, it is now clearly established that microalgae pigments have a great potential as health nutrients to prevent cancer, as biotechnological probes for cancer diagnosis and as anticancer drugs to trigger cancer cells apoptosis, prevent tumor angiogenesis, reduce the risk of metastasis, sensitize cancer cells to chemotherapy, destroy cancer cells by tumor phototherapy and filter UV to limit cancer cells initiation. Numerous studies aiming to identify antiproliferative molecules from microalgae extracts led to the isolation of carotenoids and to the demonstration of their high antiproliferative, cytostatic, cytotoxic, and/or pro-apoptotic activity in cancer cell cultures. As an example, our research team performed the bioguided isolation of pigments from Duniella tertiolecta and found that violaxanthin was the most antiproliferative molecule contained in Dt dichloromethane extract. We also recently reported the strong antiproliferative activity of zeaxanthin and cryptoxanthin in human invasive melanoma cells, after their bioguided isolation from Cyanophora paradoxa ethanolic extracts.

Besides the purification of antiproliferative pigments, additional studies established that multiple cellular and molecular processes are affected by microalgae carotenoids. It was first demonstrated that their antioxidant activity protects against ROS-induced DNA mutations, and tissue and animals studies confirmed that most carotenoids effectively reduce the inflammatory processes initiated by carcinogenic agents, and thus prevent the risk of cancer initiation. As an example, adding peridinin in the dietary water of mice limits the development of carcinogens-induced skin tumors. Some authors also demonstrated that in addition to the cancer preventive effect associated to their antioxidant activity, some carotenoids such as fucoxanthin could trigger apoptosis in cancer cells by ROS generation. Pharmacological studies performed with fucoxanthin and various epoxycarotenoids confirmed that they affect a multitude of molecular and cellular targets dysregulated in cancer cells (cyclin-dependent kinases, DNA polymerases, connexins, pro and antiapoptotic proteins, MAP kinases, NF-kB, …), and numerous studies are currently undertaken to precise the pharmacology of carotenoids, according to their high interest for human oncology. Beyond their cytotoxic activity, some carotenoids also inhibit the tumor neovascularization and the metastatic potential of cancer cells, reverse multi-drug resistance in cancer cells treated with chemotherapeutic agents, and alleviate the deleterious side effects of cytotoxic drugs. Fucoxanthin was particularly well studied and its clinical efficacy to cure cancers from various tissular origin is validated in animal models, and should soon be validated in humans.

As a conclusion, it is obvious that microalgae carotenoids have a key role to play in clinical oncology and cancer nutritional prevention, according to their remarkable pharmacological activities. In addition to synthesizing most of the oxygen that we breathe, microalgae could thus have a great potential to help us fighting the second cause of death in developed countries.