FR3134185A1 - METHOD FOR DETECTING THE POINT-SURFACE INTERACTION FORCE WITHOUT LOSS OF SENSITIVITY TO INCREASE THE ACQUISITION BANDWIDTH OF ATOMIC FORCE MICROSCOPY MEASUREMENTS IN NON-CONTACT MODE - Google Patents

Abstract

Method for measuring (100) the interaction force between a portion (3) and a tip (10) of a force probe (9) of an AFM (1) secured to an excitation means (12 ), the excitation means (12) being capable of vibrating as a function of an excitation signal (e(t)), the method comprising a step of energizing (112) the force probe, a signal excitation being applied configured to generate, a free deflection (s0(t)) forming a sinusoidal signal modulated in amplitude by a pulse, the force probe entering into resonance at a free resonance frequency (f0), a measurement step (113) of the deflection (s(t)), the force probe entering into resonance at an interaction resonance frequency (fT), a processing step (114) in which a frequency shift (ΔfT) is evaluated, difference between the interaction resonance frequency of the force probe, and the free resonance frequency. Figure for abstract: [Fig. 3]

Description

METHOD FOR DETECTING THE POINT-SURFACE INTERACTION FORCE WITHOUT LOSS OF SENSITIVITY TO INCREASE THE ACQUISITION BANDWIDTH OF ATOMIC FORCE MICROSCOPY MEASUREMENTS IN NON-CONTACT MODE INTRODUCTION

The invention relates to the field of atomic force microscopes, and more particularly to a method making it possible to increase the speed of image acquisition while maintaining a sensitivity for detecting the tip-surface interaction force equal to or greater than those of the most advanced techniques in the field as described below.

Like the scanning tunneling microscope, the atomic force microscope (“Atomic Force Microscope”, AFM) belongs to the family of local probe microscopes, that is to say microscopes using a probe to carry out measurements. physicochemical at the nanometric, or even atomic, scale on the surface of a sample.

More particularly, atomic force microscopes are equipped with a micrometric force probe, equipped with a tip of nanometric dimensions sensitive to the interaction force that it develops locally with the surface. An excitation means capable of putting the force probe into oscillation and moving it on the surface of the sample as well as a means of detecting the tip-surface interaction forces make it possible to form, among other things, a topographical image of the surface. An electronic module makes it possible to control the surface excitation and scanning means.

The expressions “force probe”, “probe” refer identically throughout the application to the mechanical device supporting the tip of the AFM, or to the tip alone.

The expression "interaction force" refers throughout the application to the vertical resultant of the tip-surface interaction forces as represented in from attractive or repulsive interatomic potentials, for example of the Lennard-Jones type, of short or long range, when the tip of an AFM is facing a portion of a sample.

The expression "portion of a sample" refers throughout the application to a surface of the sample small enough to be considered as a single point of measurement of the tip-surface interaction force (hereinafter referred to as "pixel "). Atomic force microscopes thus make it possible to determine the local interaction force between the tip and the sample, making it possible to produce an image of the surface of the sample at the atomic scale. The image is made up of a succession of pixels acquired periodically during scanning of the surface and whose contrast reflects the values of the local tip-surface interaction force.

In the remainder of the application, the force probe is characterized by its natural resonance frequency f ₀ (units: Hz) and its natural stiffness k ₀ (units: N/m) as measured at a distance from the sample for which the interaction forces which develop between them are not detectable by the AFM detection device, typically several hundred nanometers. In such a case, the tip is defined throughout the description as being free.

Typically, atomic force microscopes feature three different modes of operation: contact mode, amplitude modulation mode, or “intermittent contact,” and frequency modulation mode, or “non-contact AFM.”

In so-called “contact” mode, the tip is permanently in contact with the surface of the sample. A repulsive force is created between the surface and the tip, causing the force probe to deflect. During scanning of the surface, a plurality of such deflections are detected. Said deflections are linked to the topographic variations of the surface of the sample. It is common to scan the sample by continuously adjusting the distance between the tip and the surface, so as to keep the interaction force constant. This tip-surface distance, depending on the horizontal position of the probe, thus reproduces the topography of the sample measured at constant interaction. Constant force control is carried out using a feedback loop, typically of the proportional-integral type, working around a force setpoint chosen by the operator.

In so-called “amplitude modulation” (“Amplitude Modulation AFM”, AM-AFM), or “intermittent contact” (“ tapping ”) mode, the probe is set into oscillation by sinusoidal excitation at a fixed predetermined frequency, close to its natural resonance frequency f ₀ , and at a predetermined amplitude. The tip comes into intermittent contact with the surface. The repulsive force which develops at the moment of each contact is at the origin of the reduction in amplitude of the oscillations. This amplitude reduction is measured by the AFM detection system, which provides the local topography of the surface during scanning. Again, it is common to produce an image with constant interaction, and therefore to control the tip-sample distance to an amplitude reduction setpoint chosen by the operator.

The so-called “frequency modulation” mode (“Frequency Modulation AFM”, FM-AFM), or non-contact AFM (nc-AFM), is an alternative to the AM-AFM mode to overcome its intrinsic limitations (detailed below). -After). The nc-AFM mode is currently generally used with AFMs operating in ultra-high vacuum conditions (pressure ~10 ^-10 mbar), at low temperature (liquid helium, 4 K) or at room temperature. In nc-AFM, the force probe is oscillated at its natural resonance frequency f ₀ . Near the surface, that is to say at a distance of the order of 100 nm and below, the attractive interaction forces which develop between the tip and the surface reduce the natural resonance frequency of the probe interacting with the surface. The modified resonance frequency is thus designated throughout the remainder of the application by resonance frequency f _T.

The resulting interaction frequency shift Δf _T (units: Hz) is expressed as follows:

,

Such an interaction frequency shift Δf _T is detectable from a measurement over time of the deflection of the tip in interaction with the surface. Such a deflection is designated throughout the remainder of the description by interaction deflection s(t). Measuring the interaction frequency shift Δf _T allows, in nc-AFM mode, to determine, via the interaction deflection s(t), the tip-surface interaction force with a precision of a few piconewtons and a spatial resolution of the order of angstrom (atomic resolution). It is known to produce an image with constant interaction, and therefore to control the tip-sample distance to an interaction frequency offset setpoint Δf _T chosen by the operator.

The invention relates more particularly to the field of atomic force microscopy in non-contact mode.

Throughout the rest of the application, we designate by AFM, microscopy, or atomic force microscope, an AFM operating in non-contact mode.

1. une détection continue de la fréquence de résonance en interaction f_T,décalée par l’interaction pointe-surface
2. un calcul du décalage en fréquence en interaction Δf_T
3. un maintien de l’excitation de la sonde de force à la fréquence de résonance en interaction f_T.

Typically, an AFM includes an electronic control unit including a Phase-Locked Loop (PLL). The PLL allows:

1. continuous detection of the interacting resonance frequency f _T, shifted by the tip-surface interaction
2. a calculation of the interaction frequency shift Δf _T
3. maintaining the excitation of the force probe at the interacting resonance frequency f _T .

As the tip approaches the surface, the PLL continuously produces a probe excitation signal (or input signal), e(t) (units: m), in the form of a sinusoidal signal at the interacting resonance frequency f _T (units: Hz) and amplitude A _e (units: m), noted:

The input signal e(t) generates an interacting deflection s(t) in the form:

where A _s (units: m) is the amplitude of the interacting deflection and a static phase shift φ _s of the force probe 9 relative to the input signal e(t) lying in an interval belonging to [-180 °;0°].

The amplitude of the deflection A _s and the static phase shift φ _s are generally measured by a synchronous detection integrated into the PLL and synchronized to the interacting resonance frequency f _T . If the influence of the interaction forces is weak (free tip), the excitation takes place at the natural resonance frequency f ₀ of the free probe. The response of the force probe 9 is then in quadrature delay with respect to the excitation, φ _s ~90°. In the presence of an interaction with the surface, the interaction frequency shift Δf _T is non-zero. If the driving frequency f _d remained fixed, this would result in a phase φ _s different from -90°.

The PLL has the particular function of continuously detecting the static phase shift φ _s and of continuously adjusting the interacting resonance frequency f _T to lock the static phase shift φ _s at -90°. If no dissipative process develops between the tip and the surface, which is presupposed throughout the rest of the request, the amplitude of the oscillation A _s remains that of the resonance, maximum, during the approach or scanning the surface with the tip.

For oscillation amplitudes of the order and less than 100 pm, it can be considered that the interaction frequency shift Δf_Tdetected by the PLL is proportional to the vertical gradient of the tip-surface interaction force F_int. This last quantity also represents the variation of the stiffness in interaction k_Tof the probe in relation to its own stiffness k₀, which is formalized by the equations:

where Δk _T (units: N/m) is the variation in interaction stiffness, corresponding to the difference between the stiffness of the probe interacting with the surface k _T , and that of the free probe k ₀ .

The variation in stiffness Δk _T is represented under the interaction force F _int in the . In the previous equation, the "-" sign reflects the fact that for an attractive force, the interaction frequency shift Δf _T , like that of the stiffness, is negative, while the force probe is in a situation of " non-contact” with the portion of surface to be measured.

Measuring the interaction frequency shift Δf _T by the PLL therefore makes it possible to indirectly measure the interaction force F _int and to set its value as the operating point of the tip-surface distance regulation during scanning of the surface to form a non-contact AFM image. A topographic image in nc-AFM is therefore representative of an iso-frequency shift mapping of the surface, therefore “iso-gradient of force”.

If it is today possible with commercial AFMs used in nc-AFM mode to detect forces of the order of a few tens of piconewtons and spatial resolutions of the order of an angstrom, a major limitation remains the speed image acquisition. With available commercial probes whose natural resonance frequencies vary in a range from approximately 10 kHz to 300 kHz, the acquisition time for a single image is on the order of 30 minutes. A measurement campaign, which requires the acquisition of several dozen images, can therefore take up to several days.

For carrying out measurements on the atomic scale, this is a long and restrictive time interval for the experimenter. Over such a time interval, it is desired that the temperature of the instrument be kept as constant as possible with great precision. In addition, in order to limit the effects of drift in the position of the sample, the experimenters seek to ensure that the instrument has high mechanical stability.

The origin of the long acquisition time of a pixel comes from an effect linked to the dynamics of the tip. The characteristic equilibration time τ ₀ (unit: s) of the force probe under harmonic excitation is estimated as follows:

where Q ₀ (units: dimensionless) is the quality factor of the probe, defining the way in which it dissipates the energy supplied to it by excitation in the surrounding environment.

The higher the quality factor Q ₀ , the lower the energy dissipated by the probe in the surrounding environment, the longer its equilibration time from one state to another, and vice versa.

This makes it possible to define an adiabaticity criterion for the force probe: any force measurement carried out over a time interval equal to or greater than τ ₀ could be considered as representative of a stationary state of the probe, therefore compliant. According to an example of a probe: f ₀ ~ 100 kHz, placed in atmospheric conditions: Q ₀ ~ 400, then τ ₀ ~ 8 ms. With such values, the electronic control unit needs to integrate the measurement of the interaction force F _int leading to a pixel of the image over a duration equal to or greater than the balancing time of the force probe in order to respect the aforementioned adiabaticity criterion and to allow a consistent measurement of the interaction force F _int at this point. For an image with a resolution of 512x512 pixels, the total image acquisition time, product of the number of pixels and the characteristic equilibration time τ ₀ , is estimated at approximately 35 min. By placing an identical probe in ultra-high vacuum conditions, where dissipation is made very low by the absence of air, the quality factor reaches Q ₀ = 40,000 or even beyond. In this case, the equilibration time τ ₀ is estimated at 800 ms and the acquisition time is 3500 min or 58 h.

Such a value does not make sense from the point of view of experimental measurement time. One might therefore think that it is not possible to carry out a non-contact AFM measurement under ultra-high vacuum. Even if it clearly illustrates the problem of the long image acquisition time in AFM, the preceding reasoning is in fact only valid for the intermittent contact mode. It is true that to date, it is impossible to produce an intermittent contact image under ultra-high vacuum. On the other hand, the reasoning is incorrect for the non-contact mode. We can in fact show that the limiting element in nc-AFM is not the balancing time of the probe, but the time taken by the PLL of the electronic control unit to detect the signal corresponding to the offset of frequency Δf _T. Depending on the probes used, it is necessary to consider detecting frequency shifts Δf _T in a range from -1 Hz to -100 Hz, with the best possible frequency resolution (0.1 Hz and below). In terms of order of magnitude, this assumes a characteristic response time of the PLL of the order of 10 ms per pixel. We are therefore confronted in nc-AFM with temporal constraints substantially comparable to those of the intermittent contact mode under atmospheric conditions.

Reduce the acquisition time of measurements in nc-AFM, therefore increase the acquisition bandwidth (for example by increasing the scanning speed of the images), without this causing a loss of sensitivity to the detection of the force of interaction F _int , in other words without loss of spatial resolution in the formation of images, or even with gain in sensitivity, constitutes a technical challenge to which the invention aims to respond.

A first object of the invention is to propose a method for measuring the interaction force between a sample and a force probe of an AFM, the method making it possible to reduce the measurement time compared to known methods, without the measurement accuracy is affected.

A second object of the invention is to propose a measurement method making it possible to improve the quality of measurement of the tip-surface interaction force, by reducing noise.

A third object of the invention is to propose such a method which can be carried out without it being necessary to significantly modify the general structure of the AFM.

A fourth object of the invention is to propose such a method which maintains a certain level of ease of use for the user of the AFM.

A fifth object of the invention is to propose a program capable of implementing the method as presented above, as well as a control device comprising said program.

• une étape de pré-mesure au cours de laquelle est définie la forme du signal d’excitation, le signal d’excitation étant configuré pour engendrer, lorsque la pointe est libre de toute force d’interaction, une déflexion libre, la déflexion libre formant un signal sinusoïdal modulé en amplitude par un signal d’enveloppe appartenant à la classe des fonctions régulières et définissant une impulsion, la sonde de force entrant en résonance à une fréquence de résonance propre,
• une étape de mesure comprenant
  • une sous-étape de mise en excitation de la sonde de force, au cours de laquelle le signal d’excitation est appliqué au moyen d’excitation
  • une sous-étape de détection au cours de laquelle est évaluée un décalage de fréquence en interaction, différence entre la fréquence de résonance en interaction de la sonde de force lorsque soumise à la force d’interaction, et la fréquence de résonance propre, par l’intermédiaire d’une sous-étape de détection synchrone, permettant d’obtenir la mesure de ladite force d’interaction.

For this purpose, a method is proposed for measuring the force of interaction between a portion of a sample and a tip of a force probe of an AFM head, the force probe being integral with a means of excitation, the excitation means being capable of vibrating as a function of an excitation signal, the method comprising:

• a pre-measurement step during which the shape of the excitation signal is defined, the excitation signal being configured to generate, when the tip is free from any interaction force, a free deflection, the free deflection forming a sinusoidal signal modulated in amplitude by an envelope signal belonging to the class of regular functions and defining a pulse, the force probe entering into resonance at a specific resonance frequency,
• a measurement step comprising
  • a sub-step of energizing the force probe, during which the excitation signal is applied to the excitation means
  • a detection sub-step during which an interacting frequency shift is evaluated, the difference between the interacting resonance frequency of the force probe when subjected to the interaction force, and the natural resonance frequency, by via a synchronous detection sub-step, making it possible to obtain the measurement of said interaction force.

Thanks to the measurement method, the time for the probe to enter a steady state is reduced by approximately ten times compared to known methods using a PLL. Such a process also makes it possible to completely dispense with the use of a PLL.

Other additional features can be provided alone, or in combination:

• nulle en dehors d’un intervalle de temps prédéterminé ayant pour borne inférieure une première valeur de temps, et pour borne supérieure une quatrième valeur de temps,
• croissante entre la première valeur de temps et une deuxième une deuxième valeur de temps supérieure à la première valeur de temps, mais inférieure à la quatrième valeur de temps,
• constante entre la deuxième valeur de temps et une troisième valeur de temps, supérieure ou égale à la deuxième valeur de temps,
• décroissante entre la troisième valeur de temps et une quatrième valeur de temps, la troisième valeur de temps étant inférieure à la quatrième valeur de temps.

According to one implementation mode, the envelope is:

• zero outside a predetermined time interval having as its lower limit a first time value, and as its upper limit a fourth time value,
• increasing between the first time value and a second a second time value greater than the first time value, but less than the fourth time value,
• constant between the second time value and a third time value, greater than or equal to the second time value,
• decreasing between the third time value and a fourth time value, the third time value being less than the fourth time value.

• sinusoïdale par morceau,
• polynomiale par morceau,
• ou une fonction s’exprimant comme suit :

According to one implementation mode, the envelope defines one of the following functions:

• sinusoidal by piece,
• piecewise polynomial,
• or a function expressed as follows:

the second time value being different from the first time value and the fourth time value being different from the third time value.

According to one embodiment, the pre-measurement step and the measurement step are carried out only once, the excitation signal being a simple excitation signal and the deflection of the force probe being a deflection simple free, the form of the simple excitation signal being obtained by applying an inverse Fourier transform to the following formula:

E _{ESE, 0} (ω) being the Fourier transform of the input signal applied to the excitation means, S _ESE , ₀ (ω) the Fourier transform of the free deflection, G(ω) the transfer function of the resonator (8), the resonator (8) integrating the excitation means (12) and the force probe (9).

According to one embodiment, the shape of the excitation signal is obtained by applying an inverse Fourier transform to the following formula:

E _{ESE, 0} (ω) being the Fourier transform of the input signal applied to the excitation means, S _{ESE, 0} (ω) the Fourier transform of the free deflection, G(ω) the transfer function of the resonator , the resonator integrating the excitation means and the force probe, and W(ω) the Fourier transform of a Blackman–Harris window.

• un signal d’entrée virtuel est défini,
• au moins une fréquence de plusieurs fréquences de résonance virtuelle différente de la fréquence de résonance propre est sélectionnée, la fréquence virtuelle ayant une valeur sélectionnée dans une plage de valeur pouvant être atteinte par la fréquence de résonance en interaction,
• le signal d’entrée virtuel choisi est appliqué à la sonde de force initialement immobile et libre de la force d’interaction, de sorte à générer une réponse vibratoire impulsionnelle de la pointe définissant une déflexion de calibration, sinusoïdal et modulé en amplitude par un fonction d’enveloppe régulière ,
• des mesures de la déflexion de calibration, permettant de déterminer un coefficient de calibration, le coefficient de calibration étant proportionnel à une variable de traitement.

According to one mode of implementation, the pre-measurement step includes a calibration sub-step during which:

• a virtual input signal is defined,
• at least one frequency of several virtual resonance frequencies different from the natural resonance frequency is selected, the virtual frequency having a value selected in a value range that can be reached by the interacting resonance frequency,
• the chosen virtual input signal is applied to the force probe initially immobile and free from the interaction force, so as to generate an impulse vibration response of the tip defining a calibration deflection, sinusoidal and modulated in amplitude by a function regular envelope,
• measurements of the calibration deflection, making it possible to determine a calibration coefficient, the calibration coefficient being proportional to a processing variable.

According to one mode of implementation, the virtual input signal is defined as being the inverse Fourier transform of the following formula:

E'(ω) being the Fourier transform of the calibrating signal, S _{ESE, N} (ω) the Fourier transform of the repeated free deflection, H(ω) being a modeling of an ideal harmonic transfer function of the probe force entering into resonance at the free resonance frequency, H'(ω) being a fictitious harmonic transfer function of the force probe entering into resonance at the fictitious resonance frequency, G(ω) the transfer function of the resonator, and W(ω) a Blackman–Harris window.

• une multiplication d’une déflexion avec un signal harmonique complexe permettant l’obtention d’un signal de sortie traité par détection synchrone,
• une représentation de la partie réelle et de la partie imaginaire du signal de sortie traité sur un plan complexe, le signal de sortie traité ainsi représenté formant une boucle,
• une mesure de l’aire de la boucle, l’aire de la boucle étant proportionnelle au décalage de fréquence.

According to one mode of implementation, during the calibration sub-step, and during the measurement step; it includes a processing sub-step during which it is carried out

• a multiplication of a deflection with a complex harmonic signal making it possible to obtain an output signal processed by synchronous detection,
• a representation of the real part and the imaginary part of the processed output signal on a complex plane, the processed output signal thus represented forming a loop,
• a measure of the loop area, the loop area being proportional to the frequency offset.

According to one mode of implementation, the pre-measurement and measurement steps are repeated over time, so as to allow the measurement of the interaction force on N portions, the deflection being a repeated free deflection which is expressed as following :

n corresponding to the number of times the pre-measurement step is carried out.

According to one aspect, the invention relates to a computer program product installed on an electronic module comprising a storage medium comprising instructions which, when executed on a computer device, implement the control of the measurement method according to any of the modes implementation described above.

According to one aspect, the invention relates to a memory medium comprising the computer program product described above.

According to one aspect, the invention relates to an AFM comprising the memory medium described above.

Other characteristics and advantages of the invention will appear more clearly and concretely on reading the following description of embodiments, which is made with reference to the appended drawings in which:

represents a graph illustrating the interatomic interaction forces, as well as the force gradient acting on a force probe as a function of the distance between the sample and the tip,

represents a schematic view of an AFM head when carrying out a measurement of the interaction force between the force probe and a sample,

represents a synoptic diagram illustrating a method for measuring the force of interaction between the probe and a sample,

represents a synoptic diagram illustrating a pre-measurement step a method of measuring the force of interaction between the probe and a sample, without repetition

represents a synoptic diagram illustrating a pre-measurement step of a method for measuring the force of interaction between the probe and a sample, when the deflection is repeated,

represents a schematic graph of the time trace of the deflection of the probe under classical harmonic excitation (gray), as well as that of the deflection of the probe used in the process (black) to carry out the measurement of the interaction force for the acquisition of a pixel,

represents a schematic graph of the time trace of the repeated free deflection of the probe when the method of measuring the interaction force is repeated,

illustrates a schematic graph of the time trace of the frequency shift linked to the application on the force probe of an interaction force (top), of the corresponding time trace of the deflection of the probe s(t) in the framework of the process (middle) and the time trace of the real and imaginary parts of the quantity “P” constructed on s(t), during a processing step (bottom),

illustrates a schematic graph of the deflection of the probe after synchronous detection represented on a complex plane,

illustrates a curve obtained after carrying out a calibration sub-step.

We refer to the , illustrating an example of an AFM head 1, comprising a support 2, a force probe 9 and an excitation means. Such an AFM head 1 belongs to an AFM (not shown). Such an AFM structure is known from the state of the art.

The excitation means (not shown) linked to the support 2 is intended to vibrate, which allows the force probe to oscillate.

Advantageously, the AFM includes an electronic module MCU, making it possible in particular to control the excitation means, and to regulate and control the distance between the tip 10 and the surface of the sample 4.

As shown on the , the force probe 9 is opposite a portion 3 of a sample 4, and in such a position, undergoes an interaction force F _int from this sample 4.

When the force probe 9 is sufficiently far from the sample 4, so as not to experience an interaction force F _in , the force probe 9 is said to be free.

Advantageously, the force probe 9 comprises a first end part 5, free. In this way, the force probe 9 is cantilevered, such an arrangement allowing the force probe 9 to be directly facing the sample 4.

Advantageously, the force probe 9 comprises a second end part 6, integral with the excitation means 12, so that the vibrations of the excitation means 12 can make it possible to set the force probe 9 into oscillation.

Advantageously, the excitation means 12, itself secured to the support 2, is linked to the second end part 6 by means of a spring 7. In this way, the excitation means 12 is capable of transmitting its movement to the force probe 9. Also, the force probe 9 and the excitation means 12 form a resonator 8.

Advantageously, the first extreme part 5 of the force probe takes the form of a lever, but more generally any mechanical oscillator can be suitable. The force probe 9 can for example be compared to a mass-spring system, of mass m, and stiffness k ₀ , having a natural resonance frequency f ₀ .

In the embodiment shown, the first end part 5 is provided with a tip 10 intended to be oriented towards the sample 4, and more particularly towards a portion 3 of the sample 4.

Advantageously, the force probe 9 forms an angle typically of 10° with the surface of the sample 4, allowing the tip 10 to be placed close to the sample 4 without another part of the AFM head 1 is not in contact with sample 4.

• un axe X, définissant une direction longitudinale, horizontale, confondue avec la projection de la direction d'extension du levier 9 dans le plan de l’échantillon,
• un axe Y, définissant une direction transversale, horizontale, qui avec l'axe XY définit un plan XY horizontal,
• un axe Z, définissant une direction verticale, perpendiculaire au plan XY horizontal.

We define with respect to sample 4 an orthogonal reference frame XYZ forming a direct trihedron, comprising three perpendicular axes in pairs, namely:

• an axis
• a Y axis, defining a transverse, horizontal direction, which with the XY axis defines a horizontal XY plane,
• a Z axis, defining a vertical direction, perpendicular to the horizontal XY plane.

The deflection of such a lever 9 is generally measured using an optical system (not shown) comprising a laser and a photosensitive detector. More specifically, the laser radiation 11 is reflected on the upper surface of the lever 9 to the right of the tip 10, so as to strike the photosensitive detector. Any modification of the deflection of the lever 9 has the effect of modifying the positioning of the laser radiation on the photosensitive detector.

In embodiments not shown, the force probe comprises a U-shaped quartz tuning fork (“quartz tuning fork”, QTF), one branch of which comprises a tip interacting with the sample 4. If necessary, the deflection of the branch of the QTF supporting the tip 10 does not require an optical detection device. This is done by a charge amplifier which generates a voltage proportional to the charge generated by the piezoelectric effect in the QTF during deflection of the tip.

In other implementations not shown, the force probe 9 is a micro-electro-mechanical system, for example in the shape of a disc or ring, or taking the form of more complex structures.

Advantageously, the excitation means is an element capable of causing the force probe 9 to oscillate at a frequency close to its resonance frequency.

In the embodiment shown, the excitation means consists of a piezoelectric plate 12. Such a piezoelectric system is advantageous because its vibrations are easily controllable using an electrical excitation voltage e(t).

In other embodiments not shown, the excitation means is optical, operating for example by radiation pressure, or thermal effect, caused by the absorption of light.

In other embodiments not shown, the excitation means is magnetic or electrostatic.

We now describe a method of measuring 100, the interaction force between a portion 3 of the sample 4 and the tip 10. Such a measuring method 100, also entitled "Engineered Swift Equilibration" protocol (ESE protocol ) is schematized .

Without this being limiting, the measurement method 100 is described with a force probe 9 taking the form of a lever, and an excitation means 12 which is a piezoelectric system.

The measurement method 100 comprises a pre-measurement step 101, and a measurement step 102. Examples of implementation of such steps are illustrated in Figures 3, 4 and 5.

More precisely, the pre-measurement step 101 is carried out when the force probe 9 is free. The purpose of such a pre-measurement step 101 is in particular to predefine a simple excitation signal e _ESE,0 (t) of the force probe 9 so that its simple free deflection s _ESE,0 (t) has a predefined shape. over a time interval equivalent to a measurement pixel, typically a few periods of oscillation of the force probe 9.

During the measurement step 102, the interaction force F _int is measured in particular. The measurement step 102 is repeated as many times as there are pixels to measure, so as to acquire an image with a wide bandwidth.

More specifically, and as explained further below, during the measuring step 102, the tip 10 is placed at a measuring distance from the portion 3 of the sample, the measuring distance being defined such that the force probe 9 experiences an interaction force F _int of portion 3, the force probe 9 entering into resonance at an interaction resonance frequency f _T.

Thus, during the measurement step 102, the variations in amplitude of the deflection s(t), which are proportional to the interaction frequency shift Δf _T , are advantageously used as a reference signal for the control of the distance between the force probe 9 and the surface of the sample 4.

We describe the pre-measurement step 101 more precisely, with reference to Figures 3 and 4.

As mentioned above, the pre-measurement step 101 makes it possible to determine the shape of the simple excitation signal e _ESE,0 (t) to be supplied to the force probe 9, to impose on it a deflection of predefined shape on a predefined time interval. Such a time interval applies for example to a measurement pixel.

Advantageously, such a pre-measurement step 101 uses the transfer function of the AFM head 1, making it possible to generalize such a step to an arbitrary number N of measurement pixels, and to establish a linear relationship between the offset in frequency Δf _T and a processing variable A(t), functional to the deflection s(t).

Such operations are carried out before the measurements, therefore before the acquisition of the images. They are carried out while tip 10 is located opposite sample 4, but being free.

During the pre-measurement step 101 an excitation signal e _{ESE, 0} (t) is applied, generating when the tip (10) is free of any interaction force, a simple free deflection s _ESE,0 (t), for example over an arbitrary time interval T _p , advantageously sufficiently short, typically a few periods of oscillation of the force probe 9, authorizing the point measurement of the interaction force F _int , like that of a sinusoidal signal modulated in amplitude by an envelope signal η(t).

The shape of the simple excitation signal e _ESE,0 (t) required to obtain such a simple free deflection shape s _ESE,0 (t) is detailed later in the present description.

The envelope signal η(t) (unit: dimensionless) is in the form of a pulse.

An example of such a pulse is shown . In other words, the envelope function η(t) is initially constant (in the example of , zero), then evolves towards a non-zero value, and becomes constant again (in the example of the , nothing).

In other words, the simple free deflection s _ESE,0 (t) (units: m) takes the form of a sinusoidal signal modulated in amplitude by the envelope signal η(t).

Such a free deflection s _ESE,0 (t) is expressed for example as follows:

where A _s (units: m) and f _d (units: Hz) respectively designate the maximum amplitude and the driving frequency of the simple free deflection s _ESE,0 (t).

Advantageously, the control frequency f _d is chosen equal or substantially equal to the natural resonance frequency f ₀ of the probe, while respecting |f _d -f ₀ |<f ₀ /Q ₀ .

The envelope signal η(t) can be represented in the form of a function belonging to the class of regular functions, i.e. it is continuous and continuously differentiable.

We refer to the , giving an example of an envelope signal η(t). Such an envelope signal η(t) is zero outside a time interval [t1; t4] predetermined, increasing between a first time value t1 and a second time value t2 greater than the first time value t1, constant between the second time value t2 and a third time value t3, greater than or equal to the second time value, and decreasing between the third time value (t3) and a fourth time value t4. Then after the fourth time value t4, the envelope signal η(t) offers a zero value. The measurement time of a pixel T _p thus corresponds at least to the time interval [t1; t4], but can extend before or after such an interval.

In this way, the force probe9 subjected to the simple excitation signal e _ESE,0 (t) ensuring this form of the simple free deflection s _ESE,0 (t), evolves between a first state of equilibrium, then is excited , and returns to a second equilibrium state, identical to the first equilibrium state.

The fact of animating the force probe 9 with a sinusoidal signal modulated by a pulse makes it possible to force the force probe 9 to reach a state of equilibrium much more quickly than if the force probe 9 were excited by means of 'a simple sinusoidal signal, that is to say without modulation. We thus free ourselves from the aforementioned adiabaticity criterion imposed by the quality factor of the force probe 9. In this way, it is possible to considerably reduce the time to carry out the measurement of the interaction force F _int exerted by the sample 4 on the force probe 9, and thus detect a pixel. The image acquisition bandwidth is considerably increased.

Several examples of functions representing the envelope signal η(t) are described below.

An example of an envelope signal η(t) is:

Such an expression completely defines that of the simple free deflection s _ESE,0 (t).

A second example of an envelope signal η(t) is:

A third example of an envelope signal η(t) is:

According to yet another example, the envelope signal η(t) is a polynomial taking a zero value between 0 and the first time value t1, and a value 1 between the second time value t2 and the third time value t3 , of zero derivative at the first, second, third and fourth time value t4, and monotonic between the first and second time value t2, t4 as well as between the third time value and fourth time value t4.

We now describe a sub-step 1010 of determining the mechanical transfer function G(ω) of the AFM head 1. Advantageously, G(ω) is obtained by injecting the AFM head 1 with an excitation “white” e _n (t) in the form of white noise and by measuring the associated “white” free deflection s _n (t) of the force probe 9.

Preferably, such a measurement of the free deflection s _n (t) is repeated a predetermined number N of times in order to obtain a more precise average measurement of G (ω).

We notice :

Math [15]

And

the ^nth realizations of the Fourier transforms of the white excitation _n (t) and the associated free deflection s _n (t) among the N carried out.

Thus, the transfer function of the entire AFM head 1 G(ω) is expressed as follows:

Once determined, the transfer function G(ω) of the AFM head 1 is assumed unchanged throughout the rest of the measurement method 100.

The pre-measurement step (101) comprises a sub-step 1011 for determining the excitation signal for a measurement pixel.

The Fourier transform of the simple excitation signal e _ESE,0 (t), is expressed as follows:

In temporal form, the simple temporal excitation signal e _ESE,0 (t) is advantageously determined by an inverse Fourier transform operation:

with W(ω), the Fourier transform of a Blackman-Harris window applied to the simple excitation signal E _ESE,0 (ω) before the inverse Fourier transform operation making it possible to smooth out its local divergences.

The expressions for the simple free deflection s _ESE,0 (t) and the simple excitation signal e _ESE,0 (t) represent the deflection and excitation of the tip during a single cycle of the measurement process, ie over an acquisition duration T _p reflecting the acquisition time of a single pixel of the image, which is sufficient for the point measurement of the tip-surface interaction force.

In order to envisage the continuous acquisition of pixels over time and so as to form an image, the pre-measurement step 102 is advantageously repeated over time.

More precisely, the expression for the simple free deflection s _ESE,0 (t) is repeated over time. Considering an acquisition of N pixels over a total duration NT _p , we define a repeated deflection s _ESE,N (t) an excitation signal repeated over such a period of time. The repeated free deflection s _ESE,N (t) is expressed for example as follows:

The coefficient (-1) ⁿ in the previous equation makes it possible to maintain the maximum amplitude of the repeated deflection s _ESE,N (t) constant and to avoid its divergence over time.

Taking up the aforementioned elements and noting the Fourier transform of the repeated free deflection s _ESE,N (t), the repeated excitation signal e _ESE,N (t) to be supplied to the probe to obtain the repeated free deflection s _ESE,N (t) is given by the following relation:

Advantageously, during a processing sub-step (1020), a synchronous detection is applied to the simple deflection s _ESE,0 (t) or to the repeated deflection s _ESE,N (t) obtained after excitation by the signal excitation e _ESE,N (t). Such a step makes it possible to obtain an output functional of the signal s _ESE,N (t), called “processed deflection” and denoted p(t).

More particularly, synchronous detection is advantageously carried out as described below:

The treated deflection equation p(t) (units: m) is then expressed for example as:

The processed deflection signal p(t) is calculated continuously over time. Advantageously, such a processed deflection signal p(t) is averaged via a low pass filter, for example by integration over a period of time which is a multiple of the control period 1/f _d , for example Δt _{int =} 10 / f _d to form the signal P(t) (units: m) such that:

P(t) is a complex signal, expressed as follows:

P(t) is also calculated continuously over time.

The signal A(t) (units: m ² ) is expressed as follows:

The processing variable A(t) is calculated continuously over time and implicitly reflects the temporal variations of the deflection of the tip over a duration T _p reflecting a measurement pixel, possibly influenced by an interaction force with the surface.

For a fixed interaction value, that is to say when the interaction frequency shift Δf _T is constant, the repeated deflection s _ESE,N (t) is periodic with period T _p , therefore the processing variable A (t) is stationary and takes a fixed value A _sta characteristic of Δf _T , and therefore of interaction between the tip 10 and the surface.

Examples of implementation of the measurement method 100 are illustrated in Figures 8 and 9.

There schematically represents the time trace of the evolution of the repeated deflection s _ESE,N (t), with N = 9 measurement pixels under an imposed frequency shift step Δf _T = -16 Hz. The real parts Re(P) and imaginary Im(P) of the quantity P(t) are also reported.

There represents the trace of the quantity P(t) in the complex plane Re(P(t)), Im(P(t)) (cf. ), which forms a loop.

Preferably, and as can be observed in the figure, the processing variable A(t) represents the area of the loop formed. Thus, a measurement of the area makes it possible to obtain the interaction frequency shift Δf _T.

The processing variable A(t) makes it possible to connect the amplitude of the probe to the beginning (time t1 of the ) and at the end of each pixel of the process (time t4).

In the absence of interaction force F _int , as is the case in such a pre-measurement step 101, no variation in amplitude is expected between time t1 and time t4. The value of the processing variable A(t) is then constant and zero.

When the interaction force F _int is not zero between t1 and t4 as is the case during the approach of the force probe 9 towards the surface, or during the scanning of the tip 10 on the surface, then an interaction frequency shift Δf _T is induced relative to the free resonance frequency f ₀ of the force probe 9.

When the excitation of the force probe 9 is carried out at the control frequency f _d , this results in an amplitude variation (identical to what happens in AM-AFM). Such a variation in amplitude has the effect of modifying the repeated deflection s _ESE,N (t) at time t4, causing the quantity A(t) to vary.

We describe a calibration sub-step 1012 of the frequency shift making it possible to connect the processing variable A(t) to the interaction frequency shift Δf _T. During such a calibration sub-step 1012 of the shift of frequency, tip 10 remains free.

The force probe 9 is advantageously modeled as a simple harmonic oscillator, of mass m, expressed as follows:

,

of natural resonance frequency f ₀ , of stiffness constant k ₀ , and of quality factor Q ₀ .

The proper pulsation is expressed as follows:

,

Such a harmonic oscillator has a mechanical transfer function H(ω) expressed as follows:

Considering that the resonance frequency of the force probe 9 is set at a virtual frequency f _E , associated with the pulsation:

we define a virtual mechanical transfer function H’(ω), which is expressed as follows:

To obtain a repeated deflection signal s _ESE,N (t) with such a force probe 9 which resonates at a virtual frequency f _E , the excitation means 12 is excited by a virtual input signal e ' _{ESE, N} (t), which is calculated using the following equation:

The response of the force probe 9 to such a virtual input signal e' _{ESE, N} (t) is the repeated deflection signal s _{ESE, N} (t) if the interacting resonance frequency f _T is effectively equal to the virtual frequency f _E , and leads to obtaining a measurement via the previous equations of a zero processing variable A(t).

When the force probe 9 is free, the real resonance frequency is the natural resonance frequency f ₀ , therefore the measurement of the processing variable A(t) for such a virtual input signal e' _{ESE, N} (t ) is not zero, but gives the measurement associated with a virtual frequency shift which is expressed as follows:

.

The use of a virtual frequency shift Δf _E which does not result from a real tip-surface interaction force, but imposed in relation to the natural resonance frequency f ₀ , with equivalent consequences is particularly advantageous. In fact, we can carry out a calibration of the AFM 1 without it being necessary to use a sample.

Thus, the application of the virtual input signal e' _{ESE, N} (t) advantageously applied to the force probe 9 initially immobile and free from the interaction force F _int , makes it possible to generate an impulse vibration response of the tip 10, sinusoidal and modulated in amplitude by a regular envelope function. The virtual input signal e' _{ESE, N} (t) is defined so that the calibration deflection s' _ESE,N (t), output signal in response to the impulse vibration of the tip enters into resonance at the value of the calibration resonance frequency (f _E )

The virtual frequency shift Δf _E being independent of time, in steady state, the value of the processing variable A(t) is then constant, equal to a stationary value A _sta .

We vary the virtual frequency f _E in an interval [f ₀ -δf ₀ ; f ₀ +δf ₀ [ around the natural resonance frequency f ₀ , making it possible to determine the correspondence between the stationary processing variable A _sta and the frequency shift Δf _T.

Advantageously, a set of M values of the virtual frequency f _E uniformly distributed in the interval [f ₀ -δf ₀ ; f ₀ +δf ₀ [ around the natural resonance frequency f ₀ . A measurement of the calibration deflection for each of the values of stationary processing variable A _sta is carried out.

Preferably, the stationary processing variable values obtained A _sta are plotted as a function of the virtual frequency shift Δf _E. A linear (or at least monotonic) relationship is obtained between these two quantities of the form:

Δf _E (t) = β A(t)

with β (units: Hz/m ² ), which is a calibration coefficient.

Such a calibration coefficient makes it possible to connect, at the time of approach or when scanning the tip 10 on the surface, the processing variable A(t) measured in real time to the virtual frequency offset Δf _E (t) , and consequently to the frequency shift Δf _T (t) and thus to ensure the correct control of the vertical position z of the surface.

An example of carrying out such a calibration sub-step 1012 is shown . We find for such an example of realization a calibration coefficient β = 477.45 Hz/nm ² .

During the measurement step 102, the force probe 9 is excited during an excitation sub-step 1021. The excitation is carried out with the parameters defined during the pre-measurement step 101 described above.

Advantageously, optionally, the excitation is carried out continuously, so as to be able to measure the interaction force on several pixels. During a detection sub-step, frequency shift Δf _T is carried out from the preferably continuous measurement of the processing variable A(t).

The regulation of the tip-surface distance is preferentially carried out on the basis of the frequency shift Δf _T.

During an approach sub-step (not shown in the figures), the force probe 9 is approached to the surface, giving rise to a continuous modification of the processing variable A(t), the variation of the processing variable being a consequence of the frequency shift Δf _T.

During an offset detection sub-step 1022, the interacting frequency offset Δf _T is evaluated from the preferably continuous measurement of the processing variable A(t).

When the value of the frequency offset Δf _T reaches a setpoint value set by the user, for example -3 Hz, the approach is stopped. Scanning of the surface is then initiated.

Given that the detection of the interaction force F _int is carried out quickly (a few periods of oscillation of the probe) for each pixel, the scanning speed (ie the acquisition bandwidth) can be significantly increased by compared to the traditional nc-AFM mode using a PLL.

The method thus described also has the advantage of being able to be adapted to existing AFMs by adapting the electronic module MCU, a storage medium comprising instructions which, when executed on a computer device, implement the control of the measurement process (100). .

Claims (12)

Method for measuring (100) the interaction force (F_int) between a portion (3) of a sample (4) and a tip (10) of a force probe (9) of an AFM head (1), the force probe (9) being secured to a excitation means (12), the excitation means (12) being able to vibrate as a function of an excitation signal (e_ESE,0(t),e_ESE,N(t)), the process comprising:

• a pre-measurement step (101) during which the shape of the excitation signal (e _ESE,0 (t), e _ESE,N (t)), the excitation signal (e _ESE,0) is defined (t), e _ESE,N (t)) being configured to generate, when the tip (10) is free of any interaction force, a free deflection ((s _ESE,0 (t), s _ESE,N ( t)), the free deflection (s _ESE,0 (t), s _ESE,N (t)) forming a sinusoidal signal modulated in amplitude by an envelope signal (η(t)) belonging to the class of regular functions and defining a pulse, the force probe (9) resonating at a natural resonance frequency (f ₀ ),
• a measuring step (102) comprising
  • a sub-step of excitation (1021) of the force probe (9), during which the excitation signal (e _ESE,0 (t), e _ESE,N (t)) is applied by means excitement (12)
  • a detection sub-step (1022) during which an interaction frequency shift (Δf _T ), difference between the interaction resonance frequency (f _T ) of the force probe (9) is evaluated when subjected to the interaction force (F _int ), and the natural resonance frequency (f ₀ ), via a synchronous detection sub-step, making it possible to obtain the measurement of said interaction force.

Measuring method (100) according to the preceding claim,characterized in thatthe envelope (η(t)) is:

• zero outside a predetermined time interval having as its lower limit a first time value (t1), and as its upper limit a fourth time value (t4),
• increasing between the first time value (t1) and a second a second time value (t2) greater than the first time value (t1), but less than the fourth time value (t4),
• constant between the second time value (t2) and a third time value (t3), greater than or equal to the second time value (t2),
• decreasing between the third time value (t3) and the fourth time value (t4), the third time value (t3) being less than the fourth time value (t4).

Measuring method (100) according to the preceding claim,characterized in thatthe envelope (η(t)) is defined by one of the following functions:

• sinusoidal by piece,
• piecewise polynomial,
• or a function expressed as follows:

the second time value (t2) being different from the first time value (t1) and the fourth time value (t4) being different from the third time value (t3). Measuring method (100) according to any one of the preceding claims, characterized in that the pre-measurement step (101) and the measurement step (102) are carried out only once, the excitation signal is a simple excitation signal (e _ESE,0 (t)) and the deflection of the force probe (9) is a simple free deflection (e _ESE,0 (t)), the form of the simple excitation signal ( e _ESE,0 (t)) is obtained by applying an inverse Fourier transform to a function defined by:

E _{ESE, 0} (ω) being the Fourier transform of the input signal applied to the excitation means, S _ESE , ₀ (ω) the Fourier transform of the free deflection, G(ω) the transfer function of a resonator (8), the resonator (8) integrating the excitation means (12) and the force probe (9). Measuring method (100) according to the preceding claim, characterized in that the inverse Fourier transform is applied to a result obtained by an application to the function of a Blackman–Harris window, said result being defined by the following formula:

E _{ESE, 0} (ω) being the Fourier transform of the input signal applied to the excitation means, S _{ESE, 0} (ω) the Fourier transform of the free deflection, G(ω) the transfer function of the resonator (8), the resonator integrating the excitation means (12) and the force probe (9), and W(ω) the Fourier transform of a Blackman–Harris window. Measuring method (100) according to any one of the preceding claims,characterized in thatthe pre-measurement step (101) includes a calibration sub-step (1012) during which:

• a virtual input signal (e' _{ESE, N} (t)) is defined,
• at least one frequency of several virtual resonance frequencies (f _E ) different from the natural resonance frequency (f ₀ ) is selected, the virtual frequency (f _E ) having a value selected from a value range that can be reached by the frequency interacting resonance (f _T ),
• the chosen virtual input signal (e'(t)) is applied to the force probe (9) initially immobile and free from the interaction force, so as to generate an impulse vibration response of the tip (10) defining a calibration deflection (s' _ESE,N (t)), sinusoidal and modulated in amplitude by a regular envelope function,
• measurements of the calibration deflection (s' _ESE,N (t)), making it possible to determine a calibration coefficient (β), the calibration coefficient (β) being proportional to a processing variable A(t).

Measuring method according to the preceding claim, insofar as it depends on one of claims 4 or 5, characterized in that the virtual input signal (e' _{ESE, N} (t)) is defined by the following formula :

in which denotes the inverse Fourier transform, S _{ESE, N} (ω) the Fourier transform of the repeated free deflection, H(ω) being a modeling of an ideal harmonic transfer function of the force probe (9) entering into resonance at the free resonance frequency (f ₀ ), H'(ω) being a fictitious harmonic transfer function of the force probe (9) entering into resonance at the virtual resonance frequency (f _E ), G(ω) la transfer function of the resonator (8), and W(ω) a Blackman–Harris window. Measuring method according to any one of claims 6 or 7,characterized in thatonly during the calibration sub-step (1012), and during the measurement step (102); it includes a processing sub-step during which it is carried out

• a multiplication of a deflection (s _ESE,0 (t), s _ESE,N (t), s' _ESE,N (t)) with a complex harmonic signal making it possible to obtain a processed output signal P( t) by synchronous detection,
• a representation of the real part and the imaginary part of the processed output signal P(t) on a complex plane, the processed output signal P(t) thus represented forming a loop (13),
• a measurement of the area (A) of the loop (13), the area (A) of the loop (13) being proportional to the frequency shift (Δf _E, Δf _T ).

Measuring method according to any one of the preceding claims , characterized in that the pre-measurement (101) and measurement (102) steps are repeated over time, so as to allow the measurement of the interaction force (F _int ) over N portions, the deflection being a repeated free deflection (s _ESE,N (t)) which is expressed as follows:


(n) corresponding to the number of times the premeasurement step is carried out and T _p corresponding to a duration of the measurement. Computer program product installed on an electronic module (MCU) comprising a storage medium comprising instructions which, when executed on a computing device, implement the control of the measurement method (100) according to any one of the preceding claims. Memory medium comprising the computer program product according to the preceding claim. AFM comprising the memory medium according to the preceding claim.