4 Atomic Force Microscopy C. Fr´ etigny The atomic force microscope (AFM) is undoubtedly the most widely used of the local probe devices. It gives quick access to a wide range of surface prop- erties, including mechanical, electrical, magnetic, and other properties, with good spatial resolution. Furthermore, it can operate in air, vacuum, or solvent. There are certainly many reasons why it can be found in such a large number of research establishments. Not only are the images it provides an invaluable aid in the study of materials, chemistry and physical chemistry, but it is often used for fundamental research, wherein it has contributed to the emergence of nanoscale physics. This type of device also has applications in industry and technology. Due to the relative simplicity of the underlying principles, it is easily integrated into the microelectronic production line, where it fulfills a quality control function. Finally, it constitutes a basic element in promising data storage techniques or the fabrication of miniaturised electronic compo- nents. Here too, the AFM has a key role to play in the rise of nanotechnology. 4.1 The Device Figure 4.1 shows schematically how the AFM works. It illustrates a general feature of local probe microscopy, viz., a miniaturised sensor moves near the sample surface. The high degree of spatial localisation in the measured phys- ical quantity is made possible by the small size of the sensor and its close proximity to the surface. The sensor used in AFM is a spring-loaded can- tilever, equipped with a tip which interacts with the sample surface. A laser beam reflects off the back of the cantilever, whose deformations under the effects of interaction forces can be measured. The displacement of the spot on a photoelectric cell divided into four dials indicates the deflection and torsion of the cantilever. Displacements are achieved by the deformation of a piezo- electric tube. In Fig. 4.1, the sample moves and the sensor is fixed. In practice, one also finds the opposite system, in which the sensor scans a fixed surface.
Transcript
4
Atomic Force Microscopy
C. Fretigny
The atomic force microscope (AFM) is undoubtedly the most widely used ofthe local probe devices. It gives quick access to a wide range of surface prop-erties, including mechanical, electrical, magnetic, and other properties, withgood spatial resolution. Furthermore, it can operate in air, vacuum, or solvent.There are certainly many reasons why it can be found in such a large numberof research establishments. Not only are the images it provides an invaluableaid in the study of materials, chemistry and physical chemistry, but it is oftenused for fundamental research, wherein it has contributed to the emergenceof nanoscale physics. This type of device also has applications in industry andtechnology. Due to the relative simplicity of the underlying principles, it iseasily integrated into the microelectronic production line, where it fulfills aquality control function. Finally, it constitutes a basic element in promisingdata storage techniques or the fabrication of miniaturised electronic compo-nents. Here too, the AFM has a key role to play in the rise of nanotechnology.
4.1 The Device
Figure 4.1 shows schematically how the AFM works. It illustrates a generalfeature of local probe microscopy, viz., a miniaturised sensor moves near thesample surface. The high degree of spatial localisation in the measured phys-ical quantity is made possible by the small size of the sensor and its closeproximity to the surface. The sensor used in AFM is a spring-loaded can-tilever, equipped with a tip which interacts with the sample surface. A laserbeam reflects off the back of the cantilever, whose deformations under theeffects of interaction forces can be measured. The displacement of the spot ona photoelectric cell divided into four dials indicates the deflection and torsionof the cantilever. Displacements are achieved by the deformation of a piezo-electric tube. In Fig. 4.1, the sample moves and the sensor is fixed. In practice,one also finds the opposite system, in which the sensor scans a fixed surface.
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MirrorLaser diode
Diode
Piezotube(x. y.z )
AC
D
BCantilever
Fig. 4.1. Schematic diagram of the atomic force microscope. A piezotube displacesthe sample located just below the tip carried by a cantilever. Deformations of thebolted cantilever beam are determined by measuring the displacement of the lightspot from a reflected laser beam by means of a system of photoelectric diodes. Theopposite kind of system also exists, in which the sample is fixed and the cantileveris displaced
The system can work in air, vacuum, or liquid, and it can make measurementsat different temperatures.
An image can be formed by recording one or more characteristics of theinteracting cantilever beam, e.g., deflection, torsion, amplitude of vibration,etc., at each point of the sample. By means of a servo-system involving the zdisplacement of the piezotube, one may also control the distance between thecantilever and the surface in such a way as to hold one of these characteristicsat a constant value. The height values then give an image of the sample.
The cantilever and tip are obviously key components of the device. Fig-ure 4.2 shows several images of these components obtained by scanning elec-tron microscope (SEM). As the spatial resolution of measurements is relatedto the radius of curvature of the tip apex, one seeks to miniaturise the dimen-sions of the cantilever beam and the tip. Microfabrication processes developedfor microelectronics are used to produce them. Consequently, they are usuallymade from silicon or silicon nitride. Cantilevers with different characteristicsare used, depending on the operating mode of the AFM. Reflecting, conduct-ing, or magnetised films are deposited in certain cases. We shall also see thatthe tip can be chemically modified by tethering or depositing self-assembledlayers. Likewise, diamond tips can be used for nanoindentation experiments.In specific applications, silicon beads or carbon nanotubes can be bonded ontobare cantilevers. Table 4.1 summarises typical cantilever characteristics.
4.2 The Various Imaging Modes
The main operating modes of the AFM will be described briefly in this section.Later we shall see how to extract, apart from topographical images of the
4 Atomic Force Microscopy 93
Fig. 4.2. Scanning electron microscope images of different cantilevers and tips
Table 4.1. Typical cantilever characteristics
Shape Rectangular or V-shaped
Length 50–250 µm
Stiffness 0.01–100 N/m
Resonance frequency 10–500 kHz
Radius of curvature of tip apex 2–50 nm
Tip shape Conical or pyramidal
sample surface, supplementary information regarding the physicochemical orphysical properties of the surface.
Suppose to begin with that the sample is made from a non-deformablematerial. The tip–sample interaction is represented in Fig. 4.3 by a Lennard-Jones-type interaction force, in which the interaction is attractive at largedistances (typically, beyond a few tenths of a nanometer) due to van derWaals forces, and repulsive at very short range due to the impenetrability ofthe electron clouds associated with the two surfaces.
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Resonant modeMagnetism
Electrostatic(Topography)
Tapping modeTopographyMechanical
F
z
Contact modeTopographyMechanical
Friction
Fig. 4.3. Three AFM operating modes are located by black dots in a diagramshowing the interaction force between tip and sample. In the tapping and resonantmodes, the cantilever vibrates close to its resonance frequency, whereas the contactmode is quasi-static
(A+B)–(C+D) : Topography(A+C)–(B+D) : Friction
A B
C D
Fig. 4.4. To measure friction forces, the sample is scanned perpendicularly to itslongitudinal axis. Torsion induced by these forces is revealed by a horizontal shiftof the laser spot on the photodiodes
Contact Mode
Historically the first form of AFM, this mode operates close to the repulsiveedge of the potential. In this sense, the tip can be said to actually touchthe surface. However, with certain samples, wear and tear, and deformationscaused by the tip impair image quality. This mode is quick and easy to use,and it is often combined with simultaneous measurements of friction, adhesionor contact stiffness, described below.
Friction Mode
This is friction force microscopy (FFM) or lateral force microscopy (LFM).In contact mode, the sample can be scanned perpendicularly to the cantileveraxis. In this case, friction forces cause torsion at the end of the cantileverbeam, thus displacing the laser spot in the horizontal plane (see Fig. 4.4). Byreading this displacement, one obtains a measure of the friction forces whichcan be a true measure if the system has been suitably calibrated.
4 Atomic Force Microscopy 95
Resonant Mode
In this operating mode, which could be described as the linear resonant mode,the cantilever is made to oscillate at its resonance frequency, ‘far’ from the sur-face and with ‘small’ amplitude. The terms ‘far’ and ‘small’ are of course rel-ative and will be specified more precisely in the section dealing with resonantmodes. The gradient of the interaction force shifts the resonance frequency ofthe cantilever. As the tip oscillates relatively far from the surface, a certaindegree of spatial resolution is lost, so that this mode is not generally used fortopographical studies. However, it can serve to analyse long-range electric ormagnetic forces, by using conducting or magnetic tips, respectively.
Tapping Mode
This mode, also known as intermittent contact mode, is a nonlinear resonantmode in which oscillation amplitudes are larger and the mean position of thetip is closer to the surface. In each cycle, the tip can be said to brush againstthe repulsive wall of the surface. It is more difficult to analyse this operatingmode, which is widely used to determine sample topography. Forces applied tosample surfaces can be extremely small and contact times so short that almostno friction force occurs. One can therefore avoid deformation of the sampleand the kind of wear that is always possible in contact mode. Moreover, due tothe brevity of contact, there is no time for adhesive effects to arise. The size ofthe contact region is very small, even on highly deformable samples, and thisleads to good lateral resolution. When the sample height is servo-controlledat a constant amplitude, the phase difference between the excitation and theoscillation of the cantilever beam characterises dissipation from the system.Phase images can thus reveal slight heterogeneities in the sample surface,corresponding to different viscoelastic, adhesive or wetting properties.
4.3 Image Resolution
Very early on, images obtained by contact mode AFM were able to show thecrystallographic periodicity of certain surfaces, and this contributed signifi-cantly to the success of the method. In this case, the mechanism underlyingcontrast formation, probably caused by the jerky rubbing motion of the tip,would only appear to be possible on rather special kinds of sample with acertain degree of surface roughness on the atomic scale. It is precisely theperiodic arrangement of the surface that leads to the formation of the image,so that one could not pick out a one-off defect, for example.
Recently, dynamic mode resonant techniques have made it possible to visu-alise surface atoms under ultrahigh vacuum conditions. This operating mode,which yields high quality data, comparable with those obtained by scanningtunneling microscopy (STM), is still poorly understood and is currently under
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e)
a)
d)
b)
c)
Fig. 4.5. Finite size effects due to the tip. The measured trajectory of the tip isshown by black squares. Successive positions of the tip as it passes over the step (a)are shown in (b). Two objects of different heights lead to different lateral extensionsas shown in (c). A slightly sloping step (d) cannot be distinguished from the stepin (a). Finally, a data processing technique based on analysis of the tip positions in(b) optimises the image of the step that can be obtained with this tip (e)
active investigation. However, AFM is generally used to image surfaces on amesoscopic scale. It is the contact mechanics that determines the resolutionfor highly deformable materials, e.g., with Young’s modulus below 100 Mpa.In the case of only slightly deformable samples, the vertical resolution of im-ages is generally very good. It is only limited by the sensitivity with which theamplitude or deflection are detected (of the order of 0.1 nm) and the accuracywith which the vertical displacement of the piezoceramic is controlled (of thesame order of magnitude). For topographical studies, one can say that thevertical resolution with this method is better than the interatomic distance,whereas the lateral resolution has to be treated with great caution.
To simplify, we shall suppose here that the AFM operates in contact modeas a perfect tactile sensor passing over a non-deformable surface. However,similar conclusions can be drawn in other modes. What is the resolution ofthis imaging mode? We shall see that the answer is not as simple as in opticalor electron microscopy, for example.
The diagram in Fig. 4.5a shows the path followed by an AFM tip on onescan over a vertical step. The broadening and distortion of the shape of theobject are due to the bulkiness of the tip (Fig. 4.5b), which feels the presence ofthe step before its apex reaches the position vertically above the step edge.The point of contact between tip and surface remains at the step edge untilthe tip apex has passed through the vertical above this point, with the image
4 Atomic Force Microscopy 97
Fig. 4.6. Left : An island composed of a monolayer of monodispersed silicon beads.Centre: Single bead imaged with a pyramidal tip. The distortion caused by the tipgeometry is clearly visible. Right : Schematic of image formation
arising from the flanks of the tip. Beyond this, the trajectory is once againdictated by a contact between the tip apex and the surface. The picture hereis two-dimensional. In three dimensions, it is clear that other situations canarise with regard to the tip–sample contact point, according to exactly thesame principle.
The geometric effects due to the finite size of the tip complicate any discus-sion of resolution. From Fig. 4.5c, it is clear that the lateral broadening of anobject of given width will depend on its height. The lateral resolution of AFMcannot be described by an instrumental profile as it can in optical microscopy,for example. In fact, the imaging process is not linear. Figure 4.5d shows thata slightly sloping step will give exactly the same image as the vertical step inFig. 4.5a. This means that information can be completely lost by the imagingmechanism, in a way that would not happen with a convolution. Althoughthe term is not strictly applicable, one still speaks of the tip convolution.
Figure 4.5e uses the successive positions of the tip from Fig. 4.5b to deter-mine an optimal boundary beyond which the actual step surface cannot belocated. This data processing technique can be used to refine the resolution ofimages acquired by a tip of known shape. In order to discover this shape, onecarries out the opposite investigation on a rough sample surface: at each pointof the image, no part of the tip can be located in the half-space below therecorded surface. Hence, by successive elimination of known regions, one canreconstruct the shape of the tip. Several algorithms have been put forward toachieve these two aims.
Figure 4.6 illustrates the broadening effect produced by a pyramidal tip.The first image (5 µm× 5 µm) shows a monolayer island of silicon beads. It isknown by other means that each bead is perfectly spherical. The magnificationof a single bead shown by contours in the second image (900 nm × 900 nm)reveals a distinctly pyramidal shape. The diagram on the right shows themechanism leading to broadening in this case.
The resolution cannot be defined by a simple number in contact AFM.The finite-size effect due to the bulkiness of the tip increases as the tip gets
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blunter and the aspect ratio decreases. For example, a broken tip will revealall the smaller details on the sample surface by a single characteristic shape.A broken tip can often be identified through this behaviour.
4.4 Contact Mode:Topography, Elasticity and Adhesion Imaging
The contact mode can be described on the basis of the so-called force curve,which represents the variation of the cantilever deflection as a function of thesample height (tip–sample separation), as exemplified in Fig. 4.7. Without vi-brating the cantilever, the vertical position of the sample is varied and thecantilever deflections are recorded. The approach paths, moving towards con-tact (from right to left in the figure), differ from the retract paths, in whichcontact is broken (from left to right). The graph can be analysed into severalparts:
• Approach. Far from the sample surface, the interaction forces are veryweak and there is almost no deflection of the cantilever. This is the hor-izontal part of the curve on the right (return trip between A and J).In vacuum, air, and sometimes even liquids, the non-contact tip–sampleinteraction is attractive and causes a slight downward deflection of thecantilever (negative α) which is generally barely visible. During approach,this slightly deflected position becomes unstable at J and the tip jumpsto contact at B. The corresponding instability is revealed by the vertical
Approach
Retract
Jump to contact
C, C'
A, A'J
B
S'
D'
Zsample
Adhesion
Fig. 4.7. Force curve on an ideal non-deformable material. The deflection of thecantilever beam is graphed as a function of the vertical position of the sample duringan approach–retract cycle. Once contact is established at B, the deflection increasesin proportion with the rise of the sample surface, where BC corresponds to approachand C′S′ to retraction. As the tip moves away, contact is only broken when theadhesive forces can no longer withstand the separating force exerted by the cantilever(point S′S)
4 Atomic Force Microscopy 99
Indentation
Zsample
Fig. 4.8. Force curve for a deformable material. The dashed straight line segmentshows the slope of the linear contact region which would be measured on a perfectlyrigid material like the one represented in Fig. 4.7
jump JB of the curve during approach. If the sample is further raised to-ward the tip, for a very rigid material, the deflection will increase linearlywith the sample height (BC).
• Retraction. The force curve begins by retracing the approach path. How-ever, it goes beyond the zero force position and even the point where thejump to contact occurred. This is due to adhesion and is indicated bythe curve C ′S′. One must in fact exert a separating force on the contactto break it. Until the breaking point is reached, the trajectory continuesalong the straight line characterising contact. When the breaking point S′
is reached, the cantilever moves back to the very slightly deflected positionat D′.
Adhesion is thus manifested through hysteresis in the force curve. It is causedby several factors: van der Waals forces, as one would expect, but also elec-trostatic forces and capillary forces in liquids, etc. These interactions are thenaffected by the pH, ionic forces, and so on. On the basis of these comments, itis easy to see that AFM is highly sensitive to the physicochemical propertiesof surfaces.
Operation of this instrument can also be affected by the mechanical prop-erties of the sample. Figure 4.8 shows deformations one might expect froma rather deformable material. Once contact has been established, the tip ispressed against the material by the elasticity of the cantilever. It may thenpenetrate into the material, so that the recorded deflection will be less thanwould be obtained on a perfectly rigid sample, as indicated by the dashedstraight line segment in the figure. As the sample is raised, the increase in thedeflection is thus slower and is characteristic of the stiffness of the contact.(A simple model can be made by adding the stiffness of the cantilever beamand the stiffness of the contact in series.) Likewise, the contact may not breakabruptly, since the material may exhibit some degree of creep before completerupture occurs. Moreover, if the sample is viscoelastic, the curve will distort
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when one modifies the operating frequency, tip shape, maximal penetrationin the material, and so on.
Consequently, if the sample height is periodically modulated, the deflec-tion response to the modulation frequency is a measure of the contact stiffness.This is called elasticity mode. Likewise, a periodic modulation can be imposedin such a way that the tip goes through the minimum characterising adhesion.The characteristics of the deflection signal obtained in this way then consti-tute a signature for adhesion. One can also record the whole force curve ateach point of the image and then process the data, but the huge amount ofinformation acquired in this way makes this approach too time-consuming.It is preferable to retain only the main characteristics of the curves throughappropriate signal processing. This is what is done in the pulsed force mode.
An important characteristic of all these operating modes is that they canoften be used at the same point of the sample, without changing the can-tilever beam and sometimes even simultaneously. For example, an elasticityor friction image combined with a height image often reveals sufficient detailto understand the physicochemical structure of a sample, without necessarilyneeding to be concerned with the dissipation or tribological properties in theirown right. Shapes and sizes brought out by the contrast are used directly. It isnot then necessary to calibrate the instrument or to provide a precise modelof operation. To avoid certain experimental artifacts or obtain more quantita-tive measurements, on the other hand, the operation of these different modesmust be analysed in more detail.
4.4.1 Friction Mode
We have seen that the lateral force is measured directly from the cantilevertorsion, which is in turn detected as a displacement of the laser spot in thehorizontal plane.
Figure 4.9 shows a direct application of friction mode imaging. A sampleof latex beads has been heated above the glass transition temperature of the
Fig. 4.9. AFM images of the surface of a latex film. Left : Surface topography.Right : Simultaneously acquired image revealing local friction variations associatedwith physicochemical properties of the material
4 Atomic Force Microscopy 101
polymer in order to create an apparently homogeneous material by interpen-etration of molecular chains. The height image shows nothing more than acertain roughness on the micrometer scale, as would be found for many poly-mers. However, the simultaneously acquired friction image shows a highlyregular arrangement of domains measuring some 100 nm across, the size ofthe latex beads. The material thus has a memory of its preheated structure,and we have the demonstration that homogenisation has not been perfect.Another point is worth noting: two types of bead can be made out in thefriction image. Their tribological behaviour differs and their physicochemicalproperties too (viscoelastic modulus, surface energy, or other). Even thoughone cannot proceed much beyond the simple observation without further ex-periments, the friction mode has at least brought out an important feature ofthe material in a simple and direct way.
4.5 Resonant Modes
The cantilever, effectively a beam clamped at one end, is a good mechanicaloscillator with a low level of dissipation, mainly caused by the viscosity of theambient medium. Quality factors of around 400 are common in air, several tensof thousands in ultrahigh vacuum, and 10 in water. The resonance frequency ismodified by tip–surface interactions. This idea is applied in the AFM resonantmodes. Two techniques are used:
• One can drive the cantilever at a fixed frequency close to its resonancefrequency and follow variations in the amplitude, and possibly also thephase. This is the method most commonly used for experiments carriedout in air. In this case, the sample height is servo-controlled at a constantamplitude. This technique is sometimes called amplitude modulated AFM(AM-AFM).
• One can also set up a phase-locked loop which holds the vibration am-plitude and phase difference at pre-assigned values. The servo-controlledsystem then begins to self-oscillate and its resonance frequency is contin-uously monitored. This is dynamic mode or frequency modulated AFM(FM-AFM).
We shall not discuss the second method in further detail here. It is mainlyused in ultrahigh vacuum experiments where the natural quality factor is veryhigh, so that the oscillator has a very long reaction time (of the order of Q/ω0).However, the discussion below applies more or less directly to this scenario.The reference [1] provides a recent and very complete review of oscillatingmethods in force microscopy.
4.5.1 General Principles
In order to describe the operation of resonant modes, the general problem thathas to be solved concerns the vibrational modes of a beam that is clamped at
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one end, whilst the other is subjected to a force field. Although a completestudy is possible, we shall not present one here. The high quality coefficientsobserved experimentally allow one to restrict the analysis to a single mode forwhich the equation of motion reduces, to a very good approximation, to thatof a harmonic oscillator subjected to a force field. The appropriate equationfor this system is
x + 2βx + ω20x = γ cos ωt +
f(D, t)m
, (4.1)
where x is the position coordinate of the oscillator, which is in the presentsituation the displacement of the tip from its equilibrium position, ω0 is theresonance frequency of the oscillator, γ is the amplitude of the excitationat frequency ω, β is a dissipation term such that the quality factor is givenby Q = ω0/2β, and m is the effective mass of the oscillator, determined byω0 = k/m, where k is the cantilever stiffness. The function f(D, t) is the tip–sample interaction, where D is the tip–sample separation when the cantileveris not deflected.
The simplest case occurs when the interaction force depends only on thetip–sample separation D+x. More complex behaviour is observed if dissipativebehaviour comes into play due to adhesion, viscoelasticity, or capillarity.
Even in the very simple case described by f(D, t) ≡ F (D+x), the equationgoverning the system is not generally linear:
x + 2βx + ω20x = γ cos ωt +
F (D + x)m
. (4.2)
4.5.2 Linear Resonant Mode
It is easy to describe the linear resonant mode, which corresponds to a non-dissipative interaction and a very low amplitude oscillation far from the samplesurface (x D). Expanding the interaction to first order in x, (4.2) gives
x + 2βx + ω20x = γ cos ωt +
F (D)m
+F ′(D)
mx ,
where F ′(D) is the gradient of the force at the central position of the oscilla-tion. The constant term shifts the rest position of the tip F (D)/k, which isgenerally negligible compared with the oscillation amplitude. We thus obtainthe new equation, expressed relative to the new mean position,
x + 2βx +[ω2
0 − F ′(D)m
]x = γ cos ωt .
This is the equation of a harmonic oscillator whose resonance frequency ω′0
satisfies
ω′20 = ω2
0
[1 − F ′(D)
k
].
4 Atomic Force Microscopy 103
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.001.632 1.636 1.640 1.644 1.648 1.652
123
4
Frequency (KHz)
Am
plitu
de (
a.u.
)
Fig. 4.10. Resonance spectra of a cantilever at various distances from an MgOsurface. Curves numbered from 1 to 5 correspond to tip–sample separations of 80,60, 50, 40 and 10 nm, respectively. Note the asymmetry of the peak at shorterdistances [2]
The resonance frequency of the cantilever is thus shifted by the gradient of theinteraction. As the magnitude of the interfacial forces decreases with distance,an attractive (repulsive) interaction causes a reduction (increase) in the reso-nance frequency of the system, as one would expect qualitatively.
This linear resonant method has been used to carry out measurements oflong-range interfacial forces. It is now commonly used to obtain electrostatic ormagnetic images of surfaces. Several applications will be described in Sect. 4.6.
The above analysis assumes that a first order expansion of the interactionis adequate. This interpretation is borne out by recordings of the resonancespectrum, which is a characteristic feature of a harmonic oscillator. In prac-tice, the tip must oscillate rather far away from the sample surface for thislinear approximation to be valid. At shorter distances, the resonance peak isdistorted and a more complete analysis of the equation is in order (see be-low). Figure 4.10 shows the resonance spectra of a tungsten tip close to anMgO surface [2]. The tip–sample separations are 80, 60, 50, 40 and 10 nm inspectra 1–5. One first observes a simple shift of the peak, but then at smallerseparations the resonance spectrum becomes asymmetrical. This behaviour,characteristic of nonlinear oscillators, is discussed in the next section.
4.5.3 Nonlinear Resonant (Tapping) Mode
If the tip vibrates close to the surface, or if the amplitude of vibration is large,a first order description of the interaction is no longer adequate: the oscillator
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is nonlinear. In practice, however, the deflection remains quasi-sinusoidal. Infact, due to the high quality factor of the resonance, nonlinearities do not leadto anharmonicity in the response. Instead, they manifest themselves in the waythe response depends on the control and interaction parameters. Returningto (4.1), viz.,
x + 2βx + ω20x = γ cos ωt +
f(D, t)m
,
we seek a solution of the form x = A cos (ωt − ϕ). In this case, the inter-action is a periodic function of time that can be represented by its Fourierseries expansion. A complete analysis of this operating mode would go be-yond the scope of this discussion. However, one important consequence of thisdescription concerns the analysis of dissipation due to the tip–sample inter-action. The energy dissipated by the interaction over one cycle, denoted byUdiss =
∮f(D, t) dx, can be expressed as
Udiss = Amγ
(2Aβω
γ− sinϕ
). (4.3)
Hence, if the amplitude is held constant when an image is scanned, the phaseoffset ϕ between the excitation and the oscillation of the cantilever constitutesa measure of the local dissipation Udiss due to the tip–sample interaction, allother parameters in (4.3) remaining constant.
The experimenter can draw several useful conclusions from this analysis ofthe tapping mode. The nonlinearity of the system makes it highly sensitive tolocal variations in the physicochemical properties of the sample surface. Onesuggestion has been to use the slope of approach–retract curves (analoguesof the force curves for this mode) to estimate the elastic modulus of thematerial beneath the tip. One may discuss the local properties of the materialon the basis of a point-by-point study of the image. However, once again, theconverse problem is difficult to solve: the various models tend to provide onlyqualitative answers to questions concerning the underlying causes of imagecontrast.
Although the tapping mode is a nonlinear operating mode, the linear be-haviour of the shift in resonance frequency can be very useful for a roughanalysis of what is happening. Hence, if we seek to understand the effect ofthe choice of working frequency on the tip–sample interaction, we may refer tothis model. Figure 4.11 compares experimental situations for the same servo-controlled amplitude (the set point) Asp and working frequencies ω0±∆ω sym-metrically placed with respect to the resonance frequency of the free oscillator,leading to the same free amplitude Af . For a rather repulsive interaction, theresonance peak is shifted towards higher frequencies. An adjustment of the freeresonance leftwards to ω0 −∆ω will satisfy the condition for servo-controllingthe amplitude for a smaller shift than the one that would be required for anadjustment to the right. The corresponding interaction will thus be reduced,
4 Atomic Force Microscopy 105
Fig. 4.11. Two symmetrical adjustments on either side of the resonance frequencylead to very different operating interactions. For the same working amplitude, anexcitation to the right of the resonance peak requires a bigger interaction than anexcitation of the same amplitude to the left of the peak
and one can image in a ‘softer’ repulsive mode. This should help to reduce tipdamage.
Simplifying somewhat, the free vibration amplitude of the cantilever beamconstitutes a measure of the energy contained in the oscillator. The smallerthis amplitude is, the more the operation of the system is affected by non-contact interactions or adhesion effects. On the other hand, large amplitudeswill be more sensitive to local mechanical differences at the sample surface.These deliberately simplified considerations have to be adapted to suit thecase. For example, the radius of the tip apex is an important factor whichfixes both the strength of non-contact forces and the contact stiffness.
Given the high sensitivity of the oscillator with regard to the details ofthe tip–sample interaction, phase images tend to incorporate a great deal ofinformation. Of the two parameters which describe the oscillation (amplitudeand phase), the amplitude is held constant by the feedback loop and tends tocharacterise the topography, whilst the phase evolves in a way that dependson the physical parameters of the interaction, such as viscoelasticity, pres-ence of contamination, adhesion, and so on. The precise cause of the phasecontrasts generally remains unknown, but the information gathered can use-fully contribute to one’s understanding of a sample surface. Figure 4.12 showsheight and phase images of a block copolymer sample. Although the topogra-phy is rather flat, differences between local mechanical properties are clearlyrevealed in the phase image.
Another consequence of this high sensitivity to the details of the interac-tion is the possibility of artifacts in height measurements. For materials withdifferent viscoelastic moduli or adhesion properties, the change in the form ofthe interactions does not guarantee that the feedback device will exactly cor-rect for the height differences. Hence, height measurements on DNA molecules
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Fig. 4.12. Topography (A) and phase offset (B) measured simultaneously on asample of poly(butadiene-b-ethyleneoxide) (500× 500nm2). B and C show the samequantities after partial crystallisation. 12-nm spheres crystallise independently intoa hexagonal structure. Dark regions correspond to regions of melt polymer in whichviscoelastic dissipation is high [14]
deposited on mica vary significantly depending on the operating conditionschosen for the observation (set point amplitude, drive amplitude, working fre-quency, etc.): the major differences in chemical nature between the substrateand adsorbate give rise to very different oscillator behaviour, and these inturn lead to the same oscillation amplitude for very different tip altitudes.In practice, these effects are relevant when measuring very small heights andwhen dealing with highly dissimilar materials. In other situations, they arenegligible. Such measurements can be confirmed by analysing the contrast asa function of the operating point (or approach–retract curves, which amountsto the same thing).
Tapping Mode in Liquid and in UHV
In a liquid, resonance conditions of the cantilever are modified by the sur-rounding medium. The added mass, i.e., the mass of liquid that the beammust displace in order to vibrate, reduces the resonance frequency, e.g., by afactor of about 2 in water, while the viscosity diminishes the quality factorin proportion to the viscosity ratio, e.g., by a factor of about 50 in water.In order to work in such conditions, several excitation methods have beensuggested, including vibration of the cell for a liquid, and mechanical or mag-netic excitation. These techniques have found many applications to problemsin biology.
In ultrahigh vacuum (UHV), resonance quality factors are very high(∼ 105). Characteristic times for the oscillator to reach equilibrium would
4 Atomic Force Microscopy 107
Fig. 4.13. 10 × 7 nm2 image of the surface of a silicon (111) sample with 7 × 7reconstruction, as obtained by dynamic mode AFM in ultrahigh vacuum [4]
10
1
0.1
0 10 20 30 40
F/R
(m
N m
–1)
Distance (nm)
10–1M 10–2M10–3M
10–4M
Fig. 4.14. Interaction force vs. distance for a silicon sphere and a surface of thesame material, as obtained by analysing force curves. The bead was bonded to thecantilever at the position of the tip (right) [5]
prevent excitation at a fixed frequency. For this reason, and apparently alsofor reasons of sensitivity, most work has been done using a phase-locked loop.In fact, it is often by using this method that genuine atomic resolution hasbeen obtained. An example is shown in Fig. 4.13, which is an image of the7 × 7 reconstruction of a silicon (111) surface.
4.6 Force Measurements
The ability of AFM to make local measurements of weak forces, either con-tact or non-contact, quickly drew the attention of a wide cross-section of thescientific community: from cellular biologists concerned with questions of me-chanical stimulation, to physicochemists faced with fundamental problems ofinterfacial interaction potentials. The first approach was to use force curvesfor these explorations, as discussed above. To illustrate this application andalso the use of the linear resonant mode, several examples will be selectedfrom the literature.
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4.6.1 Non-Contact Measurements
Force curves can be used to find a formula for the forces between the tip apexand the sample surface. In a liquid, electrostatic interactions arising from thepresence of ions are revealed by analysing such curves. The DLVO model,due to Derjaguin, Landau, Verwey and Overbeek, which purports to describethis effect, has been confirmed for a great many combinations of tip andsample. Figure 4.14 shows the interaction between a silicon bead of diameter3.5 µm and a surface of the same material, for various concentrations of NaCl.Observations have been fitted to the DLVO model (continuous curves).
4.6.2 Elasticity and Adhesion Measurements on a Single Molecule
Having treated the tip, specific molecules tethered or adsorbed onto the sam-ple surface can be selectively adsorbed by a kind of fishing technique. Thepoint on the force curve where the adhesion force gives way corresponds toaccidents relating to typical features of the elongation and the rupture ofadsorption of individual molecules. Figure 4.15 shows the results of an exper-iment carried out with dextran polymers. Specific bonds can form betweenmolecules attached to the tip and functional groups in the dextrans.
Since the end-to-end length (or contour length) of the polymer is arbi-trary, the curves have been renormalised to a fixed length. The experimentally
Conformation under high tensile stress3000 pN/Å1400
700
0
3 4 5 6
Conformation under low tensile stress800 pN/Å
Entropic elasticity
Forc
e (p
N)
Normalised polymer length (Å)
Fig. 4.15. A chemically modified tip can selectively adsorb molecules attached tothe sample surface. With dextran polymers, the force–distance curves thus obtainedreflect the elasticity of the molecules. As the end-to-end length of the polymers is ar-bitrary, the curves must be renormalised to a single length to reveal the mechanismsgoverning molecular elasticity [6]
4 Atomic Force Microscopy 109
determined relationship between the force and the elongation is shown inFig. 4.15. In the same figure, a numerical simulation leads to an interpreta-tion of the two observed regimes in terms of entropic and enthalpic elasticity.The fact that the measurements can be repeated on the same molecule con-firms the elastic character of the mechanism. These experiments require ahighly stable instrument and extremely good signal-to-noise ratio in order tomeasure forces of the order of 100 pN in a repeatable manner.
4.7 Magnetic and Electrical Measurements
As we saw above, long-range forces can be investigate in linear resonant mode.One generally uses a double scanning technique, where the first scan in inter-mittent contact determines the surface topography, and a second scan carriesthe tip over the same points but at a constant distance above the surface,typically 10–20 nm, which eliminates van der Waals forces. In some cases, thecantilever deflection can simply be recorded in static mode.
4.7.1 Magnetic Measurements
Magnetic tips can be magnetised either along or perpendicularly to their axisusing a magnet. Magnetic AFM is used to analyse domain structures of al-loys, small particles, or magnetic contacts, not to mention vortex lattices insuperconductors. It is also commonly used to study recording systems, suchas magnetic tapes, hard or magneto-optical disks, recording heads, and soon (see Fig. 4.16). The energy of interaction between tip and sample can beviewed as the energy of the field produced by the sample on the dipoles of thetip, although this is of course the same as the energy of the field producedby the tip on the sample. There is therefore a mutual interaction which canperturb measurements when magnetic rigidities are low in either the tip orthe sample. We should also mention attempts to obtain mechanical detectionof nuclear magnetic resonance (NMR) for a small number of spins.
4.7.2 Electrical Measurements
Electrical measurements are important in microelectronics. However, like theother forms of local characterisation, they can prove invaluable for revealingphysicochemical differences in sample surfaces. They can also help with diffi-cult problems such as triboelectricity, when geometric conditions allow it, i.e.,no roughness on the scale of the tip–sample contact.
Several types of electrical measurement are possible with AFM. For res-onant modes, one uses a conducting tip which interacts with the sample viaCoulomb forces. Doping of semiconductors, presence of localised charges, elec-trical polarisation, or work functions for removing electrons from the surface
110 C. Fretigny
Fig. 4.16. Simultaneously acquired 14 × 14 µm2 images of a hard disk. Left :Topography. Right : Magnetic image obtained by measuring the phase shift ofthe oscillation when the tip is at a mean height of 50 nm above the surface. Seewww.ntmdt.ru/applicationnotes/MFM
can all be revealed by a change in the resonance characteristics of the can-tilever. One can directly measure the gradient of the force on the tip when itis held at a reference potential, as we saw in Sect. 4.5.2, or when it is excitedby an alternating voltage.
In the first case, electric force microscopy (EFM) is particularly useful forstudying components, such as working transistors. The passivation layer doesnot prevent the electric field produced by polarised parts from interactingwith the tip. This brand of AFM is also widely exploited in microelectronics,where it serves to analyse defects such as cut tracks, charge accumulationregions, and so on. Figure 4.17 illustrates the method for carbon nanotubeselectrically connected to two electrodes.
When a potential difference is applied between the tip and sample, it is theelectrical characteristics of the interaction that create the resonance. This isknown as Kelvin force microscopy (KFM). Since the applied force is capacitive,it depends quadratically on the potential. For a potential difference of the formV + v sin ωt, the force will include, apart from its mean value, a contributionat the excitation frequency ω, with strength proportional to the product vV ,and another at twice the excitation frequency with strength proportional tov2. For example, the cantilever can be driven at a frequency equal to half itsresonance frequency (ω ≈ ω0/2) and the component of the vibration at ω0
is then measured. One thus has access to the value of the direct voltage V ,which includes the direct voltage applied to the tip, but also certain electricalcharacteristics of the sample. On a metal, the tip–sample contact potentialcontributes to this potential difference. By this method, one can producecontact potential images. Due to their high sensitivity towards contamination
4 Atomic Force Microscopy 111
Fig. 4.17. (a) Topographic image of three carbon nanotubes connected between twogold electrodes on an oxide surface. (b) EFM image before cutting when a potentialdifference is applied across the electrodes. (c) EFM image after cutting the tubesby applying a voltage step at the points indicated by arrows in the first two images.In these images, lighter regions represent higher voltages [7]
layers, such images can prove useful in fields other than microelectronics. Onan insulating layer, one can also detect the field created by localised charges.Triboelectric phenomena have also been investigated by this technique. On asemiconductor, the voltage will depend in a nonlinear way on the potentialdifference applied between the tip and sample, whereupon it may reveal dopinglevels in the material.
Let us mention another method that serves to measure the doping charac-teristics of semiconductors: scanning capacitance microscopy (SCM). In thistechnique, the tip is held in contact with the semiconductor sample and thecharge of the metal tip–semiconductor capacitance across the passivation layeris modulated at several kHz by applying an electric field. (The depth of thedepletion layer is modified.) A UHF resonance circuit is connected to thiscondenser in order to measure the small variations in the capacitance, with asensitivity as high as 10−18 farad.
Yet another technique for electrical analysis consists in measuring the con-tact resistance of the tip in scanning spreading resistance microscopy (SSRM).
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4.8 Measuring Mechanical Properties
AFM can be used to describe the mechanical properties of the tip–samplecontact. From here, one can then identify the Young’s modulus, the hardnessand local tribological properties of the surface. The corresponding methodsare described in this section. Many techniques employed on a macroscopicscale can thus be implemented locally with an AFM, e.g., studies of adhesionor scratch tests, which are not discussed here.
4.8.1 Nanoindentation
With its force sensor and displacement control system, the AFM can be used asa nanoindenter. Force curves are considered here as instrumented indentationcurves. For hardness measurements, very stiff cantilevers and hard tips aregenerally used, the tip being made of diamond, or coated with diamond.
The geometry of the system leads to some difficulties in interpreting thedata. The cantilever axis is tilted with respect to the plane of the surface.Hence a vertical displacement is always accompanied by a lateral force on thecontact. To rectify this problem, the sample can be shifted laterally duringthe experiment in such a way as to cancel the displacement of geometricalorigin.
The results obtained by this approach are often rather qualitative. Fig-ure 4.18a shows experimental curves for different tip–sample pairs. When aquantitative interpretation is required, it must be obtained by modelling thecontact. For plastic deformations, one can also measure the image size of theimprint left by the indenter. Figure 4.18b thus shows two groups of indenta-tions obtained with loads of about 15, 20 and 25 µN on a 15-nm diamond film.The deepest indentation is less than 10 nm.
4.8.2 Measuring Contact Stiffness
With the help of the force curves, one can also measure the elastic or viscoelas-tic indentation of the tip in materials whose plastic threshold is not too low,e.g., elastomers. Although the method has sometimes been used for highly de-formable materials, there are several drawbacks. To achieve an acceptable levelof sensitivity, the experiment must be carried out with a cantilever stiffnessclose to the contact stiffness. But under such conditions, the measurement ismade neither by displacement nor by imposed force, and this complicates theanalysis in the case of viscoelastic samples. Moreover, as in the case of inden-tation, the lateral component of the force on the contact, arising due to thegeometry of the experiment, contributes to the recorded deflection. Finally,adhesion phenomena, which are often significant on the nanometric scale andbring in their own time constants, are liable to push the tip–sample systeminto a non-equilibrium state during the force curve measurement. It is thenimpossible to analyse results in terms of Young’s modulus.
4 Atomic Force Microscopy 113
Diamond / Diamond
Diamond / Graphite
Gold / Nickel
10 nm
1 nm
a.
Fig. 4.18. Indentation experiments. Left : Force curves obtained with different tipson nickel, graphite and diamond [8]. Right : Indentations obtained with loads ofabout 15, 20 and 25 µN on a 15-nm diamond film. The imprints are all less than10 nm deep (see www.veeco.com)
It is better to work at a fixed point, without scanning, by laterally modify-ing the sample position. Figure 4.19 shows the results obtained on an elastomerin viscoelastic phase. It has been possible to determine the Young’s modulusof the material from measurements of the lateral stiffness of the contact un-der sinusoidal excitation. The results are compared here with a macroscopicmeasurement of this property.
4.8.3 Contact Resonance Frequency
A useful alternative to these methods consists in measuring the resonancefrequency of the cantilever when the tip is actually in contact. Indeed, thebeam vibrates in a characteristic way for the boundary condition imposedby contact. By virtue of extremely accurate frequency measurements, theamplitude of the deflections can be maintained at a very low level. In this way,one can greatly reduce the lateral effects mentioned above. This technique hasbeen successfully used to measure the Young’s modulus of metallic nanotubeswith diameters of a few tens to a few hundred nanometers (see Fig. 4.20).
114 C. Fretigny
Frequency (Hz)
10
1
0.1
0.0110.1 10 100 1000
E' E
" (M
Pa)
E' AFM
E' macroE" macro
E" AFM
Fig. 4.19. Complex viscoelastic modulus of a latex film deduced from measure-ments in the static friction regime. Values are compared with those obtained on amacroscopic scale. (Courtesy of C. Basire and C. Fretigny)
Diameter (nm)
35
30
25
20
15
1050 100 150 200 250
Ela
stic
mod
ulus
(G
Pa) Pb
Fig. 4.20. Young’s moduli of lead nanotubes as determined by measuring the res-onance frequency of the cantilever when the tip is in contact. The macroscopicmodulus, indicated by the horizontal dashed line, corresponds for the larger diame-ters, but not for smaller ones. In fact, the modulus increases significantly as the sizedecreases. The nanotubes are scattered on a porous membrane and measurementsare carried out on structures that are caught in the pores. (Courtesy of S. Cuenot,C. Fretigny, S. Dumoustier-Champagne and B. Nysten)
4.8.4 Friction Forces
We have seen that the friction force recorded through the torsion of the can-tilever during lateral scanning can be considered as a simple way of diagnosingthe distribution of physicochemical parameters on a surface. Moreover, AFMcan tackle more fundamental questions concerning friction. In particular, theabsence of roughness effects on the scale of the AFM tip means that one canseek the underlying microscopic laws.
4 Atomic Force Microscopy 115
Applied force (nN)
25
20
15
10
5
0
Fric
tion
forc
e (n
N)
0–20–40 20 40 60 –60
Mica
Fig. 4.21. Friction law measured on a mica surface. Results have been fitted to a2/3 power law in the force [9]
In some situations, a 2/3 power law is measured with respect to the load(shifted by an adhesive force). Such a case is represented by the data shownin Fig. 4.21 for a mica sample [9]. This relationship suggests a constant shearrate at the interface between the sample and the spherical tip, where the sizeof the contact zone varies according to the DMT theory.
4.9 Applications in Nanotechnology
With a view to developing the various branches of nanotechnology, manyresearch teams are engaged in the search for experimental protocols to etch, towrite or to manipulate molecules and small structures on surfaces. The contactand tapping modes have been successfully used in a wide range of experimentalcontexts. A very complete review of these nanofabrication techniques can befound in [16]. There are many examples: displacement of carbon nanotubes orC60 molecules, local evaporation of gold deposited as a thin film on the tip,anodic oxidation of a surface in a damp atmosphere, wear, species desorptionby heating, to name but a few. The tip can function as an electrode in anelectrochemical reaction, a local heat source, and so on. A pierced cantilevercan behave as a moving mask for an atomic beam, the so-called nanostencil. Inthe present section, we shall restrict ourselves to three examples to illustratethis diversity.
By oxidising silicon in air under the AFM tip in intermittent contact, onecan produce linear oxide structures of width 15 nm and height about 1 nm. Bychemical etching, silicon nanowires can then be fabricated. Figure 4.22 showsa 0.7 × 0.7 µm2 image of oxide lines with modulated width.
Another form of writing is obtained by using the AFM tip like the nib of afountain pen. After dipping the tip into a thiol solution, the gold sample–tip
116 C. Fretigny
Fig. 4.22. 0.7 × 0.7 µm2 image of silicon oxide nanowires obtained by chemicaletching under the AFM tip [10]
Fig. 4.23. Drawing obtained by so-called dip-pen lithography, in which thiol mole-cules are deposited on a gold surface. The line thickness is 30 nm [11]
contact is used to transfer molecules through the water meniscus, whence theyform a self-assembled lattice. A line thickness of 30 nm has been obtained bythis very simple technique. Figure 4.23 shows an example.
Several investigations have shown that it is quite feasible to build orderedstructures from a random deposit of small objects. The example in Fig. 4.24(5-nm gold particles on silicon treated with poly-L-lysine) was obtained innon-contact mode by taking advantage of operating points in different regionsof the tip–sample interaction potential.
One important aim in this kind of study is the reading or writing of infor-mation using AFM tips, with a view to increasing storage density. An IBMteam in San Jose has perfected a micromechanical system on a rotating disk onwhich, in read-only, the tip recognises structures of 100 nm. The integrationdensity is 10 gigabits per square centimeter, 100 times greater than a CD-ROM. Another device, still in the experimental phase, this time single-writeand read, uses an AFM tip heated by electrical pulses. This creates dips in a
4 Atomic Force Microscopy 117
Fig. 4.24. Starting from a random deposit, 5-nm gold particles are displaced so asto form a straight line [12]
Fig. 4.25. Printed circuit obtained using 10 cantilevers in parallel (2× 2 mm2) [13]
polymer layer which it can then interpret as bits in read mode. The densityobtained here is 5 gigabits per square centimeter with a read rate of 10 Mb/s.The series write restriction when using an AFM tip makes it particularly slowand hence economically unviable. In order to go beyond the 200–300 Mb/srange of existing magnetic systems, research teams therefore build parallelarrays of cantilevers. A 2D device comprising 32 × 32 tips over an area of3× 3 mm2 has been achieved. If structures and spacings of the order of 40 nmcan be produced, this should lead to speeds as great as a few hundred Mb/s.
The technology of cantilever beams acting in parallel has also been devel-oped by Quate at Stanford University for the purposes of imaging or actingvery quickly over large areas. Figure 4.25 shows a 2 × 2mm2 printed circuit(image 25 million pixels), obtained using 10 cantilevers in parallel for 30 min.
To end this section, we should mention the development of a force feedbacknanomanipulator, which is a system coupling the microscope to a virtual real-ity interface in such a way as to give the user the feeling of actually being on
118 C. Fretigny
the sample surface, with a magnification scale factor of one million [15]. Thistool may prove useful in the development of complex fabrication processes.
4.10 Conclusion
Force microscopy is now widely used for routine analysis in many research es-tablishments. However, the AFM is far from being a simple imaging system. Itcan also be considered as the basic component of a wide range of instruments,with applications ranging from fundamental physics to nanolithography, andit is in this context that it plays such an important role in the developmentof nanotechnology. This technique is still moving forward rapidly. Apart fromthe examples chosen here to illustrate the main operating modes of the AFM,there are many other uses, each fulfilling specific requirements in fundamentalor applied science.
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