Aspects of scanning force microscope probes and their effects on dimensional measurement

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2008 J. Phys. D: Appl. Phys. 41 103001

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 41 (2008) 103001 (46pp) doi:10.1088/0022-3727/41/10/103001

TOPICAL REVIEW

Aspects of scanning force microscopeprobes and their effects on dimensionalmeasurementAndrew Yacoot1 and Ludger Koenders2

1 National Physical Laboratory, Teddington, Middlesex TW11 0LW, UK2 Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany

E-mail: andrew.yacoot@npl.co.uk

Received 7 August 2007Published 8 April 2008Online at stacks.iop.org/JPhysD/41/103001

AbstractThe review will describe the various scanning probe microscopy tips and cantilevers usedtoday for scanning force microscopy and magnetic force microscopy. Work undertaken toquantify the properties of cantilevers and tips, e.g. shape and radius, is reviewed together withan overview of the various tip–sample interactions that affect dimensional measurements.

(Some figures in this article are in colour only in the electronic version)

List of symbols used in the paper

Rtip radius�z displacement or bending of a cantilever

in the vertical directionα tip angleα1, α2 ‘half’ angles of an asymmetric tipFts force acting on cantilever due to interaction

between tip and samplek spring constant of cantileverr distance between the atom at the end of the the

tip and highest atom on the surfacedirectly above the tip

Fvdw van der Waals forceQ Q-factor of the cantileverf0 resonance frequencyA0 free amplitude of the cantileverω eigenfrequency of the cantilevertAM time scale for amplitude

changes in cantilever oscillationtFM time scale for changes in a frequency

modulated cantileverz vertical distance between tip and sample surfaceK signal-to-noise ratio� decay length of the interaction

E Young’s modulusI moment of inertiaF(x, t) force applied to the cantileverw(x, t) deflection of beamm mass/unit lengthL length of the cantileverbc cantilever widthdc cantilever thicknesskb Boltzman constantT temperature in kelvinG modulus of rigidityt length of the tipν Poisson’s ratioρ densityνs speed of soundαexp thermal expansion coefficientL[f (x)] Legendre transform of a function f (x)

i(z) STM currentφ work function in eVIs image of the sample characterizerS actual surface topography of

the characterizerP tip shapeε depth of the minimum potentialσ position of the minimum potential

0022-3727/08/103001+46$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK

J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

H Hamaker constantzc rest tip–sample separationε0 vacuum permittivityE∗ reduced elastic modulus of the tip and samplea0 intermolecular distanceV0 potential difference between the tip and the sampleg(z) takes into account the geometry of the tip and is

proportional to the gradient of the capacitancebetween the tip and sample

1. Introduction

Since their invention (Binnig et al 1982, Binnig et al 1986)scanning tunnelling (STM) and atomic force/scanning forcemicroscopes (AFM/SFM) have had a major impact on manyareas of science and have found applications for imaging,metrology and manipulation at the nanometre level. Indeedthe SFM is seen as an essential tool for nanotechnology andis regarded the window into the nano-world. Recently therehave been several reviews that have addressed various aspectsof these instruments: dynamical force methods by Garcia andPerez (2002), theoretical modelling by Hofer et al (2003) anduse in ultra-high vacuum by Giessibl (2003). Compared withthe STM whose operating principle is based on a quantumeffect, the SFM is a unique multi-purpose microscope whoseoperating principle is based on both mechanical and atomicinteractions. The SFM can be characterized by the followingfeatures:

• sensitivity to atomic forces,• ability to examine on nearly all surfaces: metals,

semiconductors and insulators,• works in air, liquid or vacuum,• can measure mechanical response (force–distance curves),• can be used for mechanical manipulation of single atoms,

molecules, polymers and surfaces.

Besides the SFM there are other classes of scanningprobe microscopes (SPMs) that are used to measurevarious properties of surfaces. These include the lateralforce microscope (LFM), friction force microscope (FFM),magnetic force microscope (MFM) and near field scanningoptical microscopy (NSOM). For traceable dimensionalmeasurements there is a class of so-called metrological SFMs,which has been developed by national metrology institutes(Danzebrink et al 2006). These metrological SFMs are mostlypurpose-built and are often equipped with laser interferometryto enable traceable measurement of displacement along thescanning axes of the instrument (x, y and z) to the SIunit of length. With such instruments samples can becalibrated for use by others to calibrate their instruments.Several comparisons of SFM measurements between nationalmetrology institutes have been performed. The NANO2comparison, Koenders et al (2003), and EUROMET707comparison, Koenders et al (2006), have shown measurementuncertainties achieved are in the sub-nanometre range for stepheight and the NANO4 (Meli (2000a)) showed that for pitch

measurements the uncertainty of measurement is of the orderof a few picometres.

The interaction of the SFM tip with the sample due tothe various forces between them gives rise to effects at thenanometre or sub-nanometre level that can affect an SPMimage and hence the dimensional measurements inferred.Consequently, these interactions have to be investigated andunderstood if measurement uncertainties are to be improved.This makes understanding of the characteristics of the tip,the cantilever and the detection system crucial for achievingmeasurements with small uncertainties.

In this review we will concentrate on the cantilevers andtips used in scanning force microscopes and the effect they haveon dimensional measurements. In order to set the scene for thisand to make the paper self-standing it is useful to provide somebackground information. While reviewing literature for thispaper the authors noticed that often more than one researchgroup was working on the same topic and published similarresults independently. It is hoped that this review will givean overview of various aspects of scanning force microscopywithout unnecessary repetition of previously published workand without excessive mathematical detail. It was felt thatthere was no point in repeating an excessive amount of detailfrom other papers. Appropriate references are given to enablethe reader to obtain further information about specific areas ofinterest.

There are many factors affecting the images obtained withan SFM and hence the quantitative information (dimensionalmeasurements) that can be inferred from an SFM image. SFMcantilevers are used in two different modes of operation, staticand dynamic measurement mode (section 1). Usually thesensing of the cantilever motion is done at the end of thecantilever rather than the end of the sensing tip. Thereforeit is important to understand the cantilever properties in detail,since they determine the response of the cantilever to forcesacting on the AFM tip. This is done in section 2. Fromthe point of view of metrology the main weakness of theAFM is that the point at which motion is detected by thedetection system is not the point that probes the surface. Thelength of the tip which could be between 6 and 10 µm and itsmovement which includes rotation due to torsion and bendingdue to lateral forces as well as unknown change of the lengthby tip wear is beyond our present measurement capabilities.Nevertheless, the very end of the tip and the shape of the tipdetermine the achievable lateral resolution. Consequently avariety of fine, sharp and stable tips are necessary for differentapplications and in section 3 we describe the various kindsof tips available. Since the tip dilates the sample topographyleading to possibly erroneous measurements, it is necessaryto determine the tip shape to evaluate ‘true’ topography or toget a better estimation of surface topography. In section 4 wegive an overview of techniques to characterize the tip. For awell-defined quantitative measurement the measurement mustbe made as precisely as possible and sometimes the maintask is estimating the measurement uncertainty. The partsdescribed above as well as contribution from the scanningsystem and its calibration must be taken into account. Thevarious scanning systems and their effects on dimensional

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

Figure 1. Schematic diagram showing surface forces as a functionof distance and their relation to the modes of SFM operation(Danzebrink et al 2006).

measurements are described in more detail in a CIRP Keynotepaper by Danzebrink et al (2006). In order to achieve sub-nanometre measurement uncertainty the physical effects ofprobing a surface by a mechanical tip, the finite forces, thevariation of surface forces due to different surface materialsacting on the tip and the effect of water layer have tobe considered as well. To do this a better knowledge of surfaceforces is necessary. Therefore in section 5 we discuss the forcesthat may be acting on the tip and thus affecting the measuredprofile. Section 6 gives the conclusions of the paper andhighlights the problems still to be solved in future for moreprecise measurements and smaller uncertainties.

1.1. Modes of SFM operation

Figure 1 shows the force between atoms at the end of an SFMtip and the surface as a function of distance of the tip from thesurface. As the sample is approached there is an attractive force(van der Waals) that increases to a maximum before a gradualdecrease until the force becomes repulsive (Pauli). Theseforces are discussed in more detail in section 5. The graph ofsurface forces also has regions marked as contact, intermittentcontact and non-contact referring to the different modes ofoperation of the SFM, which are discussed in the next section.The definition of a tip ‘coming into contact’ with a surfaceis very difficult when observing physical and other effects onan atomic scale because, in contrast to the macroscopic world,there really is no discrete boundary between the tip and sample.Therefore it is easier to separate SFM modes into static anddynamical categories (see later): the static is referred to inthe literature as the contact mode, the dynamical as the non-contact mode. Depending on the oscillation amplitude anddistances between tip and sample, terms such as ‘non-contactmode’, ‘tapping mode’, ‘intermittent mode’ or ‘dynamic forcemicroscopy’ are used.

1.2. Static or contact mode

Figure 2 is a schematic diagram showing the major componentsof a typical scanning force microscope. A very simplisticexplanation of the workings of such an SFM is as follows. A

Figure 2. Schematic diagram showing major components of ascanning force microscope with an optical beam deflection system.

cantilever supports a very fine tip having a typical end radiusof a few nanometres that is used as a probe to investigate asample. The cantilever is attached to a piezoelectric transducer(pzt). Light from a laser diode is reflected from the cantileveronto a quadrant photocell. As the cantilever and tip arescanned over the surface the cantilever bends and the light beamreflected from the cantilever is deviated by the bending, henceits position on the quadrant photocell changes. The signalsfrom the quadrant photocell are used as the input to a servosystem that applies a voltage to the pzt to move the cantileverso as to maintain a constant bending of the cantilever and hencereturn the reflected beam’s point of incidence on the quadrantphotocell to the centre. This ensures that the force exerted bythe tip on the surface remains constant and that the tip followsthe surface. A measure of changes in the servo signal gives anindication of the surface topography.

The cantilever, tip, detection system and servo are atthe heart of the SFM and their properties must be carefullycontrolled in order to achieve optimum vertical and horizontalresolution. The SFM’s high lateral resolution is achieved byusing sharp tips and low forces. Parameters used to describethe tip properties are the radius of the apex of the tip Rtip andthe cone angle α or the aspect ratio, AR.

The displacement or bending of a cantilever �z isproportional to the forces acting on it. The factor describingthe proportionality is the stiffness or spring constant of thecantilever k. In the contact or static mode of operation thefeedback loop controls the displacement or more accurately,the bending of the cantilever and keeps it constant. Thismeans that the deflection �z = Fts/k of the cantilever iskept constant. The interpretation of the images obtained inthis mode is quite simply z(x, y) at grad(Fts) = 0, Fts = constwhere Fts is the interaction force between the tip and sample.Typical scanning speeds for contact mode are of the order of75 µm s−1, (Stark et al (2004)).

1.2.1. Resolution in contact mode. Binnig (1986) succeededin imaging the lattice of a graphite surface. Later on Fujisawaet al (1993), using an optical beam detection method, foundthat an AFM works as a two-dimensional friction forcemicroscope (FFM) and that the atoms’ stick–slip motion due

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

to friction with lattice periodicity was not one-dimensional buttwo-dimensional in contrast to macroscopic friction. Abrahamet al (1988) performed molecular dynamical calculation for theSi(0 0 1)-2×1 surface reconstruction. They found that imagesobtained with forces below 1 nN should represent the normalsurface, whereas with large repulsive forces the image woulddeviate from the real surface. However, true atomic resolutionwith contact SFM on inert samples has been reported byOhnesorge and Binnig (1993) who revealed point-like defectswhen the net repulsive loading force on the sample was lessthan 10−10 N. With higher forces they found that monoatomicsteplines were slowly wiped away and images showed perfectlyordered surfaces. However imaging reactive surfaces such asSi(1 1 1) in ultra-high vacuum by contact SFM has been provento be impractical because of strong chemical forces and wearon atomic scale. A drastic example has been shown by Kizukaand Hosoki (1999) who measured the movement of a silicontip on a silicon surface in a transmission electron microscope.They found that the forces between tip and surface atoms areso strong that the tips shear off during the scan. Only theintroduction of new techniques based on non-contact forcemicroscopy allows one to resolve the atomic structure.

2. Dynamic mode

In this mode the cantilever oscillates above the surface duringthe scan and its position is servo controlled (see below) tomaintain a constant distance between the tip and the sample.The exact nature of the mode of operation depends on thedistance between the tip and sample. If the distance is suchthat the tip actually touches or ‘taps’ the surface the mode isreferred to as intermittent contact and the tip experiences therepulsive force from the sample surface. If the tip does nottouch the surface the mode is referred to as non-contact modeand depending on the minimum tip–substrate separation, thetip will experience both the repulsive and attractive forces orjust the attractive force. In all cases, the tip is in the region of afew nanometres from the surface. In high resolution modes thetip interacts strongly with the surface, i.e. feels the repulsiveforces, whereas for observation of van der Waals forces andmagnetic forces the average tip position is much larger thanthe oscillation amplitude.

Two main techniques are used to servo control thecantilever: amplitude modulation (AM) and frequency (FM)modulation. The cantilever is excited by an actuator thatis driven at either the resonant frequency f0 or close to thecantilever’s resonant frequency with fixed amplitude A0. Theexcitation is in the vertical direction for the non-contact,intermittent or tapping modes or in the horizontal directionfor the torsion mode. As the tip interacts with the samplethere is a change in amplitude of oscillation and in the AMmode, the position of the cantilever is varied to maintain aconstant amplitude and tip–sample distance. In the case of thefrequency modulation, the change in frequency of oscillationthat occurs as the tip interacts with the sample is used as theinput for the servo system. The position of the pzt tube is variedto maintain a constant tip–sample distance and to restore thecantilever’s oscillation frequency to the set frequency.

The non-contact mode was first used in the attractiveregion (true non-contact) to measure the effects of van derWaals, magnetic, electrostatic and other forces (Robrock 1990)and to investigate weak materials, polymer and molecules withlow surface adhesion. Due to the larger distance (larger decaylength of the force, see below) the lateral resolution is reduced.On the other hand, measurements of magnetic properties haveto be made with a larger tip–sample distance because otherwisethe interaction force is too small compared with the van derWaals forces (Fvdw) that are proportional to 1/r7, where r is thedistance between the atom at the end of the tip and the highestatom on the surface directly above the tip.

2.1. Resolution and speed in the dynamic mode

The dynamical scanning has the advantage of reducing theinfluence of low frequency noise, such as the 1/f noise.The changes in amplitude in the AM mode do not occurinstantaneously with a change in the tip–sample interactions,but on a time scale of

tAM ∼ 2Q/f0,

where Q is the quality factor of the cantilever and f0 is theresonance frequency.

Giessibl (2002) points out that for the AM mode,increasing the Q-factor reduces the cantilever’s sensitivitywhile reducing the noise. Albrecht et al (1991) combinedthe advantages of high Q and high speed by introducing thefrequency modulation (FM) technique. Here the change in thecantilever eigenfrequency settles on a time scale

tFM ∼ 1/f0.

For example: cantilever in air Q = 1000, f0 = 200 kHz →tAM ∼ 0.01 s equivalent to 100 Hz, whereas tFM ∼ 1/200 000 sis equivalent to 200 kHz, which would allow measurement witha higher speed.

For the FM mode the resolution was improved further byGiessibl (1994), Giessibl and Trafes (1994), leading to atomicresolution of a silicon (1 1 1) 7 × 7 surface in UHV (Giessibl1995). The main point introduced was the decrease of thetip–sample distance and operation in UHV. Both resulted in astronger influence of the short length interaction forces on thetip (see below). In vacuum this is now the preferred methodof operation.

Initially it was believed that for the non-contact mode thenet force between the front atom of the tip and sample hadto be attractive to achieve atomic resolution (Sugawara et al1996). This view has been challenged by Sokolov et al (1997,1999). In their calculation they included repulsive as well asattractive forces described by different power laws (10−n). Theresults indicate that the repulsive component of the contactforce gradient is the main contributor to the image contrastmechanism as long as the power of n of the attractive potentialis smaller than 9. For most cases, such as van der Waals,covalent bonding, magnetic, the range of n is between 7 and9, meaning that the forces are no longer long range.

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

Morita (2002) explained this in the following way by usingan exponential approximation for the interaction forces: highvertical resolution depends on the interaction forces and thetip–sample distance f (z) ∼ e(−z/�), where � is the decaylength of the interaction and z the distance between the tipand the sample surface. Changing the tip’s position from z

to z + δz changes f (z) to f (z + δz) = f (z)e(−δz/�) . Ifthe signal-to-noise ratio of f (z) is defined by K = (S/N)

the smallest measurable change of f (z) would be given byδf (z) = f (z)/K , which is equivalent to the noise level N . Ifthe smallest step change controlled by the feedback is δz, thenthe vertical resolution can be defined as δz from

K = S

N= f (z)

δf (z)= f (z)

[f (z) − f (z + δz)]= [1 − e(−δz/�)]−1

to be

δz = � ln

(K

(K − 1)

)

as a function of the decay level �. If the (S/N ) ratio becomeshigh or if � decreases then the vertical resolution increases.For example: the tunnelling current in an STM has a very shortinteraction length � and since the tunnelling current can bemeasured with a high signal-to-noise ratio atomic resolutionis achievable. For the AFM atomic resolution is achievablefor short tip–sample interaction forces for small amplitudes.Large amplitudes average over a different interaction length �.Also sharp and clean tips are necessary, because of the volumeintegration effect of the interaction around the tip apex.

In the case of the frequency modulated non-contact mode,FM-NC-SFM, atomic resolution was achieved close to thesurface by applying small oscillation amplitudes. Giessibl et al(2001) prepared a tip by breaking a silicon wafer in air alongits preferential cleavage planes and showed an unprecedentedvertical distinction of the six different surface atoms sites onthe Si(1 1 1)-7 × 7 surface. Tips can be orientated in anycrystallographic orientation, which allows detailed study of thebonding symmetry between silicon tips and samples. Giessiblet al assumed that the bulk symmetry extended up to thesurface, i.e. has a 1 × 1 unit cell structure. The measurementdata observed with their SFM were compatible with thissimple model. The availability of a measured tip–sampleinteraction over a specific surface atom means that simulationscan provide qualitative agreement with experimental results.However, even beginning to model ‘real’ tips is a very difficultprocess. In contrast to cleaving silicon in UHV, fabricationby cleaving silicon in air results in a native oxide layer,and probably a layer of adsorbed water or contamination.Foster et al (2004) and others set up models based ona ‘realistic’ approach to tip simulation. They found thatdifferent tips showed characteristic features, which could beidentified in experiments. However, they concluded that itwould be an impossible task to provide an exact model forthe tip used in an experiment. Considering more seriouslyhow the tip has been made and the processes it undergoesduring scanning would help to establish some fundamentalfeatures of tips which should be observable in experiments andhence provide signatures which would help to identify the tipproperties. They proposed setting up a database of tip–surface

interactions, which help greatly in closing the gap between thewealth of idealistic theoretical calculations and unexplainedexperiments. Typical scanning speeds can be between 180and 250 µm s−1 due to the time constant of the pzt actuator.However as with contact mode, the scan speed is to some extentgoverned by the flatness of the sample. For a more detailedoverview of the non-contact mode the reader is referred to thereviews by Hofer et al (2003) and Giessibl (2003).

2.2. Metrological SFMs

Metrological SFMs developed by national measurementinstitutes are usually capable of working in both the staticand dynamic modes and although capable of making traceablemeasurements have limitations. The measurement of thelateral distance between two points on a surface is limitedby the measurement capability of the laser interferometer,any Abbe offsets, noise, stray light effects of the detectionsystem used, and by the tip shape and tip stability. Thelast point also includes the effects of the interaction ofthe SFM tip with the sample and is still an area thatis often ignored, but requires further research in order toachieve smaller measurement uncertainties. Since calibrationstandards usually have homogeneous surfaces, e.g. step heightsare coated by a thin metallic film, effects due to differentmaterials can in this case be neglected. As in the case of opticalmicroscopy, a technique that also becomes more complicatedwhen heterogeneous samples are examined, material effectshave to be taken into account for SFM as well. The scaleon which those effects are estimated is much smaller than inthe case of optical microscopes, but should nevertheless beconsidered.

3. Cantilevers

3.1. Introduction

Some of the initial cantilevers used in SFM were made fromthin metal wires electrochemically etched or thin metal foils onwhich a small diamond was glued (Binnig et al 1986). The firstmass-produced cantilevers were made from silicon dioxide(SiO2) or silicon nitride (Si3N4) (Albrecht et al 1990). Later,cantilevers with integrated tips were machined from silicon-on-insulator (SOI) wafers, see Wolter et al (1991). The techniquesused to achieve this have been developed from semiconductorindustry and cantilevers with small radius tips are now availablewith very high reproducibility. The sharpness of the silicon tiphas been improved from 10 to 20 nm down to 1 nm radiusby more sophisticated production procedures, e.g. thermaloxidation and etching using HF (Marcus et al 1990), although5–10 nm is the more used value for commercially availabletips. For more information about the history the reader isreferred to Giessibl (2003). Since most detection systemsused in SFMs are optical systems, the upper surface of thecantilever is usually coated with a metal film to increasethe cantilever’s reflectivity. However there are an increasingnumber of self-sensing cantilevers that have been developedwhich do not require an external detection system, someof which are described here. For a more comprehensive

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review of self-sensing systems, the reader is referred toOesterschulze (2001).

Cantilevers used in the SFM should meet the followingcriteria:

• low normal spring constant k (stiffness), to ensure highsensitivity to surface forces,

• a high resonant frequency f0, to decrease sensitivity tomechanical disturbances,

• a high quality factor of the cantilever Q, for non-contactforce modes,

• high lateral spring constant (stiffness), for stability,• a highly reflective smooth opaque upper surface to avoid

transmission of light through the cantilever which couldlead to an erroneous signal being detected.

In the static or the contact mode, the cantilever shouldbe much softer than the bonds between the bulk atoms in tipand sample. Interatomic force constants in solids are in therange from 10 to about 100 N m−1—in biological samples, theycan be as small as 0.1 N m−1. Thus typical values for k, thespring constant, in the static mode are 0.01–5 N m−1 (Giessibl(2003)). On the other hand instabilities are more likely tooccur than in the dynamic mode, because if the cantilever is soft(small k) the point at which the gradient of the interaction forcebecomes equal to k is reached earlier than for a stiff cantilever.At that point the tip jumps into contact to the sample surface.Since the deflection should be larger than the deformation ofthe tip and the sample this further restricts the useful range of k.

When the dynamic mode is used for normal applications,i.e. topographic measurements, cantilevers with a resonantfrequency in the range 10–1100 kHz are used, whereas highresonance cantilevers are needed for high speed applications,for writing or reading information on disks. These applicationshave also led to the development of multiple cantileveror cantilever arrays. With a weak spring constant ofk = 0.1 N m−1 and a detection sensitivity of 0.01 nm forcesof 10−12 N can be detected. For a rectangular cantilever thefirst resonance frequency is given by f derived in the nextsection. g stiffer cantilever is generally needed in the dynamicmode; typical values for k are around 15–40 N m−1.

3.2. Cantilever characterization

The analysis for the deflection of a cantilever has beenpresented in detail elsewhere. See, for example, Butt andJaschke (1995) who calculated thermal noise of a cantilever,Sader (1998) who looked at the frequency response ofcantilevers immersed in viscous fluids or Stark and Heckl(2000) and Green and Sader (2005). The partial differentialequation that describes the bending of the cantilever is given by

EI∂4w(x, t)

∂x4− m

∂2w(x, t)

∂t2= F(x, t),

where E Young’s modulus, I the moment of inertia, w(x, t)

the deflection of beam as a function of length and time, m themass/unit length, F(x, t) is the force applied to the cantileverand w(x, t) can be split into spatial and temporal terms as

w(x, t) = �(x)Y (t)

leading to differential equations for Y and � as functionsof x and t , respectively. This leads to a term for the ntheigenfrequency (resonant frequency) of the nth eigenmode:

ω2n = (knL)4 EI

m,

where L is the length of the cantilever and

ω = 2πfn.

For rectangular cantilevers of length Lthat are free at one end,knL = 1.875, 4.694, 7.855, 10.996, 14.137 (Chen 1993) forthe first five flexural modes.

The compliance/stiffness/spring constant k of the beamdefined as,

k = 3EI

L3,

yielding the relations for the eigenfrequency w1 and k

ω1 = 1.0149d

L2

√E

ρand k = 1

4E

bcd3c

L3,

where ρ is the density, bc the cantilever width and dc thecantilever thickness.

Additionally, the stiffness in the longitudinal axis ofthe cantilever (y axis) and the torsion is of interest. Fora rectangular cantilever these values can be found inBhushan (2007):

ky = 1

4E

b3cdc

t3and kyT = 1

3G

bcd3c

L3t2,

where G is the modulus of rigidity G = E/2/(1 + ν), t thelength of the tip and ν Poisson’s ratio.

3.3. Cantilever noise

The noise level is given by the product of the cantilever springconstant k and the noise level at the deflection detection system(usually the quadrant photocell). For a given system a smallerkwould reduce the noise and therefore the sensitivity of thesystem to forces. The noise is dominated by the 1/f noisethat can be minimized by reducing the temperature or use ofmaterials with low thermal expansion coefficients (Ohnesorgeand Binnig 1993). In the static mode, instabilities are morelikely to occur than in the dynamic mode, because if thecantilever is softer (smaller k) the point at which the gradientof the interaction force becomes equal to k is reached earlier.In the normal SFM set-up, ambient vibrations and acousticalnoise can act on the cantilever and the detection system, thelatter also being affected by electrical noise. The cantileveritself is very sensitive to ambient vibrations and acousticalnoise that will result in noise on the detected signal. Inorder to reduce ambient vibrations and acoustical noise theresonance frequency of the set-up should be as high as possibleto minimize the influence of mechanical disturbances.

Achieving a high resonance frequency is possible byreducing the mass of the cantilever, i.e. the dimensions haveto be chosen to be as small as possible. Commercially

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

available cantilevers for the static mode have resonancefrequencies above 10 kHz and non-contact cantilevers typicallyhave frequencies in the range from 50 to 300 kHz. Specialcantilevers used for high speed applications can have valuesof 1 MHz and higher (Minne et al 1999). However, Hosakaet al (1993) reported cantilevers developed for high speedapplications with a length of less than 20 µm and a thicknessof less than 0.3 µm and a resonant frequency of 6.6 MHz. Todetect such high frequencies a newy developed laser beamdeflection detection system with a high numerical apertureobjective lens was used (Hosaka et al 1993). Yang et al (2005)fabricated single crystal silicon cantilevers in a batch processwith lengths between 10 and 35 µm, a width of 4 µm and athickness of 0.2 µm giving a resonant frequency between 0.2and 2 MHz. These cantilevers can be used for high speedapplications and have an improved force sensitivity of between2 and 5 better than conventional cantilevers. The thermalnoise of the cantilevers could be reduced to 10−18 N Hz−1/2

by annealing in UHV and cooling down to liquid heliumtemperatures.

As mentioned above, Butt and Jaschke (1995) used theanalysis to estimate the thermal noise in the cantilever. All theeigenmodes contribute to cantilever noise; however, the majorcontribution will be due to the first eigenmode. Clearly, thenoise in a free cantilever will be greater than that in a supportedcantilever (in contact with the sample surface). When inthermodynamic equilibrium, the mean square displacement ofa cantilever supported at only one end from its neutral positionat a temperature of 22 ◦C is given by

√Z∗2 =

√kbT

k= 0.64 × 10−10

√k

= 0.064√k

nm,

where k is in N m−1, kb is the Boltzman constant and T is thetemperature in kelvin.

However, when the deflection of the cantilever asmeasured by an optical beam deflection system is taken intoaccount the cantilever measures a change in inclination ratherthan a vertical movement. Butt and Jaschke showed that themean ‘virtual’ vertical deflection of a free cantilever, whenmeasured using beam deflection, is given by

√Z∗2 =

√4kbT

3k= 0.74 × 10−10

√k

= 0.074√k

nm.

With the cantilever in contact with a surface, they calculatedthat the mean vertical deflection is given by

√Z∗2 =

√kbT

3k= 0.37 × 10−10

√k

= 0.037√k

nm,

i.e. a factor of 2 smaller. Typically at 22 ◦C the vibration dueto thermal noise is at most a few tens of picometres. Stark et al(2001) extended this work to calculate the thermomechanicalnoise in a free V-shaped cantilever and showed that thethermomechanical noise for the first eigenfrequency was

√Z∗2 = 0.627 × 10−10

√k

= 0.064√k

nm

and the thermomechanical noise on the photodiode was

√Z∗2 = 0.558 × 10−10

√k

= 0.056√k

nm,

indicating that there is a systematic deviation in thethermomechanical noise of V -shaped cantilevers from that ofrectangular-shaped cantilevers.

3.4. Material properties

As was mentioned above cantilevers have been made frommetal, silicon and silicon nitride. However, in some casesother materials have been used to obtain specific cantileverand tip properties. For example, diamond or chemical vapourdeposited diamond is used to reduce tip wear and quartz anddiamond are less sensitive to temperature change than silicon.Recently polyimide probes have been developed for examiningsoft samples such as biological and polymeric structures,Gaitas and Gianchandani (2006). Some materials and theirproperties are listed in table 1 below. Compared with silicon,diamond has some outstanding properties.

3.4.1. Effects due to temperature changes. Any distortion ofthe cantilever has an effect on measurement results. Changesin temperature will also affect the cantilever. For thecontact mode,

F = kz = k(T )z(T ),

z being the position of the cantilever.Changes of the forces are in first approximation:

�F = ∂

∂zF (z, T )�z +

∂

∂TF(z, T )�T + · · · ,

�F = k�z +

[∂k

∂Tz + k

∂

∂Tz

]�T + · · · .

The first term k�z is kept constant by the feedback circuit.The second term (∂k/∂T )z�T describes the temperatureinduced changes in the cantilever properties and the last termk(∂/∂T )�T describes the changes of the cantilever positiondue to temperature, i.e. changes in the measurement loop.The properties of the cantilever change with the temperaturesince the spring constant k is affected by thermal expansionand changes of the Young modulus with temperature. Innon-contact measurements this would change the resonancefrequency. Since the speed of sound in a material is givenby

√(E/ρ) where ρ is the density, these dependences can be

described (Chen 1993) by

1

f0

∂f0

∂T= 1

vs

∂vs

∂T− αexp

as change of speed of sound by temperature and thermalexpansion. Since the change of speed of sound for quartz is lessthan that for silicon, cantilevers made from quartz should bemore insensitive to temperature changes than silicon (Giessibl2002). This theoretical approach was also extended to studythe mechanical behaviour of cantilever probes from layeredmaterials, see Johansson et al (1989), Lai et al (1997) and

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

Table 1. Physical properties of some common materials used for SFM cantilevers based on Oesterschulze (2001).

Material Si (0 0 1) GaAs(0 0 1) Quartz Alproperty [a] Si3N4 [b] [c] Diamond [d] Polyimide

MechanicalDensity 2.329 3–3.3 [d] 5.317 2.6 3.5 2.7 1.43 [d]

(g cm−3)Young’s 168.4 for [1 1 0] 260–320 [f] 121.5 for [1 1 0] 73.1 1163.6 for [1 1 0] 3.0–3.2 [d]

modulus (Gpa) 129 for [1 0 0] 85.5 for [1 0 0] for [1 1 0] 1050.3 for [1 0 0]577.4

Torsional 80 60 31.2 10300 27.8modulus G (Gpa)

Poisson ratio 0.27 [b] 0.25 [f] 0.17 [f] 0.1–0.29 [f] 0.34 0.41 [d]Hardness 1150 (25) 680 (100) 6360 (long)

(load/g)Speed of sound 8430 (long) [b] 9900 [c] 5980 [d] 17.520 [d] 6.360vs (m s−1)

ThermalThermal expansion 2.92 [b] 2.5–3.3 [b] 6.86 0.54 1.0 23.03 50–60 [b]

coefficient(µm K−1)

Heat capacity cp 700 45.8 J mol−1 K−1 500 670 518 880(J kg−1 K−1) [b]

Heat conductivity 156 [b] 15–45 [b] 45.5 1.46 600–2000 237 0.29–0.35 [b](k W−1 mK−1)

Melting point Tm (K) 1687 [b] 1800–1950 [b] 1513 4100 [b] 933.47 335–345 [b]Change of speed of −5.5 × 10−5 K −1 [a]sound ∂vs/∂T

(m s −1 K −1)Optical and electricalRefractive 3.4 3.878 ∼1.46 5.7

index nat 633 nm

Static dielectric 11.97 [b] 13.18 3.8 5.70 [b] 3.4 [b]constant e

Gap energy (eV) 1.12 (ind.) 1.42 9 1017

Electrical 103 109 (undoped) 7 × 109

resistivity ρ(� m)

Note: for mechanical properties see the following.[a] Schulz M and Blacknik R 1982 Landolt-Bornstein vol 111/17a (Berlin: Springer) pp 61–83[b] GaAs(0 0 1)—Blakemore J 1980 J. Appl. Phys. 53 R123–83[c] www.quartz.com[d] Martienssen W (ed) 2005 Springer Handbook of Condensed Matter and Materials Data (Berlin: Springer)[e] Bhushan B 2004 Springer Handbook of Nanotechnology (Berlin: Springer)[f] MatWeb http://www.matweb.com/

Hazel and Tsukruk (1999). For the influence of the propertiesof crystalline materials, see Oesterschulze (2001).

Two important properties of cantilevers are their resonantfrequency (ω = 2πf0) and spring constant k. It isstraightforward to determine the cantilever resonant frequency.However, determination of the cantilever spring constant ismore complicated. Initially it was derived from measurementsof the geometric and material properties of the cantilever. It hasbeen shown that the spring constants given by manufacturerscan often be in error by a factor of 2, Walters et al (1996),and often a manufacturer supplies a range of values for aspring constant rather than an actual value. The range cancover several orders of magnitude. For the investigationof molecules it is important to measure the forces appliedand in dimensional metrology, especially for roughness orhardness/nanoindentation measurements, the knowledge of the

tip shape and the applied force is mandatory. Several methodshave been proposed for determining the spring constant ofcantilevers. Rather than describing them in detail, the readeris referred to two recent comprehensive reviews, Burnhamet al (2003) and Clifford and Seah (2005). The latterfound that the model by Neumeister and Ducker (1994),although needing a revision to take into account the bendingof the triangular portion of a V-shaped cantilever, was themost accurate model. Cumpson et al (2004) used bulkmicromachined silicon to produce a compact and easy to usereference artefact for the calibration of contact cantileverswhich they call a cantilever microfabricated array of referencesprings (C-MARS). The C-MARS device spans the rangeof spring constants from 25 down to 0.03 N m−1 allowingalmost any contact-mode AFM cantilever to be calibratedeasily and rapidly. Subsequently, Cannara et al (2006)described a method for the calibration of lateral force constants

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

for AFM cantilevers. Gibson et al (2005) compared threedifferent methods for the calibration of the spring constantof two different types of silicon beam-shaped atomic forcemicroscope (AFM) cantilevers to determine each method’saccuracy, ease of use and potential destructiveness. A review oftraceable calibration for indentation AFM is given by Pratt et al(2005), who also describe a low force balance that providedtraceable measurement of low force based on capacitancesensors.

A good introduction of error analysis of lateral forcecalibration is given by Schwarz et al (1996). They foundthat small oscillations due to increased gain of the feedbackloop cause a reduction of the friction. Therefore, the gain ofthe feedback loop should be selected below a critical value.Note that in order to obtain a full calibration of the cantilever,it is important to consider its geometry within the AFM, e.g.if light from the beam deflection is not focused onto the endof the cantilever, then it is necessary to take into account thedistance between the point of incidence and the tip. Directconversion between optical signal and displacement is easierif the tip displacement is measured using optical interferometrytechniques (described later) rather than beam deflection sinceit is possible to directly convert the measured changes in theinterferometer signal into displacement.

3.5. Cantilever types and detection methods

In the early days of SFM, the cantilevers used were rectangular(diving board). However as the technique developed, the useof V-shaped cantilevers with the tip at the apex of the V becamemore widespread since it was thought that their stiffness washigher than that of the rectangular varieties. Sader (2003) andSader and Sader (2003) compared the sensitivity of V-shapedcantilevers to lateral forces with that of rectangular cantileversand found that surprisingly V-shaped cantilevers were moresensitive to lateral forces than the rectangular. CommercialV-shaped Si3N4 cantilevers have a typical width-thickness ratioof 10 to 30 which results in 100 to 1000 times stiffer springconstants in the lateral direction compared with the normaldirection. Therefore these cantilevers are not well suitedfor torsional measurements. For friction measurements, thetorsional spring constant should be minimized in order to makethe cantilever sensitive to the lateral forces. Long cantileverswith small thickness and large tip length are most suitable.Rectangular beams have lower torsional spring constants incomparison with the V-shaped cantilevers (Bhushan 2007,table 1). There is continued interest in the area of SFMcantilevers and probing systems. In addition to conventionalcantilevers, there are self-sensing probes with integrateddetection, tipless cantilevers and cantilever arrays for specialapplication that if calibrated can be used to calibrate otherSFM tips.

3.5.1. Detection methods for non-self-sensing cantilevers.The prime requirement of any detection method is that itprovides a signal that can be the input to a servo control systemfor keeping the cantilever–sample distance constant. The firstattempts to realize an AFM were made using a tunnelling tip

at the back of an aluminium cantilever to detect the bendingof the cantilever (Binnig et al 1986). Since the tunnellingcurrent depends exponentially on the distance between tipand sample (or in this case tip and cantilever) a decrease of0.1 nm in distance can, for small distances, increase the currentby an order of magnitude. Experimentally a resolution of0.001 nm can be achieved with a bandwidth of a few thousandhertz. However, the set-up is very difficult to realize; the sharptunnelling tip has to be kept very close to the cantilever withthe use of an additional piezo and owing to the high stiffness ofthe cantilever the interaction forces between the tunnelling tipand the cantilever which are of the order of 10−9 N are difficultto measure (Durig et al 1988).

In order to overcome these difficulties, optical systemswere quickly developed for detecting cantilever displacement.They are based either on the reflection of a laser beam fromthe back of the cantilever, known as beam deflection, or onoptical interferometry. Initial work concentrated on usingoptical interferometry; Martin et al (1987) and Erlandsson et al(1988) independently presented detection systems based usinghomodyne and heterodyne interferometry. Schonenberger andAlvarado (1989) and Anselmetti et al (1989) developed adifferential interferometer using the Nomarski principle. Onebeam was reflected from the end of the cantilever above thetip and the other from the base of the cantilever that remainedfixed. Sasaki et al (1994) used a differential interferometerwhere one beam was focused on the cantilever and the otherwas collimated and focused on the sample. This meant thatthe sample had to be highly reflective. Both designs ofinterferometer required the use of a polarizing beam splitter,complex optics and were complicated to set up. An alternativeinterferometric method uses a fibre interferometer that is morestable and compact (Rugar et al 1989, Moser et al 1993, Oral2003). The light output from a fibre coupler is incident on thecantilever and the interferometer adjusted to be working aroundthe zero-crossing point of an optical fringe. This region isapproximately linear and the interferometer signal can be usedas the input to the servo system. A dual fibre interferometer wasintroduced by McClelland and Glosli (1992); two optical fibreinterferometers of the type developed by Rugar et al (1989)were used to measure the cantilever deflection along the twoorthogonal directions angled 45◦ with respect to the surfacenormal. Olsson et al (1996) presented a fibre-optic baseddetection system for a combined STM/SFM in UHV. As atip/cantilever they used a polished tungsten wire with etched tipand high spring constant k between 100 and 200 N m−1. Thisenabled a switching between the STM and the SFM modes.The fibre was attached to a piezoelectric transducer to aidalignment of the fibre to the cantilever. The fibre interferometerworked in dc and ac modes with high sensitivity and a spectralnoise density of less than 200 fm (Hz)−1/2 above 40 Hz, wellbelow the limit of detection of thermally induced oscillationof the cantilever. Investigations in the STM mode on flashedSi(1 1 1) surfaces resolved the well-known 7 × 7 structure,whereas the resolution in the SFM mode was not sufficientlyhigh. More recently, Ruf et al (1997) reported a cantilever withintegrated Fabry–Perot cavitities for the use in UHV AFM. Asmall Fabry–Perot cavitiy of 7 µm gap is achieved by high

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

precision mechanics and a piezotube. The small gap reducesthe effects of drift and noise on the interferometer signal.The combined cantilever/sensor allowed easier handling andalignment in a UHV environment.

Although a detection system based on interferometericprinciples has the advantage of providing a direct measure ofthe cantilever’s normal displacement and potentially a directroute to traceability, the majority of SFMs now use a beamdeflection system for detection of cantilever displacement, denBoef (1989), since they are easier to align and their sensitivityis similar to that of the interferometric methods (Putman et al1992a, 1992b). The first optical beam deflection systems weredeveloped by Marti et al (1986) and Meyer et al (1988). Thelaser beam reflected by the cantilever was detected by a positionsensitive detector. In the contact mode a resolution of 0.01 nmhas been achieved that was sufficient for atomic resolution.The detection system was improved by Marti et al (1990) andMeyer and Amer (1990) with the use of a quadrant photodiodeto detect simultaneously the bending and the torsion of thecantilever. Kawakatsu and Saio (1997) incorporated twooptical beam deflection systems in their SFM to detect thedisplacement of the tip end in thex, y and z directions. Schafferet al (1998) characterized the detection sensitivity of a beamdeflection system using an adjustable aperture to optimize thediameter of the light beam incident on the cantilever and foundthat with careful adjustment, the signal-to-noise ratio of thecantilever deflection measurements could be increased by afactor of between 1.5 and 3.

Although theoretical studies suggested that the limits ofthe deflection sensitivities obtained by these two methodsare nearly the same (Putman 1992a, 1992b), the deflectionnoise densities in the practical beam deflection sensors aretypically between 100 and 1000 fm (Hz)−1/2 and are worsethan those in the optical interferometers. The polarizing opticalinterferometer described by Schonenberger et al (1989) had adeflection noise density of 6 fm (Hz)−1/2.

To aid the development of a low noise cantilever deflectionsensor, Fukuma et al (2005) investigated both theoreticallyand experimentally the major factors that limit the deflectionsensitivity of the optical beam deflection detection system. Theforce sensitivity of the conventional FM–AFM is limited notonly by the cantilever thermal motion, but also by electronicnoise arising from the cantilever deflection sensor, photodiodeshot noise and Johnson noise. Deflection noise arising fromlaser intensity fluctuation is mostly eliminated as a commonmode noise at the differential amplifier if its common moderejection is high enough. However the fluctuations of thespatial distribution of the laser spot on the photodiode dependon laser diode intensity and other optical components and theseproduce a differential mode noise that is amplified. Fukumaet al found a good agreement between theoretically calculatedand experimental photodiode shot noise for a laser power ofless than 2.5 mW. Above this value the deflection noise densityincreases drastically due to the contribution from the lasermode-hop noise. Any reflected or scattered light incidenton the photodiode will increase signal noise. Furthermoreany light that is reflected from the photodiode back into thelaser will increase laser mode hopping and make an additional

contribution to the optical signal noise. The optical noisewas reduced by modulating the laser power by using a radiofrequency in the range 300–500 MHz (Arimoto et al 1986).The rf modulation forces the laser diode from single modeto multimode operation and this reduces the optical feedbacknoise by reduced mode hopping and optical interference noisedue to smaller coherence length.

The other components of beam deflection systems are thefocusing elements and the cantilever itself. A collimating lenswith high numerical aperture (NA) is necessary to collect asmuch of the light emitted from the laser diode as possible; afocusing lens with relatively large NA is necessary to havea small spot on the cantilever and to minimize the beamdivergence between the cantilever to the photodiode. Ametallized upper surface of the cantilever improves the signal-to-noise ratio. If the laser spot size exceeds the cantileverwidth, some of the laser beam will be incident on the sample,which will lead to an increased signal-to-noise ratio as wellas erroneous measurements (Mendez-Vilas et al 2002, Huanget al 2006, Yacoot et al 2007). With an optimized beamdeflection system Fukuma et al (2005) achieved deflectionnoise densities of 16.7 fm (Hz)−1/2 and 38.9 fm (Hz)−1/2,when using laser powers of 2 mW and 0.5 mW, respectively.

For a rigorous mathematical treatment of the geometricalaspects of beam deflection systems, see Beaulieu et al (2007)who showed how the movement of the light beam on thequadrant photodiode depended on the orientation of theincident light, cantilever and photodiode. Their model isparticularly helpful when designing new SFMs.

Buh and Kopanski (2003) investigated the effect of laserillumination on a sample by using a scanning capacitancemicroscope (SCM) with which changes in capacitance due tofree carriers generated by light on a semiconducting samplecould be detected. They found that it is not practicable inan SFM to achieve dark illumination conditions owing tothe finite size of the depth-of-focus of a laser beam, lightspillage over the cantilever edges, light directly transmittedthrough a cantilever or light backscattered from other reflectingsurfaces. As mentioned above, for dimensional measurementsthese are very critical points, since any light reflected from asample pattern could have an effect on the sample topographymeasured. Figure 3 shows the effects of stray light falling ona sample and being reflected back into a detection system. Animage of an 80 nm step height standard is shown, together witha line profile of part of the substrate. The line profile shows anapparent undulation in surface topography of approximately18 nm peak to peak due to stray light. Subsequent removal ofthe stray light eliminated the apparent undulations in substratetopography.

3.5.2. Non-optical readout and self-actuating cantilevers.Although the traditional optical methods for detectingcantilever movement are widely used, they do have somedisadvantages, namely, they can be awkward to set up, takeup a large amount of space and as mentioned above, straylight on the sample from either the beam deflection or fibreinterferometer system can lead to erroneous measurements.Local heating of the cantilever due to the laser spot

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

(a)

(b)

Figure 3. (a) An image of an 80 nm step height standard wherestray light from the fibre interferometer was incident on the sampleand reflected back into the interferometer measurement system and(b) the apparent undulation in substrate flatness due to the stray light.

causing thermal noise can also lead to reduced sensitivity ofmeasurement. An alternative method is to use a self-sensingcantilever, a cantilever that measures its own displacement.Several types have been developed, all of which have theirrelative merits (Oesterschulze and Kassing 2003). Self-sensing cantilevers can also be combined to form multiplearrays of cantilevers.

3.5.3. Quartz tuning fork and quartz cantilevers. Followingthe development of the AFM and its use in non-contact orintermittent contact mode, cantilevers made from quartz havebeen used to oscillate the tip. Bartzke et al (1993) attachedeither diamond needles or electron beam produced carbontips to a quartz rod to produce an oscillating AFM tip systemknown as a needle sensor (Grunewald et al 1995). Subsequentdevelopments have used quartz oscillators identical to thoseused in digital wristwatches. The tip is attached to one tine ofthe oscillator. Such tuning fork oscillators are mass producedand have a high quality factor (Q) since the two tines of the forkoscillate out of phase with each other. However, the mass of thetip attached to the tine is sufficient to destroy the symmetry ofthe two tines. The effects of this have been overcome by fixingthe upper fork tine and thereby turning the fork into a cantileversystem where the cantilever is attached to a high mass substrate.

Gunther et al (1989) first used such oscillators with scanningnear field acoustic microscopes (SNAM). Karrai and Grober(1995) used a quartz tuning fork to mount an optical fibrein a scanning near field optical microscope (SNOM) and thesignal from the tip was used to position the tip to within 25 nmof the sample surface. Edwards et al (1997) subsequentlydemonstrated how an AFM tip could be mounted onto a quartzoscillator tine and used in an AFM in both open and closedloop control. Giessibl (1998, 2000) extended this work andachieved atomic resolution of a silicon (1 1 1)-(7 × 7) surfaceusing an AFM with a quartz tuning fork cantilever. Tyrrell et al(2003) developed a compact AFM head with a quartz tuningfork cantilever that had a diamond tip. The head was integratedinto a conventional microscope head and had the potential tobe converted to a SNOM.

More recently much smaller quartz cantilevers have beenproduced. Peng and West (2005) developed a self-sensingquartz sensor onto which either tungsten etched tips orcommercial cantilevers could be glued. The resonating armof the device was 2.1 mm long, 0.17 mm thick, 0.09 mm wideand had a resonance frequency of 650 kHz, much higherthan the usual 33 kHz associated with tuning forks. It wassufficiently compact to be integrated into an SPM system withan optical microscope that facilitated positioning of the sample.Lin et al (2005) described the fabrication process of somemicromachined quartz crystal cantilevers designed specificallyfor force sensing. They were made from an AT-cut quartzcrystal with the cantilever width and thickness varying between150 µm and 1500 µm and 150 µm and 250 µm, respectively.The spring constants were of the order of 10 N m−1 as opposedto those of quartz oscillators that are of the order of severalthousand N m−1. Ono et al (2005) used the cantilevers inthe non-contact mode. They measured the vibration of thecantilever using a laser Doppler vibrometer to compare theactual vibration with that measured by the piezoelectric effectof the quartz cantilever. Self-actuation of the cantilever waspossible but only with very slow amplitude. Consequently thecantilever was oscillated using a piezo-‘shaker’ as is done withconventional cantilevers.

3.5.4. Piezoresistive cantilevers. The first discreteself-sensing cantilevers (Tortonese et al 1993) used thepiezoresistive properties of silicon (Tufte and Stelzer 1963).The cantilever was fabricated from p-type silicon and formedone resistor in a Wheatstone Bridge. Deflection of thecantilever was determined by measuring the change in thecantilever’s resistance as it bent. Using these cantileversit was possible to obtain atomic resolution when the AFMwas used in the non-contact mode. Jumpertz et al (1998)attached piezoresistive components (again p-type doped) ontocommercially available cantilevers from Nanosensors GmbHand imaged single steps on a graphite (HPOG) surface.Su et al (1997, 1999) developed V-shaped piezoresistivecantilevers with the piezoresistors located near the root ofthe cantilever, rather than on the arms of the V, therebyenabling the sensitivity to be higher since the stiffness could bereduced. Indeed, the sensitivity (40 × 10−7 nm−1) was higherthan that previously reported for rectangular piezoresistive

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

cantilevers (30 × 10−7 nm−1), Giessibl and Trafas (1994).The scanning rate was increased to 1 mm s−1 with no loss ofimage quality. Gotszalk et al (2000) developed piezoresistivesensors for a variety of applications. A conductive tip allowedcapacitive microscopy and scanning tunnelling microscopy tobe performed. Changes in the positioning of the piezoresistorsenabled lateral force microscopy to be performed while amodification to the fabrication process resulted in a bimaterialcantilever that was sensitive to temperature changes. Thistogether with the addition of a heating element on the cantilevermeant that the cantilever could be used for calorimetry inthe range of 50 pJ. Piezoresistive cantilevers have also beendeveloped for magnetometry, the measurement of microscopicmagnetic samples (Takahashi et al 2002).

Minne et al (1995a) developed an array of two cantileverswith both a piezoresistive film for movement detection and apiezoelectric film for actuating. Pedrak et al (2003) developedcantilevers with a piezoresistive sensor and a thermal bimorphactuator that comprised a sandwich of layers of aluminiumsilicon dioxide and silicon. An ac driving voltage was appliedto the bimorph to provide the heating to the aluminium resistorthat generated the oscillation of the cantilever at the resonancefrequency (19 kHz) and a dc bias was used for positioningof the tip. As a demonstration of its performance for CDmeasurements, 5 nm thick platinum islands on chrome patternswere imaged.

Although offering advantages over optical detectionmethods, piezoresistive cantilevers have two disadvantages;that their power consumption is high (mW), which cancause problems, particularly for dimensional metrology, andgenerally they require an external actuator. Consequently,more effort has been directed towards developing cantileversthat utilize the piezoelectric effect although recently lateralatomic resolution of silicon surfaces using a low temperatureAFM with piezoresistive cantilevers was achieved by Shirakiet al (2006).

3.5.5. Piezoelectric cantilevers. The piezoelectric cantileveroffers the potential to be both self-actuating and self-sensing.The feasibility of piezoelectric self-sensing cantilevers wasfirst demonstrated by Itoh and Suga (1993). They developedtwo films that could be used for self-sensing cantilevers. Thefirst was a zinc oxide (ZnO) film and the second a PZT filmprepared using sol–gel technology. In each case, the undersideof a cantilever was coated to form a cantilever that was self-sensing in the non-contact mode. An ac signal was appliedto the PZT film to provide the oscillating signal for excitingthe cantilever. As the oscillation amplitude is changed, due tointeraction with the sample that also caused some bending ofthe cantilever, the cantilever’s admittance changed resulting ina change in the current flowing through the PZT film that wasused as a signal for detection of cantilever position.

Further work by Itoh et al (1996) and Watanabe andFujii (1996) that increased the number of either ZnO orPZT layers resulted in cantilevers with piezo films that couldbe used for excitation of the cantilever, deflection sensingand tip–sample spacing. In both cases the cantilever wasfabricated from silicon onto which metal electrodes were

deposited that sandwiched a thin film (approximately 1 µm)of PZT material. An ac voltage was applied to the PZTto provide the actuation and changes in the current flowingthrough the PZT which were then converted into a voltage anddetected using the lock-in technique to provide a measure ofthe tip’s displacement. The cantilevers produced by Watanabeand Fujii also had an anisotropically etched pyramid on thecantilever that formed the tip with which the authors imagedmonoatomic silicon (1 1 1) steps. Lee et al (1997a) presenteda detailed study of the properties of piezoelectric cantileverssuggesting that with careful design it should be possible toproduce cantilevers for a variety of applications. Chu et al(1997a, 1997b) developed cantilevers that could be used inhigh vacuum SFMs in both the amplitude slope detection andthe frequency modulation methods. They showed that self-actuating cantilevers were much easier to handle, align anduse in vacuum than the conventional cantilevers that reliedon optical methods for detection. Not long afterwards, theuse of piezocantilevers in liquid was demonstrated (Lee et al1997b) together with the use of self-sensing cantilevers forlow temperature scanning probe microscopy (Beck et al 1998).Rogers et al (2004) summarized the features of piezoelectriccantilevers and contrasted the performance of ZnO and PZTpiezofilms as actuating materials.

3.5.6. Cantilever arrays. Effort has been directed towardsthe use of arrays of self-sensing cantilevers in order toincrease scanning speeds and reduce the time required for dataacquisition or for lithography since an array of cantileversallows adjacent parts of a sample to be probed in parallel(Minne et al 1999). The piezoresistive and piezoelectriccantilevers do not require the bulky optics associated withconventional cantilevers so they lend themselves well tomultiple cantilever applications. A first attempt by Minneet al (1995b) produced an array of five piezoresistive siliconcantilevers. Although they were able to image an area at 4 timesthe speed of a single cantilever, it was recognized that it wouldbe necessary to servo control all of the cantilevers individuallyrather than just one. Consequently the work was extendedto produce an array of two servo-controlled cantilevers withpiezoresistive sensing elements and piezoelectric actuatingelements suitable for both imaging and lithography (Minnieet al 1995b, 1996). Improvements to the cantilever design toeliminate cross talk between the sensor and actuating elementsled to a row of 50 cantilevers with a 200 µm pitch with anequivalent scanning speed of 4 mm s−1 (Minne et al 1998).

Parallel atomic force microscopy with a novel opticaldetection method has been demonstrated. An array of 5interdigital cantilevers (Manalis et al 1996) has been usedtogether with optical interference detection or simultaneousscanning and servo control of the cantilevers (Sulchek et al2001). The cantilevers contain a series of interleaved fingersthat form a diffraction grating. Alternate fingers move whenthe cantilever bends causing the intensity of the diffractedorders to vary and thereby providing a measure of thecantilever’s deflection. Hafizovic et al (2005) developed astandalone single chip (7 mm × 10 mm) AFM with an array ofself-sensing piezoresistive cantilevers together with integrated

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Figure 4. Ratio of the measured depth Dmeas to the true depth of the groove Dgrove plotted as a function of groove aspect ratio for several tipswith different aspect ratios ARtip.

electronics that could be used for force–distance measurementsand imaging, in both dry and wet environments. Two-dimensional arrays of cantilevers have also been produced fordata storage applications (Lutwyche et al 2000). A 32 × 32array of parallel working SFM cantilevers has been used toread and write data on a polymer storage medium. Morerecently, this has been extended to a 64×64 array on a 100 µmpitch. Currently work is in progress to develop a 128 × 128array (PRONANO).

3.5.7. Capacitive detection system. Cantilever deflection hasbeen detected using capacitance based methods, Sarid (1991).This detection system consists of at least two electrodes:one of which is defined by the metallized top surface of thecantilever. Neubauer et al (1990) described an SFM thatused changes in capacitance to measure displacement changesand was sensitive to both vertical and lateral movements ofthe cantilever. Although their instrument had a theoreticalsensitivity of 5 × 10−7 F m−1, this was not achieved due tothe surface roughness of the capacitance sensor. Miller et al(1991) and Joyce and Houston (1991) introduced a rockingbeam sensor, an inherently unstable system stabilized by aservo loop. It comprised a small beam held balanced on aweak pivot by two capacitors. The force applied was calculateddirectly from the dimensions of the sensor and the bias. Blancet al (1996) demonstrated the suitability of the detector formicro-fabrication where measurements of small capacitancesare achieved by using high frequency modulation techniques.

4. Tips

4.1. Shape and material

The cantilever with the tip is the essential part of the AFMand as with any probing instrument, it is important to knowthe properties of the probe and its interaction with the sampleunder investigation. Furthermore, different types of probesare suitable for different applications. Originally tips weremade from either silicon dioxide or nitride and had a pyramidalstructure. Subsequent developments led to the large scalefabrication of silicon tips having a variety of properties that

can be adjusted to suit the specific application. These tipscan also be functionalized with a chemical or biologicalcoating designed to encourage specific interaction betweenthe tip and the surface that can be used for intermolecularforce measurement, chemical sensing, surface mapping andpromoting a hydrophobic (Luginbuhl et al 2000) or hydrophilicinteraction between the tip and the sample. More recently,carbon nanotubes have also been functionalized. A detaileddescription of functionalized tips is beyond the scope of thisreview.

SPM techniques considered here are restricted to normalcontact and non-contact SFM, CD measurements and MFMmeasurements. The measurement of small dimensions onnanostructures and the roughness of surfaces depends criticallyon the tip shape, size and the stability during scanning. The tipshape can be measured before and after measurement using tipcharacterizers. Ideally tip wear should be minimized by designand/or using special materials. Although the characterizationof the tip is important, the majority of techniques for tipcharacterization only yield a relative tip shape. Absolute valuesfor tip shape together with an uncertainty are very seldomobtained because of the difficulty of measurement. Thereforethe tip characterizer can only really be used to measure therelative stability of the tip.

The flatter a surface, the easier it is for a tip to followthe surface and achieve a higher lateral resolution. Howeveron rough surfaces a blunt tip would be insensitive to the finersurface features leading to an incorrect representation of thesurface profile. For the quantification of surface roughness(Griffith 1993) or line width there is a strong dependence ontip properties. Additionally, a sharp tip is necessary to measurethe structure of deep trenches. The aspect ratio of the tip shouldbe higher than those of the structures to be measured. This isshown in figure 4. Here the measured depth normalized to theactual groove depth is plotted as a function of the aspect ratioARgroove of the structure. The shapes of the tips are calculatedfor a radius of 10 nm with different opening angles α, theangle between the two sides of a two-dimensional projectionof the tip. The ability to measure the right depth of the groovedepends on the opening angle of the tip, i.e. the aspect ratio.The bandwidth of the transfer function becomes larger if the

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Figure 5. Measured width Wmeasured normalized to the groove width Wgroove = 400 nm as a function of the groove depth Dgroove for severaldifferent shaped tips: normal silicon tip, tip with single-walled nanotubes and multi-walled nanotubes, CD tips with different diameters anda FIB shaped tip.

Table 2. The dimensions of typical SFM tips are shown.

Radii of Cone Lengthapex (nm) half-angle (◦) (µm)

Silicon tip 7 <10 10–15Super sharp 2 <10 at last 10–15

silicon tip 200 nm of tipFIB <15 <3 >1.5EBD ∼10 ∼4 up to ∼5

aspect ratio increases. A transfer function for an ideal delta-like tip is plotted for comparison and appears as a horizontalline in figure 4.

In addition to the measurement of the height or depth ofstructures, the width of grooves or lines has to be measuredcorrectly. Figure 5 shows the transfer function for the width ofa rectangular groove with width Wgroove for increasing depthDgroove. The plotted curves show that measurements with anormal cone-shaped silicon tip would lead to an increasinglyerroneous measurement of groove width as the groove depthDgroove increases. Use of a finer cone produced, e.g. by focusedion beam etching, would reduce this error. A cylindrical tipshape as with a typical CD tip or nanotube tip shows a constantoffset over a range of groove depths Dgroove. In this case thecalibration of such tips would allow one to correct the measuredresults.

Several techniques have been developed to produce highaspect-ratio tips or CD tips. The techniques described belowdepend on the material used. Also, to improve the tip’s stabilityand wear resistance the tips can be coated; see later.

4.2. Type of tips and methods for ‘fine tuning’ of tips

Instruments for resolving small structures generally requiretips with very small radii and tip angles. In addition to thenormal or sharpened silicon tips that are mass produced, thereare other techniques available for producing extremely finetips. Table 2 lists some of the tips available together with theirtypical dimensions.

4.2.1. Focused ion beam tips. Focused ion beam etching(FIB) can be used to sharpen conventional tips making themsuitable for the examination of high aspect-ratio structures.Vasile et al (1991) etched conventional tungsten STM tips witha 50 nm radius to reduce the diameter to 3–4 nm and the coneangle to 12◦–15◦. Subsequently conventional silicon tips wereetched using FIB tips and are now commercially available. Thespecial tips used for CD measurements are produced by thistechnique.

4.2.2. Electron beam deposition. Electron beam deposition(EBD) is another method of producing high aspect-ratio tipsfor SFM. Carbon or other material from the residual gasesin a vacuum chamber is deposited on the end of a tip. Firstdeveloped for STM tips (Keller and Chih-Chung 1992), EBDtips were introduced for AFM by focusing an SEM electronbeam onto the apex of a pyramidal tip arranged so that itpointed along the electron beam axis, Schiffmann (1993). LikeFIB tips, EBD tips were found to achieve improved imagingof steep features as shown in figure 6. By controlling theposition of the focused beam, the tip geometry can be furthercontrolled. Tips were fabricated with lengths of over 5 µm andaspect ratios greater than 100 : 1, however, these tips were veryfragile (Keller et al 1992, Ichihashi and Matsui 1988).

4.2.3. Diamond tips. Diamond has several interestingproperties that make it a candidate for SFM tips. Its hardnessmeans that it is not subject to wear and its high thermalconductivity ensures that it will rapidly come into thermalequilibrium when in contact with a sample. Also since it is aninsulator that can be doped with boron and nitrogen to make itsemiconducting or conducting, this opens up the possibilityof conducting tips that can be used for nanopotentiometryand capacitance microscopy. For applications where ahigh resistance to tip wear is necessary, diamond coatedtips are commercially available that can also be used fornanoindentation measurements. Coating a tip with diamondhas the disadvantage of broadening the tip and thereby reducingits resolution. Oesterschulze et al (1997) described the

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Figure 6. Measurement of a step height standard using a normal silicon tip (left) and a silicon tip with an EBD deposited tip on the end.

fabrication of diamond cantilevers and tips on a silicon wafersubstrate, and Malvae and Oesterschulze (2006) describedanother production method that resulted in tips with a radiusof less than 10–13 nm.

4.2.4. Carbon nanotube tips. Recently there has been alot of interest in using carbon nanotubes as SFM tips. Suchtips have many advantages, including superior resolution andlow wear. They are hydrophobic and possess the ability tobe functionalized. Carbon nanotubes come in two forms;the single-walled nanotube (SWNT) and the multi-wallednanotube (MWNT). For more information about nanotubes,see the book by Harris (1999). The SWNTs have diametersin the range of 0.6 nm to a few nanometres and the multi-walled nanotubes have a maximum diameter of 50 nm. Thesegeometric properties of nanotubes, as well as other propertiesdiscussed later, make them desirable candidates as probing tipsfor AFMs; however, it is necessary for the tubes to be attachedto an existing SFM tip, something that is not trivial.

Several methods are available for attaching nanotubesto AFM tips. Essentially they fall into two groups; gluingmethods in which the tubes are bonded to the existing tipand growing methods in which the nanotube is grown ontothe end of an AFM tip using chemical vapour deposition(CVD). The first carbon nanotube AFM probes (Dai et al1996) were fabricated with a gluing method using techniquesdeveloped for assembling single-nanotube field emission tips(Rinzler et al 1995). A variety of techniques has been usedto aid attaching the nanotube to the tip including alignmentusing a magnetic field (Hall et al 2003), dip coating anddielectrophoresis (Brioude et al 2004). The method has beenrefined and is still used; however, a problem associated withthe method is the manual positioning of the tip with respect tothe cantilever. This is discussed later.

Alternatively, nanotubes can be grown onto AFM tips.The first successful method used the pore growth method(Hafner et al 1999a). Standard silicon tips were flattened andanodized to create pores of 50–100 nm diameter along the tipaxis into which a FeSO4 catalyst was deposited prior to CVDof the tip. An alternative is the surface grown method (Hafneret al 1999b), in which iron–molybdenum and colloidal ironoxide are electrophoretically deposited onto a tip prior to CVDof the carbon nanotube.

Another method of creating nanotube tips, something ofa hybrid between assembly and CVD, is called the pick-upmethod (Hafner et al 2001, Gibson et al 2007). Verticallyaligned tips are grown using CVD on a silicon substrate. Thetips are then imaged using silicon AFM tips that are coatedwith an adhesive so that they can pick up the SWNTs.

There have also been attempts at mass production ofAFM tips with carbon nanotubes. Franklin et al (2001) andYenilmez (2002) reported on the growth of CNTs onto existingcantilever tips on 4 inch diameter wafers and Ye et al (2004)used a plasma enhanced CVD method to deposit nanotubeson a 4 inch diameter array of cantilevers. Their methoddiffered from the previous ones in that they were able to definetip locations and diameters using electron beam lithography.Furthermore, no shortening of the tip was necessary. Nguyenet al (2005) discuss some of these methods in more detail.The initial work of Dai et al (1996) demonstrated the improvedresolution achievable using nanotube tips and since his originalpublication there have been many more demonstrating thesuperior resolving capabilities of CNT tips.

Both types of nanotubes can have a length of up to a fewmicrometres; however, owing to their greater diameter, theMWNTs have a larger bending force constant than the moreflexible SWNTs. Nevertheless, both types of nanotubes willbuckle under load rather than breaking, making them suitable

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for examining soft samples such as biological samples withoutcausing damage. From the point of view of maximizingresolution for dimensional measurements, the SWNT is abetter candidate for SFM metrology because of its smallerdiameter. However, its lower bending constant means that itmust be almost perpendicular to the cantilever for accurateimaging. The MWNT has a lower lateral resolution butcan work when its geometry is not ideal with respect to thecantilever and is less susceptible to thermal fluctuations. Akitaet al (1999) and Chang et al (2004) suggested that in order toavoid mechanical instabilities the length of SWNT should beless than 50 nm and that of MWNT less than 500 nm. Howeverto achieve atomic resolution, the length must be even shorter.Barwich et al (2000) calculated that for a typical MWNT(40 nm outer diameter and 1 µm long) the thermal noise wouldbe of the order of 0.1 nm so as to achieve atomic resolution,i.e. a factor of 10 better, a 40 nm diameter nanotube should beno longer than 210 nm. If a nanotube is too long, it can betrimmed using an electron beam (Martinez et al 2005). Akitaet al (2000) enhanced the stiffness of a CNT by bundling itwith others such that it protruded from a 600 nm high bundleby 50 nm. The diameter of the bundle was 28 nm diameter andthat of the nanotube 10 nm. When imaging a DVD surface, thebundled tip produced more stable images around the trenchesthan a single nanotube. However it was noted that when ina deep structure, a nanotube, with its straight profile, whilehaving higher resolving capabilities, will be more susceptibleto lateral force between the tip sidewall and the structure than aconventional tip. This can be attributed to the reduced contactarea caused by the tapered profile and hence larger tip angle ofthe conventional tip.

Dai et al (1996) and Nguyen et al (2001) have shownthat single-walled nanotube (SWNT) probes can achievea lateral resolution as small as 2 nm when measuring thetopography of thin film coatings and what is more importantfor measurements on rough surfaces is that multi-wallednanotubes (MWNT) show no detectable degradation in lateralresolution after more than 15 h of continuous scanning on asilicon nitride (Si3N4) surface. This has been confirmed byLarsen et al (2002) and Yasutake et al (2002) who found thatCNT have at least 20 times the life of etched silicon probes.The increased lifetime of CNT tips can be attributed to twofactors: their higher wear resistance and their parallel-sidedwalls in contrast to normal etched silicon tips that are tapered.As a tapered silicon tip wears, its radius of curvature becomeslarger, i.e. it becomes blunt and the achievable image resolutiondecreases, whereas for a CNT, as it wears, the tip shape(rectangular) will stay constant (Chang et al 2004). Wade et al(2004) correlated the morphology of SWNT tips with imageresolution and found that in some cases the resolution achievedappeared to be greater than the diameter of the nanotube.Modelling by Shapiro et al (2004) showed that this apparentincrease in resolution was due to elastic deformation of both thetip and the sample resulting in a decrease in apparent samplewidth. Although SWNTs have very high Young’s modulus(1.25 TPa along the tube axis), they can be subject to lateraldeformation.

This long life, the lack of debris from the tip depositedonto the sample and the ability to probe deep holes and

Figure 7. A MWCNT attached to a tungsten spike (D Cox NPL).

valleys makes CNT tips ideal for surface metrology and criticaldimension (CD) metrology (Schlaf et al 2002). However,both Schlaf and Struss et al (2005) discussed some of theimaging artefacts in SFM images of deep trenches obtainedwhen using MWNT tips that had been glued to conventionaltips. Normally the CNT is bonded to the side of a cantilever,thereby putting it in a non-perpendicular orientation withrespect to the sample surface. Schlaf et al (2002) showedhow this causes a distortion in the image. They imaged both alinewidth structure and an undercut chessboard (MikromaschTGX01—see section 5.5.2) in a series of different orientationsand showed how the resulting image was a distortion of theobject. Struss et al (2005) extended the work and providedexamples of images that appeared to have chunks missingfrom trench walls and ringing at the bottom of trenches ofhigh aspect-ratio structures. The chunks could be eliminatedby reducing the amplitude of oscillation and ringing could beeliminated with better alignment of the nanotube and choiceof servo parameters.

The problem of alignment can to some extent be overcomeby using a support for the nanotube. Figure 7 shows a multi-wall carbon nanotube tip produced at NPL. The tip shownin the micrograph was made by first depositing a tungstenspike, using ion beam deposition of W(CO)6 onto the end ofthe cantilever tip. This spike serves two purposes: it extendsthe end of the tip giving an even better aspect ratio and moreimportantly provides something to which the nanotube can beattached. This spike automatically aligns the tube down thecorrect axis, eliminating one of the biggest problems, namely,the tube has to be attached and aligned in three dimensionswith only a two-dimensional field of view.

CNT tips have also been used for imaging biologicalsamples, as was first demonstrated by Wong et al 1998.Stevens et al (2002) imaged a sample of marine sponge andshowed how additional artefacts appeared on the image whena conventional tip was used where the sides of the tip cameinto contact with the sample. When a CNT was used theseartefacts disappeared. Chang et al (2004) and Bunch et al(2004) imaged DNA. Images obtained using PtIr coated tipsand silicon tips were compared with those obtained with CNTs.

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Figure 8. Assembled cantilever probe (ACP) structure comprisingtwo cantilevers glued together.

The conventional tips provided an image with a width of11 nm compared with 3.5 nm with the CNT. Since CNTs arehydrophobic, the effects of the capillary force (see section ontip–sample interactions) are reduced and the tips can be usedto image samples in water.

The conductivity of nanotubes makes them suitable forcharge imaging (Tzeng et al 2002) and functionalization(Wong et al 1998a, 1998b, Seo et al 2003).

4.2.5. High aspect-ratio probes and side wall probes. Anincreasing industrial requirement is the ability to examine highaspect-ratio structures such as ink jet nozzles, CD structures(see next section) micro-gears, MEMS devices and otherartefacts with deep holes and side wall angles approaching90◦. Examination of these structures is not possible usingconventional cantilevers and probes. Alternative methods havebeen devised, using both FIB etched tips and using carbonnanotubes, both mentioned above. FIB tips with aspect ratiosof typically 5 : 1 have a typical length of 125 µm so their rangeof application is limited. To overcome this other approacheshave been adopted.

Clearly carbon nanotubes are suitable candidates forstudying high aspect-ratio structures. Chen et al (2006)produced carbon nanocone tips on tipless cantilevers. Theydeposited a nickel catalyst film on the cantilever followedby a carbon dot, etched the remaining nickel away and theygrew the nanocone using plasma enhanced CVD. The resultingnanocones had an aspect ratio better than 10 : 1; base diameterof ∼200 nm and a length of ∼2.5 µm. Such tips are morerobust than a conventional nanotube while offering the samelow wear rate and comparable resolution.

Another approach to investigating sidewalls of highaspect-ratio features was presented by Dai et al (2006,2007). Here a novel probe module, an assembled cantileverprobe (ACP), has been proposed, which comprises a secondcantilever glued perpendicularly onto a conventional cantilever(see figure 8) with a tip at the free end.

If this horizontally pointing tip (on the vertical cantilever)is used to scan sidewalls the deflection induced in the verticalcantilever will be transferred to the horizontal cantilever asthe vertical cantilever assembly usually has a higher stiffnessthan the horizontal cantilever. Since the horizontal cantileverdeflects, the conventional detection methods such as beam

deflection or fibre interferometry can be used. Side wallscould be scanned by using a z–x or z–y scanning, however it isnecessary to adjust the servo control of the system so the controlis done in the x or y direction respectively. Investigations byDai et al were performed on the sidewalls of microtrenches,microgears and line edge roughness samples. They showedthat the ACPs have a measurement noise at the sub-nanometrelevel.

The ACPs have the advantages of conventional AFMprobes; high sensitivity, high vertical and lateral measurementresolution and low probing forces. In addition, ACPs have twomajor advantages over conventional AFM probes. First, thetips of ACPs may extend in a horizontal direction for probingsidewalls with high sensitivity. Second, there is a large spacing(up to 600 µm or even more) between the tips and cantileverswhich allows the tips to measure micro-structures withoutbeing hindered by cantilevers. Although the current ACPs aremainly designed and fabricated using commercially availableAFM cantilevers, they are not limited to such parts. Othermicrostructures and probes can also be micro-assembled onto acantilever for measurement purposes, providing the possibilityof developing new kinds of true 3D coordinate measuringprobes and advancing AFMs to ‘micro-nano-CMMs’.

Bauza et al (2005) presented a virtual probe tip to measurehigh aspect-ratio structures, They fixed a carbon fibre probeshank of 7 µm diameter and 700 : 1 aspect ratio to the endof an oscillator. The oscillator is used to produce standingwave oscillation in the probe shank, i.e. the free end of theshank generates amplitudes of oscillation larger than the shankdiameter, thereby obviating the need for a spherical ball onthe end of the rod. A scan of a steel surface revealed surfacefeatures of less than 5 nm. When the probe was used to measurea ruby ball (a conventional probing tip) a thin water film onthe workpiece did not significantly influence the measurementresults.

4.2.6. CD Metrology tips. An important application of theAFM is the measurement of sub-micrometre track widths inelectrical devices. Indeed one could say that the electronicsindustry through the International Technology Roadmap forSemiconductors (www.itrs.net) is a driving force for ever moreaccurate measurement of smaller linewidth structures. Severalnational metrology institutes have devoted considerable efforttowards traceable metrology for SFMs in an effort to fulfilthis need, (Bosse and Wilkening 2005, Orji and Dixson 2007).For such measurements the tip shape can greatly affect themeasurements made with the SFM. When a conventionalpointed tip is used to measure line width, the width and theshape of the tip must be taken into account since they notonly result in a measurement of the linewidth being wider thanthe actual width, but also affect which parts of the sidewall(if any) will be probed. Nyyssonen et al (1991) describedan AFM with two-dimensional interferometric metrology tomeasure the tip’s position in x and z. The AFM tip wasvibrated in two dimensions and had protrusions to measuresidewalls. Martin and Wickramasinghe (1994, 1995) pointedout that, at that time, a measurement precision of better than10% was difficult to achieve when using conventional tips

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Figure 9. SEM picture of a flared CD tip (Team Nanotech GmbH).

and described an AFM with tip servoing in the horizontal andvertical directions that used so-called flared or ‘boot shaped’tips which allowed the sidewalls of a line width structure to beprobed and are referred to as CD-tips. An example of a flaredtip is shown in figure 9. This AFM was specifically designedfor CD metrology and is known as a CD-AFM. Compared withconventional AFM imaging, which is more or less a projectionin the z direction, the CD-AFM servos enable bilateral imagingof vertical side walls and at the same time of re-entrant regions,too. In addition, geometric issues associated with tip tilt, Meli(2000b), are effectively eliminated and the effect of the tip onthe image obtained is determined only by the shape and sizeof the tip.

Over the years the performance of flared tips has beencompared with that of ordinary tips, Nelson et al (1999), andthe sources of error associated with these flared tips have beenstudied. Orji et al (2005) used the CD-AFM to measure thesidewall roughness of some linewidth structures with a squarewave pattern on the sidewalls. Both a conventional AFMand a CD-AFM were used; the CD-AFM being fitted witha flared tip and the conventional AFM being used with eithera conventional tip or a carbon nanotube tip. As expected,the carbon nanotube tip had the highest resolution and couldresolve spatial features with a 200 nm periodicity. The poorerperformance of the CD tip was attributed to its larger flattip width of 117 nm and hence a larger dilation of features.Although it is possible to reconstruct complex probe tip shapes,determine their width and remove their dilating influence fromthe measurement Dahlen et al (2005) this was not performedhere (see next section for more on dilation). Clearly the lackof very sharp flared tips is limiting the performance of theCD-AFM; the attachment of carbon nanotubes onto flared tipswould clearly enhance the performance of such an instrument.In subsequently reported work by Dixon et al (2005) absolutetip width was calibrated with an uncertainty of 1 nm.

More recent work by Orji and Dixon (2007) highlightssome of the higher order tip effects when using the CD-AFM.Not only is it important to remove the dilation caused by thetip width, but also when using CD-AFM tips, the shape of

the flare has to be taken into account. Because of roundingof the flare there is a height offset that means that the lowestpart of a raised linewidth structure will not be imaged. Thisis not a problem if the walls of the linewidth are parallel andperpendicular to the substrate, but frequently the walls have aslight taper. Furthermore the protrusion of the flare from thestem of the tip needs to be accounted for, since if the width ofthe linewidth structure is wider at the top than at the bottom,there may well be a region at the top of the structure that isinaccessible to the probe.

An alternative approach has been to use a dual probesystem. Two cantilevers are used that are aligned so that thetips are horizontal and facing each other. Each tip is used toexamine one side of a linewidth structure, thereby enabling anabsolute measurement to be made that is independent of tipshape (Mancevski and McClure 2002).

4.3. Magnetic force microscopy tips

In a magnetic force microscope (MFM) a magnetic tip is usedto probe the magnetic stray field above the sample surface.The magnetic tip is mounted on a cantilever whose deflection isrelated to the strength of the magnetic field beneath the tip. Thesample is scanned in order to map the magnetic forces (contactmode) or force gradients (non-contact mode) above the surface.The first MFM cantilevers were made from nickel (Saenz et al1987, Mamin et al 1988), or iron (Martin and Wickramasinghe1987) wires which were etched and bent to define a tip. Thesetips were not very sharp and had a high magnetic momentthat interacted with the sample. To enhance the imagingresolution by minimizing the volume of the magnetic materialin the tip Rugar et al (1990) coated the sides of tungsten tipswith a magnetic CoPtCr layer. The coating technique wasalso applied to silicon microfabricated cantilevers, openingthe possibility of batch fabrication and tailoring the magneticproperties of the tips by choosing appropriate coatings (Grutter1990). Very sharp tips with high aspect ratios have beenproduced by coating electron beam deposited carbon needleswith appropriate magnetic thin film materials (Ruhrig 1994,1996, Skidmore and Dahlberg 1997). The needles were grownonto AFM tips by EBD.

4.3.1. Magnetic coating of AFM tips. The most commonMFM cantilevers are silicon batch fabricated cantilevers witha coated tip optimized for magnetic force microscopy. It isnot possible to give a short comprehensive overview over alltypes of coatings used for magnetic force microscopy. For thisreason only a few examples will be mentioned. Conventionalmagnetic coatings are made of high coercivity CoCr, CoPtCr(Grutter et al 1990, 1991a, 1991b), or low coercivityferromagnetic materials Co (Grutter et al 1991a), NiFe (Grutteret al 1991b), iron (Sueoka et al 1991). A high coercivitymakes the tip resistant against reversal. On the other hand,the stray field of the tip causes an interaction in between tipand sample. To reduce the tip stray field interaction, a varietyof coatings can be employed. Hopkins et al (1996) and Liouand Yao (1997) coated tips with a superparamagnetic granularFe(SiO2) film. Other authors demonstrated diamagnetic

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behaviour in platinum (Teschke 2001), antiferromagneticbehaviour of ((CoCrPt/Ru/CoCrPt) Wu (2003), multilayers(FM/spacer/FM) (Shen and Wu 2004, Liu et al 2002b) andnovel hardmagnetic (CoPt , Liou (1998)) coatings of MFM tips.

4.3.2. Improving resolution in MFM. Over the last fewyears different approaches have been used to improve theresolution of MFM tips. High resolution imaging of the in-plane components of stray field was reported by Folks et al(2000): a hole with a diameter as small as 20 nm was milledthrough the magnetic layer at the apex of each tip using afocused ion beam. With these tips 50 nm transitions on a harddrive were imaged. Several groups reported the fabricationof advanced probes by the confinement of the magneticallyactive tip volume. A magnetic refinement was achieved bybroad area deposition of a magnetic layer followed by thedeposition of a mask and sputter etching (Bauer et al 1996,Jumpertz et al 1997). Tips which imaged features with a fullwidth half maximum resolution of 42 nm were produced bydepositing metallic structures onto atomic force microscopetips by evaporation through nanoscale holes fabricated in astencil mask (Champagne et al 2003). In addition to theconfinement of the magnetically active volume efforts weremade to achieve a high shape anisotropy of the tips. Philipset al (2002) coated AFM tips on one face with a 30 nm thickcobalt thin film which was then modified by focused ion beam(FIB). Such tips possess a planar magnetic element with highmagnetic shape anisotropy due to an extremely high aspectratio of greater than 30 : 1. Sub-30 nm wide one-dimensionalfeatures could be resolved. Another approach using FIB isto sharpen tips before coating; Litvinov and Khizroev (2002)made FIB trimmed MFM tips in the shape of a 10 nm tallcylinder with a 50 nm diameter

The electron beam induced deposition (EBID) depositionprocess for the production of carbon needles was modified byUtke et al (2002). The EBID technique was used to depositcubiccobalt clusters dispersed in a stabilizing carbonaceousmatrix on a silicon tip to avoid the additional step of coating.Deng et al (2004) reported the production of metal-coatedcarbon nanotube tips. The carbon nanotubes with diameters of1–5 nm were deposited on a pyramidal tip by CVD and coatedwith (Ti/Co/Ti). Van den Bos et al (2002) demonstrated anew tip geometry; the cantilever was made by the depositionof magnetic material on the side of a free hanging, very thinlayer called the tip plane and magnetic features down to 30 nmcould be observed. A comprehensive analysis of the resolutionof different microscopes, tips and measurement methods hasbeen published by Abelmann et al (1998). The analysis wasbased on a set of reference samples and the obtained resolutionvaried between 30 and 100 nm. Recently Winkler et al (2006)used iron filled MWCNTs as tips for MFM although morework is necessary before the resolution of the probes can bequantified.

4.3.3. Calibration of MFM tips. The interpretation ofthe MFM signal in terms of quantitative stray field valuesrequires the knowledge of the magnetization distribution orthe stray field of the tip, i.e. a tip calibration. Different

experimental approaches are used. The tip stray field canbe analysed directly using Lorentz microscopy (McVitie et al2001), electron holography (Streblechenko et al 1996) andHall effect microsensors (Thiaville 1997). However, thesetechniques are time consuming and complicated. Alternativelythe tip magnetization Mtip can be restored from the measuredsignal of a predefined stray field distribution Hsample. The mostwidespread approach is to describe the tip by a point probemodel, Hartmann (1989). In the point probe model the tipis described as a single point monopole q or dipole m. Theposition dm and dq and magnitude of q and m are extracted fromcalibration measurements. Usually current carrying strip lineswith calculable stray field distributions are used as referencesamples. One can use straight strip lines (Gomez et al 1998,Youngsthon 2001, Kebe and Carl 2004, Liu et al 2002a) orcurrent rings (Lohau et al 1999, Kong 1997a). Kong and Chou(1997a, 1997b) found that at different lift heights the monopoleand the dipole moment contribute unequally. In a recentlypresented systematic study Kebe and Carl (2004) showed thatthe point probe parameters mz , q, dmz and dq are dependenton the width and the distance of the current carrying parallelwires that are used for the calibration. This was attributedto the characteristic decay length of the z component of themagnetic field.

The observed dependence of the calibration parameters onthe dimension of the calibration sample is a limitation of thesimple tip model. Hug et al (1998, 2002), van Schendel et al(2000) presented a new procedure for calibrating the tip thatcan overcome the problems of the point probe approach. Theirtechnique yields a feature-wavelength dependent calibrationfunction based on a transfer function approach. The process forobtaining the transfer function consists of several consecutivesteps. The calibration process starts with making severalmeasurements of different magnetization patterns. Next, themagnetization pattern of the sample is estimated. Fromthe magnetization pattern, the sample stray field or field iscalculated. The calibration function in the Fourier domain,the so-called tip transfer function TTF, can be calculated bycomparing the stray field values and the measured signal.The error in the estimated magnetization pattern is reducedby an iterative process and the sensitivity is found to bein the micro-tesla range. The iterative process is necessarybecause the magnetization of the samples that are used forthe characterization is not known a priori. This problemcan be overcome by using reference samples with well-defined magnetization distribution. Such calibration samplesare being developed by the PTB in cooperation with theUniversity of Gottingen (Dreyer et al 2007). They are based onartificially patterned hard magnetic films and are characterizedby combined quantitative magneto-optical indicator filmsMOIF and MFM.

4.3.4. Ideal tip shape for MFM. Porthun et al (1998)and Saito et al (2003) discussed the achievable resolution inmagnetic force microscopy. Their analysis is based on the tiptransfer function, as derived in Hug (1998) and the minimumdetectable wavelength is used as a measure for the lateralresolution. From their analysis the authors conclude that for

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the highest signal at high spatial frequencies the tip should beneedle like and as fine as possible. Materials with high Ms givethe highest sensitivity, but the tip volume has to be minimizedto lower the influence of the tip on the sample. While Porthunet al favoured a flat tip end, Saito et al demonstrated that tipswith an ellipsoidal end are better candidates for optimizingthe resolution. Fine needle like tips have been manufacturedusing electron beam deposition, (Skidmore and Dahlberg 1997,Ruhrig et al 1996) or focused ion beam techniques (Philips2002, Liu et al 2002b, Litvinov and Khizroev 2002).

5. Tip characterization and reconstruction

5.1. Introduction

As stated at the beginning of the paper, the uncertainties for stepheight and pitch measurements are now in the nanometre andpicometre range respectively. In both cases the measurementsare independent of the tip shape as long as the structureis not too small and the tip is stable. On the other hand,for measurements of line width (CD), particle shape, surfaceroughness and the shape of structures in the nanometre range,the tip shape has a significant effect on the measured profiles.To estimate the true shape, width and profile of a surface it isnecessary to know the tip shape or, more correctly, the effectof the tip shape on the image. For scanning force microscopythe ideal fine tip is made from infinitely sharp, stiff, durableand non-reactive material. However it must be accepted suchtips do not exist. Thus knowledge of the geometry and thephysical characteristics of the probe together with the natureof the interaction between probe and sample are of crucialimportance for form measurement. The measurement ofproperties like shape, width or CD by SFM is conceptuallysimilar to approaches developed for optical or SEM metrology.These approaches use procedures for modelling the specimenshape and the interactions of the probe with the specimen,and solve the problem using a priori knowledge. The rangeof interaction forces on an SFM probe tip with the surfacelies in the sub-nanometre range (see next section). Therefore,the accuracy of SFM for measuring CD depends on having acapability for measuring the probe shape and it is importantthat the probe and the feature on the sample are not distortedby the contact or interaction forces.

By using silicon and related etching techniques sharp tipscan be reproducibly produced and although these tips go someway towards tip-artefact free images, as do the carbon nanotubetips, the effects of the tip shape on a topographic image canstill be significant. Figures 10(a) and (b) show schematicallyhow the finite size of a tip can affect an image. In the first casethe tip is too broad and tapered to be able to probe fully thevery base of the object and in the second the tip is unable toscan the dovetail shaped sidewalls of the sample. In both casesthe profile obtained (shown in figure 10 as well) will not be atrue representation of the sample.

5.2. Observable artefacts in images

Most SFM users proceed with their measurements under theassumption that the actual tip is stable during the measurement

(a) (b)

Figure 10. (a) and (b) Schematic representation of how a finitesized tip shape can influence an image. The solid lines represent theprofiles measured.

and can be described by a few parameters, i.e. tip radius Rtip andopening angle α. Several studies in the past have shown that thedata obtained by SFM has to be critically examined since mostof the image artefacts are due to geometrical convolution of thesurface structures with the tip shape. Clearly when attemptingto make dimensional measurements from SFM images, it isessential to be able to recognize tip artefacts in SFM images.Figure 11 shows images of a gold surface. At first sight thereappear to be square features on the surface with an edge lengthof about 2 µm. Inspection of the tip used to acquire theseimages shows that it is blunt and the square features can beattributed to the tip shape. Grutter et al (1992), presented acomparison of SEM images and AFM images. On the AFMimages there were artefacts of the order of 20–600 nm thatwere related to the tips being used to image the samples.More examples of tip induced artefacts were presented bySchwartz (1994), and Markiewicz (1995) presented somefurther examples together with simulations of the tip–sampleinteraction to try to illustrate how the tip induced artefacts couldbe formed. The effects caused by this geometric interactionare critical for surface dimensions similar to or smaller thanthose of the tip radius (Rfeature < Rtip) or features with slopesgreater than those of the tip (αtip < αfeature), e.g. sharp edges,or structures with high aspect ratio ARtip < ARfeature aswas illustrated by Westra et al (1993), who when attemptingto image some thin film samples found that very often themicrostructures in the film had a smaller radius of curvaturethan the tip resulting not in an image of the film but imagesof the tip. This point is discussed later in the section on theuse of spheres to characterize tips. In all cases there remains aforbidden space or unreconstructable region on the surface thatcannot be detected by the tip. The challenge is to minimize this.

5.3. Techniques to get information about the tip shape

A brief comparison with scanning tunnelling microscopesSTMs is worthwhile in passing. For STM tips it is clearfrom the quantum mechanical foundation of the tunnellingcurrent that the properties measured by the tip depend onthe electronic structure of both the tip and the sample.

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Figure 11. Examples of how an image is distorted by a blunt tip; the tip is imaged by the sample.

Tersoff and Hanmann (1985) set up a model to explain theprofiles measured by a tunnelling tip for constant currentas the measurement of the electron density of states of thesample surface at the position of the tip atom. The highresolution surface profile obtained using the STM is basedon the exponential behaviour of the tunnelling current by(Sarid 1991).

i(z) ∝ e(−2κz),

where i(z) is the STM current, 2κ = 1.025√

φ is a constantin angstroms, φ the work function in eV and z the distancebetween tip and sample surface.

Based on this, nearly 90% of the tunnelling current goesover the atom nearest to the surface. This is the differencebetween the STM and the SFM where the repulsive forcesdepend on the distance by a power of 1010–1012 and all theatoms at the end of the tip are involved. This is illustratedin figure 12. Consequently, since the STM tunnelling currentflows only when the distance between the tip and the surface isvery small, STMs are used mainly for high resolution studies offlat crystalline surfaces in UHV to investigate physical effectson surfaces. SFMs, on the other hand, are much more flexible,because they are not restricted to the examination of conductivesurfaces.

The methods for determining SFM tip shape broadly fallinto two categories, in situ and ex situ, i.e. imaging the tipout of the SFM and imaging the tip while still in the SFM.Additionally there are various mathematical methods that canbe used, together with the in situ and ex situ methods.

5.4. Ex situ techniques (geometrical/non-interaction tipshape)

Since the apex of an SFM tip is below the resolution limit ofan optical microscope, optical techniques are mainly used asa rough check for blunt tips or to characterize NSOM tips.Consequently other techniques have to be used for proper tipcharacterization.

5.4.1. Electron microscopy (SEM and TEM). The electronmicroscope is clearly a means by which one can obtain imagesof a tip with high spatial resolution. The TEM has theadvantage of higher resolution down to the atomic columns incrystalline silicon tips, but the amount of preparation is higherthan for normal SEM. For quantitative agreement between thetwo instruments it is necessary to determine the boundary ofthe tip shape with high precision. Although the interaction ofelectrons with matter is very strong due to scattering and/orby generation of secondary electrons, it is difficult to modelwhat happens in very small volumes at the end of the tip. InMonte Carlo simulation of very fine tips Frase (to be publishedelsewhere) found that the beam current falls long before the endof the tip is reached. He calculated the profile for an SEM scanline that went across the tip surface. The tip had an openingangle of 45◦ and was tilted by 67.5◦ so that the sloping edgeof the tip was parallel to the raster line, as shown in figure 13;the results are shown in figure 13(b); with a simulation using aprimary energy of 10 keV, the signal maximum was observedapproximately 10 nm from the apex of the tip. Between thispoint and the apex , the signal rapidly decreased to 0, as shownin figure 13(c).

The determination of the boundary of the tip shape couldbe a problem for TEM as well. The intercomparison by Seah(2004) of silicon dioxide thickness measurements made bymultiple techniques found a large scatter for measurements ofthe thickness of thin SiO2/Si films measured with differentinstruments and by different users. This suggests thatestimating the profile of the end of a silicon tip with native oxidewhere the thickness is very thin could also be problematic.Also the measurements of diameters of nanoparticles usingTEM have a large spread of values, something that is notfully understood (Wang et al 2005). Nevertheless, both SEMand TEM allow one to determine a projection shape of thetip—not the 3D shape (de Rose and Revel 1997). Dongmoet al (2000) and others have shown that a good agreement,albeit qualitative, between tip shape and image shape can be

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Figure 12. For an STM only the atoms at the very end of the tip interact with the sample whereas for the SFM many more atoms interactwith the sample.

(a)

(c)

(b)

Figure 13. (a) Orientation of AFM tip for Monte Carlo simulation, (b) the path of the secondary electrons and (c) the secondary emissionsignal. Note horizontal axis is in nm and shows the distance from the AFM tip apex.

obtained. Stereo photogrammetric techniques have been usedby Gleichmann et al (1994) to determine the shape of tips. Forthe method to be successful it is necessary to correlate the samepoint in two images obtained under different viewing angles.

5.5. In situ techniques for tip shape determination

These techniques should be considered as two parts: the firstis to measure a sample in which structures are well defined orhave some well-known features, e.g. small radii. The secondpart is the evaluation of tip shape from the measured datausing mathematical procedures, such as convolution, Legendretransformation, morphological or neural network algorithms.The reconstruction of tip shapes without any knowledge about

the sample, i.e. the tip blind estimation/reconstruction, isconsidered as a separate technique later.

5.5.1. Using a tip characterizer sample. Special sampleshave been developed for in situ measurement of the tipshape, so-called tip characterizers. The idea is based on areconstruction of the tip shape from the measurement data;consequently exact knowledge of the geometrical shape of thecharacterizer is required. The disadvantage of this methodis that the characterizers must be used relatively often forprecise measurements that are rather time consuming. Thesetip characterizers consist of features on a substrate that areeither sharper or of comparable sharpness to the SFM tips.When the SFM tip is scanned over the sample, one obtains a

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Figure 14. AFM scan of a TipCheck calibration surface showingsharp randomly orientated peaks that provide the optimum surfacefor characterizing AFM tip geometry.

set of reverse images of the tip since the sample can be thoughtof as a bed of tips for probing the topography of the SFM tip.There are several types of tip characterizers available. Somecharacterizers can only determine tip radius or the tip openingangle and some can determine both.

Artificial or spiked structures. The essential aspect of thistechnique is to use sample surfaces containing delta-likefeatures or well-known structures. Based on the ideaof reversed imaging, several attempts have been made todetermine tip shape using special samples; tiny holes formedby nuclear tracks (Fraundorf and Tenschert 1991), and thecutting edge of a razor blade (Song and Vorburger 1991).Several groups have developed spike-like structures in anattempt to image SFM tips, Montelius and Tegenfeldt (1993)and Montelius et al (1994) deposited silver drops on an indiumphosphide substrate. The drops were approximately 50 nm indiameter and 120 nm high and were used to image Si3N4 tipsand high aspect-ratio tips. Seah et al (1999, 2000) described indetail the mechanism for sputter depositing polymeric coatingson InP that formed a variety of structures: caps, cones andfilaments of various heights. Their samples were designed toexamine the final 20-50 nm of a tip unlike the commerciallyavailable silicon cone arrays, Bykov et al (1998) that, dueto their height (700 nm) are not ideal for examining the tipextremity. In addition Seah et al (2000) developed softwareto allow common features from a scan image to be extractedand the dilated tip shape to be determined. More recently,Chen et al (2006) produced a bed of nanocones, identical tonanocone tips, that was used to characterize silicon nitride tips.

A ‘TipCheck’ sample (see figure 14) has been produced(Aurora Devices) that consists of a random structure with sharpfeatures [www.aurora.com]. In combination with data analysissoftware, like SPIP (Image Metrology Inc.), it can be usedto assess the shape of a tip (flank profile and opening angle)and the degree of wear the tip has suffered. This sample iscomplemented by the ‘Nioprobe’ that consists of dense sharp

needles of niobium with very small vertical height makingthe sample ideal for determining the profile of the end of thetip; the tiny peaks exhibit imaging radii of less than 5 nm.Both samples allow one to obtain the apex radius desired formedium—to small—scale work. The random orientation ofthe needles is suitable for applying blind tip reconstructionmethods.

Molecules, spheres and nanotubes. Initial work was doneusing DNA molecules of known dimensions, (Allen et al 1992,Thundat et al 1992a). These were measured with the SFM andsince the dimensions were known, a correction was made forthe finite tip shape. Subsequent work has been done usinggold and polystyrene spheres and nanotubes that are morerobust. Small colloidal gold spheres with diameters in therange 10–40 nm (Vesenka et al 1994, Xu and Arnsdorf 1994,Czerkas et al 2004) and 172 nm diameter polystyrene latexspheres (Odin et al 1994) have been used to determine tipshape. If the diameter of the sphere is known, and the sphereis not compressed when scanned, then measurement of thesphere will yield a trace similar to those shown in figure 15from which it is possible to extract the tip shape. Dependingon the relationship between the diameter of the tip and theparticle, different models should be used. If the tip radius islarger than the radius of the particle, then the situation is asshown in figure 15(a). Only the circular part of the tip comesinto contact with the particle and

W 2 = 2[(Rtip + Rparticle)2−(Rtip − Rparticle)

2](Rtip > Rparticle),

W 2 = 16RtipRparticle.

This model is sometimes referred to as a Zenhausern model(Zenhausern et al 1992).

In the case where the radius of the tip is less than that ofthe particle (figure 15(b)), then the model proposed by Garciaet al (1997, 1998) should be used since not only the curvedpart of the tip but also the side or flank of the tip comes intocontact with the particle.

W = A1Rtip + A2Rparticle,

whereA1 = 2 − tan α1 − tan α2

and

A2 = tan α1 + tan α2 −(

1

cos α1+

1

cos α2

),

where α1 and α2 are the angles of the tip as shown infigure 15(a).

In order to determine which model should be used, acritical value is given by

RC = Rtip

(B − 1

B + 1

),

whereB =

√1 + [tan(90 − α1)]2.

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(a) (b)

Figure 15. (a) An SFM tip with a radius larger than the particle on the substrate and (b) a tip with a smaller radius than the particle (afterWang and Chen et al (2007)).

Although the tip radius is to be determined, a nominal valueis usually known and so it is possible to chose which formulato use. Garcıa et al (1998) also gives a summary of previousmodels used for examining tip profile using spheres.

Assuming that every part of the tip is at some time incontact with the sphere, it is possible to reconstruct the tipsurface profile. Colloidal gold spheres are readily availablein different sizes of the order of a few nanometres and canoften be mounted onto the same substrate as the sample underinvestigation thereby facilitating tip characterization withoutchange of sample. Possible sources of error in this methodinclude non-sphericity of the gold particles, residual substratemedia around the particle and tip structures smaller than thecurvature of the sphere. This last point was illustrated byCzerkas et al (2004) who compared different methods of SFMtip shape characterization.

Nanotubes. Wang and Chen (2007) fixed single wallnanotubes to a substrate and used them to measure the radiiof SFM tips. Experimental data was evaluated using theZenhausern model. The sharpness of the tip affected onlythe ‘width measurements’ and not the ‘height measurements’.The authors pointed out that if the larger diameter multi-wallednanotubes were used (diameter 20–40 nm) the method couldbe used, together with the Garcia model (1997), to measure theradius of tips where the sides of the tip also came into contactwith the sample, thereby facilitating a full characterization ofthe tip (flank angle and radius).

5.5.2. Structures produced by lithography and etching.Several types of tip characterizers have been developed ortested. Most of them have spike-like structures or geometricalwell-defined shapes with nominal dimensions.

Sharp structures. Many of the tip characterizers comprisehigh aspect-ratio spiked structures. Figure 16 shows an array ofsharp tips with curvature of less than 10 nm and height between0.3 and 0.6 µm. A triangular ‘Nanoedge’, Griffith et al(1997) has been developed for characterizing flared tips for CDmetrology (Martin and Wickramasinghe 1995). This specimenconsists of a row of sharp knife edges with nominal tip radii of5 nm. Due to its 2D structure it is likely to be stronger and morestable than spike array. However as is the case for the other2D characterizers, it can only be used to evaluate the probe inone direction. Martin and Wickramasinghe (1995) estimated

Figure 16. Spiked structure from NT-MDT height 0.3–0.5 µm, tipangle 50◦, period 2–3 µm.

Figure 17. Spiked rows of TGG01 from NT-MDT, similar toNanoedge that can be used for determining tip radius.

that the Nanoedge allows measurements of the width to anuncertainty of approximately 3 nm. Figure 17 shows a gratingof sharp structures, similar to the Nanoedge.

Under-etched structures. Unlike the samples mentionedabove, undercut structures allow not only the radius ofcurvature R but also the flank angle α of the tip to bedetermined. Such samples are commercially available and areoften used for assessing CD metrology tips. Figures 18(a)and (b) show a section of chessboard array of square undercutpillars (TGX01) in two different orientations. Typicaldimensions for these are a period of 3 µm, a height of 1.0 µmand an edge curvature radius of less than 10 nm.

Hubner et al (2003) produced very precise line widthstructures with very sharp edges (�10 nm) that were further

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(a) (b)

Figure 18. Two views of a chessboard array of square undercut (TGX01 from ND-MDT) structures; typical dimensions are period 3 µm,height 1.0 µm and edge radius less than 10 nm.

(a) (b)

Figure 19. (a) Grating structure and (b) 2 dim. rod and hole (inverse rod) structures for tip characterization developed by Hubner et al(2003).

Figure 20. How a chirped grating can be used to determine tipwidth at various points on the vertical axis of the tip is shownschematically.

sharpened using chemical etching. The line widths werecalibrated with an uncertainty of 10 nm. Since the sidewalls ofthe structures were almost 90◦ and much steeper than the half-cone angle of the tip, the measured profiles reflect the tip flankangle. Neglecting the small radius at the edges of the line, themeasured profiles reflect the tip radius. Measurements of tipradius and flank angle derived from these structures agreed wellwith measurements made in a scanning electron microscope.The minimum tip radius observed was determined to be 30 nm.In addition, the substrate also had some rods and holes thattogether with the image analysis software SPIP could beused for surface characterization. These structures are nowcommercially available (www.supracon.com) and are shownin figures 19(a) and (b).

Chirped gratings/heterostructures. For surface roughnessmeasurements one technique to determine the radius of the

probe stylus is to use a grating with a varying period, i.e. thespacing of the grating gradually decreases. Such gratings areknown as chirped structures. These chirped structures havethe advantage that the width of the tip can be determined atdifferent parts of the tip as is shown schematically in figure 20.

In the case of the stylus measurements the structuresused are in the range of a few micrometres down to 500 nmand are made using lithographic techniques. For AFMtips such structures should be much smaller. Dwir et al(1996) and Wullner et al (1997, 1998) investigated layeredheterostructures of InGaAs/GaAs grown by metal organicvapour phase epitaxy, MOVPE. They varied the periods ofthe heterostructures in the range from 40 to 100 nm. Aftercleavage of the wafer and etching the cleaved surface, thematerial contrast is transformed into a topographical structure.Originally these grating structures were developed as lateralcalibration standards. However they have subsequently beenused to characterize AFM tips. During scanning with the tipperpendicular to the layer, this structures acts like a chirpedgrating and allows one to apply a simple geometrical modelto estimate the tip radius. Itoh et al (2006) designed a similarSFM tip characterizer based on multilayer thin films. Figure 21shows an SFM image of a chirped grating and figure 22 is theaverage profile of several lines around the centre of the image.

A simple analysis of such an ideal structure would providesome information about the radii of the tip apex. Sincethe height is relatively small compared with other tip check

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Figure 21. Image of a chirped grating structure with decreasingperiod.

Figure 22. Averaged profile of central lines of the chirped gratingshown in figure 21.

structures effects due to the feedback circuit of the SFM canbe neglected.

However these gratings have two disadvantages. First it isbroken and afterwards etched to achieve a small topographicstructure. However this etching procedure will interactselectively with both materials in a different way. Afterwardsboth materials will oxidize in air. This could lead to somerounding of the sharp edges assumed after cleaving. Comparedwith the more ideal cases of rectangular grooves, this roundingof edges would lead to deviation of the ideal curve (see the insetof figure 23). The deviation is indicated in figure 23 by thedotted curve that shows how the rounding of the edges couldlead to a false measurement of tip width in comparison withthe ideal case with no rounding of the grating edges.

5.5.3. Mathematical techniques to extract the tip shape fromimages. In addition to the simple techniques based purely ontip characterizers, there are several mathematical approachesthat have been developed to aid tip reconstruction. AFMimages are often referred to as a convolution of the tip andsample leading to an image that contains information aboutboth the tip and the sample, although it should be noted that

Figure 23. Plot of the apparent depth as a function of width W of achirped grating with ideal and non-ideal edges (r = 2 nm) asindicated in the inset.

convolution is not strictly the correct term since the process isnon-linear and a better term to use is dilation of the sample anderosion of the dilated image to obtain a better estimate of theactual sample profile. Also, erosion is not the exact inverse ofdilation, since there are points on the tip that may not comeinto contact with the sample and points of the sample that maynot come into contact with the tip. Figure 24(a) shows howa surface topography can be dilated by the finite shape of thetip resulting in the loss of surface features that are of a similarsize to the tip. The dilated surface can then be reconstructedand is shown in figure 24(b); clearly not all of the features canbe reconstructed. With the exception of blind reconstructiondiscussed later, all the mathematical methods require use of aknown characterizer sample as described above.

Slope-matching techniques and Legendre transform. Math-ematical solutions based on a process analogous tode-convolution have been derived in the form of differen-tial equations (Reiss et al 1990) or Legendre transformations(Keller 1991). Reiss et al reconstructed the surface of a roughAu–Nb–Au triple layer film and a gold stepped surface. Keller(1991) presented a method to reconstruct images of finite-sizetips based on a non-linear transform. This was an extension ofthe work of Chicon et al (1987) who developed an algorithmfor tip reconstruction of spherically shaped tips. The slope-matching technique, also known as envelope reconstruction,starts on the basis that the slope of the contacting tip and sam-ple at the point of contact point is identical. This is illustratedin figure 25, reproduced from Keller (1991). The tip surfacet (�x) is shown in contact with the true surface s(x). There isa discrepancy or offset in x and y between the apparent pointof contact (the end of the tip) and the actual point of contact ofthe tip with the surface. These two offsets �x and �s must bedetermined in order to calculate the true surface topography.Keller showed that the local curvature of the true, undistortedsample is the sum of the curvatures of the tip and the dis-torted image surface, which he then related to the LegendreTransform.

The Legendre transform of a function f (x) is defined asL[f (x)], the intercept on the y axis of the line tangent to f (x)

at point x by

L[f (x)] ≡ b(m) = f (x(m)) − mx(m),

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

Figure 24. (a) How the finite shape of a tip leads to a dilated image of the surface is shown; crosses represent the measured profile, spheresrepresent the tip (b); the reconstructed profile is shown by the line with circular points.

Figure 25. The difference between the apparent point of contact ofthe tip with the surface and the true point of contact is shown(Keller 1991).

where m = df (x)/dx is the slope at x and b, the LegendreTransform, is the intercept of the y axis.

From this Keller showed that the Legendre transform ofan image i(x ′) and tip profile t (�x) could be related to thesample surface s(x) by

L[s(x)] = L[i(x ′)] + L[t (�x)],

i.e. the Legendre transform of the genuine surface is the sumof Legendre transforms of the image and tip surfaces (Keller1991). The relationship between the reconstructed surface,image and the true surface is shown in figure 26. The LegendreTransform is given by b, the intercept of a tangential line onthe y axis. The figure shows the intercepts of tangents to thesurface, at the true and apparent points of contact of the probewith the surface (image point). If the true shape of the sampleis known, then the Legendre transform of the tip is given by thedifference between the Legendre Transforms of the true andimage surfaces. Compared with the Fourier transform methodused before (convolution and de-convolution), which is a linearbut a non-local method, the Legendre Transform is local, butnon-linear. On the other hand, with the knowledge of the tipsurface it is possible to calculate the sample surface using theinverse Legendre transform. The analysis showed too that

Figure 26. Tangential point of contact between tip and sample andthe relationship between the Legendre transforms of the true, imageand tip surfaces are shown (Keller 1991).

the reconstruction does not give information about the wholesurface and since it is a slope based method, it is susceptibleto noise in the image signal. At points where the curvatureof the surface feature is larger than that of the tip, i.e. regionswhere the tip is not able to contact each point on the surfaceor is in contact with two points simultaneously, as is shown infigure 27(a), incomplete information about the sample surfaceis available. Knowledge of this deficiency is important incases in which dimensional properties are calculated from theimages.

Envelope or morphological reconstruction. To overcomethe sensitivity of the Legendre transform method to noise,an alternative method was developed known as envelopereconstruction or morphological reconstruction (Keller andFranke 1933). It is not based on slopes and is therefore lesssensitive to noise. The principle is illustrated in figure 27.The tip surface is defined by t (x, y) and the image surface byz(x, y). If (x, y) and (x ′, y ′) are defined as two points in thexy-plane, then a function W is defined as W(x, y; x ′y ′) =z(x ′y ′) + t (x − x ′, y − y ′). W can be thought of as atip function with the end of the tip raised to z(x ′y ′). Thereconstructed surface is given by r(x, y) = minimum of the

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

(a) (b)

Figure 27. Tip profiles (a) with double contact points and unreconstructable region and (b) how envelope reconstruction works (Keller andFranke 1933).

set {z(x ′, y ′) + t (x − x ′, y − y ′)}. If the shape of the sampleis known, the method can also be used to reconstruct the tipshape as well, as was demonstrated by Markiewicz and Goh(1995) who characterized AFM tips using de-convolution ofimages of calibration array structures. When scanning anobject, regions that do not come into contact with the tip will berepresented by a segment of the AFM tip. This is in contrast tothe Legendre method that simply leaves these regions blank.In a noiseless image, this is the only difference between thetwo methods. However data are seldom free of noise. Kellerand Franke showed how upward spikes on the data could besmoothed out and downward spikes would appear as gouges inthe data.

Further work by Castle et al (1998) developed routinesto smooth out downward spikes as well. Leung et al (1997)developed routines that were able to identify the ‘questionable’regions, i.e. regions where the tip was not in contact withthe surface and the reconstructed image is indeterminate, andremove them from the reconstructed image.

Neural nets. Wang and Whitehouse (1995) used neuralnetwork techniques to reconstruct the sample surface. Themethod aimed to correct the integrating effect of a finite stylustip and assumed that the tip properties were known. A neuralnetwork, trained with simulated data, was used to reconstructimages. Although the method showed some success andis flexible, there were limitations, namely the stylus shapewas limited to those that had a unique slope at every pointin the tip and training of the neural net was difficult tooptimize.

5.6. Blind tip estimation

The above methods are essentially imaging the tip using themathematical process of dilation and erosion. Expressedmathematically, the image can also be thought of as the dilation

of the sample by the tip

Is = S ⊕ P,

where Is is the image of the sample characterizer, S is the actualsurface topography of the characterizer, P is the tip shape and⊕ denotes the mathematical operation of dilation.

If the genuine surface topography and the shape of thecharacterizer are known, then by erosion

Pr = Is � S,

where Pr is the reconstructed surface of the tip and � denotesthe mathematical operation of erosion.

However, in the case where the sample’s shape andtopography are unknown, it is still possible to obtaininformation about the shape of the tip using blind tipreconstruction. This technique was first suggested by severalgroups working independently (Bonnet et al 1994, Villarrubia1994, Williams 1996a, 1996b). If the sample were an infinitelysharp spike then the image obtained from a scan would not bethat of the sample but would be an inverted image of the tip. Asthe sample becomes blunter more information about the sampleis contained in the information. Consequently, all points in animage of a surface can be regarded as an image of the tip thathas been broadened by the surface. The goal of the blind tipreconstruction is to separate the two sources of information inorder to obtain an image of the tip. Since the genuine surfacetopography is unknown, the method cannot estimate the actualtip shape but only the shape of bluntest tip that could have beenused to record the image. When one considers the sharpestfeature in an image, this gives an estimate of the least blunttip that could have been used and if a sample scan line hasmany features more information is yielded about the possibletip shape and a good estimate of the tip shape can be made.

Three inputs are required for blind reconstruction to work,an image of a tip characterizer, a starting outer bound forthe tip shape and a threshold value for the noise on the scan

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signal. Villarrubia (1966) reported on the performance of blindreconstruction in combination with tip characterizer samplesand showed that it was a viable alternative to the conventionalmodels that require detailed knowledge of the sample. In 1997Villarrubia (1997) published his tip reconstruction algorithmsfor general use. Dongmo et al (2000) subsequently carriedout an experimental test of the blind reconstruction method inwhich they compared the reconstructed profile of two diamondtips from a surface profiler with measurements of the tips’profiles made in a scanning electron microscope and showedthat the differences between the two measurements werecomparable to the combined uncertainty of the measurements.A further demonstration of the validity of the blind tipestimation routines was given by Nie et al (2002). They imagedbiaxially oriented polypropylene (BOPP) film to evaluate blindreconstruction routines. BOPP films have suitably sharpfeatures for characterizing tips and in addition when the tip ispushed 50 nm into the film, the film cleans the tip by removingany contamination. The film surface was imaged both withthe tip contaminated and the tip free of contamination. In bothcases the tip profile was constructed. Dilation of the imageobtained with the clean tip and erosion of the image obtainedwith the dirty tip gave similar results thereby supporting thevalidity of the routines.

As mentioned above, the noise on the SFM signal canaffect the performance of tip reconstruction routines andVillarrubia included a noise threshold in his algorithms. Toddand Eppell (2001) improved this with an optimization routinethat allowed different thresholds to be set for the fast andslow scanning axes. Tranchida et al (2006) made furtherimprovements by showing how noise causes an underestimateof tip radius and an excessive spatial sampling interval leads toan overestimate of tip radius. They presented some guidelinesto obtain more reliable data by using blind tip estimation ontip radius and therefore on sample topography.

Machleidt et al (2005) and Machleidt and Franke (2006)modified the routines of Villarrubia by using some priorknowledge of the tip shape. Since the number of idealtip shapes available is limited, usually conical, tetrahedral,pyramidal or a parallel walled nanotube, this information canbe used for the tip shape reconstruction. For example, apyramidal tip can be described using four planes γ1 to γ4

and a rotation angle φ, and a cone-shaped tip can be fittedusing parametrized second order surfaces. Consequently theminimizing function knows to which type of function it istrying to find a fit and has more boundary conditions thanin the generic case. This results in a non-linear problemfor the fitting algorithm and can be solved using numericaloptimization algorithms so long as they are sufficiently stableand can reach a convergence. The latter depends very criticallyon the starting parameters. A typical non-linear algorithmis the Levenberg–Marquardt algorithm that is based on theGauß-Newton method. The advantage of the Levenberg–Marquardt method is the improved convergence in the case ofsaddle points and local minima. Local minima are a problemin cases where the global minima are needed. Thereforeseveral techniques have been developed to overcome thiseffect. Machleidt et al (to be published elsewhere) used

Figure 28. Machleidt’s revised scan routine.

the simulated annealing algorithm that was introduced byKirkpartick et al (1983). Herein the Monte Carlo technique isused to generate start parameters for the fit. The distributionof the range of starting parameters is broader at the beginningof the iteration to enable the region that contains the globalminimum to be found. With increasing iterations the width ofthe distribution function is reduced. Typically the width of thedistribution function is described using a parameter referred toas temperature (T ); the higher the temperature, the broader thewidth of the distribution. In the case of the simulated annealingalgorithm the width of the distribution functions follows anexponential decreasing behaviour.

The convergence of the algorithm is very likely, but notguaranteed and due to the necessary number of iterations ismore time consuming than the normal blind reconstruction.However, the additional time required is outweighed bythe additional knowledge about the tip such as tip shapeand symmetry that can be obtained leading to a moreaccurate reconstruction. To overcome the longer scanningand computational time, Machleidt does not perform aconventional raster scan of the sample when gathering data.Instead he performs a four-line scan as shown in figure 28.These four scans (each 50 µm long), together with priorknowledge of the ideal tip shape, provide sufficient informationfor the shape to be reconstructed and have the advantage thatthe tip wear is less than using conventional methods. Thedisadvantages are that the 3D reconstruction is not for all pointson the tip, the measurement strategy is complicated (a diagonalscan) and the effects of torsion on the tip due to changes in thescan direction are unknown. For a further improvement in thetip estimation, a conventional high resolution 5 µm × 5 µmscan must be made; however, this not only takes longer butincreases the risk of tip wear.

Clearly when combined with a sharp tip characterizersample, blind tip reconstruction has great potential since thesharp sample will allow a reconstructed tip profile close tothat of the actual tip without specific knowledge of the tipbeing necessary. However, as Machleidt showed, if there issome rudimentary information about the tip shape available,the number of line scans required to reconstruct the tip can bereduced. Tip reconstruction routines have been incorporatedinto commercial software for SPM image analysis (SPIP).

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

(a)

(b)

Figure 29. Modes of imaging (a) x–y imaging and (b) z imaging,after Dahlen et al (2005).

Figure 30. Silicon overhang characterizing structure (SOCS).

5.7. Tip reconstruction for critical dimension (CD) metrology

All the above methods are suited for use with conventionalAFM tips. Dahlen et al (2005) showed how the shape of CDtips could be reconstructed from images obtained using theCD-AFM that servos in the x and z directions. Prior to thedevelopment of their method, the usual method of removing theeffect of the tip shape from the measurement of a linewidth was‘tip width subtraction’ (Martin and Wickramasinghe 1995);a measurement of the tip width was subtracted from themeasured width of the line feature. The weakness of thismethod is that it does not remove any distortion of the imagein Z due to tip size, something that is particularly importantwhen measuring re-entrant structures.

With a tip of infinitely small tip apex radius, the point ofcontact remains fixed at the tip apex. However, as mentioned

earlier, tips have a finite size, even a CD tip has a finite edgeheight. The point of contact of a CD tip will be translatedacross the full tip width; see figure 29 which shows how botha conventional AFM tip and a CD tip contact an undercutstructure. Both x and z (2D-scanning) tip dimensions result inimage dilation. The method presented by Dahlen et al (2005)required a reconstruction of the CD tip. This was achievedusing two known structures. The first was the ‘improvedvertical parallel structure’ (IVPS) (Klos and Yedur 2000) andwas essentially a line feature with very smooth sidewallsand a uniform width. The second was the silicon overhangcharacterizing structure (SOCS), shown in cross section infigure 30 and similar to the array of undercut structures shownin figure 18.

Two algorithms were used for extracting the actual imagefrom the dilated image. The first was an extension of thetangent slope algorithm (Keller 1991) using a re-entrant surfaceand a vector surface description rather than a scalar slopedescription. The second method used an erosion algorithm.The combination of these methods resulted in a much moreaccurate reconstructed tip profile than when the tip widthsubtraction method was used. This method was favoured overblind tip reconstruction due to the complex nature of the tipshape.

5.8. Comparison of methods

Dongmo et al (2000) validated the performance of blind tipreconstruction using tips from a profiler and subsequently moregeneral comparisons of the performance of different methodshave been undertaken. Meli used a metrological AFM (Meliand Thalmann 1998) to calibrate tip radii and the cone anglesof standard silicon tapping mode sensors and Super-SharpSilicon™ sensors using two different tip characterizers fromIBM Sindelfingen (Meli 2000b). First, two Point Probe™

AFM tips (labelled tip 1 and 2) were used on the Nanoedgecharacterizer leading to two independent results for the sumof the tip and characterizer radii of 1.1 and 1.6 nm. Finallytip 2 was used to image tip 1. Considering all results it waspossible to evaluate all three radii individually. The radiusof the nano edge characterizer was within 1 ± 1 nm. Thetip radii of standard point probes were the same and within2 ± 1.5 nm. For the overall shape of super-sharp silicon tipsthe flared silicon ridge characterizer (similar to that shown infigure 18) was used at different tilt angles. The edge radii forthis standard are also within 1 nm±1 nm and the characterizer’sside wall undercuts start right at the top edges. This standardis therefore well suited to characterize the overall tip shape aswell as the tip radius.

Figure 31 shows determined tip profiles for three supersharp silicon tips in the y and the x directions (labelled x cutand y cut). At the very end of the tip the cone angle is quitelarge until a tip width of about 20 nm is reached. From thereon the cone angle is very small over a height of more than200 nm. Subsequent critical dimension measurements (CD)were corrected for these shapes. The interaction shape ofthe AFM tip seems to depend also on the sample topography.This interaction should be better understood through modelling

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

200

100

0

heig

ht [n

m]

0.20.10.0-0.1-0.2x [µm]

21nm

tip By-cut

tip Cy-cut

tip Ay-cut

tip Ax-cut

Figure 31. Measured profiles of three super-sharp silicon tips(A–C). Three profiles for the Y direction and one for the X direction(Meli (2000b)).

of the tapping mode process as a function of the tip and thesample shape.

Czerkas et al (2004) used different sample characterizers,triangular, rectangular, and undercut sidewall, as well asgold nanospheres and the TipCheck sample in an attempt todetermine an absolute measurement of the tip shape. Blindtip reconstruction was used for analysis of the image of theTipCheck sample and erosion for analysis of other images.Using gold nanospheres and a TipCheck sample they obtaineda tip angle of about 40–50◦. Other characterizers gave a valueof 60–70◦. A possible reason for this deviation was the lackof high angled features on the spheres and TipCheck sample.The experiments showed that the geometry of the sample mustbe taken into account for a precise determination of tip shape.The values of tip width determined by different methods werecompared with the width of a new tip measured in an SEM, all ata distance of 20 nm from the tip end. The reconstructed valuesvaried between 17 and 35 nm as compared with the 8 nm for thenew tip. Discrepancies might be attributable to the interactionforces that can either temporarily or permanently modify thetip shape by elastic or plastic deformation. Additionally, strongadhesion may cause tip wear, and capillary forces due to waterfilms may strongly influence line width measurements. In anycase the scatter of values is too large for precise line widthmeasurement. Therefore more investigations with tips arenecessary to achieve a better understanding of the differencein tip shape from ex situ and in situ evaluations.

Although the mathematical techniques allow one to seewhich region cannot be measured owing to the tip’s finite size,they provide only an outer bound of the tip shape, and neglecteffects due to the interaction between tip and sample. Thislatter point is also true for SEM and TEM measurements ofgeometric shape and size of a tip and can lead to interactionsmanifesting themselves as changes in the effective tip shape,i.e. tip radius and opening angle. Reconstruction of the tipradii and the shape would be possible with measurements onthe two ideal surface structures shown in figure 32.

Therefore, as discussed below, this more geometrical‘image’ of reconstruction of tip shape has some uncertaintiesdue to known width W and height H , and the radii of thestructures R1 to R3.

Figure 32. Schematic reconstruction of the tip shape and radius atthe tip apex from measurement on an undercut and a triangularstructure.

Measurements on the undercut edges in figure 32 wouldgive information about the shape of both sides of the tipand measurements on the triangular edge yield informationabout the tip radius. However there are uncertaintiesassociated with the dimensions of the two characterizers(see R1, R2, R3, R4, h1, h2, h3, and h4 in figure 32) and theselead to uncertainties in determination of tip radius. Theknowledge of the width W of lines has an uncertainty of a fewnanometres when the problems due to the radii at both sidesof the line are ignored, e.g. NIST claims 1 nm based on TEMmeasurements (Dixon 2005). The triangular sharp structurewould provide some information about the radii of the apexof the tip. However a non-ideal sharp structure would have afinite radius, which would influence the uncertainty of the tipradii. Additional the parameters of the feedback circuit wouldhave an effect on the measured profiles at the edges of the lineas well as on the tip of the sharp structure. Therefore the tipradii and the shape could only be fitted/measured with a finiteuncertainty based on these geometrical models. However,these models do not take into account dynamical properties dueto the interaction between the atoms of the tip and the atomsof the sample atoms. In order to model these, one should usemolecular dynamic calculations.

6. Tip–sample interactions

This section describes several of the interaction forces actingbetween the SFM tip and the sample. Attempts have beenmade to model their effects on the various modes of operationof an SFM. In order to avoid unnecessary repetition, only anoverview is presented here and as dimensional metrologists,the authors are interested in the effect of these forces ondimensional measurements. Traditional surface metrology isregarded as a mechanical interaction; a stylus probe is scannedacross a sample and one is not too concerned about surfaceforces. However in the case of scanning force microscopythe metrology is defined by the interaction of interatomicforces. The actual effects of all these forces on dimensionalmeasurements, that can to some degree be expressed bymodelling, are difficult to quantify. Although the effects ofthese forces on dimensional measurements are likely to besmall, as the requirements for dimensional metrology at thenanometre and sub-nanometre level increases, it will be nolonger possible to ignore tip–sample interaction forces. While

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

the effects of the forces are likely to be elastic, in contrastto tip wear mentioned later, they will still affect dimensionalmeasurements and their effect is going to be different acrossa sample; figure 33 shows schematically the end view of acantilever in contact with a sample, together with light fromthe beam deflection system. The tip will be subject to bendingand deformation. In addition, the tip’s position is measuredusing the beam deflection system that is typically about 6 µmabove the probing point of the tip and is unlikely to be sensitiveto bending of the tip.

Figure 34(a) shows schematically a tip above a sampleand figure 34(b) shows it in contact with the sample. Infigure 34(b) there is a small vertical distortion of the tip δland sample δh. This deformation could be due to elastic orinelastic compression.

Figure 35 shows a schematic diagram of the end of a SFMin contact with a sample. As the tip approaches the vertical wallon the right-hand side of the sample there will be an interactionbetween the right-hand side of the tip apex and the sample thatcould distort the tip by an amount δx that would lead to a shiftin the reference axis of the tip.

For a better understanding of dimensional measurementsthe effects of surface forces have to be taken intoaccount, especially on inhomogeneous samples, e.g. heightmeasurements of chromium lines on glass (linewidthstandards) or metal electrodes in a silicon oxide matrix. Thisis essential not only for non-contact measurement where theattractive, material dependent van der Waals force acts on the

Figure 33. End view of cantilever with reference to the tip andmeasurement axis.

(a) (b)

Figure 34. (a) and (b) How tip–sample interaction forces could lead to a vertical deformation of the tip and sample is shown.

tip, but also the other modes of operation may be influencedby such a material dependent interaction, especially during‘contact’ with the sample surface. Robrock et al (1990)calculated the effect of van der Waals forces on the change tothe tip to sample working distance for a 90 nm photoresist stepon silicon and therefore the effect of the change in material onthe measurement of step height compared with a 90 nm siliconstep on silicon. His results showed that the measured stepheight had to be corrected by 3%.

Figure 36 shows the profile produced by a tip that issubject to different interactions forces across its scan overan imaginary surface. When the tip is in contact with thesurface the repulsive component of the Lennard-Jones potentialrepels the tip, whereas in non-contact mode attractive forcessuch as the short-range van der Waals force and the long-range capillary force may act on the tip. The modelling ofvan der Waals forces has been described by Hartmann (1991)and Goodman and Garcıa (1991). Local electrical chargeson the surface may lead to attractive or repulsive electrostaticforces on the tip (electrostatic force microscopy). In a similarway, as described in the section on magnetic force microscopy,magnetic forces can be imaged using a tip that has been coatedwith a magnetic material and magnetized along the tip axis(magnetic force microscopy). There will also be some elasticor plastic deformation of the sample and unless measurementsare performed in UHV, there will be a water film on the surfacethat will affect the measurement results. Adhesion to thesurface will also play a role. An extreme case was illustratedby Paredes et al (2000), who showed images of aramid fibres(bundled to make Kevlar®). The sample was scanned in theforward and reverse directions and the orientation of the scanaxis was changed with respect to the cantilever’s longitudinalaxis. Adhesion of the tip to the sample caused a twisting of thecantilever and the image contrast was different in the forwardand reverse directions.

6.1. van der Waals force and repulsive forces

The predominant forces that are present in all tip surfaceinteractions are the van der Waals force, an attractive force andthe repulsive (Pauli) force. The van der Waals force potentialcomprises three attractive terms all related to 1/r6, where r isthe distance between the tip and the sample. The componentsare the orientation force between polar molecules each having

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

Figure 35. How tip–sample interaction forces could lead to a horizontal deformation of the tip is shown.

Figure 36. Possible interaction forces in SFM between tip and sample.

a dipole moment, the induction or Debye force between polarand non-polar molecules and the dispersion or London forcebetween non-polar molecules. The London component is themost dominant of these three attractive forces. The van derWaals force is typically active over a tip–sample separationof 100 nm, the force potential being of the order of 30 eV or∼10 nN. At smaller distances there is the repulsive Pauli forcesdue to overlapping electron orbitals. These two forces are oftenexpressed using the Lennard-Jones potential, Lennard-Jones(1931) which between two atoms is given by

V (r) = −4ε

[(σ

r

)6−

(σ

r

)12]

,

where ε is the depth of the minimum potential, σ the positionof the minimum potential and r the distance between atoms.

The Lennard-Jones potential forms the basis for the forcedistance curve shown at the start of the paper in figure 1.

In the case of a tip and flat sample geometry, the van derWaals forces are given by

FvanderWaals(zc, z) = − HRtip

6(zc + z)2,

where H is the Hamaker constant andzc is the rest tip–sample separation and z the instantaneous

tip position.When using the DMT theory (see later) the repulsive

forces are given by

Frepulsive = HRtip

6a20

+4

3E∗√Rtip(a0 − z − zc)

3/2,

where a0 is an intermolecular distance term designed to avoiddivergence of the first part of the above equation, Ciraci et al

Figure 37. Calculation of apparent height difference for a flatsurface containing SiO2 and Au as a function of thickness of a waterlayer for four amplitudes of cantilever oscillation (Dejima et al2008).

(1992) and E∗ is reduced elastic modulus of the tip andthe sample.

Robrock (1990) calculated the effect of van der Waalsforces on height measurement of two different materials (90 nmphotoresist on silicon) in the non-contact mode. Assuminga typical working distance of 10 nm, the measured heightobtained from the profile has to be corrected by 3%. Dejima(2008) calculated the differences in height of the tip abovevarious materials homogeneously covered with an additionalwater layer. He found differences less than a nanometre thatwere attributed to different van der Waals forces (figure 37).

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

van der Waals Effects SiO2-Au

y = 0.097x + 0.3553

R2 = 0.9901

0

2

4

6

8

10

1000 20 40 60 80Amplitude /nm

Cri

tica

l th

ickn

ess

of

wat

er la

yer

/nm

Figure 38. Critical thickness of water layer at which the effect ofvan der Waals forces disappears as a function of amplitude ofcantilever oscillation.

Assuming that one had a perfectly flat substrate withregions of both silicon and gold, on a clean surface the ‘height’difference due to the van der Waals forces was approximately0.6 nm; the height above the SiO2 region would appear higherthan above the gold area. This height difference varies as afunction of the amplitude of oscillation of the cantilever anddisappears abruptly as the thickness of the water film increased.The thickness of the water layer at which the effect of the Vander Waals forces disappears varies linearly as a function of theamplitude of cantilever oscillation and is shown in figure 38.The effect of the water layer is discussed in more detail in thenext section.

6.2. Capillary forces

Under ambient conditions humidity usually leads to a thin filmof water being adsorbed on the surface of both tip and sample.The presence of this water film on surfaces has been detectedusing SFM; see for example Binggeli and Mate (1994), whostudied the effects of humidity friction; on hydrophilic surfaceslarge capillaries formed at humidities above 70% and adhesiveforces and friction coefficients decreased substantially for asliding AFM tip. Thundat et al (1992b) imaged DNA strandson mica. DNA is hydrophilic and mica hydrophobic. Theyfound that the measured width of the strands was proportionalto relative humidity; at low relative humidity, the DNA strandswere thinner (∼45 nm) and the contrast was positive, i.e. theDNA strands appeared higher than the substrate, whereaswhen the relative humidity approached 30% a transition wasmade and the strands not only appeared to be wider, up to80 nm at 50%, but also the contrast was reversed, i.e. thestrands appeared to be lower than the substrate on which theywere mounted. When the relative humidity was reduced, thecontrast returned to positive and the widths of the strandsreduced to their previous values. Various designs of chambersto control the relative humidity during measurement have beenproposed (Stukalov et al 2006, Lievonen et al 2007).

If the AFM cantilever comes close to the sample surfacea capillary neck may form between tip and sample. Colcheroet al (1998) measured force distance curves on graphite andgold surfaces. These were compared with theoretical forcedistance curves and it was noted that the jump to contactposition occurred at a larger than expected theoretical tip tosample distance (3.5 nm as opposed to 0.7 nm). They attributed

Figure 39. The three pull off regions are shown (He et al 2001).

this difference to the formation of a water droplet on either thesurface or the cantilever or both. As the tip approachesthe surface a small liquid neck forms between the tip andthe cantilever that results in the cantilever being pulled to thesurface. This effect dominates van der Waals forces or anypotential difference between the surface and the tip due tomaterial properties. The presence of a water film clearly hasan effect on force distance curves as well as acquisition oftopographic data and is not restricted to biological samples.Thundat et al (1993) showed how the experimental contrastof atomic resolution images of mica varied as a function ofrelative humidity. At high relative humidity of 70% theyeasily achieved atomic resolution with a contrast of 0.2. Asthe relative humidity was reduced, the contrast increased to amaximum value of 0.3 at a relative humidity of 20%. Furtherreduction of the relative humidity then resulted in a sharp dropin contrast and at a relative humidity of 14 % the contrast wasreduced to zero. In order to regain atomic resolution at theselow humidity values it was necessary to increase the cantileverforce.

Stifter et al (2000) investigated theoretically the distancedependence of the capillary and van der Waals forcesapplicable to the AFM in the contact mode. Their calculationswere for a fixed sample to tip distance as well as forcedistance curves. They considered humidity, surface tension,tip geometry, starting distance (for force distance curves) andcontact angles. Experimental results agreed well with thetheory and they quite logically concluded when comparingmeniscus forces and van der Waals forces, three possiblearrangements can occur. The two extremes are either vander Waals forces or capillary forces are dominant. In theintermediate region both forces play a role. The capillaryforces can to some extent be reduced by tip geometry or design,e.g. a smooth tip with a hydrophobic coating.

He et al (2001) presented both theoretical andexperimental results showing how the ‘pull off force’ of acantilever from a surface varied as a function of humidity.They showed how a graph of pull off force against relativehumidity for hydrophilic tips and surfaces has three regions(see figure 39) corresponding to the three arrangementsmentioned by Stifter et al. In region I the van der Waalsforces are the dominant component of the pull off force. Asregion II is approached (the relative humidity increased), the

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

minimum water film thickness is reached for the formation ofa capillary neck between the tip and the surface and causesthe discontinuity in the curve. In region II, the pull offforce comprises both van der Waals and capillary components.In region III, the pull off force is purely due to capillaryforces. Experimental results presented in the paper verifiedthe theoretical predictions.

The effects of capillary forces are not confined to thecontact mode of operation. Zitler et al (2002) derived a modelto show the effect of capillary forces on force curves obtainedin a dynamic mode of AFM operation that provided qualitativeagreement with their results. Their model is an extension ofthe model derived by Burnham et al (1997a). Essentially if thefree amplitude of oscillation of the cantilever is greater than a‘critical’ amplitude the amplitude- and phase-distance curvesmove from a region with a net attractive tip–sample forceto a net repulsive tip–sample force. If the tip is hydrophilicthe critical amplitude increases with humidity. This effectwas attributed to the intermittent formation and breaking ofa capillary neck between the tip and the sample.

The work of van Noort et al (1997) is particularlyinteresting from the point of view of dimensional metrology.They reported height anomalies in measurements made usingthe intermittent mode that they attributed to adhesion. Theirexplanation for the effect is as follows: when the cantilevertouches a water film, its amplitude of oscillation will bereduced and the interaction time with the sample will beincreased, resulting in a phase lag that will reduce the efficiencyof cantilever excitation, thereby resulting in a further phaselag. After the interaction with the surface/water film, thecantilever will have less energy and it may be necessary for thecantilever to oscillate through several periods until it interactswith the surface again. Consequently there will be a periodicmodulation of the cantilever’s oscillation amplitude resultingin reduced average amplitude of oscillation. The amplitude ofoscillation is the input for the servo system, which will reactas though there is a height change and withdraw the cantileverfrom the sample to increase the amplitude of oscillation.Consequently a surface with a large adhesion force will appearhigher than one with a lower force. This was confirmedby their results. A freshly cleaved mica surface was partlycoated with gold to form a step approximately 80 nm high.When the step height was measured using the contact and non-contact mode it appeared to be 9 nm smaller when measuredusing the non-contact mode. When a Langmuir–Blodgett filmof lignoceric acid on silicon was examined, with differentamplitudes of oscillation, the results showed that the imagecontrast was strongly dependent on the oscillation amplitude;with heavy damping of the oscillation the contrast was reversedand only with low damping were a reliable topographic andphase images obtained. From these results it is clear that theeffects of humidity and the water film need to be consideredwhen performing dimensional measurements using SFMs inboth contact and non-contact modes.

6.3. Mechanical deformation

The earliest attempts to describe the mechanical deformationof two surfaces in contact are those of Hertz (1881). His

description does not include surface forces or adhesion and isreally designed for macroscopic surfaces. Two other modelshave been developed; the JKRS model (Johnson et al 1971) and(DMT) model (Derjauin et al 1975). Using the DMT modelthe deformation is as for the Hertzian model, however, theforces between the tip and the sample are taken into account.This model is suitable for systems with low adhesion and smalltip radii. For highly adhesive systems with low stiffness andlarge tip radii, use of the JKRS model is more appropriate. Thework of Muller et al (1980) provided an analysis that allowsa smooth transition between the JKRS and DMT models byassuming that the adhesion forces do not alter the Hertziangeometry. However, the analytical solution using a Dugdale(1960) model by Maugis (1992) is probably the most genericand can be used for all materials.

All tip–sample interaction forces depend strongly not onlyon the separation between tip and sample but also the tip andsample material properties and the area of contact. When theSFM is operated in a dynamic mode the tip can probe theelastic properties of a surface (elastic modulus spectroscopy),however, increasing the force of the tip can lead to plasticdeformation or nanoindentation. For dimensional metrologyan understanding of the area of ‘contact’ between tip andsample is particularly necessary since in the case of a largecontact area the Pauli forces are not relevant and the mechanicaldeformation of the surface must be considered. The range offorces applied in SFM is comparable to forces investigated insurface force instruments (Israelachvili 1994). Deformation ofthe tip and the sample will be dependent on the elastic moduliof the tip and the sample and also adhesion. The spatial rangeover which surface forces act depends on the chemistry ofthe materials in contact and may or may not be long-rangecompared with the scale of elastic deformations due to theseforces.

6.4. Electrical forces

There can be an electrostatic force between a tip and sample.When this is the dominant force, there is the possibility ofelectrostatic force microscopy. The electrostatic force betweenan AFM tip and surface is given by

Felec = πε0V2

0 g(z),

where ε0 is the vacuum permittivity, V0 is the potentialdifference between the tip and the sample and g(z) takes intoaccount the geometry of the tip and is proportional to thegradient of the capacitance between the tip and the sample.

Several models have been written to describe the electricalforce between tip and sample. Bonaccurso et al (2006)reviewed the models and identified those of Hudlet et al(1998) and Colchero et al (2001) as being particularly gooddescriptions of the electrical forces between the tip and thesample. Boaccurso et al extended these models to takeinto account the effects of the cantilever and its orientationwith the sample. They verified their model with a series ofmeasurements using both V-shaped and rectangular-shapedcantilevers, both with and without tips, metal coatings andunder different experimental conditions (temperature and

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J. Phys. D: Appl. Phys. 41 (2008) 103001 Topical Review

humidity). Sadewasser et al (2004) investigated heightdifferences in AFM images of C60 deposited on HPOGobtained using the non-contact mode of operation. Theyshowed how the differences in work function led to anomalousheight differences. By biasing the tip and the sample withan externally applied voltage, they were able to minimize theeffect of the electrostatic forces so that the topography wasdominated by the van der Waals force.

6.5. Models of tip–sample interactions

When attempting to model the cantilever deflection in thepresence of forces on the cantilever in the static mode, oneshould start with the equation for the bending of a beam.Recall that

F = −k�z,

where �z is the the deflection of the cantilever, k the springconstant and F the forces acting on the tip.

The starting point for most models of the AFM in thedynamic modes is to treat the cantilever as an oscillating springor harmonic oscillator with the interaction forces acting as asource of damping on the oscillations. Taking the equation forforces in the static mode, together with Newton’s second lawand terms to take into account the damping leads to a secondorder non-linear differential equation whose solution describesthe motion of the cantilever; see for example Burhnam et al(1997a). The equation is generally of the form

mz = −kcz − mω0

Qz + Fts + F0 cos(ωt),

where m is the mass of the cantilever, z the tip–sample distance,kc the force constant of the cantilever, ω0 the resonant angularfrequency of the driving force of the free cantilever, Q thequality factor, Fts the tip–sample interaction force, comprisingseveral or all of the terms discussed above and F0 the amplitudeof the driving force of the free cantilever, and

Fts = FLennard-Jones + Fcapillary + Felectrical + Fmagnetic + Fchemical

and further

FLennard-Jones = Frepulsive + Fvan der Waals.

It can be seen that the formation of the term to represent the tip–sample interactions will be complicated. Furthermore, someof the terms are interrelated; Bonaccurso et al (2006) showedexperimentally that for metallized cantilevers, the electrostaticforce increases with relative humidity and, therefore, a genericequation will be very complicated.

The equation can be solved to give a solution of the form

ZH(L) = Z0 + AH(L) cos(ωt − φH(L)),

where the subscripts L and H represent two solutions withdifferent amplitudes.

Garcıa and San Paulo (1999, 2000) and San Paulo andGarcıa (2002) have studied extensively the dynamic modesof operation of an AFM. They have discussed the cause

and effects of two stable solutions; when using amplitudemodulation (using the amplitude signal for servoing). Since thetip–sample interaction component has repulsive and attractivecomponents (short and long range) they lead to two stablesolutions. However when the cantilever is perturbed eitherby an internal or external disturbance, it can switch betweenthese solutions. Garcıa and San Paulo (2000) presentedexperimental results that showed these jumps could bemisinterpreted as being due to contamination or a poor tipshape. In their detailed review of dynamic atomic forcemicroscopy methods Garcıa and Perez (2002) concludedthat many of the harmonic models, although conceptuallyuseful, fail to provide quantitative agreement with experiments,somewhat disconcerting for metrologists who want not onlyquantitative results but also traceability of measurements.Rutzel et al (2003) have subsequently extended the work ofGarcıa et al to produce more quantitative results for siliconprobe-sample systems. A more recent paper by Solares (2007)presents a theoretical analysis for a method to remove thebistability of the cantilever through a combination of frequencyand amplitude modulation in atomic force microscopy.

6.6. Molecular dynamics

An alternative (bottom up) approach for modelling tip–sampleinteractions is based on molecular dynamical theory for idealtip surface geometries. Abraham et al (1988) have investigatedthe relaxation of a Si(0 0 1)-2 × 1 surface in the presenceof a silicon tip by using molecular dynamics (MD) basedcalculations. They modelled the images of a silicon surfaceusing AFM tips with different numbers of atoms at the endand concluded that the surface profiles that would be obtainedwith the different tips would be quite different. Furthermore,for the imaging to be non-destructive, the force exerted by thetip should be less than 1 nN (for silicon). Forces at this levelinduced surface relaxation, but did not change the qualitativefeatures of the analysis. Ciraci et al (1990) report on a similaranalysis of aluminium tips on graphite surfaces and Landmanet al (1989) applied MD to investigate the interaction betweena unreconstructed Si(1 1 1) surface and a silicon tip and ofAu tip and Ni surface and vice versa. The results of theseinvestigations were summarized by Meyer et al (1998); forforces of the order of 10−9 N the SFM image reflects thesurface topography, but tip induced atomic relaxations lead to areduction of the corrugation whereas larger forces of the orderof 10−8 N can lead to a re-arrangement of the tip or substrateatoms, e.g. creation of interstitials in silicon occurs. Orderedas well as disordered tips, having realistic dimensions, led toforce variations on an atomic scale, but the image can be alteredby multiple tip effects.

6.7. Use of SFM to investigate tip–sample interactions

The SFM itself can be used to measure tip–surface interactions.If the instrument has been properly calibrated one canquantify some material properties. Attractive and adhesiveforces can be measured with very high spatial and forceresolution as well as the hysteresis between loading andunloading, plastic or hardness, interaction stiffness, and if the

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tip properties are known, the elastic modulus of the sample(Burnham et al 1998). Some models for SFM were developedto explain experimental observations (Spatz et al 1995).

Mechler et al (2005) studied anomalies in the AFM-basedheight measurement of tungsten oxide nanoparticles on micaand graphite substrates and made a comparative analysis ofexperimental results and numerical calculations. They foundthat height measurements of the nanoparticles by dynamicAFM imaging were sensitive to the free amplitude of theoscillating probe. Their results were substrate specific andrepeatable. It appeared that the measured height reached aminimum for a free amplitude of oscillation of around 10 nm;a further decrease in the amplitude resulted in an increase ofthe measured height.

Chen et al (2007) measured the height of goldnanoparticles on freshly cleaved mica surfaces using SFM. Bychanging the cantilever free oscillation amplitude and varyingthe set-point in a range between 80% and 30% they found inan intermediate range of amplitudes a change of the height of30%. For amplitudes of 40, 30 and 10 nm the measured heightof 5 nm for the Au particles agreed well with the vendor’sspecifications. However, for an amplitude of 20 nm they founda height of only 3.5 nm that was independent of the set-point.They conclude that at this amplitude the cantilever’s regionof operation changes between the attractive and repulsiveregions of the substrate and the Au particle, respectively. Forhigher (40, 30 nm) or smaller (10 nm) amplitudes the cantileverremains within the attractive or repulsive range. This showsthat extreme care must be taken when operating the SFMwith the intention of obtaining dimensional measurements. Tohelp quantify some of the effects of the interaction forces,the authors have developed an SFM specifically for theexamination of tip–sample interactions (Yacoot et al 2007).

The measurement of forces as a function of the tip–sample separation (force–distance curves) allows one todraw conclusions regarding the material characteristics ofsurfaces and their chemical properties, see Burnham et al(1997b) and Capella and Dietler (1999) and Butt et al(2005) for very comprehensive reviews of the use of AFMsto record force distance curves. For a very recent paperon chemical identification of surface atoms using AFM seeSugimoto et al (2007).

6.8. Inelastic effects

Figures 34 and 35 showed how a tip and sample could beelastically distorted when in contact. In some cases thedistortion could be inelastic due to either plastic deformationor a chemical reaction between the tip and the sample. Thiscould be either unwanted or induced by a functionalized tip. Inboth cases it is likely to have a small effect on any dimensionalmeasurements.

6.8.1. Tip wear. As mentioned earlier in the paper, mostusers assume that the tip shape remains constant during ascan, however this is not the case and the wear of a tip issomething that should be taken into account when makingdimensional measurements. With continued use a tip will wear

and become blunt and, as Edwards and McGlothlin (1998)showed, this can affect the value of a step height measurementon a ‘rough’ sample by up to 1%. Su et al (2003) managedto show experimentally a correlation between tip speed beforeimpacting a surface in the intermittent contact mode and tipwear; for hard samples tip wear is dominated by momentumexchange between tip and sample and that the speed of impactof the tip determines the wear. This led to the conclusion thathard tapping and a low set-point of amplitude of oscillationshould be chosen for reducing tip wear.

Tip wear also results in an inability to resolve finestructures as is shown in figure 40 that shows a series oflinescans made on a 100 nm pitch calibration grating. Theline profiles at the end of the scan (bottom) are clearly not asdeep as those at the start of the scan. The height of the gratingwas measured using electron microscopy as being greater than50 nm. Later measurements with a new tip showed that thegrating had not suffered any significant wear.

If one is measuring height or surface form, it is thereforeimportant to check regularly the quality of a tip using one ofthe tip characterizers. Bakucz et al (2008) have measured thechange in shape of an AFM tip during the course of its life asit scanned a TipCheck sample and have used artificial neuralnetworks to predict further wear. Dahlen et al (2005) showedthe importance of correcting for the effects of tip wear whenmaking CD measurements; errors in a measurement of theorder of several nanometres could be obtained if the tip profilewas not regularly checked. Orji and Dixon (2007) estimatedthat wear of their CD flared tips was less than 0.5 nm for 500scan lines. The first flared tips used had a typical width of a fewhundred nanometres, and more recently wear resistant tips witha reduced width of 70 nm (Liu et al 2005) have been reported.These wear resistant tips have a thin silicon nitride cap thatreduced wear by a factor of 2.6 from 0.11 nm per measurementto 0.042 nm per measurement of a sidewall structure. Analternative coating to silicon nitride was a self-assembledmonolayer (0.5–5 nm) of either perfluorodecyltrichlorosilaneor dimethyldichlororosilane, both of which are hydrophilic.Wear was reduced to between 0.007 and 0.045 nm permeasurement; a factor of up to 17 improvement over anuncoated tip. As mentioned earlier, carbon nanotubes arehighly resistant to wear, having up to 20 times the life ofconventional probes and their use should be considered whentip wear is an issue.

7. Discussion and conclusions

The preceding sections have discussed various cantilevers, tips,detection systems, how the shape of the tip can be determinedand some of the factors affecting dimensional measurements.It is clear that the fundamental limits to the performance ofcantilevers in terms of their noise and signal detection arenot currently the limiting factors for dimensional metrology.Although the calibration of transfer standards is at the sub-nanometre and picometre level for height and pitch standards,the uncertainties for measurement of irregular shaped objectssuch as MEMS structures or inhomogeneous samples are not

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Figure 40. The effect of tip wear on measurement of a 100 nm pitch standard is shown.

so small and will be increased by the effects of tip–sampleinteractions and tip shape and operating parameters.

Self-sensing cantilevers offer possibilities for morecompact systems, however to our knowledge, they have notyet been used on metrological atomic force microscopes.Furthermore, the achievable resolution is generally lower thanthat which is achievable using conventional optical detectionsystems.

There is a large variety of different cantilevers andtips available for many applications. For dimensionalmeasurements the main points are stability of tip shape andgood aspect ratio. FIB and EBD produced tips have enabledthe resolution of SFMs to be enhanced, as have the more recentapplications of CNTs as tips. However, CNTs are not thepanacea for SFMs; they have their own problems with regardto orientation of the tip and use on a day-to-day basis. Someattempts of mass production were described; however, suchtips still remain primarily in the research sector and are notwidely available on a commercial basis. More work is neededbefore they can be used on a regular basis for dimensionalmetrology.

The various methods for tip characterization have enabledmore SFM images to be reconstructed to provide a moreaccurate estimate of surface topography and feature size. Thedisadvantage of all methods is that they are time consuming,often involve specimen changes, mathematical analysis andstill do not provide a full solution since they allow oneto determine the bluntest shape of tip that could havebeen used.

Throughout the paper examples have been cited ofanomalous measurements, both on biological and non-biological samples, where the dimensions measured dependheavily on the mode of operation of the instrument andthe measurement conditions. Obtaining accurate traceabledimensional measurements with uncertainties at the nanometreand sub-nanometre level with an SFM is far from simple.While there has been a vast amount of work leading

to improved knowledge of the theory and practice ofscanning force microscopy, more work is still needed torealize the full potential of scanning force microscopyas a tool for dimensional metrology. This will becomeincreasingly important as the need to examine structured andinhomogeneous surfaces grows and will affect SFM users fromall disciplines. It is hoped that the AFM developed by theauthors specifically for studying tip–sample interactions willhelp address some of the issues described in this paper.

Acknowledgments

Part of this work was funded by the Department of Trade andIndustry as part of the National Measurement Systems PolicyUnits’ Programme for Engineering Measurement 2005-2008.

The following people are thanked for their help:

Dr Sibylle Sievers (PTB) and Dr Martin Albrecht (PTB)are thanked for their contribution on magnetic forcemicroscopy,Dr Frase (PTB) for details of the Monte Carlo simulationof SEM measurements of tip profile,Dr Felix Meli (METS) for information on CD Metrologyat METAS,Dr T Machleidt for comments on simulation annealingmethod, Dr Hubner for tip check images,Dr David Cox (NPL) for CNT probe image,Dr Thomas Bayer from Team Nanotech GmbH for animage of a CD AFM tip.Mr Thomas Ahbe (PTB) for SEM images of the tipcharacterizers and preparation of some line drawings,Mr Thorsten Dziomba (PTB) for various SFM images andinformation about SNOM probes,Dr Shuichi Dejema for preliminary information about tip–sample interaction simulations,Dr G Dai (PTB) for information about ACP and providingan image of the ACP probe,

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Mr Keith Jackson (NPL) and Dr Harald Bosse (PTB) forcareful reading of the manuscript,Dr Gunter Wilkening (PTB) for continued interest andsupport of our work.

Disclaimer

The naming of any manufacturer or supplier by NPL or PTBin this scientific journal shall not be taken to be either NPL’sor PTB’s endorsement of specific samples of products of thesaid manufacturer; or recommendation of the said supplier.Furthermore, NPL and PTB cannot be held responsible for theuse of, or inability to use, any products mentioned herein thathave been used by NPL or PTB.

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