Принципы нанометрологии (1027623), страница 44
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For a beam deflection system, for example, thecantilever deflection is recorded in volts. An additional problem is that thedistance (or separation) between the tip and the sample is not measureddirectly [30]; the AFM measures the displacement of the piezoelectric scannerthat supports the sample. A force curve graph of cantilever deflection (in volts)and corresponding piezoelectric scanner displacement (in metres) (seeFigure 7.6a) must be interpreted to give a force–distance curve (i.e. force ofinteraction in units of force against separation between the sample and thecantilever in units of length (see Figure 7.6b)). With reference to Figure 7.6a,when the tip and sample are far apart (i) they exhibit no interaction (zeroTable 7.1Overview of guidance deviations, standards to be used and calibration measurements [12]CalibrationArtefact requiredWhat is measuredCross-talk of the lateral movementsto the z axisOrthogonality deviationOrthogonality deviationCx and Cy deviations(non-linearities)Cross-talk of the lateral axesCz deviations (non-linearities)flatness artefactout-of-plane movement of xy scan system2D artefact3D artefact1D or 2D lateral artefact2D lateral artefactstep height artefactangle formed by the two axes, on orthogonal structuresNeed description of what is measured for a 3D artefactpitch measurement, rotation, linearitypitch measurement, rotation, linearitystep height measurement, linearity189190C H A P T ER 7 : Scanning probe and particle beam microscopyFIGURE 7.6 Schematic of a force curve (a) and force–distance curve (b).force).
As the sample approaches the tip, inter-molecular forces between thetip and the sample cause the cantilever to deflect upwards (ii) due to repulsiveforces (in this case between a charged substrate and tip, but attractive forcesare commonly observed as well). Eventually the tip makes contact with thesample (iii) and their movement becomes coupled (region of constantcompliance). The sample is then retracted from the tip (iv) until the tip/cantilever and sample return to their original positions completing one cycle.Hysteresis, shown here, may occur upon retraction due to adhesion forces.Interfacial forces are measured on approach and adhesion forces are measuredupon retraction; repulsive forces are positive and attractive forces are negative.Atomic force microscopyTo obtain the force part of the force–distance curve, the photodiode valuesare converted to force using F ¼ kcd, where F is the force, d is cantileverdeflection and kc is the cantilever spring constant.
To convert the cantileverdeflection measured by the photodiode in volts to metres, a displacementconversion factor (also called the optical lever sensitivity) is obtained fromthe region of the force curve where the sample is in contact with the cantilever. For an infinitely hard contact, every displacement of the piezoelectricscanner displaces the sample or the tip; the cantilever is pushed upwards,which is recorded as a voltage output on the photodiode.
The slope of theforce curve in the region where the cantilever is in contact with the sampledefines the optical lever sensitivity. This part of the force curve is called theregion of constant compliance or region of contact.It is important to note that using the constant compliance region of theforce curve to convert photodiode response to deflection will overestimatethe force of interaction if the cantilever is not the most compliantcomponent of the system. This is often the case when soft, deformablesubstances such as polymers are used in force measurements (either asa sample or linked to the tip/cantilever).
If a compliant substrate is used,other methods are needed to accurately convert the measured deflection ofthe cantilever into a force of interaction [31]. In this case the optical leversensitivity is determined by pressing the tip/cantilever against a hardsample (for example, mica), before and after it is used on a soft sample.However, often this method does not work as the optical lever sensitivity isstrongly dependent upon a number of factors.
These factors include theposition and shape of the laser spot and the difficulty in precisely aligningthe laser spot on the same position on the cantilever from experiment toexperiment. Also, the use of a hard sample cannot be applied if it is the tip/cantilever that supports the most compliant component of the system (forexample, a molecule attached to the cantilever). Another method thatrelies on the ‘photodiode shift voltage’, a parameter that is very sensitive tothe position and shape of the laser of the photodetector, can be used toconvert volts of cantilever deflection into metres of deflection [32]. Thismethod ensures that forces can be determined regardless of the complianceof the cantilever relative to any other component in the AFM, and alsoensures the preservation of fragile macromolecules, which may be presenton the sample or attached to the cantilever.7.3.6 AFM cantilever calibrationAFMs are sensitive to very small forces in the piconewton range.
In order tomeasure these forces accurately, the stiffness of the probe must be191192C H A P T ER 7 : Scanning probe and particle beam microscopydetermined. Stiffness calibration procedures rely on either imposing knownforces on the probe, measuring the geometrical and material properties of theprobe, or measuring its thermal fluctuations.The cantilever’s spring constant is essentially dependent upon itscomposition and dimensions [33]. Nominal values listed by manufacturers may be incorrect by an order of magnitude and it is, therefore,necessary to determine the spring constant for each cantilever or for eachbatch of cantilevers from a wafer [34].
Parameters such as Young’smodulus (related to composition), and cantilever length and thickness,can be used in theoretical equations to calculate a spring constant [35].However, calculated values can be inaccurate due to the unknownmaterial properties of the cantilever (the stoichiometry of silicon nitride,for example, can vary from Si3N4 to Si5N4 [36]). Furthermore, themeasurement of cantilever thickness, which is a dominant parameter intheoretical equations, is extremely difficult.
The spring constant dependson the cantilever thickness to the third power, so even small uncertaintyin the thickness measurement will result in large variations in thecalculated spring constant [37].An accurate, but often destructive, way to measure spring constant is theadded-mass method [38]. In this method beads of known mass are attachedto the end of the cantilever.
The additional mass causes the cantileverresonant frequency to decrease proportional to the mass. A graph of addedmass against resonant frequency yields a straight line with a slope corresponding to the spring constant.A further method to determine the spring constant is the measurementof the force that an AFM imparts onto a surface by measuring the thermalfluctuations of the cantilever – in this method the cantilever is modelled asa simple harmonic oscillator (usually only in one degree of freedom) [39].With knowledge of the potential energy of the system and applying theequipartition theorem, the spring constant of the cantilever can be calculated from the motion of the cantilever and its surrounding heat-bathtemperature.
The thermal method has three major problems [40]:(a) higher vibration modes cannot be ignored, (b) the method to measuredeflection usually measures the inclination rather than the displacement,and (c) only the first modes are accessible due to the bandwidth limitationsof the experiments.For directly traceable measurements of the force an AFM cantileverimparts on a surface, electrostatic balances can be used, but they are verycostly and inconvenient (see section 10.3.3). Many of the devices discussedin section 10.3.4 can also be used to measure spring constant when used aspassive springs.Atomic force microscopy7.3.7 Inter- and intra-molecular force measurement using AFMAs discussed previously, the AFM images a sample by sensing andresponding to forces between a tip and the sample.
Because the force resolution of the AFM is so sensitive (0.1 pN to 1 pN), it is a powerful tool forprobing the inter- and intra-molecular forces between two substances.Researchers have taken advantage of this sensitivity to quantify fundamentalforces between a sample and some substance linked to the AFM cantilever ortip [41].
The AFM has enabled some truly remarkable advances in thephysical sciences due to the sensitivity and ranges of force it can measure. Afew examples will be discussed here. A basic understanding of the forcesbetween the AFM tip and the sample is essential for a proper use of theinstrument and the analysis of the data. A variety of forces that come intoplay between the tip and the sample are summarized in Table 7.2. Thediscussion that follows will focus on contact-mode AFM, which is the mostcommonly used imaging mode. A recent review highlights the effect ofsurface forces on dimensional measurements [30].The total force between the tip and the sample results from the sum ofvarious attractive and repulsive forces, as described below.
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