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Marti, O. Ò"AFM Instrumentation and Tips"Ó Handbook of Micro/Nanotribology. Ed. Bharat Bhushan Boca Raton: CRC Press LLC, 1999 © 1999 by CRC Press LLC 2 AFM Instrumentation and Tips Othmar Marti 2.1 2.2 Force Detection The Mechanics of Cantilevers Compliance and Resonances of Lumped Mass Systems • Cantilevers • Tips and Cantilevers • Materials and Geometry • Outline of Fabrication 2.3 Optical Detection Systems Interferometer • Sensitivity 2.4 Optical Lever Implementations • Sensitivity 2.5 Piezoresistive Detection Implementations • Sensitivity 2.6 Capacitive Detection Sensitivity • Implementations 2.7 2.8 Combinations for Three-Dimensional Force Measurements Scanning and Control Systems Piezotubes • Piezoeffect • Scan Range • Nonlinearities, Creep • Linearization Strategies • Alternative Scanning Systems • Control Systems 2.9 AFMs Special Design Considerations • Classical Setup • StandAlone Setup • Data Acquisition • Typical Setups • Data Representation • The Two-Dimensional Histogram Method • Some Common Image-Processing Methods Acknowledgments References Introduction The performance of AFMs and the quality of AFM images greatly depend on the instruments available and the sensors (tips) in use. To utilize a microscope to its fullest, it is necessary to know how it works and where its strong points and its weaknesses are. This chapter describes the instrumentation of force detection, of cantilevers, and of the instruments themselves. © 1999 by CRC Press LLC 2.1 Force Detection Atomic force microscopy (AFM)(Binnig et al., 1986) was an early offspring of scanning tunneling microscopy (STM). The force between a tip and the sample was used to image the surface topography. The force between the tip and the sample, also called the tracking force, was lowered by several orders of magnitude compared with the profilometer (Jones, 1970). The contact area between the tip and the sample was reduced considerably. The force resolution was similar to that achieved in the surface force apparatus (Israelachvili, 1985). Soon thereafter atomic resolution in air was demonstrated (Binnig et al., 1987), followed by atomic resolution of liquid covered surfaces (Marti et al., 1987) and low-temperature (4.2 K) operation (Kirk et al., 1988). The AFM measures either the contours of constant force, force gradients or the variation of forces or force gradients, with position, when the height of the sample is not adjusted by a feedback loop. These measurement modes are similar to the ones of the STM, where contours of constant tunneling current or the variation of the tunneling current with position at fixed sample height are recorded. The invention of the AFM demonstrated that forces could play an important role in other scanned probe techniques. It was discovered that forces might play an important role in STM. (Anders and Heiden, 1988; Blackman et al., 1990). The type of force interaction between the tip and the sample surface can be used to characterize AFMs. The highest resolution is achieved when the tip is at about zero external force, i.e., in light contact or near contact resonant operation. The forces in these modes basically stem from the Pauli exclusion principle that prevents the spatial overlap of electrons. As in the STM, the force applied to the sample can be constant, the so-called constant-force mode. If the sample z-position is not adjusted to the varying force, we speak of the constant z-mode. However, for weak cantilevers (0.01 N/m spring constant) and a static applied load of 10–8 N we get a static deflection of 10–6 m, which means that even structures of several nanometers height will be subject to an almost constant force, whether it is controlled or not. Hence, for the contact mode with soft cantilevers the distinction between constantforce mode and constant z-mode is rather arbitrary. Additional information on the sample surface can be gained by measuring lateral forces (friction mode) or modulating the force to get dF/dz, which is nothing else than the stiffness of the surfaces. When using attractive forces, one normally measures also dF/dz with a modulation technique. In the attractive mode the lateral resolution is at least one order of magnitude worse than for the contact mode. The attractive mode is also referred to as the noncontact mode. We will first try to estimate the forces between atoms to get a feeling for the tolerable range of interaction forces and, derived from them, the compliance of the cantilever. For a real AFM tips the assumption of a single interacting atom is not justified. Attractive forces like van der Waals forces reach out for several nanometers. The attractive forces are compensated by the repulsion of the electrons when one atom tries to penetrate another. The decay length of the interaction and its magnitude depend critically on the type of atoms and the crystal lattice they are bound in. The shorter the decay length, the smaller is the number of atoms which contribute a sizable amount to the total force. The decay length of the potential, on the other hand, is directly related to the type of force. Repulsive forces between atoms at small distances are governed by an exponential law (like the tunneling current in the STM), by an inverse power law with large exponents, or by even more complicated forms. Hence, the highest resolution images are obtained using the repulsive forces between atoms in contact or near contact. The high inverse power exponent or even exponential decay of this distance dependence guarantees that the other atoms beside the apex atom do not significantly interact with the sample surface. Attractive van der Waals interactions on the other hand, are reaching far out into space. Hence, a larger number of tip atoms take part in this interaction so that the resolution cannot be as good. The same is true for magnetic potentials and for the electrostatic interaction between charged bodies. A crude estimation of the forces between atoms can be obtained in the following way: assume that two atoms with mass m are bound in molecule. The potential at the equilibrium distance can be approximated by a harmonic potential or, equivalently, by a spring constant. The frequency of the vibration f of the atom around its equilibrium point is then a measure for the spring constant k : © 1999 by CRC Press LLC k = ω2 m 2 (2.1) where we have to use the reduced atomic mass. The vibration frequency can be obtained from optical vibration spectra or from the vibration quanta hω 2  hω  m k=   h  2 (2.2) As a model system, we take the hydrogen molecule H2. The mass of the hydrogen atom is m = 1.673 × 10-27 kg and its vibration quantum is hω = 8.75 × 10–20 J. Hence, the equivalent spring constant is k = 560 N/m. Typical forces for small deflections (1% of the bond length) from the equilibrium position are ∝5 × 10–10 N. The force calculated this way is an order of magnitude estimation of the forces between two atoms. An atom in a crystal lattice on the surface is more rigidly attached since it is bound to more than one other atom. Hence, the effective spring constant for small deflections is larger. The limiting force is reached when the bond length changes by 10% or more, which indicates that the forces used to image surfaces must be of the order of 10–8 N or less. The sustainable force before damage is dependent on the type of surfaces. Layered materials like mica or graphite are more resistant to damage than soft materials like biological samples. Experiments have shown that on selected inorganic surfaces such as mica one can apply up to 10–7 N. On the other hand, forces of the order of 10 to 9 N destroy some biological samples. 2.2 The Mechanics of Cantilevers 2.2.1. Compliance and Resonances of Lumped Mass Systems Any one of the building blocks of an AFM, be it the body of the microscope itself or the force measuring cantilevers, is a mechanical resonator. These resonances can be excited either by the surroundings or by the rapid movement of the tip or the sample. To avoid problems due to building or air-induced oscillations, it is of paramount importance to optimize the design of the scanning probe microscopes for high resonance frequencies; which usually means decreasing the size of the microscope (Pohl, 1986). By using cubelike or spherelike structures for the microscope, one can considerably increase the lowest eigenfrequency. The eigenfrequency of any spring is given by f= 1 k 2π meff (2.3) where k is the spring constant and meff is the effective mass. The spring constant k of a cantilevered beam with uniform cross section is given by (Thomson, 1988) k= 3EI , l3 (2.4) where E is the Young’s modulus of the material, l the length of the beam, and I the moment of inertia. For a rectangular cross section with a width b (perpendicular to the deflection) and a height h, one obtains for I I= © 1999 by CRC Press LLC bh 3 12 (2.5) Combining Equations 2.3 through 2.5, and we get the final result for f : f= 1 2π 3 EI l 3meff = Ebh 3 4l 3meff 1 2π (2.6) The effective mass can be calculated using Rayleigh’s method. The general formula using Rayleigh’s method for the kinetic energy T of a bar is 1 T= 2 ∫ ( )  dx  m ∂z x  l  ∂t l 0 2   (2.7) For the case of a uniform beam with a constant cross section and length L, one obtains for the deflection z(x) = zmax (1 – (3 x)/(2 l) + (x3)/(2l3). Inserting zmax into Equation 2.7 and solving the integral gives T= ∫ 0 l ()   m  ∂z max x  3x   x 3   1 dx − +    2l  l  ∂t    l3     (2.8) ( ) 1 = meff z maxt 2 2 and meff = 9 m 20 for the effective mass. Combining Equations 2.4 and 2.8 and noting that m = ρlbh, where ρ is the density of mass, one obtains for the eigenfrequency  1 5 f =  2π 3  E h  ρ  l2 (2.9) Further reading on how to derive this equation can be found in the literature (Thomson, 1988). It is evident from Equation 2.9, that one way to increase the eigenfrequency is to choose a material with as high a ratio E/ρ. Another way to increase the lowest eigenfrequency is also evident in Equation 2.9. By optimizing the ratio h/l 2 one can increase the resonance frequency. However, it does not help to make the length of the structure smaller than the width or height. Their roles will just be interchanged. Hence, the optimum structure is a cube. This leads to the design rule, that long, thin structures like sheet metal should be avoided. For a given resonance frequency the quality factor should be as low as possible. This means that an inelastic medium such as rubber should be in contact with the structure to convert kinetic energy into heat. 2.2.2 Cantilevers Cantilevers are mechanical devices specially shaped to measure tiny forces. The analysis given in the previous chapter is applicable. However, to understand better the intricacies of force detection systems we will discuss the example of a straight cantilevered beam (Figure 2.1). © 1999 by CRC Press LLC FIGURE 2.1 A typical force microscope cantilever with a length l, a width b, and a height h. The height of the tip is a. The material is characterized by Young’s modulus E, the shear modulus G = E/(2(1 + σ)), where σ is the Poisson number, and a density ρ. FIGURE 2.2 Moments and forces acting on an element of the beam. The bending of beams with a cross section A(x) is governed by the Euler equation (Thomson, 1988): d2  d2 EI x 2  dx  dx 2  z = p x  () () (2.10) where E is Young’s modulus, I(x) the flexure moment of inertia defined by ( ) ∫ ( ) z dydz I x = 2 (2.11) A x Equations 2.10 and 2.11 can be derived by evaluating torsion moments about an element of infinitesimal length at position x. Figure 2.2 shows the forces and moments acting on an element of the beam. V is the shear moment, M the bending moment, and p(x) the position-dependent load per unit length. Summing forces in the z-direction, one obtains () dV − p x dx = 0 (2.12) Summing moments on the right face of the element gives ( )( ) 2 dM − Vdx − 12 p x dx = 0 (2.13) Finally, one obtains for the shear and bending moments () dV =px dx dM =V dx © 1999 by CRC Press LLC (2.14) Combining both parts of Equation 2.14, one obtains the following result d2 M dx 2 = () dV =px dx (2.15) Using the flexure equation to express the bending moment, one obtains M = EI d2z dx 2 (2.16) Combining Equations 2.15 and 2.16, and one obtains the Euler Equation 2.10. Beams with a nonuniform cross section are difficult to calculate. Let us, therefore, concentrate on straight beams. These cantilever beams are widely used for friction mode as well as for noncontact experiments. A force acting on the cantilever at a position x0 can be handled by the Dirac function δ(x – x0), for which one has ∫ f (x )δ(x − x )dx = f (x ) ∞ 0 −∞ 0 (2.17) Hence, one sets () () p x = Fδ l (2.18) where l is the length of the cantilever. Integrating M twice from the beginning to the end of the cantilever, one obtains () ( ) M x = l−x F (2.19) since the moment must vanish at the end point of the cantilever. Integrating twice more and observing that EI is a constant for beams with an uniform cross section, one gets () d2z M x = EI dx 2 2 l  x  x  ⇒z x = −3 F 6EI  l   l  () 3 (2.20) The slope of the beam is () z′ x = dz l2 x  x  lx  x  = − 2 F = −2 F  2 EI  l  dx 2 EI l  l  (2.21) Evaluating this and Equation 2.20 at the end of the cantilever, i.e., for x = l, one gets () l3 F 3EI () 2 z l =− z′ l = − © 1999 by CRC Press LLC () 3z l l F= 2EI 2 l (2.22) z′(l) is also the tangent of the deflection angle. Using the definition of the moment of inertia for a beam with a rectangular cross section, I = 121 bh 3 (2.23) where b is the width and h the thickness of the lever, one gets for the deformation z at the end of the cantilever is related to the applied normal force F by 3 4  l z= F Eb  h  (2.24) Hence, the compliance kN is F Eb  h  = z 4  l  kN = 3 (2.25) and a change in angular orientation of the end of 2 ∆α = 6  l 3 ∆z FN =   2 l Ebh  h  (2.26) We can ask ourselves what will, to first order, happen if we apply a lateral force FL to the end of the cantilever. The cantilever will bend sideways and it will twist. The sideways bending can be calculated with Equation 2.24 by exchanging b and h k L ,b = FL Eh  b  = ∆z 4  l  3 (2.27) Therefore, the compliance for bending in lateral direction is larger than the compliance for bending in the normal direction by (b/h)2. The twisting or torsion on the other side is more complicated to handle. For wide, thin cantilevers (b  h), we obtain k L,tor = Gbh 3 3la 2 (2.28) The ratio of the torsion compliance to the bending compliance is (Colchero, 1993) k L,tor k L ,b 1  ab  =   2  hl  2 (2.29) where we assumed a Poisson ratio s = 0.333. We see that thin, wide cantilevers with long tips favor torsion while cantilevers with square cross sections and short tips favor bending. Finally, we calculate the ratio between the torsion compliance and the normal mode-bending compliance. k L,tor kN © 1999 by CRC Press LLC  l = 2   a 2 (2.30) FIGURE 2.3 The effect of normal FN and frontal forces FFr on a cantilever. Equations 2.28 to 2.30 hold in the case where the cantilever tip is exactly in the middle axis of the cantilever. Triangular cantilevers and cantilevers with tips not on the middle axis can be dealt with by finite-element methods. The third possible deflection mode is the one from the forces along the cantilever axis. Their effect on the cantilever is a torque. The boundary condition for the free end of the cantilever is M0 = a*FFr (see Figure 2.3). This leads to the following modification of Equation 2.19: () ( ) M x = l − x FN + FFra (2.31) Integration of Equation 2.31 now leads to () z′ x =  dz 1  lx  x  =   − 2 FN + axFFr  dx EI  2  l   (2.32) A second integration gives the deflection () z x =   1  2 x 2 lx  − 1 FN + ax FFr  2EI   3l   (2.33) Evaluating Equations 2.32 and 2.33 at the end of the cantilever, we get the deflection and the tilt due to the normal force FN and the force from the front FFr () z l =− l3 al2 l2  a l  FN + FFr =  FFr − FN  3EI 2EI 3  EI  2 l2 al l  l  z′ l = − FN + FFr =  aFFr − FN  2EI 2  EI EI  () (2.34) These equations can be inverted. One obtains the two: FN = − FFr = − © 1999 by CRC Press LLC ( )   lz ′ l 12EI z l − 3  2 l  () ( ()   ( )) 2EI 3z l − 2lz ′ l al2 (2.35) A second class of interesting properties of cantilevers is their resonance behavior. For cantilevered beams one can calculate that the resonance frequencies are (Colchero, 1993) ω nfree = λ2n h 2 l 2 3 E ρ (2.36) with λ0 = (0.596864 …)π, λ1 = (1.494175 …)π, λn → (n + ½)π. A similar Equation 2.36 as holds for cantilevers in rigid contact with the surface. Since there is an additional restriction on the movement of the cantilever, namely, the location of its end point, the resonance frequency increases. Only the λn’s terms change to (Colchero, 1993) with λ′0 = (1.2498763…)π, λ′1 = (2.2499997…)π, λ′n → (n + ¼)π(2.37) The ratio of the fundamental resonance frequency in contact to the fundamental resonance frequency not in contact is 4.3851. For the torsion mode, we can calculate the resonance frequency to ω 0tors = 2π h G lb ρ (2.38) for thin, wide cantilevers. In contact, we obtain ω 0tors ω 0tors,contact = ( ) 1 + 3 2a b 2 (2.39) The amplitude of the thermally induced vibration can be calculated from the resonance frequency using ∆z therm = k BT k (2.40) where kb is Boltzmann’s factor and k the compliance of the cantilever. Since force microscope cantilevers are resonant structures, sometimes with rather high qualities Q, the thermal noise is not evenly distributed as Equation 2.40 suggests. The spectral noise density below the peak of the response curve is z0 = 4k BT kω 0Q in m Hz (2.41) 2.2.3 Tips and Cantilevers The key to the successful operation of an AFM is the measurement of the interaction forces between the tip and the sample surface. The tip would ideally consist of only one atom, which is brought in the vicinity of the sample surface. The interaction forces between the AFM tip and the sample surface must be smaller than about 10–7 N for bulk materials and preferably well below 10–9 N for organic macromolecules. To obtain a measurable deflection larger than the inevitable thermal drifts and noise the cantilever deflection for static measurements should be at least 10 nm. Hence, the spring constants should be less than 10 N/m for bulk materials and less than 1 N/m for organic macromolecules. Experience shows that cantilevers with spring constants of about 0.01 N/m work best in liquid environments, whereas stiffer cantilevers excel in resonant detection methods. © 1999 by CRC Press LLC