Scanning Probe Microscopy

Scanning Probe Microscopy

172 Scanning Probe Microscopy Scanning Probe Microscopy J Gomez-Herrero, Universidad Autonoma de Madrid, Madrid, Spain R Reifenberger, Purdue Univers...

377KB Sizes 3 Downloads 160 Views

172 Scanning Probe Microscopy

Scanning Probe Microscopy J Gomez-Herrero, Universidad Autonoma de Madrid, Madrid, Spain R Reifenberger, Purdue University, W. Lafayette, IN, USA & 2005, Elsevier Ltd. All Rights Reserved.

Introduction The scanning probe microscope (SPM) is an extremely versatile instrument that has steadily evolved from its invention in the early 1980s. SPMs are now routinely available in many research labs throughout the world and are widely acknowledged for ushering in the study of matter at the nanoscale range. The underlying principles of an SPM are quite simple but yet completely different in many significant ways from traditional microscopes. Essentially, the SPM works by positioning a sharp tip (often called a proximal probe) B1 nm above a substrate. The highly local information provided by the microscope is achieved by a combination of the sharpness of the tip, as well as the small separation between the tip and substrate. The critical feature of any SPM is the ability to maintain a constant tip–substrate distance (with a precision approaching a few picometers), while the tip is rastered across the substrate in a highly controlled way. To achieve this precision, a signal must be acquired that is very sensitive to the tip–substrate separation. The exact physical origin of this signal then determines the property of the substrate that is mapped by the SPM. A key discovery during the development of the SPM was the realization that with a sufficiently sharp tip, a quantitative three-dimensional image of surfaces can be obtained, often with atomic resolution. The worldwide interest in scanning probe instruments was ignited by the research accomplishments of G Binnig and H Rohrer, at the IBM Zurich Research labs in Switzerland. These two individuals shared the Nobel prize in physics in 1986 for their seminal work in scanning probe microscopy. A reading of the published literature reveals relevant prior art that resembles the implementation of SPMs in the early 1980s. As examples, work on surface profilers (using optical deflection techniques similar to those used in current scanning force microscopes) can be found in the published work of G Shmalz in 1929. In 1972, R Young, J Ward, and F Scire developed an instrument (called a topografiner) designed to measure the surface microtopography of a substrate. This latter work used a controllable metal–vacuum–metal separation to maintain a fixed tip–substrate distance,

in some sense foreshadowing by some 10 years the tunnel gap approach developed independently by Binnig and Rohrer. In what follows, the general principles underlying all SPMs are discussed first. Then the two widely used families of SPMs are discussed – the scanning tunneling microscope (STM) for studying the surface topography of ‘‘electronically conducting’’ substrates and the scanning force microscope (SFM), also known as the atomic force microscope (AFM), developed to investigate the surface topography of ‘‘electronically insulating’’ substrates. The proliferation and development of seminal SPM technology has produced a wide variety of dual-probe implementations of SPMs (often called SxMs; where x stands for some physical variable of interest), which have led to simultaneous measurements with high lateral and vertical resolution, not only of surface topography but also of other local properties of substrates.

Generic SPM Components Virtually all SPMs share a number of common features that are schematically illustrated in Figure 1. This figure shows a sharp tip placed in proximity to a substrate. A vibrationless coarse approach mechanism (not illustrated) is required to initially position the tip close to the substrate without damage. After this approach, the substrate is scanned (rastered) in a controlled way under the tip by a computer program which oversees the operation of the instrument. A specially designed signal transducer forms a central part of the microscope as it measures (with high gain) a particular physical property related to the tip– substrate interaction, producing a measurable voltage that can be monitored by the computer. The key feature of the microscope is a feedback loop, which acts to keep the output of the signal transducer constant during the scanning procedure. This is usually accomplished with a controller, based on the general principle of a proportional–integral– differential (PID) controller. By recording the zmotion of the substrate necessary to keep the signal transducer output constant, a point by point ‘‘image’’ of the substrate’s surface can be generated. The image consists of N  N ordered values of (x, y, z) coordinates where N is usually of the order of a few hundred points. A one-dimensional ordered array of N points is usually referred to as one scan. An SPM image is therefore made up of N individual scans. Typically, the microscope can complete from B1 to B10 scans in one second; the speed is largely

Scanning Probe Microscopy 173

Signal conditioning

SPM frame Signal transducer Z

Substrate

z Feedback

Tip

Scanner X Y Z

Tip control

Set point

Computer control; image processing

Tip signals

Y X,Y scan control

X

Scan signals

Figure 1 A schematic of a generic SPM in which a sharp tip interacts with a substrate. The design of the signal transducer allows different local properties of the substrate to be probed. The relative tip–substrate separation is controlled via a Z-feedback mechanism, producing an image of the substrate as it is rastered in a controlled way beneath the tip.

determined by the substrate roughness and the time constant of the feedback loop, which in turn, is related to the internal mechanical resonances of the SPM frame. Once the digital image is acquired, it can be displayed in a wide variety of two- or threedimensional formats and further analyzed using a wide variety of sophisticated processing software. Because of the close proximity between the tip and substrate (typically B1 nm), the SPM head must be carefully isolated from uncontrolled wall and floor vibrations. Adequate vibration isolation is an absolute must and is often achieved by the use of a spring system designed to isolate the SPM from surrounding environmental vibrations. Immunity to vibration is also achieved by designing the SPM frame to be small, compact, and rigid thereby ensuring that the instrument’s resonant vibrational frequencies lie in the kHz range, far removed from building vibrations which tend to occur in the 1–20 Hz regime. Very often, acoustic shielding surrounds the microscope to prevent the interaction of unwanted sound waves (audible noise) with the microscope. Care must be taken to minimize temperature gradients, since a small gradient between different pieces of the instrument can cause uncontrolled expansion or contraction of B50–100 nm. If the substrate reacts spontaneously and uncontrollably with ambient environmental conditions, then further precautions must be taken to slow down or eliminate these uncontrolled surface modifications before a faithful image of the substrate’s surface can be obtained. Very often, this requirement dictates either an ultrahigh vacuum environment or the submersion of the substrate under a protective liquid. The controlled and vibrationless rastering of the substrate under the tip is critical to the operation of any SPM. This is often achieved through the use of piezoelectric ceramics which bend, contract, and expand in an appropriate manner by applying voltages

+x

+y

z −y

−x

Figure 2 A schematic diagram of a segmented piezotube often used in SPMs to perform a controlled rastering in the x, y, and z directions.

in the 100–500 V range. Piezoelectric ceramics, that are fashioned in the shape of tubes, seem to be the current choice because of their high performance-tocost ratio. Piezotubes (see Figure 2) are hollow piezoceramic cylinders (typically having a length LD224 cm, an outside diameter DD0:5 cm, and a wall thickness wD0:1 cm), electroplated with thin metal films to form continuous inner and outer electrodes spanning the entire length of the tube. The inner electrode is continuous while the outer electrode is often segmented into four orthogonal quadrants or sectors to enable motion in an xy plane. When a voltage is applied between the inner electrode and all four outer electrodes, the piezotube expands or contracts, depending on the polarity of the applied voltage. A bending motion closely resembling a pure lateral translation is achieved by applying a voltage between the inner electrode and one quadrant of the outer electrode. In order to achieve maximum translations, the applied voltage is often inverted and simultaneously applied to the opposing quadrant of the piezotube. In this way, orthogonal translations in an x–y plane can be achieved

174 Scanning Probe Microscopy

by utilizing all four segmented sectors along the outer wall of the piezotube. The displacements achieved by this action can be calculated from a knowledge of the d31 piezoelectric constant of the piezoceramic material comprising the piezotubes. Typical values of d31 (at room temperature) lie between  100 and  300  10  12 mV  1. For an applied voltage V, the elongation (or contraction) of the piezotube along its axis is given by Dz ¼ LðV=wÞd31 . The displacement of the scanner in thepffiffiffix–y plane can be estimated from Dx ¼ Dy ¼ ð2 2=pDÞ ðV=wÞL2 d31 . In practice, intrinsic nonlinearities, hysteresis, aging, and creep all conspire to limit the performance of piezoceramics, which in practice often require constant calibration against established standards if high-fidelity metrology is a requirement.

region of space formed by the physical gap between the tip and substrate, in which case Vo ¼ j þ mt where j is the work function of the tip and mt is the characteristic Fermi energy of the tip. The presence of this gap prevents the transit of a classical electron, but within the context of quantum mechanics, the electron has a finite probability of penetrating the barrier. The basic physics required to understand how an STM operates, begins by considering electrons incident upon a barrier at an energy EoVo. Such electrons can quantum mechanically tunnel through the barrier with a transmission probability T that can be obtained from a time-independent solution to the Schro¨dinger equation. For a square barrier, the transmission probability is given by T¼

The Scanning Tunneling Microscope The STM was historically the first scanning probe microscope, and was introduced in 1982 by G Binnig and H Rohrer with the demonstration that a controllable vacuum tunneling gap could be achieved between a sharp metallic tip and a conducting substrate. The vertical resolution of the microscope is a few picometers while the lateral resolution is B0.1 nm. STM images typically span an area ranging from a few nanometers to few hundred nanometers. To understand tunneling through a vacuum gap, it is useful to consider what happens when an electron wave with incident energy E encounters a barrier with a characteristic width d and a characteristic height Vo (Figure 3a). Such a barrier is present in the Vt

Substrate Tip

It

d U(z)

 t (a)

Tip

Substrate



E

d eVt z

s (b)

Figure 3 (a) A schematic of a potential barrier of width d between a metallic tip and metallic substrate. In equilibrium, the electrochemical potential of the tip (mt) and that of the substrate (ms) are aligned. The height of the potential barrier is Vo ¼ j þ mt , (b) The situation that develops when a bias (Vt) is applied between tip and substrate.

4EðVo  EÞ 4EðVo  EÞ þ Vo2 sinh2 ðkdÞ

½1

valid when EoVo, where k

2ppffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mðVo  EÞ h

m is the electron mass, and h is Planck’s constant. When kdc1, an approximation appropriate for STM experiments, eqn [1] reduces to the well-known result that TE

16EðVo  EÞ 2kd e Vo2

½2

If EDmt , as is the case for low applied bias, then Vo  EDj. Since j is B5 eV (typical values of the work function in metals and semiconductors), the coefficient 2k in eqn [2] is B23 nm  1. To pass an electrical current It between the tip and substrate, a bias voltage Vt must be applied between the tip and the substrate (Figure 3b). This bias voltage distorts the shape of the square barrier, which is also rounded and lowered in height by many-body electron correlation effects not illustrated here. The electric current between tip and substrate is proportional to the transmission probability displayed in eqn [2]. After integrating over the appropriate range of energies, the tunnel current It is given by an expression of the general form It Ef ðVt ; jÞe2kd

½3

where f ðVt ; jÞ is a function that depends on the applied voltage and the exact details of the barrier under consideration. For applied voltage differences of 1 V, typical tunnel currents encountered in STM experiments lie between 0.01 and 1 nA.

Scanning Probe Microscopy 175

The strong exponential dependence of It with distance d is the important conclusion drawn from this calculation. Rough estimates using eqn [2] indicate that a change in the barrier width d by 0.1 nm causes a change in It by roughly a factor of 10. This large amplification implies that small tip motions can be easily detected, measured, and hence controlled. The exquisite sensitivity of It to tip–substrate separation can be used as the signal transducer referred to in Figure 1 to control the vertical tip position above a substrate. Two modes of imaging can be contemplated: (1) constant-height imaging in which the tip is moved at a fixed height above the substrate and where variations in the tunnel current due to height variations are recorded and (2) constantcurrent imaging in which the tip position is continually adjusted by the z-feedback loop to produce a constant tunnel current. These two modes of imaging in STM are illustrated in Figures 4a and 4b, respectively. The constant-height mode (Figure 4a) is of limited use, since it is only appropriate for substrates that are essentially flat at the atomic length scale. Furthermore, since variations in the tunnel current measured in the constant-height mode depend exponentially on the distance, they cannot be directly interpreted as height profiles. The three-dimensional imaging process in STM is most easily understood by considering the constant-current imaging mode (Figure 4b). The STM image is formed by recording the relative motion of the substrate to maintain a constant tunnel current as the tip is swept across a preselected area of the substrate. To achieve this, a high-gain current

amplifier (typical gain is B108 to B109 VA  1) is required. Sharp tips are necessary to produce images with high lateral resolution. Common ways of producing STM tips from metal wires (such as W or Pt) with diameters of B1  10  4 m rely on electrochemical etching or physical cutting. The reliable formation of sharp tips may seem like a daunting venture, but ultimately every tip formed must end with one or possibly a few atoms which ever so slightly protrude from the apex, forming a small mini-tip at the tip’s apex. The presence of such mini-tips, along with the strong exponential drop-off of current with distance, provides a reasonable way to understand why the total tunnel current between tip and substrate may be dominated by an atomically small protrusion from an otherwise large tip. More complete theories of STM clearly have shown that the tunnel current can be related to the wave function overlap between electron states in the tip with electron states in the substrate. This implies that the images obtained from an STM not only contain surface topographic information, but also information about the variation of the local density of electronic states. This complication provides a caveat against the direct translation of relative tip– substrate separation into surface topographic features. With this caution in mind, a few representative STM images from a variety of different surfaces are given in Figure 5. STM images are notable for the amazing detail they reveal about the atomic periodicity and surface morphology of clean, electronically conducting substrates.

The Scanning Force Microscope

d

d

d+h

d

h

h

Tunnel current

Tunnel current

Tip motion

Tip motion Scan direction

(a) Constant height

Scan direction (b) Constant current

Figure 4 A schematic illustrating two modes of imaging employed in STMs. (a) The tip–substrate separation is held constant and the variations in tunnel current are measured; (b) the tunnel current is held constant by a feedback loop and the relative tip–substrate separation is varied to maintain a constant tunnel current.

While STMs provide a quantitative map of surface topography with atomic resolution, they suffer from a fundamental limitation that the substrate studied must be sufficiently conducting to support a tunnel current. In order to overcome this difficulty, an AFM was first demonstrated in 1986 by Binnig, Quate, and Gerber. The operation of this microscope relied on the surface forces acting on a sharp tip in close proximity to a nonconducting surface. For sufficiently small tip–substrate separations, these interaction forces can range from tens of pN (10  12 N) to tens of mN (10  6 N), with typical values of a few nN (10  9 N). An understanding of these interaction forces is central to understanding how an AFM (also called an SFM) functions. Most importantly, these forces are not predicated on the fact that either the tip or substrate is electrically conducting. Because of the long-range nature of the interaction forces, the vertical resolution of an SFM is typically a few

176 Scanning Probe Microscopy Microcantilever, kc

c zset z s Substrate

(a)

(b)

(c)

Figure 5 Typical STM images. Different color schemes are employed in each image to better render the z-height information. (a) An image of step edges in a Si(1 1 1) substrate showing atomic periodicity. The atoms on the surface undergo a 7  7 reconstruction, creating a new unit cell different from that observed in the bulk. The field of view is 31  31 nm. (b) An image of a Au(1 1 1) surface showing atomically flat plateaus terminated by abrupt steps that are one atom in height. The small corrugation (B0.01 nm in height) pffiffiffi observed on each flat terrace is a manifestation of the 22  3 surface reconstruction on this facet of Au. The field of view is 100  100 nm. (c) An atomically resolved image of Au(1 1 0) surface viewed from the top in false color format (bright is high, dark is deep). Rows of individual Au atoms forming a (2  2) reconstruction on different terraces of the substrate are clearly visible. The field of view is 9  9 nm. (Courtesy Go´mez-Rodrı´guez JM and Mendez J, Departmento Fı´sica de la Materia Condensada, Universidad Autonoma de Madrid.)

Figure 6 A schematic illustrating the sequence of events when a tip on a microcantilever is brought into close proximity to a substrate. Initially, the tip is located a distance zset from the substrate. Attractive interaction forces between the tip and substrate bend the tip toward the substrate until a deflection dc of the cantilever brings the system into equilibrium. If the interaction forces are sufficiently strong, the substrate may also experience a distortion ds, which may be appreciable if the substrate is soft. The final tip–substrate separation is indicated by the parameter z.

picometers (comparable to an STM) while the lateral resolution is B10–20 nm, somewhat larger than for STMs. SFM images typically span an area ranging from B100 nm to around tens of micrometers. In practice, the operation of an SFM relies on a sharp tip which is usually supported on the end of a microcantilever whose minute deflections can be carefully monitored. As shown in Figure 6, a microcantilever with spring constant kc, when positioned at a distance zset from a substrate, will deflect toward the substrate by an amount dc due to interaction forces that exist between the tip and substrate. In addition, it is possible that the surface will distort by an amount ds due to the action of the same forces. In general, it is difficult to determine the exact tip– substrate separation z due to a lack of knowledge about the distortion ds as well as the inability to accurately determine the initial tip–substrate separation zset. For sufficiently small deflections, the cantilever motion can be well approximated in terms of Hook’s law, which predicts a restoring force F given by F ¼ kc dc. Table 1 provides some dimensions and relevant properties of typical microcantilevers that are commercially available. An uncertainty in cantilever thickness causes a considerable spread in the resulting spring constants. Sharp tips, with effective radius R (typically, R is between 5 and 30 nm), are routinely formed onto these cantilevers using lithographic techniques developed by the semiconductor industry. The widespread availability of microcantilevers means that interaction forces B1 nN between the tip and the substrate can be monitored and that cantilever deflections B1 nm or less can be readily detected.

Scanning Probe Microscopy 177 Table 1 A few representative silicon cantilevers commercially available with their characteristics Length ðmmÞ

Width ðmmÞ

Thickness ðmmÞ

kc ðN m1 Þ (min, typical, max)

fo ðkHzÞ (min, typical, max)

12575 23075 9075 12575 9075 30075

3573 4073 3573 3573 3573 3573

4.070.5 7.070.5 2.070.3 2.070.5 1.070.3 1.070.3

20, 40, 75 25, 40, 60 6.5, 14, 28 1.8, 5.0, 12.5 0.45, 1.75, 5.0 0.01, 0.05, 0.1

265, 325, 400 150, 170, 190 240, 315, 405 110, 160, 220 95, 155, 230 9.5, 14, 19

Vout=VA−VB Focused laser Vout = 0

Photodiode

Vout > 0

Figure 7 A common method employed to measure the deflection of a cantilever is a beam bounce technique in which a diode laser beam is reflected from a microcantilever onto a segmented photodiode. By monitoring the voltage from each segment of the photodiode, the relative motion of the reflected laser spot can be monitored and information about sub-nanometer motion of the cantilever can be inferred.

To measure cantilever motion while scanning, a high-gain transducer of cantilever deflection (the signal transducer referred to in Figure 1) plus a feedback mechanism is required. A variety of techniques – capacitance, optical interferometry, piezoelectric microcantilevers, and optical beam deflection – have been successfully implemented to accurately detect cantilever deflection. Each technique seems to have its own advantages. Currently, the technique most often implemented is an optical deflection scheme shown schematically in Figure 7. Using this approach, a focused laser beam is deflected from a microcantilever and the reflected light is directed onto a segmented photodiode. Fine positioning of the reflected spot allows for a null condition characterized when the voltages from the appropriate photodiode segments are made to sum to zero by an external operational amplifier (not shown). A small cantilever deflection disrupts this null condition, giving rise to a voltage proportional to beam deflection. The origin of the high amplification for this particular system follows from simple geometrical considerations. For a cantilever displacement Dz, the reflected laser spot moves a distance

Ds ¼ Dzðd=lÞ, where d is the distance of the cantilever from the photodiode and l is the cantilever’s length. Typically, the ratio of d=l for a microcantilever can easily be a factor of 100–500. When discussing the nature of the interaction force between tip and substrate, it is often convenient to approximate the tip as a sphere with radius R. This sphere then interacts with the substrate via a number of possible forces, which can cause the cantilever to deflect as shown in Figure 6. The exact details of the relevant interaction forces, as well as their variations on z, depend to a large extent on the composition of the tip and substrate. For the ideal case of a clean, electrically neutral tip positioned above an electrically neutral, clean substrate in ultrahigh vacuum, the interaction forces might be well-approximated by a superposition of a short-range, hard-wall repulsion (effective when the tip–substrate separation is less than B0.3 nm) plus a longer range surface interaction due to the van der Waals (vdW) force acting between dipoles induced on the individual atoms comprising the tip and substrate. The z-dependence of this vdW force is related to the detailed shape of the substrate and tip. If the substrate/tip are studied under ambient conditions, hydration forces due to adsorbed water, as well as long-range electrostatic forces due to uncontrollable charging of the tip or substrate, may well dominate. Without a detailed knowledge of the system under study, it is difficult to accurately specify a force versus distance relationship. In general, such a curve would have the approximate shape shown in Figure 8. This figure qualitatively illustrates (1) the attractive regime ðFo0Þ in which the interaction forces cause the tip (and microcantilever) to bend toward the substrate and (2) the repulsive regime ðF40Þ that causes the microcantilever to bend away from the substrate when the tip comes into contact with it. A qualitative appreciation of the important features of this force curve is critical to understanding how an SFM obtains an image of a substrate. By rastering the substrate beneath the tip, all the while maintaining a constant force between the tip and the

178 Scanning Probe Microscopy A Interaction force Repulsive regime Tip−sample separation Attractive regime B

C

Figure 8 A schematic illustrating how the interaction force between tip and substrate varies as a function of separation. Three regions (A, B, C) are indicated. Different modes of imaging are achieved when the tip is positioned in each region.

substrate, an image closely resembling the surface topography of the substrate’s surface can be obtained. There are a variety of methods that have been developed to achieve this task. The exact method employed depends on the distance between the tip and substrate. If the tip is in region A in Figure 8, then imaging is performed in a ‘‘contact mode’’; the tip exerts a force directly on the sample as it is scanned across it. In contact mode imaging, the direct up and down motion of the cantilever is measured while scanning. This motion can be used to produce a three-dimensional image of the substrate in much the same way as a conventional profilometer, except that now the applied force lies in the nN range and the radius of the stylus is in the 5–30 nm range. This mode of operation can be damaging, especially for soft substrates and stiff microcantilevers since significant lateral forces develop during the scanning process. If the tip is in region C of Figure 8, the interaction forces are sufficiently weak so that very small deflections of the cantilever result. Since the substrate– tip separation is large, imaging in this region is often referred to as the ‘‘noncontact mode.’’ Under these circumstances, indirect detection schemes are usually employed. As an example in noncontact mode imaging, the tip is often driven sinusoidally at a frequency near its mechanical resonance. Small position-dependent shifts in the resonance frequency occur when the substrate is rastered beneath the tip. These frequency shifts can then be used as a sensitive measure of tip–substrate separation, thereby providing the transducer signal for the feedback controller. Because of the noncontact feature of this mode, it is preferred when studying soft substrates. If the tip is placed in region B of Figure 8, the interaction forces become comparable to the restoring force of the microcantilever, implying that

static tip displacements, although measurable, cannot be reliably measured because of resulting instabilities. In simple terms, the instabilities arise because of the double-valued nature of the interaction force as illustrated by the horizontal dotted line in Figure 8 which indicates that for the same value of the interaction force, there are two possible tip–sample separations. These instabilities are often referred to as jump-to-contact because the tip spontaneously snaps into contact with the substrate no matter how carefully the procedure is employed. To scan in region B, the tip must assume a time-dependent behavior which is carefully controlled by the SFM computer. During the tip’s motion, it is possible that the tip might periodically come into contact with the substrate, giving rise to what is known as ‘‘intermittentcontact’’ or ‘‘tapping mode’’ imaging. The boundaries between the different regions in Figure 8 are not necessarily well defined, so a precise distinction between the different imaging regimes is difficult to provide. When operating an SFM in either region B or C, the SFM is often referred to as a dynamic force microscope (DFM) and the imaging process is often referred to as dynamic mode imaging. To better appreciate the information contained in an SFM image obtained in regions B and C of Figure 8, it is useful to discuss in more detail the appropriate imaging modes. If the tip hovers at some distance above the substrate, then a sinusoidal modulation of the cantilever’s position is usually required to monitor the interaction forces. This is often accomplished by mounting the cantilever assembly directly onto a small piezoelectric slab, which is driven at a preset angular frequency o. During the oscillatory motion of the cantilever, the tip feels two forces: a restoring force due to the cantilever and an interaction force due to the forces acting between the tip and substrate. Under these circumstances, the tip motion can be visualized by considering the two spring systems shown in Figure 9. One spring, with spring constant kc, accounts for the restoring force of the cantilever, while the other spring, with effective spring constant kinter, accounts for the interaction forces. Effectively, the tip is acted on by a single spring with effective spring constant keff ¼ kc  kinter . If the interaction force curve is accurately known (rarely the case), kinter could be estimated according to kinter ¼ dFinter =dz, where it is understood that the derivative must be evaluated at the equilibrium separation between the cantilever and substrate. When the tip is far from the sample, kinter D0 and the equation of motion of a tip with mass m can be analyzed using the equations for a damped oscillator (with damping coefficient b), driven by a sinusoidal

Scanning Probe Microscopy 179

tip–substrate separation. The effective resonant frequency oe is given approximately by  1=2 dFinter  oe Doo 1  ½7 dz z¼zeq

Cantilever support

kc

Tip

k inter

Substrate

Figure 9 A simple way to understand the different forces acting on a tip when it is positioned in close proximity to a substrate. The tip is acted upon by two springs, one due to the restoring force of the cantilever, the other due to the interaction force between tip and substrate.

force Fo cosðotÞ: m¨z þ b’z þ kc z ¼ Fo cosðotÞ

½4

The steady-state tip motion zðtÞ ¼ Reðzo eia eiot Þ is specified by Fo zo ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 ðo2o  o2 Þ2 þ b2 o2

½5

ob tan a ¼ mðo2o  o2 Þ pffiffiffiffiffiffiffiffiffiffiffi where the resonance frequency oo ¼ 2pfo  kc =m. Clearly, the amplitude of the oscillating cantilever is maximu‘m when the driving frequency equals the resonance frequency (i.e., when o ¼ oo ). As the tip moves closer to the substrate, the interaction force Finter shown schematically in Figure 8 comes into play and the equation of motion becomes m¨z þ b’z þ kc z ¼ Fo cosðotÞ þ Finter

½6

The details of the tip motion now depend on Finter . A general analysis of this problem usually starts by expanding Finter in a Taylor series about the equilibrium tip–substrate separation. This analysis shows that the resonance frequency (which can be accurately measured) becomes a function of the

As a consequence, the amplitude of the cantilever’s oscillation, driven at a frequency o slightly off-resonance, will vary with position as the substrate is rastered beneath the oscillating microcantilever. The amplitude change resulting from variations in oe can be measured using a conventional phase-sensitive detection of the signal from the microcantilever/photodiode assembly. The substrate–tip separation can be continuously adjusted by a feedback loop to maintain either a constant resonance frequency or a constant amplitude of oscillation. By measuring the required motion of the substrate in the z-direction required to achieve this condition, it is possible to render an ‘‘image’’ of the surface of the substrate. Such an SFM image reflects relative changes in the tip–substrate separation required to maintain a constant resonant frequency, or equivalently a constant amplitude of the tip’s oscillation. As the equilibrium separation between the cantilever and substrate decreases and the tip passes from region C to region B, the intermittent mode of operation comes into play. The situation becomes considerably more complicated. The interaction forces are now comparable to the restoring forces of the cantilever. If kc ¼ kinter at some zset, instabilities will result. If the amplitude of the tip oscillation drives the tip from region B to region A, the tip will intermittently ‘‘tap’’ the surface. This could produce a substrate damage, especially if the cantilever has a high spring constant. Also, nonlinearities and instabilities in the tip motion can produce a chaotic rather than a periodic motion of the cantilever. Considerable operator skill and insight is often required to produce artifact-free images under these conditions. Another way of implementing intermittent contact mode imaging is to use an alternative approach often referred to as the ‘‘jumping mode.’’ In this procedure, the cantilever does not undergo sinusoidal motion but instead follows a motion determined by the software controlling the SFM. In practice, the software is programmed to retract the tip, position it at the beginning of a selected scan range, and then drive the substrate toward the tip under feedback control until the cantilever bending reaches a preset loading force. At this point, the z-displacement of the substrate required to meet this condition is measured and the tip is withdrawn, moved to a nearby adjacent location above the substrate where the process is again repeated. After completing a scan, the relative

180 Scanning Probe Microscopy

z-motion of the substrate at each point in the scan is plotted for further analysis. The advantage of this technique is that the force applied to the substrate can be carefully monitored during the imaging process. Furthermore, the lateral force imparted to the substrate while scanning is eliminated. The disadvantage is that the scan proceeds at a somewhat slower rate than when the cantilever is sinusoidally driven. Figure 10 provides a schematic diagram of these different imaging modes. Table 2 summarizes the above discussion by listing a few very general guidelines for cantilever selection during each mode of SFM operation. By way of summary, in SFM, no matter which mode of imaging is employed, the motion of the z-piezo is used to form an image to keep a required signal constant. In contact mode imaging, the static deflection of the cantilever is used. In noncontact

Common scanning modes

(a) Contact

(b) Noncontact

(c) Intermittent contact

(d) Jumping

Figure 10 A schematic illustrating the different scanning modes commonly employed in SFM. (a) The contact mode of imaging where the tip is in constant contact with the substrate; (b) the noncontact mode of imaging, where the tip oscillates sinusoidally while maintaining a fixed difference between tip and substrate; (c) the intermittent contact mode where the tip ‘‘taps’’ the substrate during the scanning process (Note that the frequency of tip oscillation in (b) and (c) is much greater than the scanning frequency of the microscope.); (d) the jumping mode where the tip is moved into contact with the substrate, then lifted and moved to another location before contact with the substrate is re-established.

mode imaging, the amplitude of the cantilever oscillation is employed. The SFM is characterized by a lateral resolution of B10–20 nm, somewhat higher than that for STM since the tip radius R plays a central role. Atomically resolved images in SFM have been demonstrated, but usually this requires an SFM operating under ultrahigh vacuum conditions. The vertical resolution (typically better than 0.1 nm) rivals that of STM, while a force sensitivity of the order of 1 pN can be achieved. A few typical SFM images taken in noncontact mode are given in Figure 11.

The SxM Family of Microscopes The basic techniques described above have been extended in a number of very clever ways, producing a large family of SPMs, often referred to as SxMs, each designed to detect the local variation in some quantity of interest. This extension of SPM is often referred to as dual-probe microscopy because the tip not only measures topography but also some other physical parameter of interest with high lateral resolution. A few examples include an electrostatic force microscope (EFM or scanning Kelvin probe), a magnetic force microscope (MFM), a photon scanning tunneling microscope (PSTM), a scanning electrochemical microscope (SECM), a scanning near-field optical microscope (SNOM), a scanning capacitance microscope (SCM), scanning tunneling spectroscopy (STS), and a frictional force microscope (FFM).

Summary The rapid evolution of SPMs since their first demonstration in the early 1980s has truly been remarkable. Largely because they are versatile and relatively inexpensive, SPMs have ushered in a worldwide interest in nanotechnology. With precise engineering, SPMs are capable of very high resolution (subatomic scale) metrology. Also, it is now clear that SPM tips can be used as tools capable of nanometer manipulation and fabrication. Active research in nanolithography is underway to controllably use the

Table 2 A qualitative comparison between the three commonly used modes in scanning force microscopy Scanning mode

kc ðN m1 Þ

fo , Resonance frequency ðkHzÞ

Approx. tip–substrate separation ðnmÞ

Comments

Contact Intermittent contact

o1 10–100 1–5 10–100 1–5

– 100–300 10–70 100–300 10–70

0 B3 B3 45 45

Tip wear, contact mechanics Stable Unstable Stable Stable

Noncontact

Scanning Probe Microscopy 181

100 nm

100 nm

(a)

2 × 2 µm (b)

6× 6 µm

(c)

Figure 11 Representative examples of SFM images. Different color schemes are employed in each image to better render the z-height information. (a) An image of l-DNA deposited on a mica substrate. The left edge of this image shows a gold contact pad deposited on top of the DNA molecule. After manipulation, the DNA molecule has been cut by the tip, at the position shown by the arrows. The size of the image is indicated by the scale bar. (b) An image of three gold contact pads deposited onto a multiwalled carbon nanotube and then imaged with an SFM in the noncontact mode. The field of view is 15  15 mm. (c) An example of nanolithography performed by a scanning force tip viewed in false color (bright is high, dark is low). A periodic array of silicon oxide nanostructures has been created by applying a voltage between the tip and a Si substrate under ambient conditions. Local electrochemistry between the tip and substrate produces the protrusions which are then imaged using the same tip that created them. (Courtesy DNA image – J Gomez-Herrero, Departmento Fı´sica de la Materia Condensada, Universidad Autonoma de Madrid; Carbon nanotube – Reifenberger R, Department of Physics, Purdue University; Nanolithography – C Martin and F Perez-Murano, Instituto de Microelectronica de Barcelona; Centro Nacional de Microelectronica.)

SFM tip to locally modify a substrate in a very precise way at the nanometer length scale. SPM operation has been extended to scanning under liquids, allowing a window into the biological world. Linear parallel arrays of cantilevers have been designed and fabricated to work in a massively parallel fashion, and efforts to independently control individual cantilevers in the array have also been reported. Current indicators are that technology underlying these proximal probe microscopes will continue to improve, and the SPM class of instruments will continue to

become ever more commonplace as a tool of choice to probe the properties of nanoscale objects. See also: Biomolecules, Scanning Probe Microscopy of; Confocal Optical Microscopy; Fermi Surface Measurements; Low-Energy Electron Microscopy; Metals and Alloys, Electronic States of (Including Fermi Surface Calculations); Optical Microscopy; Photoelectron Spectromicroscopy; Scanning Near-Field Optical Microscopy; Superconductivity: Tunneling; Surfaces and Interfaces, Electronic Structure of; Surfaces, Optical Properties of;

182 Scattering Techniques, Compton Transmission Electron Microscopy; Treated Surfaces, Optical Properties of; van der Waals Bonding and Inert Gases.

PACS: 07.79.  v; 68.37.Ef

Further Reading Binnig G, Quate CF, and Gerber C (1986) Atomic force microscope. Physics Review Letters 56: 930–933. Binnig G and Rohrer H (1991) In touch with atoms. Review of Modern Physics 71: S324–S330. Binnig G, Rohrer H, Gerber Ch, and Weibel E (1982) Tunnelling through a controllable vacuum gap. Applied Physics Letters 40: 178–180. Binnig G, Rohrer H, Gerber C, and Weibel E (1983) 7  7 Reconstruction of Si(1 1 1) resolved in real space. Physics Review Letters 50: 120–123. Cappella B and Dietler G (1999) Force distance curves by atomic force microscopy. Surface Science Report 34: 1–104. Giessibl FJ (2003) Advances in atomic force microscopy. Review of Modern Physics 75: 949–983.

Gracia R and Perez R (2002) Dynamic atomic force microscopy methods. Surface Science Report 47: 197–301. Guntherodt HJ and Wiesendanger R (eds.) (1994) Scanning Tunneling Microscopy I: General Principles & Applications to Clean & Adsorbate-Covered Surfaces. New York: Springer. Guntherodt HJ and Wiesendanger R (eds.) (1997) Scanning Tunneling Microscopy II: Theory of STM & Related Scanning Probe Methods. New York: Springer. Israelachvili J (1991) Intermolecular and Surface Forces, 2nd edn. London, New York: Academic Press. Julian Chen C (1993) Introduction to scanning tunneling microscopy, Oxford Series in Optical and Imaging Sciences. New York: Oxford University Press. Marti O and Amrein M (eds.) (1993) STM and SFM in Biology. New York: Academic Press. Martin Y, Williams CC, and Wickramasinghem HK (1987) Atomic force microscope – force mapping and profiling on a ˚ scale. Journal of Applied Physics 61: 4723–4729. sub 100 – A Meyer G and Amer NM (1988) Novel optical approach to atomic force microscopy. Applied Physics Letters 53: 1045–1047. Meyer E, Hug HJ, and Bennewitz R (2004) Scanning Probe Microscopy. Berlin: Springer. Sarid D (1991) Scanning force microscopy with applications to electric, magnetic, and atomic forces, Oxford Series in Optical and Imaging Sciences. New York: Oxford University Press.

Scattering Techniques, Compton P E Mijnarends, Northeastern University, Boston, MA, USA and Delft University of Technology, Delft, The Netherlands A Bansil, Northeastern University, Boston, MA, USA & 2005, Elsevier Ltd. All Rights Reserved.

Introduction ‘‘Compton scattering’’ refers to a collision between a photon and a charged particle, often an electron, in which the photon loses a substantial fraction of its energy. The use of words ‘‘photon’’ and ‘‘collision’’ suggests a corpuscular nature of light, that is, that light may be viewed as a stream of particles with well-defined energies and momenta. The first indications that light can lose a part of its energy when scattered in this manner appeared at the beginning of the twentieth century. Convincing evidence was provided by the pioneering experiments in the 1920s by Arthur Compton after whom the effect has been named. This was not the first time the suggestion had been made that light may behave as a particle. The photoelectric effect, in which a material irradiated by light can be seen to emit electrons, was earlier explained by Einstein by postulating that light of frequency n consists of particles called light quanta (and later photons), each with energy hn, where h is Planck’s constant. For a given material, the

photoemitted electrons possess energies, which depend on the frequency (or equivalently the wavelength, l ¼ c=n, where c is the speed of light), but not on the intensity of the incident light. Above a certain wavelength no electron emission is possible, as is observed to be the case, because a photon does not carry enough energy to overcome the binding energy of the electron to the material. Compton scattering, which views the photon–electron scattering to behave rather like billiard balls colliding with other billiard balls, thus reinforces Einstein’s explanation of the photoelectric effect. An outline of this chapter is as follows. The following section gives an overview of the theory of Compton scattering. The equations for energy and momentum transfer and the cross section for the Compton scattering process are given. Various forms of Compton scattering are discussed: (1) Compton scattering from the charge distribution of electrons in a material; (2) magnetic Compton scattering; and (3) ðg; egÞ scattering, in which additional information is obtained by measuring the properties of the recoil electron in addition to those of the scattered photon. The third section addresses issues of instrumentation. In keeping with current trends, the emphasis is on experiments using synchrotron radiation facilities around the world, which provide bright sources of X-rays. The forth section touches upon the effect of multiple scattering of a Compton scattered photon