An introduction to Atomic Force Microscopy Modes

The Atomic Force Microscope (AFM) is a scanning probe imaging and sensing device, useful for physical and chemical studies. In its basic configuration, it measures the microscopic surface profile of a near-planar target by mechanically scanning a tiny probe across it in a raster pattern. The probe rises and falls in accordance with the surface profile. As it does so, its position is sensed and captured digitally. That captured topogram can then be rendered as a photograph-like image, reminiscent of an optical micrograph.

AFM is far more versatile and capable than the others, viz:

• The spatial resolution of an AFM is far better than that of an optical microscope. A typical AFM probe has resolution better than 50 nm. Optical microscopy, on the other hand, is limited by the finite wavelength of visible light - roughly 400 to 700 nm. Its resolution is thus, at best, about ten times worse than the AFM.
• Scanning electron microscopes can give high resolution - of the order of a nanometer or so - but they operate only in high vacuum. They also require that the sample be conductive, thus precluding use on living objects. The targets are typically plated or frozen for imaging.
• The AFM operates under ambient or near-ambient conditions, not the high vacuum required by SEM. It can even be used under water.
• The AFM operates with little or no special surface preparation, such as the metal coatings required by SEM.
• The AFM can be used on a wide variety of surface types, ranging from hard and crystalline to soft and pliable, even living entities.
• Other types of scanning probe microscopes, such as the scanning tunneling microscope (STM) and the near-field scanning optical microscope (NSOM) have comparable resolution, but they are both quite limited in the surface properties to which they are sensitive, and they require careful target preparation. The STM, in particular, requires a conductive target.
• The AFM can be made sensitive to a variety of surface properties, such as conductivity, dielectric constant, magnetization, and others, through the use of suitably chosen probes and signal processing.
• The AFM can also act as an active tool, useful for manipulating objects at the nanometer scale.

The three dimensional profile of a surface is measured with the AFM by monitoring the position of the probe in three dimensions as it scans. A feedback control system maintains the force between the probe and the surface sensibly constant during the scan. The probe tip thus follows the surface profile, as one might do with a finger tip at a more macroscopic scales.

Soon after the invention of the AFM it was realized that these instruments were capable of measuring far more than surface topography. It is possible to measure any physically phenomenon at the nanometer scale for which a suitable sensor can be incorporated in the probe. Demonstrated examples include magnetic fields, electric fields, contact potential difference, temperature, and hardness. It is also possible to use the AFM probe as a tool to modify surfaces.

Such non-topographic uses have come to be called AFM "modes". It is the purpose of this monograph to elaborate upon these modes, both in principle and in practice.

AFM "Modes"

This is a white paper on AFM modes. There's a problem with the notion of "modes", as this is really a misnomer. The "mode" terminology is, however, deeply entrenched amongst workers in this exciting field as a sort of shorthand name for one of the myriad AFM techniques or applications. Therefore, with an English teacher's caveat regarding possible bad usage, we present the list in Table Table 2: as a practical user's map.

AFM Basic implementation
Figure 1-1 shows the elements of a basic atomic force microscope. The probe tip is positioned in three dimensions by three mutually orthogonal actuators. Each actuator is a piezoelectric ceramic transducer (PZT). PZTs are well suited to the small motions required by the AFM. Their travel is approximately linear in the applied voltage when used in the AFMs.

We adopt the convention that the X and Y coordinate axes are nominally parallel to the surface under test (SUT), and the Z axis completes a right handed Cartesian coordinate system. The Z axis positioner thus moves the probe toward (negative Z) or away (positive Z) from the SUT, while the X and Y positioner move the probe in a nominal object image plane. In normal operation, the X and Y position is programmed in a raster scanning path, while the Z position is tightly controlled relative to the SUT by a feedback control loop. The Z-axis error signal is derived from a force transducer that measures the force on the probe tip. The transducer output is differenced against a fixed voltage, which corresponds to a setpoint (target) force. The error signal is amplified and drives the Z-axis PZT. This control loop acts to reduce the error signal, and hence the probeto- SUT distance to near-zero. The feedback loop thus causes the probe to closely follow the undulations of the SUT while the X-Y PZTs scan the probe over a rectangular image area. Meanwhile, the control voltage applied to the Z-PZT provides a convenient representation of the elevation of the probe and surface is used to generate and image or topogram of the surface.

Light lever force sensing
High sensitivity in the force transducer of an AFM can be achieved by a simple geometric optical device known as the light lever (see Figure 1-2). A low-power laser beam is reflected from the top of the cantilever to a distant photo detector. Small displacements of the cantilever result in large displacements of the laser beam at the location of the detector due simply to the large "lever arm" of the light path. The detector is bifurcated, and the halves are differenced, giving a force-proportional error signal.

Probe-surface interactions
While it might be thought that the force between an AFM probe and a hard surface might have an abrupt brick wall character, this is an over-simplification at the nanoscopic scale. Not only are fundamental interatomic forces finite in extent, there is also the problem of surface contamination. AFMs are usually operated in ambient environmental conditions (room temperature, atmospheric pressure, ambient air). As a result, there is invariably a surface layer comprised of water and miscellaneous hydrocarbons. This layer is thick enough relative to the nominal probe operating height that the probe tip is almost always immersed in it (Figure 1-3).

The forces between probe and surface, to the extent that they are position-dependent only, i.e. are lossless, can be represented by an effective potential energy (see, for example, Figure 1-4)

As shown in Figure 2-9, the interaction between probe and surface falls into one of three zones:

• Large separations, where the probe experiences no significant influence from the surface
• An attractive zone, caused by capillary attraction between the probe and the contaminant layer, which is thick compared to the range of the interatomic forces from the surface. This zone is usually associated with the formation of a meniscus.
• A "hard" repulsive region, where the interatomic forces from the surface dominate those from the contaminant layer

There are two primary methods for measuring the force between a probe and a sample: contact mode and vibrating mode.
Contact mode entails a direct quasi-static force versus distance measurement, that is, a rather intimate contact between probe and surface. While producing accurate results, it tends to rapidly damage both probe and surface.
Vibrating mode may be either large amplitude, in which the probe contacts the surface on every cycle, or small amplitude non-contact. The latter is generally less stressful on both the probe and surface, and tends to give superior results. It measures more subtle properties of the surface.

As originally conceived, the AFM (see Figure 2-1) was a surface profiling device. A nanometer- scale probe, mounted at the end of a tiny cantilever, is held in contact with the surface using piezoelectric positioning. Deflection of the cantilever, easily measured optically, indicates force between probe and surface. That force is held nominally constant by a feedback control system while the probe is scanned raster- fashion across the surface. The control voltage indicates elevation as a function of transverse position. The elevation can be displayed or recorded in various ways, providing a topogram of the surface. The resolution, limited by the probe tip radius, is of the order of a few tens of nanometers, with fields-of-view of the order of a few tens of microns.

Topography by quasi-static probe contact
In contact mode, the cantilever is scanned over a surface at nominally constant force. The Z axis PZT feedback loop, if properly implemented, ensures near constancy of cantilever deflection, which corresponds to constant force.

Contact mode is typically used for scanning hard samples and when a resolution of greater than 20 nanometers is required. The cantilevers used for contact mode are usually silicon or silicon nitride. Typical resonant frequencies are about 50 KHz and force constants are below 1 N/m.

Probes
AFM probe-cantilevers assemblies are readily available from commercial vendors. They are generally made of silicon or silicon nitride, taking advantage of the highly-developed techniques of micro lithographic for fabrication. Figure 2-3 shows SEM images of some representative tips.

The sharpness of the probe is a primary determinant of the resolution of the AFM. Probe tips are approximately spherical, and are characterized by an effective radius. A typical probe radius is in the neighborhood of 10-20 nm. Special "supersharp" tips of even smaller radii, approaching 1 nm, are offered, these being more suitable for probing narrow structures.

Probe coatings. Probes are often used with various coatings for sensing particular phenomena. For use with the light lever detection system, the cantilever backside is aluminized for high optical reflectivity. Available probe coatings include:

• Polycrystalline diamond - Improves wear and breakage resistance, especially in contact mode operation
• Hard ferromagnetic - Permanently magnetized alloy, usually cobalt, for magnetization sensitivity
• Chromium and platinum iridium on both tip and cantilever to enhance electrical conductivity for electrical measurements
  

Figure 2-4 shows some SEM images of probe tips at very high magnification.

Cantilever stiffness
For small deflections of the probe relative to its equilibrium (non-contact) position, the force versus distance characteristic is, to a good approximation, linear. The force can thus be approximated by Hooke's law:

Because of the strong analogy to the simple mass-spring oscillator so familiar from elementary physics classes, we will often call k the cantilever "spring constant", tho the terminology is not rigorously appropriate.
k can be calculated if the dimensions and composition of the cantilever are known. Most commercially available cantilevers are supplied with a value for k , but these are often unreliable. For best results, Sader's method is recommended. The length and width of the cantilever are measured with an optical microscope and an approximate mechanical Q is measured by scanning the excitation frequency while observing the mechanical response. From those the effective spring constant can be calculated.

As illustrated in Figure 2-5 the cantilever in an AFM can bend, and twist as it is scanned across a surface. The horizontal forces are dependent upon the so-called "asperity" of the surface (its abrasive nature) relative to the probe, that is, the sliding friction between probe and surface. Typically the compliance for twisting is much less than the compliance for bending. The twisting sensitivity, of course, is to the forces that exert a moment around the cantilever's major axis, that is, forces perpendicular to the axis of the cantilever and parallel to the SUT.

Probe position sensing
Light lever. High sensitivity in the force transducer of an AFM can be achieved by a simple geometric optical device known as the light lever (see Figure 2-6). A low-power laser beam is reflected from the top of the cantilever to a distant photo detector. Small displacements of the cantilever result in large displacements of the laser beam at the location of the detector due simply to the large "lever arm" of the light path.

Position sensitive detector. Cantilever torsion is measured by an enhancement of the light lever. A four-quadrant position sensitive detector (PSD) is used with the signals added and subtracted as shown in Figure 2-7 to give equivalent vertical and horizontal displacements. Because the vertical and lateral deflections are substantially independent of one another, it is possible to measure topography and lateral friction force independently (see §). Because the physical dimensions of commercially available cantilevers are not well characterized, the elasticity coefficients must be calculated individually for each cantilever from measured dimensions and material properties.

Optical interferometry. Some practitioners have used optical interferometry on the cantilever for extreme sensitivity to Angstrom-scale deflection (see, for example, Martin et al, (1987) for a heterodyne technique, Erlandsson et al, (1988) for non-heterodyne). While the potential sensitivity of interferometry is higher (sub-A) than that of the light lever, its cost and complexity discourage its use for all but the most demanding applications.

Probe-surface interactions

While it might be thought that the force between an AFM probe and a hard surface might have an abrupt brick wall character, this is an over-simplification at the nanoscopic scale. Not only are fundamental interatomic forces finite in extent, there is also the problem of surface contamination. AFMs are usually operated in ambient environmental conditions (room temperature, atmospheric pressure, ambient air). As a result, there is invariably a surface layer comprised of water and miscellaneous hydrocarbons. This layer is thick enough relative to the nominal probe operating height that the probe tip is almost always immersed in it (see Figure 2-8), forming a meniscus that complicates near-surface operation.

The forces between probe and surface, to the extent that they are position-dependent only, i.e. are lossless, can be represented by an effective potential energy (see Figure 2-9).

There are three basic regions of interaction between the probe and surface:

• Free space far from the SUT, where the probe experiences no significant influence from the surface
• An attractive region, caused by capillary attraction between the probe and the contaminant layer, which is thick compared to the range of the interatomic forces from the surface
• A repulsive region, where the interatomic forces from the surface dominate those from the contaminant layer

Piezoelectric transducer (PZT) positioning The X-Y resolution of the measurements possible in an AFM are of the order of the tip radius, which is typically are a few tens of nm, and can be as small as just a few nm. If one postulates an image area of 1000 x 1000 pixels (a rather high resolution digital image), at, say, 20 nm per pixel, then the actual imaged area is 20 µ m by 20 µ m. This is far too small for simple mechanical positioning. A far better solution is the use of piezoelectric ceramic positioners.

Piezoelectrics are materials that expand or contract in the presence of internal voltage gradients (that is, bulk electric fields). The magnitude of the piezoelectric phenomenon makes these devices quite appropriate as nanometer-scale actuators (see, for example, Gallégo-Juarez (1989)). Piezoelectric actuators can produce, under electronic control, extremely fine position control down to the nanometer scale required by an AFM. The motion is smooth, with no thresholds or "stickiness" characteristic of electromechanical devices. The are non-magnetic, making them suitable for applications that are magnetic-field sensitive. They are also capable of extremely high accelerations, as much as thousands of g 's, in combination with high forces, upwards of thousands of newtons. There is no observable mechanical wear, in most applications.

Traditionally the PZT positioning in AFMs is done by a one-piece tubular actuator. Four electrodes in a quadrant configuration cover the outside, while a full-circumference electrode covers the inside. Voltages applied to opposite quadrant pairs causes tilt along the axis of symmetry. Simultaneous voltage on all four quadrants causes elongation or contraction. The major virtue of this design is its mechanical simplicity: one instrument provides motion in all three coordinate axes. It does, however, have some drawbacks:

• The relationship between the control voltages and the motion is non-orthogonal and nonlinear. Considerable correction is needed to achieve truly orthogonal axes.
• The effective image surface is warped, i.e. non-planar.
• Mechanical resonances complicate the dynamics of the X-Y scanning.
• The mass of the sample and its substrate slow the frequency response of the Z-axis positioning.

Of these, the most objectionable is perhaps the scan rate limitation. Achievable rates in typical units are so slow that a full frame scan can take many minutes. Various speedup schemes, such as the dual feedback controller scheme of Egawa et al (1999) have been tried, with mixed success. This particular scheme used, as have others, a supplementary PZT actuator on the cantilever, in addition to the principal tube-style 3D actuator. This sort of dual actuator configuration is especially applicable to vibrating cantilever mode (VCM). The PZT tube actuator provides coarse Zaxis positioning, but with slow dynamics, while the cantilever actuator provides fast, fine-grained dynamics for both Z-axis positioning and cantilever drive.

The Proportional-Integral-Differential (PID) Feedback Controller.
Most AFM schemes for generating topography images make use of some sort of feedback control system in conjunction with their error measurement and Z-axis actuator. The design and implementation of such a feedback control system is non-trivial. The design is driven by unavoidable trade-offs between tracking accuracy, speed, and stability. The simple, naive design will almost always result in instability, entailing not only failure to track properly, but also damage or destroy) the probe and sometimes damage the SUT. While a simple solution is possible, optimized solutions require considerable sophistication in both analysis and implementation.

The AFM, when tracing SUT topography, might be regarded as a special case of the age-old generic process control model shown in Figure 2-10 (upper).

The traditional industrial process control mechanism for a poorly-defined and/or inherently nonlinear process is the so-called proportional-integral-differential (PID) controller. With reference to Figure 2-10, the process drive y ( t ) is generated as a sum

Such a control mechanism is easily analyzed using LaPlace transforms and well-known techniques if the overall system, that is the "plant" plus feedback amplifier, is linear and time invariant. In the case of the AFM, however, the probe plus cantilever plus PZT actuator system is only approximately linear. Moreover, even the dynamics in the linear approximation are not easily characterized. Cantilever resonances are usually at tens of kHz and unimportant in topography following. The typical tubular PZT actuator, however, has resonances in the neighborhood of a few kHz. These are significant and do affect topography tracking.

The goals of the feedback controller design are straightforward:

• Error - The probe should follow the surface with minimum error.
• Bandwidth - The scan speed should be as fast as possible.
• Noise rejection - Measurement noise should be rejected as much as possible.
• Stability - The loop should be unconditionally stable, that is, it should not oscillate no matter what the characteristics of the SUT, other than it be mechanically passive.
• Robustness - Loop design parameters should not be sensitive to the inevitable small errors and uncertainties in modeling of components, nor to the character of the SUT.

The first and second of these are generally trade-offs against one another. Arbitrarily tight following can be accomplished at the expense of extremely slow transient response, that is, of low closed loop bandwidth. Rapid scanning can be achieved at the expense of large tracking error. The tradeoff is, of course, a consequence of the stability requirement, which is absolute. A loop is either stable or unstable, the latter being intolerable.

The usual solution in the practice of AF microscopy is to tune the PID coefficients semi-heuristically. Such techniques are well-established in the literature of industrial process control. While they do result in a stable system, its bandwidth is not large, thus limiting the usable scan speed. Scanning speed is an especially important issue. AFMs using a simple PID style of feedback controller can take many minutes to complete an image.

As noted above, the limiting factor in the loop design is usually the PZT actuator, which usually has internal mechanical resonances in the neighborhood of a few khz. This is orders of magnitude slower than the cantilever resonances. The cantilever itself thus is generally not significant in limiting the closed loop bandwidth.

Schitter et al. (2001), recognizing the importance of the PZT in limiting system performance, carefully characterized their PZT actuator dynamics through the mathematical tool known as system identification. System identification is the creation of a linear system model using only observations of input and output, rather that physical modelling. This is especially useful for the PZT, physical modelling of which would be a very complex mathematical problem. System identification, on the other hand, is relatively simple, producing directly a useful pole-zero model.

Using that model Schitter et al. then generated a controller using the H_∞ control system criterion (see, for example, Skogestad and Postlethwaite (1996)). This design was implemented, and displayed significant improvement in transient response (Figure 2-11).

Details of this sort of analysis and design are beyond the scope of this paper, but they are well-known and are documented in modern control theory textbooks (e.g. references found in the Schitter et al. (2001) paper).

Subsequent sections of this paper show a simple "feedback amplifier." These should always be understood as the classic PID controller, with the proviso that more sophisticated designs may yield substantial performance improvements.

Observing the Elevation. Note in Figure 2-10 the presence of an elevation estimator whose inputs are the PZT control voltage and the cantilever deflection. This is a correction that is required of all AFM feedback schemes, though we may, for simplicity, omit it in system diagrams. The reason is simple. Even if we assume that we know the actuator position with complete precision, the actual position of the probe tip is the sum of that actuator position and the cantilever deflection. That is to say, the actuator sets the equilibrium position of the probe, while the surface interaction deflects the probe away from that equilibrium position. One must add (or subtract, depending on mechanical design details) the two to obtain the true probe tip position. In more sophisticated designs, that is, ones that account for the PZT dynamics, one must also account for the fact that the actuator position lags its control voltage because of the mechanical resonances. While we will, in subsequent sections, show the amplifier output as an analog of position, suitable for recording, this is an over-simplification if one is trying to do precision work.

Quasi-static (QS) operation
"Quasi-static" here denotes operation where the relationship between applied force and displacement is not significantly affected by inertia (mass) of the probe; the probe position thus represents only a static force balance. "Quasi" is prefixed to show that probe motion is possible, tho scanning is slow. Such operation is primarily applicable to surface following.

"Contact mode" might be considered to be simply a high-force form of quasi-static operation, though all QS operation is not necessarily contact mode. Care should be used in reading too much into the terminology, as it is often used loosely.

A typical quasi-static system architecture is shown in Figure 2-12. A feedback control loop adjusts the equilibrium probe position so that the deflection matches a setpoint, thus keeping the force constant. The actuator control voltage thus represents the surface profile as defined by constant static force. That profile may vary slightly, depending on the exact value of the setpoint and the steepness of the force profile.

A Q-S topogram may be generated simultaneously with measurements of specific physical surface properties, or it may be itself the primary objective.
Vibrating Cantilever Mode (VCM)

In order to make more sensitive measurements, requiring better signal/noise ratios, in scientific instruments it is common to modulate the signal being measured and use phase or amplitude detection circuits. Use of modulated techniques shifts the measurement to a higher frequency regime where there is less 1/f noise. Vibrating probes also can be sensitive to near-surface force gradients, such as arise from surface polarization or magnetization.

In the AFM, the probe is vibrated as it is scanned across a surface. As shown in Figure 2-13, the probe is vibrated in and out of surface potential. The modulated signal can then be processed with a phase or amplitude demodulator.

The cantilever oscillatory excitation is normally provided by a piezoelectric ceramic, similar to those used for X-Y positioning.

A typical vibrating system architecture is shown in Figure 2-14. The Z-axis position of the probe

is the sum of a fast oscillating drive component and a slowly-varying average position that is controlled by a feedback control system. The feedback system adjusts the Z position of the SUT in accordance with the error relative to a setpoint. The measurement which is compared to the setpoint is amplitude, frequency, or phase, depending on the particular mode of operation in use. Care is needed in the detailed design of the low pass filter and feedback control system, as there is a potential for instability. That instability, moreover, is sensitive to details of the probe-surface force relationship. A system that is stable for one value of setpoint can become unstable for a different value of the setpoint. (The mathematics of this design are beyond the scope of this paper.)

Probe-cantilever dynamics. Vibrating modes all make use of the natural mechanical resonance of the probe-cantilever system. For most purposes of AF microscopy, these dynamics can be adequately modelled by a simple damped massspring system (Figure 2-15). The mass and spring identify readily with the actual mass and stiffness, tho the physical source of the damping is not entirely obvious. Such a system displays the familiar Lorentzian resonance (Figure 2-16). The amplitude response is

It is noteworthy that the amplitude at resonance can be written



where is the quasi-static amplitude response, that is, the amplitude that would result if the force acted only against the spring. The amplitude on-resonance is thus enhanced by a factor of . This is, of course, the low frequency limit of (2-4). Also noteworthy is the exact 90° phase at resonance, that is, the displacement response lags the driving force by 90°.

Probe interaction with surface. When the probe tip interacts with a surface, the resonance frequency generally shifts to a lower value, and there is a corresponding change in the phase. When scanning in the vibrating modes, a constant relationship is maintained by the feedback electronics, which keeps either the phase shift or amplitude constant at a given frequency, while scanning.

As discussed above, there is a contamination layer on surfaces in ambient air with a thickness between 1 and 50 nanometers. Capillary forces, which are attractive, strongly affect probe behavior near the contamination layer.

The probe may be used in three fashions as it is scanned across the surface (see Figure 2-17).

Regime 1 - The probe is vibrated across the surface of the contamination layer. The vibration amplitude must be very small and a very stiff probe must be used. Typically, the images of the surface contamination layer are very "cloudy" and appear to have low resolution. This is because the contamination fills in the nanostructures at the surface.

Regime 2 - "Near Contact Mode" - The probe is scanned inside the contamination layer. Very high resolution images can be produced by this technique, but great care is required. The cantilever must be stiff so that capillary forces do not snap the probe to the surface, and the amplitude must be very small.

Regime 3 - "Intermittent contact" or "tapping" mode - The probe is vibrated in and out of the contamination layer. The energy in the vibrating cantilever is much greater than the depth of the capillary attraction potential well. The probe thus moves easily in and out of the contamination layer. This mode is conceptually simple, and is the easiest to implement, but it often results in broken probes due to the surface crashes that occur on every cycle.

A side benefit of vibrating modes is that "stickiness" due to friction ("lateral") forces, if any, tends to be released on each cycle as the probe moves away from the surface.

Typically, vibrating methods are used when the highest resolution is required or if very soft samples are being scanned. The probes used for vibrating mode are often less than 10 nm in diameter.

Small amplitude VM topography. While both QS topography and VM topography use force feedback, VM can operate at a much smaller force than QS, and operates in a fundamentally different way.

Though the interaction force vs. displacement relationship is fundamentally nonlinear, for sufficiently small displacements it can be considered approximately linear, as shown in Figure 2-18. Moreover, an approximate straight line force relationship, tangent to the actual force curve, is equivalent to a linear spring, whose stiffness

Δk = -ΔF/ΔA That equivalent spring adds to or subtracts from the basic cantilever elasticity, altering keff.and thus its resonant frequency. The shift is up in a repulsive regime (larger k_eff ), or down in an attractive regime (smaller k_eff ). Although the magnitude of the change can be quite small, it is readily detected due to the generally high Q of the resonance. Viscous losses in the SUT may also affect the parameters of the resonance, though the specific relationship is less obvious, and generally is quite small compared to the elastic force contribution.

Any of several techniques can be used to detect the resulting change in the resonant frequency, changes in amplitude and phase being most common.

When scanning in vibrating modes, a constant relationship is maintained by the feedback electronics, which keeps either the phase shift or amplitude constant at a given frequency, while scanning.

Typically, small amplitude vibrating methods are used when the highest resolution is required or if very soft samples are being scanned. Some examples are shown in.Figure 2-19

The probes used for vibrating mode are often less than 10 nm in diameter. As vibrating mode imperils the probe more than simple contact modes, it is prudent to periodically check its integrity with a resolution reference sample that contains fine-scale features, such as that shown in Figure 2-20.

Large amplitude VM topography. Large amplitude VM, often called "tapping", is conceptually simpler than small amplitude VM, and easier to implement. It has the advantage of being less affected by the surface contamination layer. The probe is detached from the surface meniscus on every cycle. As the effects of the surface on the probe motion is more dramatic than in small amplitude VM, it is more easily extracted from the deflection signal. It has disadvantages, however:

• Not suitable for high resolution due to the need for large radius, durable probes
• More prone to damage the probe, due to the hard contact on every cycle
• Not suitable for soft surfaces, especially biological materials
• Less amenable to quantitative interpretation

The system architecture for large amplitude VM is essentially the same as small amplitude VM. As before, care is needed in design of the feedback control amplifier that positions the SUT in the Z coordinate, as there is a hazard of instability. The design is complicated by the fact that the value of the coordinate setpoint can change the small-signal gain, thus altering the stability properties.

Force Gradient Modes

Nanoscopic-scale electrostatic and magnetostatic material properties are particularly accessible to AFM measurement. Any interaction force for which a suitable probe exists can be mapped. The mechanism of the mapping is the spatial gradient of the force. That spatial gradient adds or subtracts from the effective spring constant of the AFM cantilever, as discussed above. The resulting changes in resonant frequency are readily measured. The fields are sensed through the changes in probe-cantilever resonance as the probe is moved slowly above the SUT.

Principles. For very small displacements of the probe, the z-axis gradient of the force modifies the effective spring constant of the cantilever as discussed above. Assuming that the field, and thus the force, varies only with Z, then the resonance of the probe-cantilever system is approximately:

and we have implicitly assumed that the force can be adequately modeled by the first term of its Taylor series expansion in ΔZ.

(The negative sign arises because F = -kZ for a simple mass-spring oscillator.)

The shift in resonant frequency is sensitive to the first spatial derivative of the spring constant, and thus to the second spatial derivative of the force, that is:



Measurements of the resonant frequency at two elevations thus yields an approximation to the second derivative of the surface field strength.

Force gradient instrumentation techniques. Several schemes are used for such measurements:

• Dual scan - The SUT is scanned twice. The first scan establishes the topography; the second
  is carried out at a constant elevation.
• Oscillating force setpoint (Hosaka Method) - Single scan, in which elevation and differential
  frequency are measured simultaneously point-by-point, rather than sequentially.
• Alternating setpoint method - Similar to the oscillating force method, but at two fixed
  elevations rather than two force points.

Application to permanently polarized materials. The force gradient technique is often applied to permanently polarized materials (called electrets when electric, or magnets when magnetic). It requires a probe that is either conductive or ferromagnetic. The coatings lead to several problems:

• Very much reduced resolution relative to topography-only measurements, due to the larger probe tip radii.
• Difficulties in fabricating high quality coatings
• Probes are easily damaged due to inadvertent surface contact

Application to more general forces. The nature of the forces sensed by the force gradient technique can be quite general. The only constraint is that the force be a function of height above surface. For example, a constant voltage bias between probe and surface produces a vertical force due to the distance-dependent capacitance between probe and surface (see §3.4, for example).

Any mutual force interaction between surface and AFM probe is subject to AFM measurement. Properties to which the AFM has been applied include:

• Mechanical properties (§3.1)
• Thermal properties (§3.2)
• Electrical properties (§3.3 and §3.4)
• Magnetic properties (§3.5)

Many of these are simply special cases of small-amplitude field gradient sensing, as discussed in §4.

Mechanical properties

When an AFM probe is scanned across a surface, in close proximity, either by closed loop tracking, or by open loop placement, the interaction forces between probe and surface alter the parameters of the mechanical resonance of the probe-cantilever system, as discussed in §2.9. Elasticity alters the resonant frequency ω₀, while losses affect primarily Q. Those changes can be detected through either amplitude or phase modulation of the cantilever motion relative to its exciting force.

In either case, the average probe Z-axis elevation is usually maintained at some preset value. That elevation can be maintained in one of two ways:

• Either a topography feedback loop operating in parallel with the VCM measurement loop,
• or simply set open-loop, in accordance with previously measured & stored topographic map

Force-distance curves. A force/distance (F/D) curve is a measure of the forces on the probe as a function of distance of the probe from a SUT. It can be measured directly or indirectly.

Direct force-distance curves. A direct force-distance curve is a record of cantilever deflection, which is proportional to force, while controlling the distance between the probe and the SUT. Typically a F/D measurement is initiated with the probe in free space, then lowering it until it interacts with the hard force in the repulsive regime. The probe is then pulled away from the surface until it is in free space again. This process is illustrated in Figure 3-1.

While the F/D relationship ideally should not depend on the direction of probe motion, in practice it often does, due to small dissipative (i.e. viscous) forces.

As the probe begins to interact with the surface, it is pulled into the surface by capillary forces. This process is called jump to contact ( JTC) and often this step breaks the tip off the probe. The position of the jump to contact is dependent on the thickness of the contamination layer on the surface. When retracting as the probe is pulled from the surface, the adhesion causes the probe to "stick" to the surface, and then disengage from the surface.

Errors can occur in the F/D curve if a position sensor is not used to measure the motion in the Z axis. The error occurs because the Z ceramic generating the motion is not linear and has hysteresis. It is also necessary to correct the PZT position for the changes in probe position to determine

the actual height of the probe above the surface. Failure to properly correct these errors leads to an artifact displacement in the F/D curve when the probe is in the repulsive region.

Indirect force-distance curves. Indirect force-distance curves are generated not by measuring cantilever displacement directly, but rather by measuring the Z-derivative of the force, as discussed in §2.10. That measurement tends to be less noisy, and thus more accurate, than direct force measurements because it is carried out in vibrating mode. Numeric integration of the measured dF/dZ can then produce the desired F(Z) curve. Some creativity, of course, is needed in determining the constant of integration for that calculation (see, for example, Martin et al. (1987). The integration constant is, in effect, the origin of coordinates - the location of the SUT's "surface".

Indirect F/D curves are subject to most of the same difficulties as the direct F/D curves, other than the absolute calibration of the Z-axis motion. Its relative linearity is important, but the origin is determined from the measurements, not the PZT control voltage. Also, if the force is not a conservative one, that is, if it is not representable as the gradient of a potential energy, then it may not be a single-valued function of distance.

Pulsed force mode. Pulsed force mode (PFM) is an "all in one" combination of topography, adhesion, and stiffness measurements, carried out simultaneously. PFM is implemented by placing a sinusoidal voltage on the Z PZT and recording the force profile. These properties can be deduced from key inflection points in the profile (Figure 3-2).

Surface loading ("nano-indenting"). The resistance of a surface to normal force distortions can be measured through the technique that might be called "nano-indenting." A nanoindent is generated by forcing the probe into a surface, causing surface material to be dislodged, creating a nano-crater. Surface distortion is measured as a function of normal force. This relationship is called a "loading curve."

Nanoindenting is possible only if the cantilever is stiff enough and the probe material hard enough. Figure 3-3 shows a nano-crater created by a silicon probe in a surface of PMMA.

It is difficult to get quantitative data when nano-indenting with an AFM because:

• The probe geometry and dimensions are neither well controlled nor well known.
• The probe-to-sample angle changes during the indentation.
• Absolute X-Y positioning accuracy of the AFM is poor, in the event that some specific, isolated location is being addressed.

Lateral (friction) force mode - nanotribology. As shown in Figure 2-5. the cantilever in an AFM can twist, or rotate as it is scanned across a surface. Use of the twisting response has opened a new avenue for the study of friction at the nanoscopic level.

The study of interacting surfaces in relative motion from a macroscopic perspective is known as tribology, and has been pursued for centuries; its pursuit at the nanoscopic scale has come to be called nanotribology, and is only a few years old.

It was long thought that there were two fundamental laws of tribology:

• Shear resistance between two bodies is independent of the apparent area of contact, and
• The shear resistance is proportional to total load only.

While these laws of friction were well-established at the macroscopic level, they prove to be quite subtle and complex at the small scale accessible to lateral force AFM. The friction force is found to depend not only on the surface asperity (i.e. gross roughness, as a piece of sandpaper) but also on intermolecular forces even when the asperity is absent or insignificant. Reitsma et al. explain it well:

"The fundamental difference between the classical laws of friction and the laws applying to nanometer scale friction is surface roughness. For contact between two solid bodies, surface roughness at the interface means that there are two ways to interpret contact. First is the type that refers to regions of the interface that are in true (or 'perfect') contact, often referred to as interfacial contact. The true contact area (A_t) of an interface is the sum of all interfacial contacts. The other is the apparent or gross contact area. This is the macroscopic surface area of the interface (A_g). Nanotribology attempts to 'scale out' the multiple asperity contact that describes tribology and the classical laws of friction.

"With reverence to the second classical law of friction (i.e. the proportionality between friction and load recently both SFA and in particular, AFM experiments, measuring interfacial friction have demonstrated a clear non-linear relation between friction and load .."

The frictional coefficient of a probe moving over a surface is defined as the loading force divided by the horizontal force:

With both vertical and lateral deflections (and thus estimates of the forces) available (see §2.5), it is possible to determine a friction coefficient. Recalling that the sense of the lateral force is always in opposition to the scan direction, measurements usually record a curve with hysteresis, as shown in Figure 3-4. With modest simplifying assumptions, and some dimensional information about the probe, an effective coefficient of friction may be calculated. Accuracy of the result is often questionable due to uncertainties in those assumptions.

Figure 3-5 shows lateral force images from a very flat gold test sample. The topography image is featureless, but the LFM image shows the drawn pattern. It is noteworthy that the appearance of the pattern is different, depending upon the scan direction.

Thermal modes

Special thermal sensing probes incorporating thermocouples or resistors (Figure 3-6) permit nano-scale measurement of:

• Temperature
• Thermal conductivity - With the resistive probe, it is possible to measure the cooling of the probe as it traverses a sample. The cooling is proportional to thermal conductivity.
• Melting point - A current ramp may be applied to a resistive probe. The melting point can be deduced from the vertical motion as the probe sinks into the surface below.

Electrical Modes

Electrical modes involve measurements of localized surface responses to various electric or magnetic stimuli. Some require direct probe-surface metallic contact and others only near proximity.

Parametric (contact) measurements. By "contact measurements" we mean those that treat the probe-plus-surface as a two-terminal circuit device. Some sort of V-I characteristic is measured point-by-point as the probe scans the surface.

Although direct contact is simple in principle, it is difficult in practice. The interface between probe and surface is complicated by:

1. Surface contamination, which typically is a lowconductivity dielectric
2. Surface oxidation
3. Abrasion of the probe coating, if used

A conductive diamond coat can mitigate issues (b) and (c), though it increases the diameter of the probe significantly. The contamination problem is largely independent of probe material.

Scanning versus static measurements. Although direct contact seems suitable for measuring electrical properties of nanostructures such as nanotubes or quantum dots, it proves problematic. Positioning error is a problem. Such structures as nanotubes and quantum dots are extremely small and difficult to locate. Although smooth scanning at the nanometer scale is easily achievable, it requires only relative positioning. Absolute, repeatable positioning at that scale, however, is very difficult.

Ohmic conductivity. Local conductivity of a surface or structures located thereon can be measured by an AFM. The probe should have high conductivity compared to the SUT, through either its bulk or through a suitable coating.

Dielectric constant. A conductive probe above a surface, with a constant voltage applied between them can be considered a simple capacitor. Any capacitor having an adjustable mechanical coordinate, say Z, held at constant voltage, can be shown from energy principles, to exert a force against that coordinate

where the negative sign signifies the direction that will reduce the capacitance. Our prior analysis of force gradient sensing (§2.10) remains valid, and equation (2-15) is once again applicable, so that

SHARK (simultaneous conductivity-topography). In the animal world, the elasmobranchs (the subclass of cartilaginous fishes that includes skates, rays and sharks), have a unique ability to sense electric fields (see, for example, Adair et al (1998)). That sense is thought to have played an important role in their evolution. They use it both for locating prey and for orientation, via motion-induced sensitivity to the earth's magnetic field. Threshold sensitivity is said to be of the order of microvolts per meter or less.

SHARK technology®, whimsically named for these remarkable animals, brings similar electric field sensing to atomic force microscopy. As an acronym it is said to stand for "surface height and resistivity camera" (if one accepts the German spelling "kamera" .). It is an AFM imaging technique that produces simultaneous topographic and conductivity maps.

The spatial map of electrical conductivity, as measured with SHARK, once again uses the stimulus- response apparatus shown in Figure 3-7. The stimulus in this case is constant voltage applied between the sample and the probe, while the response is current. Measurement of current yields a conductivity map, while topography is sensed in the conventional contact fashion. The advantage of SHARK is that the topogram and conductivity map are captured simultaneously during one scan, reducing test time. Figure 3-9 is a SHARK image of a carbon nanotube held perpendicular to a conductive substrate.

Contact potential difference (Kelvin probe). Contact potential difference (CPD) is unfortunately named, as it is something that is invariably measured without contact. It is defined as the potential difference between dissimilar conductive materials whose mobile electrons are in thermal equilibrium. It is the difference in work functions of the two materials. That equilibrium leads to small charge imbalance, and thus a potential difference between the surfaces if they are electrically connected. Two surfaces, considered as a capacitor, behave as though there were an internal potential source polarizing the capacitor. That internal potential is equal to the tiny charge difference divided by the capacitance.

The so-called Kelvin probe method is an age-old technique for measuring CPD. If the two electrically connected conductors are brought into near proximity, there is a small electric field between them due to the tiny charge imbalance.

Conventional Kelvin probe measurement places a fixed voltage across a capacitance formed of the materials to be tested (right). One of the surfaces is then mechanically vibrated. From elementary circuit theory one has

The external applied voltage is adjusted until the AC current induced by the vibration is reduced to zero. The applied is then known to be equal and opposite to the CPD. Readout of that value of V₀ thus provides the desired measurement, after a sign change.

A slight variation on this technique is eminently well-suited for the AFM, which adds imaging capability. This technique is best called Kelvin probe microscopy (KPM), avoiding the oxymoronic- seeming "contact potential difference" that involves no contact.

The AFM version (Figure 3-11) turns the macroscopic Kelvin probe method on its head: It applies an AC voltage to the capacitance formed of the materials under test, and detects the resultant vibration. It is easily shown from elementary principles that, for a capacitor whose capacitance depends on a mechanical coordinate, say, in this case, Z, then there is a force exerted by the capacitor on its supports:

Using for V the sum of a DC nulling voltage and an AC sense voltage

leads to a component of force at frequency ω that is proportional to V₀, viz:

A major advantage of this technique is that there is no need to calibrate the capacitance or its Z derivative. It is a nulling method. All that is necessary is to adjust V₀ until the force component at frequency ω goes to zero. As in a macroscopic Kelvin probe measurement, it is then known that

V₀ at null is thus equal to the negative of the CPD. If the nulling process can be carried out sufficiently rapidly, then recording V₀ while raster scanning provides a desired two-dimensional KPM image.

Several instrumental configurations for KPM are possible. There are two applied stimuli involved: the cantilever Z-axis mechanical driving force, and the applied voltage. They may be distinguished by frequency. This permits two-loop configurations in which topography is measured in conventional fashion, simultaneously with the contact potential map. One such apparatus is shown in Figure 3-12.

Unlike other imaging methods, scanning Kelvin probe microscopy is a quantitative. It measures absolute potential difference (cf EFM, §3.4, which measures electric field strength). Because KPM uses noncontact VCM to track topography, there is less tendency to damage the probe. KPM, however, is more costly to implement because it requires a lock-in amplifier and PID controller.

Electric force microscopyElectric force microscopy (EFM) is the examination of surfaces via non-contact electric field interactions. Primarily this pertains to ferroelectric and piezoelectric materials. Ferroelectric ("electret") materials have a permanent electric polarization. Piezoelectric materials have a dimensional change in response to applied electric fields, and vice versa. The two properties are often both present, as the crystal symmetry group requirements are identical. An electret, in effect, is a piezoelectric material with memory.

The advantage of the AFM probe for ferroelectric measurements is that the probe has high spatial resolution and localized measurements are possible. There are two primary types of measurements made on ferro/piezo films: purely electrical properties, and electro-spatial properties.

Electrical responses to electrical stimuli are measured with an apparatus similar to that described §3.3. The probe is placed in contact with the ferro/piezo film and the electrical measurement is made.

Dimensional responses to electrical stimuli are measured through cantilever deflection in response to an applied voltage.

In both cases the response has memory (a.k.a. hysteresis), that is, the result depends on whether the applied stimulus is increasing or decreasing. Stimulus-response curves have a "butterfly" character, as illustrated in Figure 3-14, if they are recorded through a full cycle of the stimulus.

The spatial resolution of either type of measurement is governed primarily by the probe diameter,though grain size of Polycrystalline materials also plays a role.
Magnetic field microscopy
Magnet field microscopy (MFM) is the examination of surfaces via noncontact magnetic field interactions. A probe coated with a ferromagnetic film is vibrated slightly above a surface while scanning. Modelling the magnetization as sparsely distributed surface dipoles, it can be shown that the force gradient is

where
 s = tip-surface distance
  δ= nonmagnetic thickness
  R = tip radius
  m_p = magnetic moment of surface dipole
  m_t = magnetic moment of probe tip
  d_m = mean magnetic diameter
and

Though the detailed relationship, derived from (3-11) and (4-5), is somewhat complex, the general conclusion is that the frequency change is roughly proportional to a moderately high (third to fifth) power of the separation. It is thus very sensitive to the three-dimensional structure of the magnetic domains.

AFM in magnetostatic mode is especially useful for examination of the nanoscale structure of magnetic recording media. See, for example, Figure 3-15.

As with EFM, the resolution achievable with MFM depends on the thickness and quality of the probe coating. This is difficult to control, so that the results from MFM are often inconsistent. Also, when trying to measure the magnetic domains on a soft magnetic material, magnetic domains on the probe can alter the magnetization of the SUT, masking or obliterating the original properties that were to be measured.

The machinery comprising the AFM can be utilized for active manipulation as well as passive observation. That is, it can become a toolbox for altering the physical and chemical properties of the SUT, not just a microscope. The tools in that toolbox include:

• Lithography
• Chemical deposition - The AFM as generalized printer/plotter
• Electrochemistry

Lithography
The term "lithography" originally meant "writing on stone", a technique long-used for printing of hand-drawn illustrations, as in books and magazines. In modern times it has taken on a connotation of the creation of micro-structures on semiconductor devices; conventional printing technology long ago left it behind. The exact definition in the nanoscale context is still in flux, but it retains its basic meaning of image creation.

Like modern printing, nano-scale AFM lithography uses one, or the other, or both of the two basic techniques: bit-mapped imagery and vector-drawn imagery. As in computer printers, software renders the desired image described by a source file (e.g. .bmp or .jpeg). It moves the probe and indents, scratches, or deposits "ink" of some kind (Figure 4-1). Resolution of such AFM lithography, of course, usually depends upon the probe diameter and noisiness of the PZT positioning process. The primary drawback to AFM lithography is low scan rate.

Simple Scratching. What is a drawback for AFM microscopy can be an asset for surface etching. If the force exerted by the probe is high enough, and the SUT is softer than the probe, the probe will scratch the surface.

Anodic Oxidation. In 1989 it was demonstrated that the localized current from a small probe, as in a scanning tunneling microscope, can modify a surface through localized electrochemistry. The most common example is anodic oxidation. In this particular example, the width of the oxidized line that is created depends on the number times it is traced.Chemical deposition - The AFM as generalized printer/plotter. The AFM can be used as a sort of generalized plotter, applying a variety of "inks" at the nanometer scale, for a variety of applications. The vehicle for the deposition can be:

• pyramidal scanning probe tips
• hollow probes

Many applications have been envisioned in which precise deposition or fabrication or lithographic mask creation at the nano-scale are needed.

The fluid deposited may also be a so-called "molecular glue" which forms a template or mask for subsequent chemical nano-scale fabrication.

Dip Pen Nanolithography^® (DPN^®) is a proprietary scanning probe lithography technique invented in Dr. Chad Mirkin's laboratory at Northwestern University, in which a microscopic pen (e.g. the tip of an atomic force microscope cantilever) is coated with an "ink" (a chemical compound or mixture) and put in contact with the "paper" (a substrate). Both the name and its abbreviation are now trademarks of the Nanoink corporation. "Dip pen" is an archaic name for a quill pen, by analogy to the procedure.

Electrochemical experiments
An electrochemical cell can be added to an AFM enabling electrochemistry experiments in situ. The AFM can then be used to study electrochemical changes in surface properties without thesurface being exposed to air. With the addition of a galvanostat, the surface topography can be investigated as function of the surface potential relative to a reference cell.

The AFM is its extremely high effective magnification, several orders of magnitude better than optical microscopy. Imaging of surface features as small as one atom have been reported (see, for example, Rugar et al.(2004)). One may, naturally, ask what this really means. Can the "resolution" be characterized quantitatively? The answer, of course, is yes, but doing so is difficult and complex. It depends on the specific phenomena being sensed, i.e. on the nature of the probe in use. Only the most rudimentary AFM applications actually relate to topographic imaging. Most interesting are applications to various non-visual physical phenomena.
Resolution
Resolution in images is generally taken to mean in-plane (X-Y) resolution. In the AFM one Zaxis resolution is also of interest. the latter involves both positioning and noise.

X-Y Resolution. Tip effects. One of the most important factors influencing the resolution which may be achieved with an AFM is the sharpness of the scanning probe. Typical probes have radii of a few tens of nm; the best tips may have radii less than 10 nm.

The need for sharp tips is normally explained in terms of tip convolution. This is a broad term, often used incorrectly, to identify several effects on transverse AFM resolution due to the finite size of the probe tip and the non-ideal interactions between probe and SUT.

The main influences are

• Geometrical broadening
• Longitudinal compression
• Finite range of interaction forces
• Non symmetric aspect ratio

References

 

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