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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:

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:

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:

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:

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:

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.

An introduction to Atomic Force Microscopy Modes

Material Property Modes