Pacific Nanotechnology Inc.
An introduction to Atomic Force Microscopy Modes
Paul West and Arthur Ross
Principles and Components
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.
Subsequent to those original demonstrations, many variations on this basic design have been developed. Oscillatory sensing was introduced for improved performance and for field gradient
sensitivity. While all of them measure cantilever deflection as a positioning aid, the feedback control
error signal can be derived in several ways. And exactly what constitutes a surface "profile" can
be defined variously, depending upon ones objectives.
This chapter described the basic AFM instrument and some of its possible variations.
Coordinate system.
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, as shown in Figure 2-1. The Z axis positioner thus moves the probe perpendicular to the
SUT, while the X and Y positioner move the probe in a nominal object image plane.
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:
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.
Lateral compliance
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:
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:
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:
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
Δ
k = -Δ
F/Δ
A 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
keff ), or down in an attractive regime
(smaller
keff ). 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:
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:
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:
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).
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