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Effect of tip radius on apparent feature size

Geometrical broadening. Geometrical broadening arises when the radius of curvature of the tip is comparable with, or greater than, the size of the feature being imaged. Figure 5-1 illustrates this problem. As the tip scans over the specimen, the sides of the tip make contact before the apex, and the microscope begins to respond to the feature. The effect illustrated assumes, perhaps naively, that the probe and the SUT are incompressible. This effect is sometimes called tip convolution.

Tip deconvolution/blind reconstruction. Several researchers have reported methods of partially mitigating the deleterious effects of the so-called "tip convolution" on image resolution. Particularly interesting are the techniques denoted "blind reconstruction" or "blind deconvolution," which require no information about the tip other than the image itself. Villarubbia (1997) describes avariety of algorithms for this in sufficient detail for programming. See Bottomly (1998) for an extensive bibliography.

Longitudinal compression. Longitudinal compression is distortion caused by z-axis compression of the SUT by the probe. The importance of this effect is obviously material dependent, and difficult to quantify. However studies on some soft biological polymers (such as DNA) have shown the apparent feature widths dependent upon tracking force.

To put this effect in perspective, suppose that a probe has contact area about (10 nm)², and operates with a tapping force of 1 nN (both are in the right ballpark). Recalling that 1 N/m² = 1 pascals, this is a pressure of about

Smaller probe contact areas and larger forces, of course, yield even larger pressures. Compression effects on soft samples thus should not be surprising.

Other resolution effects result from the physical nature of the probe-surface interaction. This is specifically important in vibrating modes when sensing force gradients. The quantitative nature of the broadening depends on the particular physical phenomenon in question, such as magnetization mapping, Kelvin probes that make use of fringing fields, surface capacitance, and such.

Z-Axis resolution and noise

Unless one is interested in absolute position, there are only two fundamental source of Z-axis uncertainty, both of them noise sources:

Although one might expect an effect of the quantized nature of the light being reflected by the light lever, that proves to be negligibly small.

There are, of course, non-fundamental environmental noise sources, such as man-made, and natural seismic vibration, ambient noise, airflow. As these are relatively low frequency (10s of Hz at most), they contribute little to the noisiness of the AFM deflection measurements, which are at 10s of kHz. They do contribute to uncertainty and lack of repeatability in X-Y positioning, tho this can be mitigated by well-known isolation techniques (very heavy work surfaces on elastomeric cushions, shock absorbers, draft-blocking enclosures, and the like).

Light lever shot noise. The fundamental limit on noise of the light lever deflection detector is shot noise in the detector. Moreover, and perhaps surprisingly, diffraction of the sense beam with increasing "lever" length causes the sensitivity to approach a constant, independent of length once a critical length is exceeded. The SNR is found to be

where:
λ = wavelength of laser light
η= quantum efficiency of the detector
h= Planck constant
c= speed of light
B= post detection bandwidth
g₁= geometrical factor accounting for possible bending of cantilever, ranging from 0 (fixed cantilever base) to 2 at the maximally moving end.
g₂= geometrical factor accounting for cross section of beam =

for a gaussian beam
D(X) =beam waist diameter as function of distance from its waist
z = displacement of end of cantilever

If the cantilever is assumed to move as a rigid body (not an unreasonable approximation), then g₁ = 1, and the beam deflection is 2Δz/λ, the factor of two accounting for the fact that the beam angle doubles the mechanical angle upon reflection.

The beam diameter as a function of distance is

This is a consequence of the hermite-gaussian beam structure that exits the confocal laser resonaters (see, for example, Boyd et al. (1961 and 1962), or Kogelnick and Li (1966)). The asymptotic SNR in the far field of the gaussian beam (where D is growing linearly with distance) is found from (5-2) and (5-3) to be

independent of X. The detector, of course, must be large enough in diameter to intercept substantially all of the beam at the distance used.

That threshold far-field distance is defined by the second term of (5-3), that is

For a rough idea of that distance, suppose D₀ ≈ 1 mm and λ = 780 nm. Then (5-5) gives the minimum distance as X > 4 m.

For distances smaller than (5-5), where the beam diameter is sensibly constant, the SNR does grow with X, as naive intuition would lead one to expect.

Putnam et al (1992) compare the sensitivity of several variations on the light lever, as shown in the table below. The parameters chosen for the theoretical analyses are reasonable practical values.

Note that, for a post detection bandwidth of 10 kHz, the values of ΔZ_min in the table range from about 0.008 Å to 3 Å. The high end of this range is only a few hydrogen atom diameters; even the poorest of implementations is remarkably sensitive.

Note further, that Putnam et al. also show that the performance of the light lever technique is comparable (i.e. within a factor of two or so) of that the interferometric technique, with comparable detectors. As the interferometers tend to be more complex (read: expensive), the light lever generally is to be preferred.

Light lever thermal noise. Perhaps surprisingly, the most serious limitation on AFM Z-axis performance is thermal vibrations of the cantilever itself. As is well known from elementary statistical mechanics, the equipartition theorem says that every degree of freedom of a electrical or mechanical system at absolute temperature T has average energy KT, where K_B is Boltzmann's constant ≈ 1.38*10^-23 joules per kelvin. In the case of the elementary mass-spring oscillator embodied in the simple probe-cantilever model that energy is equally divided between kinetic and potential. Thus the average kinetic energy is

Assuming that motion is approximately sinusoidal with random amplitude and phase, one finds easily from (5-6) and the fact that the average square of any sinusoid is ½, that

For typical values of this thermal noise level, one can consult the catalog of, for example, Nanosensors™, whose advertised values of k lead to the rms thermal noise amplitudes shown in Table 4:. These noise levels are larger by orders of magnitude than the light lever shot noise discussed

discussed above, and are thus the primary limitation on AFM Z-axis resolution. Their deleterious effect, however, can be partially mitigated through the used of vibrating mode techniques, which can use narrow post-detection bandwidths. The noise shown in the table has the spectrum of the cantilever resonance, whose bandwidth is f₀/Q, typically a few hundred Hz.

Other considerations

The above resolution and noise considerations are the primary limitations on AFM performance, there are other potential issues that sometimes arise:

An introduction to Atomic Force Microscopy Modes

Performance