2. Fundamentals of AFM: Sensitivity and Resolution

2a: Intrinsic sensitivity of the optical lever

Figure 1: Optical lever detection for the AFM. The smallest deflection angle that can be detected (d/L) is limited by shot noise in the segments A and B of the photodiode.

The intrinsic sensitivity of even the simple optical lever (see Fig. 1) is more than adequate for atomic resolution. A well designed instrument, free of thermal, electronic and vibrational noise, is limited only by shot noise on the laser probe. This determines the smallest change in signal detectable on the segments of the photodiode position sensitive detector, assuming that adequate gain is available. The shot noise can be made vanishingly small with adequate laser power. A 1mW optical laser emits about 10^{15 photons/second and the corresponding shot noise is or about 107 photons/second, so that the signal to noise ratio is about 1 part in 107. Thus relative changes in position as small as this may be detected. With a 100mm lever, a motion on the order of 10pm is detectable. It is other factors, the profile of the tip, thermal fluctuations, deformation of the sample and intervening contamination and fluid layers that limit the resolution.}

2b. Limits on interaction forces - thermal noise

Sub-Angstrom resolution is only realized at low temperatures. At finite temperature, the tip undergoes thermal motion. At its simplest, this requires that the sample does more work on the tip than thermal energy. Thus, with a tip of spring constant, k N/m, the smallest resolvable vertical displacement, Dz is given approximately by

. (1)

With k_{BT on the order of 4x10^{-21J at room temperature, the best vertical resolution obtainable with a cantilever withk=1N/m is on the order of 0.5nm. For operation at finite bandwidth, it is if some interest to consider the spectral distribution of noise. An approximate description of the displacement of the cantilever is given in terms of a harmonic oscillator model as (Albrecht, Grutter et al. 1991)}}

_{^{. (2)}}

where the cantilever is represented by a harmonic oscillator of mass m, resonant frequency w0 and mechanical Q factor, Q. The Q parameterizes the sharpness of the resonance, being the ratio of the resonant frequency to the full-width at half-height of the peak. It is also the ratio of the amplitude at resonance to that at a frequency well below resonance. In air or vacuum, very high Q (on the order of thousands) can be achieved and Eqn. 2 shows that there is a large reduction of the thermal noise away from the resonant peak because of the factor 1/Q in front of the integral. Contact-mode operation senses the static deflection of the tip and therefore a signal must be passed from zero frequency up to an upper limit, wmax set by the scan-speed and resolution of the microscope. Clearly, enhanced signal to noise can be obtained at the expense of operating speed by reducing wmax and scanning more slowly.

The noise spectrum as given by Eqns. 1 and 2, forms the basis of a non-destructive method for calibrating cantilever spring constants, a thorny experimental problem (Hutter and Bechhoefer 1993; Hutter and Bechhoefer 1993; Butt and Jaschke 1995). If the detector signal is first accurately calibrated in terms of absolute cantilever displacement, then the profile given under the integral in Eqn. 2 can be fitted over a range of frequencies for which amplitude data are available. This is conveniently obtained by Fourier transforming thermal noise recorded as a function of time. The integral is then carried out on the fitted function and Eqn. 1 used to extract the spring constant from the ratio of kBT to _^.

We are concerned almost exclusively with operation in water or aqueous buffer, in which case the effective Q of the cantilever is reduced by the viscous action of the surrounding liquid to a very small value (typically three (Butt, Siedle et al. 1993)). In this case, according to Eqn. 2, noise is almost equally distributed as a function of frequency because the resonant peak is so broad. However inspection of measured amplitude vs. frequency curves (Walters, Cleveland et al. 1996) shows that Eqn. 2 is only an approximation. Other sources of mechanical motion become progressively more important at low frequencies. Lower resonant frequency is equivalent to a smaller spring constant so that the amplitude of fluctuation increases hyperbolically as w0 (and hence k) go to zero (see Eqn. 1). This is referred to as 1/f noise and its reduction by operation at a finite frequency is one of the main reasons for the use of DFM as opposed to the conventional contact mode.

2c. Limits to resolution and imaging in fluid

There are several working definitions of resolution used in AFM. One of the more rigorous approaches is based on the spectrum of Fourier components visible in images of periodic arrays. This is often not useful in biological imaging, particularly of processes involving complexes of molecules. It is often more useful to compare the full-width of an image with the known crystallographic dimensions of a structure (though this, in turn, suffers some obvious drawbacks). If such a comparison is used, it is important that some measure is given for a representative field of molecules, for ‘best’ regions of images can be entirely spurious. For example, parts of a molecule may vanish in the noise, resulting in a ‘width’ that appears to be less than the crystallographic dimension! The molecular packing is also important. A densely packed monolayer will sample the tip in quite a different manner from an isolated molecule.

The resolution of an atomic force microscope is primarily a function of the radius of curvature of the end of the tip, as discussed by many authors (Keller 1991; Vesenka, Miller et al. 1994; Williams, Davies et al. 1994). Features on the surface of greater radius of curvature than the tip itself will be mapped out as a replica of the tip, and all information about the surface in the region of high curvature is lost. One approach to correcting for tip broadening (to the extent that this is possible) is based on detection of the regions of steepest slope, using these to reconstruct the tip geometry (Villarubia 1994; Williams, Davies et al. 1994).

The physics of the tip-sample interaction is also important in determining resolution. In fact, even the bluntest of tips has a rough surface, being covered with fine asperities (Putman, Igarashi et al. 1995), so that, on a relatively flat sample the apex can appear atomically-sharp. Thus, providing one is willing to select tips, no special preparation is needed in order to find a reasonable fraction (on the order of 10%) that give particularly high resolution. Ideal and reproducible tips may not be far-off. Nanotubes have shown promise as durable, sharp tips of extremely high aspect ratio (Dai, Hafner et al. 1996) and methods for quantity production of single-walled nano-tips are expected to evolve rapidly.

Bustamante has considered an ideal ‘single-line-of-atoms’ tip (Bustamante and Keller 1995) showing how the resolution falls off with distance as the tip is scanned at increasing height above a sample. Giessibl has shown how very local interactions can give rise to atomic resolution even in the presence of long-range interactions (Giessibl 1997). Yang and co-workers have proposed that short-range interactions with a local asperity give rise to high-resolution contrast while longer-range interactions with blunter parts of the tip help support the load of the tip in contact with the sample (Yang, Mou et al. 1996). Muller and Engel (Muller and Engel 1997) have argued that, in solution, the long-range force required for repulsion of the body of the tip is electrostatic. They adjust the supporting electrolyte so that the asperity just touches the sample lightly. Thus, provided that the sample can sustain the load of a tip in contact, and provided that it is reasonably flat (so that a single asperity can dominate contrast) sub-nm scale resolution can be obtained (Schabert and Engel 1994; Mou, Czajkowsky et al. 1995).

Contact mode imaging requires densely-packed samples, both in order to provide stability against the strong interaction with the tip and to present a flat surface to the tip. It has provided the best resolution to date. Examples of such images are given in Plates I and II. Densely-packed samples are not always suitable for experiments that probe biochemical function (although processes in dense arrays of membrane proteins have been studied (Muller, Schoenberger et al. 1997). It is often desirable to image well-separated molecules bound to an underlying surface rather weakly. In this case, the sample will not survive the intrusion of contact-mode scanning and dynamic force microscopy is required. This technique is very gentle, in part because it can be operated in a non-contact mode in solution (see below). However, this is also a disadvantage because layers of semi-mobile fluid between the tip and sample degrade resolution. It has proved possible to image with about 1nm broadening (averaged over all the molecules in a field of view) of well-separated molecules (Han, Dlakic et al. 1997; Han, Lindsay et al. 1997). Examples are given in Plate I.

There is some evidence that DFM can produce atomic-scale resolution on flat-enough surfaces when imaged under fluid (Ohnesorge 1999). Thus, with increasingly reliable production of sharp tips (Dai, Hafner et al. 1996) sub-nm resolution on soft samples may become routine.

We are primarily concerned with imaging in biological buffers, and this raises the further important question of the nature of the tip-sample interaction in the presence of a fluid. We will deal with aspects of this in some detail below, but it is important to realize that the tip can (and does) image the sample via the medium of an intervening fluid layer. Rapid motion of the molecules in this layer average out any fluid structure, so the layer is not visualized during imaging, but it does affect the overall interaction between tip and sample. We will show that it is possible for the tip to sense a surface many nm from it. Resolution is reduced substantially when the tip is far from the surface. In practice, the overall oscillation amplitude is set to the smallest value that gives stable imaging (this minimizes the potential damage to the sample) and then the set-point amplitude-change is increased (increasing the resolution) to the largest value that does not damage the sample.