3.6.2 Resonant frequency
There are two very good reasons for wanting the resonant frequency of the AFM cantilever to be as high as possible: to minimise the effect of vibrations from the surroundings, and to obtain a high image acquisition rate. Given the very high resolution of the measurements they are intended for, atomic force microscopes are bound to be susceptible to the effects of air movements and vibrations in the buildings where they are sited. Building vibrations are most significant in a frequency range from about 5 Hz to 2000 Hz, and frequencies ten times higher are still audible. If the cantilever has a resonant frequency within or below this range, it will be liable to oscillate in response to the fraction of the ambient vibrations that get through the vibration-suppression systems on which the AFM is mounted. The lowest resonant frequency of the cantilever should be at least ten times greater than the highest frequency present in the ambient vibrations – let's aim for at least 200 kHz.
What is the lowest natural frequency of the kitchen-foil cantilever?
The expression for the first bending mode resonance of a rectangular section cantilever, as can be found in an engineer's handbook, is:
You can see that the formula looks reasonable: under the square root are two material properties – Young's modulus and density – which, when combined with the geometry of the structure, look a lot like the stiffness and mass we are familiar with from the mass and spring example. The thickness and length of the beam are the right way round for the effects we expect them to have on the natural frequency.
Substituting the values from the previous example for the kitchen foil, and using 2700 kg m−3 for the density of aluminium, we get:
This does not satisfy the criterion of a fundamental resonance at no less than ten times the highest frequency of the ambient vibrations. It needs to be shorter.
Furthermore, if we want to reduce the stiffness of the cantilever, the solution is not going to be to increase its length as suggested before (as this will reduce the resonant frequency) but rather to reduce its thickness and width. In fact width does not affect the resonant frequency, because both the stiffness and the mass of the cantilever are proportional to it: the resonant frequency depends on the ratio of stiffness and mass, so the width gets cancelled out. If you don't believe this, think of twanging a ruler on the edge of a desk. If you brought another one alongside it and twanged them together, you would expect an increase in volume perhaps, but not in frequency.
Estimate the resonant frequency of the cantilever shown in Figure 10 and detailed in SAQ 5. The material density is 2.3 × 103 kg m−3.
Using equation 1.9:
Indirectly, there is another factor that favours higher frequencies: the thermal noise of the cantilever due to Brownian motion is minimised by maximising the spring constant of the cantilever.
Brownian motion is the random vibration of objects that happens because of their being repeatedly struck from all directions by gas molecules. In the macro world we inhabit, this effect is far too small to notice, but it becomes significant when the object we are interested in has a small mass (such as the AFM cantilever), and when we want to accurately measure very small deflections of that object (definitely the case with the AFM cantilever).
Random vibrations of the cantilever (whose dominant source is Brownian motion) impose an ultimate limit on the spatial resolution of the measurements that can be made with a given AFM probe. This random motion is likely to be greater in the vertical (z) direction than in any other because the cantilever is most compliant in this direction.
The amplitude of this random motion is expressed as a root-mean-square (rms) value (a simple mean value would be zero, as there is as much random downward movement as upward). The mathematical expression that links the amplitude of this thermal motion to the temperature is:
where kz is the spring constant of the cantilever in the z direction and the expression in brackets that follows it is the mean square motion in the z direction (the < > brackets denote taking the mean value of the expression within them). The symbol kB on the right is Boltzmann's constant, and T is the absolute temperature. The units of this equation are in joules. If you have done any kinetic theory of gases, you might recognise the expression on the right-hand side as one-third of the kinetic energy of the gas molecules at a temperature T. So, what this expression is saying is that the gas surrounding the cantilever gives an equal amount of kinetic energy to its motion in all three directions (there are similar expressions for the x and y components of the Brownian motion, each with their own corresponding spring constant, kx and ky).
The point to note here, though, is that the higher the value of the spring constant, the lower will be the amplitude of the random motion of the cantilever. And, as shown in Section 6 Vibrations and resonance, a large spring constant (all else being equal) means a high natural frequency.
Calculate the rms amplitude of the thermal motion of the kitchen-foil cantilever at room temperature. Compare this with that of a micromachined contact-mode cantilever with a compliance of 0.01 N m−1.
The spring constant in the z direction of the kitchen-foil cantilever is 3 N m−1. Room temperature is more or less 300 K, and Boltzmann's constant is about 1.4 × 10−23 J K−1.
Putting these values into the expression gives a value of 0.037 nm, or about a fifth of a typical interatomic spacing in a solid for the kitchen-foil cantilever, and 0.65 nm (roughly three interatomic spacings) for the much softer contact-mode cantilever.
The reason that the large thermal motion of the softer cantilever in the example is not a problem is that it is used only in contact mode, and therefore is not free to vibrate (remember, it is working in the region where those powerful repulsive forces are acting. For the soft cantilever, it's like a brick wall, preventing any motion in the −z direction). Cantilevers used in the non-contact mode, where they are free to vibrate, must therefore be of the stiffer type if they are to have good resolution in the z direction.