W. Richard Bowen and Nidal Hilal 4

8 1. BAsIC PRINCIPLEs OF ATOMIC FORCE MICROsCOPy
where M is the added mass, k the cantilever spring constant and v 0 and v 1
the unloaded and loaded resonant frequencies.
Although this method can produce a value for the cantilever
spring constant to a high degree of accuracy, there are some problems.
Attaching the bead to the cantilever, especially if attached using glue,
can be a destructive process, rendering the lever unusable for further
experiments. This means that calibration must be carried out at the end
of experimental measurements. However, with sufficient care spheres
may be attached in air using capillary adhesion forces alone. In addition
errors may occur from the incorrect placement of the sphere or uncertainties
in the mass of the sphere. If the sphere is placed a short distance
away from the end of the lever, then the value obtained from this
method will be high, as k is inversely proportional to the cube of the
cantilever length l. Again, this can be simply accounted for by utilizing
equation (1.4).
The masses of spheres ordinarily used in this technique are on the
order of a nanogram, making accurate weighing problematic. As such,
masses are generally estimated from the size and density of the spheres,
which may in turn lead to measurement errors. To allow for this, measurements
may be made using several spheres of different masses. A plot of M
versus (2�v 1) �2 can then be made, which will have a slope equal to the
spring constant of the cantilever [95, 101]. The advantages of this method
are that the measurements are independent of the cantilever geometry
and material properties, and many commercially available AFMs have the
capability to measure cantilever resonant frequencies.
Another technique commonly used to quantify cantilever spring
constants is the so-called ‘thermal method’ devised by Hutter and
Bechhoefer [102, 103]. Here the area of the fundamental resonant peak
of the cantilever under ambient thermal excitation, when not in the presence
of a surface, can be used to directly calculate the spring constant of
the cantilever. The mean square deflection of the cantilever, x 2 , due to
thermal fluctuations can be related to the spring constant thus, assuming
an idealised spring behaviour:
kBT k �
2
x
(1.7)
where k B and T are Boltzmann’s constant (1.38 � 10 �23 J K �1 ) and absolute
temperature, respectively, together representing the thermal energy
of the system. Once other noise sources are subtracted from the background,
the area of the fundamental resonance peak will be equal to the
mean square displacement. However, as the cantilever is not an ideal