W. Richard Bowen and Nidal Hilal 4

spring, equation (1.7) is insufficient to describe its behaviour, and a number
of other factors may need to be taken into account [103–107]:
2
0. 8174s kBT 1 � 3Dtan / 2l
k �
P 1 � 2Dtan
/ l
cos
⎛ ϕ ⎞
⎜
ϕ
⎝⎜
ϕ ⎠⎟
(1.8)
where D is the tip height, s the sensitivity (in V or A m �1 ), and P the positional
noise power of the fundamental resonance peak (the area under
the fundamental resonance peak in the power spectral density (PSD)
curve), obtained from a plot of the thermal power spectrum. Thus, with
a spectrum analyser and appropriate software available with a number
of commercially available AFM instruments, spring constant calibration is
relatively straightforward as well as being relatively non-destructive.
Higgins et al. [94] suggested a novel way of finding the optical lever
sensitivity by combining this thermal method with those of Sader. By
using Sader’s method to determine a spring constant for the cantilever,
the method of Hutter and Bechhoefer could then be used to back-
calculate the optical lever sensitivity without the need to make a hard contact
with a stiff surface. This method is potentially of use where a hard
contact is undesirable, for instance when the probe is chemically functionalised
or when a particle made of some deformable material, which is likely
to significantly deform under measurement stresses, is used as a probe.
A very simple and straightforward method of calibrating the cantilever
spring constant is to press the cantilever to be calibrated against another,
reference, cantilever of known k (Figure 1.6). This could be either a macroscopic
lever [108] or another AFM microcantilever [109]. Reference
cantilevers can be obtained either commercially or by using another calibration
method or combination of other methods to determine k to a high
precision. When the two levers are pressed together, the slope of the contact
region on the force curve will be the result of the deflection of both
levers. As a result, comparison of this slope with the slope obtained when
b
w
1.6 CALIBRATION OF AFM MICROCANTILEvERs
w ′
t
α
l ′
l
2
fIgure .6 Diagrammatic
representation of a
V-shaped AFM cantilever.