W. Richard Bowen and Nidal Hilal 4

l
1.6 CALIBRATION OF AFM MICROCANTILEvERs 7
where k m is the uncorrected calculated spring constant value, k c the corrected
value and �l the distance of the centre of the base of the probe
from the apex of the lever.
The problem with calculating spring constants purely from the measured
dimensions of the lever and nominal values for the Young’s modulus is primarily
that during cantilever manufacture variability in the material properties,
particularly the Young’s modulus, of cantilevers can occur largely due
to variations in the morphology of the silicon nitride [23]. In addition, accurate
determination of lever thickness is not always very practicable. Making
accurate measurements for every lever used in experiments by SEM is time
consuming and not necessarily convenient. By measuring the resonance
behaviour of cantilevers, variability in the material properties of the cantilevers
can be at least to some extent taken into account. This leads to a more
reliable calculation for the spring constant of a cantilever surrounded by a
fluid environment, such as air, from the following relationship [99, 100]:
2 2
f i f f
k � 0. 1906�
b lQΓ
( � ) �
(1.5)
where � f is the density of the surrounding fluid; Q the quality factor (a
measure of the sharpness of the resonance peak); � i the imaginary component
of the hydrodynamic function, dependent upon the Reynolds
number of the fluid; and � f the fundamental resonance frequency of the
cantilever. Although this approach is reliable for calibrating rectangular
levers, there are not currently any reliable approximations to allow this
method to be used for the commonly used V-shaped cantilevers.
Another method, developed by Cleveland et al. [95], requires the
attachment of known masses, such as tungsten spheres, to the end of the
cantilever whilst monitoring the resultant resonant frequency change.
Measuring the position of the fundamental resonance peak of the
cantilever before and after the addition of the sphere allows the following
relationship to be used to calculate the spring constant:
2 M
k � ( 2π
) (1.6)
( 1/ v ) � ( 1/
v )
1 2
w
0 2
t
fIgure .5 Diagram
showing relevant dimensions
on a beam-shaped
AFM cantilever.