W. Richard Bowen and Nidal Hilal 4

will be dominated by the deflection of the softer lever. It has been suggested
that one lever should not have a spring constant greater than the
other by more than a factor of three [109] for this method to be effective.
This is only a selection of some of the more commonly used techniques.
There are a large number of other methods used to determine
spring constants of AFM cantilevers listed in the literature. These include,
in no especial order, the measurement of the dynamic response of cantilevers
with colloidal spheres attached in a viscous fluid [110, 111]; calculating
k of levers on a chip, based on geometry, compared with a lever on
the same chip calibrated by another method [97]; an alternative ‘thermal
method’ with k calculated from the resonant frequency, Q factor and the
squared resonance amplitude [112]; and measuring the static deflection
of an AFM cantilever due to a known end-loaded mass [113].
.6.2 Calibration of Torsional and lateral spring Constants
To extract quantitative data from measurements of lateral forces experienced
by the probe when scanning across a surface, such as in friction
force microscopy, then the stiffness of the cantilever in the lateral or torsional
mode needs to be ascertained. However, this is much less straightforward
than calibrating the normal spring constant of the lever. For
frictional measurements, as well as knowledge of the normal and torsional
spring constants, knowledge of the lateral response of the deflection
sensor and the geometry, height and material properties of the probe
at the region of contact with the sample need to be ascertained.
At this point the difference between the torsional and lateral stiffnesses
of the cantilevers must be made clear. The torsional spring constant k � is the
resistance to rotation along the major axis of the cantilever. The lateral spring
constant k lat on the contrary is the resistance of the lever to forces experienced
laterally at the apex of the probe tip, producing a rotation at the base
of the probe. The two are related by the following simple formula [114]:
kΦ
klat
� 2
h
(1.11)
where h is the height of the probe (usually in the region of 3 μm for most
imaging probes).
Sader described equations to calculate the approximate k� for both
beam-shaped and V-shaped cantilevers [115] from their geometries.
Torsional stiffness for a beam-shaped cantilever is:
k
φ
1.6 CALIBRATION OF AFM MICROCANTILEvERs 2
⎧⎪
⎛l
� l
3
tanh ⎜
∆ ⎞ ⎫
⎪
⎪
� v
Et w ⎪
6( 1 ) ⎪
⎝⎜
w
⎠⎟
w ⎪
� ⎨
⎪1
�
⎬
⎪
6( 1 + v)( l − ∆l)
⎪
6( 1 � v)
l � ∆l⎪
⎪
⎪
⎩⎪
⎭⎪
�1
(1.12)