W. Richard Bowen and Nidal Hilal 4

1.5 THE AFM As A FORCE sENsOR 5
consequence, the slope obtained from contact with an unyielding surface
provides the sensitivity of the optical lever system. By dividing the raw
deflection data by this sensitivity value, it can be converted into an actual
deflection distance. This sensitivity value is essential for the calculation
of force values from raw deflection data. This deflection value (distance)
can also then be subtracted from the z-piezo displacement to give the
actual distance travelled by the probe. An alternative method of finding
the optical lever sensitivity without needing to make hard contact with
the surface was suggested by Higgins et al. [94] (Section 1.4).
Within operational limits the AFM cantilever behaves as a linear, or
‘Hookean’, spring. As a result the magnitude of the deflection of the cantilever
can be used to calculate the force which is exerted on the cantilever
using Hooke’s law:
F � � kx
(1.1)
where F is force (N), x the deflection of the cantilever (m) and k the spring
(or force) constant of the cantilever (N m �1 ), which essentially represents
the stiffness of the cantilever. This spring constant is dependent upon the
physical properties of the lever. This is apparent from the following relation,
used to describe a rectangular, ‘diving board’ shaped lever as shown
in 1. 5 [16, 95]:
3
Et w
k � 3
4l
(1.2)
where E is the Young’s modulus of the lever and t, w and l the thickness,
width and length of the lever, respectively. However, it must be borne in
mind that none of the diving board levers are perfectly rectangular, due
to shaping of the ends of the beams and imperfections in the manufacturing
process. As a result this equation will give only an approximate value
for the stiffness of a rectangular cantilever. In the next section the need
for the methods used to effectively measure the stiffness of a cantilever
to be used for force measurements, whether it is rectangular or V-shaped,
and the reasons for variability in k between cantilevers are addressed in
greater detail.
For different applications, cantilevers with different spring constants
may be needed. For instance, for intermittent contact mode in air, particularly
stiff levers are needed to overcome capillary forces, whereas for
measurement of weak interaction forces, very soft levers are needed for
their increased force sensitivity. The most convenient ways of producing
this variation are by altering the length and or thickness of levers during
production in order to increase or decrease the lever stiffness.