W. Richard Bowen and Nidal Hilal 4

6 1. BAsIC PRINCIPLEs OF ATOMIC FORCE MICROsCOPy
.6 CalIbraTIon of afm mICroCanTIlevers
.6. Calibration of normal spring Constants
For the accurate measurement of forces, the spring constant of the cantilever
needs to be known. In particular, for forces normal to the surfaces
of interest, this is the spring constant which governs the relationship
between force and deflection in the z-direction, as opposed to the lateral
and torsional spring constants (see Section 1.4.2). Although cantilevers
are supplied with a manufacturers’ ‘nominal’ value, the actual value can
vary to a high degree, mostly due to variations in the thickness of the
levers and defects in the material of the cantilevers themselves. The cubic
relationship between thickness and the spring constant seen in equation
(1.2) means that small variations in thickness can cause significant variations
in k. Because of this variability, for force experiments the cantilevers
to be used need to be calibrated to determine a more accurate value of k.
There are now a large number of methods by which the spring constant
can be calculated, each with their own advantages and disadvantages.
Four of the most extensively used approaches are described later.
A number of methods exist which involve calculating spring constants
based upon the dimensions and geometry of the lever. Whilst calculations
for rectangular cantilevers, such as in equation (1.2), are relatively
straightforward, for V-shaped cantilevers, approximations are most often
used based upon a simplification of their geometry, e.g. Sader approximated
a V-shaped cantilever to two parallel rectangular beams [96]. This
resulted in the following equation to describe a V-shaped lever:
3
Et w ⎧ 3
4w
⎫
k � cos � 1 � 3 � 2
3 ⎨
⎪ ( cos � )
3
⎬
⎪
2l
⎩
⎪ b
⎭
⎪
�1
(1.3)
where � is the inside angle between the two arms of the V-shaped lever,
w the width of each of the lever arms parallel to the base of the lever and
b the outer width of the base of the lever (see Figure 1.5 for diagrammatic
explanation of the dimensions). This equation, as well as equation
(1.2), assumes that the point of loading of force on the cantilever will be
at the very apex. As the probe tip itself, where the loading of force actually
occurs, is often sited a short distance from the very end, this needs to
be taken into account. A simple correction may be applied based on the
length of the lever and the distance of the probe from the end of the lever
[96–98]:
k � k ( l/ ∆l) 3 (1.4)
c m
�