Page 49 - Principles and Applications of NanoMEMS Physics

1. NANOELECTROMECHANICAL SYSTEMS 35

k = Ewt 3 (7)
4L 3

f = . 0 162 t E (8)
0 2
L ρ

Accordingly, various aspects, which depend on the application, must be
considered in designing the cantilever. For example, in the static AFM
mode, the spring constant must be chosen so that the beam easily deflects in
response to the tip-sample force. Thus, for k between 10N/m and
TS
100N/m, the rule is to choose k between 0.01N/m and ~5N/m, with typical
resonance frequencies of 2kHz.
On the other hand, for the dynamic AFM techniques it has been found
that, to avoid jump-to-contact, the product of the cantilever spring constant
and the vibration amplitude must exceed the maximum tip-sample attractive
force, i.e., kA > F max . This means that there is a trade-off between
response TS
cantilever stiffness and excitation drive amplitude. In other words, the spring
force pulling the cantilever away from its point of closest proximity to the
sample, must overcome the maximum attraction force. A refined criterion to
avoid jump-to-contact and which assumes the possibility of a hysteretic
F () z relationship is given by [45]:
TS

1 Q
kA > ∆ E (9)
2
2 TS π 2
where E∆ is the hysteresis energy supplied to the cantilever beam in each
TS
vibration cycle. A typical set of k, A values for FM-AFM are

=
k = 17 N / m, A 34 nm .

Typically, the AFM cantilevers are fabricated via Si or Quartz
micromachining, and the usual tip materials include Si integrated with beam,
W, Diamond, Fe, Co, Sm, CoSm permanent magnets, and Ir.