Sensing of Biomolecules - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 83

Fig. 2.24 Voltage
dependence of the normalized
deflection of a cantilever

y

max

1

2/3

0

V

V th

F es .y/ Ky D 0; (2.48)

where K D 3 EI =L 3 D Et 3 W=4L 3 is the spring constant. From Fig. 2.24 , it follows

that the equilibrium is preserved up to a normalized deflection of h=3, the balance
between the elastic and electrostatic forces being destroyed above this deflection
value, so that the cantilever falls down on the substrate electrode. This process
occurs at the threshold voltage

q 8 Kh 3 =27 WL " 0 ;

V th D (2.49)

or

q 8 EIh 3 =9" 0 L 4 W:

V th D (2.50)

More about MEMS cantilevers, in particular their fabrication methods, the deflec-
tion readout principles, functionalization, and applications, can be found in the
recent review ( Boisen et al. 2011 ).
At the nanoscale, cantilevers behave often very different in comparison to their
counterpart with microscale geometries. In Fig. 2.25 , we have displayed a CNT
cantilever indicating the forces acting on it.
The theory developed above is still valid if the van der Waals force F vdW ,which
is an attractive force significant only at the nanoscale, is included in the treatment.
Then, the equation obeyed by a CNT cantilever becomes ( Dequesnes et al. 2002 )

EI d 4 y=dx 4

D dŒF es C F vdW =dy D f es C f vdW ; (2.51)

where the electrostatic force per unit length is given by

f es D " 0 V 2 =RŒy.y C D/=R 2 log 2 f 1 C y=R C Œy.y C D/=R 2 1=2 g ; (2.52)

with R the CNT radius, D D 2R its diameter, and the van der Waals force per unit
length

f vdW / .1=h 7 /Œ1=.y C 2R/ 2

1=y 2 :

(2.53)