Sensing of Biomolecules - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 81

y

cantilever

L

tip

V

h

x

metal

substrate electrode

Fig. 2.23 The electrically actuated cantilever

For a cantilever with length L, width W , and thickness t, the electrostatic force
F es .y/ is given by

F es .y/ D dU=dy D .V 2 =2/dC=dy; (2.37)

where C is the capacitance between the cantilever and the substrate and U D CV 2 =2
is the electrostatic energy.
In terms of the cantilever dimensions, the electrostatic force can be expressed as

F es .y/ D .V 2 =2/" 0 WL =Œy C .t="/ 2 ;

(2.38)

where " 0 and " are the electrical permittivities of the free space and the cantilever
material, respectively.
Cantilevers are fabricated from a variety of materials: semiconductors such as
Si ( Stowe et al. 1997 ) and GaAs ( Harris et al. 1996 ), metals ( Chand et al. 2000 ),
nanotubes, nanowires, and even graphene ( Zhu et al. 2011 ). The cantilevers can be
functionalized with various materials for sensing applications. MEMS cantilevers
are tens-of-m long and have thicknesses and widths of a few microns, while
nanocantilevers are few-microns long and have nanoscale thicknesses and widths.
The cantilever bending is described by the equation

d 2 y=dx 2 D M.x/= EI ; (2.39)

where E is the Young modulus of elasticity, I is the moment of inertia given by

Z

t=2 W=2

Z

y 2 dy d z D Wt 3 =12;

I D (2.40)

t=2 W=2

and M.x/ is the total moment at positionx: