Sensing of Biomolecules - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 86

the CNT. The resonant frequency of CNTs varies from a few MHz to a few GHz
depending on the length and diameter of the nanotube. Many other details about
cantilevers, such as noise, quality factors, etc., can be found in ( Dragoman and
Dragoman 2009 ).
The oscillation frequencies of cantilevers and double-clamped NEMS are very
sensitive to an additional mass; in fact, NEMS are the most sensitive mass detectors.
The cantilevers working as mass sensors can be modeled, irrespective of their
geometry, as a damped harmonic oscillat or wit h a spring constant K, case in

which its oscillation frequency is 2f D p K=M,whereM is the cantilever mass.
Denoting by f D f 0 f , the shift of the reso nant fr equency f 0 for the

unloaded cantilever with mass M 0 ,givenby2f 0 D p K=M 0 , the additional mass
m D M M 0 is given by

m D .K=4 2 /Œ1=.f 0 f / 2 .1=f 0 / 2 ;

(2.59)

relation that can be simplified as

m=f D 2M 0 =f 0 ; (2.60)

when f f 0 .
There are astounding examples in literature for unprecedented small mass
detection with the help of cantilevers or other NEMS devices. For example, a
rectangular cantilever made of polycrystalline silicon with a thickness of 160 nm,
a width of 500 nm, and length of 4m, ended with a square paddle with an area of
1m 1m, is able to detect 0.39 attograms .1 attogram D 1 ag D 10 18 g/ via a
resonance frequency shift of about 3 kHz ( Ilic et al. 2004a ). Moreover, zeptogram
.1 zeptogram D 1 zg D 10 21 g/ or even yoctogram .10 24 g/ masses can be sensed
with CNT cantilevers in high vacuum and at low temperatures. The frequency of a
CNT sensor that can detect ultra low masses is given by ( Nishio et al. 2005 )

f D 0:32 ŒK=.m CNT C 4m/ 1=2 ; (2.61)

where the cantilever mass is m CNT D 1:4 fg.1 femtogram D 1 fg D 10 15 g/.This
cantilever has a length L D 3:6m and a frequency resolution of 10 Hz, so that
masses m less than 100 zg can be measured at room temperature.
The MEMS or NEMS mass sensor is easily transformed in a very sensitive
biological sensor by functionalization into a coupling substrate. For example, it was
found ( Ilic et al. 2004b ) that a single virus can be detected with a polycrystalline
silicon cantilever coated with an antibody solution that immobilizes an insect
baculovirus. This cantilever has the following dimensions: L D 6m, t D 150 nm,
and W D 0:5m, and its free end is ended with a paddle with an area of
1m 1m. Because the mass sensitivity of this sensor is 10 19 g Hz, it is large
enough to detect a single virus bound to the cantilever, which has a mass of
1.5 fg, the corresponding resonance frequency shift being 3.8 kHz. Other viruses, for
example, the vaccinia virus, were sensed with cantilevers ( Gupta et al. 2004 ). The