Imaging and Manipulation of Biomolecules - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 112

fraction of a nanometer so that the resulting resolution is of 2-5 nm. On the contrary,
the resolution is 0:8 h in the noncontact mode, where h is the distance between the
tip and the surface. Atomic resolution can be obtained when the AFM works at low
temperatures and in high vacuum. The resolution worsens at ambient temperature
because the tip must be well separated from the surface to avoid the attraction force
of the superficial water layer.
Micromachining techniques can fabricate one million single-crystal cantilevers
per square centimeter, all ended with a tip ( Kawakatsu et al. 2002 ). The thickness
and the length of these cantilevers are in the 30-100 nm and 0:5-200m ranges,
respectively. AFMs equipped with a huge number of cantilevers that work in parallel
can save time when topographies of large surfaces are required.
By far, the most common AFM operation mode is the noncontact mode
because:

• It has atomic resolution.

• It is nondestructive.

As already mentioned, the AFM in the noncontact mode can work in the frequency
modulation, FM, or amplitude modulation, AM, modes. In the FM mode, the
cantilever signal is shifted in phase with ' and is injected further into the
piezoelectric actuator of the cantilever. The mechanical vibration of the cantilever
is modulated in frequency, and the whole AFM loop system vibrates when the
frequency f equals the resonant mechanical frequency of the cantilever f 0 ,if
' D =2. A constant cantilever amplitude A is maintained by an automatic gain
unit, which also memorizes frequency shifts. Denoting the tip-sample force gradient
as F ts , the shift in frequency is

f D f 0 . @F ts =@ z /=2K: (3.9)

Atomic resolution surface imaging was obtained in the FM mode for a large number
of materials of different kinds and even for single molecules.
In the AM mode, the cantilever is excited at a frequency f , equal or close to its
mechanical resonance frequency f 0 , and the feedback loop maintains the excitation
amplitude constant. The signal determined by the cantilever deflection is measured
and injected into a lock-in amplifier, which uses it as reference signal for the
piezoelectric actuator. The output of the lock-in amplifier acts as the feedback signal
for controlling the tip-sample distance, whereas the output phase of the amplifier is
the detected parameter. The gradient of the tip-sample force is now given by

@F ts =@ z D K.1 A 0 =A cos'/; (3.10)

where A 0 is the free oscillation amplitude and A denotes the cantilever amplitude at
frequency f .
The mechanical movement of the cantilever in the noncontact mode is treated as
that of a damped harmonic oscillator, the deflection satisfying the equation

m B d 2 z =d 2 C d z =d C Kz D F appl ; (3.11)