7.5 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Atomic force microscopy | 7/9 | https://en.wikipedia.org/wiki/Atomic_force_microscopy | reference | science, encyclopedia | 2026-05-05T10:03:47.482245+00:00 | kb-cron |
f
m
=
−
2
π
R
γ
L
(
cos
θ
+
cos
ϕ
)
(
1
−
d
h
d
D
)
{\displaystyle f_{m}=-2\pi R\gamma _{L}(\cos \theta +\cos \phi )\left(1-{\frac {dh}{dD}}\right)}
for dry on wet,
f
m
=
−
2
π
R
γ
L
d
r
0
d
D
{\displaystyle f_{m}=-2\pi R\gamma _{L}{\frac {dr_{0}}{dD}}}
for wet on wet, where θ is the contact angle of the dry sphere and φ is the immersed angle, as shown in the figure For a conical tip, the formula becomes:
f
m
=
−
2
π
R
γ
L
tan
δ
cos
δ
(
cos
θ
+
sin
δ
)
(
h
D
)
(
1
−
d
h
d
D
)
{\displaystyle f_{m}=-2\pi R\gamma _{L}{\frac {\tan \delta }{\cos \delta }}(\cos \theta +\sin \delta )(hD)\left(1-{\frac {dh}{dD}}\right)}
for dry on wet
f
m
=
−
2
π
R
γ
L
(
1
cos
δ
+
sin
δ
)
(
r
0
)
(
d
r
0
d
D
)
{\displaystyle f_{m}=-2\pi R\gamma _{L}\left({\frac {1}{\cos \delta }}+\sin \delta \right)(r_{0})\left({\frac {dr_{0}}{dD}}\right)}
for wet on wet where δ is the half cone angle and r0 and h are parameters of the meniscus profile.
== AFM cantilever-deflection measurement ==
=== Beam-deflection measurement ===
The most common method for cantilever-deflection measurements is the beam-deflection method. In this method, laser light from a solid-state diode is reflected off the back of the cantilever and collected by a position-sensitive detector (PSD) consisting of two closely spaced photodiodes, whose output signal is collected by a differential amplifier. Angular displacement of the cantilever results in one photodiode collecting more light than the other photodiode, producing an output signal (the difference between the photodiode signals normalized by their sum), which is proportional to the deflection of the cantilever. The sensitivity of the beam-deflection method is very high, and a noise floor on the order of 10 fm Hz−1⁄2 can be obtained routinely in a well-designed system. Although this method is sometimes called the "optical lever" method, the signal is not amplified if the beam path is made longer. A longer beam path increases the motion of the reflected spot on the photodiodes, but also widens the spot by the same amount due to diffraction, so that the same amount of optical power is moved from one photodiode to the other. The "optical leverage" (output signal of the detector divided by deflection of the cantilever) is inversely proportional to the numerical aperture of the beam focusing optics, as long as the focused laser spot is small enough to fall completely on the cantilever. It is also inversely proportional to the length of the cantilever. The relative popularity of the beam-deflection method can be explained by its high sensitivity and simple operation, and by the fact that cantilevers do not require electrical contacts or other special treatments, and can therefore be fabricated relatively cheaply with sharp integrated tips.
=== Other deflection-measurement methods === Many other methods for beam-deflection measurements exist.
Piezoelectric detection – Cantilevers made from quartz (such as the qPlus configuration), or other piezoelectric materials can directly detect deflection as an electrical signal. Cantilever oscillations down to 10pm have been detected with this method. Laser Doppler vibrometry – A laser Doppler vibrometer can be used to produce very accurate deflection measurements for an oscillating cantilever (thus is only used in non-contact mode). This method is expensive and is only used by relatively few groups. Scanning tunneling microscope (STM) — The first atomic microscope used an STM complete with its own feedback mechanism to measure deflection. This method is very difficult to implement, and is slow to react to deflection changes compared to modern methods. Optical interferometry – Optical interferometry can be used to measure cantilever deflection. Due to the nanometre scale deflections measured in AFM, the interferometer is running in the sub-fringe regime, thus, any drift in laser power or wavelength has strong effects on the measurement. For these reasons optical interferometer measurements must be done with great care (for example using index matching fluids between optical fibre junctions), with very stable lasers. For these reasons optical interferometry is rarely used. Capacitive detection – Metal coated cantilevers can form a capacitor with another contact located behind the cantilever. Deflection changes the distance between the contacts and can be measured as a change in capacitance. Piezoresistive detection – Cantilevers can be fabricated with piezoresistive elements that act as a strain gauge. Using a Wheatstone bridge, strain in the AFM cantilever due to deflection can be measured. This is not commonly used in vacuum applications, as the piezoresistive detection dissipates energy from the system affecting Q of the resonance.