AFM lecture 2
AFM lecture 2
Rigid beam mechanics
Free end
Point mass approximation
think of cantilever as a mass
on a spring
Fixed end
spring constant kN
effective mass me
Gz
F
Hooke’s law
kN = F /Gz
Simple Harmonic Motion
..
me z = - kN z
This is beam mechanics, standard in engineering textbooks.
For a rectangular cross section we find
me
z
Implies resonant frequency of oscillation
l
kN = E w t3 / 4 l3
kN
Z0 = (kN / me)1/2
t
Effective mass ~ 1/3 actual mass
need to solve beam equation to work it out.
w
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AFM lecture 2
AFM lecture 2
Cantilever constraints - I
Cantilever constraints - II
Spring constant
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Resonant frequency
~ 10 times higher than scan rate * no.
of points (bandwidth),
~ 10 times higher than building
vibrations
i.e. > kHz
Interatomic bond - spring constant
Hydrogen bond - spring constant
Force between two electrons?
…
Given a density of Silicon of
U = 2330 kg m-3
this gives a volume of
Conclusion
spring constant kN ~ nN nm-1
V ~ …. Pm3
So microfabrication required:
‘standard’ dimensions are
t ~ 2 Pm, w ~ 30 Pm, l ~ 100 Pm
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So for the spring constants
kN ~ nN nm-1
effective mass should be
me <
to keep the resonance frequency
high enough
Thickness of the cantilever is
the critical parameter:
hardest to control and
measure, but has a big effect
kN = E w t3 / 4 l3
Q0 = (kN / 0.24 mc)1/2 / 2 3
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AFM lecture 2
AFM lecture 2
Driven oscillator
Steady state response
In reality we have driving force in TM, and damping
z
This is easily solved (1st year physics problem)
transient term
Experimental Results
steady state term
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AFM lecture 2
AFM lecture 2
Transient response
Low and high Q
Typical Q in
water < 10
air ~ 100
vacuum > 10000
Amplitude responds on a time scale of 1/J
Low
High
for low Q dynamics of oscillation poorly defined,
sensitivity poor and interaction forces high
for Q > 50 the amplitude of oscillation on resonance is given by
Acant = Adrive Q
For a cantilever with frequency 100 kHz and Q of 300 this is a time of ~ 1ms.
Bandwidth of measurement is thus ~ 1 kHz.
Scan speed is thus limited to about 1 Hz
for high Q bandwidth is too low
for resonance frequency 100 kHz, and Q 30000,
transient decay time is 0.1 s, bandwidth 10 Hz
Phase and frequency respond on a time scale of 1/Q
implies scan rate of < 0.01 Hz or image time ~ 10 hrs (512 by 512)
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AFM lecture 2
AFM lecture 2
Effect of Fts
Tip-surface forces
• Strength, range and direction of forces important
• Linear superposition so all forces important
• AFM detects total force (and force gradient)
• If long range force is just a perturbation it effectively
alters the spring constant
• Force gradient leads to a shift in resonance frequency
force
• Shift in resonance frequency alters phase (at set Q)
separation
as in MHB
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AFM lecture 2
AFM lecture 2
Fts - short range
Fts - van der Waals
• ‘chemical forces’ attractive (bonding) and
repulsive (ion cores)
• Decay length in the angstroms
• Forces ~ nN per interacting atom
• Model interactions such as
Lennard Jones
• short range and ‘long’ range (retarded)
• for AFM can approximate by a sphere
approaching a semi-infinite body giving
A tip of radius 30 nm, 0.5 nm from the surface
the force in vacuum is Fts ~ 2nN
Morse
• Very dependent on medium, for example greatly
reduced in water
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AFM lecture 2
AFM lecture 2
Fts - electrostatic / magnetic
Fts - capillary
• Both long range forces
• Sharp point acts as condensation nucleus
• Tip to surface water meniscus formed
• Force depends on partial pressure of
water vapour and tip and surface contact
angles
• Typical forces in ambient ~ 10 - 100 nN
• Trapped charges, work function differences,
applied potentials, surface charges all sources
of potential differences between tip and surface.
Typical values ~ 10-10 N
• Magnetic force – requires magnetic tip
Typical values ~ 10-11 N
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AFM lecture 2
AFM lecture 2
Contact mechanics
Fts nonconservative
• For conservative forces no energy dissipated in the sample
Contact areas
• Nonconservative forces result in hysteresis in Fts(z)
• Examples include …..
Contact pressures
If motion is sinusoidal, oscillation is stable, and the amplitude is fixed
the phase response is determined by
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Deformation of the tip – Hertz model, DMT,
JKR
energy dissipation
sample stiffness
adhesion
topography
See Garcia 2002 for more details
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AFM lecture 2
AFM lecture 2
Oscillation constraints
Forces and tm amplitudes
PROBLEM DUE TO
NONMONOTONIC DISTANCE
DEPENDENCE OF FORCE
• jump to contact occurs if
‘Average’ Fts over oscillation
• avoided by high enough oscillation amplitudes
Phase -> ~ average force gradient
• if energy is dissipated (hysteresis in Fts(z))
Large A => more likely to have a
larger repulsive interaction - more
stable in this regime
Small A can be more stable in
attractive regime
• typical oscillation amplitudes ~ 5 – 50 nm, k ~ 1 – 100 nN nm-1
Veeco SPM guide
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AFM lecture 2
AFM lecture 2
tm force curve
Phase bistability
• what it means
• how can you see it
• effect – inaccurate
topography
• what to do if you see it?
Garcia and San Paolo, PRB 61, R13381 (2000)
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AFM lecture 2
AFM lecture 2
Thermal noise
Sensitivity / Resolution
Ultimately will be limited by thermal oscillations in the cantilever
• Harmonic oscillator =>Thermal noise
• Equipartition theory
• Contact mode –
force sensitivity thermal noise at best, position ~ k-1/2 nm
• Dynamic mode –
position as above, force gradient sensitivity
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AFM lecture 2
AFM lecture 2
Beyond point mass
Calibrating the spring constant
• only first resonance
• Beam mechanics gives higher order
resonances (1, 6.25, 17.5, 34.4 ..)
• Analytical methods – continuous
approximation
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• FEMLAB
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Thermal noise
Added mass
Sader method
Reference cantilever
NEMS based
Standardised cantilevers
…..
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AFM lecture 2
AFM lecture 2
Thermal noise
Sader Method
• John Sader University of Melbourne
• Principle – frequency and spring constant
depend on thickness, measure both
• Method – measure length, width,
frequency and Q
• http://www.ampc.ms.unimelb.edu.au/afm/c
alibration.html
• Advantages
• Accuracy
• Sources of error
• Accuracy
• Caution on using the in-built thermal tune
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AFM lecture 2
AFM lecture 2
To be aware of
References and links
• spring constant – force at tip or end
• higher modes
• relation between deflection measured and
deflection in theory
• temperature effects
• tip movement in force curves – force direction
• boundary conditions
• tilt of cantilever
• ….
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SPM – The lab on a tip
SPM and Spectroscopy, Wiesendanger
Giessibl 2003
Garcia 2002
Dietler 1999
• John Sader’s website
• Veeco application note on spring constant
calibration AN94
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