Atomic force microscopy
Jiří Boldyš
Outline

Motivation

Minisurvey of scanning probe microscopies

Imaging principles

Ideas about application of moment invariants

Alternative reconstruction approach

Image artifacts
Classification of blur kernel
symmetries

n-fold circular symmetry (Cn symmetry)
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Dihedral symmetry (Dn)

Radial symmetry
Common blurs
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Atmospheric – radial symmetry
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Out-of-focus – radial, cyclic or dihedral symmetry
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Motion – central symmetry
What invariants we have
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Model: g = f * h
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I(f)=I(g)

Invariance x discriminability

Invariants to kernels with Cn and Dn symmetry

Potentially we can, we have not done that - arbitrary symmetry,
arbitrary dimension
Other potential applications

Electron microscopy? – N-fold symmetrical correction elements

Atomic force microscopy (AFM)?

…

But – Is there a convolution???
Scanning probe microscopy classification

Scanning tunneling microscopy - STM

Atomic force microscopy - AFM

Electric force microscopy - EFM

Magnetic force microscopy - MFM

Scanning near-field optical microscopy - SNOM

...
Scanning tunneling microscopy
Mironov: Fundamentals of scanning probe microscopy, 2004
Scanning tunneling microscopy

1981 – Swiss scientists Gerd Binnig
and Heinrich Rohrer

Atomic resolution, simple

1986 – Nobel prize
Chen: Introduction to scanning tunneling microscopy, 1993
Scanning tunneling microscopy

The first demonstration of the atomic-resolution capability of
STM – Si(111)-7x7, Binnig, Rohrer, Gerber, Weibel, 1983
Chen: Introduction to scanning tunneling microscopy, 1993
Scanning tunneling microscopy
Chen: Introduction to scanning tunneling microscopy, 1993
Atomic force microscopy

1986, Binnig, Quate, Gerber

1989 – the first commercially available AFM
Mironov: Fundamentals of scanning probe microscopy, 2004
Magnetic force microscopy


Local magnetic properties
AFM + tip covered by a layer of ferromagnetic material with
specific magnetization
Mironov: Fundamentals of scanning probe microscopy, 2004
Atomic force microscopy in detail

Forces can be explained by e.g. van der Waals forces –
approximated by Lennard-Jones potential
Mironov: Fundamentals of scanning probe microscopy, 2004
Tip – sample force

Energy of interaction

Force – normal + lateral component

Corresponds to deflections of an elastic cantilever
Mironov: Fundamentals of scanning probe microscopy, 2004
STM vs. AFM

STM




Tunneling current drops off exponentially  spatially
confined to the frontmost atom of the tip and surface
Distance dependence is monotonic  simple feedback
scheme
Modest experimental means, excellent SNR
AFM


Force – short range + long range – less tractable as a
feedback signal
Not monotonic with distance
Giessibl, Quate: Physics Today, 2006
Beam-bounce technique
Mironov: Fundamentals of scanning probe microscopy, 2004
Feedback system
Mironov: Fundamentals of scanning probe microscopy, 2004
Examples of cantilevers

Si3N4, Si

Different spring constants and resonant frequencies
Images: Mironov, Fundamentals of scanning probe microscopy, 2004
Methods used to acquire images
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Contact vs. non-contact modes
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Contact modes
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attractive or repulsive

Balance between atomic and elastic forces
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Small stiffness – high sensitivity, gentle to the sample

Tip breakage, surface damages

Not suitable for soft samples (biological)

Constant force

Constant average distance
Mironov: Fundamentals of scanning probe microscopy, 2004
AFM image acquisition at constant
force
Mironov: Fundamentals of scanning probe microscopy, 2004
AFM image acquisition at average
distance
Mironov: Fundamentals of scanning probe microscopy, 2004
Force-distance curves – elastic
interaction
Mironov: Fundamentals of scanning probe microscopy, 2004
Force-distance curves – plastic
interaction
Mironov: Fundamentals of scanning probe microscopy, 2004
Forced oscillations of a cantilever

Better for soft samples

Reduce mechanical influence of the tip on the surface

Possible to investigate more surface properties
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Piezo-vibrator

Motion equation
Mironov: Fundamentals of scanning probe microscopy, 2004
Forced oscillations of a cantilever
Mironov: Fundamentals of scanning probe microscopy, 2004
Contactless mode of AFM cantilever
oscillations
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Small forced oscillations amplitude – 1nm
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Close to surface – additional force
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Small oscillation around z0


Presence of a gradient in the tip-surface interaction force 
Additional shift of the amplitude and phase response curves
Additional phase shift
 phase contrast AFM image
Mironov: Fundamentals of scanning probe microscopy, 2004
Contactless mode of AFM cantilever
oscillations
Mironov: Fundamentals of scanning probe microscopy, 2004
Semi-contact mode of AFM
cantilever oscillations

Before – high sensitivity and stability feedback required
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In practice often semi-contact mode

Excited near resonance frequency, amplitude 10-100nm

Working point:
Mironov: Fundamentals of scanning probe microscopy, 2004
Semi-contact mode of AFM
cantilever oscillations
Mironov: Fundamentals of scanning probe microscopy, 2004
Frequency modulation AFM

Si(111) Reactive surface →

Dynamic mode

Ultrahigh vacuum

FM-AFM – frequency modulation – introduced in 1991

First with commercial cantilevers with a limited range of
spring constants

Strong bonding energy Si-Si  large amplitude of vibrations
34nm  no atomic resolution

 small amplitudes  stiff cantilevers  dramatic
improvement in spatial resolution
Giessibl, Quate: Physics Today, 2006
From static to dynamic mode


Static approach still in use

Materials in liquids

Tip subject to wear

Large lateral forces

Absolute force measurements are noisy
Amplitude modulation AFM

Driven near fundamental resonance frequency

Less noise

Sensing variations in amplitude

Lateral forces minimized – broken contacts
Giessibl, Quate: Physics Today, 2006
From static to dynamic mode

Frequency modulation AFM

Even less noisy

Fixed amplitude

Frequency as a feedback signal

Lateral forces minimized – broken contacts

Average tip-sample force gradient
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Frequency shift

Further improvement – exploiting signal proportional to higher-order
derivative – better spatial resolution

And next – reconstruction using the frequency shift and higher-harmonic
components of the cantilever vibrations

Higher harm. can be viewed as a convolution of the nth-order derivative
of the force with some weight function
Giessibl, Quate: Physics Today, 2006
Revealing angular symmetry of
chemical bonds


Combined STM and FM AFM
Angular dependence of chemical bonding forces between
CO on copper surface Cu(111) and the terminal atom of
metallic tip

Forces depend also on angles between atoms
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Other opinions: feedback artifact or multiple-atom tips
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3D force spectroscopy used
Welker, Giessibl, Science, 2012
Revealing angular symmetry …
Welker, Giessibl, Science, 2012
Silicon (111)-(7x7) surface
Giessibl, Hembacher, Bielefeldt, Mannhart, Science, 2000
Non-expert ideas in the field of AFM

Two ways of imaging
–
Tip imaging
–
Surface imaging
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Surface symmetry

Tip symmetry


Do we need to register (align) two blurred images, or one sharp
and one blurred?
Images with different class of blur – generates new
mathematical task for us
Registration of images blurred by
kernels with different symmetry


Example:
–
Tip imaging by surface with 4-fold (C4) symmetry …
–
Followed by tip imaging (the same one) by surface with
3-fold symmetry
–
Both kill different frequencies – together we might
reconstruct them more easily and precisely
Another example:
–
Scanning the same surface with 4-fold symmetrical and
3-fold symmetrical tip (due to crystallic structure)
Is there any convolution?

We are not sensing surface height (z-coordinate) - we are
sensing force / potential energy
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Can potential energy be calculated as a 3D convolution?

What we measure is a 2D surface in the 3D potential map

Probe influences atom distribution

We are sensing force through approx. linear dependence F(z)

What if we do not measure pure force but rather frequency shift
or higher order force derivatives?
Mathematical morphology based
reconstruction


Intuitively degradation corresponds to the dilation operation in
mathematical morphology
Why not if the region where the forces are significant is << tip
size
Villarrubia, J. Res. Natl. Inst. Stand. Technol., 1997
Critical dimension AFM

Higher throughput during quality control

Not so ambitious resolution
Dahlen et al., Veeco
AFM imaging artifacts
West, Starostina, Pacific Nanotechnology, Inc.
AFM imaging artifacts
West, Starostina, Pacific Nanotechnology, Inc.
AFM imaging artifacts
West, Starostina, Pacific Nanotechnology, Inc.
AFM imaging artifacts
West, Starostina, Pacific Nanotechnology, Inc.
AFM imaging artifacts
Mates, Summer school of SPM microscopy, 2007
Thank you!
Recommended reading: Giessibl, Quate: Physics Today, Exploring
the nanoworld with atomic force microscopy, 2006