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# Non-contact atomic force microscopy in ultra-high vacuum: concepts & recent trends

Laurent Nony

► To cite this version:

Laurent Nony. Non-contact atomic force microscopy in ultra-high vacuum: concepts & recent trends. Doctoral. École d'été "NANOSUM" - Carry le rouet, France. 2022, pp.168. hal-04068413

HAL Id: hal-04068413

https://cnrs.hal.science/hal-04068413

Submitted on 13 Apr 2023

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Institut Matériaux Microélectronique Nanosciences Provence

# Non-contact atomic force microscopy in ultra-high vacuum : concepts & recent trends

Laurent Nony

laurent.nony@univ-amu.fr

Aix-Marseille Université

CNRS, Institut Matériaux Microélectronique et Nanosciences de Provence, UMR 7334, 13397, Marseille, France

NANOSUM, June 13th 2022, Carry le Rouet, France

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Aix*Marseille
université
Initiative d'excellence

UNIVERSITÉ
DE TOULON

ISEN
ALL IS DIGITALI
MEDITERRANEE

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Association
INSTITUTS
CARNOT
# Before starting...

* Documents & references :

* Many documents (including those slides) and references cited in that lecture are available on the AMUBox server:

https://amubox.univ-amu.fr/s/Xo5Wiqy7kPc3ddS

* Slides including the symbols :

<figure><img src="image_6.png" /></figure>

refer to **important regular articles, review articles, or books**, some of which are provided in the above link

<figure><img src="image_7.png" /></figure>

won't be addressed during that lecture due to lack of time. They remain included in the flow of slides as they are thematically relevant and may be usefull to you when you reread that lecture with a clear head.

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>
<figure><img src="image_11.png" /><figcaption>Pentacene, chemical structure</figcaption></figure>
# Pentacene: a protoypical organic semi-conductor

## Fundamental physics

Charge manipulation

<figure><img src="image_12.png" /></figure>

« Probe-based measurement of lateral single-electron transfer between individual molecules »
W. Steurer et al., Nature Comm. **6**, 8353 (2015)

## Excitonic processes

### Exciton Band Structure of Pentacene Molecular Solids: Breakdown of the Frenkel Exciton Model

R. Schuster¹,² and M. Knupfer¹

¹IFW Dresden, Postfach 270116, D-01171 Dresden, Germany

²Institut für Physik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany

H. Berger

Institute of Physics of Complex Matter, Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
(Received 28 June 2006; published 16 January 2007)

Employing inelastic electron scattering we demonstrate here the determination of the exciton dispersion in a molecular solid. The failure of the applied tight-binding description provides strong evidence for a necessary reconsideration of the traditional, Frenkel-exciton based, understanding of the lowest-lying electronic transition in organic semiconductors.

R. Schuster et al. PRL **98**, 037402 (2007)

## Applied physics

### Organic (field effect) transistors

<figure><img src="image_13.png" /></figure>

« Orientation analysis of pentacene molecules in organic field effect transistor devices using polarization-dependent Raman Spectroscopy »
B.S. Bhardwaj et al., Sci. Rep. **9**, 15149 (2019)

### Organic photovoltaics

<figure><img src="image_14.png" /></figure>

« Pentacene-assisted planarization of photo-active layers for high performance tandem organic photovoltaics »
F. Yang et al., Solar Energy **163**, 434–442 (2018)
<figure><img src="image_15.png" /><figcaption>Pentacene, chemical structure</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Pentacene, as imaged by Scanning Tunneling Microscopy</figcaption></figure>

« *The chemical structure of a molecule resolved by atomic force microscopy* »
L. Gross et al., Science **325**, 1110 (2009)
<figure><img src="image_17.png" /><figcaption>Pentacene, as imaged by non-contact Atomic Force Microscopy (nc-AFM)</figcaption></figure>

« *The chemical structure of a molecule resolved by atomic force microscopy* »
L. Gross et al., Science **325**, 1110 (2009)
<figure><img src="image_18.png" /><figcaption>Pentacene, as imaged by non-contact Atomic Force Microscopy (nc-AFM)</figcaption></figure>

« *The chemical structure of a molecule resolved by atomic force microscopy* »

L. Gross et al., *Science* **325**, 1110 (2009)

<figure><img src="image_19.png" /></figure>
Examples of high-resolution nc-AFM imaging
of molecules since 2009

<figure><img src="image_20.png" /><figcaption>L. Gross *et al.* *Nature Chem.* **2**, 821–825 (2010).</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>F. Mohn *et al.* *Nat. Nanotechnol.* **7**, 227–231 (2012).</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>A.M. Sweetman *et al.* *Nature Comm.* **5**, 3931 (2014)</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>C. Rogers *et al.* *Angew. Chem. Int. Ed.* **54**, 15143–15146 (2015).</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>N. Pavlicek *et al.* *Nat. Chem.* **7**, 623–628 (2015).</figcaption></figure>
Examples of high-resolution nc-AFM imaging
of molecules since 2009

<figure><img src="image_25.png" /><figcaption>L. Gross *et al*. *Nature Chem.* **2**, 821–825 (2010).</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>F. Mohn *et al*. *Nat. Nanotechnol.* **7**, 227–231 (2012).</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>A.M. Sweetman *et al*. *J. Am. Chem. Soc.* **136**, 5816–5820 (2014).</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>C. Rogers *et al*. *Angew. Chem. Int. Ed.* **54**, 15143–15146 (2015).</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>N. Pavlicek *et al*. *Nat. Chem.* **7**, 623–628 (2015).</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>M. Commodo *et al*. *Combustion and Flame* **205**, 154–164 (2019).</figcaption></figure>
<figure><img src="image_31.png" /></figure>

Reference / review papers & books

* "Resolving Intra- and Inter-Molecular Structure with Non-Contact Atomic Force Microscopy"

S. Jarvis, Int. J. Mol. Sci. **16**, 19936-19959 (2015)

* "Generation, manipulation and characterization of molecules by atomic force microscopy"

N. Pavlicek & L. Gross, Nature Reviews Chemistry **1** (2017)

<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /></figure>
High-resolution nc-AFM imaging: molecules,
but bare surfaces too (earlier results)...

<figure><img src="image_35.png" /><figcaption>CaF<sub>2</sub></figcaption></figure>

C. Barth *et al.*
J. Phys.: Condens. Matter **13**, 2061(2001)

<figure><img src="image_36.png" /><figcaption>NaCl islands on
Cu(111)</figcaption></figure>

R. Bennewitz *et al.*
Phys. Rev. B **13**, 2061 (2001)

<figure><img src="image_37.png" /><figcaption>KBr + e⁻</figcaption></figure>

R. Bennewitz *et al.*
Surf. Sci. **474**, L197 (2001)

<figure><img src="image_38.png" /><figcaption>NiO(001) @7K</figcaption></figure>

R. Hoffmann *et al.*
Phys. Rev. B **67**, 085402 (2003)

Insulators (ionic crystals, metal oxides)
High-resolution nc-AFM imaging: molecules,
but bare surfaces too (earlier results)...

<figure><img src="image_39.png" /><figcaption>CaF<sub>2</sub></figcaption></figure>

C. Barth *et al.*
J. Phys.: Condens. Matter **13**, 2061(2001)

<figure><img src="image_40.png" /><figcaption>NaCl islands on
Cu(111)</figcaption></figure>

R. Bennewitz *et al.*
Phys. Rev. B **13**, 2061 (2001)

<figure><img src="image_41.png" /><figcaption>KBr + e⁻</figcaption></figure>

R. Bennewitz *et al.*
Surf. Sci. **474**, L197 (2001)

<figure><img src="image_42.png" /><figcaption>NiO(001) @7K</figcaption></figure>

R. Hoffmann *et al.*
Phys. Rev. B **67**, 085402 (2003)

<figure><img src="image_43.png" /><figcaption>Cu(001)</figcaption></figure>

C. Loppacher *et al.*
Phys. Rev. B **62**, 16944 (2000)

Metals
# High-resolution nc-AFM imaging: molecules, but bare surfaces too (earlier results)...

<figure><img src="image_44.png" /><figcaption>CaF$_2$</figcaption></figure>

C. Barth et al.
J. Phys.: Condens. Matter **13**, 2061(2001)

<figure><img src="image_45.png" /><figcaption>NaCl islands on
Cu(111)</figcaption></figure>

R. Bennewitz et al.
Phys. Rev. B **13**, 2061 (2001)

<figure><img src="image_46.png" /><figcaption>KBr + e⁻</figcaption></figure>

R. Bennewitz et al.
Surf. Sci. **474**, L197 (2001)

<figure><img src="image_47.png" /><figcaption>NiO(001) @7K</figcaption></figure>

R. Hoffmann et al.
Phys. Rev. B **67**, 085402 (2003)

<figure><img src="image_48.png" /><figcaption>Cu(001)</figcaption></figure>

C. Loppacher et al.
Phys. Rev. B **62**, 16944 (2000)

<figure><img src="image_49.png" /><figcaption>InAs(110) @14K</figcaption></figure>

A. Schwarz et al.
Phys. Rev. B **61**, 2837 (2000)

<figure><img src="image_50.png" /><figcaption>Si(111)7x7 @RT & 7K</figcaption></figure>

M. Lantz et al.
Phys. Rev. Lett. **84**, 2642 (2000)

<figure><img src="image_51.png" /></figure>

F. Giessibl
Science **267**, 68 (1995)

## Semi-conductors
# Beyond « imaging »...

* Promoting / tracking chemical synthesis at the atomic scale :

<figure><img src="image_52.png" /></figure>

TBTBP

<figure><img src="image_53.png" /><figcaption>b</figcaption></figure>

$T_2$

C-C coupling

<figure><img src="image_54.png" /><figcaption>c</figcaption></figure>

<figure><img src="image_55.png" /></figure>

N-doped porous GNR

<figure><img src="image_56.png" /></figure>

Debrominated TBTBP

<figure><img src="image_57.png" /></figure>

<figure><img src="image_58.png" /></figure>

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<figure><img src="image_63.png" /></figure>
# Beyond « imaging »...

* Chemical identification at the atomic scale :

<figure><img src="image_64.png" /></figure>

Y. Sugimoto et al. *Nature* **446**, 64 (2007)

<figure><img src="image_65.png" /></figure>

## Sublattice Identification in Scanning Force Microscopy on Alkali Halide Surfaces

R. Hoffmann,<fnref n="1" />,<fnref n="2" />,* L. N. Kantorovich,<fnref n="3" /> A. Baratoff,<fnref n="1" /> H. J. Hug,<fnref n="1" />,<fnref n="4" /> and H.-J. Güntherodt<fnref n="1" />

<footnote n="1">¹National Center of Competence in Research on Nanoscale Science, Institute of Physics, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerland</footnote>

<footnote n="2">²Physikalisches Institut, Universität Karlsruhe, 76128 Karlsruhe, Germany</footnote>

<footnote n="3">³Department of Physics, School of Physical Sciences and Engineering, King's College London,
London WC2R 2LS, United Kingdom</footnote>

<footnote n="4">⁴Swiss Federal Laboratory for Materials Testing and Research, CH-8600 Dübendorf, Switzerland
(Received 17 March 2003; published 9 April 2004)</footnote>
# Beyond « imaging »...

* Atomic / molecular manipulation :

PRL 98, 106104 (2007)

PHYSICAL REVIEW LETTERS

week ending
9 MARCH 2007

## Mechanism for Room-Temperature Single-Atom Lateral Manipulations on Semiconductors using Dynamic Force Microscopy

Yoshiaki Sugimoto,<fnref n="1" /> Pavel Jelinek,<fnref n="2" /> Pablo Pou,<fnref n="3" /> Masayuki Abe,<fnref n="1" />,⁴ Seizo Morita,<fnref n="1" /> Ruben Perez,<fnref n="3" /> and Oscar Custance<fnref n="1" />

VOLUME 90, NUMBER 6

PHYSICAL REVIEW LETTERS

week ending
14 FEBRUARY 2003

## Direct Determination of the Energy Required to Operate a Single Molecule Switch

Ch. Loppacher,<fnref n="1" />,* M. Guggisberg,<fnref n="1" /> O. Pfeiffer,<fnref n="1" /> E. Meyer,<fnref n="1" /> M. Bammerlin,<fnref n="2" /> R. Lüthi,<fnref n="2" /> R. Schlittler,<fnref n="2" />
J. K. Gimzewski,<fnref n="2" />,† H. Tang,<fnref n="3" /> and C. Joachim<fnref n="3" />

<footnote n="1">¹Institute of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland</footnote>

<footnote n="2">²IBM Research Division, Zurich Research Laboratory, CH-8803 Rüschlikon, Switzerland</footnote>

<footnote n="3">³CEMES/CNRS, 29, rue Jeanne Marvig, 31055-Toulouse Cedex, France</footnote>

<footnote>(Received 11 October 2002; published 14 February 2003)</footnote>
# Beyond « imaging »...

* Atomic / molecular charge control :

« Measuring the charge state of an adatom with non-contact atomic force microscopy »
L. Gross et al., Science **324**, 1428 (2009)

<figure><img src="image_66.png" /></figure>

« Probe-based measurement of lateral single-electron transfer between individual molecules »
W. Steurer et al., Nature Comm. **6**, 8353 (2015)

<figure><img src="image_67.png" /></figure>
# Focus of that lecture

* Introduction to non-contact Atomic Force Microscopy (nc-AFM) :

– Physical concepts

– Technical concepts (instrumentation, ultra-high vacuum environment)

– Examples of recent results

– Worldwide nc-AFM groups leaders

<figure><img src="image_68.png" /></figure>

* How does nc-AFM allow such a spatial resolution?

<figure><img src="image_69.png" /></figure>
# Outline

A. Scanning Probe Microscopies: historical context

B. Ultra-High Vacuum environment

C. AFM in 4 steps (contact mode concept)

D. Dynamic AFM

E. Non-contact AFM

<figure><img src="image_70.png" /></figure>

<figure><img src="image_71.png" /></figure>
# Outline

A. Scanning Probe Microscopies: historical context

B. Ultra-High Vacuum environment

C. AFM in 4 steps (contact mode concept)

D. Dynamic AFM

E. Non-contact AFM

<figure><img src="image_72.png" /></figure>

<figure><img src="image_73.png" /></figure>
# A – Scanning Probe Micr. : historical context

## Atomic Force Microscopy (AFM) :

* Developed in 1986 by the contrivers of the **Scanning Tunneling Microscope** (STM, 1982 at IBM Zurich research lab., G. Binnig & H. Rohrer, Physics Nobel Price)

* Images of the surface of **any kind of sample** in real space with **atomic vertical and lateral resolution** by sensing **interatomic interaction forces** between a nano-tip and the sample surface

## STM

### Tunneling through a controllable vacuum gap

G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel

IBM Zurich Research Laboratory, 8803 Rüschlikon-ZH, Switzerland

(Received 30 September 1981; accepted for publication 4 November 1981)

We report on the first successful tunneling experiment with an externally and reproducibly adjustable vacuum gap. The observation of vacuum tunneling is established by the exponential dependence of the tunneling resistance on the width of the gap. The experimental setup allows for simultaneous investigation and treatment of the tunnel electrode surfaces.

PACS numbers: 73.40.Gk

Applied Physics Letters 40, 178 (1982)

### Surface Studies by Scanning Tunneling Microscopy

G. Binning, H. Rohrer, Ch. Gerber, and E. Weibel

IBM Zurich Research Laboratory, 8803 Rüschlikon-ZH, Switzerland

(Received 30 April 1982)

Error!

Surface microscopy using vacuum tunneling is demonstrated for the first time. Topographic pictures of surfaces on an *atomic scale* have been obtained. Examples of resolved monoatomic steps and surface reconstructions are shown for (110) surfaces of CaIrSn₄ and Au.

PACS numbers: 68.20.+t, 73.40.Gk

Physical Review Letters 49, 57 (1982)

## AFM

### Atomic Force Microscope

G. Binnig(a) and C. F. Quate(b)

Edward L. Ginzton Laboratory, Stanford University, Stanford, California 94305

and

Ch. Gerber(c)

IBM San Jose Research Laboratory, San Jose, California 95193

(Received 5 December 1985)

The scanning tunneling microscope is proposed as a method to measure forces as small as $10^{-18}$ N. As one application for this concept, we introduce a new type of microscope capable of investigating surfaces of insulators on an atomic scale. The atomic force microscope is a combination of the principles of the scanning tunneling microscope and the stylus profilometer. It incorporates a probe that does not damage the surface. Our preliminary results in air demonstrate a lateral resolution of 30 Å and a vertical resolution less than 1 Å.

PACS numbers: 68.35.Gy

Physical Review Letters 56, 930 (1986)

<figure><img src="image_74.png" /></figure>

<figure><img src="image_75.png" /></figure>

<figure><img src="image_76.png" /></figure>
# A – Scanning Probe Micr. : historical context

* Non-contact Atomic Force Microscopy (nc-AFM), so-called Frequency Modulation AFM :

- Developed in 1991 by the Almaden IBM research Lab. (USA)

- Regular AFM (contact mode) spin-off

## Frequency modulation detection using high-Q cantilevers for enhanced force microscope sensitivity

T. R. Albrecht, P. Grütter, D. Horne, and D. Rugar

IBM Research Division, Almaden Research Center, 650 Harry Road, San Jose, California 95120

(Received 17 September 1990; accepted for publication 28 September 1990)

A new frequency modulation (FM) technique has been demonstrated which enhances the sensitivity of attractive mode force microscopy by an order of magnitude or more. Increased sensitivity is made possible by operating in a moderate vacuum ($< 10^{-3} \text{ Torr}$), which increases the *Q* of the vibrating cantilever. In the FM technique, the cantilever serves as the frequency determining element of an oscillator. Force gradients acting on the cantilever cause instantaneous frequency modulation of the oscillator output, which is demodulated with a FM detector. Unlike conventional "slope detection," the FM technique offers increased sensitivity through increased *Q* without restricting system bandwidth. Experimental comparisons of FM detection in vacuum ($Q \sim 50\,000$) versus slope detection in air ($Q \sim 100$) demonstrated an improvement of more than 10 times in sensitivity for a fixed bandwidth. This improvement is evident in images of magnetic transitions on a thin-film CoPtCr magnetic disk. In the future, the increased sensitivity offered by this technique should extend the range of problems accessible by force microscopy.

J. Applied Physics 69(2), 668 (1991)

* First nc-AFM image with atomic resolution reported in 1995 on the Si(111)7x7 surface :

« Atomic resolution of the Si(111) 7x7 surface by atomic force microscopy »

F. Giessibl, Science 267, 68 (1995)

<figure><img src="image_77.png" /></figure>
# Outline

A. Scanning Probe Microscopies: historical context

B. Ultra-High Vacuum environment

C. AFM in 4 steps (contact mode concept)

D. Dynamic AFM

E. Non-contact AFM
# B- Ultra-High Vacuum environment

* Investigating surface physics processes down to atomic scale requires :

- To have sample preparation methods with atomic-scale control

- To perform the sample preparation within a clean environment to keep it clean over a long-enough duration (no humidity, no external contaminants...)

- To perform SPM experiments within that laps of time
# B- Ultra-High Vacuum environment

* Investigating surface physics processes down to atomic scale requires :

- To have sample preparation methods with atomic-scale control

- To perform the sample preparation within a clean environment to keep it clean over a long-enough duration (no humidity, no external contaminants...)

- To perform SPM experiments within that laps of time

<figure><img src="image_78.png" /></figure>

<figure><img src="image_79.png" /></figure>
<figure><img src="image_80.png" /></figure>

# B- Ultra-High Vacuum environment

* A good starting point : ideal gas law

<figure><img src="image_81.png" /></figure>

Gas molecules confined within a tank with a random speeds distribution defining an averaged temperature T

<figure><img src="image_82.png" /></figure>
<figure><img src="image_83.png" /></figure>

# B- Ultra-High Vacuum environment

* A good starting point : ideal gas law

<figure><img src="image_84.png" /></figure>

Gas molecules confined within a tank with a random speeds distribution defining an averaged temperature T

$$PV = nRT \quad (\text{Equ.01})$$

<figure><img src="image_85.png" /></figure>

Pressure within the tank [Pa]

<figure><img src="image_86.png" /></figure>

Volume of the tank [m³]

<figure><img src="image_87.png" /></figure>

Number of moles of gas within the tank [mol]

<figure><img src="image_88.png" /></figure>

Temperature within the tank [K]

<figure><img src="image_89.png" /></figure>

Universal gas constant: 8.314 J/(mol·K)

<figure><img src="image_90.png" /></figure>

<figure><img src="image_91.png" /></figure>

<figure><img src="image_92.png" /></figure>

<figure><img src="image_93.png" /></figure>
<figure><img src="image_94.png" /></figure>

# B- Ultra-High Vacuum environment

* A good starting point : ideal gas law

<figure><img src="image_95.png" /></figure>

Gas molecules confined within a tank with a random speeds distribution defining an averaged temperature T

$$PV = nRT \quad (\text{Equ.01})$$

Upon standard conditions of pressure and temperature:

* $P = 1013 \text{ hPa} = 1 \text{ atm.} = 1013 \text{ mbar} = 770 \text{ Torr}$

* $T = 25 \text{ °C} = 298 \text{ K}$

One mole of gas (n = 1, i.e. N = n×N_A=6.02 10²³ atoms) occupies a volume:

$$V = 24.4 \text{ l} = 24.4 \text{ dm}^3$$

Density : ~ 2.5 10¹⁹ at./cm³
<figure><img src="image_96.png" /></figure>

# B- Ultra-High Vacuum environment

* Pumping down the tank at **constant volume** and **constant temperature** :

<figure><img src="image_97.png" /></figure>
<figure><img src="image_98.png" /></figure>

# B- Ultra-High Vacuum environment

* Vacuum regimes :

Rough vacuum

Atm. pressure – $10^{-4}$ mbar

<figure><img src="image_99.png" /></figure>

density : $2.5 \times 10^{12}$ at./cm$^3$

High vacuum

$10^{-4}$ – $10^{-8}$ mbar

<figure><img src="image_100.png" /></figure>

density : $2.5 \times 10^{8}$ at./cm$^3$

Ultra-high vacuum

$10^{-8}$ – $10^{-11}$ mbar

<figure><img src="image_101.png" /></figure>

density : $2.5 \times 10^{5}$ at./cm$^3$
<figure><img src="image_102.png" /></figure>

# B- Ultra-High Vacuum environment

* Langmuir (L) : residual gas monolayer

$$1 \text{ L} := 10^{-6} \text{ Torr} \times 1 \text{ s} \sim 10^{-6} \text{ mbar} \times 1 \text{ s}$$

## High Vacuum (HV)

4 $10^{-6}$ mbar

1 residual gas monolayer
every 4 SECONDS

<figure><img src="image_103.png" /></figure>

Sample to analyze

## Ultra High Vacuum (UHV)

4 $10^{-11}$ mbar

1 residual gas monolayer
a DAY

<figure><img src="image_104.png" /></figure>
<figure><img src="image_105.png" /></figure>

# B- Ultra-High Vacuum environment

* UHV allows the samples to be kept clean at the atomic scale over couple of days, which permits SPM experiments to be performed

* From **room down to low temperature (RT/LT):**

$$PV = nRT$$

Assume a UHV & **room temperature (T₁=298 K) regime:**

* P₁ = 4 10⁻¹¹ mbar → **1 L / day**
# B- Ultra-High Vacuum environment

* UHV allows the samples to be kept clean at the atomic scale over couple of days, which permits SPM experiments to be performed

* From **room down to low temperature (RT/LT):**

$$PV = nRT$$

Assume a UHV & **room temperature (T₁=298 K) regime:**

* P₁ = 4 × 10⁻¹¹ mbar → **1 L / day**

At **liquid nitrogen (T₂ = 77 K):**

* P₂ = P₁×T₂/T₁ = 10⁻¹¹ mbar → **1 L / 4 days**

At **liquid helium (T₃ = 4 K):**

* P₃ = P₁×T₃/T₁ = 5.3 × 10⁻¹³ mbar → **1 L / 20 days**
<figure><img src="image_106.png" /></figure>

# B- Ultra-High Vacuum environment

* A prototypical UHV SPM setup:

<figure><img src="image_107.png" /></figure>

<figure><img src="image_108.png" /></figure>

<figure><img src="image_109.png" /></figure>

<figure><img src="image_110.png" /></figure>
# UHV environment : summary

* UHV ~ $10^{-10}$ mbar (NB: ambient pressure = 1013 mbar)

* Allows the samples to be kept clean at the atomic scale over couple of days, which permit SPM experiments to be performed with atomic sensitivity

* SPM experiments in UHV can either be carried out at **room temperature** ($298$ K), or **low temperature** ($LN_2$ 77 K, LHe 4 K)

* Requires a dedicated setup (UHV chambers, cryostats, pumps, preparation stages, manipulators...)
# Outline

A. Scanning Probe Microscopies: historical context

B. Ultra-High Vacuum environment

C. AFM in 4 steps (contact mode concept)

D. Dynamic AFM

E. Non-contact AFM
# C – AFM in 4 steps (contact mode concept)

<figure><img src="image_111.png" /></figure>

<figure><img src="image_112.png" /></figure>

<figure><img src="image_113.png" /></figure>

<figure><img src="image_114.png" /></figure>

<figure><img src="image_115.png" /></figure>
# C – AFM in 4 steps (contact mode concept)

<figure><img src="image_116.png" /><figcaption>## 1- The probe</figcaption></figure>
# C – AFM in 4 steps (contact mode concept)

<figure><img src="image_117.png" /></figure>

1- The probe

2- Tip-surface

interaction forces
# C – AFM in 4 steps (contact mode concept)

<figure><img src="image_118.png" /></figure>

1. The probe

2. Tip-surface
interaction forces

3. Detecting the
interaction forces
# C – AFM in 4 steps (contact mode concept)

<figure><img src="image_119.png" /></figure>

1. The probe

2. Tip-surface
interaction forces

3. Detecting the
interaction forces

4. Scanning
& feedback
# C – AFM in 4 steps (contact mode concept)

<figure><img src="image_120.png" /></figure>

1. The probe

2. Tip-surface

interaction forces

3. Detecting the

interaction forces

4. Scanning

& feedback

<figure><img src="image_121.png" /></figure>

<figure><img src="image_122.png" /></figure>

<figure><img src="image_123.png" /></figure>

<figure><img src="image_124.png" /></figure>

<figure><img src="image_125.png" /></figure>
# C – AFM in 4 steps (contact mode concept)

<figure><img src="image_126.png" /></figure>

1. The probe

2. Tip-surface
interaction forces

3. Detecting the
interaction forces

4. Scanning
& feedback

<figure><img src="image_127.png" /></figure>

<figure><img src="image_128.png" /></figure>

<figure><img src="image_129.png" /></figure>

<figure><img src="image_130.png" /></figure>

<figure><img src="image_131.png" /></figure>
# C – AFM in 4 steps (contact mode concept)

1. The probe:

a. Cantilever

b. Quartz tuning fork

2. Tip-surface interaction forces

3. Detecting the tip-surface interaction force:

a. Cantilever

b. Quartz tuning fork

4. Surface scanning and feedback
# C1- The probe

* **AFM = scanning probe technic** (nano-tip close to a surface): tip is located at the end of a micro-lever that can either be a **cantilever**, or a **Quartz Tuning Fork (QTF)**

<figure><img src="image_132.png" /></figure>

**Manufacturers:**

* Nanosensors: https://www.nanosensors.com

* Nanoandmore: https://www.nanoandmore.com

* Nanoworld: https://www.nanoworld.com/

* ...

Silicon wafer → lithography methods →
Integrated tip-cantilever mechanical system

<figure><img src="image_133.png" /></figure>

<figure><img src="image_134.png" /></figure>

<figure><img src="image_135.png" /></figure>

<figure><img src="image_136.png" /></figure>
# C1- The probe

* **AFM** = scanning probe technic (nano-tip close to a surface): tip is located at the end of a micro-lever = this can either be a cantilever, or a **Quartz Tuning Fork (QTF)**

<figure><img src="image_137.png" /></figure>

Quartz crystal (piezoelectric material) → cutting according to watch industry methods → QTF

QTF applied to nc-AFM:

* The tip consists in a PtIr or W chemically-etched wire that is glued at the end of one of the QTF's prongs

* qPlus™ sensor concept<fnref n="1" />

<figure><img src="image_138.png" /></figure>

<footnote n="1">¹F.Giessibl, Appl.PhysLett. **76**, 1470 (2000); *Science* **267**, 68 (1995)</footnote>

<footnote n="2">²A. Castellanos-Gomez et al. Scanning Probe Microscopy in Nanoscience and Nanotechnology **3**, 23-53 (2013)</footnote>
<figure><img src="image_139.png" /></figure>

Reference / review papers & books

* "The qPlus sensor, a powerful core for the atomic force microscope" F. Giessibl, Rev. Sci. Instrum. **90**, 011101 (2019)
# C1- The probe

* **AFM** = scanning probe technic (nano-tip close to a surface): tip is located at the end of a micro-lever = cantilever / QTF

* **Mechanical analogy** for the tip-microlever system : **oscillator** of stiffness k₀, resonance frequency of the fundamental bending mode f₀ subject to an interaction force F int with the sample surface

<figure><img src="image_140.png" /></figure>
# C1- The probe

* **AFM** = scanning probe technic (nano-tip close to a surface): tip is located at the end of a micro-lever = cantilever / QTF

* **Mechanical analogy** for the tip-microlever system : **oscillator** of stiffness $k_0$, resonance frequency of the fundamental bending mode $f_0$ subject to an interaction force $F_{int}$ with the sample surface

<figure><img src="image_141.png" /></figure>
# C1- The probe

* **AFM = scanning probe technic** (nano-tip close to a surface): tip is located at the end of a micro-lever = cantilever / QTF

* **Mechanical analogy** for the tip-microlever system : **oscillator** of stiffness $k_0$, resonance frequency of the fundamental bending mode $f_0$ subject to an interaction force $F_{int}$ with the sample surface

* Thus, measuring $F_{int}$ "resumes to measuring the tip deflection $z_{int}$" (upon $k_0$ calibration)

<figure><img src="image_142.png" /></figure>
# C1- The probe

* **AFM** = scanning probe technic (nano-tip close to a surface): tip is located at the end of a micro-lever = cantilever / QTF

* **Mechanical analogy** for the tip-microlever system : **oscillator**
  frequency

* **Technical implementation** of the probes for nc-AFM applications in UHV :

## Cantilevers

<figure><img src="image_143.png" /></figure>

* Mostly used at room temperature (298 K)

* Wide range of resonance frequencies

* Wide range of stiffnesses

* Detection of the tip deflection demanding in UHV
# C1- The probe

* **AFM** = scanning probe technic (nano-tip close to a surface): tip is located at the end of a micro-lever = cantilever / QTF

* **Mechanical analogy** for the tip-microlever system : **oscillator**
  frequency

* **Technical implementation** of the probes for nc-AFM applications in UHV :

## Cantilevers

<figure><img src="image_144.png" /></figure>

* Mostly used at room temperature (298 K)

* Wide range of resonance frequencies

* Wide range of stiffnesses

* Detection of the tip deflection demanding in UHV

## Quartz tuning fork

<figure><img src="image_145.png" /></figure>

* Mostly used at low temperature (4 K)

* Narrow range of resonance frequencies

* Narrow range of stiffnesses

* Ease of detection of the tip deflection in UHV

<figure><img src="image_146.png" /></figure>
# C1- The probe

* **AFM** = scanning probe technic (nano-tip close to a surface): tip is located at the end of a micro-lever = cantilever / QTF

* **Mechanical analogy** for the tip-microlever system : **oscillator**
  frequency

* **Technical implementation** of the probes for nc-AFM applications in UHV :

## Cantilevers

<figure><img src="image_147.png" /></figure>

**Orders of magnitude:**

* Material: Silicon (mostly)

* Size: (100 x 20 x 3) µm³

* Stiffness: 30 N/m ... 100 N/m

* Frequency: 50 kHz ... 1 MHz

**The tip:**

* Aspect ratio upon manufacturers

* Height: 10 µm

* Curvature radius: 5 nm (at best)

## Quartz tuning fork

<figure><img src="image_148.png" /></figure>

**Orders of magnitude:**

* Material: piezoelectric quartz

* Size: (2.0 x 1.5 x 0.2) mm³

* Stiffness: 2000 N/m

* Frequency: 30 kHz

**The tip:**

* W, or PtIr etched wire

* Height: 100 µm

* Curvature radius: 5 nm (at best)

<figure><img src="image_149.png" /></figure>

<figure><img src="image_150.png" /></figure>

<figure><img src="image_151.png" /></figure>
# C – AFM in 4 steps (contact mode concept)

1. The probe:

a. Cantilever

b. Quartz tuning fork

2. Tip-surface interaction forces

3. Detecting the tip-surface interaction force:

a. Cantilever

b. Quartz tuning fork

4. Surface scanning and feedback

<figure><img src="image_152.png" /></figure>

<figure><img src="image_153.png" /></figure>

<figure><img src="image_154.png" /></figure>
# C2- Tip-surface interaction forces

* The micro-lever deflection is induced by tip-surface interatomic forces:

* Orders of magnitude:

* ✓ Strength: 0.1 pN..10 nN

* ✓ Distances: 1 Å .. 10 nm

* Derive from conservative interaction potentials

* Nature:

* ✓ **Van der Waals** (London, Debye, Keesom) : attractive, long-range (> 1 nm, classical description)

* ✓ **Electrostatic** (due to charges or dipoles) : attractive/repulsive, long-range/short-range, but primarily attractive & long-range by capacitive influence (> 10 nm, classical description)

* ✓ **Chemical** (interatomic) : attractive/repulsive, short-range (~ Å, classical / quantum description)

* ✓ **Magnetic** : attractive/repulsive, long-range (> 10 nm, classical description)
<figure><img src="image_155.png" /></figure>

Reference / review papers & books

* "Scanning probe microscopy: the lab on a tip" (E.Meyer, H.J. Hug, R. Bennewitz, Springer-Verlag Berlin Heidelberg 2004)

* "Intermolecular and surface forces, 3rd edition" (J.N.Israelachvili, Elsevier 2011)

<figure><img src="image_156.png" /></figure>

<figure><img src="image_157.png" /></figure>

<figure><img src="image_158.png" /></figure>

<figure><img src="image_159.png" /></figure>

<figure><img src="image_160.png" /></figure>

<figure><img src="image_161.png" /></figure>
C2- Tip-surface interaction forces

<figure><img src="image_162.png" /></figure>

¹Adapted from PhD thesis S.Burke, Mac Gill Univ., 2008
C2- Tip-surface interaction forces

<figure><img src="image_163.png" /></figure>

<figure><img src="image_164.png" /></figure>

<footnote n="1">¹Adapted from PhD thesis S.Burke, Mac Gill Univ., 2008</footnote>
C2- Tip-surface interaction forces

<figure><img src="image_165.png" /></figure>

¹Adapted from PhD thesis S.Burke, Mac Gill Univ., 2008

<figure><img src="image_166.png" /></figure>
# C2- Tip-surface interaction forces

* The micro-lever deflection is induced by tip-surface interatomic forces:

* Orders of magnitude:

* ✓ Strength: 1 pN..10 nN

* ✓ Distances: 1 Å .. 10 nm

* Derive from conservative interaction potentials

* Nature:

* ✓ **Van der Waals** (London, Debye, Keesom) : attractive, long-range (> 1 nm, classical description)

* ✓ **Electrostatic** (due to charges or dipoles) : attractive/repulsive, long-range/short-range, but primarily attractive & long-range by capacitive influence (> 10 nm, classical description)

* ✓ **Chemical** (interatomic) : attractive/repulsive, short-range (~ Å, classical / quantum description)

* ✓ **Magnetic** : attractive/repulsive, long-range (> 10 nm, classical description)
C2- Tip-surface interaction forces

<figure><img src="image_167.png" /></figure>
C2- Tip-surface interaction forces

* VdW forces are attractive forces and arise between face to face surfaces in the nm range and beyond

<figure><img src="image_168.png" /></figure>

<figure><img src="image_169.png" /></figure>
C2- Tip-surface interaction forces

* VdW forces are attractive forces and arise between face to face surfaces in the nm range and beyond

<figure><img src="image_170.png" /></figure>

<figure><img src="image_171.png" /></figure>
C2- Tip-surface interaction forces

<figure><img src="image_172.png" /></figure>

* VdW forces are attractive forces and arise between face to face surfaces in the nm range and beyond

* They stem from permanent (Debye, Keesom), and/or fluctuating (London = dispersive term), atomic dipoles

* They are universal (always occur, any type of atom)

* They result from collective effects (pairwise integration over the atoms of both surfaces), leading to the concept of Hamaker constant H of an interface

* Long-range character

* E.g. sphere (tip of radius R)/planar surface interaction force:

$$F_{vdW}(z) = -\frac{HR}{6z^2} \quad (\text{Equ.02})$$
<figure><img src="image_173.png" /></figure>

# C2- Tip-surface interaction forces

* The micro-lever deflection is induced by tip-surface interatomic forces:

* Orders of magnitude:

* ✓ Strength: 1 pN..10 nN

* ✓ Distances: 1 Å .. 10 nm

* Derive from conservative interaction potentials

* Nature:

* ✓ *Van der Waals* (London, Debye, Keesom) : attractive, long-range (> 1 nm, classical description)

* ✓ **Electrostatic** (due to charges or dipoles) : attractive/repulsive, long-range/short-range, but primarily attractive & long-range by capacitive influence (> 10 nm, classical description)

* ✓ **Chemical** (interatomic) : attractive/repulsive, short-range (~ Å, classical / quantum description)

* ✓ **Magnetic** : attractive/repulsive, long-range (> 10 nm, classic description)
C2- Tip-surface interaction forces

<figure><img src="image_174.png" /></figure>

<figure><img src="image_175.png" /></figure>

<figure><img src="image_176.png" /></figure>

<figure><img src="image_177.png" /></figure>
C2- Tip-surface interaction forces

* Electrostatic forces are mostly attractive forces and arise between charged objects / surfaces in the nm range and beyond

* They stem from local charges (atomic adsorbates, )

<figure><img src="image_178.png" /></figure>

<figure><img src="image_179.png" /></figure>

<figure><img src="image_180.png" /></figure>

<figure><img src="image_181.png" /></figure>

<figure><img src="image_182.png" /></figure>

<figure><img src="image_183.png" /></figure>
C2- Tip-surface interaction forces

<figure><img src="image_184.png" /></figure>

* Electrostatic forces are mostly attractive forces and arise between charged objects / surfaces in the nm range and beyond

* They stem from local charges (atomic adsorbates, )

<figure><img src="image_185.png" /></figure>

<figure><img src="image_186.png" /></figure>

<figure><img src="image_187.png" /></figure>

<figure><img src="image_188.png" /></figure>

<figure><img src="image_189.png" /></figure>
C2- Tip-surface interaction forces

<figure><img src="image_190.png" /></figure>

* Electrostatic forces are mostly attractive forces and arise between charged objects / surfaces in the nm range and beyond

* They stem from local charges (atomic adsorbates, defects)

<figure><img src="image_191.png" /></figure>
C2- Tip-surface interaction forces

<figure><img src="image_192.png" /></figure>

* Electrostatic forces are mostly attractive forces and arise between charged objects / surfaces in the nm range and beyond

* They stem from local charges (atomic adsorbates, defects) and/or dipoles (heteroatoms, molecules) and are tuned by the bias voltage applied in the interface

<figure><img src="image_193.png" /></figure>
C2- Tip-surface interaction forces

<figure><img src="image_194.png" /></figure>

* Electrostatic forces are mostly attractive forces and arise between charged objects / surfaces in the nm range and beyond

* They stem from local charges (atomic adsorbates, defects) and/or dipoles (heteroatoms, molecules) and are tuned by the bias voltage applied in the interface

* They influence the interface by inducing image charges in the tip and the surface yielding capacitive forces as well as additional components that are accounted for in the framework of electrostatic multipolar analysis

* Mostly long-range character, but may also exhibit short-range effects

* Require fine modelization

<figure><img src="image_195.png" /></figure>
<figure><img src="image_196.png" /></figure>

Reference / review papers & books

* "Polarization effects in noncontact atomic force microscopy: A key to model the tip-sample interaction above charged adatoms"

F. Bocquet et al., Phys. Rev. B **83**, 035411 (2011)

* "Multiscale approach for simulations of Kelvin probe force microscopy with atomic resolution"

A. Sadeghi et al., Phys. Rev. B **86**, 075407 (2012)

* "The weight function for charges—A rigorous theoretical concept for Kelvin probe force microscopy"

H. Söngen et al., J. Appl. Phys. **119**, 025304 (2016)
# C2- Tip-surface interaction forces

* The micro-lever deflection is induced by tip-surface interatomic forces:

* Orders of magnitude:

* ✓ Strength: 1 pN..10 nN

* ✓ Distances: 1 Å .. 10 nm

* Derive from conservative interaction potentials

* Nature:

* ✓ *Van der Waals (London, Debye, Keesom)* : attractive, long-range (> 1 nm, classical description)

* ✓ *Electrostatic (due to charges or dipoles)* : attractive/repulsive, long-range/short-range, but primarily attractive & long-range by capacitive influence (> 10 nm, classical description)

* ✓ *Chemical (interatomic)* : attractive/repulsive, short-range (~ Å, classical / quantum description)

* ✓ *Magnetic* : attractive/repulsive, long-range (> 10 nm, classical description)
C2- Tip-surface interaction forces

<figure><img src="image_197.png" /></figure>

<figure><img src="image_198.png" /></figure>

<figure><img src="image_199.png" /></figure>

<figure><img src="image_200.png" /></figure>
C2- Tip-surface interaction forces

* Chemical interatomic forces start prevailing few Å above the surface

<figure><img src="image_201.png" /></figure>

<figure><img src="image_202.png" /></figure>
C2- Tip-surface interaction forces

* Chemical interatomic forces start prevailing few Å above the surface

* They derive from interatomic interaction potentials (Morse, Lennard-Jones) with a short-range character

<figure><img src="image_203.png" /></figure>
C2- Tip-surface interaction forces

* Chemical interatomic forces start prevailing few Å above the surface

* They derive from interatomic interaction potentials (Morse, Lennard-Jones) with a short-range character

* They are **attractive**

* E.g. Morse interaction:

$$F_{\text{Morse}}(z) = -U_0 \frac{\partial}{\partial z} \left( e^{-(z-r_c)/\lambda} \right)$$

<figure><img src="image_204.png" /></figure>
C2- Tip-surface interaction forces

* Chemical interatomic forces start prevailing few Å above the surface

* They derive from interatomic interaction potentials (Morse, Lennard-Jones) with a short-range character

* They are **attractive** & **repulsive** (Pauli repulsion)

* E.g. Morse interaction:

$$F_{\text{Morse}}(z) = -U_0 \frac{\partial}{\partial z} \left( e^{-(z-r_c)/\lambda} \right)$$

(Equ.03)

<figure><img src="image_205.png" /></figure>
C2- Tip-surface interaction forces

* Chemical interatomic forces start prevailing few Å above the surface

* They derive from interatomic interaction potentials (Morse, Lennard-Jones) with a short-range character

* They are **attractive** & **repulsive** (Pauli repulsion)

* E.g. Morse interaction:

$$F_{\text{Morse}}(z) = -U_0 \frac{\partial}{\partial z} \left( \begin{aligned} & e^{-(z-r_c)/\lambda} \\ & -e^{-2(z-r_c)/\lambda} \end{aligned} \right)$$

(Equ.03)

<figure><img src="image_206.png" /></figure>
C2- Tip-surface interaction forces

<figure><img src="image_207.png" /></figure>

* Chemical interatomic forces start prevailing few Å above the surface

* They derive from interatomic interaction potentials (Morse, Lennard-Jones) with a short-range character

* They are **attractive** & **repulsive** (Pauli repulsion)

* E.g. Morse interaction:

$$F_{\text{Morse}}(z) = -U_0 \frac{\partial}{\partial z} \left( \begin{aligned} & e^{-(z-r_c)/\lambda} \\ & -e^{-2(z-r_c)/\lambda} \end{aligned} \right)$$

(Equ.03)

* Those forces are primarily responsible for the high-resolution imaging in nc-AFM!
C2- Tip-surface interaction forces

<figure><img src="image_208.png" /></figure>

¹Adapted from PhD thesis S.Burke, Mac Gill Univ., 2008

<figure><img src="image_209.png" /></figure>
# C2- Tip-surface interaction forces

* Tip-surface interatomic forces : figuring out the scaling

<figure><img src="image_210.png" /></figure>

G.Binnig et al., Phys. Rev. Lett. 56, p930 (1986)

<figure><img src="image_211.png" /></figure>
# C2- Tip-surface interaction forces

* Tip-surface interatomic forces : figuring out the scaling

<figure><img src="image_212.png" /></figure>

G.Binnig et al., Phys. Rev. Lett. 56, p930 (1986)
# C2- Tip-surface interaction forces

* Tip-surface interatomic forces : figuring out the scaling

<figure><img src="image_213.png" /></figure>

G.Binnig et al., Phys. Rev. Lett. 56, p930 (1986)
# C – AFM in 4 steps (contact mode concept)

1. The probe:

a. Cantilever

b. Quartz tuning fork

2. Tip-surface interaction forces

3. Detecting the tip-surface interaction force:

a. Cantilever

b. Quartz tuning fork

4. Surface scanning and feedback
# C3- Detecting the interaction force: cantilever

<figure><img src="image_214.png" /></figure>

<figure><img src="image_215.png" /></figure>

<figure><img src="image_216.png" /></figure>

<figure><img src="image_217.png" /></figure>

<figure><img src="image_218.png" /></figure>

<figure><img src="image_219.png" /></figure>

<figure><img src="image_220.png" /></figure>
<figure><img src="image_221.png" /></figure>

# C3- Detecting the interaction force: cantilever

## 4-quadrants PhotoSensitive Detector : PSD

- 4 PN junctions side-by-side

- 4 anodes + 1 common cathode

<figure><img src="image_222.png" /></figure>

<figure><img src="image_223.png" /></figure>

<figure><img src="image_224.png" /></figure>

Ex.: PDB c-203 (Advanced photonix, Inc)

<figure><img src="image_225.png" /></figure>

TO-5 PACKAGE

<figure><img src="image_226.png" /></figure>

<figure><img src="image_227.png" /></figure>

<figure><img src="image_228.png" /></figure>

<figure><img src="image_229.png" /></figure>

Aix+Marseille
université
Initiative d'excellence

<figure><img src="image_230.png" /></figure>

UNIVERSITÉ
DE TOULON

ISEN
U. ECOL.T.
PHOTONIX

<figure><img src="image_231.png" /></figure>

<figure><img src="image_232.png" /></figure>

Association
INSTITUTS
CARNOT
<figure><img src="image_233.png" /></figure>

# C3- Detecting the interaction force: cantilever

## 4-quadrants PhotoSensitive Detector : PSD

* 4 PN junctions side-by-side

* 4 anodes + 1 common cathode

<figure><img src="image_234.png" /><figcaption>Preamplifiers</figcaption></figure>

<figure><img src="image_235.png" /></figure>

Ex.: PDB c-203 (Advanced photonix, Inc)

<figure><img src="image_236.png" /><figcaption>PACKAGE DIMENSIONS INCH [mm]</figcaption></figure>

<figure><img src="image_237.png" /><figcaption>CHIP DIMENSIONS INCH [mm]</figcaption></figure>

<figure><img src="image_238.png" /><figcaption>TO-5 PACKAGE</figcaption></figure>

<figure><img src="image_239.png" /></figure>

<figure><img src="image_240.png" /></figure>
<figure><img src="image_241.png" /></figure>

# C3- Detecting the interaction force: cantilever

* Detection process :

<figure><img src="image_242.png" /></figure>

<figure><img src="image_243.png" /></figure>
<figure><img src="image_244.png" /></figure>

# C3- Detecting the interaction force: cantilever

* Detection process :

<figure><img src="image_245.png" /></figure>
<figure><img src="image_246.png" /></figure>

# C3- Detecting the interaction force: cantilever

* Detection process :

<figure><img src="image_247.png" /></figure>

<figure><img src="image_248.png" /></figure>
<figure><img src="image_249.png" /></figure>

# C3- Detecting the interaction force: cantilever

* Detection process :

<figure><img src="image_250.png" /></figure>

$$ \text{Cantilever deflection} = \text{« Normal force »} = (A + B) - (C + D) $$

$$ \text{Cantilever torsion} = \text{« Lateral force »} = (A + D) - (B + C) $$
# C3- Detecting the interaction force: cantilever

* Detection process : the **deflection** & the **torsion** calculation is performed out of a further analog operational stage

<figure><img src="image_251.png" /></figure>
C3- Detecting the interaction force: QTF

<figure><img src="image_252.png" /></figure>

<figure><img src="image_253.png" /></figure>

* Here a simple I/V converter is enough to convert the piezoelectric current produced by the quartz upon deflection into a usefull measurable voltage

* Detection of the QTF deflection is made much easier compared to cantilevers!

<figure><img src="image_254.png" /></figure>
# C – AFM in 4 steps (contact mode concept)

1. The probe:

a. Cantilever

b. Quartz tuning fork

2. Tip-surface interaction forces

3. Detecting the tip-surface interaction force:

a. Cantilever

b. Quartz tuning fork

4. Surface scanning and feedback

<figure><img src="image_255.png" /></figure>

<figure><img src="image_256.png" /></figure>
# C4- Surface scanning & feedback

* Concept, along a single line<fnref n="1" /> :

Surface Sensing | Detection Method | Imaging

Imaging

<figure><img src="image_257.png" /></figure>

<figure><img src="image_258.png" /></figure>

<footnote n="1">¹ Movie adapted from Park systems: https://parksystems.com</footnote>
# C4- Surface scanning & feedback

* Concept, along a single line :

<figure><img src="image_259.png" /></figure>

Park
SYSTEMS

www.parkAFM.com
# C4- Surface scanning & feedback

* Image topography :

<figure><img src="image_260.png" /></figure>

<figure><img src="image_261.png" /></figure>

<figure><img src="image_262.png" /></figure>

Image: n x m pixels
(512x512 typ.)

<figure><img src="image_263.png" /></figure>

<figure><img src="image_264.png" /></figure>

<figure><img src="image_265.png" /></figure>

<figure><img src="image_266.png" /></figure>
# AFM (contact mode concept): summary

* A probe (cantilever / QTF) detects the interaction force that develops between the nanoscopic tip and the surface

* A good mechanical analogy for the probe is the damped harmonic oscillator

* The resulting force ranges typically from 0.1 pN to 10 nN. It consists in a superposition of several components (van der Waals, electrostatic, interatomic) with various ranges of distances (>10 nm, ~ nm, 0.1 nm), first attractive (non-contact), then repulsive (contact).

* Upon the sample surface topography, that force gives rise to a deflection of the probe that can be detected and used as a feedback signal to control the position of the surface while scanning and ultimately form a « topographic » image of the surface
# Outline

A. Scanning Probe Microscopies: historical context

B. Ultra-High Vacuum environment

C. AFM in 4 steps (contact mode concept)

D. Dynamic AFM

E. Non-contact AFM
# D – Dynamic AFM

<figure><img src="image_267.png" /></figure>

1- The probe

2- Tip-surface
interaction forces

3- Detecting the
interaction forces

4- Scanning
& feedback

<figure><img src="image_268.png" /></figure>

<figure><img src="image_269.png" /></figure>

<figure><img src="image_270.png" /></figure>
# D – Dynamic AFM

1. Specificities

2. Force sensitivity

3. Non-linear dynamics and modelization

4. Towards non-contact AFM

<figure><img src="image_271.png" /></figure>

<figure><img src="image_272.png" /></figure>

<figure><img src="image_273.png" /></figure>

<figure><img src="image_274.png" /></figure>
# D – Dynamic AFM

1. Specificities

2. Force sensitivity

3. Non-linear dynamics and modelization

4. Towards non-contact AFM

<figure><img src="image_275.png" /></figure>

<figure><img src="image_276.png" /></figure>

<figure><img src="image_277.png" /></figure>

<figure><img src="image_278.png" /></figure>
# D1- Dynamic AFM specificities

* **Question :**

What is the pressure that applies on a 1 Å² surface loaded with a 1 nN force?

<figure><img src="image_279.png" /></figure>

<figure><img src="image_280.png" /></figure>

<figure><img src="image_281.png" /></figure>

<figure><img src="image_282.png" /></figure>
# D1- Dynamic AFM specificities

* **Question :**

What is the pressure that applies on a 1 Å² surface loaded with a 1 nN force?

<figure><img src="image_283.png" /></figure>

$$P = \frac{F}{s} = \frac{10^{-9} N}{(10^{-10} m)^2} = 100 \text{ GPa}$$

NB: Steel's Young modulus: 200 GPa
# D1- Dynamic AFM specificities

* **Question :**

What is the pressure that applies on a 1 Å² surface loaded with a 1 nN force?

<figure><img src="image_284.png" /></figure>

$$P = \frac{F}{s} = \frac{10^{-9} N}{(10^{-10} m)^2} = 100 \text{ GPa}$$

NB: Steel's Young modulus: 200 GPa

* **Answer :** tip and/or surface damaging due to contact shear forces while scanning.

<figure><img src="image_285.png" /></figure>

G.Binnig et al., Phys. Rev. Lett. 56, p930 (1986)
# D1- Dynamic AFM specificities

* Sinusoidal excitation of the probe at a frequency equal or close to the resonance frequency of its fundamental bending mode :

* Pros:

* ✓ Lowering the shear force between tip and surface

* ✓ Increasing force sensitivity w.r.t. the contact mode

* ✓ Non-contact (attractive force mapping only)

* Cons:

* ✓ Understanding the cantilever dynamics

* ✓ Detection of the oscillation (detection bandwidth> MHz)

* ✓ Static deflection: how does the image contrast reflects $F_{int}$? Origin of the contrast?

* Measures with the tip-surface distance:

1. Oscillation frequency $f$

2. Oscillation amplitude $A$

3. Excitation amplitude $A_{exc}$

4. Phase shift between excitation and oscillation $\varphi$

* The mechanical system behaves as a forced damped harmonic oscillator (to be checked experimentally)

Fundamental bending mode

Cantilever

<figure><img src="image_286.png" /></figure>

**Orders of magnitude:**

Amplitude: 2 nm ... 30 nm

Frequency: 100 kHz ... 1 MHz

QTF

<figure><img src="image_287.png" /></figure>

**Orders of magnitude:**

Amplitude: 50 pm ... 500 pm

Frequency: 30 kHz
# D – Dynamic AFM

1. Specificities

2. Force sensitivity

3. Non-linear dynamics and modelization

4. Towards non-contact AFM

<figure><img src="image_288.png" /></figure>

<figure><img src="image_289.png" /></figure>

<figure><img src="image_290.png" /></figure>

<figure><img src="image_291.png" /></figure>
<figure><img src="image_292.png" /></figure>

# D2- Force sensitivity

* **Question** : Relevant forces in AFM are in the nN range. Are these below the minimum force an AFM can detect?

* Conceptually, the **minimum force** the apparatus may detect is the force due to the **thermal fluctuations** of the probe

* The corresponding expression is :

$$F_{\min} = \sqrt{\frac{4kk_{B}T}{\omega_{0}Q}} B \quad (\text{Equ.08})$$

* k : probe stiffness

* $k_B T$ : thermal energy (temp. T)

* $\omega_0=2\pi f_0$ : resonance pulsation

* Q: probe quality factor

* B: measurement bandwidth

* Orders of magnitude (meas. bandwidth B ~ 250 Hz) :

<table><thead><tr><th rowspan="2">Parameter</th><th colspan="2">Cantilever</th><th colspan="2">QTF</th></tr><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Temperature [K]</td><td>298</td><td>4</td><td>298</td><td>4</td></tr><tr><td>Stiffness k [N/m]</td><td>30</td><td>30</td><td>1800</td><td>1800</td></tr><tr><td>Resonance f<sub>0</sub> [kHz]</td><td>150</td><td>150</td><td>32</td><td>32</td></tr><tr><td>Q-factor (UHV)</td><td>40000</td><td>40000</td><td>40000</td><td>40000</td></tr><tr><td>Thermal force [pN]</td><td>0.06</td><td>0.007</td><td>1</td><td>0.1</td></tr></tbody></table>
<figure><img src="image_293.png" /></figure>

# D2- Force sensitivity

* Tip-surface interaction force :

<figure><img src="image_294.png" /></figure>

<figure><img src="image_295.png" /></figure>
<figure><img src="image_296.png" /></figure>

# D2- Force sensitivity

* Tip-surface interaction force :

<figure><img src="image_297.png" /></figure>
<figure><img src="image_298.png" /></figure>

# D2- Force sensitivity

* Tip-surface interaction force :

<figure><img src="image_299.png" /></figure>

<figure><img src="image_300.png" /></figure>

<figure><img src="image_301.png" /></figure>

<figure><img src="image_302.png" /></figure>
<figure><img src="image_303.png" /></figure>

# D2- Force sensitivity

* Tip-surface interaction force :

<figure><img src="image_304.png" /></figure>

<figure><img src="image_305.png" /></figure>
# D – Dynamic AFM

1. Specificities

2. Force sensitivity

3. Non-linear dynamics and modelization

4. Towards non-contact AFM

<figure><img src="image_306.png" /></figure>
# D3- Non-linear dynamics and modelization

* 5 variables : **Excitation amplitude** $A_{exc}$, **Excitation frequency** $f$ (pulsation $\omega = 2\pi f$), **Oscillation amplitude** $A$, **Phase** $\varphi$ (excitation / oscillation) and **Tip-surface distance** $D$

<figure><img src="image_307.png" /></figure>

* The probe behaves as a forced damped harmonic oscillator

* Theoretical framework for the free oscillator ($D \to \infty$, no interaction force) :

$$ 
\left\{ 
\begin{array}{l}
\text{Differential equation :} \\
\ddot{z}(t) + \frac{\omega_0}{Q}\dot{z}(t) + \omega_0^2 z(t) = \omega_0^2 A_{exc}(t) \quad (\text{Equ.04}) \\
\\
\text{Harmonic solution :} \\
z(t) = A(\omega) \cos(\omega t + \varphi(\omega)) \quad (\text{Equ.05})
\end{array}
\right. 
$$

$Q$ : Probe quality factor (hint on intrinsic dissipation)

$$ Q = 2\pi \frac{\langle E_{stored} \rangle_T}{\langle E_{diss} \rangle_T} = \frac{2\pi f_0}{\Gamma_0} \quad (\text{Equ.06}) $$
# D3- Non-linear dynamics and modelization

* 5 variables : **Excitation amplitude** $A_{exc}$, **Excitation frequency** $f$ (pulsation $\omega = 2\pi f$), **Oscillation amplitude** $A$, **Phase** $\varphi$ (excitation / oscillation) and **Tip-surface distance** $D$

<figure><img src="image_308.png" /></figure>

* The probe behaves as a forced damped harmonic oscillator

* Theoretical framework for the free oscillator ($D \to \infty$, no interaction force) :

$$ \text{Amplitude vs. frequency} \qquad a \left( u = \frac{f}{f_0} \right) = \frac{A(u)}{A_0} = \frac{1}{\sqrt{Q^2 (1-u^2)^2 + u^2}} \quad \text{(Equ.07a)} $$

$$ \text{Phase vs. frequency} \qquad \varphi(u) = \arctan \left\{ \frac{u}{Q(u^2 - 1)} \right\} \quad \text{(Equ.07b)} $$
# D3- Non-linear dynamics and modelization

* Experimental illustration<fnref n="1" /> (cantilever, f₀= 270 kHz, Q = 45000, UHV) : no interaction force with the surface (D → ∞)

<figure><img src="image_309.png" /></figure>

<footnote n="1">¹ Polesel et al., Nanotechnology 14, 1036 (2003)</footnote>
# D3- Non-linear dynamics and modelization

* Experimental illustration<fnref n="1" /> (cantilever, f₀= 270 kHz, Q = 45000, UHV) : no interaction force with the surface (D → ∞)

<figure><img src="image_310.png" /></figure>

**Notes:**

* Most of the time, the assumption according to which the probe behaves as forced damped harmonic oscillator is well-valid

* Q = 45000 & f₀ = 270 kHz → Γ₀ = 6 Hz : sharp resonances ! (UHV-forced)

<figure><img src="image_311.png" /></figure>

<footnote n="1">¹ Polesel et al., Nanotechnology 14,1036 (2003)</footnote>
# D3- Non-linear dynamics and modelization

* **Question** : how does the interaction force change the probe dynamic?

<figure><img src="image_312.png" /></figure>

<figure><img src="image_313.png" /></figure>

* **Theoretical framework** :

$$ (\text{Equ.09}) \qquad \ddot{z}(t) + \frac{\omega_0}{Q}\dot{z}(t) + \omega_0^2 z(t) = \omega_0^2 A_{\text{exc}}(t) + \left[ \frac{\omega_0^2 F_{\text{int}}(t)}{k_c} \right] : F_{\text{int}} \text{ is non-linear in } z \\ z(t) = A(\omega) \cos(\omega t + \varphi(\omega)) \quad \text{: non solution} $$

$$ A(\omega), \varphi(\omega) \rightarrow A(\omega, D), \varphi(\omega, D) \quad : D \text{ dependence} $$

<footnote n="1">¹Pérez et al., Phys Rev. B 58, 10835 (1998)</footnote>
# D3- Non-linear dynamics and modelization

* Experimental illustration<fnref n="1" /> (cantilever, f₀= 270 kHz, Q = 45000, UHV) : interaction force with the surface (D - A₀ ~ 1 nm)

<figure><img src="image_314.png" /></figure>

**Notes:**

* Hysteresis = bistable structure (typical of a non-linear dynamic)

* The interaction force shifts the resonance frequency of the probe (negatively for an attractive force and vice-versa)

<footnote n="1">¹Polesel et al., Nanotechnology 14:1036 (2003)</footnote>
# D3- Non-linear dynamics and modelization

* Modelization : variational method & least action principle<fnref n="1" />

$$ \delta S = \delta \left[ \int_{0}^{1/f} L(z, \dot{z}, t) \right] = 0 \qquad (\text{Equ.10}) $$

$$ \begin{aligned} L[z(t), \dot{z}(t)] &= T - V + W \\ &= \frac{1}{2} \frac{k_c}{(2\pi f_0)^2} z(\dot{t})^2 - \left[ \frac{1}{2} k_c z(t)^2 - z(t) k_c A_x \cos(2\pi ft) + V_c^{\text{int}}(z(t)) \right] \\ &\quad - \left[ \frac{\Gamma_0 k_c}{(2\pi f_0)^2} z(t) \underline{z}(\dot{t}) + V_d^{\text{int}}(z(t), \underline{z}(\dot{t})) \right] \end{aligned} \qquad (\text{Equ.11}) $$

* Harmonic trial function : $z(t) = A \cos(2\pi ft + \varphi)$

$$ \delta A \Leftrightarrow \cos(\varphi) = \frac{A}{A_x} \left(1 - \frac{f^2}{f_0^2}\right) + \frac{2}{k_c A_x} \langle F_c \rangle \qquad (\text{Equ.12a}) $$

$$ \delta \varphi \Leftrightarrow \sin(\varphi) = -\frac{f}{f_0} \times \frac{A}{QA_x} - \frac{2}{k_c A_x} \langle F_d \rangle \qquad (\text{Equ.12b}) $$

<footnote n="1">¹L.Nony et al. J.Chem.Phys.111,1610 (1999)</footnote>
# D3- Non-linear dynamics and modelization

* Modelization : variational method & least action principle<fnref n="1" />

$$ \delta S = \delta \left[ \int_{0}^{1/f} L(z, \dot{z}, t) \right] = 0 \qquad (\text{Equ.10}) $$

$$ \begin{aligned} L[z(t), \dot{z}(t)] &= T - V + W \\ &= \frac{1}{2} \frac{k_c}{(2\pi f_0)^2} z(\dot{t})^2 - \left[ \frac{1}{2} k_c z(t)^2 - z(t) k_c A_x \cos(2\pi ft) + V_c^{\text{int}}(z(t)) \right] \\ &\quad - \left[ \frac{\Gamma_0 k_c}{(2\pi f_0)^2} z(t) \underline{z}(\dot{t}) + V_d^{\text{int}}(z(t), \underline{z}(\dot{t})) \right] \end{aligned} \qquad (\text{Equ.11}) $$

* Harmonic trial function : $z(t) = A \cos(2\pi ft + \varphi)$

$$ \delta A \Leftrightarrow \cos(\varphi) = \frac{A}{A_x} \left(1 - \frac{f^2}{f_0^2}\right) + \frac{2}{k_c A_x} \langle F_c \rangle \qquad (\text{Equ.12a}) $$

$$ \delta \varphi \Leftrightarrow \sin(\varphi) = -\frac{f}{f_0} \times \frac{A}{QA_x} - \frac{2}{k_c A_x} \langle F_d \rangle \qquad (\text{Equ.12b}) $$

<footnote n="1">¹L.Nony et al. J.Chem.Phys.111,1610 (1999)</footnote>
# D3- Non-linear dynamics and modelization

* Coupling terms (Fourier components of the interaction force @ f) :

$$ \langle F_{\text{int}}^{c} \rangle = -f \times \int_{0}^{1/f} \frac{\partial}{\partial A} V_{\text{int}}^{c}(z(t)) dt = \frac{1}{2\pi} \int_{0}^{2\pi} F_{\text{int}}^{c}(D - A \cos(u)) \cos(u) du \quad \text{(Equ.13a)} $$

$$ \langle F_{\text{int}}^{d} \rangle = -\frac{f}{A} \int_{0}^{1/f} \frac{\partial}{\partial \varphi} V_{\text{int}}^{d}(z(t); \dot{z}(t)) dt = \frac{1}{2\pi} \int_{0}^{2\pi} F_{\text{int}}^{d}(D - A \cos(u)) \sin(u) du \quad \text{(Equ.13b)} $$
# D3- Non-linear dynamics and modelization

* Coupling terms (Fourier components of the interaction force @ f) :

$$ \langle F_{\text{int}}^{c} \rangle = -f \times \int_{0}^{1/f} \frac{\partial}{\partial A} V_{\text{int}}^{c}(z(t)) dt = \frac{1}{2\pi} \int_{0}^{2\pi} F_{\text{int}}^{c}(D - A \cos(u)) \cos(u) du \quad \text{(Equ.13a)} $$

$$ \langle F_{\text{int}}^{d} \rangle = -\frac{f}{A} \int_{0}^{1/f} \frac{\partial}{\partial \varphi} V_{\text{int}}^{d}(z(t); \dot{z}(t)) dt = \frac{1}{2\pi} \int_{0}^{2\pi} F_{\text{int}}^{d}(D - A \cos(u)) \sin(u) du \quad \text{(Equ.13b)} $$

* Interaction potential :

$$ V_{\text{int}}^{c} = V_{\text{Lr}} + V_{\text{Sr}} + V_{\text{Elec}} + V_{\text{Magn}} \quad \text{(Equ.14a)} $$

$$ V_{VdW}(t) = -\frac{HR}{6(D - z(t))} \quad \text{(Equ.14b)} $$

Long-range:
Dispersive van der Waals

$$ V_{\text{Morse}}(t) = -U_0 \left[ 2e^{-\frac{D-z(t)-r_c}{\lambda}} - e^{-\frac{2(D-z(t)-r_c)}{\lambda}} \right] \quad \text{(Equ.14c)} $$

Short-range:
Morse
# D – Dynamic AFM

1. Specificities

2. Force sensitivity

3. Non-linear dynamics and modelization

4. Towards non-contact AFM

<figure><img src="image_315.png" /></figure>
# D4- Towards non-contact AFM

* 5 variables : <u>Excitation amplitude</u> A<sub>exc</sub>, <u>Excitation frequency</u> f (pulsation ω = 2π f), <u>Oscillation amplitude</u> A, <u>Phase</u> φ (excitation / oscillation) and <u>Tip-surface distance</u> D

* The non-linear dynamics of the probe is accounted for by the system {cos(φ), sin(φ)} (cf. equs. 12)

* The variables that are kept constant in this system set the operation mode :

non-contact AFM<fnref n="1" />

<figure><img src="image_316.png" /></figure>

Phase φ (-π/2 : resonance) and
Oscillation amplitude A are constant

<figure><img src="image_317.png" /></figure>

system {cos(φ), sin(φ)} solved w.r.t f and A<sub>exc</sub>...

<footnote n="1">¹ T. Albrecht et al., J.Appl.Phys.69, 668 (1991)</footnote>
Dynamic AFM : summary

* To avoid damaging the surface and/or the tip upon contact, it is worth using *dynamic* rather than *static* AFM

* Then, the probe is brought into oscillation at a frequency close, or equal, to the one of its fundamental bending mode

* Owing to the non-linear character of the tip-surface interaction force, the probe exhibits a non-linear dynamics that may be well-understood when assuming that it behaves as forced damped harmonic oscillator

* Controlling the oscillation parameters of the probe such as its **excitation amplitude**, **oscillation amplitude**, **phase** and **frequency** set the dynamical operation mode

* In non-contact AFM, the **phase** and the **oscillation amplitude** are kept constant. The phase is set to $-\pi/2$ (resonance condition)
# Outline

A. Scanning Probe Microscopies: historical context

B. Ultra-High Vacuum environment

C. AFM in 4 steps (contact mode concept)

D. Dynamic AFM

E. Non-contact AFM

<figure><img src="image_318.png" /></figure>

<figure><img src="image_319.png" /></figure>
# E – Non-contact AFM

1. Concept

2. Connection with the modelization

3. Experimental illustrations

4. Instrumentation: PLL & amplitude regulation

5. Main worldwide nc-AFM groups (experimental)
# E – Non-contact AFM

1. Concept

2. Connection with the modelization

3. Experimental illustrations

4. Instrumentation: PLL & amplitude regulation

5. Main worldwide nc-AFM groups (experimental)
# E1- Concept

Phase (≡-π/2 : resonance) and
oscillation amplitude A₀ (resonance)
are constant

* **Measures :**

* resonance frequency shift : $\Delta f = \tilde{f}_0 - f_0$

* excitation amplitude required to keep the oscillation amplitude A₀ constant

* **Probed interaction :** most generally,
attractive interaction force (non-contact)

* **z feedback (Topography image) :**
performed to keep $\Delta f$ constant (more
conceptual than the probe deflection)

<figure><img src="image_320.png" /></figure>

<figure><img src="image_321.png" /></figure>
# E1- Concept

* Back to the feedback :

<figure><img src="image_322.png" /></figure>

<figure><img src="image_323.png" /></figure>
<figure><img src="image_324.png" /></figure>

Reference / review papers & books

* "Noncontact Atomic Force Microscopy" (S.Morita,F. Giessibl, E. Meyer, R. Wiesendanger Editors), Springer-Verlag Berlin Heidelberg, Vol.1 (2002), Vol.2 (2009), Vol.3 (2015)

<figure><img src="image_325.png" /></figure>

<figure><img src="image_326.png" /></figure>

<figure><img src="image_327.png" /></figure>

<figure><img src="image_328.png" /></figure>
# E – Non-contact AFM

1. Concept

2. Connection with the modelization

3. Experimental illustrations

4. Instrumentation: PLL & amplitude regulation

5. Main worldwide nc-AFM groups (experimental)
# E2- Connection with the modelization (1)

* Oscillation amplitude ($=A_0$) and phase ($=-\pi/2$) are forced : solving the system {$\cos(\varphi)$, $\sin(\varphi)$} (cf. equ.12a & b) w.r.t. $f \rightarrow \tilde{f}_0$ and $A_{exc}$.

* Regarding conservative interactions $\rightarrow$ frequency shift : $\Delta f = \tilde{f}_0 - f_0$

$$ \frac{\Delta f}{f_0} = - \frac{1}{2\pi k_c A_0} \int_0^{2\pi} F_{\text{int}}^{c}(D + A_0\cos(u)) \cos(u) du \quad (\text{Equ.15}) $$

$$ \text{with : } F_{\text{int}}^{c}(z) = -\frac{\partial}{\partial z} V_{\text{int}}^{c}(z) \text{ and } \left\{ \begin{array}{l} V_{\text{VdW}}(t) = -\frac{HR}{6(D-z(t))} \\ V_{\text{Morse}}(t) = -U_0 \left[ 2e^{-\frac{D-z(t)-r_c}{\lambda}} - e^{-\frac{2(D-z(t)-r_c)}{\lambda}} \right] \end{array} \right. \quad (\text{Equ.16}) $$

* Hence the analytic expression for $\Delta f$ :

$$ \frac{\Delta f}{f_0} = - \frac{1}{k_c A_0} \left[ \frac{HRA_0}{6(D^2 - A_0^2)^{3/2}} + \frac{2U_0}{\lambda} (Y_{1,1/\lambda} - Y_{1,2/\lambda}) \right] \quad (\text{Equ.17}) $$
<figure><img src="image_329.png" /></figure>

Reference / review papers & books

* "Forces and frequency shifts in atomic-resolution dynamic-force microscopy"

F.J. Giessibl, Phys. Rev. B 56(24), 16010–16015 (1997)

* "Relations between interaction force and frequency shift in large amplitude dynamic force microscopy"

U. Dürig, Appl. Phys. Lett. 75(3), 433 (1999)

* "Nonlinear dynamical properties of an oscillating tip-cantilever system in the tapping mode"

L. Nony et al., J. Chem. Phys. 111(4), 1615–1627 (1999)

* "Noncontact atomic force microscopy simulator with phase-locked-loop controlled frequency detection and excitation"

L. Nony et al., Phys. Rev. B 74, 235439, (2006)

<figure><img src="image_330.png" /></figure>

<figure><img src="image_331.png" /></figure>
# E2- Connection with the modelization (1)

* $\Delta f / f_0$ as a function of the normalized tip-surface distance $d = D / A_0$ (van der waals interaction force only):

<figure><img src="image_332.png" /></figure>

* Rapid change of $\Delta f$ with the distance (« exponential-like »)

* Similar to tunneling current in STM

* $\Delta f$ always negative with an attractive interaction force (= non-contact situation)

**Orders of magnitude (cantilevers):**

$A_0 = 5$ nm

$f_0 = 150$ kHz

$k = 30$ N/m

$\Delta f(D_2=5.55\text{nm}) = -150$ Hz

$\Delta f(D_3=5.06\text{nm}) = -4800$ Hz!

<figure><img src="image_333.png" /></figure>
<figure><img src="image_334.png" /></figure>

# E2- Connection with the modelization (2)

* **Question** : How is the information of the tip-surface interaction force concealed in the $\Delta f$ ?

* The calculation of the frequency shift can be simplified upon linearization of the interaction force by assuming small oscillation amplitudes (A ~ 50 pm << D).

$$F_{\text{int}}(t) = F_{\text{int}}(s(t)) = F_{\text{int}}(D - z(t)) \approx \left. \frac{\partial F_{\text{int}}(s)}{\partial s} \right|_{s_{\min}=D-A_0} s(t) \quad (\text{Equ.18})$$

* In this case, we can prove that the interaction-shifted resonance frequency simplifies to :

$$\tilde{f}_0 = f_0 \sqrt{1 - \frac{1}{k} \left. \frac{\partial F_{\text{int}}}{\partial s} \right|_{s_{\min}}} \quad (\text{Equ.19})$$

* Assuming $k \gg \Delta F_{\text{int}}/\partial s$, we get the linearized formula for the frequency shift :

$$\frac{\tilde{f}_0 - f_0}{f_0} = \frac{\Delta f}{f_0} = -\frac{1}{2k} \left. \frac{\partial F_{\text{int}}}{\partial s} \right|_{s_{\min}} \quad (\text{Equ.20})$$
<figure><img src="image_335.png" /></figure>

# E2- Connection with the modelization (2)

* **Question** : How is the information of the tip-surface interaction force concealed in the $\Delta f$ ?

* The calculation of the frequency shift can be simplified upon linearization of the interaction force by assuming small oscillation amplitudes (A ~ 50 pm << D).

$$F_{\text{int}}(t) = F_{\text{int}}(s(t)) = F_{\text{int}}(D - z(t)) \approx \left. \frac{\partial F_{\text{int}}(s)}{\partial s} \right|_{s_{\text{min}}=D-A_0} s(t) \quad (\text{Equ.18})$$

* In this case, we can For small oscillation amplitudes and with stiff probes such as QTF, the frequency shift is proportional to the interaction force gradient. frequency simplifies to :

$$f_0 = f_0 \sqrt{1 - \frac{k}{\partial s}} \bigg|_{s_{\min}} \quad (\text{Equ.19})$$

* Assuming $k \gg \partial F_{\text{int}}/\partial s$, we get the linearized formula for the frequency shift :

$$\frac{\tilde{f}_0 - f_0}{f_0} = \frac{\Delta f}{f_0} = -\frac{1}{2k} \left. \frac{\partial F_{\text{int}}}{\partial s} \right|_{s_{\min}} \quad (\text{Equ.20})$$
# E2- Connection with the modelization (3)

* Vertical sensitivity in the short-range regime (order of magnitude):

$$ \delta z / \delta \Delta f \sim 10 \text{ pm} / \text{Hz} $$

* The experimental control on $\Delta f$ is better than 0.1 Hz (around $f_0$ ~ 150 kHz), i.e. 1 pm vertical resolution !

* This is what is achieved at low temperature (LHe, 4K).

* At **room temperature**, the noise level of the instrument (amplitude detection...) exceeds that value. The resolution is rather 10 pm.

* This sensitivity is enough to achieve atomic resolution at room temperature too

<figure><img src="image_336.png" /></figure>

<figure><img src="image_337.png" /></figure>
# E – Non-contact AFM

1. Concept

2. Connection with the modelization

3. Experimental illustrations

4. Instrumentation: PLL & amplitude regulation

5. Main worldwide nc-AFM groups (experimental)

<figure><img src="image_338.png" /></figure>

<figure><img src="image_339.png" /></figure>

<figure><img src="image_340.png" /></figure>
E3- Experimental illustrations

* $\Delta f$ as a function of the tip-surface distance measured on Cu(111) surface (QTF, $A_0$= 50 pm, $f_0$= 23 kHz, Q= 40000, $k$= 1800 N/m):

<figure><img src="image_341.png" /></figure>

<figure><img src="image_342.png" /></figure>
# E3- Experimental illustrations

* $\Delta f$ as a function of the tip-surface distance measured on Cu(111) surface (QTF, $A_0$= 50 pm, $f_0$= 23 kHz, Q= 40000, k= 1800 N/m) :

<figure><img src="image_343.png" /></figure>

<figure><img src="image_344.png" /></figure>
# E3- Experimental illustrations

* $\Delta f$ as a function of the tip-surface distance measured on Cu(111) surface (QTF, $A_0$= 50 pm, $f_0$= 23 kHz, Q= 40000, k= 1800 N/m) :

<figure><img src="image_345.png" /></figure>

<figure><img src="image_346.png" /></figure>

<figure><img src="image_347.png" /></figure>

<figure><img src="image_348.png" /></figure>
E3- Experimental illustrations

* $\Delta f$ as a function of the tip-surface distance measured on Cu(111) surface (QTF, $A_0$= 50 pm, $f_0$= 23 kHz, Q= 40000, k= 1800 N/m) :

<figure><img src="image_349.png" /></figure>

<figure><img src="image_350.png" /></figure>
# E3- Experimental illustrations

* Back to basics...

<figure><img src="image_351.png" /></figure>

« *The chemical structure of a molecule resolved by atomic force microscopy* »
L. Gross et al., Science **325**, 1110 (2009)

<figure><img src="image_352.png" /></figure>
# E3- Experimental illustrations

* Back to basics...

<figure><img src="image_353.png" /></figure>

« *The chemical structure of a molecule resolved by atomic force microscopy* »
L. Gross et al., Science **325**, 1110 (2009)

<figure><img src="image_354.png" /></figure>
# E3- Experimental illustrations

* Back to basics...

<figure><img src="image_355.png" /></figure>

« *The chemical structure of a molecule resolved by atomic force microscopy* »
L. Gross et al., Science **325**, 1110 (2009)

<figure><img src="image_356.png" /></figure>
# E3- Experimental illustrations

* Back to basics...

<figure><img src="image_357.png" /></figure>

« The chemical structure of a molecule resolved by atomic force microscopy »
L. Gross et al., Science **325**, 1110 (2009)
# E3- Experimental illustrations

* Back to basics...

<figure><img src="image_358.png" /></figure>

« The chemical structure of a molecule resolved by atomic force microscopy »
L. Gross et al., Science **325**, 1110 (2009)
# E – Non-contact AFM

1. Concept

2. Connection with the modelization

3. Experimental illustrations

4. Instrumentation: PLL & amplitude regulation

5. Main worldwide nc-AFM groups (experimental)
<figure><img src="image_359.png" /></figure>

# E4- Instrumentation : PLL & amplitude feedback

* **Question** : how is $\Delta f$ experimentally detected / what instrumentation ?

* Answer : a Phase Locked Loop (PLL) is used:

* The PLL ensures the detection of $\Delta f$ w.r.t. a reference frequency which is fixed equal to the **resonance frequency of the free cantilever $f_0$**, therefore matching the phase condition $\varphi = -\pi/2$

<figure><img src="image_360.png" /></figure>

* Non-contact AFM = Frequency demodulation around $f_0$!
# E4- Instrumentation : PLL & amplitude feedback

* Principle for the retention of the excitation at resonance (no interaction):

<figure><img src="image_361.png" /></figure>

<figure><img src="image_362.png" /></figure>
# E4- Instrumentation : PLL & amplitude feedback

<figure><img src="image_363.png" /></figure>

* Principle for the retention of the excitation at resonance (no interaction):

<figure><img src="image_364.png" /></figure>
# E4- Instrumentation : PLL & amplitude feedback

* Principle for the retention of the excitation at resonance (with interaction $f_0 \rightarrow \tilde{f}_0$) :

<figure><img src="image_365.png" /></figure>

<figure><img src="image_366.png" /></figure>
<figure><img src="image_367.png" /></figure>

Reference / review papers & books

* "Fast digital electronics for application in dynamic force microscopy using high-Q cantilevers" Ch. Loppacher et al., Appl. Phys. A **66**, 215 (1998)

* "Noncontact atomic force microscopy simulator with phase-locked-loop controlled frequency detection and excitation" L. Nony et al., Phys. Rev. B **74**, 235439 (2006)

<figure><img src="image_368.png" /></figure>
# E – Non-contact AFM

1. Concept

2. Connection with the modelization

3. Experimental illustrations: back to basics

4. Instrumentation: PLL & amplitude regulation

5. Main worldwide nc-AFM groups (experimental)

<figure><img src="image_369.png" /></figure>

<figure><img src="image_370.png" /></figure>

<figure><img src="image_371.png" /></figure>

<figure><img src="image_372.png" /></figure>
# E5 - Main worldwide nc-AFM groups (exp.)

<figure><img src="image_373.png" /></figure>
# E5 - Main worldwide nc-AFM groups (exp.)

* Ch. Loppacher's group at IM2NP<fnref n="1" /> (AMU) : UHV RT-SPM setup

<figure><img src="image_374.png" /></figure>

<figure><img src="image_375.png" /></figure>

## VT-AFM (Scienta Omicron, 2008)<fnref n="2" />:

* RT - 77 K (flux cryostat)

* STM / Non-contact AFM:

* Beam deflec. (cantilever)

* *in situ* 4-quads preamps.

* $A_{0,min} \sim 2$ Å

* Mean Z (topo.) noise ~ 10 pm

* RHK Rev9 control unit

<footnote n="1">1. https://www.im2np.fr/fr/equipe-nanostructuration-nano</footnote>

<footnote n="2">2. DOI: 10.13140/RG.2.2.28075.39200</footnote>
# E5 - Main worldwide nc-AFM groups (exp.)

* Ch. Loppacher's group at IM2NP<fnref n="1" /> (AMU) : UHV RT-SPM setup

<figure><img src="image_376.png" /></figure>

LT-SPM Infinity (Σ Surface Science, Dec.2019)<fnref n="2" />:

* 8.7 K (cryo-free closed cycle, pulse tube cooler)

* STM / nc-AFM:

* Quartz tuning fork

* $A_{0,min} \sim 10$ pm

* Mean Z (topo.) noise ~ 1 pm

* SPECS MIMEA control unit

<figure><img src="image_377.png" /></figure>

<figure><img src="image_378.png" /></figure>

<footnote n="1">1. https://www.im2np.fr/fr/equipe-nanostructuration-nano</footnote>

<footnote n="2">2. a poster is available upon reasonable request</footnote>
# Non contact-AFM : summary

* Demodulation technique of the frequency of the probe when it oscillates close to the surface ($\tilde{f}_0$) around a central frequency which is the resonance frequency of the free probe ($f_0$), which forms the so-called frequency shift signal :

$$ \Delta f = \tilde{f}_0 - f_0 $$

* The oscillation amplitude ($A_0$) and the phase ($\varphi = -\pi/2$) are kept constant : resonance values

* True non-contact method :

* Attractive interaction regime with the surface

* $\Delta f < 0$

* Very large z sensitivity in the short-range interaction regime upon accurate detection of $\Delta f$, which allows for large vertical and lateral contrasts (atomic / sub-molecular resolution)
A last (but truly not least!) word...
<figure><img src="image_379.png" /><figcaption>Pentacene, as imaged by non-contact Atomic Force Microscopy (nc-AFM)</figcaption></figure>

« *The chemical structure of a molecule resolved by atomic force microscopy* »
L. Gross et al., Science **325**, 1110 (2009)
The « trick »: CO tip functionnalization

<figure><img src="image_380.png" /></figure>
Open question / theme to think about:

"Tracking and sorting interatomic forces down to the fN regime :
find recent examples reporting very weak bonds formation
and/or transition from physisorption to chemisorption, as
measured by non-contact atomic force microscopy."
<figure><img src="image_381.png" /></figure>

Institut Matériaux Microélectronique Nanosciences Provence

Thanks for your attention

<figure><img src="image_382.png" /></figure>

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<figure><img src="image_383.png" /></figure>