Dynamic Atomic Force Microscopy:

Basic Concepts

Rubén Pérez

Nanomechanics & SPM Theory Group

Departamento de Física Teórica de la Materia Condensada

http://www.uam.es/spmth

Curso “Introducción a la Nanotecnología”

Máster en física de la materia condensada y nanotecnología

References

R. García and R. Pérez, Surf. Sci. Rep. 47, 197 (2002)

F.J. Giessibl, Rev. Mod. Phys. 75, 949 (2003)

W. Hofer, A.S. Foster & A. Shluger , Rev. Mod. Phys. 75, 1287 (2003)

• C. J. Chen. “Introduction to Scanning Tunneling

Microscopy”. 2nd Edition. (Oxford University Press,

Oxford, 2008).

• S. Morita, R. Wiesendanger, E. Meyer (Eds). “Noncontact

Atomic Force Microscopy”. (Springer, Berlin, 2002).

• S. Morita, F.J. Giessibl R. Wiesendanger (Eds).

“Noncontact Atomic Force Microscopy”. Vol. 2 (Springer,

Berlin, 2009).

Outline

• Static vs Dynamic AFM: AM-AFM & FM-AFM.

• Amplitude Modulation AFM

• Frequency Modulation AFM

Static vs Dynamic AFM:

Amplitude Modulation (AM) &

Frequency Modulation (FM).

ATOMIC FORCE MICROSCOPY (AFM)

http://monet.physik.unibas.ch/famars/afm_prin.htm

G. Binnig, C. Gerber & C. Quate, PRL 56 (1986) 930

2nd most cited PRL: +5000 citations !!!

Scanner

Piezo XYZ

Electronics and

feedback:

constant

amplitude

Computer

and display

Cantilever

Tip

Sample

Piezo

oscillator

5 m

AM-AFM

Fixed excitation

frequency

constant oscillation

amplitude

Limitations of static AFM

Contact

F

Deformation, Friction

No point defects observed

Atomic Resolution?

Non-contact

F

Detection of small forces:

soft cantilevers.

“Jump to contact” :

stiff cantilevers

AFM: G. Binnig, C. Gerber & C. Quate, PRL 56 (1986) 930

Dynamic AFM

http://monet.physik.unibas.ch/famars/afm_prin.htm

Dynamic AFM: Our Goal

Why changes observed in the dynamic

properties of a vibrating cantilever with

a tip that interacts with a surface make

possible to:

•Resolve atomic-scale defects

in UHV.

• Obtain molecular resolution

images of biological samples

in ambient conditions.

AM-dAFM

FM-dAFM

R. García and R. Pérez, Surf. Sci. Rep. 47, 197 (2002)

Dynamic description

Cantilever-tip ensemble as a point

mass spring described by a non-

linear 2nd order differential equation

Amplitude

Resonance Frequency

Phase shift

link the dynamics of a

vibrating tip to the tip-surface

Fts interaction.

(t)A z exc

2

0c

2

02

00

z(t)F

kz(t)(t)z

Q(t)z ts

Why do A and f () depend on Fts?

(simple quasi-harmonic argument)

-kz

Fts

New new resonance curve New amplitude for given exc

m

k k ω ts

czzd

d

z

Fk ts

ts

For small amplitudes and large distances

BUT: Large amplitudes Force gradient varies considerably

during oscillation Non-linear features in the dynamics

2k

k

ω

Δω ts

0

tsk k

Two major modes: AM-AFM and FM-AFM

• Excitation with constant

amplitude Aexc and frequency

exc close or at its FREE

resonance frequency 0.

• Oscillation amplitude A as

feedback for topography.

• Phase shift between

excitation and oscillation:

compositional contrast.

• Air and liquid environments.

Amplitude Modulation

AFM

Y. Martin et al, JAP 61, 4723 (1987)

Q. Zhong et al, SS 290, L688 (1993)

• Constant oscillation

amplitude at the current

resonance frequency

(depends on Fts).

• Frequency shift f as

feedback for topography.

• Excitation amplitude Aexc

provides atomic-scale

information on dissipation.

• UHV (now also liquids !)

T.R. Albrecht et al, JAP 69, 668 (1987)

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

Frequency Modulation

AFM

Amplitude Modulation (AM) AFM

Outline: AM-AFM

(or Tapping mode AFM)

• Operation Parameters.

• Non-linear dynamics: Existence of two oscillation states

(L & H): implications for imaging.

• Understanding amplitude reduction.

• Imaging materials properties: phase shifts and dissipation.

• Summary: things to remember...

van der Waals forces

Restoring force cantilever

Fc=-kz

Excitation force

F0cos t

Adhesion forces

Fa=4R

Short range repulsive forces (DMT)

Hidrodynamic forces

2vdwd6

HRF

2/3*

DMT REF

dt

dz

Q

mF 0

h

Capillary forces

Forces in AFM

Laboratorio de Fuerzas y Túnel

Instituto de Microelectrónica de Madrid

- 40

0

40

- 80

Forc

e (n

N)

Separation (nm)

0 2 4 6 8 10

PM

ts

Forced damped harmonic oscillator

)cos( 0

tkAkz(t)(t)zmQ

(t)zm excexc

m

k0

Q2

0 Q Quality factor (cantilever damping)

)cos()exp( ttCtz

)cos(

/2

0

222

0

2

0

t

Q

Aexc

excexc

exc

(transient)

22

0

0tanexc

exc Q

0=exc A = QAexc (resonance)

BUT Fts is nonlinear

anharmonic effects

0.998 1.000 1.002

0.9

1.0

1.1

1.2

A/z

c

/0

SIMULATION

R= 10 nm, A0=10 nm, zc=8

nm, E=1 GPa, k=40 N/m,

f0=325 kHz

0 1 2 3 4 5

0

2

4

6

8

10

12

14

16

Am

pli

tud

e (

nm

)

f/f0

EXPERIMENT

Silicon, A0=15 nm, A=13

nm, f0=295.64 kHz

Low to high

high to low

AM-AFM: Two stable oscillation states

Amplitude curves: AH(L) vs zc

(two steady state solutions)

)cos()( )()()( LHexcLHcLH tAztz

H: high amplitude state

L: low amplitude state

• Collection of L and H solutions gives rise to L and H branches.

• AH(L) decreases linearly with zc for both branches.

•Ambiguity in the operation: both branches can match the set

amplitude Aset .

Aexc = 10 nm

Aset

6 9 12 15 18 21 24

6

9

12

15

18

Am

plit

ude

(nm

)

z piezo displacement (nm)

40 nm

H

L A1

A2

A3

A1 low amplitude branch

A2

A3 high amplitude branch

Sample: InAs quantum dots

Experimental implications of the coexistence of states (I):

Noise and stability

García, San Paulo,

PRB 61, R13381 (2000)

6 9 12 15 18

6

9

12

15

18

low oscillation solution (L)

high oscillation solution (H)

Am

plitu

de (

nm

)

Tip-surface separation (nm)

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

z/A

0

Zc=14.5 nm

Phase space diagram with

significant H and L

contributions=unstable

operation

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

∙

∙

∙

(c)

z/A

0

z/A0

Zc=7.5 nm

Phase space diagram

dominated by the H

state basin of

attraction=stable

operation

Zc=16 nm

Phase space dominated by

the L state=stable operation

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

V/A0ω

Z/A0

Are both solutions

equally accessible ?

García and San Paulo, Phys. Rev. B

61, R13381 (2000)

Phase space diagrams:

Representation of the tip final

state as a function of the initial

velocity and positions

Tip should stay always on the same branch (deterministic) BUT…

NOISE: Implications for scanning

Mechanical, electronical,thermal and feedback perturbations...

Vscan Finite time response of the

feedback ( 10-4 s)

Change in separation can

lead to transitions before

the feedback takes over

• AM-AFM would operate properly if initial (unperturbed) and

intermediate state belong to the same branch, otherwise

instabilities and image artifacts will appear.

• Stable operation when one of the states dominates the

phase space (tip oscillates in the state with the largest

attraction basin).

0

2

4

6

8

10

-1.0

-0.5

0.0

0.5

1.0

0 2 4 6 8 10 12

0.0

0.5

1.0

(a)

High amplitude solution Low amplitude solution A

mp

litu

de

(n

m)

(b)

<F

in

t > (

nN

)

(c)

Co

nta

ct

tim

e

Tip-surface separation (nm)

Simulation data: R=20 nm

f0=350 kHz, Q=400, H=6.4x10-20,

E*=1.52 GPa

H and L states have

different properties

García, Pérez, Surf. Sci. Rep. 47, 197 (2002)

dttFT

F tsts )(1

Characterizing the physical properties of the

two states....

Does resolution depend on

the oscillation state chosen?

a-HSA

antibody

on mica

L state

H state

Morphology and dimensions of

fragments clearly resolved

No domain

structure

Irreversible

deformation

after imaging on

H state

2/12

0

410

Fts

FAA

2/12

00

0·

16112

AF

zFAA ts

000

2sin

AAk

QP

A

A

c

ts

2

0

2

2

2

0

12

1·

2cos

AkzF

k

F

AAk

Qcts

c

ts

c 0

·2cos

AAk

zFQ

c

ts

Analytical Approximations

San Paulo and García,

PRB 64, 193411 (2001) The virial theorem and energy consideration allows

to derive an analytical approximation

ω=ω0 and A>>z0

2/12

00

0·

164

12

12

AF

zF

P

P

P

PAA ts

med

ts

med

ts

Negligible power dissipation

(Understanding the amplitude reduction…: related to Fts??)

Cantilever response:

z(t) = z0+ Acos(t-)

Driving signal:

F(t) = F0cos(t)

SAMPLE

Piezo

oscillator

The dynamic response of the cantilever is modified

by the tip-surface interactions

Phase Imaging

Polymer morphology and structure as a function of

temperature. Hydrogenated diblock copolymer

(PEO-PB). Crystallisation of PEO blocks occurs

individually for each sphere (light are crystalline,

dark amorphous). Reiter et al., Phys. Rev. Lett.

87, 2261 (2001)

Polymers: Morphology and Structure

Phase Image, size

1m2

Amplitude image Phase image

sin)·(AkA)Q/1(dtdt

dz)tcos(FE

00EXT

0

2

0)(

Q

Akdt

dt

dz

dt

dz

Q

mE

med

dtdt

dzFE

TSdis

Steady solution

) t cos( ) ( A ) t ( z

Cleveland et al. APL 72, 2613(1998)

Tamayo, García APL73, 2926 (1998)

García et al. Surf. Int. Anal. 27, 1999)

sp

dissp

AkA

QE

A

A

000

sin

Dynamic equilibrium in AM - AFM (tapping mode)

dis med EXT E E E energy per period

PHASE SHIFT AND ENER GY DISSIPATION IN AMPLITUDE MODULATION AFM

At Asp=constant phase shifts are linked to

tip-surface inelastic interactions

CONTRIBUTIONS TO CONTRAST IN PHASE IMAGES

PHASE

CONTRAST

ELASTIC

CONTRIBUTIONS

INELASTIC

CONTRIBUTIONS

TOPOGRAPHIC EFFECTS

TAPPING NON CONTACT

TRANSITIONS

VISCOELASTICITY

ADHESION HYSTERESIS

CAPILLARY FORCES

HIDROPHILIC/HIDROPHOBIC

INTERACTIONS

YOUNG MODULUS (In presence of dissipative channels)

Continuous Model for the Cantilever

10 m

tsmedext

FFFt

txwbhtxw

xL

EI

2

2

4

4

4

),(),(

z

zc d(x,t)

w(x,t)

x

0),(),(),(

),(

1

3

3

1

2

2

00

xxxx x

txw

x

txw

x

txwtxw

Rodríguez and García, Appl. Phys. Lett. 80, 1646 (2002)

Point - mass model

0 2 4 6 8 10 12 0

4x10 -4

8x10 -4

1x10 0

low amplitude

(a)

0 2 4 6 8 10 12 0

4x10 -4

8x10 -4

1x10 0

high amplitude

no

rma

lize

d a

mp

litu

de

frequency (normalized to 350.6 kHz)

(b)

Continous model

0 2 4 6 8 10 12 0

4x10 -4

8x10 -4

1x10 0

low amplitude

(a)

0 2 4 6 8 10 12 0

4x10 -4

8x10 -4

1x10 0

high amplitude

no

rma

lize

d a

mp

litu

de

frequency (normalized to 350.6 kHz)

(b)

Experimental results

( Triangular cantilever )

high amplitude

low amplitude

Stark et al. APL 77, 3293 (2000)

z c

0 5 10 15 20 0

5

10

15

20

H.A. (continous) L.A. (continous) H.A. (point-mass) L.A. (pont-mass)

tip-sample distance (nm)

am

plit

ude (

nm

)

-15

0

15

L A

0 15

-15

0

15 H A

time ( s)

w (

x=

1,t

)

(nm

)

Parámetros de la simulación:

f0,= 350.6 kHz k= 40 N/m, A0= 18.22 nm Q= 400 (masa puntual, ajusta el primer modo libre).

l=119 m, h=3.6 m, b=33 m, E=170 Gpa,rc =2320 kg/m3, F=1.85 nN, a0=1.28·10-3 kg/m·s,

a1=0.2 ns,a2=10.037 (modelo continuo que ajusta la curva A vs. f experimental libre)

R=30 nm, H= 6.4·10-20 J, E*=1.51 Gpa, d0= 0.165 nm

Bimodal FM-AFM on Antibodies(IgM) Noninvasive Protein Structural Flexibility Mapping

D. Martinez et al, PRL106, 198101(2011)

AM-AFM: Things to remember...

• Operation Parameters (OP): Aexc, exc, zc.& Aset

• Two stable oscillation states: L (H) = low (large) amplitude.

• Chose OP to ensure that one state dominates phase

space stable imaging.

• Image soft materials with L state (avoid damage).

• Image stiff materials with H state (improved contrast).

• Amplitude reduction related to Fts·z.

• Imaging material properties: Phase imaging.

• Phase shift related to Ptsdiss= Fts·dz/dt.

• Nanometric resolution (both amplitude and phase images).

Frequency Modulation (FM) AFM

Outline: FM-AFM

• Dynamic AFM: AM-AFM vs FM-AFM.

• Cantilever dynamics: f vs Fts.

Perturbation theory for the frequency shifts

Normalized frequency shift

• Atomic scale contrast and Fts: tip as the key player.

• Separation of long- and short-range interactions.

• semiconductors, alkali halides, oxides, metals, nanotubes,…

• Recent developments.

Tuning forks: small amplitudes to enhance atomic contrast

Force spectroscopy: Chemical identification.

Single-atom manipulation, atomic-scale magnetic imaging

Operation in liquids

• Summary: things to remember...

1. Dynamic AFM:

AM-AFM vs FM-AFM.

Two major modes: AM-AFM and FM-AFM

• Excitation with constant

amplitude Aexc and frequency

exc close or at its FREE

resonance frequency 0.

• Oscillation amplitude A as

feedback for topography.

• Phase shift between

excitation and oscillation:

compositional contrast.

• Air and liquid environments.

Amplitude Modulation

AFM

Y. Martin et al, JAP 61, 4723 (1987)

Q. Zhong et al, SS 290, L688 (1993)

• Constant oscillation

amplitude at the current

resonance frequency

(depends on Fts).

• Frequency shift f as

feedback for topography.

• Excitation amplitude Aexc

provides atomic-scale

information on dissipation.

• UHV (now also liquids !)

T.R. Albrecht et al, JAP 69, 668 (1987)

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

Frequency Modulation

AFM

Why not AM-AFM in UHV?: transient terms!!

0

21

Q

)cos()exp( ttCtz

)cos(

/2

0

222

0

2

0

t

Q

Aexc

excexc

exc

(transient)

Q (air) = 102 –103 small

Q (UHV) = 104 –105 large

(Q=50000, 0=50 kHz = 2 s !!!)

We have to wait 2 s to record a single pixel... (small bandwidth)

Increase Q to improve resolution

BUT... Q and B (bandwidth)

linked in AM-AFM

AM-AFM vs FM-AFM set-ups

Aexc ,exc (constant)

A

FM-AFM: cantilever regulated by

electronics stable and fast response.

0

FMω

1τ

Mechanical Stability Conditions

Atomic resolution in FM-AFM:Si(111)-7x7

AFM

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

faulted half unfaulted half

12 adatoms

6 rest atoms

corner hole

dimers

STM

“Classical” FM-AFM operation conditions

k ~ 30 N/m

f0 ~ 100 kHz

Q ~ 30000

A0 ~ 200 Å

f ~ -(50-100) Hz

Stability Conditions

kA0 ~ 600 nN >> Fts ~ 1-10 nN

1/2kA02 ~ 3.75 x 104 eV >> Ets

(prevents cantilever instabilities)

(stable oscillation amplitudes)

FM-AFM: Contrast sources

f: frequency shift

Aexc: damping (excitation)

It: mean tunneling current

f It

Aexc

f Aexc It

2. Cantilever dynamics : relation

between the frequency shift and

tip-sample interaction.

Contrast source: frequency shift vs Fts

d

f

z0

2A

(t)kA z(t)zFkz(t)(t)zQ

mω(t)zm excots

0

0 z(t)zFkz(t)(t)zm ots

d cos )cos(1AdFkA

f

2π

1

zFkA

f- )f,Ak,(d, Δf

2π

0

ots

0

0

ts2

0

000

Electronics cancels

damping exactly

Perturbation theory

d z0 d+2A

E

V(z)=kz2/2 + Vts(z)

F.J. Giessibl, PRB 58, 10835 (1998)

M. Gauthier, R.P., T. Arai, M. Tomitori & M. Tsukada, PRL 87, 096801 (2001)

Confirmed by numerical simulations

including the control electronics

Normalized frequency shift

00,

0

23

0 fAk,d,Δff

kAdγ

(Si tip on Graphite)

extracts the intrinsic

contribution coming from Fts

(d)F

(d)V(d)F

2π

1dγ

ts

tsts

Not accurate for small tip-

sample distances (2-3 Å) !!

3. Atomic-scale contrast and tip-

sample interaction:

tip as the key player

• Separation of LR and SR interactions

• Semiconductors

• Alkali halides & oxides

• Metals, weakly bonded systems & carbon-

based materials (graphite, nanotubes, …)

Tip-sample Interaction: FV + FvdW + Fchem

Fchem wfs overlap

Exp: A = 340 Å !!!, R = 40 Å

Sensitivity to Short-Range Forces?

Weak singularity at

turning points !!!!

Characterizing the “macroscopic” tip:

Separation of interactions

Si tip on

Cu(111)

Electrostatic VdW

M. Guggisberg et al,

PRB 61 (2000) 11151

Computational approaches for SR Fts

OK for ionic bonding

Weakly bonded systems??

Necessary for covalent and

metallic bonding (semiconductors

and metals.)

Role of SR Covalent Bonding Interactions?

DFT-GGA plane wave pseudopotential calculations

R. P. et al, PRL 78, 678 (1997) R. P. et al, PRB 58, 10 835 (1998)

Charge

density

difference

between

tips

Si tips

Atomic scale contrast in reactive semiconductor surfaces:

chemical tip-surface interaction (between dangling bonds)

tip+surface –

(tip +surface)

Charge

acumulates in

the adatom

dangling bond

d=5Å

Contrast dependence on tip preparation

T. Uchihashi et al, PRB 56, 9834 (1997)

Force-distance curves & Atomic relaxations

atomic relaxations due to

tip-surface interactions!!

R. P. et al, PRB 58, 10835 (1998)

Force vs distance curves prediction for f vs distance

d cos )cos(1AdFkA

f

2π

1 (d) Δf

2π

0

ots

0

0

Comparison between theory and low-

temperature FM-AFM experiments

M. Lantz et al, PRL 84, 2642 (2000)

faulted half unfaulted half

M. Lantz et al, Science 291, 2580 (2001)

Separation of VdW and chemical

interaction: substracting the corner

hole contribution. R. Pérez et al , PRL 78, 678 (1997)

R. Pérez et al , PRB 58, 10835 (1998)

Tip-surface interactions

4. Recent developments…

• Tuning forks: small amplitudes to enhance

atomic contrast.

• Force spectroscopy: Chemical identification

• Single-atom manipulation at RT

• AFM detection of spin

•True atomic resolution in liquids

Other operating conditions: qPlus sensor Smallest Noise for Å-size amplitudes!!!

qPlus sensor made from a tuning

fork (k ~ 2000 N/m)

Operating under

repulsive SR

forces (stabilize

by LR

electrostatics) !!!

F.J. Giessibl et al,

Science 289 (2000) 422

The Chemical Structure of a Molecule

Resolved by Atomic Force Microscopy

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

Ad1

Ad1

Ad1 Ad2

Ad2

Ad2 Re1 H3

Dynamic Force Spectroscopy: Access to Fts

Inversion

algorithms

U. Durig, APL 76, 1203 (2000)

F.J. Giessibl, APL 78, 123 (2001)

J. E. Sader & S. P. Jarvis, APL 84,

1801 (2004). 1 2 3 4 5 6 7 8 9

-1,8

-1,6

-1,4

-1,2

-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

Exp. Adatom 2

Exp. Restatom 1

Exp. H3

Sh

ort

-ra

nge F

orc

e (

nN

)

Tip-surface Distance (Å)

SR forces amenable to

ab initio calculations

Developments based in Force

Spectroscopy

3. CHEMICAL IDENTIFICATION:

1. DISSIPATION: Characterizing the tip

structure and identifying a dissipation channel

due to single atomic contact adhesion.

N. Oyabu et al. Phys. Rev. Lett. 96, 106101 (2006).

2. IMAGING: changes in topography: access to the real surface structure?

Y. Sugimoto et al Phys. Rev. B 73, 205329 (2006).

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

1 2 3 4 5 6 7 8 9

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Exp. Adatom 2

Exp. Restatom 1

Exp. H3

Dis

sp

ati

on

(eV

per

Cycl

e)

Tip-surface Distance (Å)

Ad1

Ad1

Ad1 Ad2

Ad2

Ad2 Re1 Re2 H3

-6.3 fNm -7.3 fNm -8.4 fNm

based on the relative interaction ratio of

the maximum attractive force measured

by dynamic force spectroscopy

Magnetic exchange force microscopy with

atomic resolution

U. Kaiser, A. Schwarz & R. Wiesendanger, Nature 446, 522 (2007)

High-Resolution FM-AFM Imaging in Liquid

True Atomic Resolution (2005)

FM-AFM Image of Mica in Water

1 nm

Fukuma et al. APL 87 (2005) 034101

Cleaved Mica Surface

True Molecular Resolution (2005)

Polydiacetylene Single Crystal in Water

bc-plane of Polydiacetylene Crystal 0.5 nm

1 nm

Fukuma et al. APL 86 (2005) 193108

FM-AFM: Things to remember...

• Frequency shift as the contrast source.

• True atomic resolution. (UHV & Liquids !!!)

• self-driven oscillator: More complicated operation and

electronics, but simpler behaviour (amplitude feedback

“linearizes” the behaviour).

• Short-range (chemical, electrostatic) interactions are

responsible for the atomic resolution.

• Separation of interactions + inversion formulae

spectroscopic capabilities (in combination with theory).

• Different channels (frequency shift, tunneling currents,

energy dissipation) recorded simultaneously