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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