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
64
Embed
Dynamic Atomic Force Microscopy: Basic Conceptswebs.ftmc.uam.es/ruben.perez/master/nanotecnologia/SLIDES/nano_… · Restoring force cantilever F c =-kz Excitation force F 0 cos t
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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)