Transcript A study of laser-induced self

Imaging of flexural and torsional resonance modes of
atomic force microscopy cantilevers using optical
interferometry
Michael Reinstaedtler , Ute Rabe , Volker Scherer , Joseph
A.Turner , Walter Arnold
Surface Science 532-535(2003) 1152-1158
Date : 13th October 2005
Presenter : Ashwin Kumar
Background - Operation of the AFM
 A sharp tip is scanned over the sample surface
 the tip is maintained at a constant force (to obtain height
information), or height (to obtain force information) above
the sample surface
 Tips are typically made from Si3N4 or Si, and extended
down from the end of a cantilever
 An optical detection system is used, in which a diode laser
is focussed on the back of a reflective cantilever
 As the tip moves up and down with the contour of the
surface, the laser beam is deflected off the attached
cantilever into a dual element photodiode
AFM Schematic
Background - AFM Modes
Contact Mode
the tip scans the sample in close contact with the surface
 The force on the tip is repulsive with a mean value of 10 -9 N
 the deflection of the cantilever is sensed and compared in a
DC feedback amplifier to some desired value of deflection
Non-Contact Mode (used when tip contact might alter the
sample surface)
 In this mode the tip hovers 50 - 150 Angstrom above the
sample surface
 Attractive Van der Waals forces acting between the tip and
the sample are detected
 topographic images are constructed by scanning the tip
above the surface
Background - AFM Modes
Tapping Mode:(sample surfaces that are easily damaged )
The cantilever assembly is oscillated at or near the
cantilever's resonant frequency
 the cantilever is oscillated with a high amplitude when the
tip is not in contact with the surface
 The oscillating tip is then moved toward the surface until it
begins to lightly touch, or tap the surface.
 During scanning, the vertically oscillating tip alternately
contacts the surface and lifts off
 The reduction in oscillation amplitude is used to identify
and measure surface features.
Motivation
 Earlier Work involved determination of contact stiffness
and localized elastic modulus measurement of the surface
 The vibrational spectrum of the cantilever is used to
discern local elastic data.
 It becomes imperative to understand the vibrational
spectra completely to perform the above mentioned
measurements
 The free vibrational response would help to characterize
the cantilever or the probe
 Moreover, since the boundary conditions are also changed
during the contact mode resonance, Free vibrational
response and imaging the mode shape would help as a tool
for calibration or standard.
* Ultrasonics 38(2000) 430-437
* Journal of Applied Physics, 82(1997) 966
* Review of Scientific Instruments 67(1996) 3281
In a Nutshell
 Excite and Detect the torsional vibrations of the AFM
cantilevers.
 Examine the features of the torsional vibration
spectrum
 Image the flexural and torsional resonance modes
 Use a model based approach to explain the spurious
modes in the spectrum
b
Theory: Problem Statement
a
L
Boundary Conditions :
 Flexural Vibrations
Clamped end:
Free End:
y (0, t )  0
y '(0, t )  0
y "( L, t )  0
y "'( L, t )  0
 Torsional Vibrations
Clamped End:
Free End:
(0, t )  0
 '( L, t )  0
L - length of the beam (m)
a - width of the beam (m)
b - thickness of the beam (m)
E - Elastic Modulus of the beam (N/m2)
I - Area moment of inertia - ab3/12 (m4)
J - Polar moment of inertia - a3b/12 (m4)
G - Rigidity modulus (N/m2)
CT - Torsional Stiffness- ab3G/3 (Nm2)
Theory: Flexural Vibrations
 Equation of motion for the bending modes
4 y
2 y
EI 4   A 2  0  (1)
x
t
 The general solution of the form
y( x, t )  (a1e x  a2e x  a3ei x  a4ei x )eit  (2)
 The dispersion relation:
EI    A  0  (3)
4
2
Theory: Flexural Vibrations
 Applying the Boundary Conditions:
The Characteristic Equation -
n
cos  n L cosh  n L  1  0  (4)
 Bending-mode eigenfrequencies:
( n L)2
fn 
2 L2
EI
 (5)
A
 Amplitude Distribution:


cos  n x  cosh  n x
yn ( x)  y0  cos  nx  cosh  nx  
 sin  n x  sinh  n x  (6)
sin  n x  sinh  n x


Theory: Torsional Vibrations
 Equation of motion for the torsional modes
 2
 2
cT 2   J 2  0  (7)
x
t
 The general solution of the form
( x)  A sin x  B cos x  (8)
 Applying the boundary conditions:
2n  1 b G
fn 
 (9)
2L a 
( x)  A sin  x  (10)
* Jerry
H. Ginsberg , Mechanical and Structural
Vibrations,2001
Experimental Setup
Longitudinal Vs Shear Wave Propogation
Excitation of Torsional Vibrations
Cantilever
Sample
Shear Wave Transducer
Beam Deflection Setup
 Spatial variations of reflected beam are detected
 Transverse vibrations cause vertical movement
of the spot
 Torsional vibrations cause horizontal movement
of the spot
 If the light beam moves up or down,
Ivertical  (Iupperleft  Iupperright )  (Ilowerleft  Ilowerright )
 If the light beam moves right or left
Ihorizontal  (Iupperleft  Ilowerleft )  (Iupperright  Ilowerright )
* Handbook of Nano-Technology,Springer,2003
Experimental Results
Optical Micrograph of the cantilever
Interferometric Measuring System
Spot Size : 2-5 microns
Step Size : 2 microns
Optical Detection Of Vibration of the Beam

A=
a*ei(ωt-k(z-2δ))
Incident Beam
Reflected Beam
• Phase Information is lost during Intensity or Power Measurements
• Interferometric systems are used to convert phase change into intensity variations
Michelson Interferometer
Reference Mirror
• AR=ar*ei(t-kzR)
Laser
B.S.
Sample
• AO=ao*ei(t-k(zo-δ))
I D  A0  AR

2

2
2
I D  aO  aR
I D  aO  aR
2

2

Detector
aR aO


cos  k ( z R  zO )  2k 
1  2 2
2
aO  aR




aR aO
aR aO
1

2
cos
k
(
z

z
)

2
2
k

sin
k
(
z

z
)
 R O
 R O 

2
2
2
2
aO  aR
aO  aR


Output Intensity Vs Optical Path Length
Relative Intensity
Maximum Slope
Region of
Best Sensitivity

4

2
Path Length Difference (zr-zs)
Heterodyne Interferometry
Reference Mirror
Frequency Shifter
• AR=ar*ei((+)t-kzR)
• AO=ao*ei(t-k(zo-δ))

I D  aO  aR
2
2

Laser
B.S.


aR aO


cos


t

k
(
z

z
)

2
k



1  2 2

R
O
2
a

a


O
R


Sample
Detector
Phase locked loop demodulator
Mixer
LPF2
a1 cos[t  k ( zr  zo )  2k ]
O/p
Detector Input
aLO cos(t  LO )
VCO
O/P:
LPF1
a1aLO
{cos[k ( zr  zo )  LO  2k ]  cos[2t  k ( zr  zo )  LO  2k ]}
2
Amplitude and Phase distribution - Measured
Amplitude and Phase distribution - Calculated
Mode Coupling
 Asymmetrical shape of the modes
- Geometrical asymmetries - Tip not aligned with the center of the beam
- Tip is in force interaction with the sample surface
mt - mass of the tip
d
b
d - offset from the center of the beam
h
b - thickness of the beam
h - length of the tip
Mode Coupling
Coupling Description:
- Equation of motion:
4 y
2 y
EI 4   A 2  0  (1)
x
t
 2
 2
cT 2   J 2  0  (7)
x
t
Boundary Conditions at the free end:(x = L)
d
d ..
 ..

EIy '''( x, t )  kn y ( x, t )  kn  ( x, t )  mt  ( y ( x, t )   ( x, t )   (11)
L
L


..
 ..

ct ( x, t )  kn h  ( x, t )  kn dLy ( x, t )  mt d  L y ( x, t )  d  ( x, t )   (12)


'
2
Mode Coupling
In Case of free oscillations:
d


EIy '''( x)  mt  ( y ( x)   ( x) 
L




ct ' ( x)  mt d 2   L y ( x)  d  ( x) 


2
From Previous Results:
y( x)  (a1e x  a2e x  a3ei x  a4ei x )
 ( x)  A sin x
Coupling Parameter H :
  H 2
H
EIJ
Act L2
Mode Coupling
Calculated Amplitude distribution based on mode coupling with H=0.025
Resonance Mode at 265 Khz
- The mode does not fit into the mode coupling analysis
- Most likely occurs due to nonlinear coupling into flexural motion
d2y
 flex  (b / 2) 2  105
dx
a
 tors 
 104
L
Conclusion
 Verification of standard flexural and torsional modes in the
vibration spectrum by imaging the mode shapes and
comparing them with the model based expected pattern
 Mode Coupling due to geometrical and mass asymmetries
account for a number of resonances
 Large strain values leads to non-linear mixing of modes
Beam Deflection Setup
 Spatial variations of reflected beam are detected
 Transverse vibrations cause vertical movement
of the spot
 Torsional vibrations cause horizontal movement
of the spot
 If the light beam moves up or down,
Ivertical  (Iupperleft  Iupperright )  (Ilowerleft  Ilowerright )
 If the light beam moves right or left
Ihorizontal  (Iupperleft  Ilowerleft )  (Iupperright  Ilowerright )
Background - Operation of the AFM
 A sharp tip is scanned over the sample surface
 the tip is maintained at a constant force (to obtain height
information), or height (to obtain force information) above
the sample surface
 Tips are typically made from Si3N4 or Si, and extended
down from the end of a cantilever
 An optical detection system is used, in which a diode laser
is focussed on the back of a reflective cantilever
 As the tip moves up and down with the contour of the
surface, the laser beam is deflected off the attached
cantilever into a dual element photodiode
AFM Schematic
Background - AFM Modes
Contact Mode
the tip scans the sample in close contact with the surface
 The force on the tip is repulsive with a mean value of 10 -9 N
 the deflection of the cantilever is sensed and compared in a
DC feedback amplifier to some desired value of deflection
Non-Contact Mode (used when tip contact might alter the
sample surface)
 In this mode the tip hovers 50 - 150 Angstrom above the
sample surface
 Attractive Van der Waals forces acting between the tip and
the sample are detected
 topographic images are constructed by scanning the tip
above the surface
Background - AFM Modes
Tapping Mode:(sample surfaces that are easily damaged )
The cantilever assembly is oscillated at or near the
cantilever's resonant frequency
 the cantilever is oscillated with a high amplitude when the
tip is not in contact with the surface
 The oscillating tip is then moved toward the surface until it
begins to lightly touch, or tap the surface.
 During scanning, the vertically oscillating tip alternately
contacts the surface and lifts off
 The reduction in oscillation amplitude is used to identify
and measure surface features.
Motivation
 Earlier Work involved determination of contact stiffness
and localized elastic modulus measurement of the surface
 The vibrational spectrum of the cantilever is used to
discern local elastic data.
 It becomes imperative to understand the vibrational
spectra completely to perform the above mentioned
measurements
 The free vibrational response would help to characterize
the cantilever or the probe
 Moreover, since the boundary conditions are also changed
during the contact mode resonance, Free vibrational
response and imaging the mode shape would help as a tool
for calibration or standard.
* Ultrasonics 38(2000) 430-437
* Journal of Applied Physics, 82(1997) 966
* Review of Scientific Instruments 67(1996) 3281