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

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.

Background - AFM Modes

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

Theory: Problem Statement

Boundary Conditions : Flexural Vibrations

Clamped end: Free End:

Torsional Vibrations

Clamped End: Free End:

La

b

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)

(0, ) 0

'(0, ) 0

y t

y t

"( , ) 0

"'( , ) 0

y L t

y L t

(0, ) 0t '( , ) 0L t

Theory: Flexural Vibrations

Equation of motion for the bending modes

The general solution of the form

The dispersion relation:

4 2

4 20 (1)

y yEI A

x t

1 2 3 4( , ) ( ) (2)x x i x i x i ty x t a e a e a e a e e

4 2 0 (3)EI A

Theory: Flexural Vibrations

Applying the Boundary Conditions:

The Characteristic Equation -

Bending-mode eigenfrequencies:

Amplitude Distribution:

n

cos cosh 1 0 (4)n nL L

2

2

( )(5)

2n

n

L EIf

L A

0

cos cosh( ) cos cosh sin sinh (6)

sin sinhn n

n n nn n

x xy x y nx nx x x

x x

Theory: Torsional Vibrations

Equation of motion for the torsional modes

The general solution of the form

Applying the boundary conditions:

2 2

2 20 (7)Tc J

x t

( ) sin cos (8)x A x B x

( ) sin (10)x A x

2 1(9)

2n

n b Gf

L a

* Jerry H. Ginsberg , Mechanical and Structural Vibrations,2001

Experimental Setup

Longitudinal Vs Shear Wave Propogation

Shear Wave Transducer

Sample

Cantilever

Excitation of Torsional Vibrations

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,

If the light beam moves right or left

( ) ( )horizontal upperleft lowerleft upperright lowerrightI I I I I

( ) ( )vertical upperleft upperright lowerleft lowerrightI I I I I

* Handbook of Nano-Technology,Springer,2003

Experimental Results

Optical Micrograph of the cantilever

Interferometric Measuring System

Spot Size : 2-5 micronsStep Size : 2 microns

Incident Beam

Reflected Beam

Optical Detection Of Vibration of the Beam

A= a*ei(ωt-k(z-2δ))

• Phase Information is lost during Intensity or Power Measurements

• Interferometric systems are used to convert phase change into intensity variations

Michelson Interferometer

• AR=ar*ei(t-kzR

)

• AO=ao*ei(t-k(zo

-δ))

2

0

2 2

2 2

2 2

2 2 2 2

1 2 cos ( ) 2

1 2 cos ( ) 2 2 sin ( )

D R

R OD O R R O

O R

R O R OD O R R O R O

O R O R

I A A

a aI a a k z z k

a a

a a a aI a a k z z k k z z

a a a a

LaserSample

Detector

B.S.

Reference Mirror

Output Intensity Vs Optical Path Length

Rel

ativ

e In

tens

ity

Path Length Difference (zr-z

s)

Maximum Slope

2

4

Region ofBest Sensitivity

Heterodyne Interferometry

• AR=ar*ei((+)t-kzR

)

• AO=ao*ei(t-k(zo

-δ))

2 2

2 21 2 cos ( ) 2R OD O R R O

O R

a aI a a t k z z k

a a

Laser

Sample

Detector

B.S.

Reference Mirror

Frequency Shifter

Phase locked loop demodulator

VCO

MixerLPF2

LPF1

Detector Input

1 cos[ ( ) 2 ]r oa t k z z k

cos( )LO LOa t

1/ : {cos[ ( ) 2 ] cos[2 ( ) 2 ]}2LO

r o LO r o LO

a aO P k z z k t k z z k

O/p

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

h

b

d mt - mass of the tip

d - offset from the center of the beam

b - thickness of the beam

h - length of the tip

Mode Coupling

Coupling Description:

- Equation of motion:

Boundary Conditions at the free end:(x = L)

4 2

4 20 (1)

y yEI A

x t

2 2

2 20 (7)Tc J

x t

.. ..

.. ..' 2

'''( , ) ( , ) ( , ) ( ( , ) ( , ) (11)

( , ) ( , ) ( , ) ( , ) ( , ) (12)

n n t

t n n t

d dEIy x t k y x t k x t m y x t x t

L L

c x t k h x t k dLy x t m d L y x t d x t

Mode Coupling

In Case of free oscillations:

2

' 2

'''( ) ( ( ) ( )

( ) ( ) ( )

t

t t

dEIy x m y x x

L

c x m d L y x d x

From Previous Results:

1 2 3 4( ) ( )x x i x i xy x a e a e a e a e

( ) sinx A x

Coupling Parameter H :

2

2t

H

EIJH

Ac L

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

25

2

4

( / 2) 10

10

flex

tors

d yb

dxa

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,

If the light beam moves right or left

( ) ( )horizontal upperleft lowerleft upperright lowerrightI I I I I

( ) ( )vertical upperleft upperright lowerleft lowerrightI I I I I

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

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.

Background - AFM Modes

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