Metrology of Nanostructures Using 

Picosecond Ultrasonics 

 

by

Liwei Jim Liu

Sc.B. Brown University 2007

A dissertation submitted in partial fulfillment

of the requirements for the Degree of Doctor of Philosophy

in the Department of Physics at Brown University

PROVIDENCE, RHODE ISLAND

May 2016

© Copyright 2016 by Liwei Jim Liu

iii

This dissertation by Liwei Jim Liu is accepted in its present form

by the Department of Physics as satisfying the dissertation requirement

for the degree of Doctor of Philosophy

Date ____________________ _____________________________ Professor Humphrey J. Maris, Advisor

Recommended to the Graduate Council

Date ____________________ _____________________________ Professor James Valles, Jr., Reader

Date ____________________ _____________________________ Professor Derek Stein, Reader

Approved by the Graduate Council

Date ____________________ _____________________________ Professor Peter M. Weber

Dean of the Graduate School

iv

Vita 

Liwei Jim Liu was born in Taipei, Taiwan on January 19, 1985. He received Sc.B. in Physics and

Mathematics with honors at Brown University in May, 2007. He continued his studies at Brown

University under the 5th-year Master’s program in the Department of Physics, and transitioned

his studies to the Ph.D. program in the fall of 2008. The work for this thesis began in the spring

of 2010.

Publication 

Miao, Q., Liu, L.-W., Grimsley, T.J., Nurmikko, A.V., and Maris, H.J. “Picosecond ultrasonics measurements using an optical mask.” Ultrasonics 56, 141 (2015).

v

Acknowledgements 

I would like to express my deepest gratitude to Professor Humphrey J. Maris for

being a magnificent advisor. He is an inspiring figure who demonstrates great expertise in

research and immense patience toward guiding the students. This work could not have come

to fruition without his guidance, encouragements and support. Many thanks go to the

members of my dissertation committee, Professor James Valles, Jr. and Professor Derek

Stein, for taking their time to read this dissertation.

There have been many wonderful people who contributed to the completion of these

projects. Dr. Thomas J. Grimsley built The Z-tip-tilt stage that was central to this work, as

well as giving many helpful tips for the Planar Opto-Acoustic Microscopy measurements.

Dr. G. Andrew Antonelli provided the high quality, state-of-the-art samples, which without

the samples the project wouldn’t be possible. Dr. Joonhee Lee offered to dice samples that

were extremely difficult to cleave cleanly, on top of giving many advises regarding

cleanroom operations. Professor Alexander Zaslavsky kindly provided the chemical-

mechanical polishing machine so that it was possible to modify the samples. Dr. Anthony

McCormick helped with tricky SEM and FIB operations on many occasions. It was also a

pleasure to work with Dr. Qian Miao on a joint project before she graduated.

Lastly, I would like to extend my thanks to my family, friends, and labmates for all

the supports given in times of need. It had been a long journey, and I could not have come

this far without all the help along the way.

vi

Table of Contents 

List of Tables ........................................................................................................ viii

List of Figures .......................................................................................................... ix

Chapter 1 Introduction ............................................................................................ 1

Chapter 2 Basic Principles of Picosecond Ultrasonics ......................................... 3

2.1 Introduction ............................................................................................................................3

2.2 An Overview of Picosecond Ultrasonics ...............................................................................3

2.3 Optical Generation of Sound ..................................................................................................8

2.4 Optical Detection of Sound ..................................................................................................11

2.5 Brillouin Oscillations ...........................................................................................................14

Chapter 3 Picosecond Ultrasonics Using an Optical Mask ................................ 18

3.1 Introduction ..........................................................................................................................18

3.2 Surface Acoustic Waves .......................................................................................................20

3.2.1 Rayleigh Waves in Bulk Materials ................................................................................20

3.2.2 Rayleigh Waves in Thin Films ......................................................................................25

3.3 Experimental Implementation ..............................................................................................26

3.3.1 Precision Z-Tip-Tilt Stage .............................................................................................30

3.3.2 Rigorous Coupled-Wave Analysis (RCWA) .................................................................33

3.3 Results and Analysis ............................................................................................................38

Chapter 4 A Review of Planar Opto-Acoustic Microscopy ............................... 52

4.1 Introduction ..........................................................................................................................52

vii

4.2 The Resonant Fabry-Pérot Cavity ........................................................................................53

4.3 Metrology of 1D Periodic Nanostructures ...........................................................................58

Chapter 5 A Systematic Study on the Metrology of Planar Opto-Acoustic

Microscopy .............................................................................................................. 63

5.1 Introduction ..........................................................................................................................63

5.2 Sample Description ..............................................................................................................64

5.3 Experimental Results ............................................................................................................68

5.4 Data Analysis .......................................................................................................................71

5.4.1 Removal of Brillouin Oscillations and Thermal Signal ............................................72

5.4.2 Shape of Acoustic Echoes from Planar Sample ........................................................74

5.4.3 Finite-Difference Time-Domain Simulation for Planar Samples .............................83

5.4.4 Analysis on the Two Series of POAM Measurements .............................................87

Chapter 6 Conclusion and Future Work ............................................................. 96

 

viii

List of Tables 

Table 3.1. The frequencies found in the data measured from the three samples, by applying

the procedure as stated in the text. ..................................................................................44

Table 5.1. Catalogue of samples used in the first series of POAM measurements. ....................67

Table 5.2. Catalogue of samples used in the second series of POAM measurements. ...............67

Table 5.3. Summary of values used to calculate the sensitivity function in Figure (5.8a). See

Ref. [8] for the sources of these parameters. ..................................................................77

 

ix

List of Figures 

Figure 2.1. A schematic of the picosecond ultrasonics technique. The pump and probe

beams are focused onto the same spot. The pump beam generates an acoustic

pulse. The change in reflectivity is monitored by a time-delayed probe beam. ...............4

Figure 2.2. Schematics of the apparatus for picosecond ultrasonics. HWP – Half wave

plate. PBS – Polarized beamsplitter. EOM – Electro-optic modulator. RR –

Retroreflector. NBS – Non-polarized beamsplitter. BB – Beam blocker. M –

mirrors. L – lenses. ............................................................................................................5

Figure 2.3. The change in reflectivity plotted as a function of probe delay time, measured

from an Al film deposited onto a sapphire substrate. The thickness of the Al film is

179 nm, calculated from the difference in arrival time between successive echoes. ........7

Figure 2.4. A plot of Eqn (2.5) at different times. The solution can be divided into a time-

dependent part and a time-independent part. The time-dependent part is the strain

travelling away from z = 0. The time-independent part is the strain due to the

thermal expansion near z = 0. .........................................................................................10

Figure 2.5. The probe light reflects from the metal film (beam A) and from the

perturbation of refractive index due to the propagating strain pulse in the

transparent substrate (beams B and C). ...........................................................................15

Figure 2.6. An example of a result showing Brillouin oscillations. The measurement was

taken on an Al film deposited on quartz. The laser beams interrogated the sample

from the quartz side. .......................................................................................................16

Figure 3.1. The dispersion relation of Rayleigh waves in a structure composed of a ZnO

film on a Si substrate. R1, R2, and R3 are the dispersion of different Rayleigh

modes. Vt is the shear wave velocity of the substrate, and VR is the velocity of the

Rayleigh wave when the film is absent. ..........................................................................27

Figure 3.2. A schematic diagram of picosecond ultrasonic with optical mask. ..........................28

x

Figure 3.4. The SEM picture of the cross-section of the grating mask. The period of the

mask is measured to be 550 nm. .....................................................................................29

Figure 3.5. The degrees of freedom regarding the placement of the sample relative to the

mask. ...............................................................................................................................30

Figure 3.6. The side-view image of the fully assembled Z-tip-tilt stage. The bold-faced

letters are labels of parts as described in the text. ...........................................................31

Figure 3.7 (a). The bottom view of the top plate. (b). The top view of the main frame. The

bold-faced letters are labels of parts as described in the text. .........................................32

Figure 3.8. The RCWA simulation divides the z-axis into three regions. Region I is set up

to be SiO2 everywhere. Region II is further divided into sublayers. The distribution

of the refractive index in each sublayer is defined by a Fourier series. Region III is

set up to be Al everywhere. ............................................................................................33

Figure 3.9. The light intensity on the sample surface calculated using the RCWA for the

mask described in Fig. (3.4) at various air gap distances. The light is at normal

incidence, and is either in (a) TM polarization or (b) TE polarization. ..........................35

Figure 3.10. The light intensity on the sample surface calculated using RCWA for the

mask described in Fig. (3.4) but without the TiO2 layer, and at various air gap

distances. The light is at normal incidence, and is either in (a) TM polarization or

(b) TE polarization. .........................................................................................................36

Figure 3.11. The light intensity on the sample surface calculated using RCWA for the

mask described in Fig. (3.4) at various air gap distances. The light is at 8° from

normal incidence, and is either in (a) TM polarization or (b) TE polarization. ..............37

Figure 3.12. The plots of data measured from the 12 nm sample. (a) was taken without the

mask. (b) was taken when the mask was about 2 μm from the sample. (c) and (d)

were taken when the mask was closer but at un unreliably known distance from the

sample. ............................................................................................................................40

xi

Figure 3.13. The plots of data measured from the 120 nm sample when the mask was close

but at a distance from the sample that was not reliably known. .....................................41

Figure 3.14. The plots of data measured from the 335 nm sample. (a) was taken without the

mask. (b) was taken when the mask was about 2 μm from the sample. (c) and (d)

were taken when the mask was closer but at un unreliably known distance from the

sample. ............................................................................................................................41

Figure 3.15. The result of making a fit to the data of the 12 nm sample. This is a false

colored plot. ....................................................................................................................42

Figure 3.16. The result of making a fit to the data of the 120 nm sample. This is a false

colored plot. ....................................................................................................................43

Figure 3.17. The result of making a fit to the data of the 335 nm sample. This is a false

colored plot. ....................................................................................................................43

Figure 3.18. The frequency of Rayleigh wave plotted as function of film thickness. The

dotted line at the top (10.27 GHz) is the maximum frequency for any Rayleigh mode

without having energy radiating into the substrate. ........................................................44

Figure 3.19. Fit for the acoustic contributions in the data of all three films, using the

procedure described in the text. ......................................................................................48

Figure 4.1. A schematic diagram of POAM. ..............................................................................52

Figure 4.2. The structure of the cavity. A probe beam interrogating the cavity from the

substrate side will bounce off multiple times between the mirrors.................................54

Figure 4.3. Graphs of (a) R(γ) and (b) dR/dγ for several choices of |r|=|r1|=|r2|. ........................56

Figure 4.4. The cross-sectional SEM images of the first sample measured by POAM. The

length of the red bar is 100 nm. ......................................................................................58

Figure 4.5. The change in reflectivity contributed by acoustic echoes. ......................................59

Figure 4.6. The cross-sectional SEM images of (a) sample A and (b) sample B. .......................60

xii

Figure 5.1. A general geometry of the line structures. ................................................................65

Figure 5.2. An SEM image on the cross-section of the line structures. For this particular

patch, the pitch is 244 nm, the channel width is 80 nm, and the channel depth is 250

nm. ..................................................................................................................................65

Figure 5.3. A schematic diagram of the chemical-mechanical polishing machine in

operation. ........................................................................................................................66

Figure 5.4. The result of the POAM measurements. Series 1 is plotted in (a), and Series 2 is

plotted in (b). The plot numbers in these figures correspond to the sample number in

Table (5.1) and Table (5.2) for Series 1 and Series 2, respectively. The plots are

vertically displaced for clarity. ........................................................................................69

Figure 5.5. The results of the two series but with Brillouin oscillation and the decaying

background removed. Results from Series 1 are plotted in (a), and those from Series

2 are plotted in (b). The numbers of the plot corresponds to the sample numbers in

Table (5.1) for Series 1 and Table (5.2) for Series 2. The labels T, B, t, and etc. are as

discussed in the text. The plots are vertically displaced for clarity. ................................70

Figure 5.6. (a) A plot of data from the measurement on a planar sample when the water layer

is about 857 nm thick. (b) The plot of the data when the contribution from Brillouin

oscillations is filtered out. The plots are vertically displaced for clarity. .......................72

Figure 5.7. Acoustic signals of echoes from a planar sample at various water thicknesses. ......74

Figure 5.8. (a) The computed sensitivity function of the cavity using the parameters listed in

Table (5.1). (b) The simplified sensitivity function. The z-axis is drawn to scale. .........78

Figure 5.9. (a) The response function of the cavity. (b) The response function combined with

the additional Dirac delta pulse from signal generation of the cavity. Tsp is the transit

time for an acoustic pulse to cross the spacer layer and Tcav is single-trip time to

cross all cavity layers. .....................................................................................................79

xiii

Figure 5.10. (a) The result of picosecond ultrasonic measurement on the cavity. (b) The

derivative of the same result. The origins of the peaks A, B, C, and D are as

discussed in the text. .......................................................................................................81

Figure 5.11. The strain pulses generated inside the cavity. .........................................................82

Figure 5.12. The experimental results (red lines) are superimposed with the results (black

lines) of the FDTD simulations convoluted with the response function from

Fig.(5.13b).......................................................................................................................86

Figure 5.13. The black line is the result of a POAM measurement on a sample with pitch of

244 nm, channel width of 80 nm, and channel depth of 250 nm. For comparison, the

red line is the result of a POAM measurement on a planar sample. ...............................87

Figure 5.14. The results from FDTD simulation are compared to the experimental result.

The FDTD simulations were based on the dimensions measured using SEM. The

black lines are the experimental data from the same measurement. The green line is

the simulation result without slip boundary condition. The red line is the simulation

result assuming infinite slip length. ................................................................................89

Figure 5.15. The results of the first series of POAM measurements. The channel depths of

the samples are as listed. The red lines are the experimental results. The black lines

are results from FDTD simulations. ................................................................................90

Figure 5.16. The results of the second series of POAM measurements. The pitches of the

samples are as listed. The red lines are the experimental results. The black lines are

results from FDTD simulations. .....................................................................................92

Figure 5.17. The plot of the B/T ratio against the W/L ratio for each sample from series 2. .....93

 

1

 

Chapter 1 

Introduction 

Picosecond ultrasonics is a non-invasive technique for thin-film metrology. The technique uses

the pump-and-probe method to optically generate and detect acoustic waves. This thesis presents

the results of two variations of the picosecond ultrasonic technique. The first variation is

picosecond ultrasonics using an optical mask. The mask is a line-grating made from a transparent

material. When the mask is placed very close to the sample surface with the grating facing the

sample, the profile of the pump beam and the probe beam will be modified to give periodically

alternating stripes of bright and dark regions. This makes it possible to generate and detect

surface acoustic waves. The second variation is planar opto-acoustic microscopy (POAM).

POAM is a metrology technique that gives information about the surface profile of a sample. The

sound pulse is generated optically, and the transducer is coupled to the sample via a layer of

water. The detection of the echoes returning from the sample is enhanced by using a resonant

Fabry-Pérot cavity. Previous work has reported preliminary measurements using POAM. In this

work, we have investigated the limits of the measuring capabilities of POAM.

2

The outline of this thesis is as follows. Chapter 2 provides a basic overview of picosecond

ultrasonics. Chapter 3 presents the principles and the results of picosecond ultrasonics using an

optical mask. Chapter 4 describes the mechanism of POAM and reports the previous works.

Chapter 5 focuses on the new results of POAM. Finally, chapter 6 concludes this work and

discusses any possible points for future work.

3

 

Chapter 2 

Basic Principles of Picosecond Ultrasonics 

2.1 Introduction 

Picosecond ultrasonics is a non-intrusive technique that uses the pump-and-probe method to

optically generate and detect acoustic signals in order to measure the physical properties of nano-

structures. All experiments in this work are variants of the picosecond ultrasonics technique.

This chapter will provide an overview of the experimental apparatus and the operating principles

of picosecond ultrasonics.

2.2 An Overview of Picosecond Ultrasonics 

A schematic diagram of a picosecond ultrasonic experiment is shown in Fig. (2.1). In this

example, the sample is a thin film deposited onto a substrate. The pump beam and the probe

beam are focused onto the same spot on the film surface. The arrival of the probe pulse is

delayed relative to the pump pulse by a time t . When the energy from the pump pulse is absorbed,

the temperature near the film surface will rise by an amount typically between 1 K and 10 K,

causing a build-up of thermal stress. When the thermal stress relaxes, a strain pulse is launched

4

into the film. The strain pulse will be partially reflected at the interface between the film and the

substrate because of the difference in the acoustic impedance. When the reflected part of the

strain pulse returns to the free surface of the film, the optical properties are perturbed and there is

a small change in the optical reflectivity of the probe pulse. The change in reflectivity R is

monitored over a range of delay time in order to detect the arrival time of the returning strain

pulse.

The basic apparatus for picosecond ultrasonics measurement is shown in Fig. (2.2). The laser

source is a mode-locked Ti-Sapphire laser with operating wavelength tunable within the range

780-830 nm. The pulse width is approximately 200 fs and the repetition rate is 80 MHz. The

Fig. 2.1. A schematic of the picosecond ultrasonics technique. The pump and probe beams are focused onto the same spot. The pump beam generates an acoustic pulse. The change in reflectivity is monitored by a time-delayed probe beam.

Film Substrate

Acoustic pulse

Pump beam arriving at t

0

Probe beam arriving at t

0 + t

5

laser output is split into two beams, the pump beam and the probe beam, via a half-waveplate

(HWP), immediately followed by a polarized beamsplitter (PBS).

After the PBS, The pump beam passes through an electro-optic modulator (EOM). The EOM

modulates the intensity of the pump beam at 1 MHz, so that we can use lock-in detection to

improve the signal-to-noise ratio. The modulation of the pump beam will result in a modulation

of the reflectivity of the reflected probe beam. The lock-in will then amplify the 1 MHz

component in the intensity of the probe beam.

To vary the delay time of the probe beam with respect to the pump beam, the probe beam is

directed to a pair of retroreflectors (RR) mounted on a translational stage. The arrival time of the

Ti‐Sapphire 800 nm

Mirror Translation Stage

Photodiode

Function Generator

EOM

Lock‐In Amplifier

EOM Driver

HW PBS

NB

RR1

RR2

BB1

BB2

Reference Signal

Signal Output

Signal Input

Signal Output

GPIB

Sample

M1

M2 M3

M4 M5

L1 L2

Fig. 2.2. Schematics of the apparatus for picosecond ultrasonics. HWP – Half wave plate. PBS – Polarized beamsplitter. EOM – Electro-optic modulator. RR – Retroreflector. NBS – Non-polarized beamsplitter. BB – Beam blocker. M – mirrors. L – lenses.

6

probe pulse is controlled by translating the RR’s parallel to the probe beam path, where the

change in the arrival time t is related to the translated distance D by 4t D c . Here c is the speed

of light in air. It is extremely important that the movement of the stage is parallel to the probe

beam path, otherwise the probe beam reflected from the retroreflectors will move transversely as

the stage moves. As a result, the focused spot of the probe beam will drift, disturbing the overlap

between the pump beam and the probe beam. This effect is called “beam walk”, and it will

introduce an artificial background signal to the data.

The pump beam and the probe beam are then focused onto the sample, with the probe beam at

normal incidence and the pump beam at an angle of incidence. The pump beam and the probe

beam are arranged to have different incident angle and polarization, so that the signal from the

scattered pump beam can be reduced. The focal length of the lens is about 10 cm, and the spot

size is approximately 30 μm. A non-polarized beamsplitter (NBS) is installed before the lens of

the probe beam, so that the probe beam reflected from the sample can be directed to a photodiode.

The signal from the photodiode is fed into the high frequency lock-in amplifier running at 1 MHz.

The lock-in amplifier is synchronized to the EOM via a reference signal from the same function

generator that drives the EOM. Finally, the output from the lock-in amplifier is collected by a

data acquisition (DAQ) card installed in a computer. The computer also controls the translation

stage via a GPIB interface. By synchronizing the position of the translational stage and the data

collected from the lock-in, the computer records the change in reflectivity of the probe beam as a

function of the probe delay time.

7

Fig. (2.3) shows a plot of the change in reflectivity R as a function of the probe delay time,

from a measurement done on an Al film deposited onto a sapphire substrate. The zero of the

probe delay time corresponds to the time when both the pump pulse and the probe pulse arrive

on the sample surface simultaneously. The jump in reflectivity at zero delay time is caused by

the temperature rise due to heat deposited in the Al film from the pump pulse. After the initial

jump, the slow decay in the change in reflectivity is the result of the Al film cooling off by heat

flowing into the sapphire substrate. On top of the slow decay, there are bumps that appear every

56 ps. Each bump corresponds to the arrival of a returning strain pulse at the free surface of the

0 50 100 150 200 250

0.0

2.0x10-6

4.0x10-6

6.0x10-6

8.0x10-6

1.0x10-5

R

(V

)

Probe Delay Time (ps)

Fig. 2.3. The change in reflectivity plotted as a function of probe delay time, measured from an Al film deposited onto a sapphire substrate. The thickness of the Al film is 179 nm, calculated from the difference in arrival time between successive echoes.

8

Al film after being reflected at the Al-sapphire interface. The time t 56 ps between each echo

corresponds to the round-trip time inside the Al film, and is related to the thickness of the Al film,

Ald by 2 /Alt d , where is the longitudinal sound velocity inside the Al film. If we assume

that the sound velocity of the Al film is the same as that in the bulk material ( 6.4 nm/ps),

then the thickness of the Al film is found to be 179 nm. The amplitude of the echoes decreases

successively. This is the result of strain pulse being only partially reflected at the Al-sapphire

interface. If the bonding between the Al film and the sapphire substrate is good, then the acoustic

reflection coefficient at the interface is al sapp

al sapp

Z Z

Z Z

, where alZ and sappZ are the acoustic impedances of

the Al film and sapphire respectively. The acoustic impedance is the product of the material

density and the sound velocity. On the other hand, if the bonding between the Al film and the

substrate is poor, then one can expect a greater value for the reflection coefficient [1].

2.3 Optical Generation of Sound 

For our purposes, it is sufficient to describe the mechanism behind signal generation in terms of

a thermal expansion model [2]. When a pump pulse with energy Q arrives at the free surface of

the sample, a fraction R of the light will be reflected, while the rest is absorbed. Let A be the

area of the surface onto which the pump light is directed, d the film thickness, and ζ the

absorption length. If the film is much thicker than the absorption length, the resulting change in

temperature T of the sample material at depth z is

( ) (1 )z

QT z R e

A C

. (2.1)

9

Here C is the specific heat per unit volume. The change in temperature introduces an isotropic

thermal stress 3 ( )B T z where B is the bulk modulus and the linear thermal expansion

coefficient is . The sample material is assumed to be elastically isotropic. Since the stress only

depends on z, there is only motion along the z direction and zz is the only non-zero component

of the elastic strain tensor. Therefore the relevant equations of elasticity are

1

3 3 ( )1zz zzB B T z

, (2.2)

2

2z zzu

t z

, (2.3)

zzz

u

z

. (2.4)

Here is Poisson’s ratio, is the material density, and zu is the displacement in the z direction.

Under the condition that the initial strain is zero everywhere and that zz at the free surface z =

0 is always zero, the solution for zz while the sound pulse is propagating across the film is

// /1 12 2

1( , ) (1 ) 1 sgn

1z tz t

zz

QBz t R e e e z t

A C

. (2.5)

Here is the longitudinal sound velocity, given by

2 13

1

B

. (2.6)

The solution given in Eqn. (2.5) is plotted in Fig. (2.4) at different instances of time. At large

times, the strain can be divided into two parts. One part is the time-independent strain due to the

10

thermal expansion in the region near z = 0. The other part is a bipolar strain pulse propagating

away from the free surface at the speed of longitudinal sound. The strain pulse has an effective

wavelength of ~ 2 . The amplitude of the strain pulse generated in metal is usually on the order

of 10-5.

It has been assumed thus far that the temperature rise occurs instantaneously. Within the finite

duration 0 of the pump pulse, a strain pulse would travel a distance 0 . If this distance 0 is

2 4 6 8

0

t = 8v/t = 4v/ t = 6v/

Str

ain

(arb

. uni

ts)

z/

0

t = 2v/

Fig. 2.4. A plot of Eqn (2.5) at different times. The solution can be divided into a time-dependent part and a time-independent part. The time-dependent part is the strain travelling away from z = 0. The time-independent part is the strain due to thermal expansion near z = 0.

11

significantly shorter than the spatial length of the strain pulse, then any effects from the finite

duration of the light pulse becomes unimportant. This gives the condition 0 . For the

experiments described in this thesis, the film will always be aluminum, with longitudinal sound

velocity 64 Å/ps, and absorption depth 77 Å for light at 800 nm wavelength. The

condition then becomes 0 1.24 ps, which is satisfied in the experiments. On the other hand, it

has also been assumed that the temperature at all points remains constant after the rise. However,

the diffusion of heat results in a large change in the shape of the sound pulse as discussed in ref

[2].

 

2.4 Optical Detection of Sound 

We use a time-delayed probe pulse to measure the change in reflectivity R of the surface of the

film. The propagating strain causes a change in the optical constants of the film and this results

in a change in the reflection of the probe.

Consider first the reflection of light in the absence of the strain due to the sound pulse. For an

incoming light wave at normal incidence propagating in the z-direction and polarized in the x-

direction in the plane of the film, the electric field is

0( )0

i k z txE E e . (2.7)

The reflected and transmitted waves are

12

0( )( )0 0

i k z trxE r E e , (2.8)

( ) ( )0 0

t i kz txE t E e , (2.9)

where k0 and k are the wave numbers in air and in the film, respectively. Since the refractive

index of air is close to unity, we can set 0 02 /k where 0 is the light wavelength in vacuum,

and 0k k n i . Then the reflection and transmission coefficients are

00

0

1

1

k k n ir

k k n i

, (2.10)

00

0

2 2

1

kt

k k n i

. (2.11)

In the presence of strain, R changes because the Maxwell equation inside the dielectric medium

becomes

2 2

2 2

( )( ( , )) ( )x

x

E zz t E z

z c

, (2.12)

where 2( )n i is the dielectric constant, and

( , ) 2( ) ( , )zzzz zz

nz t n i i z t

(2.13)

is the change in the dielectric constant due to strain. Let r be the reflection coefficient in the

presence of strain. To find r , first consider the case when the strain only exists in a thin layer at

a distance 'z from the free surface, i.e., when

13

( ')F z z , (2.14)

where F gives the magnitude of the change. The light wave transmitted into the film will be

partially reflected at 'z with reflection coefficient 1r , given by

20

1 2

ikr F

k . (2.15)

This can be obtained from Eqn. (2.13), and the details were given in Ref. [2]. This reflected wave

is partially transmitted across the free surface of the film. As a result, the reflected wave outside

the film is

0

2( )( ) 2 '0

0 0 0 0( )2

i k z tr ikzx

ikE r Ft t e E e

k , (2.16)

where 0t is the transmission coefficient from the film to air, i.e.

00

2 2 2

1

k n it

k k n i

. (2.17)

If we now consider an arbitrary strain, then the reflection coefficient outside the film is

22 '0

0 0 0 0

0

( ', ) '2

.

ikzikr r t t e z t dz

kr r

(2.18)

The change in reflectivity is

2 2

0 0R r r r . (2.19)

Combining the results, we find that the change in reflectivity correct to the first order of strain is

14

2* 20

0 0 0 0

0

( ) 2Re 2( ) ( , )2

( ) ( , ) .

ikzzz

zz zz

zz

ik nR t r t t n i e i z t dz

k

f z z t dz

(2.20)

Here ( )f z is called the “sensitivity function”, and it determines how strain at different depths z

contributes to the change in reflectivity. If ,zz

n

, andzz

have constant values throughout a

half-space, i.e., if the film is taken to be infinitely thick, then the sensitivity function can be

found analytically. The result takes the form of an exponentially-damped oscillation, as ( )f z

goes to zero for z . This means that determines the range of the sensitivity function. As a

result, the sensitivity function of an infinitely thick film can be applied to any film of thickness

d . However, if, for example, there are multiple films of thicknesses id and each id is

comparable to or less than i , the absorption length of each respective film, then the form of

( )f z can become quite complicated, and an analytical solution is not guaranteed. There is a

method to calculate ( )f z numerically, and it will be discussed further in section 5.4.2.

2.5 Brillouin Oscillations 

Consider the simpler case of a thin metal film deposited onto a transparent substrate, as

illustrated in Fig. (2.5). The probe beam is focused onto the metal film through the transparent

15

substrate. A strain pulse is launched into the substrate after the metal film is excited by the pump

beam. The probe beam reflected by the metal film is labeled as “A” in Fig. (2.5). The probe

beam is also reflected by the perturbation of refractive index due to the strain pulse propagating

inside the transparent substrate; these reflected beams are labeled “B” and “C”. Only the beams

reflected once off the strain pulse are considered. The relative phase difference between beam A

and beams B and C depends on the distance that the strain pulse has traveled. The beams B and

C will interfere with beam A either constructively or destructively as the strain pulse propagates

into the substrate. As a result, the reflectivity of the probe beam will oscillate with a period T,

and it has been shown [3] that

Fig. 2.5. The probe light reflects from the metal film (beam A) and from the perturbation of refractive index due to the propagating strain pulse in the transparent substrate (beams B and C).

Metal Film Substrate

Strain Pulse

B

A

C

16

2 22 sin

Tn

, (2.21)

where is the wavelength of the probe beam in vacuum, is the sound velocity in the substrate,

n is the index of refraction of the substrate, and is the angle of incidence of the probe beam.

This oscillation in the reflectivity is called Brillouin oscillations.

Figure (2.6) is an example of data showing Brillouin oscillations. The sample was an Al film

deposited onto a quartz substrate. The probe beam was at normal incidence, and the wavelength

of the probe light was 800 nm. The period of the oscillation is 41.4T ps. The refractive index

of quartz is 1.54n , thus the sound velocity in quartz is calculated to be 6.3 nm ps-1.

-250 0 250 500 750 1000 1250 1500

0.0

5.0x10-6

1.0x10-5

1.5x10-5

2.0x10-5

2.5x10-5

R

(V

)

Delay Time (ps)Fig. 2.6. An example of a result showing Brillouin oscillations. The measurement was taken on an Al film deposited on quartz. The laser beams interrogated the sample from the quartz side.

17

Bibliography 

[1] Tas, G., Loomis, J.J., Maris, H.J., Bailes III, A.A., Seiberling, L.E. “Picosecond ultrasonics study of the modification of interfacial bonding by ion implantation.” J. Appl. Phys. 72, 2235 (1998).

[2] Thomsen, C., Grahn, H.T., Maris, H.J., and Tauc, J. “Surface generation and detection of phonons by picosecond light pulses.” Phys. Rev. B 34, 4129 (1986).

[3] Lin, H.-N., Stoner, R.J., Maris, H.J., and Tauc, J. “Phonon attenuation and velocity measurements in transparent materials by picosecond acoustic interferometry.” J. Appl. Phys. 69, 3816 (1991).

18

 

Chapter 3 

Picosecond Ultrasonics Using an Optical Mask 

3.1 Introduction 

In a conventional picosecond ultrasonics configuration, the area of the focused pump light on the

sample surface typically has a linear dimension D of some tens of microns. Because D is

usually much greater than the distance that the light penetrates into the sample, the generated

sound is a longitudinal strain pulse with particle displacement normal to the sample surface [1].

On the other hand, generating transverse sound has been possible only by using specially

prepared samples [2][3][4]. Since surface acoustic waves (Rayleigh waves) involve both

longitudinal and transverse motion, there would be many advantages to extend the technique to

study surface acoustic waves. One approach to generate surface acoustic waves using picosecond

ultrasonic has been to pattern the sample surface so that the rise in temperature due to the

application of pump light varies periodically across the surface [5][6][7]. Patterning the sample

involves modifying the geometry of the sample; as a metrology technique it cannot be considered

non-invasive. Another approach is to use the transient grating method [8], a well-established

technique in ultrafast optics [9]. The pump light is divided into two beams, which are then

19

overlapped at the sample surface at angles of plus and minus relative to normal incidence.

This gives a varying light intensity with a period of 2 sin , where is the wavelength of the

pump light. Thus this method is limited by the shortest wavelength of the light source available.

The transient grating method also requires a much more elaborated optics setup than that

required for standards picosecond ultrasonics [10].

This chapter describes an alternative method for the generation and detection of surface acoustic

waves [11]. We place a transparent optical mask very close to the sample surface. This method

can be considered non-invasive, since there is only a very light contact between the mask and the

sample. The mask has a series of grooves on the side facing the sample, and as a result the pump

and the probe light are modified to give periodically varying intensity across the sample surface.

Using this technique, we are able to generate and detect surface acoustic waves with wavelength

equal to the period w of the mask, or w divided by an integer.

The outline of this chapter is as follows. Section 3.2 reviews the general theory of surface

acoustic waves in bulk material and in a thin film on a substrate. Section 3.3 describes some

experimental details and considerations regarding the optical mask. Section 3.4 reports the

experimental results and compares the results with theoretical calculations. The work described

in the chapter was performed as a joint project with Qian Miao.

20

3.2 Surface Acoustic Waves 

Originally the studies on surface acoustic waves were started as investigations to understand

earthquakes. In 1885, Lord Rayleigh published a mathematical theory on one type of surface

acoustic wave (Rayleigh wave) [12]. In the subsequent years, other types of surface acoustic

waves were discovered with different sets of boundary conditions, e.g., Love waves [13] and

Stoneley waves [14]. In recent decades, there has been a renewed interest in the studies of

surface acoustic waves, primarily due to their applications to thin-film technologies [15]. In this

work we will focus on Rayleigh waves.

3.2.1 Rayleigh Waves in Bulk Materials 

First we start with the equations of elasticity

2

2

iji

j

u

t x

, and (3.1)

kij ijkl

l

uc

x

. (3.2)

Here is the density of the elastic medium, iu is the particle displacement vector, ij is the

stress tensor, and ijklc are the tensor components of the elasticity modulus. Take the surface of

the elastic medium to be the plane 0z Let the elastic medium fill the space where 0z and

let the medium have cubic symmetry. If we consider a wave travelling in the x-direction, its

motion confined in the x-z plane, then the equations of elasticity are

21

2

2x xx xzu

t x z

,

2

2xzz zzu

t z x

, (3.3)

11 12x z

xx

u uc c

x z

,

11 12xz

zz

uuc c

z x

,

44x z

xz

u uc

z x

. (3.4)

Combining Eqn. (3.3) and Eqn. (3.4), we get

2 2 22 2

11 12 442 2 2x x xz zu u uu u

c c ct x x z z x z

,

2 22 2 2

11 12 442 2 2x xz z zu uu u u

c c ct z x z x x z

. (3.5)

We are looking for a solution in which the motion is periodic in the x-direction and goes to zero

as z . The solution also needs to satisfy the traction boundary conditions at the surface.

First consider displacements of the form

( )( , ) i kx t zx xu x t A e e ,

( )( , ) i kx t zz zu x t A e e . (3.6)

If we plug these into Eqn. (3.5) we get

22

2 2 211 44 12 44( ) 0x zk c c A ik c c A ,

2 2 212 44 44 11( ) 0x zik c c A k c c A . (3.7)

These equations have non-trivial solutions for xA and zA if

4 2 0a b c , (3.8)

where

11 44a c c ,

2 2 2 2 2 244 44 11 11 12 44( ) ( ) ( )b c k c c k c k c c ,

2 2 2 211 44( )( )c k c k c . (3.9)

Eqn. (3.8) gives two solutions of that are real and positive as required by the boundary

condition. For each of the values of , there is an associated ratio x zA A

12 442 2 2

11 44

( )x

z

A k c ci

A k c c

. (3.10)

Thus, the solution for Eqn. (3.5) can be written as

1 2( ) ( )1 2( , ) z zi kx t i kx t

x x xu x t A e e A e e ,

1 2( ) ( )1 2( , ) z zi kx t i kx t

z z zu x t A e e A e e , (3.11)

where 1 1x zA A and 2 2x zA A are obtained by setting 1 and 2 , respectively, in Eqn.

(3.10).

23

The boundary conditions at the free surface are 0zz xz . Therefore, at 0z , we have the

conditions

11 12 0xz uuc c

z x

,

44 0x zu uc

z x

. (3.12)

These conditions set further constraints between the coefficients 1xA , 2xA , 1zA , and 2zA . As a

consequence, a non-trivial solution exists for only one particular value of . For an isotropic

medium, the constants of elasticity are related by 12 11 442c c c . As a result, Eqn. (3.8) can be

reduced to

2 2 2 2 2 211 11 44 44 0k c c k c c , (3.13)

where 2

21 2

l

kc

and 2

22 2

t

kc

are the solutions to Eqn. (3.13). Here lc is the

longitudinal sound velocity and tc is the transverse sound velocity. lc and tc are equal to 11c

and 44c

, respectively. Applying the conditions from Eqn. (3.10) and Eqn. (3.12) gives

2 21 2 1 2 22 0x xA k A ,

2 2 22 1 22 0x xk A k A . (3.14)

The solution for Eqn. (3.14) exists if and only if 22 2 21 2 24 k k . After squaring this result

and substituting the values for 1 and 2 , we get

24

22 2 22 4 2 2

2 2 22 16

t l t

k k k kc c c

, (3.15)

which gives the relationship between and k . If we let

2 2 2tc k , (3.16)

then, Eqn. (3.15) can be written as

2 2

3 22 2

8 8 3 2 16 1 0t t

l l

c c

c c

. (3.17)

Therefore only depends on the ratio t lc c , which is a constant characteristic of the given

material. The ratio /t lc c is related to Poisson’s ratio by (1 2 ) (2 2 )t lc c . The

quantity must be real and positive so that and k are real; To keep 1 and 2 real, must

also be less than one. There is only one root from Eqn. (3.17) that satisfies these conditions, so

there is only one value of for any given value of t lc c . Thus, the frequency of the surface

wave is proportional to its wave number, and the velocity U of propagation of the surface wave

is

tU c . (3.18)

   

25

3.2.2 Rayleigh Waves in Thin Films 

The same type of analysis given in the previous section can also be extended to consider surface

waves in a thin film coated on a half-space substrate. This section provides just the summary of

the analysis.

In the film, we now have to include both positive and negative real solutions of . On the other

hand, the solutions of in the substrate need to be real and positive so that the solutions go to

zero as z . As a result, we are considering 4 solutions in the film and 2 in the substrate,

giving six solutions in total. At the free surface of the film, the traction boundary condition that

0zz xz still applies. At the boundary between the film and the substrate, stress components

and particle displacements must be continuous. Thus there are six solutions and six boundary

conditions, which together give the components of a 6 6 matrix. A frequency has to be chosen

to make the determinant of this matrix vanish in order to obtain a non-trivial solution. It is

necessary to use a numerical method to find the frequency.

Unlike the dispersion of a Rayleigh wave on a half-space, where the frequency is proportional to

k , the dispersion relation is more complex when there is a film on a substrate. In order to verify

our computer program and algebra, we have calculated the dispersion of Rayleigh waves for

ZnO films on isotropic silicon; this was previously calculated in Ref.[16]. We take the ZnO to be

elastically isotropic with 1211 2.043 10c , 12

12 1.133 10c , and 1244 0.455 10c cgs units, and

5.675 g/cm3. For the silicon we take 1211 1.865 10c , 12

12 0.535 10c , and

26

1244 0.665 10c , and 2.331 g/cm3 [17]. The result of our calculation is shown in Fig. (3.1)

as a plot of the phase velocity as a function of the parameter kh , where h is the thickness of the

film. As kh increases, the phase velocity of each Rayleigh mode decreases, and higher order

Rayleigh modes appear. The results agree with Ref. [16].

 

3.3 Experimental Implementation 

In this method, the optically transparent mask is suspended immediately above the sample as

shown in Fig. (3.2). Both the pump and probe light are focused on the sample surface through the

Fig. 3.1. The dispersion relation of Rayleigh waves in a structure composed of a ZnO film on a Si substrate. R1, R2, and R3 are the dispersion of different Rayleigh modes. Vt is the shear wave velocity of the substrate, and VR is the velocity of the Rayleigh wave when the film is absent.

27

mask, and the diameters of the focused beam spots are 20 microns. The intensity of the light

varies periodically across the sample surface if the air gap between the mask and the sample is

sufficiently narrow. The sample is an Al thin film deposited on Si substrate.

The grating used in this work was made by the method described in Ref. [18]. The fabrication

steps are shown schematically in Fig. (3.3). A layer of Cr of 5 nm thickness was first deposited

on top of the SiO2 wafer, then a layer of photoresist (PR) was coated on top of the Cr layer (a).

The photoresist was patterned using holographic lithography (b). The pattern was transferred to

Fig. 3.2. A schematic diagram of picosecond ultrasonic with optical mask.

Mask

Air gap

Probe light Pump light

Al film

Si substrate

28

the Cr layer by etching Cr through the gaps of the photoresist (c), and then the photoresist was

removed (d). The pattern Cr layer then became the hard mask for etching SiO2. The exposed

portion of the silica was etched away to a desired depth (e). The Cr mask was then stripped away

(f). Lastly, a 50 nm layer of TiO2 was coated onto the structure using atomic layer deposition (g)

[19]. The TiO2 layer was added because of its high index of refraction. Fig. (3.4) shows an SEM

[20] picture of cross-section of the mask used in this set of experiments; the period w of the

mask was measured to be 550 nm.

Cr TiO2 SiO2

v

(a)

(c)

(d)

PR

(g)

(f)

(e)

(b)

Fig. 3.3 The procedure for fabricating the gratings on the mask. The details of the procedure are described in the text.

29

In an earlier attempt at an experiment similar to the experiment presented here, the mask was

simply placed resting on the sample surface [21]. The pitch of the mask was 1 micron. The

sample was ~1 micron Al film on Si substrate. It was possible to detect signals from a surface

acoustic wave from this setup. The measured frequency was 2.73 GHz, and was in reasonable

agreement with that expected from the calculations based on the mask period and the velocity of

the Rayleigh wave in Al. However, the signal-to-noise ratio in the experiment was poor, and

often times the signal due to surface waves could not be detected despite having the mask resting

on the sample. It is natural to assume that the failure to observe a signal resulted from dirt

particles stuck in between the mask and the sample preventing the mask from reaching close

proximity to the sample. In the present experiment, in order to minimize the chance of failure

due to dirt, we carefully cleaned the mask and the sample, and performed the experiment inside a

mini clean-room placed on the optical table. The area of the mask was also reduced to 2 6 mm2

in order to decrease the chance of the mask resting on a dirt particle somewhere on the sample.

550

216 nm

243 nm

190 nm

Fig. 3.4. The SEM picture of the cross-section of the grating mask. The period of the mask is measured to be 550 nm.

30

We have also mounted the mask and the sample on a precision Z-tip-tilt stage to gain better

control over the positioning of the sample and the mask.

3.3.1 Precision Z‐Tip‐Tilt Stage 

There are three critical degrees of freedom regarding the placement of the sample relative to the

mask: the gap distance d , and the tip/tilt angles x and y . d and x are shown in Fig. (3.5),

and y is the tilt about an axis perpendicular to the axis for x . It is crucial to keep these degrees

of freedom fixed throughout the course of a measurement. Thomas Grimsley, a graduate student

who completed his thesis in 2012, built a precision Z-tip-tilt stage capable of such positioning

control for his work [22]. This stage has been used in other experiments in our lab, including the

present work. A thorough description of the stage is included in Ref. [22]. A brief summary is as

follows.

d θ

x

Fig. 3.5. The degrees of freedom regarding the placement of the sample relative to the mask.

31

A photograph of the fully assembled Z-tip-tilt stage is shown in Fig. (3.6). The stage consists of

two main parts: the top plate and the main frame. Figure (3.7a) is a photograph of the top plate

from underneath, and Fig. (3.7b) shows a photograph looking down onto the main frame. The

top plate is with a holder piece H1 in its center where the mask is held. The top plate rests on the

main frame and is fixed firmly by the ball bearings B seated in the three v-blocks V bolted on the

supporting pillars P on the main frame. The position of the mask is fixed. On the main frame, the

sample is held on the sample holder H2 fixed on the sample plate K, which is supported by three

piezo-electric actuators T. The sample plate K rests firmly on the actuators via a ball bearing/v-

block mechanism like the ones supporting the top plate. The parameters d , x , and y can be

Fig. 3.6. The side-view image of the fully assembled Z-tip-tilt stage. The bold-faced letters are labels of parts as described in the text.

T

P

V

32

controlled by adjusting the actuators. The distance of the sample plate K relative to the invar

plate A is measured by three capacitive displacement sensors S1-S2. During operation, the

actuators T are continuously adjusted by a controller so that the readings from the capacitive

displacement sensors are maintained at a defined value. The stage is capable of maintaining d to

an accuracy of 1 nm, and x and y can be controlled to 410 radians.

It is essential to keep the mask parallel to the sample surface, and any tip/tilt misalignment can

cause a variation in the gap distance across the area of focused light beam. To ensure that the

mask is parallel to the surface of the sample, the direction of He-Ne light reflected from the mask

is compared to the direction of the light reflected from the sample.

Fig. 3.7(a). The bottom view of the top plate. (b). The top view of the main frame. The bold-faced letters are labels of parts as described in the text.

P

V

H1

H2 B S1

S2 K

(a (b

33

3.3.2 Rigorous Coupled‐Wave Analysis (RCWA) 

Qian Miao wrote a Fortran program based on RCWA to calculate the profile of the light intensity

on the surface of the sample based on the configuration of the mask and the incident light. The

full detail of the program can be found in her thesis [23], but a brief overview is as follows.

The computation only needs a two-dimensional space since the grating is symmetrical in one

dimension, say along the y-axis. As shown in Fig. (3.8), the vertical z-axis is divided into three

regions. Region I is set up to be SiO2 everywhere and region III is configured to be Al

everywhere. Region II is divided up into sublayers of horizontal slabs. The variation of the index

Fig. 3.8. The RCWA simulation divides the z-axis into three regions. Region I is set up to be SiO2 everywhere. Region II is further divided into sublayers. The distribution of the refractive index in each sublayer is defined by a Fourier series. Region III is set up to be Al everywhere.

x

z

Region I: SiO2

Region III: Al

Region II

34

of refraction in each slab is represented by a Fourier series. The program expands the x

components of the electric field and the magnetic field as a Fourier series. Each term in the series

has different amplitude in each slab. The fields in Region I include the contribution from both

the incident light and the reflected light, whereas the fields in Region III only have transmitted

light. The program calculates the coefficients in each slab by matching the tangential

components of the fields at all boundaries, so that Maxwell’s equations are satisfied in all regions.

By recombining the Fourier modes based on the calculated coefficients at region III, the program

recovers the spatial profile of the transmitted light.

In order to increase the accuracy of the simulation, it is necessary to increase the number of slabs

and the number of diffraction orders. As a result, the program often runs into the issue of

inverting almost singular matrices when calculating the coefficients. Inverting this kind of matrix

can easily cause computation failure or erroneous results due to truncation errors. Therefore, one

important aspect of RCWA involves finding a method to compute the inverse of almost singular

matrices. This method has been discussed in the literature on RCWA [24], but is beyond the

scope of this work, so it will not be further discussed here.

We have calculated for both TE and TM incident light. It is important to calculate both modes

because the probe light and the pump light are arranged to be in different polarizations as a

standard practice to reduce the noise from scattered pump light. Figure (3.9) shows a plot of the

electromagnetic energy density 2 2/ 8I E B at the sample surface along the axis

perpendicular to the grating lines. The result for the same grating but without the TiO2 layer is

35

Fig. 3.9. The light intensity on the sample surface calculated using the RCWA for the mask described in Fig. (3.4) at various air gap distances. The light is at normal incidence, and is either in (a) TM polarization or (b) TE polarization.

36

Fig. 3.10. The light intensity on the sample surface calculated using RCWA for the mask described in Fig. (3.4) but without the TiO2 layer, and at various air gap distances. The light is at normal incidence, and is either in (a) TM polarization or (b) TE polarization.

37

Fig. 3.11. The light intensity on the sample surface calculated using RCWA for the mask described in Fig. (3.4) at various air gap distances. The light is at 8° from normal incidence, and is either in (a) TM polarization or (b) TE polarization.

38

shown in Fig. (3.10). The light is at normal incidence for both figures. It can be seen that the

TiO2 layer greatly enhances the contrast between the maximum and the minimum intensity for

light with TM polarization. On the other hand, such contrast is small for light with TE

polarization, with or without the additional TiO2 layer. This suggests that an experiment with

both beams at normal incidence but with different polarization is unlikely to be successful.

Therefore, we have investigated the effect of changing the beam orientation. Fig. (3.11) shows

the RCWA result of light at 8° from normal incidence, where the plane of incidence is normal to

the grating lines. For the TE light, this change in the angle gives a large increase in contrast

between the maximum and minimum light intensity on the sample surface along the axis normal

to the grating. Therefore, the simulation suggests the configuration of TE pump light at 8° angle

of incidence and TM probe light at normal incidence; we used this as our standard configuration

for all of the measurements. We have also rotated the mask by 90° so that the configuration

became TM pump at 8° angle of incidence and TE probe light at normal incidence. When we did

this we were not able to detect signal from the surface acoustic waves above the noise level. We

believe this is because for TE probe light at normal incidence the contrast in intensity is small.

 

3.4 Results and Analysis 

In this section, we discuss the experimental results on three samples of Al films with different

thicknesses on a (100) Si substrate. The Al films were deposited onto silicon wafers using

plasma sputtering. The thicknesses of the Al films are 12 nm, 120 nm, and 335 nm; the

thicknesses were measured by SEM. The mask was mounted on H1 and the sample was mounted

on H2 of the Z-tip-tilt stage as mentioned in section 3.3.1. The pump light and the probe light

39

were in the standard configuration just discussed. The beam spot size was about 20 microns in

diameter and so by controlling the angle between the mask and the sample to 410 radians, the

variation in the separation distance between the mask and the sample surface across the area of

the beam spot was only about 2 nm.

We had hoped to make a series of measurements as a function of the separation distance between

the mask and the sample, but tracking the spacing became difficult when the distance was less

than 1 μm. When the mask is nowhere in contact with the sample surface, the Z-tip-tilt stage can

make very precise change to the position of the sample and maintain that position. Under this

condition, we confirmed that the change in position reported by the controller corresponded to

the actual change in position. We tested this by measuring the intensity of the probe light at

different separation distance between the mask and the sample. The intensity varied periodically

as a function of the separation distance, and the period was 400 nm as expected. However, when

the sample surface was in close proximity to the mask, the change in intensity of the reflected

probe light was smaller than what was expected from the change in the spacing as indicated by

the stage controller. As a result, we could not obtain a definite value for the separation distance

between the mask and the sample. We believe this was caused by some part of the sample

coming into contact with the mask when the separation distance was small. This could be due to

dirt stuck between the mask and the sample, or to the sample or the mask being insufficiently flat.

This problem can be reduced by decreasing the area of the mask, but we have not done this.

40

Figure (3.12) shows the result for the change in reflectivity R as a function of probe delay time

for a sample with 12 nm Al on Si. Curve (a) shows the data obtained without the mask. The

change in reflectivity is due to the change in temperature of the Al film (thermo-reflectance).

There should be an echo arriving at ~4 ps, due to the reflection of the sound pulse at the interface

between the Al and the Si. However, it is difficult to identify the echo at this time because of the

rapidly varying signal from thermo-reflectance. Curve (b) shows the data measured with the

mask at approximately 2 μm from the sample surface. Surface waves can be seen in curve (c)

and (d), which are with the mask at a closer but not reliably known distance. Figure (3.13) shows

the plot of the data for the 120 nm Al film. Figure (3.14) shows the plots of the data for the 335

nm Al film, and the distance progression from (a) to (d) follows that of Fig. (3.12).

Fig. 3.12. The plots of data measured from the 12 nm sample. (a) was taken without the mask. (b) was taken when the mask was about 2 μm from the sample. (c) and (d) were taken when the mask was closer but at un unreliably known distance from the sample.

41

Fig. 3.13. The plots of data measured from the 120 nm sample when the mask was close but at a distance from the sample that was not reliably known.

Fig. 3.14. The plots of data measured from the 335 nm sample. (a) was taken without the mask. (b) was taken when the mask was about 2 μm from the sample. (c) and (d) were taken when the mask was closer but at un unreliably known distance from the sample.

42

As a first step to analyze these data, the results are fit to the function

6

1 1 1 2 2 21

( ) exp / cos 2 cos 2n nn

F t A t B f t B f t

(3.19)

The six exponential terms with time constants 5, 20, 100, 500, 2000, and 5000 ps are included to

fit the thermo-reflectance contribution to the signal. The set of parameters nA , 1B , 2B , 1 , 2

are chosen so that the error function 2

1 2, ( ) ( )S f f R t F t dt is minimized for each dataset.

The results of this procedure applied to 12 nm film, 120 nm film, and 335 nm film are shown as

false colored plots in Fig. (3.15), Fig. (3.16), and Fig. (3.17), respectively. From these figures,

the data show evidence of two distinct frequency components. The frequencies found for each

sample are listed in Table (3.1). The listed frequencies are accurate to about 0.02 GHz.

Fig. 3.15. The result of making a fit to the data of the 12 nm sample. This is a false colored plot.

Fre

quen

cy (

Ghz

)

Frequency (GHz)

43

Fig. 3.16. The result of making a fit to the data of the 120 nm sample. This is a false colored plot.

Fre

quen

cy (

Ghz

)

Frequency (GHz)

Fig. 3.17. The result of making a fit to the data of the 335 nm sample. This is a false colored plot.

Fre

quen

cy (

Ghz

)

Frequency (GHz)

44

Film thickness 12 nm 120 nm 335 nm

f1 8.64 GHz 7.07 GHz 5.39 GHz

f2 17.07 GHz 9.20 GHz 11.37 GHz

In order to investigate the origin of the frequencies, we calculated the dispersion relation of

Rayleigh waves for Al films on crystalline Si using the method mentioned in 3.2.2. Al was taken

to be elastically isotropic with 1211 1.113 10c , 12

12 0.591 10c , and 1244 0.261 10c cgs

units, and 2.699 g/cm3 [25]. The Si was taken to have 1211 1.657 10c , 12

12 0.639 10c ,

and 1244 0.796 10c , and 2.331 g/cm3. Figure (3.18) shows the result as a plot of mode

frequency as a function of thickness h of the Al film; the wavelength of the Rayleigh wave was

Table 3.1 The frequencies found in the data measured from the three samples, by applying the procedure as stated in the text.

Fig. 3.18. The frequency of Rayleigh wave plotted as function of film thickness. The dotted line at the top (10.27 GHz) is the maximum frequency for any Rayleigh mode without having energy radiating into the substrate.

45

taken to be equal to the pitch of the mask, which was 550 nm. Higher order Rayleigh modes

appear as h increases [26]; first of these higher modes happens when the thickness of Al reaches

82 nm. The line at 10.27 GHz at the top of the plot corresponds to the maximum frequency that

any Rayleigh modes with 550 nm wavelength can attain without leaking any energy into the

substrate [27].

For the 12 nm sample, the frequency 8.64 GHz is in good agreement with the calculated

frequency 8.74 GHz of the first Rayleigh mode. The 17.07 GHz component is present, because

the mask with period w can also generate surface waves with wave number 2 /nq n w , where

2n . Since the phase velocity of the first Rayleigh mode is a function only of the parameter qh ,

it follows that the frequency of the wave with 2n is twice the frequency of a wave with 1n

propagating in a structure of a film with twice the thickness, i.e. 24h nm. The calculation of

this frequency gives 17.16 GHz, which is in reasonable agreement with the experimental value.

For the 120 nm sample, the measured frequencies are 7.07 and 9.20 GHz, which correspond to

the calculated frequencies of 7.56 and 9.82 GHz of the first and the second Rayleigh modes.

For the 335 nm sample, the measured frequency of 5.39 GHz is in good agreement with the

calculated frequency of the first Rayleigh mode, which is 5.61 GHz. However, the measured

frequency at 11.37 GHz cannot be a Rayleigh mode, because it is above the frequency limit at

10.27 GHz (see Fig. (3.18)). The origin of this frequency comes from longitudinal sound

travelling in a direction almost parallel to the surface. This give a frequency which is

approximately the longitudinal sound velocity divided by the period of the mask. Unlike the

46

Rayleigh wave, the longitudinal sound is strongly attenuated. Longitudinal sound propagating

parallel to the surface does not satisfy the boundary condition at a free surface, and thus

generates a head wave propagating away from the free surface as the longitudinal sound travels

[28]. The evidence of attenuation can be seen in curve (c) and (d) from Fig. (3.14). The strong

periodic signal extending out to at least 1000 is undamped and it comes from the Rayleigh mode

of frequency 5.39 GHz. On the other hand, the signal appearing only before 300 ps comes from

the longitudinal sound. It is interesting that when the mask is further away from the sample, as in

curve (b), the longitudinal sound signal is larger than the Rayleigh wave signal, while the

opposite is observed in curve (d).

To investigate the differences between experiment and theory, we made a more detailed fit to the

data based on the following approximations:

1) The strain pulse is generated by the stress build-up due to heating the surface of the sample.

Nearly all of the light is absorbed in the Al film, while a small part of light passing through the

film is absorbed over a large distance in the Si substrate. The energy deposited by the light pulse

in the first few nanometers of the Al generates hot electrons that diffuse a significant distance

before dissipating their energy through phonon interactions [29]. As a rough approximation, the

heating of the Al film is assumed to be proportional to exp /z with 60 nm.

2) We assume the light intensity to be the sum of three components: one independent of x , and

two varying as cos 2 /x w and cos 4 /x w .

47

3) We calculate the time-evolution of the displacement field and strain field at the surface of the

sample for the three components in 2).

4) One contribution to the detected signal comes from the normal component zu of the

displacement at the sample surface; this changes the distance between the sample and the mask.

A second component arises from the piezo-optic effect, i.e., from the change in the optical

properties of the material due to the strain field near the surface of the Al film. At the surface, the

normal component of the stress tensor has to be zero. It follows that the strain components xx

and zz are related by 12 11/zz xxc c at all times. Therefore we can assume that the piezo-optic

contribution is proportional to xx . Note that the strain arising from the component in 2) above

that is independent of x gives 0xx at the sample surface [30]. So this particular strain term is

neglected.

As a result, there are five components to the acoustic signal that we use to make a fit to each

dataset. We also include the exponential terms from Eqn. (3.19) in order to allow the thermo-

reflectance contribution in the fit. Fig. (3.19) shows the plots of the fit and the data superimposed

on top of each other for all three samples. Only the acoustic contributions are shown in the figure,

i.e., only the fit without the exponential terms and the data minus the exponential terms are

shown. The result for the 12 nm sample shows a good fit based on the thickness of the film

measured by the SEM and the literature values of the elastic constants previously given.

However, for the 120 nm sample, we were not able to obtain a good fit based on elastic constant

from the literature and the thickness measured by the SEM. Consequentially, we made a series of

fits by adjusting 11c and 44c , while keeping 12c to the elastically isotropic condition

48

12 11 442c c c . The best fit we could obtain was with 1211 0.74 10c , 12

12 0.314 10c , and

1244 0.213 10c cgs units, which gave the results shown in Fig. (3.19). Lastly, the fit for the

335 nm sample was obtained by using the film thickness measured by SEM and the literature

values of the elastic constants all reduced by a factor of 0.92. There have been reports from a

number of papers that the elastic constants of thin films are less than those of the bulk material

[31]. Since the three samples measured were prepared at different growth rates, it is possible that

the oxygen content, grain size, and stress differ significantly. We have not investigated how the

sample preparation affects the elastic constants, but this certainly is a topic of interest.

Fig. 3.19. Fit for the acoustic contributions in the data of all three films, using the procedure described in the text.

49

Bibliography 

[1] Thomsen, C., Grahn, H.T., Maris, H.J., and Tauc, J. “Surface generation and detection of phonons by picosecond light pulses.” Phys. Rev. B 34, 4129 (1986).

[2] Pezeril, T., Ruello, P., Gougeon, S., Chigarev, N., Mounier, D., Breteau, J.-M., Picart, P., and Gusev, V. “Generation and detection of plane coherent shear picosecond acoustic pulses by lasers: Experiment and theory.” Phys. Rev. B 75, 174307 (2007).

[3] Mastsuda, O., Wright, O.B., Hurley, D.H., Gusev, V., and Shimizu, K. “Coherent shear phonon generation and detection with picosecond laser acoustics.” Phys. Rev. B 77, 224110 (2008).

[4] Mounier, D., Morozov, E., Ruello, P., Breteau, J.-M., Picart, P., and Gusev, V. “Detection of shear picosecond acoustic pulses by transient femtosecond polarimetry.” Eur. Phys. J. Special Topics 153, 243 (2008).

[5] Hurley, D.H., Telschow, K.L. “Picosecond surface acoustic waves using a suboptical wavelength absorption grating.” Phys Rev B 66, 153301 (2002).

[6] Hurley, D.H. “Optical generation and spatially distinct interferometric detection of ultrahigh frequency surface acoustic waves.” Appl. Phys. Lett. 88, 191106 (2006).

[7] Sadhu, J., Lee, J.H., Sinha, S. “Frequency shift and attenuation of hypersonic surface acoustic phonons under metallic gratings.” Appl. Phys. Lett. 97, 133106 (2010).

[8] Duggal, A.R., Rogers, J.A., Nelson, K.A., and Rothschild, M. “Real-time characterization of acoustic modes of polymide thin-film coatings using impulsive stimulated thermal scattering.” Appl. Phys. Lett. 60, 692 (1992); Duggal, A.R., Rogers, J.A., and Nelson, K.A. “Real-time optical characterization of surface acoustic modes of polymide thin-film coatings.” J. Appl. Phys. 72, 2823 (1992).

[9] See, Phillion, D.W., Kuizenga, D.J., and Siegman, A.E. “Subnanosecond relaxation time measurements using a transient inducing grating method.” Appl. Phys. Lett. 27, 85 (1975), and references to early work listed therein.

[10] See, for example, the description in Rogers, J.A., Fuchs, M., Banet, M., Hanselman, J.B., Logan, R., and Nelson, K. “Optical system for rapid materials characterization with the transient grating technique: Application to nondesctructive evaluation of thin films used in microelectronics.” Appl. Phys. Lett. 71, 225 (1997).

[11] Miao, Q., Liu, L.-W., Grimsley, T.J., Nurmikko, A.V., and Maris, H.J. “Picosecond ultrasonics measurements using an optical mask.” Ultrasonics 56, 141 (2015).

50

[12] Rayleigh, J.W.S. “On waves propagated along the place surface of an elastic solid.” Proc. London Math. Soc. 17, 4 (1885).

[13] Love, A.E.H. A Treatise on the Mathematical Theory of Elasticity, 2nd Ed, Cambridge: at the University Press (1906).

[14] Stoneley, R. “Elastic waves at the surface of separation of two solids.” Proc. R. Soc. Lond. A. 106, 416 (1924).

[15] See references listed in Kino, G.S., and Matthews, H. “Signal processing in acoustic surface-wave devices.” IEEE Spectrum 8, 22 (1971).

[16] Anderson, O.L. in Physical Acoustics, edited by Mason W.P. (Academic, New York, 1965), Volume IX, p. 59.

[17] Anderson, O.L. in Physical Acoustics, edited by Mason, W.P. (Academic, New York, 1965), Volume IX, p. 52.

[18] Karvinen, P., Nuutinen, T., Rahomaki, J., Hyvarinen, O., and Vahimaa, P. “String fluorescence-signal gain with single-excitation-enhancing and emission-directing nanostructured diffraction grating.” Opt. Lett. 34, 3208 (2009).

[19] We thank Dr. Kwangdong Roh graduated from Brown University, K. Roh and S. Ahn, H. Kim and H. Jeon of Seoul National University for the help with preparing the optical mask.

[20] Zeiss (LEO) 1530 VP SEM.

[21] Antonelli, G.A., Zannitto, P., and Maris, H.J. “New method for the generation of surface acoustic waves of high frequency.” In the Proceedings of the 10th International Conference on Phonon Scattering in Condensed Matter, Physica B 316-317, 377 (2002).

[22] Grimsley, T.J. (2012) Planar Opto-acoustic Microscopy Applied to the Metrhology of Periodic Nanostructures. Ph.D. dissertation, Brown University, Providence, Rhode Island.

[23] Miao, Q. (2014) Picosecond Ultrasonic Measurement with Enhanced Sensitivity. Ph.D. dissertation, Brown University, Providence, Rhode Island.

[24] Moharam, M.G., Pommet, D.A., and Grann, E.B. “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach.” J.Opt. Soc. Am. A 12, 1077 (1995).

[25] Anderson, O.L. in Physical Acoustics, edited by Mason, W.P. (Academic, New York, 1965), Volume IIIB, p.43.

[26] See, for example, Farnell, G.W. and Adler, E.L., in Physical Acoustics, edited by Mason, W.P. and Thurston, R.N. (Academic, New York, 1972), Volume IX, p.35.

51

[27] For an elastically isotropic substrate, this critical frequency is equal to the shear wave velocity in the substrate divided by the period of the mask. The radiation first occurs through the generation of shear waves propagating into the substrate at a very small angle relative to the surface. However, for crystalline silicon, due to elastic anisotropy, the surface waves start losing energy by radiation at a finite angle (~24°) from the free surface.

[28] See. For example, Lamb, H. “On the propagation of tremors over the surface of an elastic solid.” Phil. Trans. R. Soc. Lond. A 203, 1 (1904); Richards, P.G. “Elementary solutions to Lamb’s problem for a point source and their relevance to three-dimensional studies of spontaneous crack propagation.” Bull. Seis. Soc. Am. 69, 947 (1979).

[29] Tas, G. and Maris, H.J. “Electron diffusion in metals studied by picosecond ultrasonics.” Phys. Rev. B 49, 15046 (1994).

[30] There actually will be a small piezo-optic contribution from this component, because the probe light does penetrate some small distance into the Al film, and as a result the system will relax to non-zero strain.

[31] See, for example, Ogi, H., Nakamura, N., and Hirao, M. “Picosecond ultrasound spectroscopy for the studying elastic modulus of thin films: a review.” Nondestructive Testing and Evaluation 26, 267 (2011); Nan, C.-W., Li, X., Cai, K., and Tong, J. “Grain size-dependent elastic moduli of nanocrystals.” J. Mat. Sci. Lett. 17, 1917 (1998).

52

 

Chapter 4 

A Review of Planar Opto‐Acoustic Microscopy 

 

4.1 Introduction 

Planar Opto-Acoustic Microscopy (POAM) is a method to measure the geometry of

nanostructures using picosecond ultrasonics. A schematic diagram of POAM is shown in Fig.

(4.1). A Fabry-Pérot cavity is held close to the sample surface by the Z-tip-tilt stage as

mentioned in section 3.3.1. The sample has a series of identical features such as grooves. The

Fig. 4.1. A schematic diagram of POAM.

Substrate

Cavity

Sample

Water

Probe beam Pump beam

53

cavity is coupled to the sample by water. When the cavity is excited by a pump light pulse, a

planar sound pulse is launched into the water and toward the sample. The acoustic echo returning

from the sample is detected by a probe light pulse. By analyzing the shape of the acoustic echo,

information about the geometry of the sample can be obtained.

4.2 The Resonant Fabry‐Pérot Cavity 

The Fabry-Pérot cavity is implemented in order to resolve two issues in the detection of

returning echoes. The first issue is that the transmission of sound into the fluid is very small. For

a one-dimensional strain wave travelling from medium 1 to medium 2, the reflection and the

transmission coefficients are

2 1

2 1

Z ZR

Z Z

,

2 2

1 2 1

2ZT

Z Z

. (4.1)

Here the acoustic impedance Z is related to the density and the sound velocity by Z .

In general, the acoustic impedance of common metals or oxides is one order of magnitude

greater than that of water. This results in small transmission of sound into water and causes a

problem to the signal-to-noise ratio. The second issue is that the amplitude of the sound pulse is

rapidly attenuated in water. The attenuation eliminates high frequency components and broadens

the sound pulse. In chapter 2, it was shown that the probe light can only detect strain located as

deep as the penetration depth of a metal film; usually the penetration depth is only of a few

54

nanometers. As a result, the probe light will only sample a narrow portion of a broad pulse with

small amplitude. Thus it is necessary to use a suitable nanostructure that can sample a broadened

pulse. This section will explain how a Fabry-Pérot cavity can overcome these two issues.

The structure of the Fabry-Pérot cavity is as shown in Fig. (4.2). The cavity is fabricated on a

transparent substrate, and its layer structure consists of a half-mirror, a spacer layer, a full-mirror,

and a protective cap layer. Let 1r and 2r be the complex-valued reflection coefficients of the

half-mirror and the full-mirror, respectively. The spacer layer is assumed to have a real refractive

index n and thickness w . If the wavelength of the probe light is 0 and the angle of incidence is

0 , then the reflectivity R of the probe light can be shown [1] to be

2 21 2 1 2

2 21 2 1 2

4 sin / 2

1 4 sin / 2

r r r rR

r r r r

. (4.2)

Fig. 4.2. The structure of the cavity. A probe beam interrogating the cavity from the substrate side will bounce off multiple times between the mirrors.

Substrate

Half-mirror

Spacer

Full-mirror

Probe beam

Cap layer

55

The parameter

2 zk w (4.3)

is the phase delay for a light ray that has been reflected once off each mirror and has traversed

the spacer layer twice. Here,

2 20

0

2sinzk n

, (4.4)

and is the phase change suffered by the light ray after it has been once reflected from both

mirrors, and it depends on the optical refractive indices and the structures of the mirrors.

Although changes the particular values of w which the optical resonance occurs, it does not

change the spacing w between any two optical resonances. Figure (4.3a) shows plots of ( )R

for several choices of 1 2r r r , and Fig. (4.3b) shows plots of /dR d for the same choices

of r . It can be seen from Fig. (4.3a) that higher value of r results in shaper resonance. Figure

(4.3b) shows that the reflectivity of the cavity is sensitive to perturbations in when the cavity

is tuned to be slightly off resonance.

There are three mechanisms for to be perturbed by a sound pulse traveling through the cavity.

The first is that the strain pulse can change the distance between the mirrors. The second is that

the strain pulse induces a change in the refractive index of the spacer layer. The third is the

strain-induced change in the piezo-optic coefficients of the mirrors. The first two effects both

change the optical thickness of the spacer layer. It is these two effects combined that change the

56

0.0

0.2

0.4

0.6

0.8

1.0

R(

)

r = 0.95 r = 0.8 r = 0.65 r = 0.5

(a)

-2

0

2

dR(

)/d

r = 0.95 r = 0.8 r = 0.65 r = 0.5

(b)

Fig. 4.3. Graphs of (a) R(γ) and (b) dR/dγ for several choices of |r|=|r1|=|r2|.

57

reflectivity the most for a broadened acoustic pulse propagating through the cavity. Assuming

that the probe light is at normal incidence, the unperturbed optical thickness is given by

0

0 0 0 00

wl n dz n w , (4.5)

where 0w and 0n are the unperturbed thickness and refractive index of the spacer layer,

respectively. The change in the optical thickness is

( )l nw w n n w . (4.6)

In terms of strain, the change in the thickness is

0

00( )

ww z dz w , (4.7)

and the change in the index is

( )n

n z

. (4.8)

To the lower order in strain, the change in the optical thickness is

0 0

nl w n

. (4.9)

Therefore, the change in reflectivity due to strain inside the spacer layer is

dR

R ldl

. (4.10)

In our work, we have chosen to use Al films as the mirror material and SiO2 for the spacer layer.

It has been found that the first maximum of dR

dloccurs around 0 230w nm for 0 800 nm.

Using 2.8n and 8.3 for Al, and 1.45n and 0 for SiO2 [2], it can be calculated that

58

0.11dR

dl nm-1. From Ref. [1], it was measured that 0.10

n

for amorphous SiO2. As a result,

the reflectivity change due to the returning sound pulse is

39R . (4.11)

4.3 Metrology of 1D Periodic Nanostructures 

This section presents works on POAM done by previous students, Fan Yang and Thomas

Grimsley.

Fig. 4.4. The cross-sectional SEM images of the first sample measured by POAM. The length of the red bar is 100 nm.

59

Figure (4.4) shows the cross-sectional SEM picture of the first sample measured by POAM [3].

The sample was SiO2 lines patterned on a Si wafer with a conformal layer of SiN (5 nm)

deposited on the surface of the lines and the substrate. From the SEM picture, the pitch, the

width at the bottom of the channels, and the depth of the channels were measured to be 240 nm,

38 nm, and 405 nm, respectively. The sample was then measured using POAM. The result from

the POAM measurement is plotted in Fig. (4.5); the result was processed so that only the signal

contributed by the acoustic echoes remained. The first peak was the echo from the top surface of

the lines and the later peak was the echo from the bottom of the channels. From the difference of

the two arrival times, the depth of the channel was calculated to be 440 nm.

∆R

(ar

b. u

nits

)

Probe Delay Time (ps)

Fig. 4.5. The change in reflectivity contributed by acoustic echoes.

60

In another set of measurements, an attempt was made to investigate the feasibility of using

POAM data to reconstruct the sample profile and to measure other dimensions of the sample [4],

i.e., the pitch, the depth of the channels, and the width of the channel at half of its depth. The

cross-sectional SEM images of the samples measured in this set are shown in Fig. (4.6). A

library of simulations was generated for different sample profile and dimensions. The

experimental result was then compared to the result from each simulation. For sample A, there

was a good agreement between the experimental result and the result of the simulation based on

the actual sample profile. An investigation was then made of the extent to which the profile

could be varied before the simulation ceased to agree with the experimental result. In this way, it

Fig. 4.6. The cross-sectional SEM images of (a) sample A and (b) sample B.

(a)

(b)

61

was possible to make a rough estimate of how well sample dimensions can be determined from a

POAM measurement. However, for sample B the results were inconclusive. A simulation based

on the SEM profile of sample B did not give results that agreed well with the experimental data.

In addition, when variations in the geometry were made to try to find a best fit to the data, large

changes in some dimensions of the sample made only a very small change in the results of the

simulation. This difficulty may arise because of the complicated shape of sample B. In the next

chapter we describe a series of experiments on a set of samples of simple geometry to investigate

this type of problem more systematically.

62

Bibliography 

[1] Li, Y., Miao, Q., Nurmikko, A.V., and Maris, H.J. “Picosecond ultrasonic measurements using an optical cavity.” J. Appl. Phys. 105, 083516 (2009).

[2] Palik, E.D. Handbook of Optical Constants of Solids, 1985 (New York: Academic).

[3] Yang, F. (2010) Study of Gigahertz Ultrasound Propagation in Water Using Picosecond Ultrasonics. Ph.D. dissertation, Brown University, Providence, Rhode Island.

[4] Grimsley, T.J. (2012) Planar Opto-acoustic Microscopy Applied to the Metrhology of Periodic Nanostructures. Ph.D. dissertation, Brown University, Providence, Rhode Island.

63

 

Chapter 5 

A  Systematic  Study  on  the Metrology  of  Planar 

Opto‐Acoustic Microscopy 

5.1 Introduction 

In the previous chapter, we introduced Planar Opto-Acoustic Microscopy (POAM) and

preliminary results of POAM. The samples that were studied came in different sizes and shapes

because the original intent was simply to see whether sound pulses could propagate along narrow

channels. A more rigorous study on the metrology requires measurements on a series of samples

which have a structure with only one varied parameter. In this chapter, we present work on two

series of samples to systematically study the metrology capability of POAM. The first set of

measurements was done on a series of structures in which only the channel depth was varied.

This is to verify whether the results of POAM can properly measure channel depths, and to find

out what is the shallowest channel depth that POAM can still detect the channels. The second set

of measurements was done on a series of structures for which only the pitch was varied. This is

64

to explore whether POAM measurements can give information regarding the ratio between the

surface at the top of the lines and the surface at the bottom of the channel.

5.2 Sample Description 

The samples were fabricated by Novellus Systems, Inc., and were provided to us by Dr. Andrew

Antonelli. Three thin film layers were deposited on top of the 300 mm Si wafer; a SiO2 layer of

thickness 400 nm, followed by a 40 nm thick SiN layer, and then a 250 nm layer of SiO2. A

layer of photoresist was then deposited and patterned by photolithography to produce a pattern of

trenches extending through the thickness of the photoresist layer. Anisotropic etching was then

used to remove the regions of the upper SiO2 film that were exposed by the trenches. The SiN

film was used as an etch stop. Finally, the unexposed areas of photoresist were removed.

The general cross-sectional structure of the samples is shown in Fig. (5.1). All line structures

share the same channel depth, which is 250 nm. The angle between the wall of a line structure

and a line perpendicular to the SiN surface is 0.9° ± 0.4°. An SEM [1] picture of the cross-

section of one of the line structures is shown in Fig. (5.2). The pitch and the channel width of this

particular structure were measured to be 244 nm and 80 nm, respectively, with an error of 5 nm.

The wafer has several patches of line structures with different pitches and different channel

widths; each patch is 1 mm long and 250 μm wide.

65

The channel depth of some samples was modified using chemical-mechanical polishing [2] in

order to create a series of samples with varied channel depths. A schematic diagram of a

chemical-mechanical polishing machine is shown in Fig. (5.3). The machine consists of the

polishing plate, the sample jig, and a slurry nozzle. A polishing cloth is attached to the top side

SiN

SiO2

SiO2

Si

Pitch

Channel Width

Channel Depth

Fig. 5.1 A general geometry of the line structures.

500 nm

Fig. 5.2. An SEM image on the cross-section of the line structures. For this particular patch, the pitch is 244 nm, the channel width is 80 nm, and the channel depth is 250 nm.

66

of the plate. The sample to be polished is mounted on the jig. The jig presses the sample against

the polishing cloth and maintains the orientation of the sample so that the sample is parallel to

the plate. The slurry nozzle drops slurry onto the polishing cloth. The slurry is a chemical

solution with suspension of polishing particles. The chemical solution softens the material to be

polished while the polishing particles grind the material away. During operation, the main plate

rotates at a controlled speed. The jig rests on top of the plate with the sample facing down,

grinding the sample against the polishing cloth that is soaked in slurry. The jig is held in place by

an extension arm, but is allowed to rotate freely so that the polishing rate across the sample

surface is uniform. The polishing rate is proportional to the relative speed between the plate and

the sample as well as the pressure applied to the sample against the plate. The samples were

coated by a layer of photoresist before polishing. The photoresist prevents the polishing particles

from jamming into the channels. The photoresist was stripped using acetone after the polishing.

The channel depths of the modified sample were measured by SEM [1].

Fig. 5.3. A schematic diagram of the chemical-mechanical polishing machine in operation.

Sample jig Slurry

Polishing cloth

Polishing plate

Extension arm

Slurry nozzle

67

A full catalogue of the samples used in this work is listed in Table (5.1) for Series 1 and Table

(5.2) for Series 2.

Series 1 Pitch: 244 nm Channel Width: 80 nm Error: ± 5 nm

Sample Channel Depth (nm)

1 50

2 85

3 109

4 159

5 250

Series 2 Channel Width: 75 nm Channel Depth: 250 nm Error: ± 5 nm

Sample Pitch (nm)

1 961

2 770

3 576

4 481

Table 5.1. Catalogue of samples used in the first series of POAM measurements.

Table 5.2. Catalogue of samples used in the second series of POAM measurements.

68

5.3 Experimental Results 

In this section we present the results from the two series of POAM measurements. A sample and

a cavity slab [3] were mounted onto the Z-tip-tilt stage. A few drops of distilled water were

inserted between the cavity and the sample. The sample was raised toward the cavity in

increments of 2 microns until the acoustic signals reflected from the sample arrived within the

time range of the measurement. Then the sample position was adjusted in finer increments on the

order of nanometers until the separation was about 1 micron.

The results from the POAM measurements are shown in Fig. (5.4). There are three components

in the change in reflectivity. The first component is the slowly decaying background due to

temperature change of the cavity. The second component is Brillouin oscillations with period of

44 ps. The third component is the signal of acoustic pulses that appear as bumps with widths of

around a hundred picoseconds. We are interested in analyzing the acoustic signals and to do this

need to remove the Brillouin oscillations and the decaying background. Our method for doing

this will be explained in detail in section 5.4 below. The results are shown in Fig. (5.5).

In Fig. (5.5), the peaks in the plots correspond to acoustic pulses reflected from different parts of

the line structure. An acoustic pulse can be reflected by the line structures in three different ways.

A peak corresponding to a pulse reflected from the top surface of the lines is tagged with a label

T. A peak corresponding to a pulse reflected from the bottom surface of the channels is tagged

with a label B. Lastly, after an acoustic pulse has been reflected at the bottom surface of the

69

Fig. 5.4. The result of the POAM measurements. Series 1 is plotted in (a), and Series 2 is plotted in (b). The plot numbers in these figures correspond to the sample number in Table (5.1) and Table (5.2) for Series 1 and Series 2, respectively. The plots are vertically displaced for clarity.

70

Fig. 5.5. The results of the two series but with Brillouin oscillation and the decaying background removed. Results from Series 1 are plotted in (a), and those from Series 2 are plotted in (b). The numbers of the plot corresponds to the sample numbers in Table (5.1) for Series 1 and Table (5.2) for Series 2. The labels T, B, t, and etc. are as discussed in the text. The plots are vertically displaced for clarity.

71

channels, it can be partially reflected back into the channel right at the opening of the channels; it

is tagged with label t. The change in the polarity of the peaks tagged with label t suggests that

the acoustic reflection coefficient corresponding to this “back reflection” is negative. At longer

times, there are peaks that correspond to acoustic pulses that were reflected back from the sample

for a second time. As an example of the labeling scheme, a pulse that has first been reflected

from the top of the lines and then reflected from the bottom of the channel is labeled TB. The

peaks are identified with the corresponding reflections as labeled in Fig. (5.5).

5.4 Data Analysis 

To analyze the data we need to understand how the sound is generated from the optical cavity

and how the returning echoes give rise to the measured signal. One approach to this might be to

first calculate the change in temperature of the metal films on either side of the cavity due to the

application of the pump light pulse. The stress resulting from the temperature change could then

be found. This could be followed by a calculation of the shape of the sound pulse entering into

the water, and the change in the optical reflectivity that results when the echo returns. However,

this first-principles approach is impractical. The geometry of the cavity and the optical properties

of the different elements are not accurately known. We therefore have calibrated the system via

measurement of the signal obtained when a planar Si sample is used instead of a patterned

sample.

72

5.4.1 Removal of Brillouin Oscillations and Thermal Signal 

Figure (5.6a) shows the result from a measurement on the planar sample. For this data set, the

thickness of the water layer was about 857 nm. The reflectivity change consists of components

from Brillouin oscillations with a period of 44 ps, acoustic echoes with width on the order of 50-

100 ps, and a smoothly decaying background due to thermal-reflectance. The separation time

between the echoes corresponds to the round-trip duration of an acoustic pulse travelling within

Fig. 5.6. (a) A plot of data from the measurement on a planar sample when the water layer is about 857 nm thick. (b) The plot of the data when the contribution from Brillouin oscillations is filtered out. The plots are vertically displaced for clarity.

73

the water layer. To filter out the Brillouin oscillations, the following transform was applied to the

data:

1 1 1

( ) ( / 2) ( ) ( / 2)4 2 4raw raw rawR t R t T R t R t T . (5.1)

Here R is the filtered data, rawR is the raw data, and T is the period of the Brillouin oscillation.

The filtered data is plotted in Fig. (5.6b). It can be seen that the shape of each echo consists of a

positive pulse followed by a weaker negative pulse. The filtered data were then fit to a function

( )y t of the form

2 23

0 00

1 1,1 ,2

( ) exp exp exp( / )( ) ( )

N

n i in in n

t nt t nt Ty t A R C t y

t t

, (5.2)

where the Gaussian terms model the acoustic signal and the exponential decay terms model the

decaying background. Here N is the number of echoes detected in the measurement at a

particular water thickness. The parameters nA , 0t , ,1( )n t , ,2 ( )n t , R , T , iC , i , and 0y are

all determined using nonlinear fitting. The exponential decay terms from ( )y t were then

subtracted from the filtered data to remove the contribution from the slowly decaying

background. The data were also smoothed by averaging with Gaussian weighting over the range

of 5 ps around each point. This gave the results for different thickness of water that are shown in

Fig. (5.7).

74

 

5.4.2 Shape of Acoustic Echoes from Planar Sample 

At first sight the acoustic echoes seen in Fig. (5.7) have a surprising shape. Suppose that the

cavity structure introduces into the liquid a narrow positive-going pressure pulse and that the

change in the reflectivity of the probe light is proportional to the returning pressure pulse. After

propagation through water a narrow pulse will be broadened because the high frequency Fourier

Fig. 5.7. Acoustic signals of echoes from a planar sample at various water thicknesses.

75

components will be damped out. Thus we might expect that the detected echo would be

broadened but we would not expect it to consist of a positive and a negative component. We

conclude that the positive and negative components have to be explained in terms of the sound

propagation not in the water, but in the optical cavity.

To explain the shape of the echoes we make the following two assumptions [7]. First, the strain

pulse generated inside the thin Al film is much greater than that generated in the thick Al film so

that the strain pulse generated in that film can be neglected. Second, the acoustic impedance of

the materials making up the cavity are such that the acoustic reflection coefficient is small at the

cavity/substrate interface, significant at the cavity/water interface, and negligible at every other

interface.

Upon excitation, the thin Al film generates two pulses each with a width less than 2 ps. One

pulse propagates into the substrate; it does not contribute to any strain pulses going into the water.

The other pulse will bounce back and forth within the cavity layers and be partially transmitted at

both the cavity-water interface and the cavity-substrate interface. Let watr be the reflection

coefficient of strain from the cavity to the water and subr be the reflection coefficient of strain

from the cavity to the substrate. Hence, the initial pressure pulses in water can be approximated

by a series of Dirac delta functions, as

0

( , ) 2 / 2n

sub wat cavn

p t z B r r t z nT

. (5.3)

76

Here B is a measure of the amplitude, z is the distance traveled in water, is the sound

velocity in water, and cavT is the transit time of an acoustic pulse propagating through all the film

layers. If we neglect pulses which have been reflected more than once at the cavity/substrate

interface then n only runs from 0 to 1. Since watr is negative, the sign of the second pulse is

negative. The pulses will broaden into Gaussian shapes as they propagate into the water, and so

the returning pulse shapes will have to be replaced by Gaussian functions of appropriate width.

The cavity responds to the acoustic pulses according to the response function, defined as a time-

dependent function of the change in the optical reflectivity by a Dirac delta pulse entering the

cavity from the coupling fluid. From chapter 2, the change in reflectivity R has the following

form

( ) ( ) ( , )R t f z z t dz . (5.4)

The sensitivity function ( )f z describes how strain at position z inside a thin-film structure

affects the optical reflectivity. A closed form expression for the sensitivity function of the cavity

is complicated, since the cavity involves a mix of transparent films and opaque films, each

having different piezo-optic constants. However, the sensitivity function can be calculated

numerically based on a matrix method for computing the optical reflectivity of multilayered

structures. The steps in the calculations are:

1) Each layer is first divided into several bins corresponding to different positions, and each bin

is assigned an optical transfer matrix. The optical transfer matrix is a function of the width of the

bin and the optical constants of the bin material.

77

2) The reflection coefficient R of the structure is calculated in terms of these matrices.

3) A strain of amplitude 0 is applied to the bin at position z . This changes the transfer matrix

because the optical constants change with strain and because the width of the bin is changed. The

reflection coefficient 'R of the structure is then calculated. From these results the sensitivity

function at z can be found from the relation

0

'R R

z

(5.5)

where z is the width of the particular bin considered.

The calculation is then repeated for each bin in the structure, giving the sensitivity function for

all values of z . The optical parameters used in the calculations are listed in Table (5.3). The

sources of the parameters can be found in Ref. [8]. For the purpose of this analysis, we are only

interested in the qualitative features of the sensitivity function. Therefore the parameters only

need to be sufficiently accurate to reflect the structure of the cavity. The computed sensitivity

Material n k ∂n/∂η ∂k/∂η

Al 1.38 5.61 -0.8 4.5

Amorphous SiO2 1.45 0 0.1 0

Crystal SiO2 1.54 0 0.1 0

Table 5.3. Summary of values used to calculate the sensitivity function in Figure (5.8a). See Ref. [8] for the sources of these parameters.

78

function of the cavity is plotted in Fig. (5.8a). We can simplify the sensitivity function to just the

step function, as shown in Fig. (5.8b). The oscillations inside the transparent films can be

ignored since the Brillouin oscillations have already been filtered out. As the pulses return to the

cavity from the water, their width will be much wider than the narrow feature of the sensitivity

function; any effects from the narrow feature will be washed out. According to the simplified

sensitivity function, the cavity suffers a change in reflectivity only when the Dirac delta pulse is

inside the spacer layer. From Eqn. (5.11), the resulting response is a rectangular pulse with the

Crystal SiO2 Amorphous SiO

2 Al

-1000 -800 -600 -400 -200 0 200 400

f(z)

(ar

b. u

nits

)

Position z

(a)

(b)

Fig. 5.8. (a) The computed sensitivity function of the cavity using the parameters listed in Table (5.1). (b) The simplified sensitivity function. The z-axis is drawn to scale.

79

Fig. 5.9. (a) The response function of the cavity. (b) The response function combined with the additional Dirac delta pulse from signal generation of the cavity. Tsp is the transit time for an acoustic pulse to cross the spacer layer and Tcav is single-trip time to cross all cavity layers.

80

duration equal to spT , where spT is the transit time for an acoustic pulse to cross the spacer layer.

After the Dirac delta pulse has entered the cavity from the water side, it will bounce back and

forth between the cavity/substrate interface and the cavity/water interface. However since 2subr is

assumed to be negligible, the cavity only has three detectable responses per one Dirac delta pulse.

The first two responses occur back-to-back, and the third response occurs at a time 2 cavT after the

start of the first response. The response function is as shown in Fig. (5.9a). After including the

additional pulse from sound generation, the response function becomes what is shown in Fig.

(5.9b), which also only has three responses because all the other responses involve higher orders

of subr .

In order to measure spT and cavT , a picosecond ultrasonic measurement was taken on the cavity

from the substrate side; Figure (5.10a) shows the measured change in reflectivity. The jump at 0

ps and the decay that follows are due to thermal-reflectance. The Brillouin oscillations with a

fixed period of 44 ps are present after 0 ps but are more distinguishable beyond 250 ps. The

smaller irregular bumps between 40 ps and 200 ps are attributed to strain pulses bouncing inside

the cavity. The derivative of the change in reflectivity was taken so that the signals from the

strain pulses stand out; the derivative is plotted in Fig. (5.10b).

According to the sensitivity function, the reflectivity suffers a rapid change whenever a strain

pulse crosses the boundaries of the spacer layer and the thin Al film. Upon excitation from the

pump light pulse, the cavity generates two pairs of strain pulses that travel in opposite directions,

81

Fig. 5.10. (a) The result of picosecond ultrasonic measurement on the cavity. (b) The derivative of the same result. The origins of the peaks A, B, C, and D are as discussed in the text.

82

as shown in Fig. (5.11). The pulse 1L only contributes to Brillouin oscillations. Using the values

of the bulk material for both Al and amorphous SiO2, the reflection coefficient of strain is

approximately 0.09. It is therefore sufficient to consider signal contributions from pulse 1R, 2L,

and 2R without the reflections at Al/SiO2 interfaces. These contributions manifest as the peaks

labeled as A, B, C, and D, which occur at about 48 ps, 56 ps, 103 ps, and 150 ps, respectively [9].

Peak A occurs when pulses 1R and 2L first crosses the spacer layer. Peak B occurs after pulse

2R has traveled round-trip across the thick al layer and the cap layer. Peak C occurs when both

pulse 1R and 2R have both traveled round-trip in the thick Al layer and the cap layer and have

crossed the spacer layer once. Peak D is when pulse 1R has traveled round-trip through the entire

cavity. Therefore, we have 48spT ps and 75cavT ps. Additionally, the magnitude of peak A

(contributed by 1R and 2L) is significantly greater than peak B (contributed only by 2R). This

supports the assumption of the signal generation model that the amplitude of the strain pulses

generated in the thin Al film is more significant than those generated in the thick Al film.

Strain

1R 1L 2R 2L

Cavity layers

Fig. 5.11. The strain pulses generated inside the cavity.

83

Combining the result of fitting the data of planar sample to Eqn. (5.2) with the model of the

response function, it can be deduced that 2 cavT T and 2 sub watR r r . The value of T agrees

with the result of 2 cavT . Using the fitted values of R and watr , it can be deduced that 0.296subr .

We will then take the convolution between the response function of the cavity and the simulation

results.

5.4.3  Finite‐Difference  Time‐Domain  Simulation  for  Planar 

Samples 

To test our understanding of the results we have performed finite-difference time-domain FDTD

simulations of sound propagation in water. The particle motion of a fluid is governed by the

Navier-Stokes equation and the continuity equation. In our experiments the amplitudes of the

sound pulses are small, thus we can use the linearized versions of the equations, and they are

20 '

3p

t

v

v v , (5.6)

0

'0

t

v . (5.7)

Here 0 is the density of water at equilibrium, v is the particle velocity, 'p is the excess

pressure from equilibrium, is the shear viscosity, is the bulk viscosity, and ' is the small

variation in density of water from equilibrium. Since the process is adiabatic, 'p and ' are

related by

84

0

' 'sBp

, (5.8)

where 0s

s

pB

is the isentropic bulk modulus. Then Eqn. (5.7) can be rewritten as

02

1 '0

p

t

v , (5.9)

where 0/sB is the sound velocity in water. For a plane acoustic wave of the form

( )0'( , ) z i kz tp z t p e e , (5.10)

it can be shown [4] that the attenuation is given by

2 2

23

0

2 4

3

fAf

. (5.11)

Here f is the frequency of the plane wave in Hz. In our experiments, the temperature of water

was approximately 20 °C where the measured value of A is 162.53 10 s2 cm-1 [5]. The

measured value of the shear viscosity is 21.002 10 g s-1 cm-1 [6]. From this it follows that

23.45 10 g s-1 cm-1.

In our experiment, a large area of the surface (~15 µm diameter) is illuminated by the pump and

probe pulses. Consequently, for the patterned samples we can consider the displacement of the

water and the pressure to be functions of just the coordinate z in the direction normal to the

surface and the coordinate x perpendicular to both the surface and the direction run by the lines.

Equations (5.6) and (5.9) then become

85

2

0

1 '0x zv vp

t x z

,

2 2

0 2 2

'

3x x x xzv v v vvp

t x x x z x z

,

2 2

0 2 2

'

3xz z z zvv v v vp

t z z x z x z

. (5.12)

For the planar samples all variables depend only on .z

To carry out a simulation the domain is split into a grid. The particle acceleration, velocity, and

displacement are calculated on the grid points, while the stress and the strain are calculated at the

centers of the cells that are defined by the grid lines. Periodic boundary conditions are applied in

the x-direction. The displacement at the top and the bottom rows of the grid are held fixed. The

materials of the line structure are assumed to be rigid, since the elastic constants of the solid

materials are at least 40 times larger than that of water. The top row of the simulation domain is

assumed to be the boundary of the cavity. The initial condition is that there is sudden small

displacement of the top row. The pressure at the top row is recorded over the entire duration of

the simulation.

As a first application of the simulation program we consider the planar samples. In order to

properly model the shape of the echo, we need to take the convolution between the result from

the FDTD simulation and the response function from Fig. (5.9b). Since the position of the top

86

row is held fixed after the initial displacement, there is perfect reflection at this boundary rather

than the partial reflection that should occur. To correct for this the amplitudes of successive

echoes have been manually scaled by a factor n

watr . The result of the convolution agrees well

with the experimental data, as shown in Fig. (5.12).

Fig. 5.12. The experimental results (red lines) are superimposed with the results (black lines) of the FDTD simulations convoluted with the response function from Fig.(5.13b).

87

5.4.4 Analysis on the Two Series of POAM Measurements 

In this section we analyze the two series of measurements that were conducted for this

investigation. The samples used for these measurements were as discussed in section 5.2. The

first set of measurements was done on a series of samples with variation in channel depth. We

will first explain the analysis procedure using the result from one POAM measurement.

Fig. 5.13. The black line is the result of a POAM measurement on a sample with pitch of 244 nm, channel width of 80 nm, and channel depth of 250 nm. For comparison, the red line is the result of a POAM measurement on a planar sample.

88

An example of results obtained on a sample with line structures is shown as the black line in Fig.

(5.13). The pitch of the line structure was 244 nm, the channel width was 80 nm, and the channel

depth was 250 nm. The Brillouin oscillations and the decaying background have already been

removed from the data and so it only shows the acoustic contributions to the signal. The

Brillouin oscillations were removed using the same method mentioned in section 5.4.1. The

decaying background was removed by first fitting the data to a function ( )y t similar to Eqn. (5.2)

but with more Gaussian terms to account for additional reflections from the line structures, and

then subtracting only the exponential decay terms from the data. For comparison, the red line in

Fig. (5.13) is the result of a POAM measurement on a planar sample.

FDTD simulations were done based on the sample dimensions measured by SEM [1]. The result

of the FDTD simulation convoluted with the response function is plotted with the experimental

data in the upper part of Fig. (5.14); the results were normalized to their respective magnitudes of

signal T. It can be seen that although the timing of signals T, B, and t agree well to each other,

the magnitudes of signal B and t do not agree. In an attempt to make the simulation agree better

to the experiment, slip boundary condition was implemented in the simulation by setting the

viscosity of a few layers of water along the channel walls to a lower value. A finite slip length

in the simulation is achieved by setting the viscosity of the slip layer according to

0

d

d

. (5.13)

Here d is the thickness of the slip layer and 0 is the measured viscosity of water. A slip length

refers to the depth into the boundary such that the linearly extrapolated velocity at the depth is

89

zero. The result of FDTD simulation with 32 nm shows a much better agreement to the

experimental data. This comparison is also plotted in Fig. (5.14). Since the simulations with this

slip condition sufficiently reflect the behavior of the experimental results, they will be used for

all the analysis below.

Fig. 5.14. The results from FDTD simulation are compared to the experimental result. The FDTD simulations were based on the dimensions measured using SEM. The black lines are the experimental data from the same measurement. The green line is the simulation result without slip boundary condition. The red line is the simulation result assuming infinite slip length.

90

It can be seen that for the case with the slip boundary condition, the magnitude of the second

echo (starting with the peak labeled TT) is larger in the simulation result than in the

experimental data. As already mentioned when discussing planar samples, the treatment of the

top boundary to be fixed in the simulations gives perfect reflection of a returning echo. For

planar samples it is straightforward to correct for this, but this is not possible for the patterned

samples. Therefore, we will only limit our analysis to the shape of the first echo.

Fig. 5.15. The results of the first series of POAM measurements. The channel depths of the samples are as listed. The red lines are the experimental results. The black lines are results from FDTD simulations.

91

The results of the first series of POAM measurements are plotted in Fig. (5.15); this set of data

was measured from a series of structures with variation in channel depth. The dimensions of the

structures in this series are listed in Table (5.1). A measurement on the planar sample is included

for comparison. The depths of the channel were measured using SEM after the POAM

measurements. FDTD simulations with slip boundary condition were done based on the channel

depths measured by SEM. The results of the simulations are superimposed on the experimental

results. The timings of the signals agree very well to each other. The differences in the echo

times between signal T and signal B suggest that the discrepancy between POAM and SEM is at

most 9 nm. The difference in arrival time between signal T and signal B decreases as the channel

depth decreases. Signal T and signal B overlap significantly when the channel depth is about 85

nm, and they completely merge when the channel depth is about 50 nm. If we define the critical

channel depth to be the deepest channel depth in which POAM is unable to detect the channels,

then the critical depth falls somewhere between 50 to 85 nm. However, it can be seen in the

simulation result for channel depth at 50 nm that the “back reflection” can still be distinguished

despite the merging of signal T and signal B. This suggests that, if there is a way to increase the

signal-to-noise ratio, it will still be possible to measure the channel depth, since the timing of

signal t also depends on the channel depth.

The second series of POAM measurements are plotted in Fig. (5.16); this set of data was

measured from a series of structures with variation in pitch. Table (5.2) contains the dimensions

of the structures used in this series; these dimensions were measured by SEM. The plots of the

experimental data are superimposed on top of the plots of the simulation results based on the

profile measured using SEM. The figure shows that as the pitch is increased, the magnitude of

92

signal B decreases. However, signal B remains detectable even at the largest pitch. Currently, the

results show that signal B is detectable for / 0.085W L . The channel width is W and L is the

width of the line structures. To find out how the relative magnitudes of signal T and signal B

depends on /W L , the /B T ratio is plotted against the /W L ratio in Fig. (5.17). Here /B T is

the magnitude of signal B divided by the magnitude of signal T. If we assume that signal T and

signal B are contributed by acoustic pulse reflected from all of the surface at the top of the lines

and the bottom of the channels, respectively, then we would expect that / /B T W L . Although

the plot indeed suggests a linear relationship, a line fitting the plot has a slope of 1.89. Since only

Fig. 5.16. The results of the second series of POAM measurements. The pitches of the samples are as listed. The red lines are the experimental results. The black lines are results from FDTD simulations.

93

L is varied, we can only deduce from the slope that only acoustic pulse reflected from 53% of

the surface at the top of the lines contributes to signal T. Regardless, more data points at wider

range of /W L are necessary before a proper model can be established.

Fig. 5.17. The plot of the B/T ratio against the W/L ratio for each sample from series 2.

94

Bibliography 

[1] FEI HeliosNanoLab 600 Focused Ion Beam.

[2] Logitech PM5 Precision Lapping and Polishing Machine.

[3] The cavity layers were deposited onto a 3 mm thick z-cut crystal quartz substrate using e-beam deposition for the Al films and reactive plasma deposition for the SiO2 films. The approximate thicknesses of the thin Al film, the spacer SiO2 film, and the thick Al film were 9 nm, 230 nm, and 100 nm, respectively. The cavity was fabricated without breaking the vacuum in order to ensure good adherence between the layers. The cavity was then capped by a protective layer of SiO2 that was 50 nm thick. The base pressure of the deposition system was

approximately 62 10 torr. The cavity was diced into 3 mm x 4 mm slabs.

[4] Landau, L.D. and Lifshitz, E.M. Fluid Mechanics, 2nd Ed, translated by Sykes, J.B. and Reid, W.H. (Elsevier, Oxford, 2010.)

[5] Attal, J., and Quate, C.F. “Investigation of some low ultrasonic absorption liquids.” J. Acoust. Soc. Am. 59, 69 (1976).

[6] CRC Handbook of Chemistry and Physics, 69th Edition, edited by Weast, R.C., Astle, M. J., and Beyer, W.H. CRC Press, Inc (1988-1989).

[7] Grimsley, T.J. (2012) Planar Opto-acoustic Microscopy Applied to the Metrhology of Periodic Nanostructures. Ph.D. dissertation, Brown University, Providence, Rhode Island.

[8] The indices of refraction for the Al films and the amorphous SiO2 films were measured using J.A. Woollam Co., Inc. M-2000 DI Ellipsometer. The derivatives for the Al was taken from Li, Y., Miao, Q., Nurmikko, A.V., and Maris, H.J. “Picosecond ultrasonic measurements using an optical cavity.” J. Appl. Phys. 105, 083516 (2009). The derivative for the amorphous SiO2 was extrapolated using the results from Biegelsen, D.K. and Zesch, J.C. “Optical frequency dependence of the photoelastic coefficients of fused silica.” J. Appl. Phys. 47, 4024 (1976). The index of refraction for the crystal SiO2 was taken from Ghosh, G. “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals.” Opt. Commun. 163, 95 (1999). Lastly, the derivative for the crystal SiO2 was taken from Detraux, F. and Gonze, X. “Photoelasticity of α-quartz from first principles.” Phys. Rev. B 63, 115118 (2001).

[9] Note that the measured transit time inside each cavity layer deviates by about ten percent of the transit time calculated using values of the bulk material. It is either that the actual thicknesses are more than what the thickness monitor reported, or that the elastic moduli of the thin films are different from that of the bulk materials due to the chosen deposition methods. It is unlikely that

95

the actual thicknesses differ more than a few nanometers from the reported values, since the thickness monitor was calibrated using test films right before each deposition. Due to lack of definite measurements, we can only assume that a combination of both factors has occurred. However, since peaks A, B, C, and D can be easily identified, the acoustic impedances across the cavity appear relatively well matched. Therefore it is not a problem to the experiments.

96

 

Chapter 6 

Conclusion and Future Work 

 

This thesis presented two parts of work on variations of picosecond ultrasonic technique. In the

first part, we demonstrated that we were able to generate and detect surface acoustic waves using

picosecond ultrasonics with an optical mask. It was shown that the wavelength of the surface

waves generated depended on the pitch of the mask. We were also able to generate and identify

different modes of surface acoustic waves by using samples with different thicknesses of thin Al

film deposited onto a Si substrate. In the second part of the work, we presented results on Planar

Opto-Acoustic Microscopy. We made a series of measurements on one-dimensionally periodic

nanostructures that were series of channels with nearly rectangular in shape. We were able to use

POAM to measure the channel depth quite accurately. We also found that POAM was not able to

detect the channels if their depths were shallower than a depth between 50 to 85 nm. On the

other hand, POAM was able to detect the channels if the structure had the ratio of /W L that was

larger than 0.085.

There are two areas of possible future works. The first area of future work is the possibility of

using POAM to measure slip boundary condition, as it was shown from the FDTD simulations

97

that slip boundary condition has an effect on the magnitude of signal B. Here we present a

possible systematic approach. The samples should have the same simple geometry as the ones

used in this work, but with a variation in the pitch in the way that the /W L ratio is fixed, so that

the /B T ratio is not affected by the /W L ratio. The channel depth should also be at least as

deep as the ones provided for current work so that signal B is more pronounced. Lastly, there

should also be a series of samples with the same geometry but made with different materials so

that different slip lengths can be studied.

Another area of future work will be to investigate the relationship between /B T and /W L . It

will be of great interest to confirm that /B T and /W L are related linearly, and that such

relationship holds even for / 1W L . It will also be significant to find out at what /W L ratio

that POAM ceases to detect either signal B or signal T. Samples for these series of study should

cover a wide range of /W L ratio. In addition to varying L , there should also be series that

variesW , so that the portion at the bottom of the channel that contributes to signal B can be

measured.