Uncovering new thermal and mechanical behavior at thenanoscale using coherent extreme ultraviolet lightbyKathleen Marie Hoogeboom-Pot
B.S., Calvin College, 2008
M.S., University of Colorado Boulder, 2012

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2015This thesis entitled:
Uncovering new thermal and mechanical behavior at the nanoscale using coherent extreme
ultraviolet light
written by Kathleen Marie Hoogeboom-Pot
has been approved for the Department of Physics

Prof. Margaret M. Murnane

Prof. Henry C. Kapteyn

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

Hoogeboom-Pot, Kathleen Marie (Ph.D., Physics)
Uncovering new thermal and mechanical behavior at the nanoscale using coherent extreme ultraviolet
light
Thesis directed by Prof. Margaret M. Murnane and Prof. Henry C. Kapteyn

Tremendous recent progress in nanofabrication capabilities has made high-quality singleatomic layers and nanostructures with dimensions well below 50 nm commonplace, enabling unprecedented access to materials at the nanoscale. However, tools and techniques capable of characterizing the properties and function of nanosystems are still quite limited, leaving much of thefundamental physics that dominates material behavior in the deep nano-regime still unknown.
Further understanding gained by studying nanoscale materials is critical both to fundamental science and to continued technological development. This thesis applies coherent extreme ultraviolet(EUV) light from tabletop high harmonic generation to study nanoscale systems on their intrinsic
length and time scales (nanometers and femtoseconds, and above), specifically following thermaltransport and acoustic dynamics. These studies have shown where and how nanostructured material properties can be quite different from their bulk counterparts. This has in turn allowed us todevelop new theoretical descriptions to guide further work.
By observing heat dissipation from the smallest nanostructure heat sources measured to date(at 20 nm in lateral size), this work uncovers a previously unobserved and unpredicted nanoscale
thermal transport regime where both size and spacing of heat sources play a role in determining
the heat dissipation efficiency. Surprisingly, this shows that nanoscale heat sources can cool more
quickly when spaced close together than when far apart. This discovery is significant to the engineering of thermal management in nanoscale systems and devices while also revealing new insight
into the fundamental nature of thermal transport. Furthermore, we harness this new regime todemonstrate the first experimental measurement of the differential contributions of phonons with

iv
different mean free paths to thermal conductivity, down to mean free paths as short as 14 nm forthe first time.
The same technique is then applied to the study of acoustic waves in nanostructured materials, where they are used to characterize mechanical properties at the nanoscale. This thesis
demonstrates the application of EUV nanometrology for the complete characterization of isotropic
ultrathin films down to 50 nm in thickness across a broad range of stiffnesses. By simultaneously
measuring both longitudinal and transverse waves, we are able to study trends in elastic properties
that are normally assumed to be constant because it is difficult to measure them. This work also
extends the technique to study anisotropic materials.
Finally, by observing the acoustic resonances of nanostructured ultrathin bilayers, this workis the first to apply EUV nanometrology to layers with sub-10nm thickness and to measure themechanical properties of nanostructures down to single monolayer levels. Here it is shown that thedensity ratio of the ultrathin layers is not substantially altered from the bulk material counterpart,but the nanoscale elastic properties do deviate significantly and follow opposing trends for two
different metallic materials.

Dedication

To Justin for constant love and friendship.
To my family for always encouraging deep curiosity about the world.
And to Him who is able to do immeasurably more than all we ask or imagine.

vi

Acknowledgements

My time in grad school and in JILA has been a fantastic experience thanks in no small partto the many great people in the KM group that I’ve spent this time with and the great leadership,
guidance and mentorship that Margaret and Henry provide. While I can’t possibly extend thanks
to all those who deserve it here, a few particular highlights must be mentioned. Through Margaretand Henry’s excellent examples, I learned how important it is to develop the skill of good scientificstory-telling — a lesson I will benefit from for the rest of my life. Special thanks to Tory andCraig who taught me that secret stashes of critical lab supplies must be maintained. Thanks
to Dan Adams for sharing your truly impressive optics expertise and many excellent discussions
which always furthered my understanding of the rigorous science behind my more intuitive ideas.
Mark, your friendship and willingness to help us with experimental quirks and understanding morebackground have been wonderful. Most importantly, thanks to my Boulder family: Qing, youtaught me everything I needed to be able to go on and delve deeply in these experiments, and you
were a great friend in my first years of late nights in the lab. Travis, it has been a pleasure to watch
you gaining the same kind of familiarity I learned from Qing, and you give me great confidence
in turning over the ongoing experiments to you and Nico. Damiano and Nico, I think I’ve spent
more of my life in Boulder with you than just about anyone else, and your research expertise andfriendship made this all work so well. And those crazy nights when we finally felt like we were
figuring it all out were unforgettable. Finally, thanks to Justin whose constant encouragement,
support and sanity-keeping — and not a small number of late dinner deliveries in lab made all of
this work possible.

Contents

Chapter
1 Introduction

1

2 Experimental background7

2.1

Pump-probe spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1

Tabletop EUV from high harmonic generation. . . . . . . . . . . . . . . . . 14

2.2.2

Building a signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3

Diffraction as Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3

Why use EUV? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 New regime in nanoscale thermal transport

34

3.1

Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2

Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3

Observing non-diffusive thermal transport . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4

3.3.1

Sample design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.2

Quantifying deviations from diffusive transport . . . . . . . . . . . . . . . . . 47

3.3.3

Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Understanding a new thermal transport regime . . . . . . . . . . . . . . . . . . . . . 58
3.4.1

Quasi-ballistic model for isolated heat sources . . . . . . . . . . . . . . . . . . 58

viii
3.4.2

Interacting heat sources, grey phonon approximation . . . . . . . . . . . . . . 60

3.4.3Interacting heat sources, full phonon distribution . . . . . . . . . . . . . . . . 65

3.5

Alternate understanding through effective conductivity . . . . . . . . . . . . . . . . . 67

3.6

Outlook for the study of nanoscale thermal transport . . . . . . . . . . . . . . . . . . 71

3.7

3.6.1

Extension to 2D heat sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.6.2

Opportunities with dynamic imaging . . . . . . . . . . . . . . . . . . . . . . . 73

3.6.3

Avenues for further theoretical development . . . . . . . . . . . . . . . . . . . 75

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Probing phonon mean free path spectra

78

4.1

Developments in phonon MFP spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 79

4.2

Experimental measurement of MFPs down to 14 nm . . . . . . . . . . . . . . . . . . 82

4.3

Applying MFP spectroscopy with EUV light

4.4

Conclusion. . . . . . . . . . . . . . . . . . . . . . 87

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Thin film metrology

91

5.1

Elastic properties of materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.2

Existing metrology techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3

EUV nanometrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.1

Extracting acoustic velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.2

Elastic properties from acoustic velocities . . . . . . . . . . . . . . . . . . . . 102

5.4

Anisotropic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.5

Future opportunities in nano-mechanical characterization with EUV . . . . . . . . . 113

5.6

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 Nanostructure metrology

115

6.1

Nanoscale departures from bulk properties . . . . . . . . . . . . . . . . . . . . . . . . 116

6.2

Samples for study of patterned films with sub-10nm layer thickness . . . . . . . . . . 117

ix
6.3

6.4

6.5

Measuring material density with the surface acoustic resonance . . . . . . . . . . . . 121
6.3.1

Extracting small frequency shifts with high resolution using CZT . . . . . . . 122

6.3.2

Interpreting frequency-shift data . . . . . . . . . . . . . . . . . . . . . . . . . 122

Characterizing elastic properties with the longitudinal acoustic resonance . . . . . . 125
6.4.1

Extracting LAW periods from fast damping using MPM . . . . . . . . . . . . 125

6.4.2

Interpreting LAW oscillation periods . . . . . . . . . . . . . . . . . . . . . . . 132

6.4.3

Potential mechanisms for modified nanoscale elastic parameters . . . . . . . . 138

Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7 Conclusions and Future Opportunities

143

7.1

Ongoing efforts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.2

Future opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Bibliography

148

Appendix
A MATLAB program for frequency peak-finding using chirp z-transform164

B Mathematica code for the Matrix Pencil Method

168

B.1 Set up special functions and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
B.2 Import data and extract signal components . . . . . . . . . . . . . . . . . . . . . . . 170
C MATLAB program for least-squares fitting with the acoustic transfer matrix

172

x

Tables

Table
3.1

Material parameters for thermal simulations . . . . . . . . . . . . . . . . . . . . . . . 49

6.1

Nanoscale density results from SAW under Ni/Ta bilayers . . . . . . . . . . . . . . . 124

6.2

Best-fit effective LAW velocities in ultrathin Ni and Ta . . . . . . . . . . . . . . . . . 137

Figures

Figure1.1

Exponential scaling of transistor feature size . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Hard drive data storage density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.1

Highlights in the history of time resolution . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2

Scientific photography: Marey’s falling cat . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4

Three-step model for HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5

Phase-matching for bright HHG source . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6

Timing schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7

From diffraction image to signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.8

EUV harmonics respond differently to height changes

2.9

Data separates dynamics by timescale . . . . . . . . . . . . . . . . . . . . . . . . . . 24

. . . . . . . . . . . . . . . . . 22

2.10 Diffraction peaks highlight specific SAW orders . . . . . . . . . . . . . . . . . . . . . 26
2.11 SAW selectivity observed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.12 Phase advantage from EUV beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1

Diffusive vs. ballistic heat flux predictions . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2

Contrast between transient gratings and periodic nanostructures . . . . . . . . . . . 43

3.3

Thermal sample layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4

Finite-element simulation unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

xii
3.5

Comparing inertial and quasi-static simulations . . . . . . . . . . . . . . . . . . . . . 51

3.6

Extracting thermal boundary resistivity from dynamic EUV diffraction . . . . . . . . 54

3.7

Effective thermal boundary resistivity in sapphire and silicon . . . . . . . . . . . . . 56

3.8

Illustration of three thermal transport regimes

3.9

Phonon filter model schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

. . . . . . . . . . . . . . . . . . . . . 57

3.10 Phonon suppression filter function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.11 Suppression filter model predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.12 Testing a measurement-sensitivity explanation of observed dynamics . . . . . . . . . 68
3.13 Confirming predictions for closely-spaced nanoscale heat sources . . . . . . . . . . . 68
3.14 Fitting data with effective conductivity. . . . . . . . . . . . . . . . . . . . . . . . . 70

3.15 Effective conductivity simulation cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.16 Thermal penetration depth vs. heat source linewidth . . . . . . . . . . . . . . . . . . 72
3.17 Comparison of 1D and 2D heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.1

Setting bins to extract MFP spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2

Extracted MFP spectra for sapphire and silicon . . . . . . . . . . . . . . . . . . . . . 85

4.3

Thin film thermal data for MFP extraction . . . . . . . . . . . . . . . . . . . . . . . 89

5.1

Range of E and ν for materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2

Setting SAW penetration depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3

Samples for EUV photoacoustic metrology of thin films . . . . . . . . . . . . . . . . 99

5.4

LAW echoes in a set of grating periods . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.5

Extracting SAW frequency from thin film signals . . . . . . . . . . . . . . . . . . . . 103

5.6

Layer sensitivity in the SAW velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.7

Identifying excited SAW modes with simulation . . . . . . . . . . . . . . . . . . . . . 107

5.8

Elastic characterization of thin film series . . . . . . . . . . . . . . . . . . . . . . . . 108

5.9

Extra SAW modes from 2D nanodot arrays . . . . . . . . . . . . . . . . . . . . . . . 110

5.10 Examining 1D and 2D SAWs to characterize anisotropy . . . . . . . . . . . . . . . . 111

xiii
5.11 Extracted velocities from 1D and 2D nanograting SAW . . . . . . . . . . . . . . . . 112
6.1

Ultrathin bilayer samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.2

X-ray reflectivity measurements of bilayer samples . . . . . . . . . . . . . . . . . . . 120

6.3

SAW frequency shifts with Ta thickness . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.4

Extracted SAW frequency shifts in four harmonics . . . . . . . . . . . . . . . . . . . 123

6.5

LAW resonances observed as a function of Ta thickness . . . . . . . . . . . . . . . . 125

6.6

No systematic effects due to etch depth . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.7

Preparing LAW data for MPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.8

Separating data from noise with singular values . . . . . . . . . . . . . . . . . . . . . 129

6.9

Comparing MPM output with measured data . . . . . . . . . . . . . . . . . . . . . . 130

6.10 Extracted resonant LAW periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.11 Finite-element model of the open-pipe-type resonant mode . . . . . . . . . . . . . . . 132
6.12 Resonant frequency determined by sign of acoustic mismatch . . . . . . . . . . . . . 133
6.13 Best-fit residual surface for range of vT a and vN i . . . . . . . . . . . . . . . . . . . . 137
6.14 Possible variation in Ta velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.15 Double pump for selective acoustic excitation . . . . . . . . . . . . . . . . . . . . . . 141

Chapter 1

Introduction

The rapidly growing sector of Information and Communications Technologies now consumes
up to ≈ 10% of global electricity production, powering billions of personal computing devices,
data centers and data transmission services around the world, some recent reports estimate [1, 2].
Moreover, the amount of Internet traffic occurring every hour will soon surpass the total fromthe year 2000 [2]. Much of this explosion in personal computing, connectivity, and data use andproduction has been driven by the rapid scaling of the transistors that comprise computer processors
to very small size and data storage to very high density. This in turn has made incredible computing
power very inexpensive and thus very accessible, such that the majority of people on earth carrybillions of transistors in each pocket-sized device. More people today have mobile phones than haveaccess to toilets or latrines [3], raising demand for high-volume production from semiconductor
fabrication facilities that now output more transistors every year than the world’s farmers growgrains of wheat or rice [4].
These rapid advances drove developments in nanofabrication capabilities such that nanoelectronic devices sold today are made with features smaller than 10 nm [5]. However, a host of new
physical effects become significant for materials at such small size scales. Surfaces and interfaces
take up an increasing proportion of the total number of atoms within a nanostructure volume.
Continuum models for processes like heat transfer and material elasticity no longer apply as the
dynamics of individual grains or the atomic lattices that comprise solid materials deviate fromthe macroscale approximations. Moreover, the capabilities for producing such small structures far

2Intel 4004

Intel 286
Intel Pentium

AMD K6 LittlefootIntel Xeon
Intel
Core M

Figure 1.1: Exponential scaling of transistor feature size | Moore’s Law began as an observation of the trend in the number of transistors comprising an integrated circuit, which haveapproximately doubled every two years since the 1960s. Now it is the driving guideline that sets
the research and development goals of the modern semiconductor industry. To accommodate thegrowth in number, the minimum feature size within the transistors has seen extraordinary scaling
as well. This is a large part of why the computers that used to require rooms full of equipment to
do basic math can now easily fit into a pocket and give almost instantaneous access to informationfrom around the world. A few specific CPU feature sizes are highlighted by the red points andlabeled.

3

Figure 1.2: Hard drive data storage density | Exponential growth is also evident in data storage
density. The first commercial hard drive, shown in the left inset image, was introduced in 1956 andstored 3.75 megabytes across its fifty 24-inch disks. Vast improvements in technology since thenfor both reading and writing data in smaller and more densely packed bits brought us to where we
can now hold terabytes on a drive like that shown in the right inset, which can easily be held in
one hand.

4
outstripped the development of tools and techniques that are able to characterize them, such thatthe nanoscale physics that comes to dominate these systems is still not well understood. This is
one of the grand challenges standing in the way of further technological developments at this time.
Sub-atomic precision is already needed but hard to obtain [6, 7], prompting the common refrain,“We can’t fix what we can’t measure.” But with steadily rising electricity and energy costs ofcurrent technologies, there is much that needs to be fixed — both to improve efficiency within
information and communications technologies and to apply them in ways that allow us to lower
the other 90% of electricity consumption.
This creates a unique opportunity. Few-atom layers and sub-50nm structures can now be
fabricated reliably, offering well-controlled systems for systematic study where we can uncover thefundamental processes that determine dynamics and material characteristics at the nanoscale. Inturn, the deeper understanding we achieve will inform future technological developments, as wellas provide further probes into the atomic and sub-atomic interactions that control material-levelproperties and dynamics.
While many types of dynamics demand more fundamental understanding in this nanoscale
regime, my focus is on phonons — the collective motions of atomic lattices. These dynamics will
fall into two main categories: thermal transport and acoustic wave propagation. Observing both ofthese in a single measurement with coherent extreme ultraviolet (EUV) light obtained from tabletop
high harmonic generation offers the nanometric wavelength and femtosecond light pulses that yield
the high spatial and temporal resolution necessary for measurement of nanoscale structures andtheir ultrafast dynamics.
Chapter 2 introduces the experimental techniques in more detail, discussing pump-probe
spectroscopy in general before looking at how it is specifically applied in this thesis. In particular,some unique features of using periodic metallic nanostructures as both samples of study and transducers of dynamics that will illuminate other aspects of the whole material system are presented.
Finally, it examines the unique advantages gained from probing nanoscale systems with EUV lightand how high harmonic generation creates this coherent light source in a tabletop experiment.

5
The first application of these techniques is demonstrated in Chapter 3. It begins with an introduction into how and why nanoscale thermal transport is fundamentally different from macroscaletransport, along with a summary of important previous experimental observations of this phenomenon. Then, by examining the heat dissipation from the smallest heat sources measured todate (at 20 nm in lateral size), we uncover a new thermal transport regime that dominates whenboth heat source size and spacing are small compared to the dominant mean free paths of heatcarrying phonons in the substrate material underneath the heat sources. Surprisingly, our new
findings imply that nanoscale heat sources can, under defined circumstances, cool more quickly
when spaced close together than when isolated, a prediction that we directly confirm with experimental observations of sub-50nm heat sources on a silicon substrate. This discovery has importantdesign implications for many applications of nanoscale thermal engineering, and it yields an unpredicted and previously unobserved insight into nanoscale heat transfer dynamics.
The phonon-filtering nature of this new thermal transport regime is then applied in Chapter
4 to assess the spectrum of phonon contributions to thermal conductivity as a function of mean
free path. This represents the first experimental measurement that can benchmark theoretical
calculations of such spectra to mean free paths as small as 14 nm. It also offers the first direct
experimental access to differential, rather than only integrated, phonon conductivity spectra. Since
these spectra are essential to determining nanoscale thermal transport in a given material, thismethod will be important for future tests of theoretical predictions and to measure these spectra
for more complex materials where calculations do not yet exist and may not be practical.
Chapter 5 shifts focus to the acoustic dynamics within nanostructured systems. First it introduces the physical meaning of elastic properties, and then demonstrates how an EUV nanometrology
pump-probe technique can be applied to characterize the full elastic tensor of a wide range of softand stiff isotropic ultrathin films (≤ 100 nm in thickness). In particular, the very short acoustic
wavelengths observable with EUV probes allow a wide range in measurement sensitivity to various
layers, from characterization of a substrate to full confinement within a thin film to uniquely isolate
its properties.

6
Chapter 6 extends EUV nanometrology to the mechanical properties of composite nanostructures for the first time. Here, surface acoustic waves launched below the nanostructures reveal thatthe mass density ratio between the nickel and tantalum in a sub-20nm bilayer structure is not significantly modified from its bulk counterpart. On the other hand, longitudinal acoustic resonancesof the bilayers themselves show that the elastic properties of the two materials deviate significantly
from bulk properties and with opposing trends. In particular, the tantalum layers are observed tostiffen significantly more than is typically observed for other ultrathin metallic layers.
This thesis concludes in Chapter 7 with a summary of the new findings presented and perspectives on a variety of future opportunities inspired by this work, spanning from local characterization
based on dynamic coherent diffractive imaging, to new advanced developments in nanometrology
using VUV or EUV transient grating technology. Beyond enabling continued development in nanoelectronics, clean energy and new medical therapies, the deeper understanding gained of nanoscale
thermal transport and nanoscale elastic properties builds a strong foundation for further explorationof the fundamental material processes at work in nanostructured systems.

Chapter 2

Experimental background

While at first thought the studies of nanoscale heat dissipation and of the elastic properties ofnanoscale materials may seem to have little in common, the methods used to observe the dynamics
that reveal both sets of properties are very similar. Both can be related to physical deformationsof solid materials: the flow of heat through a system will cause thermal expansion and relaxation;
elastic properties determine the propagation of acoustic waves, which cause regular expansion andcontraction of lattices as they pass. Thus, in order to study the elastic and thermal transport
properties unique to nanostructured systems, we use a pump-probe technique to directly observe
the systems’ dynamic surface deformation. This yields a signal that can be parsed to characterizeboth sets of nanoscale properties with one efficient measurement.
In this chapter I will introduce the general concepts of pump-probe spectroscopy and its
specific application in our experiments, including the use of periodic nanostructures and shortwavelength extreme ultraviolet (EUV) probes to gain strong sensitivity to materials at the nanoscale.

2.1

Pump-probe spectroscopy
Dynamic observations have a few fundamental requirements: some aspect of the system inquestion must be evolving; those changes must be detectable by some probe of the system; anddifferent times within this evolution must be isolated from one another by a limited observation
window and labeled relative to some well-defined time-zero. To rule out statistical anomalies, one

8would ideally observe the same evolving process many times, requiring that the dynamics progress
repeatably.
Because the understanding of dynamic processes and the ability to predict how they unfold
is vital to many aspects of life, civilization and progress, humans have been refining the techniques
for making such observations for thousands of years. One of the earliest recorded examples comesfrom the Ivory Tablet inscribed during the First Dynasty of United Egypt under Menes, ca. 3100
B.C. The Tablet speaks of “Sothis Bringer of the Year and of the Inundation”, referring to thebeginning of their astronomical calendar and the subsequent flooding of the Nile [8]. By observing
the regular progression of the river’s dynamics between the heliacal rising (when in the year a starfirst becomes visible above the eastern horizon for a brief moment just before sunrise [9]) of Sothis(also known as Sirius) and the annual floods, they learned to predict the precise timing of the floods
and to arrange their agricultural cycles around them. After that, it was ‘just’ a matter of refining
time resolution to gain understanding about progressively faster processes (see Fig. 2.1).
Two aspects of time resolution required improvement. First, the resolution of time-keeping
was improved to make ever more precise measurement of the passage of time. Starting around1500 B.C., sundials and water clocks broke days into hours and minutes [11]. The introductionof mechanical clocks in Europe around 1300 A.D. made the passage of seconds more visible [15].
Precision was further improved through the use of pendula (first implemented by Christiaan Huygens in 1656), then wheels and springs, before quartz crystal oscillations obviated the need forcomplicated interactions between mechanical parts in the 1920s [16]. Time-keeping precision took
another leap forward with the advent of the cesium atomic clock, first demonstrated at the National
Physical Laboratory in England in 1955. The most precise atomic clocks today use the interaction
between laser light and atoms to keep track of the passage of time while losing less than a tenth of
a second over the age of the universe [17]. However, resolution in the precise timing of the start ofobservations is not the only necessary parameter to improve.
The second time resolution improvement needed was the duration of an observation, whichmust be limited so as not to integrate over the dynamics of interest. But until the 1800s the9Figure 2.1: Highlights in the history of time resolution | Time-resolved observation of dynamics require both accurate time-keeping and short observation durations to freeze the system in
different stages of its evolution. Dynamic speeds extend over many orders of magnitude dependingon the object of observation; some examples are shown along the left: the slow movements of aturtle, the spinning of helicopter blades, the motion of a bullet through a bubble, the oscillationof nanoscale acoustic waves and the motions of atoms and the electrons within them. The filledgreen squares plotted follow the development of accuracy in clocks (measured by how many seconds
they lose or gain per day), from mechanical gears and springs to atomic clocks. The open blue
squares track the shortest time windows available for limiting observation duration. Notably bothcurves see a steep drop when scientists began to use light for both of these tasks (exploiting atomic
resonances for clocks and flashing samples with light to limit an observation window), particularly
following the invention of the laser in 1960. [10, 11, 12, 13, 14]

10
fastest ‘detectors’ were the human eye (response time around 0.1 s) and ear (0.1 ms) [11]. Thenthe mechanical-shutter snapshots by photographer Eadweard Muybridge captured images of the
sub-second regime with his famous galloping horse in 1878 [18], and physiologist Etienne-Jules
Marey led the way in developing these techniques for quantitative measurement [19] (see Fig.
2.2). Harold Edgerton introduced stroboscopic images in the 1920s, freezing the motion of bullets,
bubbles, birds and more by limiting observation duration with a microsecond flash of light [14]. Thus
measurements could be isolated to the duration of the shortest light flashes that could be generated.
Thanks to the invention of the laser in 1960 [20] and fast subsequent developments in pulsedlaser techniques from Q-switching (nanosecond pulses), to active modelocking (< 1 picosecond),
to passive modelocking (femtosecond), these incredibly short flashes of light improved minimum
observation durations by nine orders of magnitude [12].
It is also the extraordinary coherence and monochromiticity of laser light that enables thestability and sensitivity required for the accurate time-keeping of atomic clocks. Moreover, lasers
allowed the precise measurement of time (and frequency) that spurred the redefinition of length,
explicitly defining the meter in relation to the speed of light [22]. This highlights an equivalence
between precise time-keeping and distance measurement. Thus, we have the tools for isolating
femtosecond moments within dynamic processes using the flashes of ultrashort laser pulses and forordering those moments along a precisely labeled timeline by tracking the distance light travels.
Using changes in light-travel distance to set differences in time was an important aspect of
the first investigation that could be recognized as a modern pump-probe experiment. Abrahamand Lemoine used an electrical pulse to simultaneously launch a spark and trigger a Kerr-cellshutter [11]. The Kerr cell placed between crossed polarizers will rotate the polarization of incident
light enough to allow light through the second polarizer when the cell is activated by an electricfield. By varying the distance the light from the spark needed to travel before reaching the Kerrcell and observing the light that traversed the Kerr-cell shutter, they determined how long after
activation by the electrical pulse the shutter would remain ‘open’. Here we see the well-definedtime-zero attained by launching dynamics with a purposeful pump (electrical pulse) to the system,

11

Figure 2.2: Scientific photography: Marey’s falling cat | While Muybridge famously usedphotography to win an argument for California governor Stanford about how horses trot andgallop, Marey sought to use similar principles in a quantitative way to learn about the physiology
of movement. He developed cameras to capture a series of frames of moving animals, people andobjects on the same film so that physical measurements of the photos would be comparable fromframe to frame, like those shown in this series of images which offered the first definitive insight
into how a falling cat lands [19, 21]. It is also the world’s first cat video.

12
as well as stable synchronicity between the pump and probe (spark light) attained via initiating
both from the same electrical pulse. For faster dynamics and the increasingly short measurement
windows attained with ultrashort laser pulses, this latter requirement — which ensures repeatablepump-probe delay — will be even more stringent.

2.2

Experimental setup
Thus we have all the necessary ingredients for direct observation of dynamics in nanostruc-tured systems: a system evolving repeatably with time following excitation with a pump laser
pulse; and ultrashort probe laser pulses to observe the evolving system at specific moments in timeafter the pump pulse. The experimental setup is shown schematically in Fig. 2.3. We fabricate
nanostructured samples — specifically periodic arrays of metallic nanowires and nanodots on dielectric and semiconductor substrates. The metallic nanostructures strongly absorb infrared light,
so we can launch repeatable dynamics with specific time-zero by heating them with an infraredlaser pump pulse of consistent energy. The heat delivered by the pump pulse will cause a thermal
expansion of the nanostructures that will evolve as the system relaxes back toward its initial state.
Our probe pulse will measure these dynamic changes in the surface profile.
Both pump and probe pulses are derived from the same amplified Ti:sapphire laser, yielding
the ultrashort pulses which confine our observation windows to below 20 fs with the required
synchronicity between pump and probe pulses. Translation stages in the pump arm introduce
specific path-length differences relative to the probe arm, yielding controlled time delay between
the two pulses with resolution as high as ≈ 50 fs (due the step resolution of our finest delay stage)that labels the time axis of our measurements. Because the pump pulse is significantly shorter thantypical thermal-expansion response times, the initial thermal expansion of the nanostructures will
occur impulsively and launch acoustic waves in the system at the same time, giving us two classesof dynamics to focus on: heat dissipation, and acoustic oscillation and wave propagation.
As shown in Fig. 2.3, the probe pulse is coupled into a 5cm-long hollow-core waveguide with
inner diameter of 150 µm filled with argon gas to generate the short-wavelength EUV beam that

13

Ti:Sapphire Amplifier
BPS camera 2

Pump ArmBPS camera 1
BPS mirror 2

Lenses
Shutter

BPS mirror 1
Delay stage forlong delays

Probe
Arm

Lens

High-resolution delay stage
Argon-filled hollow
waveguide for HHG

Al filter

Nanostructured sample

EUV beam
Toroidal mirror

X-ray CCD

Figure 2.3: Experimental setup | The output from a Ti:sapphire amplifier with typical pulse
energy of 1.5-2.0 mJ, 3-5kHz repetition rate, 25fs pulse length and spectrum centered around 800
nm in wavelength is split into a pump and probe beam. The pump beam traverses two computercontrolled mechanical delay stages, one with high step resolution (50fs) with maximum delay around1 ns and the other with much longer travel range which can reach maximum delay times around8 ns (when the beam travels across the stage four times rather than the two shown here). Twolenses enable an adjustable focal length and spot size for setting the beam diameter at the sample.
The intensity can be further adjusted by a half-wave plate and polarizer that can reduce the pump
power. A shutter blocks the pump for every other camera exposure to enable frequent comparison
to a relaxed sample, and two 200nm aluminum filters mounted on the CCD block the pump lightthat reflects from the samples. The majority of the laser energy is left for the probe beam, which is
focused into an argon-filled hollow waveguide with inner diameter = 150 µm to drive the process ofhigh harmonic generation. This generates even shorter pulses of EUV light with wavelength centeredaround 30 nm that propagate through two aluminum filters that block the remaining infrared light.
The EUV beam is focused by a glancing-incidence toroidal mirror onto nanostructured samples,
where it reflects and diffracts to an x-ray-sensitive CCD. A small fraction of the amplifier output
is also picked off and split to two cameras separated by a large propagation distance (not to scale)for an active-feedback beam-pointing stabilization (BPS) system which uses two motorized mirrors
to compensate for drifts in the laser pointing.

14yields sensitivity to the surface dynamics of nanostructures (as discussed in more detail below).
Here is where we take advantage of another benefit of using ultrashort light pulses: the possibility
for very high peak intensity. To illustrate the significance of this advantage, consider the average
power typical of our amplifier output: 8 W at a repetition rate of 4 kHz. To couple into the
waveguide, we focus this beam to focal spot size ≈ 120 µm. If this were a continuous-wave laser,
this would imply an intensity of ≈ 1.8 × 104 W/cm2 . However, concentrating this power into 25fs
pulses increases the intensity at the peak of each pulse by ten orders of magnitude to ≈ 1.8 × 1014
W/cm2 . Put another way, the corresponding peak power of 8×1010 W is a little more than the
total capacity of the world’s largest nuclear power plant, Kashiwazaki-Kariwa in Japan.

2.2.1

Tabletop EUV from high harmonic generation

These very high laser pulse intensities are used to drive the process of high harmonic generation (HHG) [23] to obtain coherent extreme ultraviolet (EUV) light from a tabletop source.
The need for such high intensity can be considered in two complementary ways. First, HHG is anextremely nonlinear process, adding the energy from many of the fundamental infrared photons to
produce one high-energy high-order harmonic photon. Intuitively this requires a high intensity toconcentrate many IR photons in a small area at the same time to make this possible.
More precisely, the high intensity (> 1014 W/cm2 ) is necessary to make the strength ofthe laser’s electric field comparable to the Coulomb potentials experienced by the electrons in theatomic gas medium. This sets HHG apart from low-order nonlinear processes in which anharmonic
motions of bound electrons can radiate few-order harmonic photons. In HHG, the intensity of thedriving laser field is high enough to significantly modify the Coulomb potential of the electrons suchthat they can tunnel ionize away from their parent ions and spend time moving freely in the laser’s
electric field. This allows a wide range of harmonics with consistent intensity corresponding to
the region between having enough energy to ionize and the greatest amount of energy that can begained during the laser field acceleration, known as the ponderomotive energy UP . Notably, these
constraints on the necessary driving intensity for HHG can be exploited to isolate attosecond pulses

15
generated only at the peak of driving laser pulse [24, 25], further improving possible resolution fortime-resolved experiments utilizing stroboscopic measurement with short light pulses.
The high driving laser intensity implies that the free electrons with charge q and mass mcan experience very high accelerations a and pick up a significant amount of kinetic energy, since
the force F exerted by the electric field E : F = qE = ma. However, laser fields oscillate inamplitude and direction; when the direction reverses, the electrons will slow and then accelerateback toward their parent ions. Most will simply scatter away from the ions, but a few electrons will
recombine (about 1 in 105 for the 800nm Ti:sapphire driving laser used in this work [26]), emitting
the extra energy they gained from the laser field as a high-harmonic photon. For a given driving
laser wavelength λL and intensity IL (which set the magnitude of acceleration and maximum time
before recombination), and atomic species with ionization potential IP , this implies the highest
photon energy that can be generated is [26]Ecutof f ≈ IP + 3.2UP ≈ IP + IL λ2L .

(2.1)

This simple, semi-classical picture of the HHG process, also known as the Three-Step Model, wasdeveloped by Corkum in the early 1990s [27, 28]. Figure 2.4 illustrates the model. While more
complete descriptions of the full quantum-mechanical process of HHG have since been developed,
this picture remains one that captures the essential elements of single-atom HHG.
However, one atom produces one photon at a time. To generate the photon flux necessary for
applying this unique light source to material characterization experiments, we require many atoms
creating many HHG photons all adding constructively. Here is where the hollow-core waveguide
becomes important. By coupling the driving laser into the gas-filled waveguide, we can maintaina high intensity through a long interaction region. Compare this to a tightly focused beam, which
will be characterized by strong divergence, leaving only a very small interaction region with therequisite high intensity near the focus.
Moreover, to ensure the constructive build-up of HHG intensity, all the HHG photons must
be emitted in phase. This requires the phase velocity of the HHG light to equal that of the driving

16

Figure 2.4: Three-step model for HHG | The intense laser field of a femtosecond pulse focused
into a gas is strong enough to drive electrons to tunnel ionize away from their parent atoms. The
electrons are then accelerated in the field, though their wavepackets spread according to the quantum mechanical evolution of free electron trajectories. When the laser field reverses in direction,
the electron is driven back toward its parent ion. If it recombines, it can emit the extra energy
gained from the laser field as a high-energy EUV or soft x-ray photon. Figure adapted from [29].

laser light, a condition known as ‘phase-matching’. This way, those HHG photons created during
a particular phase of the driving laser pulse will travel with the driving laser to match up withthose HHG photons created at the next equivalent phase of the driving pulse. This is illustrated
in Fig. 2.5. The phase velocity of the EUV or soft x-ray HHG photons is essentially always equal
to the speed of light in vacuum. The infrared driving laser, however, is strongly affected by both
the gas and the waveguide. Phase-matching between the two beams is achieved by tuning the gaspressure P to balance the effects on the driving laser of the waveguide and dispersion due to theionized atoms (both of which increase the phase velocity) with the dispersion due to the neutral
atoms (which decreases the phase velocity) [23] such that the following phase-matching conditionis satisfied:
 2



u11 λ02π
0 = ∆k = q− P (1 − η)Natm δ − η(Natm re λ0 )4πa2
λ0= waveguide − (neutral atoms − ionized atoms)(2.2)

where q represents the harmonic order, u11 is a constant related to the lowest-order electromagnetic
propagation mode of the waveguide, λ0 is the driving laser wavelength, a gives the inner diameterof the waveguide, η is the ionization fraction of the gas with number density Natm , δ is a constant17

Figure 2.5: Phase-matching for bright HHG source | By tuning the gas pressure inside a
waveguide, the HHG process can be phase-matched. This implies that the phase velocity of thedriving laser field and the emitted x-ray fields are equal, such that x-rays produced at different
phases of the driving field add constructively (shown in the inset) to produce a bright, coherent
beam at the waveguide output.

18(for a given laser wavelength) related to the index of refraction for the atomic gas species and re isthe classical electron radius.
This phase-matching is possible only until the ionization fraction of the gas is so high thatthe balance of terms can no longer be achieved. Understanding the interplay between the phasematching cutoff and the photon energy cutoff found in Eqn. 2.1 paved the way for a unified
understanding of how choices in gas medium, driving laser wavelength and intensity can be madeto optimize the generation of high-harmonic light spanning from vacuum ultraviolet to > keV x-rays[26]. Thus, while the Ti:sapphire driving laser combined with argon is effective for generating ahigh flux at the 30nm EUV wavelength utilized in this work, the HHG source is incredibly flexible
through different choices in driving laser and gas medium. Furthermore, the ionization fraction willgrow through the progression of the driving laser pulse, such that phase-matching can be limited
to a small portion of the full pulse duration — particularly for longer driving wavelengths. Thishas been demonstrated as a way of generating isolated attosecond pulses with much higher photon
energies than were previously accessible [30].
The particular advantages of using EUV light from HHG to probe the thermal and acoustic
dynamics in nanostructured systems are discussed in more detail in section 2.3. But this wavelength
regime also presents a number of challenges. It is strongly absorbed by nearly every medium,
requiring all measurements to be conducted in vacuum. This also implies that standard lenses
and mirrors cannot be used to direct and focus the EUV beam. Where we need to focus the
EUV beam, which diverges from the output of the waveguide, we make use of a glancing-incidence
toroidal mirror to preserve as much intensity as possible (shown in Fig. 2.3). Specially-designed
multilayer mirrors can also be engineered for relatively high reflectivities (up to about 70% for
13.5nm wavelength), but the high-reflection bandwidth is generally so limited so as to exclude all
but one harmonic peak. As will be discussed in the following section, every harmonic wavelength
contributes to building our signals, so such monochromatic reflectors would needlessly sacrifice a
significant fraction of the EUV flux we generate.

19
2.2.2

Building a signal

The periodic arrays we study diffract the EUV probe light, yielding an interferometric measurement primarily sensitive to the difference in height between the substrate surface and the top of
the nanostructures. The phase difference in the light reflected from these two surfaces controls thediffraction efficiency (the ratio of diffracted peak intensity to reflected zero-order peak intensity),
and the changes in surface profile induced by the thermal and acoustic dynamics launched by thepump pulse will dynamically change this diffraction efficiency.
Like Marey in section 2.1, we make use of quantitative photography by capturing diffraction
patterns on an x-ray sensitive CCD, where in addition to quantitative information about position
across the detector, we also have quantitative information about the intensity of light incident at
each pixel. Relative changes in the diffracted peak intensities are on the order of ≈ 10−2 − 10−3 , so
to construct a clear signal, we record an integrated change in diffraction between the pumped andunpumped sample at each time step. Specifically, the shutter shown in Fig. 2.3 blocks the pumpbeam for every other camera exposure, alternating the recorded diffraction patterns between thatassociated with the pumped sample and that from the unpumped sample (see Figs. 2.6 and 2.7).
This alternation ensures that the two images are captured close together in time so they will be ascomparable as possible. The shorter the exposure times, the less that can change between the two
images. However, exposure time cannot be so short that the shutter just catching or just missing
the last pulse significantly affects the total number of laser pulses contributing to each image. In
practice, we have found that 30ms exposures usually result in high-quality signals, likely striking
this balance between variation in number of pulses and variation in the probe from one exposure
to the next. At a 4kHz repetition rate, this corresponds to 120 pulses per exposure.
The pixel-by-pixel difference image between the two patterns shown in Fig 2.7 reveals how
the change in the zero-order peak is generally opposite to that in the diffraction peaks, as should
be expected from conservation of energy. Thus, to integrate the signal in a way that constructs
rather than cancels out the contributions from different diffraction orders, we flip the changes in20

Pump pulses

Systemrelaxation

Probe pulses
at dt1
Probe pulses
at dt2
Cameraexposures
Shutter
activation

Time

Figure 2.6: Timing schematic | Every pump pulse arriving at the sample launches the same
dynamics, represented by a simple relaxation in this schematic. The dynamics are probed witha pulse train set to a particular time delay dt relative to the pump pulses, such that the camera
exposures receive stroboscopic frames of each time step within the dynamics. A shutter blocks thepump pulses for every other camera exposure, enabling comparison of pump-on to pump-off images
nearby in time, thus decreasing the influence of fluctuations in the probe intensity. Note that thisschematic is not drawn to scale: each camera exposure actually spans > 100 probe pulses and thesample dynamics (< 20 ns) decay much sooner before the arrival of the subsequent pump pulse(after 250 µs).

21

a

b
10000

Pump-off
Pump-onstnuoc VUE

8000

6000

4000

2000

c
300

Pump-onstnuoc VUE

minus Pump-off
200

100

0

Figure 2.7: From diffraction image to signal | In order to speed acquisition time, we do notsave full diffraction images like that shown in part a. Instead, we vertically bin the images to takeadvantage of the significantly faster read-out time from the CCD, creating the diffraction patterns
shown in part b. A careful comparison of the pump-on (dotted red) and pump-off (solid blue)
patterns can reveal small differences — the central peak is just slightly taller in the pump-on, thediffraction peaks are just slightly shorter. However, it’s the difference image (c.) which most clearly
reveals the differences between the pumped and unpumped case. After flipping the diffraction-peakdifferences to match the sign of the central-peak difference, we integrate across the regions with
non-zero signal to extract the change-in-diffraction point for this particular time step, 40 ps.

22
the different peaks to match in sign and then sum across the image. The same combination offlipped and unflipped regions is used for every difference image in a given signal. Also, since HHG
generates multiple harmonics simultaneously, a set of discrete harmonic peaks shows up in eachdiffraction order — and because of their differing wavelengths it is possible that the change in
diffraction for one harmonic will be opposite in sign to that for another harmonic, depending on
the initial height of the nanostructures (see Fig. 2.8). The time-dependent change in each part ofthe signal, however, will be the same such that proper summing across the difference images willin general build up the total signal. In principle, it would be possible to use the comparison of thedifferent harmonic signals to determine the initial height of the grating structures — essentiallyfinding the point along the x -axis of Fig. 2.8 where the ratios between the corresponding points on
the curves shown matches those observed. In practice, however, too many of the harmonics have
too low an intensity and the variation from shot to shot in the ratios between change-in-diffraction
signal amplitude from different harmonics is too large to generate a result with useful precision.

b
23rd

25th

egnahc thgieh mp04 rof langiS

a

0.01

0.00

21st

27th
29th

-0.01

8

10

12

14

16

18

20

Static grating height (nm)Figure 2.8: EUV harmonics respond differently to height changes | Phase-matching conditions for HHG differ from one harmonic wavelength to the next. This controls the relative intensitiesof the harmonic peaks, like those seen in the typical experimental spectrum shown in part a. Eachharmonic will also display slightly different responses to a given grating height change (like thosecaused by laser excitation), depending on the initial static grating height. In particular, part b
shows that for a given static height, each harmonic will contribute different magnitudes to thesignal; at certain grating heights some harmonic signals even have different sign.

Note that simply taking the absolute value of the difference image would lose too muchinformation since it eliminates the possibility of a signal crossing zero. Acoustic oscillations in23particular tend to oscillate around the unpumped state once most of the thermal decay has passed.
Creating an artificially non-negative signal by using an absolute value in this case can make thesignal appear to oscillate at twice its real frequency (like the absolute value of a sine wave).
To decrease the influence of exposure-to-exposure changes in the pump and probe beam
intensities, we normalize the sum of the difference image to the sum of the pump-off image. This
result is averaged tens or hundreds of times to produce the data point for one time step. Then the
delay stage is moved, changing the pump-probe time delay, and the process repeated. One exampleof the resulting change-in-diffraction signal is shown in Fig 2.9. Over the time of taking a full scanof many time delays (typically around one-half to one hour), shifts in beam alignment can shift
the pump beam location relative to the probe on the sample, and it can significantly change theprobe beam intensity since the coupling of the beam into the fiber for HHG is strongly dependent
on the beam pointing. To lessen these effects, we employ the active-feedback beam stabilization
system visible in Fig. 2.3. This system picks off a small portion of the input laser beam before it
splits into pump and probe arms. This diagnostic beam is further split in half and sent toward two
cameras that image two points in the beam separated by a long propagation distance. When the
beam moves on these cameras, motorized mirrors on the input beam adjust the beam back to its
set position. One image point is chosen just downstream of the second motorized mirror in orderto decouple corrections by the two mirrors as much as possible; the second image point is chosen tohave at least as long of propagation distance as the distance to the sample. We observe that thissystem reduces fluctuations in the HHG counts and virtually eliminates long-term drift, typicallymaintaining the EUV intensity over > 10 hours.
Generated as it is by a whole array of nanostructures, though, how should we interpret thisdynamic change in diffraction signal?

2.2.3

Diffraction as Fourier transform

Far-field diffraction is reached when the propagation distance z following an aperture is large
compared to the ratio of the square of the aperture size to the wavelength λ, and here the output

24

Thin film
LAW echoes

Nanoscale thermal decay

Nanostructure LAW
0

20

40

Delay time (ps)6080Multi-frequency SAW
0

IR Pumpfemtoseconds

picoseconds

1000

2000

Delay time (ps)

3000

4000

nanoseconds

Figure 2.9: Data separates dynamics by timescale | The acoustic and thermal dynamics
launched by the pump laser excitation naturally separate on different time scales, enabling simple
parsing of the signal to extract clear information from each component. Within the first 100 ps,
two sets of longitudinal acoustic dynamics appear: the high-frequency oscillation at the beginningcorresponds to the longitudinal resonant mode of the nanostructures; the later echoes are the resultof a longitudinal pulse that traveled down from the nanostructures into a thin film and reflected backto the surface from a buried interface between the film and the substrate. On the nanosecond timescale, we see multi-frequency surface acoustic oscillations superimposed onto a long, exponential
thermal decay.

25
electric field can be calculated using the Fraunhofer equation [31].
U (x, y, z) ∝

x

2π

0

0

A(x0 , y 0 )e−i λz (xx +yy ) dx0 dy 0

(2.3)

for aperture transmittance A. It can be seen that this is completely equivalent to a Fourier transformof the aperture function A evaluated at frequencies fx = x/(λz) and fy = y/(λz) [32].
While the ≈ 6 cm between our sample and CCD camera is not fully within this regime,
this treatment still offers a useful approximation. Therefore we can use some simple properties ofFourier transforms to gain better intuition regarding what our diffraction signals correspond to.
First, our 1D periodic grating samples with linewidth L and period P are well approximatedby the convolution of a comb of spacing P with a rect function of width L. The Fourier transformthen approximates the diffraction pattern:
F[comb(x/P ) ⊗ rect(x/L)] = comb(P fx )sinc(Lfx )(2.4)

where again fx = x/(λz) relates the incoming spatial coordinate x to the outgoing spatial frequencies. The resulting product of comb and sinc reveals that the locations of the discrete diffraction
peaks are set according to the grating period, and their relative heights are set by the sinc function
associated with the diffraction pattern from a single nanostructure as shown by the black arrows
sampling the red curve in Fig. 2.10. Thus, when we follow the changes in diffraction efficiency as
a function of time following pump excitation, this signal corresponds to the response of a single
nanostructure, averaged over all the structures illuminated by our probe beam.
This highlights the importance of ensuring a consistent pump excitation over the whole region
we probe. For this reason we use a pump beam diameter ≈ 400-600µm — much larger than theprobe spot of ≈ 100 µm such that the pump intensity (assuming a roughly Gaussian intensity profile)does not vary significantly over the area of the probe. The pump beam size can be measured on a
CCD camera placed at the same propagation distance as that to the sample location and adjusted
using the two lenses shown in the pump arm in Fig. 2.3. Note that these lenses must be placed
after the delay stages since the beam must be collimated through the stages so that changing the26

Sinc shift µ 1/L = n/P

Fundamental SAW

2nd-order SAW

Sinc widthµ1/L

Diffraction peaks at
comb spacing µ 1/P

Figure 2.10: Diffraction peaks highlight specific SAW orders | The diffraction pattern froma single rectangular-profile nanowire is given by a sinc function, shown in solid red, with width set
by the linewidth L of the nanowire. The effect of repeating that nanowire in a periodic grating is
to sample the single-structure pattern at regular intervals with spacing set by the period P. This isshown by the black arrows (using P = 2L as an example) and it results in the familiar discrete setof diffraction peaks we recognize from the patterns we capture on our CCD. Adding a cosine-shaped
SAW excitation below the nanowires adds contributions from the shifted sinc functions (shown in
dotted lines) to the original, single-structure pattern. Because the SAW wavelengths are set by the
harmonics of the grating period, the peaks of these shifted sincs line up with existing diffraction
peaks, concentrating the signal from specific SAW orders in their corresponding diffraction orders.

27time delay does not alter the pump beam size at the sample. The beam must also be carefullyaligned parallel to the stages so that it does not move on the sample for changing time delay. Pump
beam power must be chosen to balance the desire for a large signal from stronger excitation with
the need to avoid damage to the sample, or to avoid exciting it so strongly that the dynamics are nolonger linear with pump power. In practice, we typically find this balance around a pump fluenceof ≤10 mJ/cm2 .
Thus far, this intuitive analysis has only considered the dynamics of small perturbations to
the original rect of our static sample. This can basically account for the thermal dynamics andlongitudinal acoustic wave (LAW) modes within the nanostructures, but the surface acoustic waves(SAWs) launched in the substrate could be better included by adding a term where the comb andrect are multiplied by a cos wave representing the standing wave that forms below all the structures.
The diffraction pattern we calculate here is a little more complicated:
F [(comb(x/P ) ⊗ rect(x/L)) · (1 + cos(Λx))]= comb(P fx )sinc(Lfx )+ (δ(fx − 1/Λ) + δ(fx + 1/Λ)) ⊗ (comb(P fx ) · sinc(Lfx )) (2.5)
where Λ is the SAW wavelength — equal to the grating period and integer-fractions thereof (higher
harmonics of the spatial frequency) [33]. The first term in this result is the same as in Eqn. 2.4 — a
comb sampling the sinc associated with diffraction from a single structure at points corresponding
to the spatial frequency of the grating period. The second set of terms repeats the same pattern
except that it is shifted by ±1/Λ. Because any order of the SAWs we excite has spatial frequency
equal to one of the grating’s spatial frequencies, this shift moves the peak of the repeated sinconto the corresponding diffraction peak from the unshifted function as shown in Fig 2.10. For
this reason, the different diffraction orders will display increased sensitivity to the corresponding
SAW-order frequencies. This is demonstrated in Fig. 2.11 and often provides a useful method forextracting high-order SAW frequencies with increased precision by integrating the signal only overthe corresponding diffraction orders.

28

a

10000

stnuoc VUE

8000

6000

4000

2000

).bra( rewop lartcepS

noitcarffid ni egnahC

b

c

500
400
300
200
100
0

0

200

400

600

800

0

1000

Delay time (ps)

100

200

300

400

500

SAW frequency (GHz)01x

6-

Figure 2.11: SAW selectivity observed | By integrating specific sections of the diffraction patterns to build a time-dependent change-in-diffraction signal, we can directly observe that the signal
from different SAW harmonics is concentrated in their corresponding diffraction orders. Specifically, integrating over the blue sections (including only the second-order diffraction peaks) of adiffraction pattern shown in part a from a sample with nickel linewidth 200 nm and period 600
nm on a SiC:H thin film on a silicon substrate results in the signal plotted in blue in part b, while
the green section (first-order peaks) results in the green signal. It is clear in both the signal andin Fourier transforms of the two signals, shown in c, that the fundamental SAW order has its signal concentrated in the first-order diffraction peak, while the second-order SAW is displayed quite
clearly in the second-order diffraction peak. (Note that both Fourier transforms are displayed on
the same vertical scale.)29
Having established this intuitive argument for understanding the changing diffraction signals
we measure as the average behavior of a single unit cell within a full periodic grating sample, we can
now confidently assign the various signal features visible in Fig. 2.9 to specific dynamics occurring
within our samples. The time steps before time-zero (i.e. when the probe pulse reaches the sample
before the pump pulse, before the sample is excited) serve as a measure of the zero-signal level,
since pump-on and pump-off images would ideally be identical. The fluctuations around this levelshow the noise level in the signal. The average of the before-time-zero difference images can also
be used to subtract out unchanging features from all the difference images — like those caused byleak-through of scattered pump light around the filters in front of the CCD.
Within the first 100 ps following time-zero, two sets of longitudinal wave dynamics are visible.
The initial high-frequency oscillation corresponds to the longitudinal resonance excited within thenanostructures themselves. The two echo features visible later are due to the return to the surfaceof LAW pulses which are launched into a thin film layer and reflect from the buried interface withthe substrate. On the nanosecond time scale, another two types of dynamics appear. The slow,
exponential decay represents the thermal relaxation that follows the sudden sharp expansion attime-zero. The oscillations superimposed on this overall trend correspond to the various SAW
harmonics launched in the substrate. It can be seen that later times are more dominated by one,low-frequency SAW mode while early times display more higher frequencies. This is due to the factthat damping rates tend to be faster for higher-frequency SAWs. They are confined to propagate
in a region nearer the surface where they are more strongly affected by scattering from the grating
structures.

2.3

Why use EUV?
The clearest benefit of using EUV beams to probe the dynamics of nanostructured systems

is the lower diffraction limit that the short wavelength enables. Since the diffraction limit goes as
λ/2NA for wavelength λ and numerical aperture NA, using 30nm light rather than visible wavelengths (as small as ≈ 350nm) immediately offers better than a factor of ten improvement in the

30small structure sizes that we can measure. EUV is also sensitive to the very short structure periods
that launch the shortest-wavelength SAWs, which are important for confining measurement sensitivity within ultrathin film layers at the surface as we will see in Chap. 5 [34, 35]. Particularly at thistime when the nanoelectronics industry is beginning to push the smallest features into the singledigit nanometer regime [6], sensitivity to these small sizes will be mandatory for any technologicallyuseful characterization tool. At the same time, nanofabrication capabilities have progressed in pacewith the nanoelectronics goals, meaning that single-atom layers and few-nanometer structures can
now be reliably fabricated. Since this capability is such a recent development, these are precisely
the size regimes where much of the governing physics remains to be discovered.
A related benefit of the short-wavelength probe is seen in the phase difference induced bysurface displacements. A shorter wavelength implies a larger relative change in phase for a given
displacement. Because the resulting diffraction pattern serves as an interferometric measurementof the height difference between substrate and structure surfaces, this larger phase change createsa larger change-in-diffraction signal (see Fig. 2.12). In particular, our experiments see a typicalnoise level of about 10−4 relative to the total probe intensity, implying that height changes around
10 pm are detectable, a change corresponding to λ/3000 at our probe wavelength of 30 nm.
Finally, EUV photon energies avoid extraneous contributions to signal from electronic andphotoelastic effects that must be accounted for in analysis of visible-wavelength experiments (e.g.
[36]). When the pump pulse excites the metallic nanostructures, the light is initially absorbedby the electrons within the pump-photon energy of the Fermi surface. Only after the first few
picoseconds does the electronic system fully equilibrate with the atomic lattice, creating the thermal
expansion we ultimately observe. However, the excited electrons cause a very large change in theindex of refraction for visible light by significantly increasing the electron population available toabsorption of those lower-energy photons. Thus the evolving and relaxing electron population
contributes significantly to the early-time signals in visible-probe experiments [37]. This effectis largely separable by time scale from other dynamics of interest, particularly for materials withmaximal difference between electronic and thermal diffusivity, although this will be less manageable

31

Short-wavelength
probe

Wave
crest

Long-wavelength
probe

ExcitedprofilePhase difference
between surface
and structurereflections
Phase shift induced
by expansion to
excited profile

Nanostructure

Figure 2.12: Phase advantage from EUV beams | The static diffraction pattern from an arrayof structures is determined by the reflectivities of both substrate and structure and the phasedifference between the part of the beam reflected from the substrate and that reflected from thestructures. This phase difference is highlighted by following the wave crests in a probe plane wave,
particularly in the overlapping region of the two beams shown after reflection. The phase differenceis shifted slightly when the structure is expanded, as by our pump excitation, shown by the dashedlines. The advantage of using short-wavelength EUV probes can be seen in comparing this phaseshift to the full distance between wave crests for both the blue short-wavelength beam and the
red long-wavelength beam. The much larger relative shift for the short-wavelength beam creates amuch larger change in the diffraction signal we measure.

32
for smaller systems where dynamics occur on faster time scales. Notably these statements regarding
negligible reflectivity changes for EUV do not apply to EUV beams near the absorption edges due
to inner electron shell resonances in materials. Here the strong changes in reflectivity with sharp
differences in absorption from one harmonic wavelength to the next can be exploited for elementspecific measurements of material dynamics (e.g. [38]).
In contrast, the photoelastic effect refers to changes in the index of refraction caused bymaterial density changes due to thermal expansion or the passage of acoustic waves. This effectwill be a major contributor to signals on all timescales when the photoelastic coefficient of theobserved material is large [39]. However, EUV again has the benefit of very weak coupling to theseeffects. Tobey et al. calculate that the density-induced change in reflectivity for EUV wavelengths
is approximately 0.0006 — about 20 times lower than that for visible wavelengths [40]. Signalthat could be attributed to such a reflectivity change would be about 50× smaller than the signals
caused by displacements of the surface profile, usually below the level of noise in our data.
While the following chapters will discuss some ways that visible-probe experiments are used
to address similar questions to those we study, they all require special techniques or extra signal
processing to pick out the signal that is significantly smaller in magnitude than those generated by
EUV probes. However, they may also offer inspiration for future improvements to our experimental
technique.

2.4

Conclusion
In this chapter we have seen how the short wavelength and pulse duration of EUV light from

HHG enables an incredibly sensitive probe to study any of the dynamics which deform the surface
at the 10-picometer scale or above with the temporal resolution needed follow ultrafast dynamics
at the nanoscale. This will include thermal and acoustic dynamics in structures as small as 20 nm
in lateral size and layers as thin as 1 nm.
This chapter has also illustrated how the sample designs we choose serve several importantfunctions. First, the preferential pump laser absorption in the metallic nanostructures creates the33
impulsive stress needed to excite the acoustic resonances that will probe the mechanical propertiesof the substrate, thin film layers, or the nanostructures in which they are confined (as discussed
in Chaps. 5 and 6). In particular, they have been used to launch the shortest-wavelength SAWs
observed to date (< 45 nm) [41], which are fully confined within only the first ≈ 10 nm below
the surface. Second, the nanostructures serve as nanoscale heat sources as small as 20 nm inlinewidth, and their cooling rates reveal important transitions in the nature of thermal transport
over short length scales, as will be explored in detail in Chap. 3. Finally, the nanowire periodicity
allows for efficient diffraction of the EUV probe light, enabling the interferometric measurementwith sensitivity to λ/3000 displacements and averaging over the dynamics of as many as a million
individual structures to build up the strong signal we observe.

Chapter 3

New regime in nanoscale thermal transport

Critical applications including thermoelectrics for energy harvesting [42, 43, 44], nanoparticlemediated thermal therapies in medicine [45, 46], nano-enhanced photovoltaics, controlled material
processing with localized thermal annealing [46, 47] and thermal management in integrated circuits[48, 49, 50, 51] require a comprehensive understanding of energy transport at the nanoscale. Recent
work has shown that the rate of heat dissipation from a heat source is reduced significantly belowthat predicted by Fourier’s law for diffusive heat transfer when the characteristic dimension of theheat source is smaller than the mean free path (MFP) of the dominant heat carriers (phonons
in dielectric and semiconductor materials) [52, 53, 54, 55, 56]. However, a complete fundamental
description of nanoscale thermal transport is still elusive, and current theoretical efforts are limited
by a lack of experimental validation.
Detailed study of heat-carrying phonons is very challenging for reasons summarized well by
Maasilta and Minnich [57]:
Unlike electrons, phonons have no charge and cannot be controlled by external
electric or magnetic fields. Unlike photons, phonons have no benchtop source thatcan produce them at a particular frequency, nor can an arbitrary phonon mode beeasily probed. Furthermore, whereas photons interact very weakly with each other,
the nonlinear interactions between phonons are comparatively strong and responsible for thermal resistance at room temperature and above. Roughly speaking,studying phonons is akin to studying light using only light bulbs, without spectrometers, and with strong interactions between the photons. It’s no surprise that
thermal phonons have remained so poorly understood.

35This chapter will discuss the first application of the ultrafast pump-probe technique with
EUV described in Chap. 2. In particular we focus here on the thermal decay present in the changein-diffraction signal shown in Fig. 2.9 and seek to understand how the heat flow and dissipation
rates change with heat source size and geometry. By observing phonon-mediated dynamics (heatflow in non-metallic materials), we gain new understanding about their fundamental properties.
The chapter begins with a discussion of theoretical background on heat flow and the connection between macroscopic and microscopic views of thermal transport processes, as well as an
overview of past experiments which have observed size-induced transitions away from continuumlevel diffusive heat transfer. Then I discuss how we extend such observations to the smallest
nanoscale heat sources observed to date at 20 nm in linewidth. There, through both experimentand theory we show that the size of the heat source is not the only important scale that determines nanoscale heat dissipation. We identify a new regime of thermal transport that occurs when
the separation between nanoscale heat sources is smaller than the average phonon MFP [58, 59].
Surprisingly, we find that when phonons from neighboring heat sources are close enough together
to interact, more of them dissipate heat in a diffusive regime, thus counteracting the inefficiency
caused by ballistic phonon propagation seen in the case of isolated heat sources. This collective
behavior can restore heat transfer to near the diffusive limit. Most importantly, the appearance of
this new ‘collectively-diffusive’ regime mitigates the scaling problems for thermal management in
nanoelectronics, which may not be as serious as projected [6, 53, 60].

3.1

Theoretical background
Thermal transport is generally categorized into three types: convection, radiation and con-duction. Convection describes the transfer of heat by the bulk motion of material, as for magma inthe earth’s core or the motion of weather cells in the atmosphere, as well as smaller-scale phenomena
like hot water circulating in a pot heating up on a stove.

36
Radiation is typically described as the transfer of energy by the emission and absorption ofphotons. This is the only heat transfer mechanism that does not require a material medium inwhich to occur since photons can carry energy across vacuum.
Conduction is the one of primary interest here. This involves the transfer of heat by microscopic movements within a material without motion of the material as a whole. In a gas or liquid,
this may occur by the motions of atoms or molecules where collisions between more energetic andless energetic particles will tend to transfer heat energy from hot regions to cold. In solids, heat
energy manifests as the vibrations of the atomic or molecular lattice and can be transfered by the
movements of electrons in conducting materials, or by the lattice motion itself. Mathematically thelattice vibrations can be quantized as phonons populating the normal modes of oscillation of thewhole lattice. Like atoms in a gas, these phonons will spread, scatter (and decay) to dissipate heat.
Because all these descriptions involve collisions as the means to exchange, transfer and spread theheat, this is an inherently diffusive process.
Macroscopically, collisions will establish local thermal equilibria which create a smooth temperature gradient directing the flow of heat energy from hot regions to cold. This process is
described by Fourier’s law, which can be derived in one dimension by considering the phonon flux
at some point x along a thermal gradient in the x direction. For a given phonon number density nand energy E, the heat flux qx through a surface at x is given by the particle fluxes in each direction
from a distance equal to what they can move at an average velocity vx in lifetime t

qx =

1
(nE(x0 )vx )
2




−
x−vx t
1
0(nE(x )vx )2(3.1)
x+vx tsince the random motion of phonons implies half will be moving in each direction. Applying a
1st-order Taylor expansion around vx t = 0 for each component implies




1
d(nE(x)vx )1d(nE(x)vx )qx =(nE(x)vx ) − vx t−
(nE(x)vx ) + vx t
2
dx
2
dx
dU
= −vx2 tdx
2v t dU dT=−
3 dT dx(3.2)

37
using the facts that nE(x) gives the position-dependent energy density U(x), vx2 is related to theoverall average phonon velocity by vx2 = v 2 /3, and we assume this average velocity is independent of
position. Furthermore, dU
dT is the volumetric heat capacity C, and we can approximate the thermal
conductivity Kbulk = 31 CvΛ where Λ = vt, the average phonon mean free path. Combining thiswith Eqn. 3.2 gives
qx = −Cv 2 t dT= −Kbulk ∇T,
3 dx

(3.3)

the well-known form of Fourier’s law.
However, a few problems arise with this formulation once we move to thermal transport innanoscale systems. First, the Taylor expansion assumes that vx t should be ‘small’. However, sincephonon MFPs in crystalline materials can vary by orders of magnitude from a few nanometers tohundreds of microns [61, 62], heat sources in the nanoscale regime and their associated thermal
gradients will hardly be ‘large’ in comparison. Similarly, the ∇T in Eqn. 3.3 requires a continuous,differentiable thermal gradient, which can only be established by having enough collisions among
heat carriers to maintain local thermal equilibria throughout the heat transport region. But atsize scales below one MFP, phonons travel ballistically through the region without scattering,
introducing a nonlocality to the heat transport as temperature conditions and gradients ‘far’ awayfrom the heat source have an increasing role to play [63]This transition also reveals a breakdown in the simple categorization of thermal transportthat introduced this section. The ballistic motion of phonons is analogous to radiative photontransfer [64] rather than conduction, thus introducing a smooth transition between the two initially
distinguishable heat flows. Ballistic transport sets an upper bound on the heat transfer rate.
Phonons cannot carry heat away from a heat source faster than they themselves can move. Fig. 3.1
illustrates how this results in an over-prediction of the rate of heat dissipation by Fourier diffusionas the heat source size becomes comparable to or smaller than phonon MFPs in the substrate where
heat is flowing. This occurs because the heat flux predicted by Fourier’s law (Eqn. 3.3) diverges
once dx is small and the temperature gradient becomes unphysically large [37]. As we will see

38

Figure 3.1: Diffusive vs. ballistic heat flux predictions | Because heat diffusion assumes thatscaling of the heat source corresponds directly to scaling of the thermal gradient, its prediction forheat flux diverges for small heat sources (solid blue). Ballistic heat transfer represents the limitingcase for heat flux as thermal energy is transported at the unimpeded speed of phonon motion(dash-dot green). The diffusive prediction is usually below the ballistic limit, but for small heatsources, the ballistic limit takes over. The diffusive over-prediction of heat flux (red arrow) can bequantified in a diffusive simulation by adding an effective resistivity term to artificially slow thediffusive heat flow.

39
below, this over-prediction can be accounted for by introducing an extra effective resistivity at theinterface between the heat source and substrate to artificially model the lower heat flux and thereby
quantify the deviation from purely diffusive thermal transport. However, the wide range of phonon
MFPs present in the substrate material further complicates this picture. For a given nanoscaleheat source size, phonons with MFPs shorter than the hot spot dimension remain fully diffusiveand contribute to efficient heat dissipation and a high thermal conductivity (or equivalently, a
low thermal resistivity). In contrast, phonons with long MFPs travel ballistically far from the heatsource before scattering, with an effective thermal resistivity far larger than the diffusive prediction.
Phonons with intermediate MFPs fall in between; here heat transfer is quasi-ballistic with varying
degrees of reduced contributions to the conduction of heat away from the nanoscale source.
A full account of the phonon motions in a material can be obtained using the Boltzmann
Transport Equation (BTE) – as long as the size scales in the system remain large enough toneglect phonon-wavelength related effects like interference. The time-dependent BTE follows thefull evolution of the spatial and energy distributions of phonons in a system:∂f~ r f + F~ · ∇~ p f = ∂f
+ ~v · ∇
∂t
∂t

+s

(3.4)

collision

where f is the time- and space-dependent carrier distribution, p~ is the momentum coordinate, ~r isthe position coordinate, F~ is an applied force and s represents any source. The collision term must
be calculated from the scattering matrix (S) element
hf (p~0 )|S(~p, p~0 )|f (~
p)i

(3.5)

Given three dimensions for each of position and momentum, plus time, the full BTE represents aseven-dimensional integro-differential equation. This poses a significant computational challenge,
even for numerical models on modern high-performance computers. Even more importantly, a fullaccount of the collision matrix requires very complete knowledge of the phonon modes throughout the full Brillouin zone and high-order force constants describing the anharmonic interactions
between atoms in a lattice. Thus the complete collision matrix cannot be calculated from firstprinciples nor can it be directly measured experimentally [65].

40
Several types of simplifications have been employed to make such calculations more feasible[66]. One of the most significant simplifying assumptions is that of the grey model approximation,
where it is assumed that all the phonon dynamics can be approximated by a single average phononmode with average wavelength, velocity and MFP. However, it has been shown in many casesthat this assumption results in missing many important dynamics in the systems of interest today[67, 68]. Slightly less severe is to divide phonons into groups by ballistic vs. diffusive [67, 69], or
propagating vs. non-propagating [66]. Accounting only for certain elastic or inelastic scattering
possibilities allows more potential simplifications [62]. The relaxation time approximation has
been the most broadly applied, assuming that every phonon mode is associated with one effective
relaxation time to neglect the specific mechanisms and inelastic processes which cause them todecay [65]More recent Monte Carlo methods for modeling only departures from an overall equilibrium
state have offered good improvements (≈ 104 − 106 calculation speed improvements for typicalmodeling parameters) for computational efficiency [65, 70], and one study using the measured
phonon dispersion of silicon as an input used such Monte Carlo simulations to successfully simulate
the diffusive-to-ballistic transition in thermal resistance for heat sources smaller than dominantphonon MFPs [71]. However, while these newer methods are improving the computational efficiency
and the available computational power continues to grow, the complexity of these methods and acorresponding lack of physical insight available from their results motivate the continued use anddevelopment of other models and nanoscale measurement techniques as well.

3.2

Previous work
The great technological interest in nanostructured systems and devices has both motivated agreat number of experiments examining the unique dynamics that dominate small size scales anddriven the progress in nanofabrication capabilities that enables the systematic study of such dynamics. In recent years, non-diffusive thermal transport has been the subject of an increasing amountof experimental effort due to its importance to nanotechnology development and to building a more

41
fundamental understanding of the physics of thermal transport. However, our EUV nanometrology
technique continues to be the best method for observing dynamics in nanostructured systems withcharacteristic dimensions down to 20 nm.
There are two important effects that drive the appearance of non-diffusive (typically geometrydependent) thermal transport. The spectrum of phonon MFPs involved in heat transfer can be
modified by the increased importance of boundary scattering in nanostructures. Alternatively, theparticipation of phonon modes in thermal transport can be modified relative to diffusive assumptions as the heat transport distance or thermal gradient size scale becomes small compared to the
MFPs. Many of the experiments addressing these effects have been reviewed in [72] and [73], but
I will also highlight a number of important examples here.
Thermal transport through nanostructures (like nanowires) and near surfaces or interfaces(as in superlattices, for example) will be modified from that in a bulk material because the presenceof additional scattering surfaces will affect the phonon MFP spectrum [43]. Reductions in thermal
conductivity due to this mechanism have been observed in superlattices (e.g. [74, 75]), thin films[76, 77], nanowires [78, 79, 80] and graphene ribbons [81]. Engineering of the edge/boundary shapes
and roughness can also be used to tune the degree of conductivity suppression [82]. In this way,small structures can exhibit conductivity at, and even below, the Casimir limit where thermal
conductivity is dominated by boundary scattering, and is thus dependent on the size and shape ofthe medium [83].
Chen notes that boundary scattering effects are not the only ones at play in this conductivity
modification, however [74]. When the transport distance across a layer in a superlattice, for example, is small compared to the MFP, the temperatures defined at the boundaries must be treatedwith care. Similarly, when graphene ribbons were made very short, the apparent conductivity
depended on length simply because heat flux had reached the ballistic limit [81].
Whenever the transport distance becomes small compared to the MFP, heat transfer to theend location will become less efficient as long MFPs tend to pass through without interacting and‘dropping off’ their thermal energy. This was directly observed by monitoring the x-ray diffraction

42
from a buried layer of 120-250 nm of In-doped GaAs to examine its thermal expansion [84]. Whenthe layer was thin, it received less heat from an laser excitation at the surface than when it was
thick. Similarly, transient-grating (TG) experiments, which induce spatially-periodic heating byinterfering two pump laser beams at the sample surface, measure thermal transport across the
distance from interference maximum to minimum [85]. When this distance is comparable to MFPs,
heat transfer appears increasingly non-diffusive as the energy from phonons with MFPs longer than
the grating period distributes uniformly over the peaks and nulls of the thermal grating, contributing nothing to the thermal transport observable by the experimental probe [37, 67]. Notably this isa very different case from our experiment which creates the spatially periodic heating with nanostructure heat sources and directly observes their thermal expansion. As illustrated in Fig. 3.2, theTG signal will disappear once the heat is uniformly distributed across the horizontal dimension,
since the probe beam will no longer diffract to interfere with the reference beam. In contrast, we
still observe a signal at this point, since the thermally expanded nanowires will affect the diffraction
signal as long as their temperature is higher than their initial state. This means we measure thecooling rate of the nanostructures (and the full thermal relaxation of the system) with no regard
to where the heat ends up. In contrast, TG experiments measure the rate of heat transfer acrossthe specific distance from peaks to nulls, or half the thermal grating period.
This makes it clear that TG always measures the transport from effectively isolated heatsources. Most other work to date has focused on more obviously isolated micro- and nanoscale heatsources created by small thermal penetration depth (short transport distance) [53, 55, 56], laser
absorption at a surface confined by pump spot size (small source size) [54], or in nanostructures(small source size) [52, 86, 87]. In all these cases, it is proposed that the apparent thermal conductivity is reduced because the contributions of long-MFP phonons to the total thermal conductivity
are cut off by the small heat source size, an effect which can be described by a geometry-specificphonon suppression function [88, 89].
However, more care is needed in the interpretation of apparent conductivity suppression.
The distinction between experiments which observe boundary-scattering-induced effects and those43

At excitation
Probe

Later in relaxation

Reference

Probe

Reference

Transientgrating

Diffracted
probe

Probe

Probe
Reference

Reference

Nanostructure
heat sources

Figure 3.2: Contrast between transient gratings and periodic nanostructures | Transient
grating measurements rely on the periodic thermal grating in a material to diffract a portion of
the probe beam to interfere with a reference beam, thus constructing the signal of thermal andacoustic dynamics. Once heat has traveled from the peaks to the nulls and the layer thermalizes,
the signal has fully decayed away. In contrast, the equivalent reference in the case of diffraction
from nanostructure heat sources is the fully relaxed, cold system represented by the dotted lines.
Each pump-on diffraction pattern is compared directly to this pump-off reference. As long as the
structure temperature is different from their initial reference state, their thermal expansion willaffect the probe diffraction — even after the surface of the nanostructured system has thermalized.
Thus while transient gratings are sensitive to the thermal transport distance set by the period of
the thermal grating, diffraction from nanostructure heat sources tracks the whole process of heatdissipation from the structures.

44
which observe non-diffusive transport due to ballistic phonon motion across steep thermal gradientsis not as sharp as is often assumed [90]. In particular, apparent conductivity modifications should
typically be anisotropic with in-plane conductivity strongly influenced by surface-scattering effects
in addition to the break-down of Fourier assumptions. Notably Wilson and Cahill find that effective
changes in conductivity observed in those experiments relying on varying the thermal penetration
depth by the modulation frequency of a laser pump [53, 55] can be explained completely by thechanges in sensitivity to thermal transport right at the surface where it is suppressed simply by
extra boundary scattering[90].
These details will be an important factor to consider moving forward for accurate translationof effective conductivity measurements into characterization of the fundamental phonon propertiesof materials. It also emphasizes the care that is necessary when using temperature near interfaces
where highly non-equilibrium phonon processes are important for determining the true heat transfer
rates. Newer theoretical work has even suggested that a better account of the phonon flux atboundaries (rather than temperatures) could allow an accurate application of Fourier’s law todescribe ballistic thermal transport in nanostructured systems [91].

3.3

Observing non-diffusive thermal transport
The primary goal of my work on nanoscale thermal transport was to extend previous ob-

servations deeper into the nano-regime by examining heat sources that are smaller compared tothe average phonon MFP Λgrey than were possible to explore before, as well as to compare heatsources that are geometrically confined in two dimensions (nanodots) with those confined in one(nanowires). Using the techniques discussed in Chap. 2, we study two sets of sample gratings
on two types of substrate, shown in Fig. 3.3, to pursue this goal. In particular, the change-indiffraction signal we obtain can be used to directly extract the average thermal expansion andrelaxation of each individual nanowire induced by laser heating and subsequent heat dissipation
into the substrate.

45
3.3.1

Sample design

The first set of sample gratings includes both 1D nanowire and 2D nanodot arrays of nickel
nanostructures with height h ≈ 13.5 nm and linewidths L ranging from 750 to 30 nm. The duty
cycle is fixed for all gratings in this set such that the grating period P = 4L. The second set also
combines 1D and 2D arrays of nickel structures with linewidths ranging from 1000 to 20 nm (twolinewidths, L = 30 and 100 nm, are repeated from the first set in order to compare measurements
across two substrates), but the duty cycle is not fixed for all gratings. In particular, half of thegratings have the 25% duty cycle of the first set while the other half repeats each linewidth withlarger period (P = 400 nm for L < 100 nm and P =1500 nm for L ≥ 100 nm). Each set isfabricated in a single deposition on one substrate — on sapphire and on silicon. This way the
intrinsic thermal boundary resistivity at the interface between the metallic structures and thesubstrate will be constant across all gratings in a given set: any variation in efficiency of heatdissipation as the structure size and period are varied can thus be attributed to different regimesof thermal transport. The sapphire substrate is transparent to the 800nm pump wavelength, while
the silicon substrate has such a long absorption depth (≈ 10 µm) that any small, uniform heatingof the substrate can be neglected. The nickel nanostructure height is chosen to be smaller than theabsorption depth of the pump light ensuring near-uniform initial temperature distribution inside
the structures. Laser excitation of the nanostructures thus creates an array of nanoscale hot spots(lines or dots) on the surface of a cold substrate.
The use of nano-patterned structures rather than optical absorption allows us to explore thedynamics of heat sources much smaller than the diffraction limit of visible light, which represents asignificant limitation to many of the experiments discussed in the previous section. The L = 20 nm
heat sources represent a substantial improvement over the smallest structures previously observedon sapphire at L = 65 nm [52]. The silicon substrate also further extends L/Λgrey because silicon
has a larger Λgrey than sapphire [52, 37]. In fact, substantial contributions to thermal conductivity
in silicon have been observed from phonons with MFPs well above 1 µm [53, 37]. Furthermore,

46

Figure 3.3: Thermal sample layout | We place 1D nickel nanowire and 2D nanodot arrays side byside on one substrate (shown in yellow and green squares respectively) so that differences in the heatdissipation dynamics we measure can be uniquely attributed to differences in the dimensionality ofthermal transport rather than to unknown differences in the structure fabrication. The same set oflinewidth (L)/period (P ) pairs with structure heights h are fabricated on both sapphire and silicon
substrates, allowing us to compare the trends in heat flow dynamics for two different materials.
The two example SEM images of L = 100 nm, P = 400 nm gratings show that good uniformity infabrication was achieved.

47
despite silicon’s obvious importance to current nanoelectronics development, previous work on
silicon has only addressed heat sources as small as 1 µm.

3.3.2

Quantifying deviations from diffusive transport

Unfortunately, observing the onset of non-diffusive transport as heat source size is varied is
not as simple as plotting one thermal decay on top of the next and looking for slower dissipation.
Heat dissipation rates for smaller heat sources will tend to be faster for both diffusive and nondiffusive cases. Instead, the cooling rates for smaller heat sources increase by less than predictedby macroscopic heat diffusion. To quantify these deviations from diffusive transport, we model thesystems diffusively while allowing the thermal boundary resistivity (which sets the temperature
discontinuity across the boundary between the nickel nanowires and the substrate) to vary in order
to fit the thermal decay signals we measure. The intrinsic thermal boundary resistivity rT BR is notexpected to change with the size of the interface — at least not until the interface approaches thesize scale of phonon wavelengths (≈ 1−3 nm) where one should begin to expect substantial changes
in the interface transmittance of different phonon wavelengths. By artificially allowing changes inan effective rT BR , the magnitude of variation indicates a degree of deviation from normal heatdiffusion. This method is built upon that discussed by Siemens et al. [52], but we include morecomprehensive finite element modeling of the heat transfer to reduce uncertainty in the comparison
between experimental observations and the simulation output.
We model our experimental observation of the nanowire samples using a two-dimensional
simulation unit cell with the plane strain approximation to follow the full thermal expansion andcooling dynamics of the nickel nano-gratings on the sapphire and silicon substrates [92]. We use
measurements from atomic force microscope (AFM) characterization of our samples to set theheight, linewidth and period of the structures according to the actual dimensions. The nano-gratinggeometry allows for periodic boundary conditions for the temperature T and the displacement ~u on
the sides of the unit cell. All mesh points are initialized at room temperature and zero displacement(T = 293 K, ~u = 0). The top boundaries are free to move but heat flux across them is set to zero

48
(no radiative heat dissipation). Continuity in the displacement is enforced at the interface between
the nickel structure and the substrate, effectively joining the two materials such that no slipping isallowed. The bottom boundary is fixed and the heat flux across it is also set to zero. The height ofthe substrate section is set to 10 µm, much larger than the maximum thermal penetration depths
of 1.7 µm in silicon and 0.7 µm in sapphire, to ensure no excess heat reaches the bottom boundary
during the simulation time (up to 8 ns). An example of the top region of the unit cell geometry
and mesh profile for the finite element simulations is illustrated in Fig. 3.4.

Figure 3.4: Finite-element simulation unit cell | a. The simulation mesh concentrates points
inside the nanostructure and near the interface. b. The simulations output the time-dependent
surface displacements following laser heating of the nano-grating (the surface deformation shown
here is exaggerated for the purpose of visualization).

The simulations solve the coupled differential equations for the profiles of T and ~u in thetime domain including the inertial terms in the initial thermal expansion which lead to the acoustic
waves that accompany impulsive laser excitation [93]:2~ · (c : ∇(~~ u − α∆Tp )) = ρ ∂ ~u
∇
∂t2
∂Tp
~ p=∇
~ · (Kbulk ∇T~ p) + Q
ρCp+ ρCp ~u · ∇T
∂t

(3.6)
(3.7)

49
Table 3.1: Material parameters used in multiphysics simulations.

Material properties

NickelSilicon*‡

SapphireCp, Specific heat (at 300 K) [J/(kg K)]

456.8

710.0

657.5*

Kbulk, Bulk thermal conductivity [W/(m K)]

90.9‡

149.0°

41.1*

Debye temperature (K)

450§

645§

1047§

Poisson’s ratio

0.31†

0.27†

0.25†

Young’s modulus [1011 Pa]

2.00◊

1.31☼

3.45χ

α, Linear coefficient of thermal expansion [10-6/K]

12.77*

3.00

5.21*

ρ, Density [kg/m3]

8910*

2330☺

4000*

∗ Siemens et al. [52]; † Nardi et al. [92]; ‡ Desai [96]; ◦ Dean [97]; § Kittel [98];  Zacharias [99];☼ Wortman and Evans [100]; χ Wachtman et al. [101] ; : Okada and Tokumaru [102]; , Weber[103]

where c is the elastic tensor, ~u is the displacement, ρ is the density of the material, α is thelinear coefficient of thermal expansion, Tp is the phonon temperature, Cp is the specific heat of thematerial, Kbulk is the bulk thermal conductivity and Q is the heat source term accounting for thelaser heating of the nano-grating, calculated according to the two-temperature model as describedby Banfi et al. [94].
Q = −Γe−p · (Tp − Te )~ · [−ke (Te )∇T~ e ] + Ce (Te ) ∂Te = P (t, y) − Γe−p (Te − Tp )
∇
∂t

(3.8)
(3.9)

where Ce , Te and ke are the volumetric specific heat, temperature and thermal conductivity, respectively, of the electrons in the sample materials and Γ represents the electron-phonon couplingof the material. (Values for nickel come from [95].) P represents the profile in time and depth
below the surface of the power absorbed from the laser pump pulse, given by
P (t, y) = [pulse peak power] × [temporal pulse shape] × [laser penetration](3.10)All the material properties used in the finite element multiphysics modeling are listed in Table
3.1. Given the maximum substrate temperature change induced in our experiment of ≤ 40 K, the50specific heat and bulk thermal conductivity can change by ≤ 10%, and we confirm that any changeof this magnitude does not cause appreciable differences in the simulated time-dependent diffraction
signal we use to compare with experimental data. Moreover, because we observe no pump-fluence
dependence in the normalized measured signals [104], precise choices for simulation temperatures
will not affect the resulting comparison with experimental data.
The more complete account of the physical dynamics in our sample allows for a more precise
fit to our data and lowers our uncertainty, particular for small-linewidth samples. However, theadded complexity significantly increases the computation time needed to model the full time scaleof the thermal decay, especially for large linewidths, making it impractical for the large set ofsimulations with different ref f ’s required for fitting our data. Instead, we can use the comparison
between complete inertial and simpler quasi-static simulations (where the right-hand side of Eqn.
3.6 is set to zero) to understand how they relate to one another (see Fig 3.5). Of particularimportance is to understand how our data (which will include all the dynamics found only in thefull inertial simulations) should be normalized to correctly choose the quasi-static simulation thatshould be considered the best fit. This relation only needs to be determined at one value of ref f
for a given linewidth. This is particularly an issue for large-linewidth samples where the initial
inertial expansion occurs over a relatively long time scale and lateral dynamics across the widelines strongly modify the vertical dynamics, such that there is a greater mismatch between thepeak of the quasi-static simulation and that of the full inertial simulation.
The effective thermal boundary resistivity ref f , which sets the temperature discontinuity,∆T , across the boundary between the nickel nanostructures and the substrate, is introduced at themesh points that join the two materials with the equation:~ ) = − ∆Tn̂ · (Kbulk ∇T
ref fwhere n̂ is the unit vector normal to the substrate surface.

(3.11)

51

Sapphire, L = 60 nm

Silicon, L = 750 nm

noitcarffid ni egnahC

Inertial

Inertial

Quasi-static

Quasi-static

0

100

200

300

Delay time (ps)

400

500

0

100

200

300

400

500

600

Delay time (ps)Figure 3.5: Comparing inertial and quasi-static simulations | Full inertial simulations of
the thermo-mechanical dynamics in our samples allow a more accurate comparison to the data wemeasure, but they are also much more computationally expensive than simpler quasi-static simulations (which ignore the second-derivative inertial terms in Eqn. 3.6). Thus we use a comparison ofthese two simulations for each linewidth and substrate at the same peak temperature rise to inform
how the experimental data should be normalized in order to correspond to the correct quasi-static
simulation. The left panel from a small-linewidth sapphire sample provides an example of the simplest type of comparison: the normalization is such that the quasi-static curve follows the middleof the acoustic oscillations from the beginning of the scan. However, the early dynamics are more
complicated for larger linewidths, as shown on the right for one of the most difficult cases. Here thelengthy onset of the first surface acoustic oscillation and the greater impact of dynamics occurring
laterally across the wide structures mean that the quasi-static simulation must be normalized to amuch higher value than might be naively assumed.

52
The simulations provide a time-dependent surface deformation profile of one unit cell, as
illustrated in Fig. 3.4b. We tile this into a large array to represent our whole grating sample.
Using Fresnel optical propagation, we then calculate the diffraction patterns associated withthe static, unpumped surface profile and with each time step of the deformed profile. First weconstruct the E-field at the sample surface Eobject by multiplying three terms: the periodic amplitude grating formed by the different reflectivities of the substrate and grating structures; the phasegrating given by exp(2ik ∗ h(x)) where k = 2π/λ and h(x) gives the surface height at every point x(the factor of two in the exponential function is necessary for the round trip applicable to reflection
from the surface rather than transmission); and the incident EUV probe estimated by a Gaussian
beam of waist radius w0 = 50 µm,
Eprobe = w0 /w ∗ exp(−(x2 )/w2 ) ∗ exp(−i(k ∗ zf oc − arctan(2zf oc /(kw02 ))) ∗ exp(−ikx2 /(2R)) (3.12)
whereqw = w02 (1 + (2zf oc /(kw02 ))2 )(3.13)

R = zf oc (1 + (kw02 /(2zf oc ))2 )(3.14)

where zf oc gives the distance from beam focus to sample (typically assumed to be zero, which is agood assumption for the loosely-focused beam we employ) [105]. The output field incident on ourcamera at distance zcam from the sample is calculated using a Fourier transform by
Ecam =


exp(ikzcam )
π h
π i
exp iX 2
F Eobject exp ix2.
iλzcamλz
λz(3.15)

The camera-plane coordinates X are determined by the object-plane coordinates x. Both havethe same number N of evenly-spaced samples (spaced by dx in the object plane), and X extends
from -1/(2dx) · λzcam to +[1/(2dx) − 1/(N dx)] · λzcam (following MATLAB’s convention that thezero-order peak will appear at the N/2 + 1 point for even N ). The diffraction patterns are thengiven by |Ecam |2 . The difference in diffraction at each time step is integrated in the same way
as the experimental signal is built (described in section 2.2.2). This procedure builds a simulated
signal which can be directly compared to our experimental observations.

53
For each sample geometry, we further calculate the diffraction signals from a comprehensive
set of deformation profiles related to different values of ref f . The effective resistivity ref f thatthen provides the best fit to the experimental data, like the two examples shown in Fig. 3.6,
represents the sum of two terms: first, the constant intrinsic thermal boundary resistivity, rT BR ,that originates from the material difference between nickel and substrate [106]; and second, acorrection term, rCorr , due to non-diffusive transport in the substrate close to the heat source
when either L or P is smaller than phonon MFPs. We also include fitting parameters to set thetime offset (since the time-zero of the simulation could fall between the particular time steps wemeasure in the experiment) and normalization of our change-in-diffraction signals.
Notably this method assigns the effect of non-diffusive transport in the substrate near theheat source to a term which is specifically located at the interface between them. While this is asimilar effective correction for non-diffusive transport near isolated nanoscale heat sources as that
employed in previous works [37, 53, 54, 55, 56], by assigning the non-diffusive contribution to thethermal boundary resistivity rather than to changes in the thermal conductivity of the substrate,
we maintain a simple modeling geometry while automatically limiting the effective changes to anarea near the interface without having to assume a particular region of the substrate in whichan effective conductivity change should apply. However, ongoing work is further exploring theconnection between the two methods, as discussed below in Section 3.5.

3.3.3

Experimental results

The effective resistivity results of this analysis are plotted in Fig. 3.7. For large linewidths
on both sapphire and silicon substrates, the effective resistivity converges toward a constant value
which we associate with the intrinsic thermal boundary resistivity. As the linewidth approaches
the dominant phonon MFPs in the substrate, the effective resistivity rises as thermal transport
becomes quasi-ballistic and the contribution to heat dissipation of long-MFP phonon modes issuppressed [52, 54, 88]. This behavior was successfully described in past work using a simple grey
model for sapphire and fused silica, which assumes a single phonon MFP to loosely describe a

54Normalized change in diffraction signal

1.0
0.8

Sapphire data L = 60 nm

0.6
0.4

Best-fit reff0.2

Diffusive rTBR

0
0

400

800

1200

1.0
0.8

Silicon data L = 60 nm

0.6
0.4

Best-fit reff0.2

Diffusive rTBR

0
0

400

800

1200

Delay time (ps)Figure 3.6: Extracting thermal boundary resistivity from dynamic EUV diffraction |The dynamic diffraction signals from 60nm-wide nickel lines on sapphire (top) and silicon (bottom)
display a sudden rise due to impulsive thermal expansion following laser heating, a long decay due
to thermal relaxation and oscillations due to surface acoustic waves. Dashed black lines plot the
diffusive prediction for the thermal decay. Clearly it significantly underestimates the thermal decay
time for both cases shown. Green lines plot the decay obtained from the best-fit effective thermalboundary resistivity.

55
weighted average of the MFPs from all the phonon modes contributing to thermal transport in agiven material. According to this model, a ballistic correction term proportional to Λgrey /(L/2)
can be added to the intrinsic thermal boundary resistivity [52, 106]; this prediction is plotted in
red in Fig. 3.7.
However, as the linewidth (and period) shrinks further, Fig. 3.7 shows that the effective
resistivity starts to decrease rather than continuing to increase [58, 59]. The constant grating duty
cycle for our series of samples means that the smallest-linewidth nanowires are also those with thesmallest separation between neighboring heat sources. Thus, for small linewidths the separation
becomes comparable to dominant phonon MFPs, and we must develop a new theory to take this into
account in order to describe this previously unobserved and unpredicted phenomenon. Notably forsilicon, the peak in ref f is shifted toward longer linewidths/periods compared to sapphire because
the phonon MFP distribution in silicon is also shifted toward longer MFPs; i.e., silicon has a longer
average MFP than sapphire [37, 52].
Figure 3.8 illustrates the differences between the three regimes of heat transport from nanoscaleheat sources represented by the colored background in Fig. 3.7 – purely diffusive, quasi-ballistic
and collectively-diffusive. Quasi-ballistic transport (Fig. 3.8b) dominates when the size of isolated
nanoscale heat sources is smaller than dominant phonon MFPs. In the new collectively-diffusive
regime we uncovered (Fig. 3.8c), the separation between heat sources is small enough that longMFP phonons, whose contribution to heat dissipation would normally be limited by the small sizeof nano-heat sources, can once again play a significant role and restore heat transfer efficiency tonear the diffusive limit. Although these phonons travel ballistically away from each individual heatsource, they can scatter with phonons originating from a neighboring heat source, thus creating aneffectively larger heat source size. In the limiting case, the spacing between heat sources vanishes
and this regime approaches heat dissipation from a uniformly heated layer.

56Collectively-diffusive Quasi-ballistic Diffusive
1000

Period (nm)
2000

2

6
5

-9

Effective boundary resistivity (x 10 m K/W)

04Interacting multi-mode

3000
SapphireInteracting single-MFP
Isolated heat source

3
2

rTBR

5
Silicon

4

Interacting multi-MFP

3
2

Isolated heat source
Interacting single-MFP

1
0

200

400
Linewidth (nm)

rTBR

600

800

Figure 3.7: Effective thermal boundary resistivity in sapphire and silicon | The extracted
effective resistivities for each linewidth L on both sapphire and silicon substrates increase withdecreasing linewidth until the periods (equal to 4L) are comparable to the average phonon MFP. For
smaller periods (spacing), the effective resistivity decreases and approaches the diffusive limit (black
dashed line), which was determined by fitting the large-linewidth data to extract the asymptotic
value. The error bars represent the standard deviation among multiple data sets for the same
linewidth samples. Dotted red lines: predictions for isolated heat sources based on the grey model.
Dash-dot blue lines: grey model including the onset of the collectively-diffusive regime. Solid purple
lines: more complete model that includes contributions from multiple phonon MFPs.

57

L

a

Λi

Diffusive: Λi << L, P

b

P

Λi

Quasi-ballistic: L  Λi  PcΛiCollectively-diffusive: L, P  Λi

Figure 3.8: Illustration of three thermal transport regimes | Nanoscale heat transport isdetermined by the interplay between three length scales: the size of the heat sources L, the spacingof the heat sources P, and the MFPs Λi of heat-carrying phonons. Materials support a broad
distribution of MFPs, represented here by short (black) and long (white) MFP phonons. a. When
all MFPs are smaller than L, heat dissipation is fully diffusive. b. As L shrinks, long-MFP
phonons travel ballistically until they are relatively far from the heat source, decreasing the rate ofheat dissipation relative to diffusive predictions. Short-MFP phonons remain diffusive. c. Whenboth L and P shrink, long-MFP phonons originating from neighboring heat sources interact as theywould if they originated from a single, large heat source, enabling more efficient diffusive-like heattransfer.

58

3.4

Understanding a new thermal transport regime
To better understand and quantify the different regimes of thermal transport illustrated inFig. 3.8, we consider three models: 1) the model described in our previous work that assumes
isolated heat sources [52]; 2) an analytical model we develop here to account for interactions ofphonons originating from neighboring heat sources using a grey, single-phonon-MFP approximation;and 3) a more advanced interacting model that includes a distribution of phonon MFPs.

3.4.1

Quasi-ballistic model for isolated heat sources

First, though we have already observed that isolated heat source theories will not be sufficient
to explain or predict the ref f trends we find in our experiment, it is important to understand theorigin and implications of the model proposed by Siemens et al. [52] in order to implement themodifications required to account for closely-spaced heat sources. Moreover, a new and morecomprehensive model must still reproduce the trends of the isolated model for the appropriate
limiting cases.
Chen [64] and Siemens et al. [52] focused on the ratio of ballistic to diffusive heat flux nearnanoscale heat sources to develop a sense for when and how ballistic contributions would becomea significant factor to consider. This yielded a dependence proportional to Λgrey /(L/2). However,the connection between this ratio and the effective resistivity correction was not clear. Instead we
can gain more insight into this connection by examining the full expressions for the approximate
constriction resistance derived by Wexler [107] and Prasher [108].
For the 1D nanowire samples of focus in this work, the heat flux in the periodic sample canbe fully represented by the unit cell with the flux between concentric half-cylinders: one of radius
L/2, one of radius P/2, both centered at the center of the nanowire. Ballistic transport between thecylinders is phonon radiation with heat flux qballistic governed by the phonon blackbody equation
qballistic =

Cv
∆T
4

(3.16)

where C is the volumetric heat capacity, v is the phonon group velocity and ∆T represents the59temperature difference between the two cylinders [108]. The resistivity associated with this heat
flux component is given by rB = ∆T /qballistic = 4/Cv, containing no dependence on MFP, L or P.
Given that the thermal conductivity Kbulk can be approximated by 31 CvΛgrey , this can be re-writtenasrB = 4Λgrey /3Kbulk .

(3.17)

Diffusive heat flux between the cylinders from the Fourier equation is given by
qdif f usive =Kbulk ∆Tln(P/L)L/2(3.18)

so thatrD =ln(P/L)L/2.
Kbulk(3.19)

Here the Λgrey /(L/2) dependence for the ratio of rB /rD is clear, but it must also be emphasizedthat rB is not itself equivalent to the ballistic correction rBC term employed by Siemens et al. [52].
To relate rB and rD to the varying rCorr we observe, we must consider the manner in which we
represent these two constriction terms by a single diffusive term. Specifically we set the sum equal
to one diffusive term with a modified thermal conductivity, Knano .
ln(P/L)L/2
ln(P/L)L/2+ 4Λgrey /3Kbulk ≡
.
Kbulk
Knano

(3.20)

Then the resistivity correction will be related to the effective conductivity change represented here.
1
Knano

−

1
Kbulk

=

4Λgrey.
3Kbulk ln(P/L)L/2(3.21)

rCorr,iso is then related to the difference in heat flux induced by this modified thermal conductivity
at radius d which encloses the finite region in which non-diffusive effects are important.
rCorr,iso =

d
Knano

−

d
Kbulk

=

4Λgrey d
3Kbulk (L/2) ln(P/L)(3.22)The size of the region where ballistic effects are important should be determined only by the phonon
MFP distribution, so while d can depend on Λgrey of the substrate, it should have no dependence

60
on L or P. Thus the proportionality to Λ/(L/2) is maintained and the finite region where theconductivity change should apply is built in.
However, some problems do remain with the details of this approximate result. Most importantly the limiting case where P = L implies a divergent rCorr though it should imply a return
to normal diffusion. Also, as P → ∞, rCorr in this expression will always go to zero even thoughsmall L with large P should imply a large rCorr . Because of these problems with the precise formof the proportionality constant, as we develop a new model for incorporating the effects of close
spacing on the heat transfer, we will seek to reproduce the Λgrey /(L/2) dependence for isolated
heat sources and leave the rest to a scaling constant that we fit to the data.

3.4.2

Interacting heat sources, grey phonon approximation

The quasi-ballistic model for isolated heat sources clearly fails to capture the experimental
observation that heat source spacing influences the effective rCorr , and a new model for rCorr isrequired to account for the transition to the new collectively-diffusive regime. As illustrated inFig. 3.8, the transition from the diffusive to quasi-ballistic regime can be qualitatively attributed
to a loss of phonon collisions near enough to the small heat sources to be adequately describedby normal heat diffusion. This effectively suppresses the efficiency of heat transfer via long-MFP
modes relative to the diffusive prediction. But if another heat source is placed nearby such that thelong MFPs allow more overlap in the region between the heat sources, it is reasonable to observethat the scattering locations are once again ‘close enough’ to a heat source to appear more diffusivelike. Indeed the long-MFP phonons do not ‘know’ that they are propagating away from two smallheat sources rather than one large source. Note that this qualitative description does not suggest
a change in scattering rates (which would imply changes in the MFP distribution) or that phonons
originating from the neighboring heat sources must specifically interact with each other. Rather,
the scattering events for long MFPs transition from diffusive-like to non-diffusive-like and back todiffusive-like depending on the distance from the heat sources.

61

Figure 3.9: Phonon filter model schematic | Linewidth and period define a suppression filter
for phonon MFP spectra. The observed increase in effective thermal boundary resistivity for small
linewidths L is due to the suppression of the contribution to thermal conductivity of phonon modes
with MFP larger than L. Decreasing the period P can reactivate modes with MFP larger than P,
decreasing the effective resistivity. In the limiting case of a uniformly heated layer, P approaches L
and all phonon modes participate in thermal transport. We use as an example the smoothed differential conductivity distribution for silicon (top graphs, green line), calculated from first-principles
density functional theory [59].

62
Given this qualitative picture of the how closely-spaced heat sources can build the conditions
for collective diffusion, we postulate that the conductivity contributions of phonon modes aresuppressed according to the model for isolated heat sources as heat source size becomes smaller
than the phonon MFP, but then smaller spacing reintroduces those contributions in the sameway. This can be represented by a notch filter in the MFP spectrum to describe the effects ofgrating linewidth and separation, shown schematically in Fig. 3.9. The notch filter suppresses
the contribution of phonon modes with MFPs that fall between the linewidth L and period P of
the nano-gratings. Thus, if the grating period (separation) remains large while the linewidth isdecreased, one would expect the effective boundary resistivity to continue to rise, as shown in the
red dotted lines of Fig. 3.7. This is because the contributions of all phonon modes with MFPs
longer than the linewidth L are suppressed in the quasi-ballistic regime of isolated heat sources. On
the other hand, if the grating period shrinks, long-MFP phonon modes start to contribute again
since phonons originating from neighboring heat sources interact with each other as they would ina bulk system. So the effective boundary resistivity should recover toward the bulk value, as seen
experimentally in Fig. 3.7.
To build an analytical expression for rCorr based on this idea, we use the concept of a phonon
conductivity suppression function, S(L, P, Λ), similar to those described by others [67, 88, 109]. This
suppression function is applied to a bulk differential conductivity spectrum versus phonon MFP of
the substrate, k(Λi ), to calculate an effective nanoscale conductivity Knano :Knano =

Xk(Λi ) · S(L, P, Λ) .

(3.23)i

We then relate rCorr to the change in conductivity represented by this suppression.

1
1
rCorr = A−
Knano Kbulk(3.24)

where A collects geometrical constants discussed in section 3.4.1 and Kbulk is the bulk conductivityof the substrate, simply given byPi k(Λi ).

From physical arguments it is clear that for a given phonon MFP Λi , S must approach unity
in the diffusive regime when both L and P are large and at the limit of uniform heating when L = P.

63Furthermore, for the limit of small, isolated heat sources when L → 0 but P is large, S → L/(2Λ)
to reproduce the behavior of the quasi-ballistic model discussed above. Finally, the effects of L
and P should be uncoupled and the same but opposite to each other so that L suppresses phononmode contributions in the same way as P reactivates them. In addition, we assume that the filter
function should be smooth. The relevant non-dimensional variables are L/Λ and P/Λ. All of thisbehavior is captured by a special case of the generic family of logistic functions:
SL (L/Λ) = tanh(L/2Λ)(3.25)SP (L/Λ) = 1 − tanh(P/2Λ)(3.26)

Stotal (L, P, Λ) = SL + SP

(3.27)

These functions are plotted in Fig. 3.10 where we can see the similarity between the shape of Stotal
and standard notch filters.
While more rigorous methods of deriving suppression functions for various experimental geometries are currently being explored [67, 89, 110, 111], none have yet sought to account for closelyspaced heat sources or an accompanying reintroduction of phonon modes. Equation 3.27 represents
the first attempt to include the contribution of heat source spacing and offers a model which issimple enough for fast integration into existing models of heat transfer in nanoscale devices, forexample, but complex enough to capture the previously unobserved behavior and make successful
predictions.
To test this new model for rCorr , we first assume the simple single-MFP (grey) model (where
the MFP distribution is a delta function). The resulting predictions are shown in the blue curves
in Fig. 3.7. Specifically, rCorr in this case is given by:rCorr,grey (L, P ) =

A
Kbulk




1
−1 .
S(L, P, Λgrey )(3.28)

Fitting this interacting model to the ref f data for sapphire, we extract values for rT BR and Λgrey
which are consistent with previous results [52]: Λgrey = 131 ± 11 nm, rT BR = 2.58 ± 0.19 × 10−9
m2 K/W. This good agreement with the previous larger-linewidth data and the accurate fit for the

64

Figure 3.10: Phonon suppression filter function | For a phonon mode with a given mean
free path, SL describes the suppression of this mode’s contribution to thermal conductivity as the
linewidth of a heat source decreases. For large L/Λ it approaches unity as required for the diffusive
limit. As L → 0 the linear relationship to L/2Λ is clear. SP undoes this suppression, and Stotal
represents the total suppression when both small heat source size and interaction between heatsources are taken into account. Note also that each configuration for L and P can be related to
one most-suppressed MFP (minimum of Stotal ). This information is used to partition the MFP
spectrum into bins by the distribution of sensitivity to suppression of the different spectral regions
given by our set of linewidth-period pairs, as discussed further in Chap. 4.

65full range of our data validate our interacting rCorr model as an improved method to account fornanoscale size effects in heat transport – for both quasi-ballistic and collectively-diffusive regimes.
Interestingly, this single-MFP model provides a reasonable approximation for the entire range ofheat transport in sapphire.
For silicon, the interacting rCorr follows the general shape of the data and yields Λgrey = 306±
17 nm, which is consistent with previously reported values [37, 77]. However, the interacting greymodel approach, although more successful than the isolated model, fails to capture the additional
structured tail in effective resistivity that appears for very small linewidths and periods, below L= 100 nm. The failure of this approach is not surprising, since the single-MFP model is known to
be a poor approximation for silicon with its broad distribution of phonon MFPs [67, 88, 112].

3.4.3

Interacting heat sources, full phonon distribution

To successfully fit the silicon data, we must account for the full phonon distribution. To makethis extension, we simply apply the same suppression function to each individual MFP-dependentP
contribution to thermal conductivity to calculate an effective Knano = i k(Λi ) · S(L, P, Λi ).
The resistivity correction for heat transport from interacting nanoscale heat sources can again
be related to the change in conductivity imposed by the suppression function:
rCorr,int ∝1
1−Knano Kbulk(3.29)

which gives:rCorr,int (L, P ) = A

1
1
−P
k(Λ
)S(L,
P,
Λ
)
k(Λ
i
i
i)
i
i

P


.

(3.30)

This more complete model can be used in combination with calculated differential conductivity
distributions (like the one shown in Fig. 3.9) to test how well they can account for our observationsof ref f , including in particular the structured tail we observe for the silicon substrate. It can also be
inverted to allow the extraction of differential conductivity information from ref f data, as will be
discussed in Chap. 4. This more complete fit to the data results in the purple curves shown in Fig.
3.7. While the grey model assuming a single MFP yielded a reasonably good approximation for66
sapphire only, the ref f data from both sapphire and silicon are well fit by including the extension
to multiple MFPs.
While no models are fully ‘correct’, some are useful, and we can test the utility of thismodel by examining its implications. In particular, it makes the rather surprising prediction thatnanoscale heat sources placed close together will cool more quickly than those that are isolatedor spaced far apart from each other (see Fig. 3.11). This is due to the wide suppression filter
associated with large spacing compared to the narrow filter associated with small spacing.
L

0.6

L

0.4

P

ion
ss
pre erp
Su filt
Participating modes

0.2
0

0

100

200

300

MFP (nm)

400

500

600

P

1.0
0.8

9

0.8

Differential k (10 Wm-2K-1)

P

1.0

9

Differential k (10 Wm-2K-1)

L

0.6

P

L

Suppression
filter0.4

Participating
modes

0.2
0

0

100

200

300

MFP (nm)

400

500

600

Figure 3.11: Suppression filter model predictions | The narrow conductivity suppression filter
associated with small heat sources spaced closely together (left) predicts that heat sources in thisarrangement will dissipate heat more quickly than the same size heat sources with wide spacing.
Suppressing a smaller portion of the MFP spectrum maintains higher effective conductivity in thesubstrate and allows heat transfer closer to the diffusive limit.

This prediction is in opposition to what would be expected from an argument relying solely
on shifting measurement sensitivity as discussed by Cahill and Wilson [90]. Fig. 3.12 represents
this sensitivity argument schematically. It argues that when heat transfer is predominantly 1D incharacter with heat traveling primarily perpendicular to the interface between our nanostructure
heat sources and the substrate, it will appear diffusive since the heat source size should not affectthermal transport in this direction. This situation would apply both to large heat sources and to67
small heat sources closely spaced (when the lateral thermal gradient is small relative to the caseof an isolated heat source). Deviations from diffusive, bulk heat transport would arise when heatflow becomes more 2D, as from small, isolated heat sources, because the surface plays an importantrole in modifying the heat current nearby by changing the MFP spectrum, and because the lengthscale of the thermal gradient becomes small in this direction. However, for the same small heatsource with neighbors, the cross-plane heat current Jz should be identical while the in-plane heatcurrent Jx would be smaller in the case where the heat sources are closely spaced. Therefore theoverall heat current, and thus rate of heat dissipation, would be smaller for small, closely-spaced
heat sources than for isolated or widely-spaced heat sources of the same size.
Both this sensitivity argument and our suppression function model can qualitatively explain
the trends we observe in ref f — transitioning from fully diffusive to higher resistivity as heat flow
becomes more 2D in nature, and then back toward diffusive when the heat sources are close enoughto diminish the lateral temperature gradient. However, an observation which directly compares
the heat dissipation from the same linewidth heat sources at different periods will distinguishbetween the two model predictions. Using the set of periodic nickel nanowires on silicon withvaried duty cycle, we directly observe that closer spacing does result in faster cooling. For example
for both L = 20 and 30 nm (shown in Fig. 3.13) the blue curves with small period decay more
quickly than the red curves with large period. Without any appeal to model comparisons, thesedata demonstrate unequivocally that the 1D-2D transport sensitivity argument is not sufficient
to explain our observations. Furthermore, our new model successfully predicted this unexpected
outcome, lending support for the utility of the phonon-suppression filter model.

3.5

Alternate understanding through effective conductivity
In principle, our new formulation for rCorr in Eqn. 3.30 explicitly links our effective resistivity

representation of non-diffusive effects in nanoscale thermal transport with the effective conductivity
representation used by others. To test this link, we can fit our change-in-diffraction signals tosimulations in which the substrate thermal conductivity is varied rather than the thermal boundary

68

1.0

L = 20 nm on silicon

0.8

Acoustic oscillations

0.6

P = 400 nm

0.4
0.2

P = 80 nm

0
0

200

400

600

Delay time (ps)800

Change in diffraction

Change in diffraction

Figure 3.12: Testing a measurement-sensitivity explanation of observed dynamics | An
explanation based on shifting measurement sensitivity from sample to sample makes predictions
opposite to that of the phonon suppression filter model. Considering first isolated heat sources,
when the heat source is large (a.), most of the heat current J is in the z direction, so while Jx
will be governed by a modified MFP spectrum due to its proximity to the surface and lowered
compared to heat diffusion in the bulk, it does not contribute significantly to the overall heat flux
and will therefore not affect the measurement enough to be observed. When the heat source issmall (b.), Jx is a more significant fraction of the total heat current, so its lower value will makethe measurement appear to deviate from diffusive predictions as heat flow transitions to this more
2D nature. When the heat sources are no longer isolated, both large and small heat sources would
display predominantly 1D heat flow as Jx is always a small perturbation due to the small lateral
temperature gradients, particularly in (d.). Note however that Jz should be the same for both (b.)
and (d.) while Jx would be smaller for case (d.), therefore yielding a lower total heat current. But
this is contrary to the experimental results we observe.

1.0

L = 30 nm on silicon

0.8

Acoustic oscillations

0.6
0.4

P = 400 nm

0.2

P = 120 nm

0
0

200

400

600

Delay time (ps)800

Figure 3.13: Confirming predictions for closely-spaced nanoscale heat sources | Our new
suppression function model and the arguments for measurement sensitivity to 1D or 2D thermaltransport make opposite predictions for the heat dissipation rates of nanoscale heat sources which
are closely spaced compared to those far apart from each other. Here we observe that smalllinewidth heat sources on silicon cool more quickly at small periods (blue) than the same size withlarge period (red). This behavior is only predicted by our new model for the phonon suppression
filter.

69
resistivity. However, we quickly find that when we allow the conductivity to vary everywhere in thesubstrate as other works have done, it is impossible to find a good fit to the full time ranges of ourdata. One example is shown in Fig. 3.14. This observation is intuitively satisfying, since effective
conductivity accounts for the deviations from diffusive heat transfer that should only occur within
the first few MFPs from the heat source.
Instead we must explicitly include d, the length scale associated with the conversion fromconductivity to resistivity that sets the region of the substrate over which the thermal conductivity
is effectively modified. Thus far we have assumed this single parameter should apply for all MFPs,
although one possible future improvement would be to allow it to vary with MFP. Fig. 3.15 shows
how it is implemented in the simulation by adding a layer of thickness d at the top of the substrate
where the thermal conductivity is set to Knano ; below this layer the conductivity is set to Kbulk .
One can also observe that this is also an approximate way to implement anisotropic changes in
K since the full layer of modified Knano will affect in-plane thermal transport everywhere andcross-plane only before the heat crosses the lower boundary.
This procedure adds a fit parameter to the optimization algorithm since rT BR must still
also be fit using our data. However, it does introduce the opportunity for some comparisons thatreassure our physical intuition. The first thing to note is that the need to use d to achieve a good
fit to the thermal decay data is most apparent for large-linewidth samples. This can be understoodby considering the time scale of thermal decay t and the corresponding thermal penetration depth,qbulk
approximated by Fick’s Second Law as 2 KρC
t. As shown in Fig. 3.16, the longer measurement
time scale for large linewidths implies heat can travel a larger maximum distance. If this distance islarger than d, a transition in the thermal transport behavior from quasi-ballistic to diffusive at large
distances from the heat source will be required to fit the thermal decay data, requiring a transition
from effective to bulk conductivity. In contrast, the short time scales of small-linewidth decay
measurements imply the heat may only penetrate some fraction of the layer in which thermal
conductivity is effectively modified. Thus the thermal penetration depth corresponding to the70
L = 140nm on sapphire
Signalnoitcarffid ni egnahC

Resistivity best-fit

Conductivity best-fit
with limited depth
Conductivity best-fit
to early signal

Conductivity best-fit
to late signal
0

500

1000

1500

2000

2500

3000

Delay time (ps)Figure 3.14: Fitting data with effective conductivity | When we try to model our data by
varying effective conductivity rather than effective resistivity, we quickly find that reasonable fits
to the whole time range of our signals require that the effective conductivity change only occur
within a limited depth below the nanostructures. The original effective resistivity fit (solid green)
to the data (grey) can be fit by a low effective conductivity at early times (dashed red) or a higheffective conductivity at late times (dashed blue). The effective conductivity with limited depth(orange) can fit the whole range of time delays as well as the original resistivity fit.

Figure 3.15: Effective conductivity simulation cell | Implementing an effective conductivity ina limited region within the substrate below our nanostructure heat sources requires an extra layer
in the simulation cell. Within this layer of thickness d, the thermal conductivity is equal to Knano .
Outside, the thermal conductivity is set to the bulk conductivity value Kbulk taken from literature.
The data fit is also strongly affected by a third fit parameter, rT BR . The coupling of these threeparameters makes it difficult to find unique solutions to fit the thermal decays we observe.

71largest linewidth which displays no significant dependence on d when fitting the data provides alower bound for this fit parameter.
Moreover we find that an initial fitting series which allows d to vary as a function of linewidthverifies that a single d can be chosen to allow good fits for all linewidths. And this d is equivalent
to one or two times the Λgrey we find through fitting our data with the grey suppression function
model. This result is intuitively satisfying since it suggests the region of effective conductivity
modification should be confined within one or two average MFPs of propagation away from theheat source. However, it is difficult to decouple the effects of d, Knano and rT BR to find a uniquesolution. Further improvement of the fitting methods will be necessary to enable a full analysis ineffective conductivity.
If this fitting procedure can obviate the need for the proportionality constant A present in
the resistivity formulation (Eqn. 3.30), this method will provide a simpler method going forwardand extending this work to 2D nanodot heat sources, as well as to samples without fixed duty
cycle where we are not yet sure if and how the proportionality constant should vary. It can alsooffer a more direct comparison to other experiments and theoretical work in this field where theuse of effective thermal conductivity is the norm. In particular it will allow us to directly examine
the validity of the first theoretical work that begins to explore the realm of periodic rather thanisolated heat sources [113].

3.6

Outlook for the study of nanoscale thermal transport

3.6.1

Extension to 2D heat sources

While this chapter has focused thus far on 1D nanowire heat sources, we expect the quasiballistic phonon transport effects should be even more pronounced for 2D nanodot heat sources
where the spatial confinement limits transport in two directions rather than only one. To enable asdirect a comparison of these dynamics as possible, both nanowire and nanodot arrays are fabricatedon the same substrate with the same sets of linewidths and periods as shown in Fig. 3.3. The

72

Figure 3.16: Thermal penetration depth vs. heat source linewidth | The shorter time scaleof thermal decay from small-linewidth structures also implies that heat will not penetrate as deeply
into the substrate during the measurement window, as plotted in blue triangles for sapphire andorange circles for silicon. Once the region of effective conductivity change represents a substantial
fraction of the thermal penetration depth — as is true for the smaller linewidths represented here if
we consider a modified-conductivity layer thickness equal to twice the grey MFP for each material— the decay will only be sensitive to Knano and their thermal penetration depths set a lower bound
on d.

73
theoretical analysis of the 2D results is complicated by the issues discussed above — particularly thequestion of whether effective resistivity or effective conductivity will offer a more useful parameter
space for fitting the thermal decay data. However, some initial comparisons in the raw data are
illuminating (see Fig. 3.17).
Greater dimensional confinement and further extension to smaller heat sources may alsoenable the observation of quantized heat dissipation dynamics [114].

3.6.2

Opportunities with dynamic imaging

Observing the average thermal dynamics of many nanowires or nanodots has already andwill continue to reveal important new insight into the fundamental mechanisms of heat transfer at
the nanoscale. However, recent significant developments in the application of coherent EUV beams
to coherent diffractive imaging (CDI) opens the door to a more detailed look at thermal transport
on a localized scale. In particular, 22nm lateral spatial resolution was demonstrated with tabletop
HHG sources in 2011 [115], and non-isolated objects were imaged in 2013 [116]. CDI on opaqueextended objects in reflection mode was demonstrated with visible wavelengths in 2012 [117] andextended to EUV from HHG in 2014 [118]. These developments in combination with a pump-probe
setup for time-resolved measurements will enable the study of nanoscale dynamics with high spatial
and temporal resolution.
For nanoscale thermal transport, this could enable direct imaging of dynamics over arbitrarily
small distances where more details of the effects of quasi-ballistic transport will be apparent. It will
also allow the study of arbitrarily-shaped heat sources where non-uniform laser absorption and heatflow around and from nanoscale features in larger objects may be observed. The localized detailthat will be attainable with dynamic CDI will further enhance our understanding of nanoscopic
processes like heat transfer.

74

a

b
1D

1.0

0.8

1.0

500 nm

250 nm
0.4

0.2

60 nm
0

140 nm
1000

2000

3000

4000

2D

1.0

0.8

500 nm0.6

250 nm0.4

noitcarffid ni egnahc dezilamroN

noitcarffid ni egnahc dezilamroN

0.6

0.0

L = 350 nm

0.8

0.6

1D

0.4

0.2

2D
0.0

0

60 nm

140 nm
1000

3000

4000

L = 60 nm

1.0

0.8

0.6

1D

2D

0.2

0

2000

0.4

0.2

0.0

1000

2000

Delay time (ps)

0.0

3000

4000

0

200

400

600

800

1000

Delay time (ps)Figure 3.17: Comparison of 1D and 2D heat transfer | a. The change in decay time from large
linewidth (red) to small (blue) is larger for 1D nanowire heat sources (top) than for 2D nanodots(bottom) on a sapphire substrate. Since we have already seen that the small-nanowire decay is notenough faster than large nanowires to match diffusive predictions, the smaller change across the2D linewidths suggests even stronger quasi-ballistic effects at play. b. A direct comparison of thesame large linewidth (350 nm) for 1D (blue) and 2D (green) heat sources (top) suggests a slightly
faster initial decay in 2D, which has an extra spatial dimension into which to dissipate heat. Late in
the decay, both curves overlap when substrate dynamics become dominant. For a small linewidth(60 nm, bottom), the initial decays look better overlapped, again suggesting there may be stronger
ballistic effects working against the additional dissipation direction.

75
3.6.3

Avenues for further theoretical development

One clear opportunity for further improvements to our theoretical analysis of the thermaltransport from nanoscale heat sources is suggested by Wilson and Cahill [90]. They incorporate
anisotropic modifications to the effective or apparent conductivities and seek to better resolve theeffects of boundary-scattering-induced MFP changes from the effects of MFPs being longer than therelevant thermal-gradient length scales. Such details will become more important moving forward
to improve the physical interpretations of the heat dissipation measurements we make.
Currently one of the most significant limitations for the development of a more fundamental
picture of micro- and nanoscopic phonon thermal transport is the continued application of diffusive
heat flow models to characterize inherently non-diffusive thermal transport. While such effective
theories have contributed much to our intuitive understanding of how and when nanoscale dimensions affect heat transfer, more comprehensive and fundamental insight into nanoscale thermaltransport may be possible by adopting the framework created to bridge all types of anomalous
diffusion [119, 120, 121, 122]. For example, Vermeersch et al. [123] point out that Fourier diffusion
fails to capture the transition to quasi-ballistic (or super-diffusive) transport at short length andtime scales because it assumes normally distributed jumps, as for Brownian motion. They argue thetrue jump-length distribution is better characterized by a truncated Lévy flight distribution (alsoused to describe the movements of foraging animals, protein movements along DNA and fluctuations in financial markets), which has a long tail of possible long jump-lengths with a characteristic
fall-off in the probability of those long jumps. Thus they fit nanoscale thermal transport data using
such a distribution to extract two parameters for a given material: uBD to identify a characteristic
length scale of transition from super to normal diffusion and α to characterize how quickly the tailof long-jump probability should fall off [122, 123]. Dhar et al. have also demonstrated the efficacy
of Lévy flight distributions for characterizing heat transfer in the quasi-ballistic regime [124]. However, a comparison of these descriptions with the idea of phonon participation controlled by the
relationship of MFP to important length scales in the system suggests that an even better approach

76
may be to use the MFP distribution directly to set the probability of different jump sizes. It may
be that the truncated Lévy distribution is simply a better fit for real phonon MFP distributions
than the distribution assumed by normal diffusion.
A similar two-parameter model has been proposed by Ma to characterize non-Fourier conduction by a diffusive conductivity and a ballistic transport length [125]. However it is addressed
specifically to TG experiments and through nanowires of varying length, which both measure transport over particular distances rather than overall nanoscale heat dissipation.
In general, while much work is presently concentrated on improving the fundamental theoretical descriptions of thermal transport at the nanoscale, most models still fail to take into accountthe presence of multiple heat sources and their possible interactions in determining the overall
heat transfer within a full system. Given the fact that most applications of nanoscale heat sourcesand devices do not fall into the category of isolated objects, exploration of non-isolated nanoscaleheat sources will be an important contribution from our future work as our EUV-probe techniqueremains the only one that is sensitive to heat sources with such small dimensions and spacings.

3.7

Conclusion
Ongoing improvements in nanofabrication capabilities will continue to open access to nanoscaleheat sources of smaller size and engineered shape, and enable the systematic study of the new physics
that comes to dominate small size scales. Here we extended observations of heat dissipation to thesmallest nanostructure heat sources to date and uncovered unexpected transport behavior in which
the close spacing of heat sources can increase the efficiency of thermal transport from nanoscale heatsources when both size and spacing are small compared to dominant phonon MFPs. This discovery motivates new possible strategies for thermal management in nanoelectronics, informs designof nanostructured thermoelectric devices and contributes to a more fundamental understandingof phononic heat transfer which will be important for harnessing nanoscale heat sources in any
application. Furthermore, as we will see in the next chapter, this new thermal transport regime

77enables a new type of MFP spectroscopy for measuring the distributions of phonon contribution tothermal conductivity in materials.

Chapter 4

Probing phonon mean free path spectra

In Chap. 3 we saw how a new regime of nanoscale thermal transport could be described bya phonon suppression filter determined by the heat source size and spacing acting on the spectrumof phonon mean free paths (MFPs) in the material where heat is dissipated. Thus, to be able
to predict and control the nature of thermal transport in nanostructured systems, it is clear thatknowledge of such spectra of MFP-dependent contributions to thermal conductivity will be required[88]. However, theoretical calculation of these spectra requires significant computational power for
all but the simplest crystalline materials, as well as accurate knowledge of the interatomic potentials
that control the lattice dynamics, and means of measuring such spectra experimentally only began
to be developed in the past 15 years.
Here I show how the new phenomenon of collective diffusion is used to extract the contribution to thermal transport from specific regions of the phonon MFP spectrum, opening up a new
approach for thermal transport metrology and MFP spectroscopy. This is because by varying bothnanostructure size and separation, an effective phonon filter is introduced that suppresses specificsections of the spectrum of phonon MFP contributions to thermal conductivity. The larger the
resistivity correction needed to model the dissipation for a given nano-grating, the stronger the
conductivity contribution of phonon modes which were suppressed. We compare our extracted
phonon mean free path spectra with predictions from first-principles calculations and find good
agreement between experiment and theory. Looking forward, we have a unique new capability forcharacterizing phonon transport in novel materials where predictions do not yet exist.

79

4.1

Developments in phonon MFP spectroscopy
The two types of conductivity suppression discussed in Chap 3 Sec. 3.2 — namely increas-

ing the importance of boundary scattering or shortening the thermal length scale — have eachbeen explored for methods of quantitative measurement of phonon MFP spectra. The former wasdemonstrated by Ju and Goodson in 1999 as they used Joule heating and electrical resistancein aluminum bridges to measure the thermal conductivity of thin silicon films (standard SOI, orsilicon-on-insulator, systems) as a function of film thickness to extract an average MFP value of
300 nm [77]. More recently Cuffe et al. used transient grating (TG) measurements at large interference fringe spacing (≈ 21 µm) on silicon membranes varying in thickness from 15 to 1500 nm toreconstruct a more complete distribution of MFPs [126]. The large thermal grating period ensured
that heat flow remained in the diffusive regime, influenced only by the increasing importance ofboundary scattering for the thinner membranes. If this effect can be fully accounted for, this can
produce a measurement of the bulk material MFP spectrum. However, boundary scattering is aprocess that inherently involves many variables that are difficult to control, including parameterslike surface roughness and oxidation. Therefore these measurements have been distinguished from
‘MFP spectroscopy’, which seeks instead to measure an unchanging MFP spectrum by monitoring
the degree of non-diffusive heat transport observed in a macroscopic material for small thermaltransport distances or thermal gradient length scales [65].
Although the idea of using observations of non-diffusive transport to quantify phonon relaxation times was first proposed in 1971 by Simons [127], it was first tied to experimental observations
by Koh and Cahill in 2007 [55]. They observed thermal conductivity which depended on the modulation frequency of a laser pump beam heating the surface and attributed it to the suppression of
long-MFP conductivity due to the short thermal penetration depth within the time scale set by the
inverse of the modulation frequency. This idea was supported by Boltzmann Transport Equation(BTE) simulations in 2011 [62].

80
Following this development, a number of experiments sought to apply this idea to measure
the cumulative contribution to thermal conductivity of phonon modes with MFP shorter than therelevant experimental length scale. Limiting heat source size by adjusting the size of the pumplaser focal spot in time-domain thermal reflectance experiments (TDTR) yielded the first proofof-principle demonstration for studying long-MFP (> 1 µm) phonons in silicon [54]. Regner et
al. [53, 128] and Freedman et al. [56] applied frequency-domain thermal reflectance (FDTR),
again using thermal penetration depth to limit the thermal transport distance and suppress thecontributions of longer-MFP phonons. They reconstructed cumulative conductivity functions down
to MFPs of ≈ 200 nm for a number of semiconductor and dielectric materials by assuming thethermal penetration depth serves as a hard cut-off limiting the MFPs which contribute to theobserved conductivity. Johnson et al. demonstrated that TG measurements with the transportdistance limited by the TG period could be used in silicon membranes in a similar way [37]. In
particular, they note that such measurements could distinguish the cumulative conductivity ofmembranes with different thickness since the effect of boundary scattering will have different levels
of influence on the intrinsic MFP spectra, while the resonant modes of the full membrane will also
affect the phonon dispersion [129].
The influence of heat source geometry, and how an experimental length scale actually acts
to modify the contributions of phonon modes with different MFPs to thermal transport, is more
complicated than simple cut-offs can account for. Thus geometry-dependent suppression functions,like the one derived in Chap. 3 Sec. 3.4, will be an important part of improving the MFP spectra
extracted by these measurements. Suppression functions have been derived for the isolated heatsources of TG experiments [68] and of TDTR and FDTR experiments [89] from grey-approximation
solutions of the BTE applied to every MFP. They can also be extracted from Monte Carlo simulations of any given geometry [111]. Once a suppression function for a given experimental geometry
is known, Minnich proposed a convex optimization method using the suppression function as thekernel of an integral transform to convert measurements of the cumulative thermal conductivity
into cumulative MFP distributions [110]. With this technique he demonstrated that existing TG

81measurements using relatively large thermal grating periods (1 µm) could result in MFP spectra
extrapolated down to 10 nm. This is due to the very broad suppression function serving as kernel,
such that measurements at large length scales can contain information about much smaller MFPs.
However, this broad kernel function also implies significant averaging, which will wash out any
sharp features in the real MFP distribution, and he emphasizes that where the kernel function
is close to zero (as for MFPs below 100 nm in his demonstration), unique solutions for the MFP
distribution no longer exist.
It is important to emphasize that the accuracy of any reconstruction of the detailed MFP
distribution will be closely tied to the accuracy of the suppression function derived for a given
experimental geometry. Wilson and Cahill have recently highlighted some important factors thathave thus far been ignored for the construction of these suppression functions, such as the role ofchanging measurement sensitivity to in-plane vs. cross-plane thermal transport and the intrinsicallysuppressed conductivity near surfaces due to boundary scattering [90]. Continuing to improve
our understanding of the factors which control phonon MFP suppression functions will lead also
to improvements in the interpretation of effective conductivity and resistivity measurements andcorrespondingly improved details in MFP spectra. However we also note that given the current
levels of experimental precision, various proposed forms for suppression function approximations
cannot yet be meaningfully distinguished in the results.
Some works have also suggested that cumulative MFP distributions may follow universal
curves for broad classes of materials. In particular Freedman et al. find that a number of crystalline semiconductors, for which resistive scattering is dominated by Umklapp processes, fall along
the same normalized cumulative conductivity curve when the MFP cut-off (set by the thermal
penetration depth in their FDTR experiments) is scaled by the Umklapp scattering length for eachmaterial [56]. Aketo et al. suggest these curves may have even more universal characteristics for
all crystalline materials with common expressions for estimating the bounds for MFPs contributing between 10 and 90% of the total thermal conductivity [130]. These bounds can be estimatedsolely from knowledge of the bulk single-crystal thermal conductivity, heat capacity, sound veloci-82
ties and Debye temperature. More precision can be obtained through calculation of the harmonic
and anharmonic force constants, but reasonable estimates can even be obtained simply from anempirical relationship between the lower 10% bound and upper 90% bound for thermal conductivity
contributions.

4.2

Experimental measurement of MFPs down to 14 nm
All the previous work on MFP spectroscopy suffers from two significant limitations: mostimportantly, the experimental techniques discussed above cannot access MFPs below a few hundred
nanometers even though the thermal conductivity of most materials is dominated by phonons
with shorter MFPs; moreover, by limiting observations to isolated heat sources, the only direct
measurement is of the cumulative conductivity. Our technique takes advantage of the phonon
filtering effect of non-isolated heat sources to probe arbitrary segments of the MFP spectrum forany novel material, enabling more direct access to the differential, rather than only the cumulative,
MFP distribution. At the same time, combining the use of periodic nanostructures as heat sources
with the exceptional phase sensitivity of short-wavelength probes is the only way to experimentally
access dimensions far below 100 nm in order to directly resolve the contributions of phonons with
MFP down to 14 nm to date [59].
The general principle of this MFP spectroscopy technique is that the degree of non-diffusivity,
represented by the magnitude of rCorr for a given heat source size and spacing, is directly related tothe integrated MFP range of thermal conductivity contributions that has been suppressed relative
to the diffusive prediction. Because linewidth and period set the location and width of the effectivenotch filter in the phonon MFP spectrum, each nanograting configuration uniquely samples thecontribution to thermal conductivity of different MFP ranges of phonon modes with a resolution
controlled primarily by the number of configurations tested.

83
To quantitatively extract information about the differential conductivity spectrum, we use
the full multi-MFP form of rCorr , given by:
rCorr (L, P ) = A P

1
1
−
Kbulk
i k(Λi )S(L, P, Λi )


.

(4.1)

+
We partition the full sum in bins between i−bin and ibin according to the MFP-sensitivity of eachgrating configuration.

rCorr (L, P ) = A 


1 
 .

−
P i+
Kbulk bin
kbin i− S(L, P, Λi )
1


P

bins

(4.2)

bin

The upper and lower bounds of the full range of MFP contributions to which we are sensitive are
set by the suppression functions related to each of our nano-gratings. We choose the minimum(14 nm) and maximum (5 µm) MFPs of our experimental spectrum to include only MFPs that aresuppressed by at least 20% in our smallest and largest sample geometries, respectively. As can beseen in Fig. 3.10, each particular configuration for L and P can be related to one most-suppressed
MFP at the minimum of Stotal . We use this information from our set of nano-gratings to establish
the MFP bins which we use when fitting ref f data in the full interacting multi-MFP model, as
shown in Fig. 4.1: bin boundaries are chosen halfway between neighboring most-suppressed MFPs
for every other sample.
Then by fitting our set of ref f data as ref f = rT BR +rCorr we extract the average k(Λi ) = kbinwhich is associated with all modes Λi within each bin, thus assessing the relative contribution per
nanometer to the differential thermal conductivity of each region of the phonon MFP spectrum(plotted in Fig. 4.2). The error bars in the histograms are obtained by varying kbin while maintaining the residual of the fit within the range of experimental uncertainty to explore the full range
of weights allowed by the shape of the data. By limiting the number of bins to be no more thanhalf the number of data points, we ensure a conservative, well-conditioned fit, although we notethat changing the bin number does not substantially alter the trends we observe.
Although the number of experimental data points limits the number of regions we can reasonably consider in this first demonstration, this approach still offers unprecedented new experimen-

84

Figure 4.1: Setting bins to extract MFP spectrum | In order to fit our ref f data using thefull spectrum of phonon MFPs, we bin the spectrum according to the sensitivity of each linewidthperiod pair to different regions of the spectrum. The teal diamonds mark the most-suppressed
MFP for each linewidth represented in our initial sample set with constant duty cycle at P = 4L.
The boundaries between bins are set halfway between every other sample point, as shown by thevertical dashed lines.

85

Figure 4.2: Extracted MFP spectra for sapphire and silicon | By fitting ref f with multiple
bins of phonon modes, the weight kbin assigned to each bin gives the average relative contribution
to the differential thermal conductivity (purple shading). Both differential (distributions) andcumulative (lines) conductivities are normalized to the total bulk conductivity. For sapphire (toppanel), our data (solid purple line) and first-principles DFT calculations (dashed green line) indicate
there are no significant contributions from long-MFP phonons, so the cumulative curves approach
unity at 1 µm. For silicon (bottom panel), our data are consistent with large contributions fromlonger MFPs.

86
tal access to the differential thermal conductivity contributions of phonons with different MFPs.
Furthermore it provides important benchmarks for theoretical predictions, including those fromfirst-principles density functional theory (DFT) calculations shown in Figs. 3.9 and 4.2 calculated
according to the methods presented by [57, 59, 61, 131]. In particular, our experimental data acrossall MFP ranges measured in silicon are in good agreement with our DFT calculations (which also
agree with those in the literature [88]). However, some small discrepancies appear for phonon MFPs
around ≈ 100 nm, where experimental verification was not possible before. Differences between theexperimental and theoretical spectra in this region may also be exaggerated by our limited set ofsmall-linewidth gratings; increased resolution with a larger sample set can address this issue. Ourdata are also consistent with observations by others which emphasize significant contributions fromlong-MFP (> 1 µm) modes in silicon [53, 54, 37], but the limited number of data points we havefor structures much larger than the average phonon MFP results in a relatively large uncertainty inthis region. For the purpose of comparison in Fig. 4.2, we normalized the experimental spectra insilicon by assuming the integrated conductivity up to 1 µm should match that calculated by DFT.
For sapphire, both calculation and experimental data imply that phonons with MFPs shorter
than 1 µm are responsible for > 95% of the thermal conductivity. The discrepancy below 300 nm
between experimental and theoretical curves (most apparent in the cumulative distribution) is dueto two factors. First, the sharper rise in the DFT cumulative curve comes primarily from the verystrong short-MFP peak in the conductivity spectrum – a peak that lies at approximately 5 nm,
below the lower bound of our experimental sensitivity (14 nm) using 30nm nanostructures. Thus,
the experimental data simply does not include the shortest phonon MFP peak. Second, because of
the complex crystal structure of sapphire, the DFT calculations required the use of relatively smallinteraction-distance cut-offs for determining the harmonic and anharmonic force constants, which
may cause a larger error in the theoretical predictions than for silicon.
It is important to emphasize that different experimental geometries (for example, 1D- versus
2D-confined heat sources, or bulk materials compared with thin films compared with nanotubes)
will result in observed conductivities that are specific to the given geometry. Consequently, effective

87
thermal conductivity results cannot necessarily be compared in a straightforward manner. However, the phonon MFP spectrum corresponds to a physically real attribute of a material alone andtherefore provides the more appropriate tool for comparison across different experimental geometries. Furthermore, the effective thermal conductivity for any experimental geometry can then be
predicted using an experimental phonon MFP spectrum combined with the appropriate theoreticalmodel for conductivity suppression. We note that while both heat source geometry and spacing
must be included in such suppression functions, spacing effects have only recently been explored,
both by our group [132, 58], and more recently by numerical studies with a grey phonon model[113] using the phonon BTE [133].
The ability to experimentally extract a phonon MFP distribution down to such small MFPs
offers a new useful method for validating existing first-principles predictions across the whole rangeof phonon MFPs significant for heat conduction, as well as the first access to such information for
more complex materials where calculations have not yet been performed. Furthermore, combined
knowledge of both the differential and cumulative thermal conductivity may offer intriguing insight
into the full MFP spectrum with the detail necessary for accurate prediction of heat transfer innanostructured systems.

4.3

Applying MFP spectroscopy with EUV light
Having demonstrated the utility of observing collective diffusion to extract the phonon MFP

spectra of materials, we can now begin applying this new tool to the study of more complex
materials. As a first step, we study a series of amorphous low-k dielectric SiC:H thin films withvarying levels of hydrogenation, which significantly modifies the mechanical, electrical and thermal
properties of the films. Some studies have already observed reductions in thermal conductivity ofsimilar materials with increased porosity [134, 135]. Added information about the MFP spectrum
and how it is changed with porosity and composition could yield new insight into how to control
and optimize the material for the best combination of mechanical, electrical and thermal properties.

88
Moreover, the amorphous nature of these materials imply that ab initio theoretical methods forcalculating the MFP spectra will not apply.
We compare how heat dissipation rates change with linewidth and period for nickel nanostructures deposited on this series of thin films (like those shown in Fig. 4.3). A future full analysis
following the procedures outlined in this chapter will allow us to extract the contributions to thermal
conductivity of a few ranges of MFPs as a function of thin film hydrogenation. One experimental
challenge of studying these films is their very low thermal conductivity. This results in the very
long decay times visible in Fig. 4.3.
Measurements of this type will also inform the design and optimization for many applicationsof phonon engineering, either to frustrate phononic thermal transport as for thermo-electrics [44, 43]
or to take advantage of as much heat dissipation efficiency as possible as for the transistors innanoelectronics [48, 49, 50] or bits in magnetic data recording media [51]. In all cases, knowledge ofthe particular distributions of differential phonon conductivity by MFP will indicate the size scales
on which the engineering must take place. For example, mazes [136] and nanomeshes [137] on asize scale of phonon MFPs have been used to slow phonon but not electron transport, and MFPscale corrugations along the edges of graphene ribbons can be designed to preferentially backscatterphonons [82]. These efforts can be more precisely targeted with knowledge of which phonon MFPs
are most dominant in contributing to the overall thermal conductivity.

4.4

Conclusion
The differential phonon MFP contributions to thermal conductivity in materials are funda-

mental to determining how thermal transport is modified by nanoscale dimensions. They determine
the nanostructure sizes and surface roughness levels at which boundary scattering will come to dominate the resistive processes that control thermal conductivity. And they set the length scales atwhich thermal gradients drive heat transfer that can no longer be approximated by Fourier diffusion, and thus the heat source size and spacing that will see significant deviations from diffusive
dissipation behavior. Moreover, the often broad distributions of MFPs must be known in detail to

89

Figure 4.3: Thin film thermal data for MFP extraction | Comparing the linewidth-dependent
thermal decays for a series of SiC:H thin films – like the two sets shown here for a highly hydrogenated, soft film with Young’s modulus E = 13 GPa and a stiff film with E = 200 GPa – will allow
us to study how the MFP spectra may change in correlation with thin film hydrogenation levels.
The traces are approximately normalized so that the center of the acoustic oscillations starts at
one.

90
enable accurate prediction of the few-order-of-magnitude ranges in length-scales where transitionsin heat transfer behavior will take place and how they will manifest.
Harnessing the phonon-filtering effects of the collectively-diffusive regime of nanoscale thermaltransport that we discovered enables a unique new MFP spectroscopy tool. I have demonstrated
here that this offers the first experimental access to MFPs as short as 14 nm and the first direct
measurements of differential, rather than only cumulative, distributions of thermal conductivity
contributions [59]. This technique will be indispensable going forward both as a method for testing
theoretical first-principles calculations of MFP spectra and for measuring these spectra in materials
where such predictions do not yet exist.

Chapter 5

Thin film metrology

The mechanical properties of materials determine their strength and flexibility, and govern
their reliability in the structures and devices they comprise. Tremendous progress in nanofabrication capabilities, particularly in service of increasingly tiny nanoelectronic devices, means thatsingle-atom layers and sub-50nm structures have become commonplace. However, means of characterizing mechanical properties are still severely limited for dimensions below 100 nm [6]. In
particular, optimizing the thermal and electronic properties of new materials in manufacturing isoften detrimental to the elastic properties that allow reliable processing and fabrication [138]In this chapter, I give some background about the physical meaning and significance of theelastic properties of materials and then summarize a number of techniques that are used to measurethese properties in bulk and film materials. Then, to extend measurement capabilities to sub-100nm
ultrathin films, I introduce EUV nanometrology, which makes use of the techniques discussed in
Chap. 2, to fully characterize the elastic tensor of isotropic dielectric films as thin as 50 nm, andI demonstrate its applicability to a broad range of material stiffnesses. To improve the precisionof our measurements, I also introduce an improved frequency extraction method beyond Fouriertransforms to lower our frequency uncertainty by approximately a factor of 10. I then demonstrate
the first application of this technique to characterize anisotropic materials before providing an
outlook on a number of future opportunities.

92

5.1

Elastic properties of materials
The elasticity, compressibility, rigidity and hardness of materials are related fundamentallyto the interactions between the atoms that comprise them, whether arranged in an ordered crystal
lattice or packed together in an amorphous material. The strength of interatomic bonds and thedensity of atoms will determine how a macroscopic object responds to applied forces. These properties are also directly tied to how robust a material is, how much it can be stretched before failure,
how susceptible it may be to cracking, how brittle or flexible it is. Therefore, characterization ofelastic properties is important both for using materials in a wide array of applications and to learn
about the microstructure of materials.
Displacements of atoms from their equilibrium positions in a crystal lattice will result in
restoring forces, just as stretching or compressing a spring by a distance x from equilibrium results
in a restoring force F characterized in the linear regime by Hooke’s law, F = −kx. In the simple
case of a spring, k is the elastic constant related to the particular spring. In an atomic lattice(or continuum material), an analogous law applies, but because the atoms can be displaced in any
combination of the three dimensions, the full strain (or displacement) field and resulting stress(restoring force) of an object are characterized by 3 × 3 matrices, and the ‘spring constant’ relatingthem is a 3 × 3 × 3 × 3 tensor [93]. Symmetry reduces the number of independent components ofthis elastic tensor c to 21, and the symmetry of specific crystal lattice structures can reduce it evenfurther. Then, just as k sets the resonant frequencies of spring oscillations or the speed of travelingwaves, c will offer the same information for acoustic waves in materials specific to each direction of
propagation and to wave type (longitudinal and transverse).
The elastic tensor c relates the microscopic displacements of atoms to the behavior of amaterial as whole, but its specific attention to directions of stress and strain means that relatively
simple macroscopic processes require several components to fully encapsulate the material response.
For example, linear compression along the axis of a rod combines compressive stress along theaxis with resulting additional stress perpendicular to the applied force. Thus macroscopic elastic

93
responses are often related instead to elastic moduli, which are generally combinations of multiple
components of c. Specifically, Young’s modulus and Poisson’s ratio are enough to completely
characterize isotropic materials.
The Young’s modulus E characterizes the stiffness (or rigidity) of a material, quantifying
how far a material will deform along the direction of an applied force. Poisson’s ratio ν records
how much a material will expand (or sometimes contract) in one direction while it is compressedin the perpendicular direction. As shown in Fig. 5.1, E can vary over several orders of magnitude
from MPa (= 106 N/m2 ) to TPa, while ν is more narrowly confined between -1 and 0.5 for stablematerials, making it an intriguing parameter for more universal characterization of materials fromthe softest solids to the strongest liquids [140]. Indeed, by quantifying the balance between amaterial’s resistance to volume change and its resistance to shape change, ν is a quantity that canbe used to indicate the structural performance of any material and has been shown to directly
correlate with critical features like damage tolerance in metallic glasses [141].

5.2

Existing metrology techniques
While a number of techniques are available for characterizing the elastic and mechanicalproperties of thin films, challenges mount as film thickness shrinks.
Nanoindentation is a widely-used technique that follows the displacement of a speciallydesigned tip, called an indenter, into the surface of a material as precise loading forces are applied.
This direct measurement of the stiffness and hardness of a material is used to extract the stressstrain relationship and various elastic moduli. These relations require careful modeling of thecontact area between the indenter and the indented surface, and once the material layer is thin,further modeling must account for the influence of the substrate [142, 143, 144]. Accurate measurements still require relatively thick films (hundreds of nanometers) or reliable knowledge of somematerial parameters in the system, such as values for the Poisson’s ratio of both substrate and film.
Characterization of multilayers only yields average values. It is advantageous for local measurements across a film surface as its sensitivity is confined to the region directly compressed by the

94

a

b

Figure 5.1: Range of E and ν for materials | a. Young’s modulus can vary by orders ofmagnitude, describing the stiffness or resistance to deformation of materials (graph from [139]). b.
Poisson’s ratio ν is more limited, unifying all stable materials along a spectrum extending from -1
to 0.5 and quantifies the balance between a material’s resistance to volume change and resistanceto deformation. ν = 0.5 implies a perfectly incompressible material which spreads out to theside as much as it is compressed in the other direction. Negative values describe materials which
shrink perpendicular to an applied compression (graph from [140]) The two insets offer a schematic
illustration of the forces and strains related by the two parameters.

95
nanoindenter. However, it is also a destructive measurement and has contact areas that are toolarge to apply to small structures.
A newer atomic force microscopy-based (AFM-based) technique relies instead on the mechanical resonances of the AFM cantilever, which shift when the tip is moved into contact witha material surface. This contact-resonance AFM technique involves much smaller contact regions
because of the much smaller forces involved — especially for high-order resonant modes of thecantilever [145] — and is therefore less destructive to sample surfaces. Without shear components
in the surface motion, this technique is only sensitive to the Young’s modulus, leaving the Poisson’s
ratio to be assumed [146], and can still be influenced by substrate properties for films below 1 µm
[147], with the onset of substrate effects depending on the properties of the AFM tip and sample[148]. Moreover, it requires strong models of the cantilever eigenmodes for quantitative measurements of the material properties [146] or comparison with precisely known reference samples [147].
However, properties of films as thin as 7 nm have been extracted from CR-AFM measurements
when the substrate was characterized separately [149]. It further offers a very sensitive tool for
mapping local variations in elastic parameters across a surface when the differences, rather than
the absolute values, are of interest.
Non-contact thin film metrology typically relies on the interactions between light and acoustic waves or phonons in the material of study. Brillouin light scattering (BLS), for example, makes
use of direct interactions between light and the acoustic phonon modes of a material. A photonthat scatters with a phonon loses or gains the phonon energy and shifts slightly in frequency,
yielding a spectrum of scattered light that can identify many phonon modes [150]. Because of theaccess to both transverse and longitudinal acoustic velocities, BLS does allow measurement of bothYoung’s modulus and Poisson’s ratio. However, interpretation is more complex: the weak intensityof scattered light and the difficulty of proper identification of phonon modes make the film characterization strongly dependent on the experimental accuracy attained, and strong understanding ofboth the scattering cross-sections of different modes and the response of the measurement system
are required [151]. Moreover, though scattering from bulk acoustic modes tends to be stronger

96
than from surface modes and more separated from the bright elastically scattered spectral peak,they can only be observed for materials transparent to the probe light [150]. But given the rightconditions, measurements with BLS can have very high precision [152] and have been demonstrated
on films (free-standing) as thin as 8 nm [153].
The other major class of optical non-contact techniques falls under the title of picosecond
ultrasonics or photoacoustics. Traditionally this has made use of the longitudinal acoustic wave
(LAW) velocity to calculate the Young’s modulus of an isotropic thin film by assuming a valueof Poisson’s ratio. Strong absorption of a laser pump pulse in the thin film or in an additional
thin metal transducer layer launches these LAWs, which are observed by transient changes in
the reflected intensity of a subsequent probe pulse [154, 155]. When the films are transparent to
the probe and thicker than a few acoustic wavelengths, a traveling acoustic pulse can also cause
oscillation in the interference between light reflected from the surface and light reflecting off themoving acoustic pulse, allowing a measurement of the acoustic velocity without knowledge of film
thickness [156]. Because the probe signal does not require high spatial resolution to detect LAW
signals in reflectivity across a large area, these techniques have been used to study films down toa few nanometers in thickness with visible-wavelength light [156]. The dispersion of broadband
surface acoustic waves (SAWs) has also been used to evaluate the Young’s modulus of a film asthin as 5 nm [157].
Nanostructured transducers were introduced to these measurements more recently as a way
to simultaneously excite SAWs and LAWs [158, 159, 160, 161]. By accessing both the longitudinal
and transverse acoustic velocities, this alleviates the need to assume a value for Poisson’s ratio,
but the continued use of visible-wavelength probe light inherently limits the sensitivity of thistechnique. Thinner films require shorter-wavelength surface displacements – smaller than thoseeasily observable by the resolution available with visible light. Thus far, visible-light picosecond
ultrasonics with nanostructured transducers has been used to characterize films as thin as 260 nm[158]. Narrowband SAWs can also be excited by interfering two pump laser beams on an absorbing
surface to create a transient grating without the need for absorbing nanostructures and excite

97
SAWs with wavelength equal to that of the interference period [162, 163, 164]. This allows an
all-optical method for capturing both SAWs and LAWs on any absorbing surface, but visible pumpwavelengths limit the accessible transient grating periods to a minimum of ≈ 750 nm.
Two more developments have extended the capabilities of visible-probe photoacoustics. Highspeed asynchronous optical sampling (ASOPS) uses two femtosecond lasers with slightly different
repetition rates to set up a train of pump and probe pulses which scan the delay time without any
need for a mechanical delay stage. This fact combined with the high repetition rates (hundredsof MHz) of the lasers enables very low noise levels, and signals at the 10−7 level are visible evenafter only one second of acquisition time [165]. Thus, even though visible probes are significantly
less sensitive to the surface displacements of nanostructures than EUV probes, techniques like thisimprove signal detection compared to standard visible-light techniques and allow very small signals
to be resolved. Devos et al. also demonstrated a new technique interfering two probe beams where
one reflects from the sample before a pump pulse arrives and one at variable delay after. In this way,
they gain sensitivity to the out-of-plane oscillations of nanostructures smaller than the diffraction
limit of visible light [166]. These two clever techniques demonstrate how the applications of visibleprobe photoacoustics can be extended further than previously thought possible. An exciting idea tocontemplate is how similar techniques could be adapted for EUV wavelengths, potentially extending
capabilities far beyond the resolution improvements intrinsic to short-wavelength probes.

5.3

EUV nanometrology
To enable full elastic characterization of ultrathin films we make use of nanostructured metal-

lic transducers with periods as short as 90 nm to simultaneously excite LAWs and very shortwavelength SAWs in a series of 50-100nm thin films of varying elastic properties – the thinnest
films directly probed to date using photoacoustic probes to extract both Young’s modulus andPoisson’s ratio. The shortest SAW wavelength implies a small penetration depth fully confinedwithin the thin film of interest, as shown schematically in Figure 5.2. This allows us to selectively
probe the properties of the thin film material without unwanted contribution from the substrate un-

98derneath. Given the shortest achievable SAW wavelengths from nanostructured transducers within
the present capabilities of nanofabrication, this approach to mechanical nano-characterization will
scale to sub-10nm thin films [41]. Such capabilities are particularly important as a number oftheoretical studies suggest thickness dependence in the elastic properties of thin film materials[167, 168], and experiments are needed to directly test theoretical models of these systems. At thesame time, our technique enables the testing of new ultrathin film materials being developed innanoelectronics, as well as a potential in-line characterization tool for process control.

Figure 5.2: Setting SAW penetration depth | The propagation of surface acoustic waves islimited to a penetration depth equal to a fraction of their wavelength. Thus, by setting a short SAW
wavelength with a suitably small-period nano-grating, the penetration and sensitivity of the SAW
measurement can be confined to thin film layers at the surface of a sample, while long-wavelength
SAWs from large-period gratings probe the properties of the substrate material.

In these experiments we probe a series of 50-100nm low-k dielectric SiC:H thin films (i.e. withlow dielectric constant k ) deposited on silicon substrates representing a range of elastic properties set
by the levels of hydrogenation in the film material – in particular with Young’s moduli nominallyvarying from 13 to 200 GPa [169]. Using electron-beam lithography and lift-off techniques, wedeposit a series of periodic metallic nanostructures on the surface of each film to serve as transducers
of nanoscale acoustic waves in the system. A schematic of the samples and the acoustic waves thatwill characterize the films is shown in Figure 5.3. Nickel is chosen as an efficient transducer material

99

P=3L
Ni
SiC:H

L

SAW

LAW

SiliconFigure 5.3: Samples for EUV photoacoustic metrology of thin films | Our samples consistof 1D nickel nano-gratings deposited on the surface of 50-100nm SiC:H low-k dielectric thin films
on silicon substrates. An infrared pump pulse is absorbed by the nickel, and the subsequent
impulsive thermal expansion launches acoustic waves in the system. The SAW and LAW within
the film isolate the material properties of the film from those of the rest of the system. The SAW
wavelength is set by the grating period P. LAW pulses launched downward reflect back to thesurface from the buried interface with the substrate.

100
for its strong absorption of our 800-nm pump light and high coefficient of thermal expansion (at12.77×10−6 K−1 ).
As discussed in Chap. 2, an infrared pump pulse causes an impulsive thermal expansion of
the nanostructures which launches the acoustic waves: LAWs traveling downward through the thinfilm as in traditional photoacoustics; and SAWs excited due to the transverse periodicity of thestress induced at the film surface. The SAW wavelength Λ is governed simply by the period of
the nanostructures, which allows the excitation of shorter wavelengths than accessible by current
transient grating techniques [35, 41].
For each of our thin film samples, we make use of a range of periods P from 90 to 1500 nm
with constant filling fraction of 1/3 and constant nanostructure height of 10 nm. With the SAW
penetration depth ζ ≈ Λ/π (as observed in a number of simulations), full confinement inside a filmof thickness t is achieved when P ≤ 3t. Therefore, this range of periods can excite waves ranging
from those fully confined within even our thinnest 50nm film to waves propagating primarily inthe silicon substrate, allowing us to demonstrate unequivocally the layer-selective measurement ofboth film and substrate elastic properties. We achieve detection of the shortest-wavelength SAWs
for full confinement by harnessing the similarly short-wavelength probe light we have from HHG.

5.3.1

Extracting acoustic velocities

The dynamic signals revealing the SAW oscillations and LAW propagation in thin films are
directly used to extract the velocities of both types of acoustic waves. The longitudinal pulse
echoes visible in Fig. 5.4 give the time required for a round-trip through the film layer of thickness
t. We confirm this delay time τLAW by looking for these echoes in signals from all the available
grating periods. As shown in Fig. 5.4, the echoes return to the surface at the same time regardless ofgrating period since all have traversed the same thin film layer. The velocity is then calculated fromvLAW = 2t/τLAW . Note that the film thickness must be known independently; in this case we relyon accurate ellipsometry measurements of the thicknesses of each film. While we currently estimate
the echo round-trip time as shown by the dashed lines in Fig. 5.4, more precision could likely be

101

τ

LAW

L = 500 nm

Change in diffraction

L = 350 nm

L = 200 nm

L = 100 nm

L = 50 nm
0

20

40

60

Delay Time (ps)80

Figure 5.4: LAW echoes in a set of grating periods | The signatures of LAW pulse echoes
returning to the surface arrive at the same time delay τLAW regardless of grating period because
all have traveled across the same thin film layer. The high-frequency oscillations early in the signal
correspond to the resonant LAW mode of the nanostructures.

102
obtained through a more thorough understanding and model of the shape of the echo signal. Sucha model could be fit to the measured data to extract the precise delay time corresponding to thepoint in the signal that would yield the exact longitudinal velocity.
For the SAWs, we know their wavelength Λ from the period of the nano-grating that excites
them (and its higher harmonics) [33]. Then by extracting the oscillation frequencies f we observe,
the velocities can be calculated, vSAW = Λ·f . To isolate the f with high precision we employ a chirp
z-transform (CZT) [170]. Unlike the discrete Fourier transform, the CZT allows an arbitrary numberof frequency samples (up to the number of time samples) to be concentrated within a specified
frequency range. This makes it well-suited for attaining high frequency resolution over a small
bandwidth. We then fit the peak to a Gaussian function to extract the central frequency, as shown in
Fig. 5.5, and average the peak values from many scans on the same sample. The standard deviationsof these measurements generally represent a factor of 10 improvement in uncertainty compared tothe standard deviation of Fourier transform peaks among the same set of measurements.
Figure 5.6 shows that for large-period gratings, the measured SAW velocity is consistent
with literature values for bulk silicon (≈ 5000 m/s [171]) because the wave penetrates so far below
the surface that it travels mostly in the substrate. In contrast, the short-period gratings display
the slower velocities associated with all the different film materials because these waves are truly
confined within the film layer, isolating the film properties from any substrate influence. Even
different SAW orders launched by the same nano-grating can illustrate this effect; in particular,
the P = 600nm grating excites a fundamental wave which mostly represents the substrate properties and a second-order wave with wavelength Λ = 300 nm which is much more confined in andrepresentative of the thin film material.

5.3.2

Elastic properties from acoustic velocities

For isotropic materials, measurements of the longitudinal and transverse acoustic velocities
are enough to characterize the full elastic tensor c, which has only two independent components in

103a
L = 500 nm

noitcarffid ni egnahC

P = 1500 nm

L = 200 nm
P = 600 nm

0

500

1000

1500

2000

Delay time (ps)brewop lartceps dezilamroN

L = 500 nm
CZT

L = 200 nm
CZT

FT

FT
2

4

6

8

10

12

Frequency (GHz)

Figure 5.5: Extracting SAW frequency from thin film signals | a. Different nanowire
linewidths, associated with different periods at a constant relationship of P = 3L, excite particular
SAW wavelengths which appear at distinct frequencies in the diffraction signal. b. Fourier transforms (as shown by the light green and blue curves) identify the same frequency peaks as the chirp
z-transform (marked in circles) but with lower resolution that can miss the actual peak locations
by amounts smaller than the FT step size. Fitting Gaussian curves to the CZT peaks (solid curves
across the data points in the first-order peaks) then extracts the central values.

104

SAW penetration depth » Ë/ð (nm)
0

100

200

300

400

500

6000

E = 200 GPa
5000

E = 175 GPa)s/m( yticolev WAS

E = 153 GPa

4000

E = 60 GPa
E = 30 GPa

3000

E = 13 GPa

2000
Fundamental SAW
2nd-order SAW
1000
0

200

400

600

800

1000

1200

1400

1600

SAW wavelength Ë (nm)Figure 5.6: Layer sensitivity in the SAW velocity | While for long wavelengths and penetration depths all of the 100nm film samples display SAW velocities consistent with that of thesilicon substrate, the short-wavelength, small penetration depth measurements clearly separate thedifferent films with their slower velocities. The green dashed line represents the point at which the
SAW penetration depth is approximately equal to the film thickness; green shading implies thatthe effect of the film is still important even for the intermediate wavelengths. In some intermediatecases – particularly Λ = 600 nm – it is apparent that the fundamental SAW (circles) is consistent
with propagation mostly in the silicon substrate while the second harmonic excited by the samegrating (triangles) is consistent with a slower thin film velocity.

105
this case [93]. Specifically c = (c11 , c44 ) where2c11 = ρvLAW(5.1)

c44 = ρvT2 AW .

(5.2)

Here the density ρ must be known independently, as from x-ray reflectivity measurements in thiscase, and the transverse velocity vT AW is related to vSAW = vT AW σ where σ is the Landau factor,
which is generally > 0.9 [172]. These elastic constants can then be related to the Young’s modulus
E and Poisson’s ratio ν, which are more typical quantities for describing the macroscopic responses
of materials.


3c11 − 4c44
E = c44c11 − c44
c11 − 2c44
ν=2(c11 − c44 )


(5.3)
(5.4)

Of course, the presence of the nanostructures does introduce loading on the film, which will
modify the measured SAW velocities from the case of a free surface. In particular, the SAW modes
at a free surface are eigenmodes of the system and do not radiate energy into the bulk material
or decay in an ideal system [173]. In contrast, the stress at the interface between film surface andnanostructures that allows the structures to move with the surface also serves to scatter energy
into the bulk [173]. Put another way, the pure SAW modes are no longer eigenmodes of thenanostructure/film/substrate system; instead the nanostructures serve to couple the pure SAW
modes to the bulk modes of the film/substrate, leading to the radiation of energy away from thesurface [161].
To understand and account for the effect of the nanostructures on our measurement of thethin film properties, we employ finite-element simulation to model the whole system. As in thesimulations of thermal dynamics in our samples (described in Chap. 3 Sec. 3.3.2), the laser
excitation of the nickel nanostructures is modeled according to Eqn. 3.6. Following Nardi et al.
[92], the displacements umax (~r) at each point ~r at the time of maximal deformation are projected
onto the set of eigenmodes ui (~r) of the nanostructure-film-substrate system, calculated by solving

106
the acoustic eigenvalue problem,~ · [c(~r) : ∇ũ~ i (~r)] = −ρ(~r)ωi2 ũi (~r)
∇

(5.5)

and normalizing over the unit cell
s
ui =

Mc
ũi
C Vc ρũ2i dV

(5.6)

R

where Mc and Vc are the mass and volume of the unit cell, respectively and C is a unity constantwith units [m−2 ]. The projection|umax i =X

hui |umax i|ui i =iXci |ui i(5.7)

ireveals the degree to which each eigenmode contributes to the impulsively excited displacement,
and thus how strongly each eigenmode will be excited by the impulsive expansion. This techniqueunambiguously resolves which eigenmodes are excited, revealing that the main contributions to thesample dynamics are a symmetric SAW and its higher harmonics, as shown in Fig. 5.7, and we
can directly determine their frequencies. By then adjusting the Young’s modulus and Poisson’s
ratio of the simulated thin film material and seeking to match the predicted frequencies to those
we observe for the whole set of grating periods, we can more accurately extract the properties ofthe film alone, having accounted for the influence of the grating structures. In practice this isoften most readily accomplished by using the large-period sample acoustics to confirm the elastic
properties for the simulated substrate and then tuning the film properties to match the shortestperiod samples. Finally, one set of elastic parameters for the whole system is found to match theobservations from all sample periods. This procedure results in the Young’s modulus and Poisson’s
ratio for each SiC:H thin film sample, reported in Fig. 5.8.
In comparing our measurement of the Young’s moduli to the nominal values obtained via
nanoindentation on thicker samples of the same film materials, the measurements falling along theline with slope one indicates close agreement. However, it is interesting to note that for one sample(Enom = 153 GPa), the initial nanoindentation value was much higher than what we observed.

107

3

2

1H

0

19
|u1H| (m)

0

|u2H|

2H

13

0

11.88

1

6.00

Projection coeff.s |ci | ( x10

-14

)

Eigen
Fano

5.6

5.8

6.0

Frequency (GHz)

6.2 11.6

11.8

12.0

Frequency (GHz)

Figure 5.7: Identifying excited SAW modes with simulation | The initial thermal expansion
displacement of the nanostructured system is projected onto the set of acoustic eigenmodes todetermine which are most strongly excited by the impulsive pump excitation. Thus peaks in the
projection coefficient |ci | can identify the mode frequencies that will be excited in the experiment.
Here the peaks are fit to Fano profiles to extract the central frequencies and lifetimes. The two insets
show the absolute total displacement fields related to each excited eigenmode — the fundamental
and second harmonic are shown here. Figure adapted from [92].

108

a

b

300

200

0.4

oitar s'nossioP)aPG( suludom s'gnuoY

Simulated

250

0.5

Measured

Measured

0.3

150

0.2

100

Measured

0.1

50

0

Simulated

0.0
50

100

150

Nominal Young's modulus (GPa)

200

50

100

150

200

Nominal Young's modulus (GPa)Figure 5.8: Elastic characterization of thin film series | From the measurements of SAW
and LAW velocities in a series of SiC:H thin films with different degrees of hydrogenation, we
demonstrate the capability of EUV nanometrology to extract both Young’s modulus and Poisson’s
ratio across the whole range of nominal Young’s moduli (characterized by nanoindentation on
thicker samples of the same film material). a. The measurements of Young’s modulus mostly
match those indicated by the nanoindentation values (dashed line). Also note that accounting
for the effect of the nanostructures with simulation (solid symbols) generally results in a larger
E than simply calculated from the measured velocities (open symbols). b. Our nanometrology
measurements were the first to address Poisson’s ratio for these types of film materials. Again
the simulations typically shift the results compared to those calculated directly from the measured
acoustic velocities, to lower values in this case. Note that the sample with Enom = 200 GPa is
the only film with thickness 50 nm, and its particularly high Poisson’s ratio may be anomalous forthat reason. For the rest of the films, which all have thickness 100 nm, Poisson’s ratio is mostly
constant until an upward trend begins toward the soft end where the high levels of hydrogenation
may push bond coordination past a critical point [138]. The large error bar for Enom = 153 GPa isdue primarily to the similarity between the film and silicon substrate which decreases the reflected
amplitude of the LAW pulse.

109
A subsequent re-measurement of that film with nanoindentation matched our result much moreclosely.
Since we excite both transverse and longitudinal waves simultaneously, we are also able tomeasure the Poisson’s ratio. And in fact, our measurements represent the first characterizationof Poisson’s ratio for these films. While the values appear mostly constant across this range offilm stiffnesses (except for the only 50nm film at Enom = 200 GPa where the isotropic assumption
may begin to break down), we do note that the lowest-Enom films may suggest the beginning of an
upward trend in ν. This may be due to the crossing of a critical value in the bond coordination inthese highly-hydrogenated films [138]. The future study of even softer, more highly hydrogenatedfilms will allow further probing of this trend.
While our analysis thus far has only utilized the fundamental SAWs observed, similar fitting
techniques that incorporate information from the higher SAW harmonics as well could likely improve
the overall precision in the future.

5.4

Anisotropic materials
Launching SAWs with 2D arrays of nanodots increases the number of harmonics we observe.

For example, it adds one in between the fundamental f0 and second-order 2f0 frequencies of the√
1D nanowire arrays. With frequency equal to 2f0 , it corresponds to the diagonal periodicity
represented in Fig. 5.9. For a nanodot array aligned along the [110] direction of single-crystal
silicon, the fundamental SAW and higher integer orders lie along the [110] direction while thediagonal orders are along [100]. Therefore, for an anisotropic crystal like silicon, the different
modes could, in principle, be expected to display different velocities [174].
I examine the SAW frequencies observed from 1D and 2D arrays of nickel structures withlinewidths L from 30-750 nm and periods P = 4L on silicon to test this idea. Fig. 5.10 plots theobserved frequencies vs. k/2π = 1/Λ for all the observed harmonics. For both 1D and 2D arrays,
all harmonics appear to follow the same-slope line for the long-wavelength (small-k ) samples where
the influence of the gratings does not significantly affect the dispersion relation, implying they all

110

FT: L = 750 nm
P = 3000 nm on Si.dro dn2.dro ts1

1st ord.

.dro .gaid ts1

2nd ord.

2nd ord.

2D array

1D array

1st diag. ord.
1

2

3

4

5

6

7

8

-3

x10

Frequency (GHz)
1st ord.

Figure 5.9: Extra SAW modes from 2D nanodot arrays | All of the SAW harmonics launchedby 1D nanowire arrays cross the surface in the same direction. 2D nanodot arrays, on the other
hand, add periodicity in the diagonal direction. As visible in the Fourier transform of signals from
1D (dashed blue) and 2D (solid green) gratings with linewidth L = 750 nm and P = 3000 nm on
silicon, this additional periodicity adds frequency peaks√ in between the integer orders launched by
1D gratings. The first diagonal order showing here is 2 times the fundamental frequency.

111
propagate with the same velocity. While this is expected for the 1D arrays, for the 2D arrays thissuggests that we do not detect any anisotropy comparing the integer harmonics with the diagonalorders.
40

40

Fundamental

Fundamental)zHG( ycneuqerf WAS

2nd order

1st diagonal

3rd order

2nd order

30

30

20

20

10

10

2nd diagonal

1D nanowire arrays0

2D nanowire arrays
0

0

2

4

6
-3

k / 2p =1/LSAW (x10

0

-3

2

4

x10

-1

nm

8

)

6
-3

k / 2p =1/LSAW (x108)

Figure 5.10: Examining 1D and 2D SAWs to characterize anisotropy | The long-wavelength(small k ) SAWs follow a linear dispersion (black lines) where the slope gives the bare-substrate
SAW velocity. If the diagonal orders (diamond symbols) launched by 2D nanodot arrays have
speed determined by the [100] rather than [110] axis of silicon, they should lie along a distinct linefrom the integer orders (circles, squares and triangles). However, no difference is apparent in theseplots.

For quantitative comparison, linear fits to the long-wavelength data of each harmonic yields
the SAW velocity associated with each. Fig. 5.11 shows that integer harmonics compare wellfrom 1D to 2D arrays, but the diagonal orders match in velocity within error bars. However, theanisotropy in silicon is not particularly large and the expected difference in velocity is only fromabout 5100 m/s along [110] to 4925 m/s along [100] [171]. If I compare the 1D measurement of
the fundamental wave velocity (v0 = 5110 ± 40 m/s) with the first diagonal wave velocity fromthe 2D arrays (vdiag = 4850 ± 130 m/s), they are consistent with this expected difference and arestatistically distinguishable. Therefore observations of the SAW excited by 2D surface phononic
crystals could offer simultaneous measurement of the multiple propagation directions needed tocharacterize materials with anisotropy across the in-plane directions.

-3

x10

-1

nm

112

6000

)s/m( yticolev WAS

5500

1D nanogratings
5000

4500

2D nanogratings

4000

3500

1st order

1st diag.
order

2nd order

2nd diag.
order

Figure 5.11: Extracted velocities from 1D and 2D nanograting SAW | By fitting a line
to the long-wavelength SAW dispersion, we extract the velocity associated with each SAW order— the lowest two integer orders for the 1D nanogratings (blue triangles) plus two diagonal orders
from the 2D nanogratings (green open squares). The overlapping measurements from 1D and 2D
gratings agree well; the diagonal orders also match within error bars, but both appear slightly lower
than the integer orders. This matches expectations for the known anisotropy of silicon.

113

5.5

Future opportunities in nano-mechanical characterization with EUV
Two distinct limitations are apparent in the methods discussed so far. The first is the need fornanostructures to achieve the small periodicities necessary to launch SAWs that are confined within
ultrathin layers. Particularly as the film layers of interest become even thinner, the influence of thegrating structures on the SAWs we measure and on the film material itself will be more significant.
Moreover, the resonant LAW mode of the nanostructures will oscillate on a time scale much moresimilar to the longitudinal waves in the film, which may make it more difficult to extract the film’s
longitudinal properties. An adaptation of transient grating techniques that uses short-wavelength
VUV or EUV pump beams could relieve these complications, and the photon flux achievable withpresent tabletop HHG sources is now high enough to make such experiments more plausible thanthey were a few years ago. However, the lack of simple reflecting and focusing optics at thesewavelengths and the need to introduce relatively long delay times between pump and probe pulses
still pose significant challenges. If VUV/EUV TG were implemented, though, it would greatly
improve flexibility in the types of materials that could be studied without the need for strong
absorption at visible wavelengths or the fabrication of nanostructures.
The second limitation, which would also still be an issue in a transient grating geometry, is the
non-local nature of these measurements, averaging the elastic properties over the whole area thatis probed. Here too, the application of dynamic imaging with EUV CDI could offer an extension
to localized measurements of elastic properties. In particular, full maps of surface displacements
caused by acoustic modes in anisotropic materials could be used for complete characterization ofcomplex elastic tensors as in laser-based resonant ultrasound techniques [175]. Mechanical properties can also be used as a contrast mechanism for material-specific mapping of a surface whichappears uniform to other types of microscopy [176]. Local mapping of elastic constants has evenbeen used to identify the progression of malignancy in breast cancer cells [177]. In all these case,the enhanced spatial resolution available with EUV CDI would improve capabilities significantly,
thereby opening opportunities for deeper studies of elastic properties at the nanoscale.

114

5.6

Conclusion
Here I have demonstrated that EUV nanometrology can be applied to characterize a wide

range of ultrathin films, particularly emphasizing the layer-specificity which makes EUV-probed,
short-wavelength acoustic nanometrology so attractive for the characterization of the mechanicalproperties of ultrathin films. Given the shortest SAW wavelengths we have observed with ourtechnique at < 45 nm [41], this method of film characterization can extend directly to 10nm films.
The changing layer sensitivity with SAW wavelength also shows that this could be applied todepth-profiling of inhomogeneous materials and multilayer stacks. I have also demonstrated thefeasibility of using 2D nanodot arrays for a simultaneous measurement of SAW velocity in two
directions across a surface to study materials with anisotropic elastic characteristics.
One great advantage of EUV nanometrology for thin film characterization is the possibility fora wide variety of separate measurements using the same setup. EUV high harmonics are sensitive
to magnetic [178] and thermal dynamics (as discussed in Chaps. 3 and 4) and can also be used
to directly image a nanostructured sample via coherent diffractive imaging [115]. Therefore thistechnique offers the possibility for the future development of a unique, flexible nanometrology toolcapable of a wide variety of characterization modalities.

Chapter 6

Nanostructure metrology

The acoustic waves confined within ultrathin films proved a useful non-destructive tool in
Chap. 5 for characterizing the elastic properties of films as thin as 50 nm where the density andthickness were independently known. Here I will show that these techniques can be extended to thestudy of the density and elastic properties of nanostructures with nickel (Ni) and tantalum (Ta)layers below 10 nm in thickness.
In particular, the SAWs excited below the structures will reveal the structure mass anddensity, while the LAW resonances of the structures themselves are very sensitive to the layer
thicknesses and elasticity. In this case we use samples fabricated with carefully controlled layer
thicknesses to focus on how elastic properties deviate from their bulk material counterparts. To
resolve shifts in acoustic resonance frequencies below 1%, we again make use of the chirp z-transform(CZT) method introduced in Chap. 5 Sec. 5.3.1 for SAWs, and I will introduce a second frequency
extraction method beyond simple Fourier transforms. Through the use of the Matrix Pencil Method,which is designed to extract both decaying and non-stationary oscillatory signal components, thissecond method significantly improves our experimental uncertainty in resonant LAW measurement.
This study represents the first to date to examine the elastic properties of ultrathin bilayers
and extract the individual layer acoustic velocities. Here we observe their opposing trends indeviating from bulk properties and examine how one layer may influence the other. It is also thefirst demonstration of the EUV-based SAW mass sensor technique introduced by Nardi et al. [179]and proves that the density ratio of ultrathin material layers is not significantly different from bulk.

116

6.1

Nanoscale departures from bulk properties
While a significant departure from bulk mechanical properties for nanoscale materials was firstobserved in 1977 in superlattices with nanometer-scale periodicities [180], the precise mechanisms,
magnitude and even sign of these changes are still open questions. A number of the earliest
observations of superlattice in-plane stiffening > 100% were later attributed to artifacts of the bulgetesting measurement technique employed [181]. However, a number of more recent measurementswith more reliable characterization techniques (like picosecond ultrasonics, nanoindentation and
Brillouin light scattering, as discussed in Chap. 5 Sec. 5.2), and theoretical and simulation efforts
observe and seek to explain nanoscale changes in elastic moduli.
In particular, stiffening by 14-50% has been observed for the in-plane elastic constant (control2) in superlattices [182, 183, 184].
ling LAW velocity parallel to the surface and interfaces as ρvparallel

The cross-plane elastic constant (perpendicular to the interfaces) softens by ≈ 15-30% with superlattice periods below 10 nm [182, 185, 186] or 40 nm [187] compared to what would be expectedfrom the volume fraction of each material in the superlattice and bulk values for the individual
elastic moduli. By varying the composition ratio in a series of Mo/Si superlattices, Pu et al. werealso able to determine that the sound velocities in each of the 2-5nm material layers were both 2-6%
lower than the corresponding bulk sound velocities [188]. Proposed mechanisms primarily focus on
softening or weak bonding at the interfaces [182, 185, 186], or a change in the layer morphologies
with smaller grain sizes in thinner layers leading to a larger volume fraction of softer intergrain
space [187].
Results for nanoparticles and nanowires are even more mixed, with changes in elastic moduli
ranging from -60 to +200% of the bulk material values [189, 190, 191, 192]. The much larger surfaceto-volume ratio of nano-objects has led most explanations to consider effects of surface tension,
surface energy and lack of surface bond coordination. However, some nanoparticle experiments
which saw no change from bulk parameters suggest that most observed modifications may be due
primarily to interactions with a substrate or other surrounding material [193, 194].

117
Measurements of sub-100nm metallic films have also observed a broad range of modification
to the cross-plane elastic constant from -40% [195] to +15% [196] and changes which are not always
monotonic with film thickness [196, 197, 198]. Softening was attributed to film defects [156], weakinterfacial bonding [195], or lowered bond coordination contributing to differences in surface elastic
constants [199, 200, 201]. The stiffening, on the other hand, can occur when a different process
dominates the thin film behavior, whether strain in the films [196, 198] or electron redistribution
at the surface [200].
There is not yet any comprehensive theory that predicts which materials should soften or
stiffen or by what magnitude or at what size scale. The properties of the individual layers in a
multilayer system have also not been explored in detail to understand how one might affect theother. Thus, there still remain many unanswered questions regarding the mechanical properties ofmaterials in ultrathin films and nanostructures.

6.2

Samples for study of patterned films with sub-10nm layer thickness
To resolve the mechanical properties of the two components in an ultrathin bilayer film, westudy a series of samples with varying composition. Ultrathin films of Ni and Ta are depositedon SiO2 /Si substrates (where the SiO2 layer is 150-200 nm thick) using a calibrated sputtering
deposition process to achieve precise layer thicknesses [195]. Sputtering rates are first calibratedusing x-ray reflectivity (XRR) measurements of separate 30nm films of Ni and Ta. XRR employs
the changing interference conditions of short-wavelength light reflecting from thin layers at different
angles to precisely characterize film thickness and density. Even though the parameters must be
extracted through a model-fitting process, the properties of > 10nm single layers can be repeatably
and uniquely determined in most cases. Then sputtering time is chosen to precisely deposit a 10nm
layer of Ni for all samples and a Ta capping layer varying between 1-6 nm (see Fig. 6.1). This
results in a thickness uncertainty of ≈ 5% while layer thickness is varied by as little as 0.1 nm fromsample to sample.

118

EUV probe pulse
Infraredpump pulse

x nm Ta10nm Ni

2μm

1μm

SiO2/Si

Figure 6.1: Ultrathin bilayer samples | To examine the mechanical properties of ultrathinbilayers, we study a series of samples with a Ni layer at a constant thickness of 10 nm and a Ta
capping layer varying from 0 to 6 nm. As described in Chap. 2, the metallic structures absorb an
infrared pump pulse and the change in diffraction of the EUV probe pulse follows the dynamics
that ensue. Thus, while the large 1 µm linewidth implies these structures will exhibit essentiallythin-film behavior, the periodic pattern gives us the probe diffraction that ensures sensitivity to
the small surface displacements associated with nanostructure dynamics.

119
While XRR can also be applied to the ultrathin bilayers directly, the large number of model fit
parameters and variation introduced by different fitting techniques, as shown in Fig. 6.2, make thisan unreliable source of absolute thickness measurements. Particularly with such thin films, uniquesolutions are rare. To illustrate one apparent problem, consider the amount of Ta that is consumed
to create a layer of tantalum oxide. The most stable oxidation state of Ta is +5 so that the typicaloxide layer is Ta2 O5 and has about 1 nm of thickness [202]. Given the Ta2 O5 density ρoxide = 8.20
g/cm3 [203], the surface density of a 1nm layer would be 8.20 ×10−7 g/cm2 . To determine the molar
surface density, the molecular weights are required: wT a = 180.948 g/mol [202], wO = 15.999 g/mol[204] and woxide = 2wT a + 5wO = 441.891 g/mol. Thus the molar surface density is σM = 0.0186
×10−7 mol/cm2 . There is twice this number of mols of Ta per cm2 contained within the oxide,
meaning that the surface density of Ta in the oxide layer is σT a = 2σM · wT a = 6.731 × 10−7 g/cm2 .
This corresponds to σT a /ρT a = 0.403 nm of Ta lost per nanometer of oxide formed. However, XRR
measurements from the sample which nominally had 1 nm of Ta initially deposited suggests that
0.4 nm of Ta remain below the oxide layer, but the 0.6 nm of Ta supposedly lost to oxidation would
not be enough to form the 2.25nm oxide layer indicated.
On the other hand, sample-to-sample precision is strong, so that XRR characterization ofthis sample set demonstrates that the oxide and Ni layer thicknesses are essentially constant fromsample to sample, and the Ta layer thickness grows as expected. Thus, nominal thicknesses canreasonably be used for analyzing the acoustic dynamics we observe; modifications suggested by thepresence of the additional oxide layer are discussed below.
The films are then patterned using optical lithography and etched into a periodic arrayof wires that are 1 µm wide with a periodicity of 2 µm. Acoustic resonances of these arrays are
excited by the infrared pump pulse absorbed in the metallic layers, causing an impulsive thermal
expansion, as described in Chap. 2. We observe both individual longitudinal acoustic resonancesin the through-thickness direction of each wire and surface acoustic standing-wave modes in thesubstrate launched by the whole array, with wavelengths set by the array periodicity and its higher
harmonics [35, 41, 161]. The large wire width ensures that the longitudinal resonances we probe

120

XRR thickness (nm)

10

Ni

8

Ta

6
4

Ta oxide

2
0
0

1

2

3

4

5

Nominal Ta thickness (nm)6

Figure 6.2: X-ray reflectivity measurements of bilayer samples | XRR requires fitting amodel to reflectivity data to extract layer thicknesses and densities. The increased complexity ofthis model for multiple layers, along with the very small layer thicknesses represented in our sample
set (down to 1 nm), makes it very difficult to find a unique solution. Different choices of input cause
variations in the output fit parameters and uncertain levels of accuracy. For the three layers in oursystems, Ni (grey circles), Ta (green triangles) and Ta oxide (blue squares), the error bars represent
the fit uncertainty while multiple points at the same nominal thickness of Ta show the variability
due to differences in the model fitting procedure. These measurements do indicate, however, that
the Ni and Ta layers are essentially as they were designed, and the oxide layer is approximately
constant from sample to sample.

121
are essentially those of thin-film bilayers, but the periodic pattern yields the diffractive interference
that enables strong sensitivity to vertical displacements of the surface profile down to the picometerscale [40], as well as creating the periodic stress that launches the standing surface acoustic waves(SAWs).

6.3

Measuring material density with the surface acoustic resonance
SAW velocity is exquisitely sensitive to the mass of any structures on the surface, particularly

for high-frequency SAWs since shorter wavelengths imply confinement to shallower layers below thesurface where the wave velocity is most affected [179]. In Chap. 5 we needed to account for thiseffect by using finite-element simulation of the SAWs we observed experimentally to extract the‘real’ elastic properties of thin films. Here, each sample with a different Ta layer thickness (overthe constant thickness of Ni) exhibits a slightly different SAW velocity and oscillation frequency
fSAW , as shown in Fig. 6.3, and this can be used to gauge the structure mass.

Figure 6.3: SAW frequency shifts with Ta thickness | a. The SAW frequency shift induced
by adding 6 nm of Ta on top of 10 nm of Ni is clear in the raw change-in-diffraction signal. The
additional mass slows the wave propagation, lowering the oscillation frequency. b. Because thefrequency shifts are very small, we employ the chirp z-transform to generate high-resolution spectra,
plotted in crosses, near the resonance peaks (examples of the fundamental and 4th-order peaks areshown here for the two raw-data curves from part a). The center frequencies are extracted by
Gaussian fit functions (solid lines).

122
6.3.1

Extracting small frequency shifts with high resolution using CZTWith the large array periodicity present in our sample set, the SAW frequency shifts inducedby the changing Ta mass are quite small: only around 30 MHz, or about 1%, across the whole rangeof samples for the fundamental SAW order. To extract the SAW frequencies with enough precision
to resolve this small change, we again apply a CZT [170] to obtain the frequency spectrum near thefrequency peak, shown in Fig. 6.3b. More detail is given in Chap. 5 Sec. 5.3.1. The average fitted
peak values from many scans on the same sample yield the results reported in Fig. 6.4. The error
bars represent the standard deviations of these measurements. This procedure yields frequency
uncertainties of less than 0.5% for the fundamental order and better than 2% for the higher orders.

6.3.2

Interpreting frequency-shift data

The change in the volume of Ta from sample to sample can be derived from the known change
in thickness. The SAW frequency shift can be related to the implied mass change, leading to ameasure of material densities. Specifically, we see that the SAW frequency decreases linearly withTa thickness. Fitting this line yields values for two parameters: the slope γ, and intercept f0 . The
relationship between fSAW and Ta volume has slope α = γ/(Lz) where L is the linewidth of thewires and z is the length of the wires. Then the relationship of fSAW to wire mass must have slope
β = α/ρT a where ρT a is the density of the Ta layers. Since we also know v0 for when the mass of thewire is equal to the mass of just the 10nm layer of Ni mN i , β = (f0 − fs )/mN i where fs is the SAW
frequency for the same wavelength propagating along the bare substrate surface. The frequency fs
must be known a priori, as for example by using finite-element analysis and literature properties
for the materials comprising the substrate to identify the dominant excited modes [161, 205, 206].
Therefore,ρT a =

αγ mN iγ ρN i hLz==
βLz f0 − fs
Lz f0 − fs

(6.1)

where ρN i and h are the density and thickness of the Ni layer, respectively. While the densities of123

Figure 6.4: Extracted SAW frequency shifts in four harmonics | The increase of Ta thickness
from sample to sample results in a linear shift in the SAW frequency. This is visible in all the four
lowest-order SAW harmonics, but the magnitude of change is significantly higher for higher orders.
This is because the shorter wavelength implies the wave propagation is confined closer to the surface
where the material is most affected by the extra mass in the bilayer structures. The linear fits toeach set of data are used to calculate the Ta-to-Ni density ratio.

124
Table 6.1: Fit results from SAW resonances for the lowest four SAW orders.

f0 (GHz)J (GHz/nm)
UTa/UNi

1st order
2.420 ± 0.003

2nd order
4.584 ± 0.048

3rd order
6.23 ± 0.06

4th order
7.81 ± 0.11

-0.0051 ± 0.0009-0.101 ± 0.014-0.067 ± 0.019-0.092 ± 0.028

1.93

13.88

1.70

1.88

the two materials in the bilayer cannot be independently determined in this way, the ratio between
them can be found as

γh
ρT a
=
.
ρN i
f0 − fs(6.2)

Linear fits to the SAW frequency data for each of the first four SAW harmonics, shown in Fig.
6.4, yield values for the Ta-to-Ni density ratio calculated according to Eqn. 6.2. These results arereported in Table 6.1. The values obtained from the 1st, 3rd and 4th harmonics are consistent witheach other while the ratio extracted from the 2nd-order SAW data is significantly higher. Possible
causes for such a large discrepancy are the subject of ongoing discussion. Averaging the threesimilar values yields ρT a /ρN i = 1.84 ± 0.12, which is somewhat lower but statistically consistent
with the ratio of the bulk material densities, 1.92 [207]. Note that because this measurement relieson the slope of the lines with respect to the change in Ta thickness, the presence of a constant oxide
layer will not alter the conclusions.
This measurement demonstrates the first all-optical, non-destructive characterization of thedensity ratio between materials in ultrathin bilayers. Notably the slope is significantly higher for
higher SAW harmonics due to the higher mass sensitivity enabled by shorter wavelengths. Thisillustrates that greater precision for density measurements with this technique could be attainedby using shorter array periodicities.

125

6.4

Characterizing elastic properties with the longitudinal acoustic resonance
While the SAW dynamics dominate the diffraction signal over many nanoseconds, Fig. 6.5

shows that the first 20 picoseconds following excitation by the pump pulse reveal the resonant
longitudinal oscillation of the wires themselves [156]. Again the frequency shifts significantly as theTa thickness is varied. Also, by comparing three identical Ni/Ta film samples etched for different
amounts of time, we verify that the resonant frequencies we measure are not systematically affectedby potential over-etching of the wire pattern into the substrate (see examples in Fig. 6.6). This
ensures that the shifts we observe are uniquely related to the changing Ta thickness.

Figure 6.5: LAW resonances observed as a function of Ta thickness | The increase of
Ta thickness from sample to sample also results in a shift in the resonant LAW frequencies of thebilayers themselves. The dashed lines serve as guides to the eye, marking time-zero and the shifting
location of the second oscillation maximum.

6.4.1

Extracting LAW periods from fast damping using MPM

The fast decay of the LAW oscillations makes a Fourier transform, which assumes stationary
oscillations, an inappropriate and unreliable tool for extracting the resonant frequencies. Instead we
employ the Matrix Pencil Method (MPM) which projects a function onto the set of exponentials

126

Figure 6.6: No systematic effects due to etch depth | By testing three samples etched todifferent depths from the same 10nm Ni/6nm Ta bilayer, we confirm there are no systematic effects
in our observations of resonant LAW and SAW frequencies arising from potential over-etching into
the substrate. These plots compare the SAW signals with their CZTs (top) and LAW signals(bottom) from the ideal-etch (solid orange) and the most over-etched (dash-dot blue) samples.

127with complex exponents ea+bi to a specified precision limit, allowing it to reliably capture bothoscillation frequencies and damping rates of the signal components isolated from the noise inherent
to experimental data [208, 209, 210]. As shown in Fig. 6.7 we remove irrelevant parts of the data
— namely the signal before time-zero and the background rise due to thermal expansion and theonset of the first surface acoustic oscillation.

Figure 6.7: Preparing LAW data for MPM | To achieve the best results from the MPM
algorithm, we first clean out the irrelevant parts of the measured signal (solid grey) by cutting off
the points from before time-zero and subtracting out a polynomial fit (dotted grey) to remove theslow background rise. From the remaining signal data (blue) we construct the Hankel matrix foruse in MPM.

The cleaned data is used to construct a Hankel matrix. For (2n+1) data points [a1 , a2 , a3 , ...a2n+1 ],this matrix takes the form




a2
a3
 a1


 a2
a3
a4


a
a4
a5
 3

 ..
..
..
.
.
.


an an+1 an+2

...
...
...
..

.

...

an 


an+1 
an+2 


.. 
. 


a2n+1

(6.3)

The singular value decomposition of this matrix gives something analogous to eigenvalues andeigenvectors that can represent the underlying organization of patterns in the data. A plot of

128
the relative strength of the singular values, shown in Fig. 6.8a, shows a sharp transition fromsignal components to a long tail of random noise where all the modes are similarly populated. The
number of values appearing before this transition represents the number of signal components thatexist above the noise level. In this example, two are clearly visible. They represent the complex
exponentials with positive and negative imaginary parts of the same magnitude which sum into
the real oscillation signal visible in Fig. 6.7. MPM then constructs a new matrix from the singular
vectors corresponding to the signal components identified and finds the eigenvalues that give thecomplex exponentials that best represent each signal component. Figure 6.8b shows where thetwo exponential components lie in the complex plane for our signal from Fig. 6.7. The symmetricplacement on either side of the real axis again displays that they combine to form the real oscillationof the signal. Their location inside the unit circle arises because the signal decays. Stationary
oscillations would lie on the unit circle with modulus one. Pure exponential decay components(like the thermal decays we observe in long time scale data) appear as individual nodes on the real
axis inside the unit circle.
The real component gives the amplitude decay while the imaginary components give theoscillation period and initial phase. Plotting the extracted oscillatory component with the original
data in Fig. 6.9 shows that the overlap is very good. Moreover, the residual difference between
the measured data and the extracted oscillation (shown in grey) is comparable to the noise signal
recorded before time-zero (red).
While the same amplitude, period and phase parameters could be accessed via a least-squares
fit to the data, in practice, results are significantly more reliable when using MPM. In particular,
they are not strongly influenced by choices like the starting guesses for least-squares algorithms,
and in this data set MPM generally lowered the relative uncertainty in the extracted periods by afactor of about five. In this way we extract the set of oscillation periods shown in Fig. 6.10 withuncertainties below 0.7%.

129

Figure 6.8: Separating data from noise with singular values | a. A plot of the singularvalues from the data shown in Fig. 6.7 displays a sudden jump between the two strong signal
components and a long tail of similarly-valued noise components. We use this cut-off, marked by
the dashed line, to select the number of signal components to solve for using MPM. b. The output
of MPM then gives the nodes in the complex plane corresponding to the complex exponentialthat best represents each signal component. The oscillation visible in Fig. 6.7 shows up here asa pair of nodes with the same magnitudes of real and imaginary parts but opposite sign on theimaginary part. Pairs like this sum to a real oscillation signal with frequency (or period) givenby the magnitude of the imaginary part and amplitude decay time given by the magnitude of thereal part. Both lie inside the unit circle, because the oscillation is decaying rather than growing
or remaining stationary. Non-oscillatory decaying signal components would appear as individual
nodes on the real axis inside the unit circle.

130

Figure 6.9: Comparing MPM output with measured data | The residual difference (grey)
between the measured data (dotted blue) and the oscillatory signal component extracted using
MPM (green) is comparable in magnitude to the noise signal recorded before time-zero (red). Thus
it is reasonable to assert that the MPM-extracted component successfully accounts for all the signal
above the noise level in our measurement.

131

Figure 6.10: Extracted resonant LAW periods | The resonant LAW periods of the bilayer
structures extracted by using MPM clearly are not well matched by the predictions resulting frombulk material parameters (dashed red). Instead, we use a least-squares optimization algorithm to
fit the data in order to extract effective nanoscale longitudinal velocities of the Ni and Ta layers.
This fit is plotted in solid blue, while the dashed blue lines mark the fits to the diagonal extremesof the error bars and indicate the upper and lower bounds of the extracted effective velocities.
Note however that the data point at 0 nm of Ta thickness (the Ni-only sample) must be excluded
to achieve a good fit to the rest of the data. The vertical error bars on the data points (most ofwhich are too small to distinguish from the points themselves) represent the variation in the MPMextracted periods related to different choices of the data starting point and number of singular
vectors to keep as signal. The horizontal error bars represent the 5% uncertainty in layer thickness
resulting from the calibrated sputtering deposition process.

132
6.4.2

Interpreting LAW oscillation periods

The resonant frequencies of a bilayer can be calculated by enforcing the condition that (displacement, stress) = (u,0) be an eigenvector of the appropriate acoustic transfer matrix (stress-freeboundary condition), as in calculating the localized surface modes of a semi-infinite superlattice[195, 211]. Like the optical transfer matrix method for calculating the propagation of electromagnetic waves across layered media, the acoustic transfer matrix applies continuity conditions for the
stress and displacements across boundaries from one medium to the next to construct the matrix
operation that relates an input field to the output field at the next boundary.

Oscillation min

Ta
Ni

100
80
60
40
20
0
-20
-40
-60

SiO2

-80
-100

Vertical displacement (pm)Oscillation max

Figure 6.11: Finite-element model of the open-pipe-type resonant mode | Comparisonof the displacement fields calculated by finite-element-analysis [212] at the peak and subsequentvalley of the oscillation in the diffraction signal from our Ta/Ni bilayers on SiO2 (seen in Fig. 6.10)
clearly reveals the symmetric shape of the mode, placing antinodes at both the top surface and theinterface with the substrate.

In our case where the SiO2 substrate has lower acoustic impedance than the Ni layer, theresonant modes will be analogous to organ-pipe modes with two open ends (see Fig. 6.11); i.e., theresonant frequencies will be the same as those for an unsupported bilayer membrane with antinodes(and the stree-free boundary condition applied) on both sides [213]. We test this principle bysimulating the laser excitation and subsequent dynamics of excited metal lines on substrates with
Young’s modulus varying from 10 to 1000 times stiffer or softer than the metals [212].

133

Figure 6.12: Resonant frequency determined by sign of acoustic mismatch | Simulated
signals compare the resonant LAW periods for cases where the substrate is 10-1000 times stiffer(green) or softer (blue) than the metallic structures. This confirms the idea that the magnitude of
acoustic mismatch effects the oscillation amplitude, but the period only relies on the sign of themismatch — all the stiff substrates result in a period that is twice that from the soft substrates(even for the very low-amplitude oscillation seen for the substrate that is 10× stiffer). The two
insets schematically represent this difference between closed-pipe and open-pipe resonances with
anti-nodes at the open ends and a node at the closed end.

134
The simulation results in Fig. 6.12 clearly show that while the amplitude of oscillation isstrongly related to the degree of acoustic mismatch between the structures and the substrate, theresonant periods are set only by whether the substrate is stiffer or softer. Also, the periods on
the soft substrate are indeed what would be expected from free-standing membranes. Therefore
they can be calculated by enforcing the eigenvector condition for a transfer matrix corresponding
to one-way propagation across the Ta and Ni layers. This transfer matrix τ is given by



τ =


cos(qN i dN i )sin(qN i dN i )/ωZN i −ωZN i sin(qN i dN i )cos(qN i dN i )


×


cos(qT a dT a )sin(qT a dT a )/ωZT a  (6.4)−ωZT a sin(qT a dT a )cos(qT a dT a )where ω is the LAW angular frequency, dx is the thickness of the respective layers, Zx is the acoustic
impedance (Zx = ρx vx for LAW velocity vx ) and qx = ω/vx . We require








 u 
 u 
τ
 = λ

0
0

(6.5)

for any scalar eigenvalue λ. The second component of this matrix equation implies that resonant
frequencies will solve the transcendental equation


tan

ωdN i
vN i




+ ZT a /ZN i tan

ωdT a
vT a


= 0.

(6.6)

Note that these equations simplify dramatically in the long-wavelength limit (qx dx → 0),predicting an effective velocity through alternating material layers that is set by the weighted
average of the cross-plane elastic constants C = ρv 2 of each material ((d1 + d2 )/Cef f = d1 /C1 +d2 /C2 ) or of the individual acoustic velocities for the further case of low impedance mismatch

135(Z1 /Z2 → 1). But these cannot be applied to our case where the wavelength of the open-pipe
resonance is twice the total thickness of the bilayer. Furthermore, we confirm that the full transfer
matrix approach is the only one to reliably predict the resonant frequencies extracted by MPM fromsimulations with finite-element models where layer thickness and acoustic properties are precisely
known inputs for the simulation.
As shown by the red dashed line in Fig. 6.10, the resonant periods predicted by this approach
using bulk material properties clearly fail to match those we observe experimentally as the Ta layer
thickness is varied.
Before analyzing the data further to understand how layer properties may deviate from thoseof bulk materials, it is important to consider how the presence of the typical tantalum pentoxide
layer on Ta would affect our measurements. Since every nanometer of Ta2 O5 that forms consumes
approximately 0.4 nm of Ta (as calculated in section 6.2), we compare simulated dynamics of
the Ni/Ta bilayer with nominal thicknesses to those of a Ni/Ta/Ta2 O5 system with nominal Ni
thickness and nominal Ta thickness minus 0.4 nm per nm of added oxide. Given literature valuesof all material properties for the sake of comparison, we find that the resonant frequencies from thetwo simulations are the same within our experimental error bars; the effect of the excess oxide isessentially compensated by the Ta loss so that it is reasonable to conduct our analysis assuming thenominal Ni and Ta thicknesses and no oxide. Therefore, we conclude that the discrepancy between
the observed resonant periods and those predicted by bulk material properties can be attributed
primarily to changes in the elastic properties of the ultrathin layers.
We then use vN i and vT a as our only fit parameters in a least-squares optimization algorithm
using Eqn. 6.6 to find the effective nano-layer velocities that provide the best fit to the data.
Specifically, for a wide range of values for vN i and vT a , we use Eqn. 6.6 to solve for the largest
resonant periods (corresponding to the lowest-order mode that we observe) for each bilayer sample
predicted by every possible pair. To enable the best comparison to the data, we fit a low-orderpolynomial to the data points which will serve as a representation of the data over the range ofmeasurement. We then calculate the root-sum-squared difference between the polynomial fit to

136
the measured periods and each set of predicted periods and seek the minimum (see Fig. 6.13).
We also compare this residual function to the average uncertainty in our data points, marked by
the plane crossing the surface. Because the transcendental equation is very sensitive to the input
parameters, I find that using the polynomial representation, which assumes that the period shouldchange smoothly as the Ta thickness is varied, to mediate the comparison with predictions enables
a more consistent result. It also produces a lower final residual between the fit and measured data
than is obtained by fitting the data points directly with this algorithm.
However, we find that the point from the Ni-only sample (at Ta thickness = 0 nm) must be
excluded to achieve a good fit to the rest of the data. This is quantified in the comparison of theresidual surface to the error plane representing the uncertainty in the measured data: when the
Ni-only sample point is included in the fitting algorithm, the surface never drops below the error
plane.
The blue solid line in Fig. 6.10 displays the final fitting result, while the blue dotted lines
mark the outer bounds determined by fitting the diagonal extremes of the error bars on eachdata point. The effective longitudinal velocities extracted are summarized in Table 6.2. Notethat the effective velocities differ significantly from their bulk counterparts. Moreover, the best-fit
vN i is indeed significantly different from the value associated with the Ni-only resonant period τ ,calculated by vN i,only = 2dN i /τ . This implies both that the 10nm layer of Ni has different elasticity
from that expected for a bulk material and that it is further modified by the addition of the Ta
layer — though in a way that lessens the difference from bulk elasticity. Moreover, the two layers
deviate from their bulk counterparts in opposite ways such that they are more similar to eachother than bulk Ni is to bulk Ta. It should be noted that this should not be caused by significant
intermixing between the two materials since independent measurements on similar samples showthat the magnetic properties of the Ni layer remain those of pure Ni.
Fitting the data in this way presumes that the elastic properties of Ni and Ta are constantacross the entire set of samples. While this is a reasonable assumption for the constant 10 nm
layer of Ni, one could argue the Ta layer may well behave differently at 1 nm of thickness than

137

Figure 6.13: Best-fit residual surface for range of vTa and vNi | To find the best-fit effective
velocities of nickel and tantalum, we use many possible pairs of vN i and vT a to predict the periods
that would correspond to our set of bilayer samples. This surface plots the root-sum-squared
difference between those predictions and the polynomial representation of the data for each pair,and the minimum will mark the least-squares best-fit pair of velocities. The grey plane marks thelevel of data uncertainty. Where the surface drops below this plane, possible solutions are indicated.

Table 6.2: Fit results from the LAW resonances of the Ni/Ta bilayers. Bulk values from [207].

Bulk vLAW (m/s)
5600

Best-fit vLAW (m/s)
5000 +10/-10

Ni under Ta

5600

5240 +40/-30

Tantalum

4100

5180 +150/-170

Ni only138
at 6 nm. Given the lower sensitivity of the measured oscillation periods to the Ta velocity in thethinnest layers we examine, we can neither confirm nor rule out this possibility. If we assume thevN i extracted by the method outlined above and then solve for vT a with each measured period (asrepresented by the polynomial fit), we find the trend shown in Fig. 6.14, which shows that our
set of measured periods would be consistent with an effective velocity in Ta that varies from 5000
m/s at 6 nm to about 6000 m/s for < 3 nm. Notably, trying to solve for the varying vT a using
either vN i,only or the data points directly (rather than the polynomial representation) results inthere being no solutions to Eqn. 6.6.

6.4.3

Potential mechanisms for modified nanoscale elastic parameters

A number of mechanisms for nanoscale elastic changes in films and superlattices of the magnitude that we observe have been proposed. For example, the atomic layers near a surface will
exhibit different elastic constants than those deep within a volume, and these layers will make up
a significant portion of ultrathin films [199]. However, the surface elastic constants calculated by
Shenoy [214] suggest a change that is at least an order of magnitude smaller than we observe.
Interface layers and bonding across them will also influence the effective elastic constants
we extract. In particular, the fact that the ultrathin Ni layer alone behaves differently than the
Ni layers under Ta lends evidence to the importance of these effects. Mismatch between different
materials and residual stress have been used to explain some elasticity changes comparable to those
we observe [167], although the trends predicted by more rigorous models for this effect [185, 186] do
not account for what we observe. Softening due to a weak interfacial bond or, equivalently, interface
layers with different properties than a bulk-like core was modeled by Rossignol et al. [195], butnotably their single parameter describing the displacement discontinuity across the interface does
not allow enough freedom to fit our data if either bulk velocities, or vN i,only and the best-fit vT a
are assumed.
Ultrathin films are also prone to large volume fractions of defects or soft intergrain areas
between small grains which will contribute to an apparent overall softening of the film material

139

Figure 6.14: Possible variation in Ta velocity | If the constant vN i fitted to the data is assumed,
one can solve for the vT a implied by each data point. As apparent from the error bars (obtainedby solving at the upper and lower bounds of vN i ), the thinner films trend upward, but the error
bars are always consistent with constant value fit to the whole data series (blue dashed line). All
are still significantly above the bulk Ta velocity (red dashed line). The error bars also increase
drastically with thinner Ta as the layer has less effect on the measurement; indeed, below 3 nm theeffect is too low to calculate any solution.

140[156, 187, 198, 215]. Stiffening is less often observed, but it has typically been attributed to straininduced changes in third-order elasticity [156, 197, 198, 216]. In particular, Ogi et al. [197] observea correlation between measured out-of-plane strain and the stiffening of Pt and Fe films. However,to our knowledge, changes of the magnitude exhibited by our Ta layers have not been previously
observed in metallic films.
It is likely that many of these effects contribute to the changes in elastic properties that
we observe. Further characterization using multiple techniques to analyze film microstructure (aswith transmission electron microscopy) and lattice strain (as with x-ray diffraction), as well as
exploration of how fabrication techniques (for example, sputtering vs. ALD) and conditions affectthe elasticity changes, could illuminate more of the specific mechanisms involved.
Exciting higher-order LAW resonances in the bilayers could also reveal more about the importance of changes localized around interfaces [156]. The symmetric mode with anti-nodes at thesurface and substrate interface of our bilayers incorporates little displacement near the interface
between the ultrathin layers, which is close to the node of the resonant mode. In contrast, thesecond-order mode would see an anti-node in the center of the bilayer, and could therefore be moresensitive to elastic properties at the middle interface (particularly when the two ultrathin layers
have similar thickness). In our previous work, we introduced a Michelson interferometer into thepump beam line to split the pump into two pulses separated by a variable time delay (introduced
by a variable optical path length in one arm of the interferometer) [41]. There we demonstrated
that the time delay between the two pump pulses could be set to selectively suppress or enhancedifferent SAW orders (see Fig. 6.15). For example, by setting the delay time equal to half theperiod of the fundamental SAW, the fundamental oscillation is completely suppressed while thesecond-order mode is reinforced and slightly enhanced. This same principle could be applied toexcite the second-order LAW resonance of nanostructures.

141

FFT
1st order

4
0
2
-0.2
0

0 phase shift
100

200

300

400

0

0

20

40

60

2

0.1

80 100FFT
2nd order

0

1

-0.1
0

π phase shift
100

200

300

Delay time (ps)

400

0

0

20

40

60

Spectral power

Change in diffraction (arb. units)

6

Spectral power

1D Ni grating
P = 220 nm

0.2

80 100Frequency (GHz)

Figure 6.15: Double pump for selective acoustic excitation | Two pump pulses can be used
to selectively excite specific acoustic modes by setting the delay time between them to suppressor enhance particular frequencies, represented schematically on the left. When the two pumppulses (red arrows) excite the system at the same time (top), the two excitations (blue and purple)
reinforce each other and the fundamental SAW mode is primarily excited; when they are offset bya time delay equal to half the fundamental SAW oscillation period (π phase) (bottom) the secondorder mode is reinforced while the fundamental is suppressed (visible in the dotted line, whichshows the lowest-order mode that can be excited in this case). This effect is clearly observed in
the experimental signals from the same sample at two pump offsets and Fourier transforms of thesignals shown on the right. This principle could also be applied to selectively excite higher-order
LAW resonances. Figure adapted from [41].

142

6.5

Conclusions and outlook
In this chapter I have demonstrated the first application of the EUV-based SAW mass sensor

proposed by Nardi et al. [179] to show that the density ratio of Ta to Ni in sub-10nm films is
not significantly changed from that expected from bulk materials. However, the LAW resonancesof the bilayer reveal that elastic properties are significantly modified. In particular, we observethat within the same bilayer, the ultrathin films of Ni and Ta soften and stiffen relative to theirbulk counterparts, respectively. Moreover, the presence or absence of the Ta capping layer further
influences the effective properties of the Ni layer. The precision of this measurement was limited
primarily by the uncertainty in layer thickness.
On one hand the sensitivity to atomic-monolayer differences in thickness poses challenges
for precise characterization of mechanical properties. Other film deposition processes, like atomic
layer deposition with its carefully controlled layers, will allow us to address this issue in future
experiments, as well as to study how different interface types and qualities may further affect theobserved elastic properties. Importantly, this metrology technique can be applied as easily to widefilm-like structures as to narrow nanowires or nanodots, enabling unprecedented metrology forsmall-volume structures that are inaccessible to many other techniques.
On the other hand, the strong sensitivity of the acoustic resonances to the size and elasticityof the structures will allow mapping of variation in height, for example, across many nanostructures,
particularly with the possibility of extending this technique to localized measurements with dynamic
coherent diffractive imaging. Such non-destructive measurements would enable very strong tests forthe repeatability of fabrication techniques and for understanding how various processes can affectthe morphology and structure of materials at the nanoscale.

Chapter 7

Conclusions and Future Opportunities

By directly observing the thermal and acoustic dynamics of nanoscale systems with coherent
extreme ultraviolet light, this thesis demonstrates new insight into how and why material behavior
deviates from that observed in bulk systems. It represents the distillation, reduction, analysis,
interpretation and understanding of around 20,000 files of raw data collected over more than 300
days/nights, accumulating well over 3000 hours of camera exposure time collecting more than 43
billion diffracted EUV pulses to characterize a variety of nanostructured samples. To summarize
the primary results presented:• Observations of heat dissipation from the smallest nanoscale heat sources measured todate (≈ 20 nm) have revealed a new regime of nanoscale thermal transport that dominates
when the separation between nanoscale heat sources is comparable to the dominant phononmean free paths [58, 59]. In this efficient collectively-diffusive regime of thermal transport,
closely-spaced nanoscale heat sources cool faster than isolated sources of the same size,
potentially mitigating projected problems for thermal management in nano-electronics,
where the power density is likely to increase as the individual nanostructures shrink in size
[6, 60]. It also highlights important design implications for nanostructured materials anddevices for energy and biomedical applications.
• The collectively diffusive regime enabled a new approach for characterizing the relative
contributions of phonons with different mean free paths to total heat conduction [217].

144
In particular, it gave access to the small-MFP region below ≈ 300 nm which has beenpreviously inaccessible to experiment. This unique capability is important as the need for
precise knowledge of phonon MFP distributions in complex nanostructures becomes morepressing – for both fundamental understanding and to harness systems where modelingdoes not yet exist.
• The development and application of EUV nanometrology to a broad range of stiff and softultrathin films, with thickness as small as 50 nm, was demonstrated. In particular, theselective layer sensitivity of the short-wavelength surface acoustic waves observable with
EUV probes was shown to isolate thin film properties from those of the substrate [34, 35].
Improvements in measurement precision were obtained through use of the chirp z-transform(rather than the Fourier transform) to extract the acoustic oscillation frequencies observed.
I also demonstrated the feasibility of extending these techniques to anisotropic materials.
• In extending EUV nanometrology to nanostructured bilayers with metallic layers below 10
nm in thickness, this work showed how the individual layer mechanical properties within
the bilayer can be assessed. It further demonstrated sensitivity to shifts in acoustic resonances corresponding to single-monolayer changes in layer thickness, particularly throughthe introduction of a second frequency extraction technique employing the Matrix Pencil
Method. Moreover, it reported the first application of an EUV-based mass sensor proposed by [179] and revealed that the ratio of material densities within the ultrathin layersof Ni and Ta did not deviate significantly from its bulk counterpart. The elastic properties, on the other hand, were observed to deviate significantly and with opposing trends[217]. These observations represent an important step toward a deeper understanding of
how elastic properties of materials build up layer by layer, as well as a demonstrationof how EUV nanometrology enables the characterization of mechanical properties within
nanostructures.

145
The outlook is very positive for using coherent short-wavelength light to study nanoscale systems.
Since bright soft x-ray high harmonic sources can now reach wavelengths below 1 nm [26], thesestudies can be extended even further into the deep nano-regime. Moreover, their sensitivity to awide variety of dynamics, including heat transfer, acoustic waves, magnetic states, surface electronic
states, plasmon dynamics and more, creates opportunity for unique studies of many physical properties within a single nanosystem, as well as examination of coupled dynamics and element-specificmeasurements. nanoelectronics

7.1

Ongoing efforts
By implementing significant updates to our experimental setup, we will soon improve oursensitivity to smaller nanostructures by shortening the EUV propagation distance between thesamples and CCD. This will enable the collection of light at wider diffraction angles and allow
extensions of all the work discussed in this thesis to even smaller size scales to further explore
thermal transport and elastic properties in the deep nano-regime. In particular, the improved sensitivity to smaller grating periods and shorter-wavelength SAWs (with their correspondingly shortpenetration depths) will improve the study of thinner and softer films where we can systematicallyexamine the effects of thickness on elastic properties and trends in the Poisson’s ratio of very soft
dielectric materials.
Thickness-dependent elasticity will also be better explored through the use of films grownthrough atomic-layer deposition techniques. Here the sample thickness is more precisely controlledthan in the sputtering deposition process employed in Chap. 6, so that the relationship between
resonant frequencies and acoustic velocities and elastic properties will be more precisely determined.
The interfaces between materials will also likely be more ideal.
The new setup will also enable a visible-wavelength transient grating system to extend ourcharacterization capabilities for flat surfaces. In the immediate future, this will enable the studyof phonon-engineered samples where tall nanostructures coupled to a thin membrane modify thesystem’s phonon dispersion and thus thermal conductivity [129].

146
Further theoretical developments are also under way, particularly focused on the extension
from our existing focus on nanoscale thermal transport from 1D-confined nanowire heat sources
to 2D-confined nanodot heat sources and improving understanding of the connections between
effective thermal conductivity, effective thermal boundary resistivity, and the fundamental phonon
dynamics they seek to capture. Understanding the extension from 1D- to 2D-confined heat sources
should also lay a strong foundation for extending to 3D-confined heat sources, like those relevant
to nanoparticle-mediated thermal therapies in medicine [45], 3D devices encapsulated by othermaterials, or controlled thermal annealing through the use of embedded nanoparticles [46, 47].
The first successful demonstrations of combining pump-probe observations of dynamics withcoherent diffractive imaging will launch exciting new opportunities to examine localized dynamics,
such as the elastic responses of single nanostructures and the heat dissipation away from nanoscaleheat sources at all points across the surface nearby.

7.2

Future opportunities
Further opportunities for nanoscience with coherent EUV from high harmonic generation are

numerous, and a few have already been discussed in the previous chapters. In particular, dynamic
imaging could enable more exploration of complex anisotropic materials or surface defects with its
introduction of high-resolution (in both space and time) sensitivity to dynamics. It could also better
follow lateral heat flow and thermal transport from arbitrary and non-periodic structures. Thesestudies could also be aided by the development of transient grating (TG) experiments employing
VUV or EUV excitation where a uniform surface could be excited at much shorter length scales
than ever accessible through visible-light TG.
Improving sensitivity to smaller nanostructures will enable study of more fundamental questions about how elastic properties arise from the interactions between atoms or observations ofquantized thermal transport. It will also reveal more detail of phonon mean free path spectra andof the various mechanisms that affect them in nanoscale materials. Combining these studies with

147
those of magnetic systems could allow the observation of magneto-elastic coupling dynamics where
the magnetization of a system can be driven by acoustic waves.
Benefiting from the nanofabrication capabilities that can reliably manufacture nanoscale
objects for systematic investigation, these studies will uncover the nanoscale physics that is essential
for future technological development and a more fundamental understanding of materials and the
dynamics within them.

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Appendix A

MATLAB program for frequency peak-finding using chirp z-transform

This program was used to extract the SAW frequencies for both thin film samples in Chap.
5 and Ni/Ta bilayer samples in Chap. 6. It uses a chirp z-transform to calculate the frequency
spectrum of the signal, and then fits the peaks to a Gaussian to extract the central frequency. This
procedure generally improved frequency extraction precision by approximately 10× compared withlooking to Fourier transform peaks.

f u n c t i o n freqs = fSAW ( dataFile , fMin , fMax , numFreqs , fGuess )
%
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=========================================================================
The function unwraps the loops which make up a given data scan ,separately calculates the chirp z - transform and fits the frequency peaks
to a Gaussian to extract the central frequencies
Inputs : dataFile = string giving location of data file - a two - column. dat with [ time ( ps ) , signal ] and time - zero actually at 0 ps
fMin , fMax = min and max of frequency range for CZT , in GHz
numFreqs = number of frequency peaks to look for ( in descending
amplitude order )fGuess = frequency in GHz , initial guesses for CZT peak fit([ f1 , f2 ,...] , as many as requested by numFreqs )Output : freqs = matrix giving the central frequenciesof the peaks
Dependencies : Makes use of versions of chirpZTrans , andft_pad functions included in this file
Written by : Kathy Hoogeboom - Pot , 7/3/13=========================================================================% Import data
data = load ( dataFile );% Chirp z - transform of signal and plot

165
f i g u r e (11);czt = chirpZTrans ( data , fMin , fMax );% For comparison , look also at Fourier transform ( Can% change second ft_pad input to add zero - padding .)f i g u r e (13); hold on ;ft = ft_pad ( data ,0);p l o t ( ft (: ,1) , ft (: ,2) , ’ linewidth ’ ,2);% Fit frequency peaks to find centers% Initialize output frequency matrix
freqs = z e r o s (1 , numFreqs );% Find peaks to fit[ pks , pkLocs ] = findpeaks ( czt (: ,2));[~ , trLocs ] = findpeaks ( - czt (: ,2));peaksMat = [ pkLocs , pks ];sortPeaksMat = sortrows ( peaksMat , -2);
f o r idxFreq =1: numFreqs% Find bottom on either side of peak to define range for fit
troughBefore = trLocs ( f i n d ( trLocs < sortPeaksMat ( idxFreq ,1) , 1 , ’ last ’ ));troughAfter = trLocs ( f i n d ( trLocs > sortPeaksMat ( idxFreq ,1) , 1 , ’ first ’ ));fitRange = czt ( troughBefore : troughAfter ,:);%%%

% optionally uncomment to manually set fitRange and comment out prev
% three lines
fitRange = czt ( sortPeaksMat ( idxFreq ,1) -60: sortPeaksMat ( idxFreq ,1)+60 ,:);
% See range highlight on graphf i g u r e (11); hold on ;p l o t ( fitRange (: ,1) , fitRange (: ,2)/max( czt (: ,2)) , ’+ ’)hold off ;% Fit to find central frequencies
gaussian = f i t t y p e ( @ (A , f0 ,b , x ) A * exp ( -(x - f0 ).^2/ b ) );curvefit = f i t ( fitRange (: ,1) , fitRange (: ,2)./max( fitRange (: ,2)) , . . .
gaussian , ’ StartPoint ’ ,[1 , fGuess ( idxFreq ) ,1]);
f i g u r e (12); hold on ; t i t l e ( ’ Fit frequency peaks ’ );p l o t ( curvefit , fitRange (: ,1) , fitRange (: ,2)./max( fitRange (: ,2)));legend off ;
coeffvals = c o e f f v a l u e s ( curvefit );freqs ( idxFreq ) = coeffvals (2);end % stepping through frequency peaks
hold off

166

end % of function fSAW
% =======================================
f u n c t i o n transform = chirpZTrans ( data , fMin , fMax )
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takes the chirp Z - transform of data - evenly spaced in the frequency
range specified by fMin and fMax below
Input : data = [ time ( in ps ) , signal ] , with t0 = 0 ps
fMin , fMax = min and max of frequency range to explore , in GHz
Output : transform = [ freq ( in GHz ) , amplitude ]Written by : Kathy Hoogeboom - Pot , 3/4/13================================================

time = data (: ,1);
idx = f i n d ( time >= 0);time = time ( idx );signal = data ( idx (20: end) ,2);dt = mean( d i f f ( time )) * 1e -12; % signal time step
L = numel( signal ); % length of signal ( after t0 )maxFreq = 1/ dt ;f1 = fMin *1 e9 ; f2 = fMax *1 e9 ; % frequency range of interest in Hz
m = 2^nextpow2( L );w = exp ( -1 i *2* pi *( f2 - f1 )/( m * maxFreq ));a = exp (1 i *2* pi * f1 / maxFreq );winSig = signal .* hamming( L ); % Hamming Window% Find transform and associated frequency scale in GHz
z = czt ( winSig ,m ,w , a );freqs = (((0: length ( z ) -1) ’*( f2 - f1 ) / length ( z )) + f1 ) / 1 e9 ;transform = [ freqs abs ( z )];% plot resultp l o t ( freqs , abs ( z )./max( abs ( z )) , ’ lineWidth ’ , 2); a x i s tightset ( gca , ’ yLim ’ ,[0 ,1.1]);
x l a b e l ( ’ Frequency ( GHz ) ’)y l a b e l ( ’ Normalized CZT amplitude ’)hold onend % of chirpZTrans function167% ==========================================
f u n c t i o n Data = ft_pad ( data , padMult )
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Takes Fourier transform with optional zero - padding
Inputs : data = [ time , signal ] , assumes t0 is at zero
padMult = integer giving how many times the length of theoriginal signal should be added in zero - pad ; 0 means no padding
Outputs : Data = [ freq scale , FT amplitudes ] , only positive freqs , in GHz
Written by : Kathy Hoogeboom - Pot , 3/1/2013================================================

time = data (: ,1);
sig = data (: ,2);
N = numel( sig );M = ( padMult + 1) * N ;winSig = sig .* hamming( length ( sig ));sigPad = padarray ( winSig ,[ f l o o r ( M /2) - f l o o r ( N /2) ,0]);SigPad = f f t s h i f t ( f f t ( i f f t s h i f t ( sigPad )));
dt = mean( d i f f ( time ));wMax = 1/ dt ;dw = wMax / M ;w = - wMax /2: dw : wMax /2 - dw ;idx = f i n d (w >0);
Data = [1000 * w ( idx ) ’ abs ( SigPad ( idx ))];
end % of ft_Pad function

Appendix B

Mathematica code for the Matrix Pencil Method

This code uses the Matrix Pencil Method to extract the complex exponential components
of any signal with evenly-spaced sampling. Specifically it can extract the decay times of bothoscillating and pure exponentials, oscillation frequencies and starting phase for a specified numberof signal components, ignoring the noise. It was used to extract the resonant LAW periods in the
Ni/Ta bilayer studied in Chap. 6 of this thesis. For best results, irrelevant data components —especially any which are not described well by complex exponentials, like the transition from before
to after time-zero — should be removed before applying this algorithm. This code comes with great
thanks to Matthew Reynolds, who taught us this method and built the framework of this program.

B.1

Set up special functions and modules(*
This function plots the locations of the exponentials ;inputs ;z - list of exponentials ;pr - range of the plot , such as {{ -1 ,1} ,{ -1 ,1}};
co - color of the points ;outputs ;A plot of the exponentials*)
ArgandPlot [ z_ , pr_ , co_ ] :=Show [ Graphics [{ PointSize [0.015] , co ,Point / @ Transpose [{ Re [ z ] , Im [ z ]}]} , Axes -> Automatic ,AspectRatio -> 1 , AxesLabel -> { " Re [ z ] " , " Im [ z ] " }] ,PlotRange -> pr ]

169(*
This module takes an input sequence of length 2 N +1 , forms a( N +1)×( N +1) Hankel matrix , and takes the SVD of that Hankel matrix ;inputs ;input data - 2 N +1 samples ;outputs ;U ,σ ,V - SVD , where σ is a list of the singular values ;H - Hankel matrix*)
hanksvd [ inputdata_ ] :=Module [{ data = inputdata , num , U , σ , V , H , i , j ,singularvalues } ,(* Form the Hankel Matrix *)num = ( Length [ data ] - 1)/2;
H = Table [ data [[ i + j + 1]] , {i , 0 , num } , {j , 0 , num }];(* SVD *){U , σ , V } = S i n g u l a r V a l u e D e c o m p o s i t i o n [ H ];singularvalues = Diagonal [σ ];Return [{ U , singularvalues , V , H }]](*This module implements the matrix pencil method . It modifies theappropriate singular vectors , forms the matrix whose eigenvalues are
the exponentials you want , and finds those exponentials ;inputs ;U0 - singular vectors ;ind - index of the appropriate singular value ;outputs ;zz1 - Output of matrix pencil method ;*)(* Create a function to find nodes via the matrix pencil method *)findnodesMPM [ U0_ , index_ ] :=Module [{ Upp , Up1 , Up2 , vmatrix , zz1 } ,(* Construct the pencil matrix and find the eigenvalues *)(* modify the vector V from the SVD *)Upp = U0 [[ All , 1 ;; index ]];Up1 = Upp [[;; -2]];
Up2 = Upp [[2 ;;]];
vmatrix = PseudoInverse [ Up1 ]. Up2 ;(* Find the eigenvalues *)zz1 = Eigenvalues [ vmatrix ];Return [ zz1 ];
]

170

B.2

Import data and extract signal components(* Import the data , etc *)data0 = Import [" FilePath \\ FileName . extension " ]; (* 2 - column matrix , [ time , signal ] *)data = data0 [[ All , 2]]; (* We ’ ll only use the signal column *)(* The spacing of the time samples will be needed to convert
back to real time units *)dt = Mean [ Differences [ data0 [[ All ,1]]]];(* See what we have *)Show [ ListPlot [ data , Joined -> True ] ,ListPlot [ data , PlotStyle -> Hue [1]] , PlotRange -> All ](* Select the index of the starting point *)(* Note that it is important have an odd number of samples tocorrectly form the Hankel matrix !!! so Length [ hankelData0 ] must be
odd *)
startingPoint = < number after time - zero >; (* without the brackets *)hankelData0 = data [[ startingPoint ;;]];
Length [ hankelData0 ](* Optionally fit and subtract out a 2 nd - order polynomial -- as toremove the slow rising background in LAW data *)trendFit [ x_ ] = Fit [ hankelData0 , {1 , x , x ^2} , x ];hankelData =Table [ hankelData0 [[ j ]] - trendFit [ j ] , {j , 1 , Length [ hankelData0 ]}];Show [ ListPlot [ hankelData , Joined -> True ] ,ListPlot [ hankelData , PlotStyle -> Hue [1]] , PlotRange -> All ](* Use this line instead to skip background fitting *)(* hankelData = hankelData0 ; *)(* Form the Hankel matrix H and take the SVD *){U , σ , V , H } = hanksvd [ hankelData ];(* Plot the singular values ( normalized ) and select the index
corresponding to the change in behavior . *)ListPlot [ Log [10 , σ/σ [[1]]] , PlotRange -> All ](* Enter that index and find the nodes using the rootfinding algorithm *)(* Note that the pencil method outputs a number of exponentials = ind ;Expect 2* number of oscillating components + number of decaying components *)ind = < number of SVs corresponding to signal >;nodesIn = findnodesMPM [U , ind ];(* Create a vandermonde system to solve for the weights *)(* Note that since the pencil method outputs the number of nodes equal
to the index , there is no need to solve the Vandermonde system twice *)vand ermond eMatr ix =Table [ nodesIn [[ j ]]^ i , {i , 1 , Length [ hankelData ]} , {j , 1 ,Length [ nodesIn ]}];
weights = LeastSquares [ vandermondeMatrix , hankelData ];171(* Plot the exponentials with respect to the unit circle *)cir = Graphics [ Circle []];Show [ ArgandPlot [ nodesIn , {{ -1 , 1} , { -1 , 1}} , Black ] , cir ](* Select the strictly decaying exponentials and corresponding weights *)indexRe0 = Table [i , {i , 1 , Length [ nodesIn ]}];
indexRe = Select [ indexRe0 , Abs [ Im [ nodesIn [[#]]]] < 10^ -10 &];nodesRe = nodesIn [[ indexRe ]];weightsRe = weights [[ indexRe ]];(* Select the oscillating exponentials and corresponding weights *)indexIm0 = Table [i , {i , 1 , Length [ nodesIn ]}];
indexIm = Select [ indexIm0 , Abs [ Im [ nodesIn [[#]]]] >= 10^ -10 &];nodesIm = nodesIn [[ indexIm ]];weightsIm = weights [[ indexIm ]];(* Check the results . Here I look atof the output approximation *)result = van dermon deMatr ix . weights
decayingResult =Table [ Sum [weightsRe [[ j ]]* nodesRe [[ j ]]^ i ,Length [ hankelData ]}];
osci llatin gResu lt =Table [ Sum [weightsIm [[ j ]]* nodesIm [[ j ]]^ i ,Length [ hankelData ]}];
ListPlot [ decayingResult , Joined ->ListPlot [ oscillatingResult , Joined

the decaying and oscillating parts// Re ;

{j , 1 , Length [ nodesRe ]}] , {i , 1 ,{j , 1 , Length [ nodesIm ]}] , {i , 1 ,True , PlotRange -> All ]-> True , PlotRange -> All ](* Compare the output of the algorithm with the original data andcheck remaining residual *)Show [ ListPlot [ result , Joined -> True ] ,ListPlot [ hankelData , PlotStyle -> Hue [1] , Joined -> True ] ,PlotRange -> All ]ListPlot [ hankelData - result , Joined -> True ](* Convert back to time decay rates and periods *)aRe = Log [ Abs [ nodesRe ]];bIm = Arg [ nodesIm ];aIm = Log [ Abs [ nodesIm ]];(* periods and decay times in whatever units the timesteps were recorded *)P = Table [2* Pi * dt / bIm [[ i ]] , {i , 1 , Length [ bIm ]}]oscDecay = Table [ dt / aIm [[ i ]] , {i , 1 , Length [ aIm ]}]decay = Table [ dt / aRe [[ i ]] , {i , 1 , Length [ aRe ]}]Appendix C

MATLAB program for least-squares fitting with the acoustic transfer matrix

This program was used to find the best-fit velocities for Ni and Ta layers in an ultrathinbilayer from their resonant LAW oscillation periods in Chap. 6. It requires a range of velocities totest and the densities to assume (though only the ratio is actually used) for the two materials. Thenit fits a polynomial to period vs. layer thickness data, and tests every pair of material velocities
within the specified ranges to find the set of periods (calculated according to Eqn. 6.6) that havethe smallest total squared difference to the polynomial fit representing the data. This procedure
can optionally be executed in multiple sections, finding best-fit velocities for each subsection of thedata.

f u n c t i o n [ vFitTop , vFitBottom ] = transMatFit ( data , props , vRangeTop , . . .
vRangeBottom , res )
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transMatFit works to fit frequency vs . bilayer layer thickness ( as forseries of LAW samples ) with optimal vLAW values for the two materials
using the transfer matrix method of calculating resonant frequencies .
Assumes one ( ’ bottom ’) layer is of fixed thickness .
Inputs : data = [ top layer thickness in nm , LAW period in ps ]props = material properties , [ densityTop , densityBottom ,vInitTop , vInitBottom , thicknessBottom ]; densities must matchin units , v ’ s expected in m /s , thickness in nm
vRangeTop , Bottom = fraction (0 to 1) by which to vary initial
velocities to seek fit
res = number of points to test across each vRange
Outputs : vFitTop , vFitBottom = best - fit ( in least squares sense ) of twomaterial velocities from absolute minima in each section
Written by : Kathy Hoogeboom - Pot , 9/18/13

173% Modifications :%9/23/13 , KHP , add second iteration of optimization for vTop%in separate sections of the polynomial fit%
9/24/13 , KHP , try 2 D surface optimization in%sections rather than two iterations%
11/1/13 , KHP , add error bars to final plot and error plane to each 2 D%
surface% =========================================================================% Some setup
nm = 1e -9;
ps = 1e -12;pTop = props (1);pBottom = props (2);viTop = props (3);viBottom = props (4);tBottom = props (5)* nm ;tTop = data (: ,1)* nm ;wData = 2* pi ./( data (: ,2)* ps );w = sym ( ’w ’ );% Setup parameters you may wish to change :order = 2; % order of polynomial to fit to data
numPts = 80; % number of points in fit
sections = 1; % number of separate sections over which to optimize vTop ;% must divde into numPts without remainder
dataError = 1.1974 e10 ; % sqrt ( sum (( errors on wData points ).^2))
errorPoints = 9; % number of points included in the dataError calc .
errorCompare = dataError / errorPoints * numPts / sections ; % scale the average% data error to the number of points in% each vTa section to allow comparison on% plots of vTa residual curves
vTopTest = l i n s p a c e ( viTop - vRangeTop * viTop , viTop + vRangeTop * viTop , res );vBottomTest = l i n s p a c e ( viBottom - vRangeBottom * viBottom , . . .
viBottom + vRangeBottom * viBottom , res );% Fit a low - order polynomial to data points to enable more points at which% to test a fit plus smooth curve that leads to better result
tTopFit = l i n s p a c e (min( tTop ) ,max( tTop ) , numPts );[p ,S , mu ] = p o l y f i t ( tTop , wData , order );wFit = polyval (p , tTopFit ,S , mu ) ’;f i g u r e ; hold off ;p l o t ( tTop , wData , ’* ’ , tTopFit , wFit , ’ - ’ ); t i t l e ( ’ Poly fit to data ’ );x l a b e l ( ’ Top layer thickness ( m ) ’ ); y l a b e l ( ’ Angular frequency ’ );% -----------% % Optionally uncomment to use this instead of previous polynomical fit to% % fit to raw data
% tTopFit = tTop ;% wFit = wData ;% ------------

174% Break polynomial in sections (# specified at beginning )tSection = reshape ( tTopFit , length ( tTopFit )/ sections , sections );wFitSection = reshape ( wFit , length ( tTopFit )/ sections , sections );% -----------% Solve for w ( resonant frequency ) for the sweeps through vTop and vBottom% and calculate the sum of the squares of the differences between% calculated w and wFit
variance = z e r o s ( res , res , sections ); % initialize output matrix
vFitTop = z e r o s ( sections , 1); % initialize output v vectors
vFitBottom = z e r o s ( sections ,1);vBottomBounds = z e r o s ( sections , 2); % initialize output v error bar vectors
vTopBounds = z e r o s ( sections ,2);tSectionLength = length ( tSection (: ,1));
f o r idx =1: sections % for each section of polyfit
f o r ii =1: res % sweep vTop
f o r jj =1: res % sweep vBottom
wSimSection = z e r o s ( tSectionLength , sections ); % initialize the% simulated vectors of w ’ s which we ’ ll compare to dataf o r kk =1: tSectionLength % step through data point fit
wSimSection ( kk , idx ) = . . .
vpasolve ( tan ( w * tBottom / vBottomTest ( jj )) . . .
+ pTop * vTopTest ( ii )/( pBottom * vBottomTest ( jj )) . . .
* tan ( w * tSection ( kk , idx )/ vTopTest ( ii ))==0 , w ,1 e12 );end % kk step through data points% Calculate the summed squared difference between sim and data
variance ( ii , jj , idx ) = s q r t (sum(( wSimSection (: , idx ) - . . .
wFitSection (: , idx )).^2));
end % jj sweep of vBottom
ii % comment in / out to show / hide running counter
end % ii sweep of vTop
% Display resulting surface of variancef i g u r e ; s u r f ( vBottomTest , vTopTest , variance (: ,: , idx ));x l a b e l ( ’ vBottom ’ ); y l a b e l ( ’ vTop ’ ); z l a b e l ( ’ variance ’ );t i t l e ( strcat ([ ’ Section ’ num2str( idx ) ’: tTop = ’ . . .
num2str( tSection (1 , idx )/ nm ) ’ , ’ num2str( tSection (end, idx )/ nm )]));
hold on% Find v ’ s corresponding to minimum and plot the pointmini = min(min( variance (: ,: , idx )));[ locTop , locBottom ] = f i n d ( variance (: ,: , idx ) == mini );vFitTop ( idx ) = vTopTest ( locTop );vFitBottom ( idx ) = vBottomTest ( locBottom );175

plot3 ( vFitBottom ( idx ) , vFitTop ( idx ) , mini , ’r * ’ );% Add a plane representing the data error
errorPlane = errorCompare * ones ( length ( vTopTest ) , length ( vBottomTest ));mesh ( vBottomTest , vTopTest , errorPlane );% Find error bars on fit v ’ s based on the variance surface ’ s% intersection with error plane
testVariance = variance (: ,: , idx );testVariance ( testVariance > errorCompare ) = 0;i f nnz( testVariance ) > 0
locLower1 = f i n d ( testVariance ( locTop ,:) ,1 , ’ first ’ );locUpper1 = f i n d ( testVariance ( locTop ,:) ,1 , ’ last ’ );vBottomBounds ( idx ,1) = abs ( vFitBottom ( idx ) - vBottomTest ( locLower1 ));vBottomBounds ( idx ,2) = abs ( vFitBottom ( idx ) - vBottomTest ( locUpper1 ));locLower2 = f i n d ( testVariance (: , locBottom ) ,1 , ’ first ’ );locUpper2 = f i n d ( testVariance (: , locBottom ) ,1 , ’ last ’ );vTopBounds ( idx ,1) = abs ( vFitTop ( idx ) - vTopTest ( locLower2 ));vTopBounds ( idx ,2) = abs ( vFitTop ( idx ) - vTopTest ( locUpper2 ));else
vBottomBounds ( idx ,:) = [0 ,0];vTopBounds ( idx ,:) = [0 ,0];end
end % idx step through sections% Plot fit velocities according to thickness of top layer at low end of% section with the error bars found from plane intersections
figure ;errorbar ( tSection (1 ,:)/ nm , vFitTop , vTopBounds (: ,1) , vTopBounds (: ,2) , ’o ’ );hold on ;errorbar ( tSection (1 ,:)/ nm , vFitBottom , vBottomBounds (: ,1) , . . .
vBottomBounds (: ,2) , ’ ro ’ );x l a b e l ( ’ Top layer thickness starting each section ( nm ) ’ );y l a b e l ( ’ Fit velocities ( m / s ) ’ );t i t l e ( ’ What velocities appear optimum in each section ? ’ );legend ( ’ Top layer ’ , ’ Bottom layer ’ );
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Optionally display the resulting data fit when fitting in one section
wFinal = zeros ( length ( tTop ) ,1);w = sym ( ’w ’);for kk =1: length ( tTop ) % step through data point fit
wFinal ( kk ) = vpasolve ( tan ( w * tBottom / vFitBottom )...
+ pTop * vFitTop /( pBottom * vFitBottom )...
* tan ( w * tTop ( kk )/ vFitTop )==0 , w ,1 e12 );end % kk step through data points
figure (10); hold on ;plot ( tTop , wFinal , ’o ’);legend ( ’ Data ’ , ’ PolyFit ’ , ’ OptFit ’);