Extreme Ultraviolet Measurements of Thermal and Elastic
Dynamics in Nanostructured Mediaby
Travis Daniel Frazer
B.S., California Polytechnic State University San Luis Obispo, 2013
M.S., University of Colorado Boulder, 2016

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
2019This thesis entitled:
Extreme Ultraviolet Measurements of Thermal and Elastic Dynamics in Nanostructured Media
written by Travis Daniel Frazer
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 the content andthe form meet acceptable presentation standards of scholarly work in the above mentioned discipline.

iii
Frazer, Travis Daniel (Ph.D., Physics)
Extreme Ultraviolet Measurements of Thermal and Elastic Dynamics in Nanostructured Media
Thesis directed by Prof. Margaret M. Murnane and Prof. Henry C. Kapteyn

Nanofabrication today spans from atomic level precision to hierarchically organized structures
reaching up to microns. Nanoscale material properties are profoundly different from their bulk counterparts,
and when combined with metamaterial approaches, systems can be engineered with properties unavailable
in naturally occurring substances. These effects have applications from nanoelectronics, to thermoelectrics,
to nanoparticle-based cancer therapies. However, the full capabilities of these materials have not yet beenrealized due to the difficulty of studying functional nanosystems. In this thesis, I study the properties ofnanoscale materials at their intrinsic length and time scales, via the diffraction of extreme ultraviolet (EUV)
beams, generated coherently using tabletop high harmonic generation. First, using periodic nanoline
gratings, I systematically explore size- and spacing-dependence in the ultrafast cooling of nanoscale heatsources. We find that, though nanoscale heat sources generally cool much slower than the bulk diffusive
prediction, they can be brought within a factor of two of the efficient, diffusive prediction simply bybringing them closer together. I then use similar nanoline transducers to study nanoscale acoustic waves,and thus extract the elastic tensor of isotropic, amorphous films, down to 11nm thickness. We find thathydrogenating these films to a critical level of broken bonds causes a divergence towards incompressible
behavior, which could also mitigate thickness-dependent changes in the films’ mechanical properties. I next
demonstrate our technique can measure thermal and elastic dynamics in 3D silicon metalattices, a promisingthermoelectric material. To measure even more general samples, I finally extend this EUV nanometrology
technique into a non-contact modality. First, I implement an optical transient grating excitation to study
micron-scale thermal transport in 2D nanoparticle-molecular arrays. We measure an effective thermal
conductivity for the arrays that is three orders of magnitude lower than bulk gold. Second, I participate in
EUV transient grating experiments at the FERMI free electron laser to directly excite deep nanoscale
thermal and elastic dynamics, and design a similar experiment for tabletop EUV light sources.

iv

Dedication

To my family, to Bianca
To Mr. Koop, to Kat
To those who came before, to those who come behind
To those who aren’t here now, to those who are always there“The mind that is not baffled is not employed.”-Wendell Berry, from “The Real Work”

v

Acknowledgements

If I have learned one thing over the course of this degree, it’s that science is a collaborative effort.
I am deeply grateful to so many people over the last six years – I will acknowledge all that I can, but thelist will be incomplete. Margaret and Henry, thank you for your years of leadership, guidance, andmentoring, as I came from not even being sure what grad school entailed to finally completing my Ph.D.
Kathy, Damiano, and Nico, thank you for teaching me about the lab, grad life, and starting me on a great
project. Nico, Josh, and Begoña, thank you for being my team through it all. I’m so grateful that you were
not only exceptional to work with, but you were a team that supported balance and health, and I can callyou my friends on the other side of all the hard work. Josh and Brendan, I’m confident that you will haveamazing theses, and you will continue to do awesome science in B1B19 and beyond. Thank you to the restof the KM group plus Daisy and Kevin for all your assistance, particularly with our old laser! Thank you
to all of our collaborators, particularly the FERMI team for an incredible experience working and learningin Italy. I also thank for their patient instruction Xavier Alvarez, Dan Adams, Alex Maznev, Hossein
Honarvar, and Mahmoud Hussein, each of whom has taught me a great deal. Thank you to the Physics and
JILA staff, especially Jeanne, Kim, Kyle, Hans, Felix, Terry, J.R., Dave, Mark, Julie, Steven, Jen, Brian,
Lois, Lauren, Kim, Agnieszka, Beth, Cindy and all the custodial and administrative staff that keep JILA
running. Finally, I thank the friends and family who have supported me throughout my time in Boulder.
Emily, Scott, Will, and Adam, thank you for your guidance as I navigated all the changes and trials of grad
school. Amy, Adam, Torrey, and Houston, thanks for the food in the desperate final push! Mom and Dad,
thank you for always supporting me with calls, visits, prayers and cookies. Bianca, thank you for being mypartner closer than anyone else on this journey and for keeping me human, even when it got overwhelming.
I’m so grateful we supported each other on the final mile, and I’m glad you’ll be there for miles to come.

Contents
Chapter
1

Introduction

1

2

Experimental Methodology

9

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9Optical Pump – EUV Probe Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122.2.1 High Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2.2 Diffraction Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.2

2.3 Upgrades to the Experiment

2.4
3

23

2.3.1 Optical Transient Grating Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.3.2 Synchronized Chopper as a Pump Shutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.3.3 B-Integral and Chirped Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Conclusions and Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Nanoscale Thermal Transport

45

3.1

Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.2

Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.2.1 Nanoline Grating Design and Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.2.2 Probing and Measurement Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.2.3 Finite Element Analysis to Fit the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.3.1 Collectively Diffusive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.3.2 Silicon and Fused Silica Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.3

vii

3.4
4

Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68
69

4.1

Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.2

Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.2.1 Sample Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.2.2 Data Processing and Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

4.2.3 Fitting and Modeling Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.3.1 Hydrogenation-dependent Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.3.2 Thickness-dependent Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100

4.4

Extending Dynamic EUV Metrology

102

5.1

Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

5.2

Silicon Metalattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

104

5.3 Optical Transient Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

5.3.1 Data Processing and Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

5.3.2 Thermal Transport in Novel Nanosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

Extreme Ultraviolet Transient Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124

5.4.1 EUV TG with an Optical Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127

5.4.2 EUV TG with an EUV Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132

5.4.3 EUV TG on a Tabletop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

5.4

5.5
6

64

Thin Film Elastic Properties

4.3

5

3.3.3 Comparison to Other Experiment and Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions and Future Work

144

Bibliography

148

Appendix

160

viii

TablesTable
2.1

Delay box and chopper settings for specific pulse exposures . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.1

Linewidth and period of each nickel grating on silicon, measured by AFM . . . . . . . . . . . . . .

51

3.2

Linewidth and period of each nickel grating on fused silica, measured by AFM . . . . . . . . . .

51

3.3

Pump beam average power, spot size at the sample plane, and average fluence . . . . . . . . . . .

54

3.4

Parameters used in FEA multiphysics simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.1

Hydrogenation-dependent thin films nominal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.2 Thickness-dependent thin film nominal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4.3

Nanoline grating dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

5.1 Volume fraction of constituent elements in nanoparticle arrays and nanomeshes . . . . . . . . . .

119

5.2

Bulk material properties used in eigenfunction calculations . . . . . . . . . . . . . . . . . . . . . . . . . .

119

5.3

Best fit conductivities for nanoparticle molecular arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123

5.4

Example signal strength reductions for non-Bragg probing . . . . . . . . . . . . . . . . . . . . . . . . . . .

137

Figures

Figure1.1

Colloidal suspensions of gold nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2 Transistor number per chip and clock speeds since 1970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Structural metamaterials with nanoscale effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1

Example of pump probe measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2

Block diagram of the experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.3

Nanograting and optical transient grating excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.4

High harmonic generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.5

Dynamic diffraction measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.6

Experimentally measured diffraction signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

2.7 The modular optical TG setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.8

New vacuum chamber for TG and nanograting measurements . . . . . . . . . . . . . . . . . . . . . . . .

28

2.9

Optical chopper as a beam shutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.10 Timing schematic of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.11 Comparator circuit with voltage dividers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.12 Diagram of the analog to TTL converter circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.13 The one-shot K parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.14 Function of the analog to TTL converter circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.15 Chirped mirrors and vacuum redesign to reduce B integral . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

x
3.1 The EUV nanometrology laser setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.2

Diagram of the pump-probe technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.3

SEM images of the fused silica sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.4

Dynamically changing diffraction signals for 20nm linewidth gratings on silicon . . . . . . . . .

53

3.5

Measurement sensitivity to nanostructure and substrate temperatures . . . . . . . . . . . . . . . . . . .

56

3.6

Measurement sensitivity to the substrate coefficient of thermal expansion . . . . . . . . . . . . . . .

57

3.7

Best fit effective thermal boundary resistivity for nanolines on silicon and fused silica . . . . .

61

3.8

Comparison to previous measurements on silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.9

Results for the constant 1500nm period series of gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

3.10 MD temperature profiles compared to the diffusive prediction . . . . . . . . . . . . . . . . . . . . . . . .

67

4.1

Low-k dielectrics for metal interconnects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

4.2 Thin film sample design and characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.3

EUV diffraction measurement of thin film acoustic waves . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

4.4

SAW confinement for the thickness-dependent study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.5 The matrix pencil method for frequency extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

4.6

11nm Film (#155) fLAW and carbon deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.7

fLAWs for films 148 and 145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.8

fLAWs for film 503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.9

Measuring the two independent elastic constants with acoustic waves . . . . . . . . . . . . . . . . . .

87

4.10 FEA calculations of thermoelastic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

4.11 Young’s modulus and Poisson’s ratio for the hydrogenation-dependent study . . . . . . . . . . . .

92

4.12 Film 145 (44nm thick) elastic constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

4.13 Film 155 (11nm thick) elastic constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

4.14 Elastic properties of varying-thickness SiOC:H films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

4.15 TEM measurements of film 503 – double deposited 5nm SiC:H films . . . . . . . . . . . . . . . . . .

97

4.16 HAADF and EDS measurements of film 503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

xi
4.17 Elastic characterizations of SiC:H films 335 and 503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.1

Fabrication of silicon metalattices from silica templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105

5.2

Measurements of 30nm gratings on a 30nm metalattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

106

5.3

Measured signals on larger gratings for the two 30nm metalattices . . . . . . . . . . . . . . . . . . . . .

107

5.4

Comparing diffraction from a 100nm linewidth, 400nm period grating . . . . . . . . . . . . . . . . . .

108

5.5

Surface acoustic wave dispersion in silicon metalattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109

5.6

MD calculations of thermal transport in porous silicon metalattices . . . . . . . . . . . . . . . . . . . .

110

5.7

Geometry for eigenfunction expansion calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

5.8

Graphical intersection method for the determination of eigenvalues . . . . . . . . . . . . . . . . . . . .

114

5.9

Nanoparticle molecular hybrid arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

5.10 Example of nanomesh geometry and diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

118

5.11 Optical microscope images of burned-in TG patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121

5.12 Identification of sample burning and subsequent calculation of TG period . . . . . . . . . . . . . . .

122

5.13 Best fits for the nanoparticle molecular arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124

5.14 Pulse front tilt and the 4f optical TG setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

5.15 The mini-TIMER setup for optically-probed EUV TG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127

5.16 Optically probed EUV TG signal on single crystal diamond . . . . . . . . . . . . . . . . . . . . . . . . . .

130

5.17 Optically probed EUV TG signal on BK7 glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

5.18 Optically probed EUV TG signal on Bi4Ge3O12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

5.19 Schematic of the TIMER experiment with EUV-probed EUV TG . . . . . . . . . . . . . . . . . . . . . .

133

5.20 EUV TG signal on a Si3N4 membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134

5.21 EUV TG signal on a Si membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134

5.22 Design for initial demonstration of EUV TG at JILA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

5.23 Thin and thick grating diffraction conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137

5.24 Calculated TG patterns and diffraction for single- and multi-frequency beams . . . . . . . . . . . .

139

5.25 Schematic of four-mirror design for all-EUV TG at JILA . . . . . . . . . . . . . . . . . . . . . . . . . . . .

142

Chapter 1
Introduction“We’re going to see things no one has ever seen before.” In the classic science fiction film,
Fantastic Voyage, the character Cora Peterson delivers this line before being shrunken to a sub-micron size
to explore inside the human body and remove a blood clot. While shrinking humans is completely fictional,
modern medicine is closer to a Fantastic Voyage than you might think – nanoparticles can be used to treat
tumors, precisely deliver drugs, and kill bacteria [1, 2]. Unlike in the movie, however, these nanoparticles
don’t behave like “miniaturized submarines”. When confined to the nanoscale, material properties
drastically change from those expected at the macroscale. This presents both opportunity and challenge tonanotechnology in general, where fabrication is now controlled one atomic layer at a time.
As an example of the opportunity given by nanoscale changes in materials, consider gold. When it
is scaled down into particles only a few nanometers across and suspended in water, gold loses its familiar
metallic sheen and instead looks bright red. As the nanoparticle diameter is increased, the solution’s color
changes from red to purple, as shown in Fig. 1.1. These optical effects come from surface plasmon
resonances, where specific optical wavelengths resonantly excite charge oscillations in each nanoparticle[3, 4]. By increasing only the nanoparticle diameter, the resonant wavelength correspondingly increases,
leaving a blue-shifted spectrum of scattered, non-absorbed light. This size-dependence introduces a newdegree of freedom to materials engineering – using nanoscale effects to produce useful materials properties
not present in bulk samples. Indeed, the use of gold nanoparticles extends all the way back to ancient Rome,
where artisans used colloidal nanoparticles to produce impressive glasses that appear green when viewed
from the outside, but red when illuminated from within [5].

2Figure 1.1: Colloidal suspensions of gold nanoparticles. As the nanoparticle diameter is increased from
5nm to 100nm, the scattered light from the suspension changes from red to purple. This is because thesurface plasmon resonant wavelength increases with increasing nanoparticle size. Thus, for the smallest
nanoparticles, blue wavelengths are absorbed from white light, while for the largest nanoparticles, redwavelengths are absorbed from white light. Image from nanoComposix.
As an example of the challenges presented by nanoscale materials, however, you need look nofurther than your computer. All of modern computing relies on complex patterned materials at the deepnanoscale, with the critical dimensions of the latest generation of transistors now reaching below 10nm.
Accounting for changes in material properties at these scales is critically important for the successful
production and operation of devices. Indeed, materials innovation is what is now needed for the continued
scaling of computing power.
The first materials challenge for the semiconductor industry is how to manage heat. When current
runs through wires, they heat up, and computer chips are essentially a densely packed nest of currentcarrying wires. To keep the heat at tolerable levels, the industry has not been able to increase clock speeds
since 2004 [6], as shown in Fig. 1.2. To continue scaling computing power, they have instead had toparallelize processors, all running at the same, heat-limited clock speed. The heat management problem isexacerbated by the fact that thermal transport at the nanoscale is much slower than we would predict frommacroscopic behavior [7].

3Figure 1.2: Transistor number per chip and clock speeds since 1970. The number of transistors per chip
(blue points) has doubled every two years since the 1970s – an observation and prediction known as
Moore’s Law. The clock speed of processors (black points), however, has stagnated since 2004. This
capping of clock speeds is required to limit the thermal load to manageable levels. Figure adapted from [6].
At the macroscale, heat flow is governed by Fourier’s law of heat conduction, where the heat flux,q, is proportional to the temperature gradient, ∇𝑇, according to the material’s thermal conductivity, κ:𝑞⃗ = −𝜅∇𝑇.
This diffusive behavior assumes heat transfer is occurring over large enough distances for the heat carriers
to repeatedly scatter and form a smooth thermal gradient. At the nanoscale, however, this assumption breaks
down. For semiconductors, the main heat carriers are phonons (quanta of lattice vibrations), which can
travel hundreds of nanometers, up to microns without scattering [8-11]. This means that some of thephonons carrying heat away from nanoscale hot spots travel ballistically, and deposit their heat non-locally,in stark contrast to diffusion. Simply applying Fourier’s law to nanoscale systems dramatically overpredicts the rate of heat transfer, when the heat flux is in reality limited by the ballistic velocity of phonons.
Since transistors and other circuit elements essentially serve as nanoscale hot spots on top of asemiconductor, this quasiballistic thermal transport is certainly at play in the large thermal load faced bymodern electronics.

4
One further materials challenge relevant to the semiconductor industry is nanoscale mechanicalproperties. Between the metallic, current-carrying interconnects on a chip, a dielectric material is requiredthat is both low dielectric constant, and sufficiently mechanically robust to produce a functional device.
However, it is difficult to achieve a low dielectric constant material, which increases both the device’s
efficiency and its switching speed, without sacrificing its mechanical properties, which can lead to device
failure [12]. Moreover, the elastic properties of materials can change at the nanoscale, due to the large
fraction of atoms at free surfaces or at interfaces [13, 14]. To get around challenges at the nanoscale, thesemiconductor industry has often had to resort to trial-and-error, modifying the fabrication recipe until a
successful combination is found. How much better would it be, however, if we could design materials up
front with the desired properties?
One powerful materials-by-design approach is to fabricate metamaterials, where some form of
periodic structuring gives rise to novel properties not attainable by homogenous, bulk materials [15, 16].
Combined with nanoscale effects, metamaterials are an exciting avenue to solving difficult technologicalproblems, particularly by engineering new thermal and elastic properties. As one example, the Greer groupat Cal Tech produces structural metamaterials by engineering nano-truss structures out of metallic glass
tubes [17]. As shown in Fig. 1.3, by reducing the wall thicknesses of the tubes to the deep nanoscale, themetamaterial undergoes a brittle-to-deformable transition. Thus, this stiff yet ultralight truss structure has
tunable mechanical properties by taking advantage of nanoscale size effects. As a second example, theHussein group at CU Boulder has developed the concept of nanophononic metamaterials, where forests ofnanopillars are placed on the surface of a thin film [18]. The local resonances of these pillars hybridize withthe phonons in the film to slow their group velocity and reduce the rate at which they can carry heat. Thus,
through only surface engineering, this metamaterial can reduce the thermal conductivity of the underlying
film, without impacting its electrical conductivity. This is a promising route towards producing an efficient
thermoelectric material.
Thermoelectric materials produce a voltage when a thermal gradient is placed across them,according to the Seebeck effect. Using this effect, the field of thermoelectrics seeks to recover the energy

5lost as waste heat in industrial, transportation, and power generation contexts (over 60% of the energy
produced [19]). However, the barrier to recovering the vast amounts of waste heat into useable energy isthat most materials are very inefficient at this process. The efficiency of a thermoelectric material is largely
determined by its nondimensional figure of merit, ZT. ZT must be greater than three for a material to becommercially viable on a large scale. ZT is defined by
𝑍𝑇 = (𝑆2𝜎) 𝑇,𝜅

where S is the Seebeck coefficient, σ is the electrical conductivity, and κ is the thermal conductivity. Thismeans that the ideal thermoelectric material would have very low thermal conductivity, but very highelectrical conductivity, which a nanophononic metamaterial may be able to achieve.

Figure 1.3: Structural metamaterials with nanoscale effects. These truss structures are fabricated out ofhollow nanotubes of metallic glass (Zr-Ni-Al). When the tubes have 10nm wall thickness, they exhibit moreductile compression (a), and partially elastically recover to their initial shape when the stress is removed(b). When the wall thickness is increased to 88nm, however, the structures exhibit brittle fracture upon
compression (c), and cannot recover their shape (d). Figure from [17].

6Given the fundamental and technological importance of nanoscale materials, with applications
ranging from integrated circuits and thermoelectrics, to metamaterials and nano-medicine, there is a greatneed for more research. In many cases, we are still developing theories to connect first principles rigor toreal experimental geometries, and computational approaches are forced to balance sufficient complexity
with feasibility as every atom in an extended nanosystem cannot yet be simultaneously modeled. To
increase understanding of nanomaterials and benchmark these theories and computations, the KapteynMurnane group uses coherent, nanometer-wavelength beams to directly probe functional nanosystems.
For my thesis, I have used tabletop extreme ultraviolet (EUV) sources to study the ultrafast thermaland elastic dynamics of nanostructured materials. By using ultrafast pulses, this technique accesses phonontransport on its natural timescales (sub-picosecond to several nanoseconds), enabling repeatable, nondestructive measurements, unlike that shown in Fig. 1.3. Moreover, the use of coherent EUV beams enables
the direct, non-contact probing of nanostructures well below the diffraction limit of visible light (hundredsof nanometers). To further advance EUV metrology to general samples and in-situ environments, I havedevoted a substantial portion of my thesis to the development of fully non-contact methods based on thetransient grating technique, where two coherent beams are interfered to produce periodic excitations in asample. The culmination of this effort is the interference of two EUV beams to directly excite dynamics inthe deep nanoscale. This has been demonstrated at the FERMI free electron laser and is currently underconstruction at JILA.
In Chapter 2, I present details on the experimental approach I took to perform nanoscale thermaland elastic measurements. After an overview of the entire experiment, I explain in more detail highharmonic generation and diffraction-based measurements, which are unique distinguishing factors of thetechnique. I then comprehensively present the major upgrades to the experimental apparatus that were madeduring my thesis.
In Chapter 3, I present measurements of nanoscale thermal transport in silicon and fused silica. For
silicon, we observe quasiballistic reduction in heat flow away from laser-heated nanostructures into thesilicon substrate. Once these structures are brought closer together, within the scattering distance of

7phonons, thermal transport recovers back towards the faster, diffusive prediction. These results confirm oursurprising prediction made in [20] that closely spaced nanostructures can cool faster than widely spaced
ones, when period dependence in thermal transport counteracts quasiballistic effects. By independently
tuning nanostructure size and spacing, this work maps the transition between isolated quasiballistic andcollectively diffusive thermal transport. For fused silica, no deviation from diffusive transport is observed
as a function of nanostructure size or spacing. This is expected as phonons in this amorphous material arescattered at lengths smaller than those accessed experimentally. This sample serves as an important control
study, showing that our results are not dominated by effects coming from the nanostructures alone,
independent of thermal transport in the substrate material. This chapter is based on our publication by
Frazer, et al. [21]. At the end of the chapter, I also briefly discuss theoretical efforts currently underway to
explain the fundamental mechanisms behind our observations. These efforts are in preparation for
publication by Beardo, et al., and Honarvar, et al.
In Chapter 4, I present acoustic measurements of ultrathin film elastic properties. Using the same
nanostructure transducers as in Chapter 3, we are able to launch nanoscale acoustic waves whose velocities
depend on the elastic properties of the sample materials. By measuring these wave velocities, we firstcharacterize the full elastic tensor of a series of low-dielectric-constant films relevant to the semiconductor
industry. These films are hydrogenated to lower their dielectric constant, but beyond a critical level, thishydrogenation causes a transition towards incompressible behavior. This section of the chapter is based on
our publication by Hernandez-Charpak, et al. [22]. Second, we use this method to explore thicknessdependence in the elastic properties of two separate series of dielectric films, down to 5nm thickness. We
observe no thickness dependence in the series of SiOC:H films, but we do see evidence of thicknessdependent softening in the series of SiC:H films. This section of the chapter is currently in preparation for
publication by Frazer, et al.
In Chapter 5, I present efforts to extend the technique used in Chapters 3 and 4 to more general
samples. First, we use the same nanostructure transducers to study the thermal and elastic properties ofsilicon metalattices − 3D periodic inverse opal materials with hierarchical structuring from single

8nanometers up to microns. This work is being prepared for publication by Abad, et al., and Knobloch, et al.
Next, I present measurements of micron-scale thermal transport using EUV-probed, optically-generated
transient gratings. Using this approach, we successfully characterize the thermal conductivity of hybrid
nanoparticle-molecular arrays. This work is in preparation by Frazer, et al., for publication. Finally, I
present the work to date using EUV transient gratings to excite thermal and acoustic dynamics. In
collaboration with the FERMI free electron laser, we demonstrated optically-probed, EUV-generated
transient gratings for the thermal and elastic characterization of materials, as reported in [23]. Our workwith FERMI later contributed to the first demonstration of all-EUV transient grating measurements in the
deep nanoscale, as reported in [24]. I close the chapter with a discussion of the current state of the effort todemonstrate all-EUV transient gratings on a tabletop at JILA.
Chapter 6 closes this dissertation with a summary of its major results, and a brief description of thenext steps expected in the near future. The Appendix gives further details on the COMSOL modelingperformed to analyze our thermal and acoustic measurements, as well as more details on the thermal data
fitting procedure.

Chapter 2
Experimental Methodology

In this chapter, I will describe the details of the experimental apparatus used for the tabletop
measurements presented in this dissertation. Our lab first demonstrated these types of measurements in2004 with an apparatus very similar to the one I use [25]. While the basic principles of the approach havenot changed during the last 15 years, some significant additions and improvements have been made to theapparatus over the course of my PhD. After a brief description of the experimental methodology, I will
focus the bulk of this chapter on the three primary new additions. These additions are 1) rebuilding our
optical transient grating capability, 2) implementing a synchronized optical chopper in the excitation beam,and 3) taking steps to mitigate nonlinear effects during the propagation of our high-intensity laser beam.
2.1 Introduction
To measure ultrafast dynamics, we use a pump-probe approach. In pump-probe experiments, someform of “pump” is used to initiate dynamics in a sample. After a controlled delay, some “probe” is used tomeasure the state of the system at precisely that time. Provided the excited dynamics are repeatable, thisprocess may be repeated for a whole series of delay times, at a temporal resolution down to the pumpduration. The repeated measurements at various delays may then be ordered into a single temporal signal
or effective slow-motion movie of the sample dynamics. As an approachable example, consider the studyof dripping water shown in Fig. 2.1. For this experiment, the repeatable dynamics are water drops formingat the base of a glass capillary tube, stretching out a liquid bridge, and finally releasing to fall. The probe isa precisely timed 15μs flash of light, which exposes photographic film. By adjusting the timing of the flash

10from drop to drop, the authors photograph each step in the water dripping process, and use the full timeseries of images to study the fluid mechanics involved in water dripping [26].

Figure 2.1: Example of pump probe measurements. In general, high temporal resolution pump-probe
measurements carefully delay repeated “snapshots” of the system, measured by a probe, with respect to therepeated initiation of dynamics, launched by the pump. In this case, the repeatedly “pumped” dynamics are
regularly spaced water drips from a capillary tube. The probe is a 15μs camera flash, which exposes
photographic film. By slightly delaying the flash from one drop to the next, the researchers resolve eachstep of the dripping process, such as the initial release of the water bridge connecting the drop to the water
still in the tube (left), and the subsequent retraction of the water bridge back into the tube (right). Figure
taken from [26].
For the work presented in this dissertation, I study the repeatable thermal and elastic dynamics ofnanoscale materials. The natural timescales for such dynamics at nanometer dimensions range from subpicosecond to several nanoseconds. Thus, to use a pump-probe technique to study these dynamics, I requirea pump duration well below the fastest dynamics analyzed (~0.5ps in the present work). This is available
in the form of ultrafast mode-locked lasers, which are an expertise of the Kapteyn-Murnane group. In

11particular, I use a near-infrared (IR) laser with pulse durations of 30fs to launch dynamics in the sample.
To probe these dynamics directly at the nanoscale, a nanoscale wavelength ultrafast laser would be ideal.
Thankfully, this is also an expertise of the Kapteyn-Murnane group in the form of coherent EUV beams
generated by high harmonic generation (HHG). Using these coherent EUV beams, we extend establishedoptical metrology techniques into the nanoscale.
Our experiment grows out of the tradition of optical techniques that measure thermal and elastic
material properties [27]. It takes elements from transient thermal gratings [28, 29], picosecond ultrasonics[30], and surface acoustic wave (SAW) spectrometry [31]. Similar to transient gratings (TG), we excite
specific acoustic wavelengths and thermal transport distances. Similar to picosecond ultrasonics, wemeasure longitudinal acoustic pulses that reflect from buried interfaces. Similar to SAW spectrometry, wemeasure a spectrum of SAWs, and characterize their dispersion in the sample material. We differ from eachof these optical measurements, however, in our use of a coherent EUV probe.
By using our EUV probe, we have made several advancements and discoveries beyond traditional
optical probes. Beyond micron-scale TG, we have observed a previously-unpredicted period-dependence
in nanoscale thermal transport, where closely-spaced non-diffusive heat sources can cool down faster thanwidely spaced ones [20]. Building from laser picosecond ultrasonics, we have measured thin film
mechanical properties down to monolayer sensitivity, and we have also observed thickness dependence inbilayer elastic properties [32]. Extending SAW spectrometry to nanoscale SAW wavelengths, we havemeasured the full elastic properties of films down to 11nm thickness and observed a deviation towards
incompressible behavior in highly-hydrogenated films [22]. All of this has been achieved in a single
measurement platform, using an EUV probe to measure optically-excited dynamics from nanoscale metallic
gratings.
In this chapter, I present the experimental details of our measurements. I first give a broad overview
of the experimental apparatus, followed by more detailed descriptions of HHG and diffraction-based
measurements. I then present in-depth discussion of the three primary additions to the experimental
apparatus made during my PhD. First, I describe the optical TG setup I implemented to generate micron-12
scale dynamics without a grating transducer. Next, I describe the optical chopper added to the pump beamof our experiment, which allows for short exposure times and rapid alternation between pumped andunpumped measurements. Finally, I describe steps taken to reduce the B-integral along our optical path, ameasure of nonlinear effects that can cause laser mode degradation and instabilities.
2.2 Optical Pump – EUV Probe Measurements
All of the tabletop experiments described in Chapters 3-5 are performed using the same basic
optical pump – EUV probe technique. In this technique, a sample absorbs energy from an incident opticallaser pulse, launching thermal and acoustic dynamics at its surface. After a controlled temporal delay, an
EUV probe pulse encodes the dynamic changes to the sample surface as a change in diffraction efficiency.
The dynamically changing diffraction is then analyzed to extract both the thermal and acoustic contributions
to the surface deformation. Using appropriate modeling, we then determine the thermal and elastic
properties of the sample. With this general approach in mind, I now describe the experimental apparatus
shown in Fig. 2.2.

Figure 2.2: Block diagram of the experimental apparatus. The laser properties at each stage are labeledin italics. The lines between each block represent the laser wavelength there, while the block colors marksubsystems: pump lasers in green, IR sources in red, vacuum system in grey, and cryo system in blue.

13
Each experiment begins with a Ti:sapphire oscillator. In the oscillator, Kerr lens mode-locking
produces an ultrafast pulse train at 80MHz repetition rate, with 780nm central wavelength and 50nm
bandwidth. This process is pumped by a 532nm continuous wave laser. After being coupled out of theoscillator, the pulse train enters a chirped pulse amplification system.
The first step in chirped pulse amplification is to stretch the ultrafast pulses out to ~200ps width
using a grating-based stretcher. These pulses then pass through a Pockels cell, which down-samples thepulse train to a 4kHz repetition rate. The pulses are next amplified by multiple passes through a second,helium-cooled Ti:sapphire crystal that is pumped by a 532nm wavelength, 90ns pulse width Q-switchedlaser. This amplifies each pulse to 2mJ of pulse energy. The amplified pulses then pass through a gratingbased compressor, which compresses the pulse length close to its 30fs limit. The pulses are not fullycompressed until later in the setup to avoid nonlinear effects during propagation, as detailed in Sec. 2.3.3.
Having completed chirped pulse amplification, the pulse train is then steered to a beam splitter that
divides the beam with most of the energy going towards the probe line, and part of the energy going towards
the pump line. In the pump line, the optical path length is controlled by both a short, high-resolution
mechanical delay stage, and a long, low-resolution mechanical delay stage. The optical path length
difference between the pump and the probe lines sets the delay between pump and probe pulses at the
sample. The pump pulses next reflect six times off of chirped mirrors that finish compressing the pulse
width down to 30fs. The pump pulses are then periodically blocked and unblocked by a chopper that issynchronized to the laser system. This provides a constant alternation between pumped and unpumped
measurements of the sample. After the chopper, the pump may either be routed directly to excite ananostructured sample, or it may be passed through a transient grating setup, to produce periodic
interference fringes that excite the sample. These two pumping schemes are shown in Fig. 2.3. The pumpis assumed to excite the sample identically at each path length (delay time) chosen. This means it is criticalthat the pump beam be collimated through the delay stages so the spot size does not change at the sample
as a function of optical path length. The changing pump intensity would create a systematic error as afunction of delay time.

14

Figure 2.3: Nanograting and optical transient grating excitations. a) For most of our samples, gratingsof periodic nanolines are used to transduce thermal and acoustic dynamics, upon optical excitation. b) Morerecently, we have added an optical transient grating capability. Here, we interfere two pump beams todirectly sinusoidally excite an absorbing sample at the micron scale. In both (a) and (b), we probe thesurface deformation by diffracting a coherent EUV probe from the sample, into a camera.
For the probe line, the first step is to fully compress the pulses to 30fs, again using chirped mirrors.
Part of the weak leak-through from these mirrors is collected and used as feedback for an active beam
pointing stabilization system. After compression, the probe pulses enter a vacuum system before performing
HHG. This is necessary because EUV photons are strongly absorbed in air. The intense near-IR probe

15
pulses drive high harmonics in an argon-filled, 150μm inner diameter waveguide. The generated harmonics
extend up to about a 27nm cutoff wavelength and lie under a 10fs envelope. Next, the harmonic pulses anddriving laser reflect together from a toroidal mirror that focuses the diverging beam toward the sample.
Two 200nm thick aluminum filters then remove the residual driving laser and the harmonics longer thanabout 37nm wavelength. This leaves a comb of about 5 harmonics, from the 21st to the 29th, with the 25th
being dominant, impinging on the sample. The EUV probe beam diffracts in a reflection geometry from theoptically pumped sample, either from periodic nanostructure transducers, or from periodic interference
fringes of the pump light. A charge coupled device (CCD) camera collects the diffracted beams with anadditional two 200nm thick aluminum filters that prevent pump light from reaching the camera.
These two processes of high harmonic generation and dynamic diffraction are what differentiate
our technique from optical techniques, such as Brillouin light scattering or time domain thermoreflectance.
I will describe each of our unique processes in more detail in the following two sections.
2.2.1 High Harmonic Generation
The field of high harmonic generation (HHG) is at this point both well established and constantly
growing [33, 34]. I will limit the discussion here to summarizing what is important to our thermal andacoustic experiments [35, 36]. The basics of HHG include two processes – the single-atom yield of EUV
photons, and the macroscopic phase matching of multiple atoms’ emissions to obtain a coherent beam.
The single-atom picture of HHG may be most simply understood in the three-step model, shownin Fig. 2.4(a) [37, 38]. In this semiclassical model, we first consider an electron bound in the Coulomb
potential of its parent atom. We then place this atom in a high intensity laser beam. Provided the intensity
is high enough, the oscillating electric field of the laser is comparable to the Coulomb potential of the atom,
and it depresses one side upon linear superposition. This allows the electron to tunnel ionize from its parent
atom (step 1). The electron is then accelerated in the laser field, gaining extra momentum (step 2). As thelaser field oscillates, it next reverses and begins to accelerate the electron back towards its parent ion. Thereis some probability then that the electron will recombine with the ion, emitting its excess energy in the form

16
of an EUV photon (step 3). Thus, every half-cycle of the driving laser, there is a burst of EUV photons
emitted from the irradiated atoms, as shown in Fig. 2.4(c).

Figure 2.4: High harmonic generation. a) The three-step model for HHG. For an electron in its groundstate in the Coulomb potential (left), the electric field of a sufficiently intense laser will distort the potential
enough for the electron to tunnel ionize (1). The electron then accelerates in the field (2) and returns to its
parent ion when the oscillating electric field reverses. The electron then recombines (3) and emits its excessenergy as an EUV photon. b) Phase matching yields a macroscopic, coherent beam. This is achieved bytuning the gas pressure to match the velocities of the driving IR laser and the EUV beam being generated.
c) EUV photons are emitted in attosecond bursts every half cycle of the driving laser. These bursts lie undera ~10fs envelope, defined by the phase matching window. Panel (b) is from [34], and panel (c) is from [39].
The next step is to phase match the photons emitted from each atom to obtain a macroscopic,
coherent EUV beam, as shown in Fig. 2.4(b) [36]. If the phase is uncontrolled, each EUV photon will in
general be out of phase, resulting in destructive interference and negligible output beam intensity. To avoid
this, the phase velocities of both the HHG and driving laser must be matched in the HHG medium. The
phase velocity of the EUV will remain largely unaffected, as the EUV index of refraction remains very
close to 1 for all materials. For the driving laser, however, the phase velocity is reduced by neutral atoms,
but it is increased by the plasma of ionized atoms and the waveguide. Thus, by tuning the pressure of gas
in the waveguide, the atom, plasma, and waveguide contributions to the driving laser phase velocity can bebalanced at a critical ionization percentage to match the EUV phase velocity. For our case of generatinghigh harmonics in argon with a 780nm driver, we obtain an EUV beam up to the 29th harmonic with about
1nJ of pulse energy.

17Though the energy of our EUV pulses is relatively low, when the coherent beam is focused, there
is enough intensity to cause some experimental challenges. The short-wavelength EUV photons havesufficient energy to break apart long hydrocarbon chains. No matter how good a vacuum system is, it is
impossible to completely avoid hydrocarbon contamination. Thus, at the sample and at any other location
the EUV beam is focused to a sufficient intensity, a large number of carbon radicals will be generated,which then deposit on the sample as an amorphous, hydrogenated carbon film [40]. As this process isrelatively slow, measurements may be made before a significant carbon layer is developed, but in generalcarbon deposition must be considered for any experiment involving EUV.
Using an HHG probe has many distinct benefits for our measurements. First, with 30nm
wavelength and 10fs pulse duration, HHG is ideally suited to measure the ultrafast repeatable dynamics ofnanoscale systems. Second, because the EUV probe is generated from the same near-IR laser as the pump,there is intrinsic timing stability in our pump-probe measurements. Also, by using EUV photon energies
that are away from core electron resonances in our sample materials, our probe is largely insensitive to theelectron temperature. Rather, we measure the surface deformation of the lattice, down to picometer-scalesensitivity, without hot electron effects dominating the signal at early times [41].
2.2.2 Diffraction Measurements
Diffraction can be thought of as the interference of light coming from two different points on an
object, with locally varying amplitude or phase. For example, in the familiar double slit experiment, raysfrom each of the two slits have different relative phases at the image plane, yielding the classic interference
fringes. In the far field, under the Born approximation, the diffracted light at the image plane is simply a
Fourier transform of the light at the exit surface of the object. This means that Fourier intuition may be
applied to diffraction, e.g. more closely spaced slits produce more widely spaced interference fringes. For
2

our modeling, we improve the accuracy of the approximation by adding Fresnel phases (𝑒 𝑖𝑥𝑖,𝑜 π/λ𝑧 in 1D,for wavelength, λ, propagating a distance, z) at both the object (xo) and image (xi) planes, but it is still basedon a Fourier transform.

18
For the majority of our measurements, we diffract our probe from nickel nanoline gratings
fabricated on the surface of our flat sample by collaborators. Because these nanolines form a 1D diffraction
grating, I will only discuss 1D diffraction here. This process is, however, generalizable to 2D and 3D
diffraction in principle, and we have previously measured 2D diffractive objects [42]. We use nickel for thenanoline material because of its good absorption of IR light and large coefficient of thermal expansion;
however, other metals may also be used. The height of the nanolines is ~10nm, which is thin enough forthe nanolines to be uniformly heated by the pump laser, but thick enough to maximize our diffraction signal.
To maximize signal, both the static diffraction, and the change in diffraction (slope as a function of gratingheight) must be large. This is because our approach measures small, dynamic changes to the grating height,
as described below.
The gratings diffract the EUV beam by periodically modulating both its amplitude (differingreflectivity from the substrate) and phase (differing path lengths for EUV rays reflected from the top of thenanolines and EUV rays reflected from the sample surface). As each grating consists of a square wave witha particular period and duty cycle, its diffraction pattern (Fourier transform) consists of a discrete series ofdiffracted orders, as well as the directly reflected beam, or DC peak. Because there are up to 5 harmonic
wavelengths present in our EUV beam, each diffracted order consists of multiple peaks, as longer
wavelengths diffract to wider angles, for a given diffracted order. Next, we launch thermal and acoustic
dynamics in the sample by preferential absorption of IR pump light in the metallic nanolines.
Upon ultrafast laser pumping, the metallic nanolines impulsively thermally expand within 3-10ps,
launching acoustic waves in the substrate down to a temporal period roughly equal to twice the thermal
expansion time. Both the thermal expansion and the acoustic waves change the amplitude of the phasegrating by changing the nanoline height and/or the substrate surface profile, as illustrated in Fig. 2.5(a).
This changes the diffraction efficiency of the grating, which I’m defining here to be the ratio of power inthe diffracted orders versus that in the DC beam. For example, if the phase difference between uniformly
expanded gratings and the substrate is exactly 2π (path length difference of λ), then there will be nodiffraction from the phase grating. There is in general a small change in the amplitude grating, as thermal

19
expansion slightly changes the material density and thus its reflectivity, but our signal is dominated by thechange in the phase grating coming from changes in the surface profile [41].

Figure 2.5: Dynamic diffraction measurements. This is a COMSOL simulation of a 100nm linewidth,
400nm period grating on fused silica. a) Viewed in cross section, we see both the thermal expansion of thenanoline and acoustic waves in the line and substrate change the surface profile (solid versus dotted lines).
Relative changes in nanoline-substrate heights (purple arrows) change the amplitude of the phase grating.
b) Changing the phase grating amplitude changes the diffraction efficiency compared to the unpumped
pattern (dotted line). c) By subtracting the change in the DC peak from the changes in diffracted orders, we
get a temporal signal showing thermal and acoustic dynamics. The purple circle marks the point of maximal
expansion, corresponding to panels (a) and (b). Changes are amplified for clarity, 50x in (a) and 25x in (b).
More details on COMSOL simulations are given in Chapters 3, 4, and the Appendix.
The experimental signal is the relative change in diffraction efficiency upon laser pumping. As
thermal expansion and acoustic waves change the amplitude of the phase grating, the relative intensities of
the DC beam and the diffracted orders change with the diffraction efficiency, as shown in Fig. 2.5(b). To
conserve energy, all of the intensity that is gained in the diffracted orders must be taken from the DC beam,
assuming no change in absorption or transmission of the EUV probe. This means that dynamic changes inthe DC intensity encode all the information from each diffracted order, with an opposite sign. Analyzing
the DC beam thus allows us to measure dynamics down to the diffraction limit of our probe, even though
we have poor numerical aperture (NA) with our camera. Changes in the diffracted orders’ intensities, on
the other hand, each contain partial information about the dynamics of the system. In the case of surface
acoustic waves, where the spectrum of SAWs launched is the same as the spectrum of spatial frequencies
that define the grating’s diffraction, each diffracted order contains the frequency of each respective SAW.

20
To build a change in diffraction efficiency signal, we first take the difference between pumped andunpumped diffraction patterns and normalize by the integrated unpumped diffraction pattern. This
normalization is performed at every delay time to reduce the influence of probe intensity fluctuations on
our measurement. Next, we subtract the normalized difference in the DC peak from that of the diffracted
peaks. In this way, we obtain twice the signal of the DC peak alone. This final value for the change in
diffraction efficiency as a function of delay time is the signal shown in Fig. 2.5(c). Note that we currently
subtract an offset from this temporal signal so that the pre-time zero (the time when dynamics begin) points
average to zero. We should also subtract the dark-current baseline counts on the camera from our raw
diffraction patterns, as this pedestal only reduces the amplitude of our signal. This has not yet been
implemented, however. To reduce the noise in our final signal, we follow a specific experimental routinethat I describe now in more detail.
Our signal begins with CCD-recorded diffraction patterns, fully vertically binned on-chip to
produce a single count value for each of our 1024 pixels (chip width). These diffraction patterns alternate
between pumped and unpumped diffraction (via manual blocking of the pump beam) with each camera
exposure. The camera integrates over some number of pulses, depending on the selected exposure time. At
a given pump-probe delay time (stage position), we further average over several repeated exposures of both
pumped and unpumped diffraction. After recording average diffraction patterns at each delay time, we
repeat the entire process to obtain at least three “loops” across the full experimental time window. Theseloops are averaged together to obtain a final “scan” of both pumped and unpumped diffraction patterns as
a function of delay time. By averaging over both exposures and loops, we reduce the influence offluctuations and drift in our probe intensity on both short and long timescales. Note that each loop iscurrently saved as a combination of each preceding loop. To “unwrap” a loop to contain only the diffraction
measured during that loop, use the following formula:𝐷𝑢𝑛𝑤𝑟𝑎𝑝𝑝𝑒𝑑 = 𝑛𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝐷𝑐𝑢𝑟𝑟𝑒𝑛𝑡 − 𝑛𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝐷𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 ,where D is the pixel by time matrix of a scan’s diffraction patterns, and n is the loop number. This process
is done separately for the pumped and unpumped measurements. In the future, it might be possible to21normalize each pair of pumped and unpumped diffraction patterns to the unpumped total intensity, for eachpair of exposures in the set of averages. This would allow us to remove the influence of fluctuations on thescale of twice the exposure time (generally tens of milliseconds), which is faster than we currently do.
Finally, I will now discuss the dynamics present in our experimentally measured traces, as shownin Fig. 2.6. First, there are the desired, repeatable dynamics illustrated in Fig. 2.6(a)-(c). At timescales from
10ps to 8ns, depending on grating linewidth and period, we observe the oscillations of the multiple excited
SAWs, which change the relative heights of both the nanolines and the substrate in between lines. Theseoscillations are superimposed on a multiexponential thermal decay, where the expanded height of thenanolines relaxes back towards the unpumped profile as heat is transferred into the substrate. At sub-100ps
timescales, depending on sample thickness and mechanical properties, we can observe discrete bumps inthe signal. These bumps are echoes of longitudinal acoustic waves that are launched downward by theimpulsive nanoline expansion and reflected back to the surface by any buried interfaces, e.g. the interface
between an ultrathin film sample and its substrate. At sub-10ps timescales, we can observe the longitudinal
acoustic resonance of the nanoline itself, as it rings briefly upon impulsive expansion. The shortest
timescale material response we have observed is a low-amplitude ultrafast peak at ~130fs (observed in the
30nm metalattice discussed in Chapter 5 and thin film #148 discussed in Chapter 4). Though our signal islargely insensitive to electron dynamics, I believe with sufficient signal to noise ratio (as obtained for thesemeasurements) we can distinguish a small hot electron signal. The other potential explanation for a material
response this fast would be magnetic dynamics, as the nanolines are made of nickel. Because the ultrafast
peak is very small, we have not tried to analyze it further, thus far. I discuss the analysis of the other decays
and oscillations in the signal for thermal and elastic properties in Chapters 3 and 4, respectively.

22

Figure 2.6: Experimentally measured diffraction signals. a) At long timescales, we observe themultifrequency SAWs launched by the gratings, and their slow thermal decay as heat is transferred into thesubstrate. We also observe the decay of the acoustic wave amplitude as SAWs are scattered in the sample.
b) At shorter timescales, we observe the longitudinal acoustic resonance of the nanolines themselves, andthe echo of longitudinal acoustic pulses reflected from buried interfaces. c) At ~130fs after the start ofthermal expansion, we observe a low-amplitude peak, which I attribute to hot electrons. d) When there issample damage occurring during a measurement, one good indication is the normalized loops of a scan notexhibiting identical dynamics, as shown here. Panels (a) and (b) are from [22]. Panel (c) is from a 100nm
linewidth, 400nm period grating on a 30nm silicon metalattice. Panel (d) is from a 200nm linewidth, 800nm
period grating on sapphire.
In addition to the repeatable dynamics that reveal intrinsic sample properties, we also analyze ourmeasured signals for non-repeatable dynamics that indicate sample damage. For the most obvious check,we monitor the static diffraction pattern while we are taking data. If we observe the DC peak steadily growwhile the diffracted orders shrink over the course of a scan (as a function of real time, not delay stage
position), this indicates we are ablating our metallic nanolines with our pump beam, constantly reducing
their height and burning the sample. If we observe both the DC and diffracted peaks either grow or shrink
together, this does not necessarily indicate sample damage, and it is more likely a result of drifting probe
intensity or carbon deposition. The next check is to compare the signal in each loop of a scan, normalized

23
to its maximum. If the dynamics are not the same from loop to loop, as shown in Fig. 2.6(d), this can
indicate sample damage. For the measurements shown in Fig. 2.6(d), the gratings were later confirmed via
atomic force microscopy to have been ablated to a lower height. Note that normalization is required as amere loss of signal amplitude does not necessarily indicate sample damage. Degrading probe stability or
carbon deposition can each reduce signal amplitude over the course of a scan. Additionally, if the signal islow enough that many loops must be averaged to see signal in the scan, then the loop-to-loop comparison
would not be a good test for sample damage. A more rigorous test is to compare the unpumped diffraction
patterns from the beginning and end (in real time) of a scan. This is done extensively for the optical TG
data described in Chapter 5, where the appearance of TG diffracted orders in the unpumped diffraction
indicates the TG pattern has been burned into the sample. Finally, we also check for signatures of pulse
accumulation during our measurements.
Our analysis assumes all dynamics have fully relaxed before the next pump pulse arrives, which is
generally a good assumption for few-kHz repetition rates. However for some systems, such as suspended
membranes, the sample might not be fully cooled before the next pump pulse arrives. In this case, thesystem reaches an elevated temperature steady-state background once the pump pulses have been
accumulating for long enough in real time. To check for this behavior, we manually block the pump beamwith a card, and observe the diffraction pattern and signal upon unblocking the pump beam before time
zero. If the pre-time zero diffraction or signal change, this indicates there is pulse accumulation as there arestill dynamics occurring before the next pump pulse arrives.
2.3 Upgrades to the ExperimentNow that I have described the experimental methodology in some detail, I will describe the main
additions to the apparatus made in the last five years. I will first describe the optical TG setup, followed by
the implementation of an optical chopper in the pump beam, and lastly the mitigation of nonlinear effects
during IR beam transport.

24
2.3.1 Optical Transient Grating Setup
Transient gratings are an elegant approach to the study of material dynamics. By interfering two
single-wavelength beams at a defined crossing angle, this method produces pure sinusoidal excitations in acompletely non-contact modality [29]. The single-frequency excitation greatly simplifies the interpretation
and analysis of both thermal and acoustic signals, and the avoidance of transducer fabrication allows agreater variety of samples to be studied, including liquids.
By simple geometric consideration of the wavevectors of the two pump beams and the resulting
transient grating, one may derive the transient grating period, Λ, as a function of the pump wavelength, λ,and the half crossing angle, θ, asΛ=𝜆.
2 sin 𝜃It is clear that the smallest TG period possible is half the pump wavelength. However, this requires counterpropagating pump beams, which is generally impractical for exciting real materials. Thus, most optical TG
periods are confined to the micron scale in practice, with the limit being around 500nm [43]. While these
optical TG periods do not reach the deep nanoscale accessible by EUV probes, we have implemented an
optical TG setup to (a) enable the measurement of systems where transducer fabrication is prohibitive, andto (b) serve as a stepping stone to all-EUV TG using HHG beams.
Our optical TG setup is similar to that of Ref. [44], and it was first demonstrated in 2006 in our lab
using an HHG probe [45]. The 2006 setup was dismantled, however, so I have reimplemented it as amodular pumping setup on our existing EUV nanometrology apparatus. In the optical TG setup, our 780nm
pump beam is first up-converted to 390nm by second harmonic generation (SHG) in a beta barium borate(BBO) crystal. The second harmonic beam is preferentially reflected by two dielectric mirrors and thenpasses through a transmission phase mask, optimized to send most of the beam intensity into the +1 and -1
diffracted orders. Several phase masks are etched into the same transmissive optic, with a series ofcorresponding TG periods: 2, 2.5, 3.2, 4, 5, 7, 9, 13, 20, 30, 40, 60, 90, and 130μm. This allows for rapidchanging of TG periods by translation of the phase mask optic from one mask to the next. However, due to25
spherical aberration in subsequent lenses, changing phase masks will slightly change the position of theoverlap plane (the required sample position).
We next collect the beams with a 4f imaging system of two 12.5cm focal length (f), 2” diameter
lenses, where the phase mask−first lens separation is 1f, the inter-lens separation is 2f, and the secondlens−sample separation is 1f. This geometry images the phase mask period onto the sample, with a pulse
front tilt to optimize maximal interference fringes [44]. Note that ideal TG excitation requires only two
beams to be crossed at the sample, but the phase mask does not ideally produce only the ±1 diffracted
orders. Thus, we use spatial filters to block all beams reflected and transmitted from the phase mask, except
for the two desired diffracted orders. This also blocks any residual 780nm light not filtered by the twodielectric mirrors. See Fig. 2.7 for a schematic of the optical TG setup.

Figure 2.7: The modular optical TG setup. a) Top view of the setup. By adjusting the linear stage beneath
the TG breadboard, the horizontal position of the TG pump spot on the sample is adjusted. b) Side view ofthe setup. By adjusting the lab jack beneath the TG breadboard, the vertical position of the TG pump spot
on the sample is adjusted. Note that the linear stage adjusts the path length between the two periscopes, andthe lab jack adjusts the path length in an expandable periscope.
As shown in Fig. 2.7, the setup is slightly more complicated than just the 4f imaging system usedhistorically. This is because we must spatially overlap the pump and probe beams at the sample, but the
EUV optics are not adjustable to tune the probe position. Instead, we must adjust the TG pump position to
overlap it with the probe. To do this, we move the entire setup described above on a breadboard withhorizontal and vertical adjustment via manual stages. Adjusting these manual stages slightly changes the26total path length of the pump, relative to the probe, but this may be corrected using the delay stages earlier
in the pump line. To ensure the alignment of the setup is not disrupted upon adjustment of the stages beneath
the TG breadboard, the horizontal and vertical beam path segments that change length (marked in Fig.
2.7(b) by arrows) must be aligned to the stage axes. Use the following procedure to ensure this alignment.
First, place a target in the position of the bottom mirror of the expandable periscope. Move thehorizontal linear stage to either end of its travel. If the pump beam changes position on the target, adjust
the two fixed periscope mirrors until the beam no longer moves on the target, and it is aligned to the linear
stage axis. Next, replace the mirror in the bottom of the expandable periscope and place a target after the
expandable periscope, on the TG breadboard. Move the vertical lab jack through its travel range and checkif the beam moves on the target. If it does, adjust the bottom periscope mirror to align the beam to the lab
jack axis. Note that single mirror alignment will set the beam pointing, but not its position. Provided theinitial placement of the mirrors is close enough to the ideal, differences in beam position may be correctedby the subsequent mirrors. It is also important to check the beam polarization when adjusting periscopealignment. If a periscope is not aligned linearly, it will rotate the pump beam polarization. Because both
periscopes are present in the pump line whether using nanograting or TG excitation approaches, polarization
variation from the periscopes can reduce the nanograting pumping efficiency. Ideal absorption occurs whenpump polarization is aligned to the grating line direction. Polarization rotation may be simply checked usinga reference polarizer on the TG breadboard and corrected using the half waveplate earlier in the pump line.
After aligning the periscopes and expandable beam path segments, the rest of the TG setup mustbe aligned. The focus of the pump beam, set by a ±1m lens pair earlier in the pump line, should be set atthe BBO crystal for maximum conversion efficiency. The BBO tip, tilt, and rotation must also be adjusted
to optimize phase matching. Note that pushing the focus of the beam too far from the optical chopper
described in the next section will reduce the number of pulses that can be cleanly passed per exposure as
the beam will be too large at the chopper position. The two dielectric mirrors after the BBO are next used
to align the beam to the 4f lens system.

27
The 4f lens system is mounted in an optical cage. To align the beam to it, we first remove the phasemask and first 4f lens, and block the beam from entering the measurement chamber. We place an empty
lens mount in the lens’s place to maintain stability of the optical cage. Using the two dielectric mirrors anda 2” cage target, we align the beam to the cage axis. We then reinsert the lens at the previously-marked
position for 4f geometry on the cage and replace the phase mask in its mount. To check that the phase mask
is placed at the proper 4f condition, we use the cage target to verify the ±1 diffracted orders are parallel to
one another between the two lenses. To correct the diffracted orders getting closer together (farther apart)as they propagate between the lenses, we move the phase mask slightly closer to (further from) the firstlens. Finally, the sample must be placed in the TG overlap plane. Provided the initial placement of the TG
breadboard and optical cage is as close as possible to the measurement chamber, the distances should
already be close to correct, and the final adjustment of the sample may be done by the long in-vacuum
sample stage, which should be set roughly in line with the optical cage axis.
As a part of the optical TG build, I also designed a new measurement chamber in collaboration
with JILA machinists, who fabricated the final chamber. Computer-aided design (CAD) drawings of thechamber are shown in Fig. 2.8(a-b). The chamber is shaped like a rectangle with one clipped corner, whichhouses a square sapphire window that admits the TG beams, visible in Fig. 2.8(d). This design allows the
optical cage outside the chamber to approach as close to the sample as possible, without intersecting with
the CCD camera placed in the TG measurement position, shown in Fig. 2.8(d). The three vacuum flanges
on the chamber shown in Fig. 2.8(a) are currently used for the two electrical feedthroughs needed by thein-vacuum sample stages, and a dual-range vacuum gauge. The larger flange is placed in line with the beamentrance flange to enable transmission experiments by mounting the CCD camera on this flange. This has
not been attempted yet. Also note that the cathode of the vacuum gauge is known to sputter in argoncontaining environments. Over the years of use, we have observed discoloration of the chamber near thegauge, and loss of gauge accuracy, which likely indicate the residual argon from HHG is sputtering thecathode.

28

Figure 2.8: New vacuum chamber for TG and nanograting measurements. a) CAD drawing of thechamber showing the vacuum flanges on the back, and the inset pocket in the wall that decreases camerasample distances when measuring nanogratings. The groove along the bottom of the chamber is used toclamp it in position. b) Top view of the chamber, showing the in-vacuum stages. The circle marks how
close the camera filters (grey rectangle) come to the TG window (green rectangle). c) Camera and chamber
positions for nanograting measurements. d) Camera and chamber positions for TG measurements. Note thechamber is clamped to a platform that rests on two manual stages used to adjust the chamber position.
Also visible in Fig. 2.8(b) is a CAD drawing of the in-vacuum sample stages purchased as a partof the optical TG build. This stage assembly has encoded xy-translation to move the measurement position
to different sample locations, with nanoscale precision. It also has un-encoded stages for sample rotation(rotation axis points out of the page in Fig. 2.8(b)) and long-distance translation. When performing TG
measurements, this long stage should be rotated from the position shown in Fig. 2.8(b) to be perpendicular
to the TG entrance window. This allows for easy placement of the sample surface at the TG interference
plane. The rotation and long-translation stages are further used when switching between TG measurements
and nanograting measurements, which only require uniform, round beam pumping.

29
To switch between round beam (RB) and TG pumping, the system was made to be relatively
modular. Starting from the TG pumping configuration, shown in Fig. 2.8(d), vent the vacuum system andflip the CCD and its filters from the far side of the chamber to the near side of the chamber, as shown in
Fig. 2.8(c). Remove the middle segment of the string of thin bellows connected to the CCD filters. The
extra length is not needed with the camera in the RB position. Remove the block holding the chamber
platform in place and adjust the linear stage beneath it, parallel to the camera axis, to the furthest pencil
mark. This will place the incident probe beam as close to the near side of the chamber as possible. This,
combined with the camera inset in the chamber wall, increases the numerical aperture (NA) of the RB
pumping configuration, which is important for measuring nanoscale gratings, rather than micron-scale TG.
Reposition the in-vacuum sample stage assembly from its position along the optical cage axis to instead beparallel to the camera axis. Place the long stage on the row of bolt holes that places the sample holder inline with the center of the CCD window. Rotate the rotation stage 90° so that the probe reflection from thesample will go towards the new camera position. The long stage may now be used to adjust the samplecamera distance, and it should be set such that the probe beam strikes the center of the sample. Note thatthe full range of the sample’s x-stage may not be accessible without striking the chamber wall, when thesample is very close to the camera. Striking the wall should be avoided, so care must be used when adjusting
the x-stage.
Next align the pump and probe lines for RB pumping. Start with aligning a weak IR beam throughthe probe line, into the chamber. The linear stage beneath the chamber platform that is aligned along theprobe line may be adjusted to ensure the focus of the toroidal mirror lands at the sample plane. This probe
line reference beam will be important for aligning the RB pump line. To align the RB pump, begin byremoving the first mirror on the TG breadboard (just before the BBO) and the optical cage so that the beam
can freely exit the expandable periscope and travel across the breadboard. Make sure to mark the cage andmirror positions for easy replacement when switching back to TG pumping. Next place a 2” silver mirror
on the optical table just past the TG setup. Using the silver mirror’s position and pointing, adjust the pumpbeam to enter the in-vacuum mirror chamber (connected by large bellows to the measurement chamber,

30
shown in Fig. 2.8(c)) and propagate nearly co-linearly with the probe, crossing at the sample plane. Do not
adjust the top mirror of the expandable periscope as this will make it more difficult to switch back to TG
pumping. To switch from RB pumping back to TG pumping, exactly reverse the process described above,and follow the TG alignment procedure instead of the RB alignment procedure just described. Note that theperiscopes should not have to be realigned every time.
2.3.2 Synchronized Chopper as a Pump Shutter
To perform our pump-probe technique, we require a series of pump-on and pump-off measurements
at each delay time. This means that the pump beam must be periodically blocked to measure the un-pumped
sample profile via EUV diffraction. In the past, we have done this using a mechanical shutter, set to certain
exposure and cycle times. When the shutter wore out, however, we took the opportunity to upgrade it to an
optical chopper. A chopper consists of a wheel, as shown in Fig. 2.9, rotating at a set frequency and with a
set number of slots. As the slots rotate past the laser beam, they imprint their specific frequency on the
beam by repeatedly blocking and unblocking it. This is the same result as a simple mechanical shutter, but
the optical chopper has distinct advantages. First, its frequency may be locked to the repetition rate of thelaser, enabling the selection of a specific numbers of pulses, repeatably, in each diffraction measurement.
Second, the rotating wheel of the chopper is able to close across the beam faster than a mechanical shutter.
This enables "clean chopping" where the beam is blocked between laser pulses such that the first pulse isfully passed, and the next pulse fully blocked. In principle, our chopper can go down to single pulse
exposures. The upper and lower bounds on selecting exposure lengths for a given wheel are the maximum
physical rotation frequency of the wheel (106.5Hz), and when jitter and the loss of clean chopping become
too significant at slow wheel speeds, respectively.

31

Figure 2.9: Optical chopper as a beam shutter. As the chopper wheel rotates at a given physicalfrequency, it locks to a frequency set by the outer number of slots, but it chops the beam at a frequency
determined by the inner number of slots. We have a variety of chopper wheels, but the most commonlyused one is shown here: 60 outer slots and 2 inner slots. When synchronizing the chopper to the 4kHz laser
repetition rate, the outer frequency may be locked to multiples of the 250μs pulse period. This then produces
multiples of 15 pulse exposures in the inner slots, for this wheel. The number of pulses per exposure may
be further adjusted using the harmonic/subharmonic function of the chopper controller.
We synchronize the chopper to the laser repetition rate using the reference trigger output signal
from the amplifier controller (one pulse every 250μs). We pass the reference trigger signal through a DG
535 digital delay/pulse generator, or delay box, which allows multiples of 250μs pulse periods to then be
sent as a trigger signal to the chopper. To select the pulse period, we set the falling edge of the delay box’s
output pulse to the desired multiple of 250μs and the rising edge to be 1.5μs earlier. Note that the chopper
must be set to trigger on the falling edge for this to work properly. This input signal to the chopper sets its

32
fundamental frequency, though more chopping frequencies are obtained using the harmonic andsubharmonic functions of the chopper controller.
Once the chopper frequency is set, the camera must be synchronized to it such that the camera
exposures line up precisely with when the pump beam is unblocked or blocked, as the case may be. The
relative timings of pump, probe, chopper, and camera exposure are shown schematically in Fig. 2.10. To
start the camera exposure exactly when pump pulses are unblocked, we trigger the camera from the pumpbeam after the chopper. We collect the low-power leak-though from the bottom mirror of the fixed
periscope in the TG setup (see Fig. 2.7) on a photodiode for this purpose. The analog pulses from thephotodiode cannot directly trigger the camera, however. The 4ns response of the photodiode is too fast forthe camera, which requires a transistor–transistor logic (TTL) trigger. Thus, I have built an analog to TTL
converter circuit to go between the photodiode and the camera.

Figure 2.10: Timing schematic of the experiment. The chopper periodically blocks the pump beam,
which is temporally delayed from the probe beam, according to the mechanical stage position. The camera
records the probe pulses when the pump beam is both blocked and unblocked to obtain unpumped andpumped diffraction patterns, respectively, at each delay time. The length of each segment in the schematicis not quantitative, as several pulses are skipped during the 2.02ms camera readout, and an unknown numberof pulses are skipped between the triggering of the camera and the beginning of the camera exposure.
First, we must prepare the photodiode's pulses for the circuit using analog means. A 35MHz low
pass filter stretches the pulse to 20ns, and a combination of up to two Mini-Circuits ZFL-1000LN+

33
amplifiers and up to two 10dB attenuators produce a signal amplitude >400mV going into the converter
circuit. It is important to choose a combination of circuit elements that avoids clipping the pulse amplitudein the amplifiers. This relatively amplifies the noise around the main pulse. If these specifications cannot
be obtained, or the signal is jumpy, it is likely that the photodiode position needs to be adjusted to properly
catch the leak-through beam. Once we have a good analog signal, we send it to the converter circuit.
The first part of the converter circuit is a comparator that cleans up the signal and converts eachanalog pulse into a digital pulse of similar width. The comparator design is shown in Fig. 2.11(a). I ACcouple the input signal to the comparator, biased up to 1V, to avoid negative voltages that are incompatible
with the comparator chip. To AC-couple a signal, send it through a capacitor to a voltage divider that sets
the DC offset. The time constant of this RC circuit must be much longer than the pulse width, but much
shorter than the pulse period, which is achieved by setting the time constant equal to 1μs. Because we havea very low duty cycle signal (10ns pulses every 250μs), the baseline voltage of the AC coupled signal will
be only slightly below the DC offset voltage, as shown in Fig. 2.11(b). The comparator outputs a digitalhigh voltage only when the input signal is larger than a reference voltage. Using a second voltage divider,I set the reference voltage to 1.4V. This way, an 800mV amplitude pulse from the amplified photodiodesignal will be captured at FWHM. Note that the comparator chip used cannot exceed 3V input voltage. This
sets the upper limit for the input signal amplitude at 2V, though clipping occurs in the amplifiers at ~1.5V,
before this limit is reached.

34

Figure 2.11: Comparator circuit with voltage dividers. a) Circuit diagram. The signal is input through agrounded BNC, and a small load resistor impedance matches the circuit to 50Ω. The signal is AC-coupledthrough CS with a DC offset voltage, Vdivide,S, determined by the RS voltage divider. The reference voltage,
Vdivide,R, for the comparator is set by the RR voltage divider. b) When the input of the amplified, stretchedphotodiode signal is greater than the reference voltage, the comparator outputs a steady digital high voltage.
By AC-coupling the input signal, its average is exactly at the DC offset voltage, VDC. This means thebaseline voltage between pulses, Vlow, is slightly lower than VDC.
The comparator chip has a bias current, which must be taken into account to precisely set theresistor values for the input and reference voltage dividers. To do this, I use the principle of superposition,
where the voltage at the node of interest is the sum of the voltage from the voltage divider in isolation andthe voltage from the comparator's bias current in isolation. Together, I derive the following equations forthe voltage divider resistor values, where i denotes either the signal (S) or the reference (R) comparator
inputs, Vsupply=5V, Vdivide,S and Vdivide,R are at the labeled nodes in Fig. 2.11(a), Vbias=1mV, and Ibias=3.5μA:
𝑅𝑖,1 =

𝑅𝑖,2 =

𝑉𝑠𝑢𝑝𝑝𝑙𝑦 𝑉𝑏𝑖𝑎𝑠,
𝑉𝑑𝑖𝑣𝑖𝑑𝑒,𝑖 𝐼𝑏𝑖𝑎𝑠

𝑉𝑠𝑢𝑝𝑝𝑙𝑦
𝑉𝑏𝑖𝑎𝑠.
𝑉𝑠𝑢𝑝𝑝𝑙𝑦 − 𝑉𝑑𝑖𝑣𝑖𝑑𝑒,𝑖 𝐼𝑏𝑖𝑎𝑠

35
The capacitors are then determined by the following equation, with the condition that the RC time
constants are 𝜏𝑆 = 1μs, and 𝜏𝑅 = 1ms:
𝐶𝑖 = τ𝑖

𝑅𝑖,1 + 𝑅𝑖,2.
𝑅𝑖,1 𝑅𝑖,2Lastly, a load resistor must be added to the input, so the signal is impedance matched at 50Ω. This reducesreflections that can cause significant problems in ultrafast electronics. The load resistor is determined by
the condition 𝑅𝐿 //𝑅𝑝,𝑠 , where 𝑅𝑝,𝑠 = 𝑅𝑠1 //𝑅𝑠2 . Thus,
𝑅𝐿 =(50Ω)𝑅𝑆1 𝑅𝑆2.
𝑅𝑆1 𝑅𝑆2 − (50Ω)(𝑅𝑆1 + 𝑆𝑆2 )The available resistors and capacitors that most closely match the ideal calculated values are shown in Fig.
2.12, the schematic of the full analog to TTL converter circuit.

Figure 2.12: Diagram of the analog to TTL converter circuit. The voltage input first goes through a
voltage regulator to smooth out any power fluctuations and reduce the supply voltage to 5V. The capacitorson each chip’s supply voltage line further smooths the power supply to each integrated circuit (IC). The
BNC input and output are directly grounded to the metal box holding circuit, which is also the circuit’s
ground. The comparator portion of the circuit is identical to the diagram in Fig. 2.11(a), though the explicit
resistor and capacitor values used are shown here.
The digital pulse from the comparator is then sent to a 74LS122N retriggerable one-shot, as shownin Fig. 2.12. The one-shot produces a TTL pulse of a specified length, when triggered by an input signal
pulse. The length of this pulse, 𝑡𝑊 is set by36
𝑡𝑤 = 𝐾𝑅𝑒𝑥𝑡 𝐶𝑒𝑥𝑡 ,where Rext and Cext are set by external resistors and capacitors, and K varies with Cext as shown in Fig. 2.13.
This one-shot is also retriggerable, which means if it receives another input pulse before the full width ofthe output pulse has passed, it will retrigger to maintain a high voltage output. Thus, by tuning 𝑡𝑊 > 250μs,
the one-shot will output high (1V) as long as laser pulses are passing through the chopper and hitting thephotodiode, as shown in Fig. 2.14. This defines the exposure window of the camera while the pump isunblocked. The one-shot’s output TTL pulse, with 50Ω termination added, may then be used directly totrigger the camera. Note the output pulse was originally tunable from 175-250μs, but I had to add resistanceto Rext after a few months of operation to obtain pulse widths >250μs. This could have been caused bydeviations in VCC, VRC, temperature, or Cext, all of which change K. Alternatively, I recently discovered thepotentiometer that sets Rext had been degrading, which could have been changing the attainable range of𝑡𝑊 . I have now repaired the potentiometer, so the attainable pulse widths may now be 211-308μs, asnominally calculated from the resistor and capacitor values in Fig.2.12 and K=0.39. In this case, theconverter circuit can no longer resolve individual pulses at 5kHz.

Figure 2.13: The one-shot K parameter. For the range of Rext we use, K depends on the (constant) valueof Cext as shown. Note K also varies with temperature, VCC and VRC. See the IC’s datasheet for more details.

37

Figure 2.14: Function of the analog to TTL converter circuit. Panels (a)-(c) show an oscilloscope
reading of the circuit’s output while tuning Rext. When 𝑡𝑊 is below 250μs (a), each pulse of the pump laser
produces a TTL signal pulse. As 𝑡𝑊 is increased, the gap between pulses shrinks (b), until 𝑡𝑊 >250μs and acontinuous high voltage is output until the pump laser is blocked (c). (d) Picture of the completed circuit.
Note the cables are twisted, the wire lengths are kept short, and the input and output BNC’s are separated.
These steps are important when working with ultrafast electronics. The BNC/filter/amplifier chain
connecting the photodiode to the circuit (not shown) should also be kept as short as possible to avoid picking
up spurious frequencies from other lab equipment. Also note Cext is a surface mount capacitor on theunderside of the breadboard, and so it is not visible here.
Finally, now that the camera has a trigger synchronized to the pump laser, we must set the camera
exposure. To set the camera exposure, we use a kinetic series of at least two exposures, since the camera istriggered when the pump beam passes through the chopper apertures (pump on), but both pump on andpump off exposures are needed. Use the following equation to set the kinetic series cycle time and set thecamera exposure time to be 2.02ms shorter to allow for the minimum camera readout time:
𝑡𝑐𝑦𝑐𝑙𝑒 =1𝑆
2𝑓𝑜𝑢𝑡𝑒𝑟 𝑆𝑖𝑛𝑛𝑒𝑟

𝑜𝑢𝑡𝑒𝑟

.

38
Here, fouter is the exact frequency of the outer row of slots, as read out from the chopper controller, Sinner isthe number of slots on the inner portion of the wheel (where the pump beam is), and Souter is the number ofslots on the outer portion of the wheel (where the chopper’s optical encoder is). In practice, doing a kinetic
series of up to 50 leaves fewer skipped chopper cycles while waiting for the next trigger pulse to arrive.
Use Table 2.1 to get a specific number of pulses through the chopper. To check for clean chopping,use an oscilloscope to read the output from the analog to TTL converter circuit. If the pulse is not of aconstant, expected width, then clipping of the pulses is occurring. To get a specific number of pulses on thecamera, however, more work is needed. There is a finite time for the camera to begin exposing, once it
receives a trigger, and this skips some pulses. This delay time needs to be better characterized and controlled
before the system can be operated at a precise number of pulse exposures.

39
Pulses:Delay box μs
Delay box kHz
7/5 harmonic
42/30 harmonic
Pulses:Delay box μs
Delay box kHz
60/2 harmonic
Pulses:Delay box μs
Delay box kHz
60/2 harmonic
Pulses:Delay box μs
Delay box kHz
60/2 harmonic
Pulses:Delay box μs
Delay box kHz
60/2 harmonic
Pulses:Delay box μs
Delay box kHz
60/2 harmonic

1
250
4
7/10
10
250
4
3/2
21
500
2
10/7
35
750
4/3
9/7
54
1000
1
10/9
78
1500
2/3
15/13

2
500
2
7/10
11
250
4
15/11
22
500
2
15/11
36
750
4/3
15/12
55
1000
1
12/11
80
1500
2/3
9/8

3
750
4/3
7/10
12
250
4
5/4
24
500
2
15/12
39
750
4/3
15/13
56
1000
1
15/14
84
1500
2/3
15/14

4
1000
1
7/10
13
250
4
15/13
25
500
2
6/5
40
750
4/3
9/8
60
1000
1
1
90
1500
2/3
1

5
1250
0.8
7/10
14
250
4
15/14
26
500
2
15/13
42
750
4/3
15/14
65
1000
1
12/13
100
1500
2/3
9/10

6
1500
2/3
7/10
15
250
4
1
27
500
2
10/9
45
750
4/3
1
66
1000
1
10/11
105
1750
4/7
1

7
250
4
1/10
16
500
2
15/8
28
500
2
15/14
48
1000
1
15/12
70
1000
1
6/7
120*
2000
1/2
1

8
2000
0.5
7/10
18
500
2
15/9
30
500
2
1
50
1000
1
6/5
72
1500
2/3
15/12
135*
2250
4/9
1

9
2250
4/9
7/10
20
500
2
3/2
33
500
2
10/11
52
1000
1
15/13
75
1250
0.8
1
150*
2500
2/5
1

Table 2.1: Delay box and chopper settings for specific pulse exposures. The delay box is set to a multipleof the 250μs period trigger from the laser amplifier. Using the specified wheel (Souter/Sinner) andharmonic/subharmonic setting of the chopper controller, a specific number of pulses is passed by the
chopper. The asterisk means the wheel spins too slowly at these exposure times to cleanly chop the beam.
2.3.3 B-Integral and Chirped Mirrors
I will now present the steps taken to mitigate nonlinear effects during transport of our intense IR
beam. Specifically, we reduce the value B, which is a measurement of the nonlinear phase accumulated
over distance. B is in general an integral, but as an estimate of the maximum nonlinear phase accumulated,
we calculate the B integral at the peak of our laser pulse, with a constant intensity along the full optical
path. This reduces to
𝐵 = 𝑘0 𝑛2 𝐼𝐿,

(1)

40
where k0 is the wavevector of the laser, L is the propagation distance, and 𝑛2 is the nonlinear component of
the refractive index from the Kerr effect (𝑛 = 𝑛0 + 𝑛2 𝐼). The peak intensity of the Gaussian laser pulse isgiven by𝑙𝑛(2) 𝐸𝐼 = 4√,𝜋 𝜏(𝜋𝑟 2 )(2)where τ is the full width at half maximum (FWHM) pulse duration, E is the pulse energy, and r is the beam
radius. Because B varies with intensity, the nonlinear phase accumulated varies across the width of thebeam. This can apply a quadratic phase to the beam and cause self-focusing, or it can cause more generalmode distortion.
To check if there is significant B integral for a given optical setup, there are a few tests that may be
performed. First, by un-seeding the amplifier by removing the oscillator input, the IR beam is changed fromhigh intensity, ultrafast pulses to low intensity, longer pulses of amplified spontaneous emission (ASE). If
the focal plane of a short focal length lens changes when unseeding the beam, this indicates the highintensity beam accumulates nonlinear phase as it propagates. The next check is to direct a low intensity
portion of the fully seeded beam into a spectrometer. If there is a fluctuating peak on the short-wavelength
end of the spectrum, this indicates there is self phase modulation during propagation. Lastly, the predicted
B value may be calculated using Eqn. 1 for every segment of the beam path, and every transmissive optic
the beam passes through. Performing this calculation for our experiment, I estimate a maximum B of 6 forthe combined optical path up to the beam splitter plus the probe line. The pump line has much lower
intensity, and will only have a lower B integral. As B measures phase, this value is large on the scale of pi,
indicating nonlinear phase is a problem. Ideally, B should be well below 1.
Considering Eqns. 1-2, the B integral can be reduced by reducing any of the values k0, 𝑛2 , I, or L.
Assuming the beam path is already as short as possible, L is fixed. For most of the beam path, k0 is also
fixed, though waiting until the final moment possible to do second harmonic generation helps to reduce B
as nonlinear phase generally accumulates faster for the shorter wavelength beam. This primarily leaves 𝑛2

41
and I as the experimentally adjustable parameters for B integral reduction. To reduce 𝑛2 , some materials
are better than others. For example, CaF2 glass has a lower 𝑛2 than other optical glasses, so the easy firststep to reducing B in our setup is to replace the silica lens focusing the probe into the HHG fiber with a
CaF2 lens. In the extreme case, 𝑛2 may be further reduced by doing all high intensity beam transport invacuum (𝑛2 =0). This is cumbersome, however, and not a practical solution for our setup, where multiplealignments and adjustments must be performed on the optics regularly. This leaves reducing I as the primarymethod for reducing the B integral.
To reduce I, we cannot reduce the pulse energy, E, as large pulse energies are needed to perform
HHG. Instead, we ensure that τ is as large as possible where nonlinear phase accumulation is occurring. We
do this by leaving the pulse not quite fully compressed out of the amplifier, for the majority of beam
propagation. We then fully compress the probe just before entering vacuum using chirped mirrors. The
pump is compressed before entering the chopper due to space constraints and the necessity of having acompressed pulse to perform SHG in the TG setup. As the pump is lower intensity than the probe, thisshould be tolerable. There is a separate pair of identical chirped mirrors in both the pump and probe lines.
Chirped mirrors are multilayer mirrors designed to have a negative group delay dispersion (GDD).
They compress the pulse by reflecting each wavelength in the optical pulse at a different depth in themultilayer stack. In this way, each wavelength has a different optical path length, and these differences canbe carefully designed to align the phases of each component wavelength, fully compressing the pulse.
Mathematically, a pulse is stretched by a positive GDD according to the equation2

𝜏 = 𝜏0 √1 + (4𝑙𝑛(2)

𝐷2
) ,
𝜏02

(3)

where 𝜏0 is the fully compressed pulse duration (again in FWHM), and D2 is the GDD. Using Eqn. 3, wethen calculate how many bounces from our D2 = −450fs2 chirped mirrors would be needed to compress apulse of a desired stretched length. Using six bounces, we fully compress a stretched pulse of 300fs duration.

42
Propagating this longer pulse until the chirped mirrors reduces the B integral coming from our 8m of
propagation in air from 1.9 to 0.2.
To fit six bounces on a pair of 2" chirped mirrors in both the pump and probe lines, we use compact
mirror geometries, as shown in Fig. 2.15(a-b). In the pump line, the beam is picked off before entering the
chopper and directed to the pair of chirped mirrors. After 3 bounces between the two mirrors, the beam iscaught on a dielectric mirror that is tilted slightly horizontally off from retro-reflecting. This directs thebeam back through the three bounces between the chirped mirrors, for a total of six, and leaves the beamhorizontally displaced enough from the input beam to be picked off with an additional mirror. This final
mirror then directs the beam through the chopper, and onward toward the rest of the experiment. For theprobe, the design is somewhat simpler. The beam bounces back and forth between the two chirped mirrors,
three times on each. One slight complication is that these multi-bounce mirrors must be used to align the
HHG fiber line, as they are the final mirrors before entering vacuum. I have found it is easiest to use thechirped mirror closest to the fiber to align to the first iris (before the fiber position), and the chirped mirror
furthest from the fiber to align to the second iris (after the fiber position).
Lastly, we utilize a larger beam radius, r, to get the final reductions in B. It is impractical for us toexpand the beam size everywhere after the amplifier using a telescope, though this could be done in more
extreme cases. Instead, we increase the beam size at its most critical position by moving the vacuum
entrance window closer to the HHG focusing lens, as shown in Fig. 2.15(c). This position is critical because
the beam is fully compressed, and 𝑛2 for any solid material is much higher than it is for air. By moving theentrance window as close as possible to the lens, the beam radius is close to its largest value, and this change
alone reduces our B integral by 2. Using a thinner entrance window would further reduce the B integral byan additional 0.4, but we have not implemented this at the present time.

43

Figure 2.15: Chirped mirrors and vacuum redesign to reduce B integral. a) In the pump line, the beam
is diverted to the chirped mirrors before entering the optical chopper. A nearly retro-reflecting dielectric
mirror sends the beam back along an output path (yellow dotted lines) nearly retracing its input path (redlines). b) In the probe line, the beam is bounced back and forth 6 times between the chirped mirrors. Pathsegments alternate between red and dotted yellow lines, for clarity. c) The entrance window to the vacuum
system was originally at the position marked by a dotted red oval. It was extended to be closer to the HHG
focusing lens so that the beam passing through the glass would have a larger radius, and lower intensity.
With all of the above changes in place, I calculate a B integral reduced to 1.15 from the initial 6.
To try to reduce it further would take more significant changes to our experimental apparatus. We have,
however, addressed the most significant issues, such as reducing the nonlinear phase accumulation during
propagation in air. Phase differences across the beam, even if small, can build to significant issues overlong propagation distances, such as ours. Thus, it is important to have low B in air, which starts directly
after the compressor and continues until entering vacuum. Conversely, the unavoidably high B in the final
lens and entrance window can be more tolerable because the propagation after them is shorter. That said,
the lens and vacuum entrance window are now the limiting factor in further reducing B, together
contributing ~0.8 to the total B integral.

44
2.4 Conclusions and Future WorkIn this chapter I have explained the most important aspects of our experimental methodology. I
have primarily focused on the details necessary for understanding Chapters 3-5, where the setup is used tocharacterize the thermal and elastic properties of nanoscale materials. I have also taken a large portion ofthis chapter to describe the significant upgrades made to the experimental apparatus during my PhD.
In particular, I have described the modular optical TG setup implemented in the pump line, alongwith how to switch between TG and uniform round beam pumping configurations. I have also described
the optical chopper, along with a camera triggering circuit used to rapidly obtain pump-probe
measurements. Finally, I described the steps taken to reduce nonlinear phase accumulation by our highintensity IR beam, most significantly by using chirped mirrors to compress the probe and pump pulses as
close to when they enter vacuum as possible.
In the near future, more upgrades to the experiment are planned. We are expecting a new amplifier
pump laser, and we have the components for an optical isolator to avoid back-lasing into our oscillator. We
are also expecting a direct blue-diode-pumped oscillator, and a higher repetition rate amplifier system. A
higher repetition rate will give better statistics for an enhanced signal-to-noise ratio, though the experimentwill then be more likely to suffer from pulse accumulation in low thermal conductivity samples. We arealso in the process of upgrading our beam pointing stabilization system to automate some of our alignment
procedure. We could also consider reducing the change in pump beam spot size with delay stage position
by collimating the beam to a larger size before it enters the stages. For an ideal Gaussian laser beam, larger
beam waists produce longer Rayleigh ranges, and the beam radius stays approximately constant for longer.
This experimental apparatus is a valuable tool for nanoscale metrology, and it should continue toprovide useful measurements well into the future. One of the most significant capabilities of our techniqueis the direct probing of non-diffusive thermal transport in nanostructured systems. This research is the topicof Chapter 3.

Chapter 3
Nanoscale Thermal Transport

Nanoscale thermal transport is becoming ever more technologically important with thedevelopment of next generation nanoelectronics, nano-mediated thermal therapies, and high efficiency
thermoelectric devices. However, direct experimental measurements of nondiffusive heat flow in nanoscale
systems are challenging, and first principle models of real geometries are not yet computationally feasible.
In recent work, we used ultrafast pulses of short wavelength light to uncover a new regime of nanoscale
thermal transport that occurs when the width and separation of heat sources are comparable to the mean
free paths of the dominant heat carrying phonons in the substrate. In this chapter, I now systematicallycompare thermal transport from gratings of metallic nanolines with different periodicities, on both silicon
and fused silica substrates, to map the entire nanoscale thermal transport landscape – from closely spaced
through increasingly isolated to fully isolated heat transfer regimes. By monitoring the surface profile
dynamics with sub-angstrom sensitivity, we directly measure thermal transport from the nanolines into thesubstrate. This allows us to quantify for the first time how the nanoline separation significantly impacts
thermal transport into the substrate, making it possible to reach efficiencies that are within a factor of 2 ofthe diffusive (i.e. thin film) limit. We also show that partially-isolated nanolines perform significantly
worse, because cooling occurs in a regime that is intermediate between close-packed and fully isolated heatsources. This work thus confirms the surprising prediction that closely spaced nanoscale heat sources cancool more quickly than when far apart. These results are important for benchmarking new theories that go
beyond the Fourier model of heat diffusion, and for informed design of nanoengineered systems.

46
3.1 Introduction and Motivation
Advancing technology through nanoengineering often requires an understanding of how nanoscale
thermal transport differs from its bulk counterpart. In modern electronics, characteristic dimensions are
already below 10nm – thus, thermal load management is already facing challenges imposed by nanoscale
thermal transport. In medicine, nanoparticle-mediated thermal therapies are being developed to treat cancer
and tumors with localized heating of those cells [1, 46]. Additionally, nanostructuring is a promising avenue
to engineer novel materials for efficient thermoelectric devices, either through enhanced scattering [47, 48]
or through phononic metamaterial effects [18, 49, 50]. For these and many other applications, informed
design must take into account that nanoscale thermal transport can be significantly slower than bulk model
predictions.
Fourier’s law of heat conduction, which accurately describes thermal transport in bulk materials,
relies on the fundamental assumption that the distances being considered are much larger than the averagemean free path (MFP) of heat carriers in the system, therefore establishing a smooth thermal gradient thatdrives transport in a material. As a result, when the relevant length scale of thermal transport is on the orderof the MFP of the heat carriers, Fourier’s law is no longer accurate because it dramatically overpredicts theheat transfer rate [7]. In this regime, some of the heat carriers travel ballistically from the heat source anddeposit their energy non-locally, producing non-diffusive (quasiballistic) thermal transport. Since thedominant heat carriers for dielectric and semiconducting materials are phonons, which can have MFPs fromnanometers up to several microns in materials such as silicon at room temperature [11, 20, 51, 52],
quasiballistic effects are observable in nanostructured systems, even at dimensions up to microns [10].
Observations of quasiballistic thermal transport have been made in several experimental
geometries, including 1D transport using transient grating experiments, ballistic transport through a thinslab of material whose thickness is shorter than the heat carrier’s MFP, and also by measuring the transport
away from nanoscale heat sources [8-10, 51, 53]. In this last category, effectively isolated heat sources were
initially used to study thermal transport as a function of heat source size. As the size of the heat source was

47
decreased, a monotonic deviation from the Fourier prediction was observed, due to the onset ofquasiballistic transport. More recently, the study of nanoscale heat sources that are not isolated has revealed
that not only the size of, but the spacing between, heat sources impacts the nature of thermal transport [20,54-56]. In past work, we made the first observation of a new collectively diffusive regime, where thethermal transport away from nanoscale hot spots can return towards the diffusive prediction when their
spacing is reduced below the dominant heat carrying phonon MFPs of the substrate [20]. This previously
unobserved phenomenon has promising technological applications. We also developed a predictive, periodand linewidth-dependent theory for the effective resistivity, which predicted that the period, or line spacing,mattered. However, in our initial work, we did not independently vary both the heat source size and spacing,or explore the transition between isolated and close-packed regimes, motivating the need for more
systematic studies to benchmark advanced models and theories.
Here, I present a systematic study that independently controls the nanoscale heat source size andspacing with laser heated nanolines fabricated on silicon and fused silica substrates. This allows us todefinitively validate our surprising prediction that nanoscale heat sources whose separation is comparable
to the phonon mean free paths of the substrate should cool faster than identical heat sources that are spacedmuch farther apart [20]. Using an effective Fourier model, we quantify the enhancement of nanoscale
thermal transport by engineering the heat source spacing: for example, for a constant 20nm-width nanoline,
the isolated heat source model [9] predicts an effective thermal boundary resistivity, reff, of 52nKm2/W,
nanolines spaced by 400nm have a measured reff of 9.2nKm2/W, nanolines spaced by 80nm have a measured
reff of 3.7nKm2/W, while the diffusive prediction for a uniform film is 2.45nKm2/W. Thus, the closelypacked nanolines exhibit thermal transport very close to diffusive, and varying their spacing makes it
possible to significantly tune their transport properties. We also explore the transition from collective, closepacked transport to isolated, quasiballistic transport, which represents a broad parameter space that is
important for optimized thermal design and for the development of full, first principles models. Finally, we
perform the same series of measurements on nanolines fabricated on a fused silica glass substrate, which
exhibit no changes in transport as a function of either heat source size or period. This behavior is expected:

48
the phonon mean free paths in fused silica are small compared to our experimental geometries, and thus
heat transfer from these nanolines should be well described using a diffusive model. This measurement
serves as a control to confirm that plasmonic or other effects in the nanolines do not influence ourmeasurement. I also briefly discuss how these new results compare to past results using different
measurement techniques, and describe potential theories for the underlying transport mechanisms.
3.2 Experimental Methods
We measure nanoscale thermal transport in silicon and fused silica using extreme ultraviolet (EUV)
nanometrology, as described in Chapter 2. For this study, the silicon and fused silica substrates are patternedwith periodic gratings of nickel nanolines via e-beam lithography. Using an ultrafast femtosecond laser
pulse, we impulsively excite these nanostructures, which serve as transducers for the thermal dynamics wemeasure. The excited thermal and acoustic dynamics deform the nanostructures and substrates, whose
composite surface is then measured very sensitively using coherent EUV beams generated by the highharmonic generation (HHG) process. All measurements presented here are obtained at room temperatureand high vacuum. See Fig. 3.1 for a diagram of the laser setup used in these measurements and Fig. 3.2 fora diagram of our pump-probe technique.

49

Figure 3.1: The EUV nanometrology laser setup. From the same amplified, ultrafast laser source, wepump and probe the nanostructured samples. The pump remains in the near-IR wavelength region, while
the probe is converted to the extreme ultraviolet region via high harmonic generation in an argon-filled
hollow waveguide. Pump-probe delay is controlled by a translation stage in the pump beam path, and a
shutter blocks and unblocks the pump beam to provide an unexcited reference measurement at each pumpprobe delay. The probe beam is focused onto the sample with a glancing incidence toroidal mirror, and thebeam diffracted from the sample is collected on a CCD camera.

50

Figure 3.2: Diagram of the pump-probe technique. Gratings of nickel nanolines of linewidth L andperiod P are fabricated on silicon and fused silica substrates. An ultrafast IR laser pulse heats the nanolines
and they impulsively thermally expand (shown in lower right). An ultrafast EUV pulse then probes the
sample as a function of time delay by measuring diffracted light using a CCD camera. The diffraction
efficiency for a thermally excited sample (red in inset) changes compared to an unperturbed sample (black
dotted line in inset). This change in diffraction efficiency captures the thermal and elastic dynamics that are
deforming the sample surface. Note the change in the diffraction pattern shown is magnified 150x for
illustration purposes.
3.2.1 Nanoline Grating Design and Excitation
To explore the various nanoscale thermal transport regimes, we choose grating linewidths andspacings that span from much smaller, to comparable, to greater than the average MFP, which is ≈ 300nm
in silicon [20, 57]. Therefore, we compare constant 25% duty cycle gratings to constant period (varying
duty cycle) gratings with linewidths ranging from 20nm to 1000nm. For the smallest linewidths (20nm to
100nm), the constant period was chosen to be 400nm so that collective effects would still be present, but
the reducing linewidth would probe the transition from the collectively diffusive regime to the isolatedquasiballistic heat source regime. Similarly, for linewidths from 100nm to 300nm, the constant period is
1500nm, where collective effects should be much weaker. A complete list of grating L and P characterized
by AFM is given in Tables 3.1 and 3.2. Example SEM images are given in Fig. 3.3.

51
Nominal L/P (nm)1000/4000
300/1500
300/1200
200/1500
200/800
100/1500
100/400
50/40050/20030/40030/12020/40020/80

L from AFM (nm)970 ± 30300 ± 20260 ± 10190 ± 20
188 ± 999 ± 886 ± 541 ± 741 ± 225 ± 123 ± 220 ± 419 ± 1

P from AFM (nm)4110 ± 40
1580 ± 20
1240± 101570 ± 20
842 ± 91520 ± 10
412 ± 5411 ± 5205 ± 2411 ± 5121 ± 1411 ± 582 ± 1

Duty Cycle
23.5%
19%
21%
12%
22%
6.5%
21%
10%
20%
6%
19%
5%
23%

Table 3.1: Linewidth and period of each nickel grating on silicon, measured by AFM. Note that the
duty cycles for the gratings marked in grey are systematically lower than the nominal value of 25%, but
still consistent, averaging 21.4±1.5%. The two sets of constant period gratings, marked in white, vary by
no more than 2% from a constant period, though the linewidths can differ from the corresponding constantduty cycle gratings by up to 13% for the large sizes, and 8% for the small sizes. The nanoline heights are
11±1nm. AFM data was analyzed using WSxM [58]. Uncertainty is calculated by standard deviation of
measurements and AFM image pixel size.
Nominal L/P (nm)1000/4000
300/1500
300/1200
200/1500
200/800
100/1500
100/400
50/40050/20030/40030/12020/40020/80

L from AFM (nm)940 ± 40280 ± 10280 ± 20
188 ± 9190 ± 1090 ± 1096 ± 645 ± 443 ± 325 ± 224 ± 219 ± 415 ± 4

P from AFM (nm)3930 ± 40
1500 ± 101200 ± 101500 ± 10804 ± 81510 ± 10
405 ± 4407 ± 4204 ± 2406 ± 4123 ± 1406 ± 483 ± 1

Duty Cycle
24%
19%
23%
12.5%
23.5%
6%
24%
11%
21%
6%
19.5%
4.7%
18%

Table 3.2: Linewidth and period of each nickel grating on fused silica, measured by AFM. The
nominally constant duty cycle gratings are highlighted in grey. For this series of gratings, the duty cycle
systematically decreases for the sub-100nm linewidth gratings, but as we observe no trend in the effective
resistivity for these gratings, this is not significant to our results. The series of constant period gratings,
marked in white, show excellent consistency in period, and the linewidths compare well to the constantduty cycle gratings for all but the 20nm linewidth. The nanoline heights are 11.2±0.5nm. AFM data wasanalyzed using WSxM [58]. Uncertainty is calculated by standard deviation of measurements and AFM
image pixel size.

52

Figure 3.3: SEM images of the fused silica sample. a) A grating with 30nm linewidth (L) nanolines and120nm period (P). b) A grating with 30nm linewidth and 400nm period. Note the measurements made by
SEM are slightly larger than the values measured by AFM. Only AFM values for linewidth and period aretabulated above because a comprehensive set of measurements by SEM was unavailable. The slightsystematic offset between the two characterizations does not change our trends or conclusions.
By comparing the measured thermal transport from each of these gratings, we directly probe theeffect of heat source spacing on nanoscale thermal transport. The constant period set of gratings produces
increasingly isolated heat sources as the linewidth is decreased, whereas the constant duty cycle set ofgratings brings the heat sources closer together as the linewidth is decreased. Each grating covers an areaof 150μm2 and each nanoline is 11nm tall on average.
We excite the nanolines using an ultrafast infrared laser pulse with a central wavelength of 780nm,
with a 25fs pulse duration, and at a 4kHz repetition rate. The nickel gratings preferentially absorb the laser
light, with no significant heat deposition in the substrate underneath: fused silica is transparent to thiswavelength, while silicon has a >8μm penetration depth. The nanolines are thinner than the optical
penetration depth of Ni, ensuring uniform heating. After impulsive heating, the nanolines thermally expand
within ~20ps, launching acoustic waves that are visible in our raw data, as shown in Fig. 3.4. (In Chapter
4, these acoustic waves were used to characterize the elastic properties of ultrathin films.) After expanding,the Ni nanolines relax back to their unperturbed surface profile, as heat is transferred into the substrate on
few nanosecond time scales. The maximum experimental time window is 8ns, as determined by amechanical delay stage in the pump line.

53

Figure 3.4: Dynamically changing diffraction signals for 20nm linewidth gratings on silicon. The raw
data directly show that closely spaced nanolines cool faster. Multifrequency acoustic oscillations in thesubstrate visibly modulate the slow thermal decay of the signal. Left: 20nm nanolines at a 400nm period
do not fully cool, even after 1ns. Right: 20nm nanolines at an 80nm period have almost completely cooled
within 1ns.
Note that we assume the measured system response is linear with pump power. We check this bynormalizing experimental signals from the same grating at several different laser pump fluences andconfirming that the dynamics are the same. We confirmed this for all the gratings at all the pump fluences
listed in Table 3.3. We also assume there is uniform pumping of the grating over the probed region. For
this study, the pump beam diameter is 320μm on average, and the probe spot size is ~50μm radius, so weshould meet this condition and be mostly insensitive to transport along the length of the nanolines.

54
2Power (mW)

Pump diameter (μm)

Fluence (mJ/cm )

45
35
35
31
30
38
40
50
50
50
47
47
47
47
60
60
60
60
65
60
110
135
150
175
125
190

-

-

300
340
350
400
400
300
360x390400x436440x500

11.0
8.3
13.0
12.0
12.0
21.2
24.9
22.8
27.4

Table 3.3: Pump beam average power, spot size at the sample plane, and average fluence. Note thatthe spot size was not recorded every day during the experiment. When the beam was oblong, an averageradius was used to estimate the fluence. The averages of pump power, spot size, and fluence over the courseof the experiment are 70.8mW, 323μm, and 17.0mJ/cm2, respectively. Note that for all the data presented,
at all the fluences listed, there was no discrepancy between change in diffraction efficiency signals after
normalization. This indicates our observations are linear with pump fluence.
3.2.2 Probing and Measurement Sensitivity
We use coherent EUV beams to directly measure changes in the nanosystem surface profile. We
generate the EUV beams at a central wavelength of 29nm through high harmonic generation [36]. This isan extreme nonlinear quantum process that up-converts IR light into the EUV region by focusing the laser
pulse into an argon-filled waveguide. Since we use the same Ti:sapphire laser for both the excitation pulse
and also to generate the probe EUV pulse, the timing stability between our excitation and probe pulses isin the sub-femtosecond range. In addition, EUV wavelengths are far from any electronic resonances in thenanosystem, so that the EUV reflectivity is largely insensitive to the electron temperature [41]. Thus, our55
technique predominantly probes lattice expansion and surface deformation, even for very short times after
excitation (few picoseconds).
The diffracted EUV beam probes the dynamically changing sample surface, and is collected on a
CCD camera, as shown in Fig. 3.2. The diffraction efficiency, i.e. the relative power in the direct beam anddiffracted orders, is a function of the height difference between the tops of the nanolines and the substrate.
Thus, by monitoring the change in diffraction efficiency as a function of excitation-probe delay time, wemeasure the thermal and elastic dynamics in both the nanolines and substrate, with picometer scalesensitivity (see Fig. 3.4 for an example) [41].
Our measured signal closely follows the lattice temperature of an average nanoline; this differsfrom the temperature difference measured by the transient thermal grating (TTG) technique [29]. Thisdistinction is important to keep in mind, as TTG signal is lost once the peak-valley temperature differenceis lost, even if the average temperature of the sample is above the pre-excitation value. Thus, for shorter
period gratings, there is a shorter distance to transfer heat, and the signal should decay faster as the peakvalley temperature difference washes out. This effect mimics our observed period dependence, but it isdifferent in origin.
By using a separate metallic grating as our heat source, we measure the thermal expansion of thenanolines for any temperature above the pre-excitation value, even in the case that the top layer of thesubstrate has thermalized. Additionally, our heating geometry generates inherently two-dimensional
thermal transport in the substrate – one dimension cross plane and one dimension in plane. Because themetal nanolines are uniformly pumped and thermalize internally in the first few picoseconds after laser
excitation, our signal tracks this 2D thermal transport away from the resulting heat sources at the substratesurface. Traditional TTG measurements only measure the 1D thermal transport in-plane.
To confirm that a 1D TTG interpretation of our signal (peak-valley temperature difference washingout, as shown in Fig. 3.5(c)) with diffusive thermal transport does not explain our diffraction measurements,
we perform finite element analysis (FEA) calculations of the thermal transport from the L=20nm, P=80nm
and L=20nm, P=400nm gratings on silicon with identical thermal boundary resistivities. We compare the56diffraction signal for each simulation to both the peak-valley temperature difference in the substrate andthe temperature in the center of the nanostructure. As shown in Fig. 3.5(a,b), the diffraction signal nearly
identically follows the temperature of the nanostructure, not the temperature difference in the substrate.

Figure 3.5: Measurement sensitivity to nanostructure and substrate temperatures. Simulations of thetwo L=20nm gratings on silicon are shown, both with r=5nKm2/W. I compare the change in diffraction
efficiency signal to the temperature at different points in the sample. Black lines are change in diffraction
efficiency signals; cyan lines are the temperatures at the nanostructure centers; red lines are the peak-valley
temperature differences in the substrate between the two red points shown schematically. a) For the 80nm
period grating, the change in diffraction efficiency signal nearly identically tracks the temperature of thecenter of the nickel nanostructure but does not follow as closely the peak-valley temperature difference inthe substrate. b) For the 400nm period grating, the change in diffraction efficiency signal again follows thetemperature of the center of the nanostructure. c) The peak-valley temperature difference in the substrate
goes to zero faster for the 80nm period grating as there is a shorter distance to transfer the heat, comparedto the 400nm period grating. This temperature difference is similar to what is of interest to 1D TTG, butdoes not match our diffraction signal, as shown in (a) and (b). d) When both gratings have the same effective
resistivity, the change in diffraction efficiency signal for the 80nm period grating (solid line) is slightly
slower to decay than the 400nm period grating (dash-dot line). Thus, a period dependence of the thermaltransport is required to model our experimental results.

57
We further checked the sensitivity of our measurement to the temperature difference in the substrate
washing out by enhancing the substrate coefficient of thermal expansion (CTE) in the FEA calculations.
As shown in Fig. 3.6, for the L=20nm gratings, substrate CTE values up to ten times nominal do not change
our diffraction signal, confirming diffraction is indeed predominantly sensitive to the nanostructure
expansion at small sizes.

Figure 3.6: Measurement sensitivity to the substrate coefficient of thermal expansion. FEA
calculations shown are performed in the quasistatic approximation, at a constant thermal boundary
resistivity, with a uniform excitation. A two-temperature model is used to couple the electrons and phonons
in the nanostructure, which provides the rapid thermal expansion at the earliest delay times [59]. a) For the
20nm linewidth gratings, whether 80nm or 400nm period, changes to the substrate coefficient of thermal
expansion up to ten times the nominal value do not significantly affect the diffraction signal. b) For large
grating sizes, where there is a large volume of substrate between nanostructures, we observe more
dependence of the signal on substrate CTE. This cannot, however, explain the period dependence of thermaltransport we observe, as the period dependence only manifests for small gratings, where the sensitivity to
substrate CTE is minimal, as shown.

58
3.2.3 Finite Element Analysis to Fit the Data
We analyze the change in EUV diffraction efficiency signal by fitting with an effective Fouriermodel, implemented in the FEA software, COMSOL [60]. See the appendix for more details. FEA
calculations use literature values for the nickel, silicon and fused silica properties, as listed in Table 3.4.
The thermal boundary resistivity at the nickel-substrate interface is taken as the effective fitting parameter.
We simulate the thermal expansion and cooling of the nanolines for a variety of effective resistivities, andthen calculate the diffraction of a Gaussian EUV beam from the dynamically changing surface using Fresnel
propagation. The resulting calculated change in diffraction efficiency signal is then fit to the experimental
signal to determine which effective resistivity best fits our experimental data. This effective resistivity
contains both the intrinsic thermal boundary resistivity between the nanolines and the substrate, as well as
an effective correction that is larger for greater deviations from diffusive thermal transport near thenanolines. The intrinsic thermal boundary resistivity should be constant across all gratings, independent ofsize and spacing, as all gratings on a given substrate were fabricated simultaneously. Therefore, trends inthe best fit effective resistivity as a function of experimental geometry are due to the measured thermaltransport’s deviation from the diffusive prediction.
Material Properties
NickelSilicon[20]
cp at 300K (J/kgK)
456.8
710.0 [61]kbulk (W/mK)
90.9 [62]149.0 [62][64]
Poisson’s Ratio
0.31
0.27 [64][66]Young’s Modulus (GPa)200
131 [67]
α, linear coefficient of
12.77 [68]
3 [69]-6
thermal expansion (10 /K)
ρ, density (kg/m3)
8910 [62]2330 [62]Table 3.4: Parameters used in FEA multiphysics simulations.

Fused silica
830.2 [9]1.19 [63]0.17 [65]72 [65]0.54 [65]2650 [70]59
3.3 Results and Discussion
Using the approaches described above to obtain and analyze our data, we characterize nanoscale
thermal transport in silicon and fused silica. To fit and interpret the effective resistivity parameter obtained
from COMSOL, we use the suppression function approach from [20].
3.3.1 Collectively Diffusive Model
Here, I summarize the suppression function approach used to fit our measured trends, as describedin detail in [20]. The collectively diffusive model relates each grating geometry to all the phonon modes’
contributions to the thermal transport, according to the suppression function formalism [57, 71]. The
different MFP phonon modes’ contributions to the total thermal conductivity are suppressed depending on
the linewidth and reintroduced depending on the period of the grating. The suppression function is definedas
𝐿𝑃

𝑆(𝐿, 𝑃, 𝛬) = 𝑡𝑎𝑛ℎ (2𝛬) + [1 − 𝑡𝑎𝑛ℎ (2Λ)],(1)

where Λ is the MFP of a given phonon mode. Applying this suppression function to the bulk differential
thermal conductivity spectrum of the substrate, k(Λi), we obtain an effective nanoscale conductivity foreach grating geometry,
𝐾𝑛𝑎𝑛𝑜 = ∑ 𝑘(Λ 𝑖 ) ∙ 𝑆(𝐿, 𝑃, Λ 𝑖 ) ,(2)𝑖where the sum is over each MFP mode, Λi. The model for the collectively diffusive effective resistivity isthen defined by𝑃

𝑟𝑒𝑓𝑓 = 𝑑 ∗ 𝑙𝑛 ( 𝐿 ) (𝐾

1

𝑛𝑎𝑛𝑜

− 𝐾

1

𝑏𝑢𝑙𝑘

) + 𝑟𝑡𝑏𝑟 ,(3)

where rtbr is the intrinsic thermal boundary resistivity, Kbulk is the bulk thermal conductivity of silicon, andd can be interpreted as the depth into the substrate where transport is affected by the nanoscale geometry.
We additionally compare our results to the isolated quasiballistic heat source model for 1D gratings,
as presented in [9, 20], where no collective effects take place. The predictions of this model represent an60
upper limit to the transition region between collectively diffusive and isolated quasiballistic transport,
which our measurements explore. The isolated quasiballistic model gives an effective resistivity of𝑟𝑖𝑠𝑜𝑙𝑎𝑡𝑒𝑑 =4Λ𝑔𝑟𝑎𝑦 𝑑+ 𝑟𝑡𝑏𝑟 ,3𝐾𝑏𝑢𝑙𝑘 (𝐿⁄2)𝑙𝑛(𝑃⁄𝐿)(4)

where Λgray is the gray MFP approximation for the substrate (single, average MFP). Note that this model
does not take the fully ballistic limit into consideration, so it may not be valid at extremely small grating
sizes. It is presented here as a reference limiting case, using a previously-derived model.
3.3.2 Silicon and Fused Silica Results
First, we compare a series of gratings with constant period with another series of gratings with 25%
duty cycle, both on a silicon substrate. For the 25% duty cycle gratings, as the linewidth is decreased from
1000nm to 20nm, the effective resistivity first increases due to the quasiballistic deviation from diffusivetransport, then decreases due to collective effects, as shown in Fig. 3.7(a) (blue points). At the smallest
sizes, the effective resistivity approaches the diffusive, uniform heat source limit, where there is only the
intrinsic thermal boundary resistivity, which has no effective correction due to nanoscale effects (black
dashed line, Fig. 3.7(a)). The reduction in effective resistivity occurs when the period becomes comparable
to the dominant heat carrying phonon MFPs in silicon at room temperature (≈300nm). These findings arein good agreement with our previous work, as illustrated in Fig. 3.8.

61

Figure 3.7: Best fit effective thermal boundary resistivity for nanolines on silicon and fused silica.
a) For silicon (gray MFP≈300nm [20, 57]), as the linewidth is reduced below 50nm, the effective resistivityof the constant 25% duty cycle gratings (blue points) decreases, which is a return towards the diffusive
prediction (thin film case, black dashed line). In contrast, the constant 400nm period gratings’ resistivities
increase as the linewidth is reduced (red points), mapping the transition from dominant collective effects
towards isolated quasiballistic thermal transport (isolated prediction, black dotted line). The blue solid lineis the 25% duty cycle collectively diffusive prediction from [20], whereas the red dashed line is a fit withthe same model to the new data. b) For fused silica, the resistivities of constant duty cycle (blue points) and
400nm period (red points) gratings all agree within error bars. This indicates that there is no non-diffusivetransport that depends on experimental geometry for this short phonon MFP material (~2nm [9]), and that
there are no dominant systematic errors coming from the grating geometry independent of the sample
substrate material. The dotted line is the average of all grating resistivities on fused silica (6.7nKm2/W).
Error bars in (a) and (b) are calculated by the standard deviation of best fit values from multiple
measurements.

62

Figure 3.8: Comparison to previous measurements on silicon. The data of Hoogeboom-Pot, et al. [20]are shown in green and the present data are shown in blue. The two samples were fabricated from two
different silicon wafers at two different times, so we expect a constant offset between the two data setscoming from different intrinsic thermal boundary resistivities, which are determined by fabrication
conditions. To compare trends in the two data sets in this plot, we subtracted a constant 1.56m2nK/W fromall points in the current data set. This constant offset was calculated as the average difference between
previous and current resistivities, weighted by the current study’s uncertainty, of the 30nm and 100nm
gratings, which are the only identical geometries between the two data sets. We observe good agreement
between the previous and current studies. This same experimental offset was added to the collectively
diffusive model fit, intrinsic rtbr fit, and isolated quasiballistic model fit from the previous study to plot themwith the new data in Fig. 3.7(a).
As the grating period is increased for a fixed nanoline width, we would expect that thermal transport
will become less efficient, if the models developed in [20] are valid. Fig. 3.7(a) plots the effective resistivity(red data points) as the period is increased from the collectively diffusive regime (blue data points). These
values are seen to increase significantly when the period is increased, confirming model predictions.
Considering now the trend in the red data points alone, the grating linewidth is reduced from 100nm to
20nm at a constant 400nm period. Thus, the heat sources become increasingly isolated and the increasing
effective resistivity maps the transition from collectively diffusive toward isolated quasiballistic transport.
To successfully fit the 400nm period data, we use the collectively diffusive model described by Equations
1-3, applied to all the new data. As shown by the red dashed line in Fig. 3.7(a), this predictive model

63
provides good fits, even in the intermediate regime, where heat sources are neither isolated nor closely
spaced.
These measurements on silicon represent the direct observation that the more closely spaced heatsources cool more quickly for sufficiently small periods at early times. At larger periods and linewidths,
we do not observe this effect within our uncertainty, as shown in Fig. 3.9(a). Our observed trends fordifferent periods over the full range from 20nm to 300nm linewidth gratings with the full effective Fourier
analysis have not been previously reported and reveal that bringing heat sources closer together can bring
the thermal transport within a factor of 2 of the diffusive prediction, even for heat source sizes far below
the dominant heat carrying phonon MFPs. We further conclude that this effect is only dominant when
grating periods are below the substrate MFPs, which is further supported by our control study on fusedsilica. We note that our best fit effective resistivities in the present work are close to the resistivity calculated
from the corresponding thermal boundary conductance measured by time-domain thermoreflectance(TDTR): 5.1±0.3nKm2/W, converted from [72]. Some difference is expected due to differences in sample
fabrication and the effective correction we add to the intrinsic thermal boundary resistivity to capture nondiffusive effects in the substrate.

Figure 3.9: Results for the constant 1500nm period series of gratings. I compare constant period
gratings (red diamonds) to constant 25% duty cycle gratings (blue points) at large linewidths. a) For silicon,
we observe no period dependence in effective resistivity within error bars. This lack of period dependence
is likely because these grating periods are still large compared to the mean free paths of the dominant heat
carrying phonons in silicon at room temperature (≈300nm), and because the percent change in periods forlarge linewidths is not as great as in the small linewidth case, where period dependence was observed.
b) For fused silica, the 1500nm period gratings agree with both the 25% duty cycle gratings, and the averageof resistivities fit for all grating geometries on fused silica (dotted line). This indicates there is no deviationfrom the diffusive prediction, as expected for this short MFP (~2nm) material.

64
Next, we perform the same series of measurements on nanolines fabricated on a fused silica glass
substrate, where the phonon mean free paths (≈ 2nm) are small compared to our experimental geometries,
and therefore no deviation from diffusive transport would be expected. The phonon mean free paths of
fused silica can be estimated as only a few nanometers at room temperature [9], and non-propagating modes(diffusons) that contribute to the thermal conductivity should not be affected by our geometries [53, 73].
Thus, these data serve as a control to confirm that plasmonic or other effects in the nanolines do not
influence our Si measurements. Indeed, as shown in Fig. 3.7(b), the effective resistivity that best fits ourdata is constant, within error bars, for all linewidths and periodicities. It should be noted that the error bars
for the fused silica data, calculated from the standard deviation of multiple measurements, are larger thanin the silicon case due to two factors. First, experimental noise was larger, and second, our fitting has
decreased sensitivity for low conductivity samples where the thermal transport into the substrate isdominated by substrate conductivity rather than thermal boundary resistivity. The thermal boundary
resistivities we fit for fused silica are consistent within error bars with previous measurements using thistechnique, with only a small offset we attribute to differences in the glass substrates and sample fabrication
conditions [9].
3.3.3 Comparison to Other Experiments and Theory
Some care must be taken when comparing the results of the current study to TDTR measurements
reported in the past on nanostructured systems [54-56]. We note that this comparison can be beneficial for
an improved understanding of the physics that is present in all three cases. Zeng, et al., use periodic nanolinetransducers, like we do, but with a constant sub-wavelength gap to avoid directly optically exciting theirsubstrates of interest. This results in a decreasing duty cycle with decreasing heat source size, so theirmeasured monotonic deviation from the diffusive prediction is not necessarily inconsistent with our results.
Indeed, their analysis shows that higher duty cycle gratings have higher effective conductivities, which
agrees with our direct observations.

65
Oyake, et al., and Hu, et al., both also measure fused silica substrates, but both use different
approaches compared to us to obtain nanoscale heat sources. Oyake, et al. use aperiodic gold nanoislands
formed by annealing a gold film on their substrate of interest. While the authors’ results are generally
consistent with the other TDTR experiments, they observe suppressed thermal transport in fused silica as afunction of heat source size. This is inconsistent with both our results on fused silica, and those of Hu, etal., where no size-dependence is observed down to 30nm. I tentatively attribute this difference to ourdifferent transducer designs, though the effects of periodicity versus aperiodicity are beyond the scope of
the present work. Hu, et al., use a hybrid transducer to generate nanoscale 2D heat sources at a constant
50% duty cycle without directly optically exciting their substrates of interest. Their measurements of fusedsilica agree with ours – no non-diffusive transport is observed. The authors also show results for 2D(nanocube) gratings of constant period but decreasing linewidth on silicon, and they observe a similar
monotonic deviation from the bulk Fourier prediction as we observe for that case.
All three TDTR measurements differ from our results, however, in that they never observe a returntowards the diffusive prediction at constant duty cycle, when the heat source period is comparable to thedominant substrate phonon mean free paths. The reason for this difference might come from the different
experimental temporal frequencies, as suggested by recent work by Hua and Minnich [74].
In their work, Hua and Minnich cast the Boltzmann transport equation (BTE) for phonons into thespatial frequency domain, under the relaxation time approximation, and apply it to nanoscale gratingtransducer geometries. They find that, due to the interplay of cross-plane and in-plane quasiballistic effects,
thermal transport can indeed deviate from, and then return to, the diffusive prediction at constant duty cycle,
but this will happen at smaller grating sizes for higher temporal frequencies of heating. For our EUV
nanometrology measurements, we use a laser amplifier system to generate high harmonics at a few kilohertz
repetition rate. In contrast, the TDTR measurements use laser oscillators with a repetition rate of ≈80MHz,
modulated in the megahertz range as well. This difference might explain the difference between the EUV
nanometrology and the TDTR measurements. Note that the authors also predict a greater deviation fromdiffusive transport for smaller duty cycle gratings, which is consistent with both our work and Zeng’s.

66
Other theoretical approaches are in development that are very promising for explaining thefundamental mechanism behind our observed return towards the diffusive prediction. First, hydrodynamic
and super-diffusive views of the BTE show lower temperatures near nanoscale heat sources than thetemperature predicted by effective Fourier models [75-77]. This indicates these models capture physics
excluded by effective Fourier approaches. The hydrodynamic formulation is particularly useful, as itprovides a physical intuition for how quasiballistic effects appear as a vorticity of the heat flux, and it
captures sufficient complexity to be fully predictive of our data, down to L=300nm, using only firstprinciples inputs [78]. However, the present form of this model has been shown to not hold for constrictions
smaller than twice the hydrodynamic length (176nm for silicon at room temperature) [79]. This could be
the cause of the model’s loss of predictive power when the experimental measurements return towards
diffusive behavior at the smallest sizes. We are collaborating with the Alvarez group from UniversitatAutònoma de Barcelona to see if modifications to the level of approximation currently in the hydrodynamic
approach can explain our experimental findings.
Finally, it has been predicted that coherent effects such as local resonance can modify the thermaltransport in nanostructured systems, even at room temperature [18, 50]. According to lattice dynamics andmolecular dynamics (MD) calculations, nanostructures can behave as local resonators that hybridize withphonon branches in the substrate to create a lower group velocity, and thus slower thermal transport. Thisis a metamaterial effect which occurs at room temperature and tens of nanometer dimensions. In contrast,
holey-membrane-based phononic crystals rely on phonon coherence effects that require low temperatures
or few nanometer dimensions to have hole periods comparable to the phonon wavelength [80, 81]. Thus,
local resonance effects could be more plausible in our experiment, though the strongest effects are expected
for nanostructures with much higher aspect ratio than those used in our experiment [50]. Currently, themechanism behind the collectively diffusive regime is unclear, but we are pursuing MD simulations of ourgeometries in order to understand the mechanics governing our observations at an atomistic level.
In collaboration with the Hussein group at CU Boulder, we have performed non-equilibrium MD
simulations of heated silicon nanolines on a silicon substrate. This avoids the ambiguities related to heat

67
transfer across dissimilar material interfaces and focuses instead on effects related to periodic heating andnanostructuring. As shown in Fig. 3.10(b), the temperature profiles obtained by MD exhibit the same effectpredicted by hydrodynamic and superdiffusive models, where the temperature near the nanostructure islower than that predicted by diffusion. Significantly, as we reduce the nanostructure size at constant 50%
duty cycle, we observe the temperature profile first deviate from the diffusive prediction and then returntowards diffusive behavior, as calculated by FEA (Fig. 3.10(a)). This agrees with our experimental findingsof the collectively diffusive regime, and we are now further analyzing the MD data to determine what themechanism is for the return towards diffusive behavior for the smallest grating linewidths and periods.

Figure 3.10: MD temperature profiles compared to the diffusive prediction. a) Using FEA, we calculate
the temperature field in a silicon substrate beneath three different nanoline widths, assuming diffusivetransport. b) Using MD, we calculate the local temperature for bins of atoms in the substrate, using thesame geometries as FEA. Note that the pointed temperature contours in the middle of the simulation cells
are artifacts from using only three bins in the in-plane direction, but this does not change our conclusions.
Comparing (a) and (b), we observe the largest linewidth results agree well. However, for the middle
linewidth, MD shows the substrate surface furthest from the nanoline is cooler than the diffusive result.
This leads to a “lightbulb” temperature field in the substrate not obtained using FEA. At the smallest
linewidth, this effect is reduced in MD, and the temperature field bears more similarity to the diffusive
prediction by FEA. The duty cycle of all gratings shown is 50%. Simulations courtesy of Hossein Honarvarin (b), and Joshua Knobloch in (a).

68
3.4 Conclusions and Future Work
We use ultrafast, coherent EUV beams to measure the thermal transport away from periodic
gratings of laser heated nickel nanolines with varying linewidth and spacing on both single-crystal silicon
and amorphous fused silica substrates. We observe quasiballistic transport that deviates from the diffusive
prediction for nanolines on silicon and model this behavior with a higher effective thermal boundary
resistivity between the nanolines and the substrate. We observe that quasiballistically suppressed thermaltransport returns towards the diffusive prediction when the nanoline spacing is comparable to the dominantphonon mean free paths in the substrate. This period dependence confirms our previous prediction thatclosely spaced heat sources can initially cool faster than widely spaced ones and is now quantified to be asignificant effect in silicon, bringing the thermal transport within a factor of 2 of the diffusive, uniform heatsource limit. Measurements on a fused silica substrate displayed no quasiballistic behavior down to heatsource sizes of 20nm. Our observations are in general agreement with other experimental and theoreticalworks. More work is needed to describe the fundamental physics behind these collective effects, and weare pursuing first principles models to determine the fundamental mechanisms underlying these behaviors.
In the near future, we expect to have a comprehensive explanation of our experimentally observed
phenomena, using a combination of hydrodynamic thermal transport models and MD simulations. We will
also perform new measurements of nanoscale thermal transport at sizes that reach fully into the ballistic
regime, where no phonon modes behave diffusively. In this regime, we can expect thermal decay times
comparable to the lifetime of phonon modes. Here, it may be useful to consider the coherent oscillation ofphonons, which is the subject of the next chapter. Chapter 4 presents our work using coherent acoustic
waves to characterize the elastic properties of nanoscale materials.

Chapter 4Thin Film Elastic Properties

Thin films from micron to nanometer to single atomic layer thicknesses present a rich field of study,
both for practical applications and fundamental materials science. Thin films are an enabling technology
for items we use daily, such as scratch-resistant eyeglasses or protective linings in soda cans. Extending to
the nanoscale, patterned ultrathin films form the basis of all modern integrated circuits. From a fundamental
perspective, ultrathin films are a class of nanomaterials, which can exhibit significantly different properties
from bulk materials due to their nanoscale thicknesses. There is much work still to be done in exploring
how different materials change once made ultrathin, where surfaces and interfaces become a significant
fraction of the material volume. One change that takes place at the nanoscale, which is also very difficult
to measure experimentally, is deviations in the elastic properties that govern the mechanical performanceof ultrathin films.
In this chapter, I will describe our work characterizing the elastic properties of ultrathin films down
to 5nm thickness. I will first survey the main areas of interest for these films (semiconductor applications
and fundamental thickness-dependent changes) and then describe our experimental approach to characterize
the full elastic tensor of these films. I will then present results on a series of films with varyinghydrogenation levels, and a series of films with varying thicknesses.
4.1 Introduction and Motivation
The number of transistors on a chip has doubled every two years for decades, an observation known
as Moore’s Law. To continue increasing computing power at this rate, the semiconductor industry needs
materials innovation beyond simply making systems smaller. One significant challenge that requires new

70materials to solve comes from scaling down the metal interconnect structures that electrically connect thedifferent layers throughout an integrated circuit. Each pair of closely spaced metal wires in the interconnect
and routing structure, shown in Fig. 4.1(a), can be thought of as a parallel-plate capacitor, as illustrated inFig. 4.1(b). In essence, the metal interconnects and routing form an unintentional network of resistorcapacitor (RC) circuits. This decreases the efficiency of the device as power is lost to parasitic capacitance[82]. It also limits the switching speed of the device once the characteristic relaxation time for RC circuits(τ = RC) approaches the transistor speed [12]. This problem becomes substantial as device dimensions are
scaled down, because shrinking the wires increases their resistance, and bringing them closer together
increases their capacitance. To overcome the challenges of parasitic capacitance and RC time delay, there
is a great need for new low dielectric constant (k) materials to serve as the interlayer dielectric and reduce
the overall capacitance of the interconnect structure.

Figure 4.1: Low-k dielectrics for metal interconnects. a) Cross section of metal interconnect structure atthe 90nm node. b) Schematic of the resulting RC circuits. c) Schematic of SiO2 with few terminal groups
added, and the rigid network emphasized in bold. d) Schematic of SiO2 with many added terminal groups
and a less interconnected network. a) and b) from [12], c) and d) modeled after [83].

71
To lower the dielectric constant of materials, one approach is to fabricate materials with fewer polarbonds. In practice, this has meant either introducing terminal CH3 groups into the amorphous SiO2
historically used as the interlayer dielectric (shown in Fig. 4.1(c)) or using less polar materials like SiC.
Additional hydrogenation for Si-H bonds further reduces the dielectric constant. Note the other main
approach to engineering new low-k materials is to introduce porosity or air gaps. While all the results inthis chapter are for non-porous films, I will show measurements of porous silicon in Chapter 5.
The main challenge in these routes to lowering k is that hydrogenation and porosity also change theelastic properties of these materials, which in the extreme can lead to mechanical failures of the device,
such as delamination of layers. Hydrogenation terminates bonds and lowers the network connectivity of theoverall material, as illustrated in Fig. 4.1(d). Once a critical level of network connectivity is reached, thematerial loses its rigidity, and several material properties, including Young’s modulus, Poisson’s ratio,
thermal conductivity, and electrical resistivity behave fundamentally differently [22, 82, 84]. Thus, it isvery important to have reliable measurements of low-k films’ elastic properties to feed forward and informthe design of integrated circuit architectures.
The other consideration for the elastic properties of ultrathin films, beyond their chemicalcomposition, is the presence of changes that come from nanoscale dimensions alone. The fundamental
mechanisms of thickness dependence in elastic properties are generally related to the high proportion of
atoms at the surface of the material, as opposed to in the bulk volume. However, the final effect on elastic
properties varies from system to system and can make the film either softer or stiffer than the bulk prediction[13, 85]. Additionally, high residual strain in thin films [86], fabrication-dependent changes like grains andislands in thin film deposition [87, 88], or density gradients and interlayer mixing [89, 90], can all lead todifferent elastic properties from bulk. In the present work, however, I focus on elastic property deviations
coming fundamentally from surface effects.
Atoms at the surface of a material generally have a lower coordination number, or number ofnearest neighbors, than do atoms in the bulk. This can be imagined by cutting through an infinite atomic
lattice to form an outer surface, leaving dangling bonds on the surface atoms, reducing their number of

72
nearest neighbors. In reality, these dangling bond defects will be passivated, for instance by H, but suchterminal groups will still reduce the average coordination number of the surface [84, 91]. Note that surfaces
can be made that are not undercoordinated, even down to a monolayer of material, depending on how thefilm is grown [92]. Indeed, defect-free sheets of graphene have been shown to reach the intrinsic strengthof the carbon bonds – the highest strength achievable for that material [93]. For most materials, however,this will not be the case.
When surface atoms are under-coordinated, this can make the surface softer or stiffer due to twocompeting effects. Surfaces soften due to bond loss and under-coordination, making them less rigid [85].
This mechanism is conceptually similar to how terminal groups in the bulk of low-k films reduce network
connectivity and soften the entire film. Softening in thin films has been experimentally observed in systems
such as TiN films (along with increasing strain) [94], in nanoscale Si cantilevers and films [95, 96], and inamorphous polymer films [97]. Surfaces can also stiffen, however, as the electrons redistribute for theunder-coordinated surface atoms. This can lead to charging or contraction of the remaining bonds, which
stiffens the surface [83, 98]. Stiffening has been observed in Au films [99], Pt films [86], and Fe films[100]. In the latter case, Pt films were also measured, and the two films softened with decreasing thickness
first, then stiffened for the smallest sizes. Also, the measured strain in Pt and Fe films trended in opposite
directions. In addition to single films, there have been many observations of softening in multilayers [14],though in this case interface effects can be dominant over surface effects. Our group has previously
measured Ni/Ta bilayers and observed evidence of simultaneous softening of Ni and stiffening of Ta [32].
The work I present here, however, focuses on single films, aiming to isolate surface effects.
It should be noted that surface effects can also mediate large plastic (permanent) deformations to
nanoscale materials. This can be related to the surface atomic coordination and exhibits a strong sizedependence. Taking advantage of this effect enables the destructive testing of thickness dependent elastic
properties [97, 101]. For the work I present here, however, we require only elastic deformations that are
non-destructive to characterize thin film elastic properties.

73
Thickness-dependence in elastic properties can be modeled using continuum approaches
considering surface and bulk energies [96, 102, 103]. These approaches naturally account for surface stressand predict the strongest deviations in elastic properties usually for only 1-2nm films. One notable
exception is [96], where the authors derive a strain-dependent critical thickness at which surface energy
equals bulk energy. According to their calculations and nanoindentation measurements, this thickness canbe tens of nanometers in silicon, though in general sub-micron nanoindentation measurements need carefulanalysis as it is difficult to avoid substrate influence [104, 105]. Molecular dynamics and statics approaches
are also very valuable in modeling nanoscale elastic properties as they explicitly incorporate atomic
coordination effects, and long-range non-bonding effects [83, 85, 106]. Particularly in [85], the authors use
molecular statics to show only the outermost atomic layer is softened due to under-coordination, but thiscan cause softening of nanowires up to 20nm in diameter.
Considering the industrial and fundamental interest in nanoscale film elastic properties as a functionof chemical composition and thickness, I now present measurements of the elastic properties of three seriesof amorphous low-k ultrathin films. The first is a series of SiC:H films with varying levels of hydrogenationand thicknesses above 50nm, where we observe divergence towards incompressible behavior once a criticalnumber of bonds are terminated. The second series is a trio of SiOC:H films with identical chemicalcomposition, and a range of thicknesses down to 11nm. We observe no deviations in the elastic propertiesof these films. The final series is a pair of SiC:H films with nominally identical composition and thicknessesof 50nm and 5nm, where we observe significant softening for the thinner film.
4.2 Experimental Methods
In Chapter 2, I outlined the general procedure for our EUV diffraction experiments. I now present
the details of how we analyze the measured data to extract thin film elastic properties. The approach we use
is similar to picosecond ultrasonics with a structured transducer and SAW spectroscopy, which have bothbeen used previously to characterize thin film systems [94, 107-109]. However, our technique extends tothinner, device-relevant film scales, enabled by the nanoscale sensitivity of our EUV probe, and by accurate

74
finite element analysis of our systems [22, 32, 42]. In the following section, I will describe in detail how I
extend our established technique to thinner films, with special emphasis on the aspects of the analysis that
require the most expert input to correctly identify acoustic dynamics.
4.2.1 Sample Design
All the samples characterized in this chapter are amorphous, dielectric ultrathin films deposited on
silicon (100) by Intel. The films consist of either SiC:H or SiOC:H and are assumed to be isotropic. The
films are fabricated by plasma enhanced chemical vapor deposition from diluted silane precursors,
according to [82, 110, 111]. Nanoindentation and Rutherford backscattering (RBS) characterizations are
performed by Intel on thicker films of nominally the same material, while X-ray reflectivity (XRR)
measurements are performed on the ultrathin films to obtain their thicknesses and densities. Results are
summarized in Table 4.1 for the hydrogenation-dependent study and in Table 4.2 for the thicknessdependent study. The only inputs required a-priori for our analysis are thickness and density, butnanoindentation and RBS provide a nominal Young’s modulus value to compare to and reveal the films’
compositions and network connectivities, respectively. Also note that these films strain the silicon substrate,
which can be used to tune the Si electron mobility, as in [112]. Studying this is beyond the scope of thiswork, but the acoustic waves are highly sensitive to strain, so it must be included in our fitting.
Film name
63
7
5
75
65
9
49

Thickness (nm)
65.8
103.2
131.6
95.3
97.9
103.3
83.6

Young’s (GPa)
4.7
13.05
12.87
33.12
62.08
153
175

Density (g/cm3)
1.3
1.9
1.8
1.8
1.9
2.2
2.7

Network Connectivity
2.02
2.39
2.43
2.56
2.62
3.06
3.2

Table 4.1: Hydrogenation-dependent thin films nominal properties. From [22]. Thicknesses anddensities are measured by X-ray reflectivity (XRR). Young’s modulus is measured for thicker films of thesame material with nanoindentation. Network connectivity is calculated from Rutherford backscattering(RBS) measurements of elemental composition.

75
Film name
Film material
Thickness (nm)Density (g/cm3)
Young’s (GPa)Hardness (GPa)Stress (MPa)k
RBS %Si
RBS %O
RBS %C
RBS %H
Network
Connectivity

145
SiO1.2C0.3:H
43.75
1.5±0.1
19±2
3.0±0.550
3.3±0.1
20.27
34.03
15.50
30.20
2.4

148
SiO1.2C0.3:H
18.73
1.5±0.1
19±2
3.0±0.550
3.3±0.1
20.27
34.03
15.50
30.20
2.4

155
SiO1.2C0.3:H
10.9
1.5±0.1
19±2
3.0±0.550
3.3±0.1
20.27
34.03
15.50
30.20
2.4

335
SiC:H
50
2.5±0.1
210±1026.1±1.56±1
7.2±0.2
37.1
0.3
36.3
26.3
3.2

503
SiC:H
5
2.5±0.1
210±1026.1±1.56±1
7.2±0.2
37.1
0.3
36.3
26.3
3.2

Table 4.2: Thickness-dependent thin film nominal properties. The listed properties are measured via
XRR, RBS, and nanoindentation on thicker films of nominally the same material. The thicknesses listed
for films 335 and 503 are the nominal deposited thicknesses. See Section 4.3.2 for complete TEM
measurements of these samples. SiOC:H data are from [110], with further information provided by Intel,
and SiC:H data are from [82] and [111].
Once the films are deposited by Intel, our collaborators at Lawrence Berkeley National Lab (LBNL)
fabricate a set of nanoline gratings on top to serve as our transducers. The nanolines are fabricated by ebeam lithography and lift-off. As described in Chapter 2, we use nickel as the transducer material as it has
good absorption of our 780nm wavelength pump laser and good thermal expansion to launch acoustic
dynamics. The core-electron absorption edges of nickel are also well away from the 43eV photon energyof our probe beam, giving us minimal sensitivity to hot electrons, allowing us to observe acoustic dynamics
down to few picosecond timescales. The nanolines are deposited to be ~12nm tall on average, to give strong
diffraction and change in diffraction signal. The duty cycle (nanoline width divided by nanoline period) is
maintained at 33% for all gratings. Each grating covers an area of 150μm2, and is spatially separated by
200μm from the next nearest neighbor. Table 4.3 lists all of the grating linewidths, periods, and heights, ascharacterized by atomic force microscopy (AFM). Fig. 4.2 shows a schematic of a general thin film sample,
as well as representative AFM and scanning electron microscopy (SEM) characterizations of these samples.

76

Figure 4.2: Thin film sample design and characterization. a) Schematic cross-section of the thin films
samples. b) SEM image of a 100nm linewidth grating on film 145. Scale bar is 100nm. c) AFM height mapof a 30nm linewidth grating on film 145. Average height is the average for that grating.

Film name500(nm)/1500 (nm)350(nm)/1050 (nm)200(nm)/600 (nm)100(nm)/300 (nm)50(nm)/150 (nm)30(nm)/90 (nm)20(nm)/60 (nm)15(nm)/45 (nm)Average height (nm)145490/1540
350/1080
240/61090/31040/160
24/91
19/61
15/40
12.8

148
510/1540
360/1080
210/620100/31040/150
22/94
20/62
13/40
11.7

155
500/1510
360/1060
220/610110/31050/150
36/94
24/62
11.1

335
540/1530
390/1060
230/610130/300
80/150
47/97
14.8

503
480/1520
330/1070
170/61080/30040/150
24/91
26/62
18/42
11.0

Table 4.3: Nanoline grating dimensions. All measurements made by AFM, with data processing in
WSXM [58]. Values are stated as “linewidth/period” and organized by nominal sizes in the left-mostcolumn. The average height for all gratings on each film is listed in the final row. Note not every grating
size was fabricated on every film, and the gratings on film 335 were fabricated by a different researcher
than the others, and at an earlier date.

77
4.2.2 Data Processing and Interpretation
As described in chapter 2, the nanoline transducers simultaneously launch surface acoustic waves
in the film and substrate (SAWs), longitudinal acoustic waves in the film (fLAWs), and longitudinal
acoustic waves in the gratings’ nanolines themselves (gLAWs). Each of these waves is sensitive to adifferent elastic property, and all are measured via EUV diffraction. Measuring the velocities of SAWs andfLAWs is sufficient to extract the two independent elastic constants of isotropic thin films. The gLAW
velocity, on the other hand, accesses one elastic constant of the metal nanoline. Though not a completecharacterization of the nanoline’s elastic properties, measurements of these waves still provide important
information about the mechanical properties of nanoscale materials, as shown in [32]. The gLAW frequency
is also highly sensitive to the nanostructure’s environment, providing information about the relative
acoustic impedances between the nanoline and the film (open vs closed pipe modes) and about the presenceof any additional mass on top of the nanoline (like a quartz microbalance). Generally, SAWs occur on longtimescales (ns), while fLAWs and gLAWs occur on shorter timescales (ps), providing a natural separationfor easy identification of the different waves. This is illustrated in Fig. 4.3.

Figure 4.3: EUV diffraction measurement of thin film acoustic waves. a) Schematic of the EUV
diffraction measurement. A near infrared (IR) pump pulse launches acoustic waves by impulsively heating
nanoline gratings. After a controlled delay time, an EUV probe pulse diffracts from the surface into a CCD
camera, recording changes to the sample surface profile. b) The change in diffraction efficiency (as afraction of the integrated counts on the camera) shows a gLAW at early times, and a SAW at longer times.
Insets show the waves that produce the two signals. Note this is a composite trace of two back-to-backscans on film 145 (100nm linewidth) with different time steps. No normalization was needed for alignment.

78
The SAW velocity in the film is the first of two pieces of information needed to characterize thefilm’s elastic properties. From the temporal change in diffraction signal, e.g. that in Fig. 4.3(b), we extract
the SAW frequency via chirp-Z transform. This is a generalization of the discrete Fourier transform thatprovides finer interpolation in the window around the SAW frequency [32, 113]. The wavelength of the
SAW is defined by the nanoline grating period, which can then simply be multiplied by the SAW frequency
to obtain the SAW velocity. The penetration depth of SAWs is confined to the surface by a fraction of the
SAW wavelength. Because the SAW wavelength is defined by the grating period, we can tune SAW
penetration from large depths (large gratings) primarily sensitive to substrate elastic properties, to very
short depths (small gratings) that are fully confined within ultrathin films. We use this to independently
characterize the substrate (e.g. slight shifts due do the film straining the substrate) and film elastic
properties. An example of SAW velocity changing with confinement to softer films on stiffer substrates isshown in Fig. 4.4. Using an EUV probe is critical to this approach to characterizing ultrathin films. We
have diffracted our 29nm wavelength probe from 45nm period gratings (far below the diffraction limit ofvisible probes), which confines SAWs to ~15nm thick films, and provides high sensitivity to even thinner
films. Note, to extract the films’ elastic properties, we must also account for the influence of the nanolines,
which suppress the “unloaded” SAW frequency. This is discussed in the next section.
It is worth noting that all the analysis in this chapter uses only the fundamental SAW that islaunched by each grating, though higher orders are simultaneously excited. The gratings that launch thewaves can be approximated as a 33% duty cycle square wave, which is composed of a characteristic Fourier
series of sine waves. This is the series of SAWs that are launched when the gratings uniformly impulsively
expand. Because the Fourier series of each grating is also what defines the diffraction of our EUV probe(which spatial frequencies are present), the first, second, third, etc. SAW mode frequencies can be seen
most clearly in the dynamic changes to the first, second, third, etc. diffracted orders. All of these SAWs lie
on the same branch in the material’s dispersion; the higher order SAWs simply correspond to higher
wavevectors, and shorter penetration depths. It is possible that the weak presence of frequencies higher than
the fundamental SAW frequency in the first diffracted order correspond to points on different branches in

79
the material’s dispersion (different frequencies with the same wavevector), as in [114]. This is worth
investigating, but it is beyond the current scope of this work.

Figure 4.4: SAW confinement for the thickness-dependent study. Large penetration depths comparedto film thicknesses (greater than 10x), converge to the SAW velocity of Si. For penetration depths less thanor equal to the film thicknesses, the SAW velocity converges to that of the dielectric thin film.
At shorter times than SAWs, we measure a gLAW, which is each nanoline’s vertical breathingmode, excited by its rapid thermal expansion. The gLAW generally decays within the first few tens of ps,completing only a few periods of oscillation before it drops below the experimental noise. This rapid decayof the oscillation makes a Fourier transform ill-suited to extract the gLAW frequency. Instead, we use thematrix pencil method (MPM), which performs a singular value decomposition on the data, then projects
the singular values onto the family of complex exponentials [32]. The so-called scree plot, pictured in Fig.
4.5(a), shows how the singular values are naturally separated by this algorithm into single nodes that
correspond to purely decaying exponentials, pairs of nodes that correspond to decaying oscillations, and anoise tail that embodies the rest of the data. The single and paired nodes can then be separately plotted andanalyzed to isolate the frequencies present in the data. Fig. 4.5(b) gives an example of how the extracted
exponents lie in the complex plane to give exponential decays (only real component) and decaying
oscillations (pair of points with identical real components and opposite complex components).

80

Figure 4.5: The matrix pencil method for frequency extraction. a) The algorithm performs singularvalue decomposition on the data and projects the singular values onto the family of complex exponentials.
The scree plot, shown here, displays a single node that corresponds to a purely decaying exponential, a pairof nodes that correspond to a decaying oscillation, and a tail of closely-spaced nodes that embody the restof the features in the data. b) Taking the first three nodes and plotting the values of their exponents in thecomplex plane, we can see that indeed the single point has only a real component, producing a pure decay,
while the pair of points have equal real components and opposite complex components, producing adecaying oscillation. The waveform generated by these three nodes is compared to the data in Fig. 4.7(a).
The initial expansion of the gratings, in addition to launching gLAWs, generates a longitudinalpulse in the film – the fLAW mentioned above. The fLAW pulse reflects from the film-substrate interface
with an amplitude and phase determined by the acoustic impedance mismatch between the two materials.
When the fLAW returns to the surface, it causes a change in EUV diffraction efficiency. In general, thegLAW launches a pulse train of fLAWs into the film, though usually only the strongest, first pulse isobserved above experimental noise. The round-trip time for the echo’s arrival, combined with the known
film thickness, gives the longitudinal velocity of the film. This is the second piece of information, alongwith the SAW velocity, that is needed to fully characterize the elastic properties of the film. Because thefLAWs do not have a well-defined frequency, they must be identified by eye rather than extracted bymethods such as a Fourier transform. Due to the high level of expert input in their identification, the processused to analyze fLAWs must be carefully documented. It is especially important in the thickness-dependent
study, where the films are thin enough that the fLAWs are not well separated in pump-probe delay timefrom the gLAWs. That is, the fLAW arrives before the gLAW has fully decayed.

81
To correctly identify fLAWs for thinner films, both fLAWs and gLAWs must be analyzed together.
This is most evident for the 11nm thick film 155, where the first fLAW echo returns within the first gLAW
oscillation. The ideal way to unambiguously distinguish the two waves experimentally would be to have amechanism to tune the frequency of one, but not the other. Thankfully, the carbon deposition caused by the
EUV probe does this automatically. Amorphous, hydrogenated carbon is deposited on the whole irradiatedsample surface due to the high energy photons in our EUV probe beam breaking down residualhydrocarbons in our vacuum chamber [40]. The longer a given grating has been irradiated during
measurement, the more carbon will have been deposited on the top surface of that grating. The presence ofcarbon on top of the nanolines lowers their gLAW frequency, whereas the fLAW return time remains
unaffected, being dictated by subsurface interfaces. Thus, by observing which features shift with each
repeated measurement, and which stay at the same pump-probe delay time, we unambiguously identify thefLAWs for this thin film, as shown in Fig 4.6(a). Furthermore, I use the shifting gLAW frequency toestimate how thick the deposited carbon layer is, which is confirmed by AFM, as in Fig. 4.6(b).

Figure 4.6: 11nm Film (#155) fLAW and carbon deposition. a) As carbon is deposited on top of thenanolines, the gLAW frequency lowers, while the fLAW return time from the film-substrate interface
remains unaffected. Traces shown are averages of the first short time-window scan taken on each large
linewidth grating on this film, then the second short scan on each grating, then the third. This approach did
not carefully control exposure time, but simply ensured each average trace had more carbon deposition thanthe last. b) Large area AFM scan of 19m film (148), 500nm linewidth. Carbon on top of the nanolines ishighlighted in yellow, while that on the substrate is highlighted in green. A horizontal lineout through thehighest deposited area shows 7.4±0.5nm and 7±1nm of carbon on the substrate and nanolines respectively.
Note film 155 was not available for AFM due to subsequent sample damage, but film 148 is representative.

82
To estimate the carbon deposition from the shifting gLAW frequency, I use the transfer matrix
approach, following [32, 115, 116]. This approach is very precise, but the results are only approximate
because exposures were not carefully controlled during the experiment – various long time-window scans
were taken between the short time-window scans that were averaged together. To solve for the thickness ofthe carbon layer, I first define the acoustic transfer matrix for the vertical displacement of a wave travelingthrough the carbon and nickel layers from the transfer matrices of each material individually, consideringthat the displacement and stress must be continuous,
𝜔𝑑
𝜔𝑑
𝜔𝑑
𝜔𝑑
(𝜔𝜌𝐶 𝑉𝐶 )−1 sin ( 𝐶 )(𝜔𝜌𝑁𝑖 𝑉𝑁𝑖 )−1 sin ( 𝑁𝑖 )cos ( 𝐶 )cos ( 𝑁𝑖 )
𝑉𝐶
𝑉𝐶
𝑉𝑁𝑖
𝑉𝑁𝑖
𝑇=(
)(
).
𝜔𝑑𝐶
𝜔𝑑𝐶
𝜔𝑑𝑁𝑖
𝜔𝑑𝑁𝑖
−𝜔𝜌𝑁𝑖 𝑉𝑁𝑖 sin ( 𝑉 )cos ( 𝑉 )−𝜔𝜌𝐶 𝑉𝐶 sin ( 𝑉 )cos ( 𝑉 )
𝑁𝑖
𝑁𝑖
𝐶
𝐶

Here, 𝜔 is the angular frequency of the acoustic wave, 𝑑𝑁𝑖(𝐶) is the thickness of the nickel (carbon)
layer, 𝑉𝑁𝑖(𝐶) is the longitudinal acoustic wave velocity of nickel (carbon), and 𝜌𝑁𝑖(𝐶) is the density of nickel(carbon). Carrying out the matrix multiplication gives𝑇=(cos(𝜉𝑁𝑖 ) cos(𝜉𝐶 ) − 𝜂 sin(𝜉𝑁𝑖 ) sin(𝜉𝐶 )−𝜙𝑁𝑖 sin(𝜉𝑁𝑖 ) cos(𝜉𝐶 ) − 𝜙𝐶 cos(𝜉𝑁𝑖 ) sin(𝜉𝐶 )𝜙𝐶 −1 cos(𝜉𝑁𝑖 ) sin(𝜉𝐶 ) + 𝜙𝑁𝑖 −1 sin(𝜉𝑁𝑖 ) cos(𝜉𝐶 )),−𝜂 −1 sin(𝜉𝑁𝑖 ) sin(𝜉𝐶 ) + cos(𝜉𝑁𝑖 ) cos(𝜉𝐶 )where 𝜉𝑁𝑖(𝐶) ≡ 𝜔𝑑𝑁𝑖(𝐶) ⁄𝑉𝑁𝑖(𝐶) , 𝜙𝑁𝑖(𝐶) ≡ 𝜔𝜌𝑁𝑖(𝐶) 𝑉𝑁𝑖(𝐶) and 𝜂 = 𝜌𝐶 𝑉𝐶 ⁄𝜌𝑁𝑖 𝑉𝑁𝑖 have been introduced forbrevity. Assuming there is zero stress at the free surface (the carbon-vacuum interface), I now solve for the
eigenvector of T that corresponds to our mode𝑢𝑢
𝑇 ( ) = 𝜆 ( ),00where u is the displacement and 𝜆 is the eigenvalue. Substituting in the expression for T gives the equation(−𝜙𝑁𝑖 sin(𝜉𝑁𝑖 ) cos(𝜉𝐶 ) − 𝜙𝐶 cos(𝜉𝑁𝑖 ) sin(𝜉𝐶 ))𝑢 = 0,or simplifying and substituting the material properties back in explicitly,
𝜌𝐶 𝑉𝐶
𝜔𝑑𝐶
𝜔𝑑𝑁𝑖
tan (
) = − tan ().
𝜌𝑁𝑖 𝑉𝑁𝑖
𝑉𝐶
𝑉𝑁𝑖

This final formula is then solved for the carbon layer thickness at each experimentally measured 𝜔by graphical intersection as a function of 𝑑𝐶 . The values of 𝜌𝐶 and 𝑉𝐶 are taken from [117], where theauthors characterize the density and longitudinal acoustic velocity of a variety of amorphous, hydrogenated83
carbon films. According to [40], the films deposited by EUV exposure should be soft, so I use the valuesof the lower velocity films in [117], 𝜌𝐶 = 0.97 – 1.24g/cm3 and 𝑉𝐶 = 3025 – 4198m/s. I further account forthe possibility that the nickel velocity is lowered to 5000m/s from its bulk value of 5630m/s, while its
density remains nominal, according to [32]. With this range of inputs, I calculate that 0.7 – 1.1nm of carbon
is deposited between the first and second gLAW measurements, and only 0.1nm of additional carbon isdeposited between the second and third gLAW measurements. Because there was EUV exposure during
long time-window measurements before the first short time-window measurement, I calculate 5.1 – 7.2nm
of carbon was already present on the nanolines at the start of the first short time-window scan. These results
are consistent with AFM measurements of 7±1nm of carbon deposition on the nanolines, as shown in Fig.
4.6(b). Note that further EUV exposure occurred between the third gLAW measurement and the AFM
measurement, indicating that the AFM measurement is likely an overprediction of the carbon present during
the measurements shown, and the lower limit of the carbon velocity is likely more accurate. In future work,
it would be interesting to use this approach to precisely determine the carbon deposition rate as a functionof exposure time. In the present study, however, exposure time was not precisely controlled as the main
goal was distinguishing the gLAW and fLAW for film 155.
To identify the fLAWs for films 148 and 145, their gLAWs must once again be taken into account.
For film 148, the fLAW was particularly weak and difficult to identify. The best method I found was to use
MPM to fit the gLAW and look for bumps where the full data trace deviates from the pure gLAW
oscillation. This also shows up as a wider bump in the residual error between the MPM fit and the full data
trace, as shown in Fig. 4.7(a). For film 145, the fLAW was much stronger, and even showed evidence of apulse train of echoes, launched by the strong gLAW, as shown in Fig. 4.3(b). Because the film has a lower
acoustic impedance than both the nickel nanolines and the silicon substrate, we expect the pulse train that
returns to the surface to be 180° out of phase with the gLAW that launched it. This is why the dip is“downward”. For film 145, I thus identify the location of the fLAW by “missing peaks” in the gLAW
oscillation, where the fLAW has destructively interfered with it. This effect is shown in Fig 4.7(b) alongwith reference sinusoids as guides to the eye. At the location of the third peak of the gLAW in this84
measurement, we instead observe a slight dip and then rise in the signal 180° out of phase with the gLAW
oscillation. This indicates that the fLAW returns to the surface at almost exactly the same time as the third
gLAW peak, cancelling it out. Similar features are consistently repeated in other measurements of gratings
on this film, with both first and second fLAW echoes at the appropriate delay times.

Figure 4.7: fLAWs for films 148 and 145. a) Using MPM, we identify the exponential rise of the initial
thermal expansion, and the pure decaying oscillation of the gLAW for film 148 (scree plot and complex
exponents shown in Fig. 4.5). Comparing the full data trace (many grating measurements averaged together)to the curve fit by MPM, we observe an fLAW dip at approximately 6ps. Inset: the residual error between
the two curves shows a slightly broader feature, corresponding to this dip. b) Comparing to reference
sinusoids for film 145, we can see that the third peak of the gLAW is missing, due to destructive interferencewith the fLAW. A second fLAW pulse train is visible at twice the return time of the first fLAW.

85
Finally, the fLAWs for films 503 and 335 are relatively easy to identify. Film 503 is so thin, thatthe fLAW arrives before the first full oscillation of the gLAW, allowing for clear identification. Fig. 4.8
shows that the fLAW arrives on a sub-picosecond timescale. Note that this fLAW return time is still wellafter the weak electronic peak sometimes visible in our signal at ~100fs, as discussed in Chapter 2. Thus,
the off-resonance condition of our EUV probe, with minimal sensitivity to electron temperature, is criticalto characterizing such ultrathin films. Film 335 is thick enough that the fLAW is well separated from thegLAW and may be readily identified.

Figure 4.8: fLAWs for film 503. For this 5nm thick film, the fLAW return time is shorter than the periodof the gLAW, meaning that the dip in the signal is clearly identifiable in the initial rise of the signal. Two
average traces are shown, which include measurements of the largest linewidth gratings on this film, where
the fLAW signal is strongest. The blue trace exhibits a dip at 0.7ps and 1.4ps for the first and second fLAW,
respectively, while the orange trace includes only the traces that align the best in time, showing the first
fLAW most clearly. This sub-picosecond feature is not present on thicker films, indicating it is indeed thefLAW for this 5nm film. Note that this feature is well after the time of the weak electronic signal discussed
in Chapter 2. Indeed, in these traces we see no evidence of electronic sensitivity. Change in diffraction
efficiency for each trace going into the average is expressed as a fraction of the total integrated intensity on
the camera.

86
4.2.3 Fitting and Modeling Elastic Properties
The complete elastic properties of a material are expressed in a rank 4 tensor known as the elasticity
tensor [118, 119]. This is a generalization of Hook’s law, which states the force on a spring is proportional
to its extension. Similarly, the stress in a material is proportional to its strain, with the proportionality
constants organized in the elasticity tensor. Written mathematically,
𝑇𝑖𝑗 = 𝑐𝑖𝑗𝑘𝑙 𝑆𝑘𝑙 ,where 𝑇𝑖𝑗 is the stress in the i direction on the j-oriented surface, 𝑐𝑖𝑗𝑘𝑙 is the ijkl component of the elasticity
tensor, and 𝑆𝑘𝑙 is the strain in the k direction of the l-oriented surface. To give a concrete example, consider
compressing a spring placed on the x-axis. The force on the spring in the x-direction is proportional to thedisplacement in the x-direction at a ratio of 𝑐𝑥𝑥𝑥𝑥 .
Though there are 81 components to the elasticity tensor, only two of them are important for thecurrent study. It can be shown that, for any material, at most 21 of the tensor components are independent
elastic constants. For the case of an isotropic material, further simplifications show that only two of theelastic constants are independent; all the others are either 0 or simple expressions of these two constants.
The two constants will be denoted here as 𝑐11 (equal to 𝑐𝑥𝑥𝑥𝑥 =𝑐𝑦𝑦𝑦𝑦 =𝑐𝑧𝑧𝑧𝑧 ) and 𝑐44 (equal to 𝑐𝑖𝑗𝑖𝑗 ).
In experimental and engineering contexts, it is more common to express the elastic properties ofisotropic materials in terms of Young’s modulus, E, and Poisson’s ratio, 𝜈. These constants are moreintuitive, as Young’s modulus reflects how much pressure needs to be applied to a material for a given onaxis displacement, while Poisson’s ratio is the ratio of how much a material displaces to the sides relative
to its on-axis displacement. Schematics of this are shown in Fig. 4.9(b). Note that Poisson’s ratio must be
between 0 (fully compressible material) and 0.5 (fully incompressible material), excluding auxetic materials
that have negative Poisson’s ratio. Young’s modulus and Poisson’s ratio are related to 𝑐11 and 𝑐44 by
3𝑐11 − 4𝑐44
𝐸 = 𝑐44 ()
𝑐11 − 𝑐44𝜈=𝑐11 − 2𝑐442(𝑐11 − 𝑐44 )

(1)

(2)

87

Figure 4.9: Measuring the two independent elastic constants with acoustic waves. a) The SAW and
LAW velocities in the film are related to c44 and c11, respectively. b) Young’s modulus and Poisson’s ratio
are functions of c11 and c44. Young’s modulus relates an applied pressure to the materials on-axis
deformation. Poisson’s ratio is the ratio of transverse deformation to on-axis deformation, for a given
applied pressure.
Two independent measurements are needed to extract the two independent elastic constants.
Longitudinal and surface acoustic waves confined to the film very naturally provide these measurements.
The film’s LAW velocity, calculated from the known film thickness and the measured fLAW return time,is related to 𝑐11 by𝑐11
𝑉𝐿 = √,
𝜌(3)

where 𝜌 is the film’s known density. The film’s SAW velocity, calculated from the grating period and themeasured SAW frequency, is related to 𝑐44 by𝑐44
𝑉𝑆 = 𝜉 √,𝜌

(4)where 𝜉 is the ratio between a surface acoustic wave and a pure transverse acoustic wave. While 𝜉 may be
calculated [120], this expression is only true for an un-loaded surface, and our case with nanolines on topof the film is actually more complicated. To extract the film’s 𝑐44 elastic constant, and thereby calculate
Young’s modulus and Poisson’s ratio, we must account for how the nanolines change the SAWs wemeasure.

88
The presence of nanoline gratings on top of the films makes the surface into a hypersonic surface
phononic crystal, which can be modeled using finite element analysis (FEA) [64, 121]. For our FEA
calculations, the geometry of a single nanoline period is implemented in COMSOL Multiphysics withperiodic boundary conditions and a mesh of discrete calculation points, such as that shown in Fig. 4.10(a)[60]. The temperature and displacement are then calculated at each mesh point as a function of time,
according to the thermoelastic differential equations and method described in [22]. See Appendix for moredetails. This approach captures all the continuum mechanics properties of this system, including the surface
acoustic wave’s confinement to the surface, how its energy is scattered into the bulk substrate by thegratings, and how the mass of the nanolines lowers the frequency of the SAW. By diffracting a Gaussian
beam from the FEA calculated surface displacement using Fresnel propagation, we can produce a calculatedchange in diffraction efficiency signal, which compares very well with our experimentally measured change
in diffraction efficiency signal (see Fig. 4.10(b)). While this is a good indication that the approach is valid,in practice the full physics calculations are computationally expensive, and we use a faster approach thansimulating a full transient signal and fitting it to our experimental signal.
To obtain the best fit elastic properties of the films, we use a combination of full physics
calculations of the dynamic displacement field and eigenfrequency analysis of the static structure, allperformed in COMSOL. First, we perform eigenfrequency analysis on the un-deformed nanoline-filmsubstrate structure to identify all the modes it supports. Next, we apply a transient heat source to thenanoline, and the full physics displacement field is calculated only until the time when the nanoline hasreached maximum thermal expansion. This captures the amplitude and content of the initial excitation of
the acoustic waves we measure. Finally, the displacement field at maximum expansion is projected onto
the set of eigenmodes for the structure to identify which long-lived acoustic waves are excited and measured
in our experiment. The elastic properties of the system are varied until the calculated frequencies match theexperimentally observed frequencies. This method is executed for each experimental geometry we measure.
First, large linewidth gratings are used to fit the silicon substrate’s elastic properties and accountfor strain and wafer-to-wafer variation. Because these waves penetrate deep into the substrate and are only

89
minimally sensitive to the thin film, we perform FEA using the approximate thin film properties estimatedfrom the confined SAW velocity that hasn’t yet been corrected for the presence of the smaller linewidth
gratings (Eqn. 4). Next, we use the corrected substrate properties and fit the thin film Young’s modulus bymatching the measured frequency on a small linewidth grating that confines the SAW to the film. Together
with 𝑐11 calculated from the film LAW velocity, this value gives the true elastic properties of the film. We
additionally calculate the frequencies for each intermediate grating, using these extracted elastic constants,
which verifies our results as they exhibit good agreement with experiment. The intermediate grating sizes
could be used in future work to characterize sub-surface layers in a multilayer system.

Figure 4.10: FEA calculations of thermoelastic dynamics. a) FEA calculations of the temperature (right)
and displacement at a mesh of discrete points (left). b) The calculated diffraction from the dynamically
deforming surface in the FEA model agrees well with the experimentally measured diffraction signal.

90
Finally, we calculate the uncertainty on our best fit thin film elastic properties. To do this, we must
propagate the experimental uncertainties in film density, film thickness, fLAW return time, and SAW
frequency through our analysis procedure. Due to the numerical approaches used, there is no analytic
equation for error propagation in our analysis. Instead, we set the inputs to the FEA calculations at theextreme ends of their error bars that produce the lowest calculated SAW frequency. We then increase thefilm’s Young’s modulus until the calculated frequency matches the upper end of the experimental SAW
frequency error bar. We repeat this process with the inputs that give a high calculated SAW frequency andtune Young’s modulus to match the lower end of the experimental SAW frequency. In this way, our error
bars capture the maximal deviations possible, and are rather conservative as it is unlikely that all properties
would be at the limits of their ranges simultaneously. More details are provided in [22].
4.3 Results and Discussion
I now present the full elastic characterization of SiC:H and SiOC:H ultrathin films using theapproach described above. The SiC:H films have varying levels of hydrogenation to reduce their dielectric
constant, and we observe critical behavior in Poisson’s ratio when enough bonds in the network are broken.
The SiOC:H films are all made from the same material, at the critical level of broken bonds, but they varyin thickness from 50nm down to 11nm. We observe no thickness-dependent changes in the elastic propertiesof these films. In comparison with this set of films, we characterize two additional SiC:H films at 50nm
and 5nm thicknesses, which have much more interconnected networks with fewer broken bonds than the
SiOC:H film series. We do see evidence of thickness-dependent softening for this pair of films.
4.3.1 Hydrogenation-dependent Study
To understand the results of the hydrogenation-dependent study, we must first formalize our
discussion of bond termination to soften a material. This is best achieved in the Phillips-Thorpe frameworkof topological constraint theory, originally developed to describe bulk glasses [122, 123]. In this framework,a material is flexible if there are fewer constraints on each atom than the three degrees of freedom they haveto move. Similarly, a material is rigid if each atom has more than three constraints on its motion. Since the91constraints on each atom come from the network of bonds in a material, we can relate the average numberof atomic constraints, 𝑛, to the average coordination number of the atoms in the material, 〈𝑟〉. For example,
consider an atom with two nearest neighbors, 〈𝑟〉 = 2. These neighboring bonds constrain one degree of
freedom of this atom, giving 𝑛 = 1 for the bond constraint. Additionally, since a material has fixed bondangles, one more degree of freedom is constrained for the central atom, 𝑛 = 1 for the angle constraint. As
we increase the coordination number of the central atom, each new bond defines two new angles, thus
angular constraints for 〈𝑟〉 = 3 gives 𝑛 = 3, 〈𝑟〉 = 4 gives 𝑛 = 5, and so on. Taking the bond and angle
constraints together, we define the general formula𝑛=〈𝑟〉+ (2〈𝑟〉 − 3).
2(5)

The critical transition from a flexible material to a rigid material occurs when the number of atomic
constraints exactly balances the number of degrees of freedom, 𝑛 = 3. Substituting this into Eqn. 5, we
obtain the critical value 〈𝑟〉𝑐 = 2.4, known as the rigidity percolation threshold. Once the average coordination
number of the atoms is 2.4, rigid structures percolate throughout the material, creating an interconnected
network. With this in mind, the average atomic coordination number, 〈𝑟〉, is also referred to as the network
connectivity of a material.
The network connectivity of a material is calculated from the coordination numbers of each atomic
species comprising it, 𝑟𝑖 . If the mole fractions of each element, 𝑥𝑖 , in the material is known, for instance measured
by RBS, then the network connectivity is calculated by〈𝑟〉 = ∑ 𝑥𝑖 𝑟𝑖 .

(6)𝑖

For the SiC:H films in this study, 𝑟𝑆𝑖 = 4, 𝑟𝐶 = 4, and 𝑟𝐻 = 1. Thus, by increasing the hydrogenation of thefilms, we reduce their network connectivity, as calculated by Eqn. 6 and the mole fractions measured by RBS.

As shown in Table 4.1, the seven films characterized in this study span network connectivity values from2.02 – 3.2, crossing the rigidity percolation threshold.
As shown in Fig. 4.11, we extract the Young’s modulus and Poisson’s ratio of the full series of
SiC:H films. For those films with network connectivity above the rigidity percolation threshold, the elastic

92
constants behave nominally. The Young’s modulus of the thin films matches nanoindentation
measurements on thicker films of the same material, and it scales with network connectivity according to
the trend observed in [82]. Similarly, Poisson’s ratio stays at about 0.2, the nominal value for this material.
Below the rigidity percolation threshold, however, Poisson’s ratio diverges towards 0.5, that of a fully
incompressible material, like rubber. Due to the difficulty in measuring Poisson’s ratio for ultrathin films,
it is usually assumed to be constant. Our results show that this assumption is not valid once a film drops
below the rigidity percolation threshold. This divergence coincides with a number of other discontinuities
in material properties across the rigidity percolation threshold, as shown in [82]. Note that MD simulationsin the similar material, SiOC:H, did predict that adding hydrogen terminal groups for a lower network
connectivity would make the Poisson’s ratio increase [84]. However, the change they predict is smaller
than that which we measure in SiC:H, making these measurements an important insight for designing robust
nanosystems with this material.

Figure 4.11: Young’s modulus and Poisson’s ratio for the hydrogenation-dependent study. a) The
Young’s modulus for all seven films follows the trend with 〈𝑟〉 predicted in [82] for network connectivity
above the rigidity percolation threshold. Below the threshold, our measured values still agree well withnanoindentation measurements on thicker films [82]. b) The Poisson’s ratio diverges from its assumedconstant value for films with 〈𝑟〉 below the rigidity percolation threshold. This rapid change towards
incompressible behavior coincides with other material property changes at the threshold [82], and it is an
important property of these materials to consider when engineering nanosystems with thin films like thosemeasured. Figure from [22].

93
4.3.2 Thickness-dependent Study
To study thickness-dependent elastic properties, we first characterize a set of three SiOC:H films
with thicknesses of 44nm, 19nm, and 11nm. As shown in Table 4.2, these films all have 〈𝑟〉 = 2.4 (notethat 𝑟𝑂 = 2). To best represent the results, it is worth considering again the calculation process described in
Section 4.2.3. The experimentally measured quantities are SAW frequency and fLAW return time, which
both have associated uncertainties from the standard deviation of multiple measurements. We calculate 𝑐11
from the fLAW, and fit Young’s modulus using FEA. While these two quantities contain all the information
about the film’s elastic tensor, they are an odd pair to report. Using Eqn. 1 – 2, we instead convert the results
into either 𝑐11 and 𝑐44 , to report the independent components of the elastic tensor, or we convert to Young’s
modulus and Poisson’s ratio, to report the most commonly used elastic constants. As shown in Fig. 4.12,
all of these transformations are well behaved for the thicker film. Note, however, that the properties of thesefilms place their elastic constants close to a physical boundary in 𝑐11 – 𝑐44 space, at 𝑐11 = 2𝑐44 (Fig. 4.12(c),red region). Considering Eqn. 2, this boundary comes from the requirement that Poisson’s ratio be positive,
as these materials are assumed to be non-auxetic. There is an additional boundary at 𝑐11 = 4⁄3 𝑐44 , fromthe requirement that Young’s modulus be greater than zero (Eqn. 1), but this lies within the red region
shown in Fig. 4.12(c) and does not need to be shown. The 𝑐11 = 2𝑐44 boundary can then be back-propagatedinto 𝑐11 – Young’s modulus space, and finally to SAW frequency – fLAW echo space (Fig. 4.12(b) and
Fig. 4.12(a), respectively). This boundary complicates our results when the film thickness is reduced.
The thinner films have larger experimental uncertainties, which cross the back-propagatedboundary, as shown in Fig. 4.13(a). Note even a small uncertainty in return time is significant because thefilms are so thin. Crossing the boundary does not mean that the materials are auxetic, but rather the unrelated
experimental uncertainties, such as the fLAW pulse width, can make the calculated value cross theboundary that this material’s constants are naturally close to. Once the 𝑐11 = 2𝑐44 boundary is crossed, thelower bound on Poisson’s ratio is pulled down to 0 (Fig. 4.13(d)). Due to these transformations, the error bars

are more symmetric in 𝑐11 – 𝑐44 space, though Young’s modulus and Poisson’s ratio are generally preferred.

94

Figure 4.12: Film 145 (44nm thick) elastic constants. a) Measured fLAW echo return time and SAW
frequency. The central blue dot tracks the calculations performed at the measured values. The other blue
dots are calculations sampling the rest of the region within experimental uncertainty (dashed pink lines).
Calculations are then carried forward into b) 𝑐11 –Young’s modulus space, c) 𝑐11 –𝑐44 space, and d) Poisson’s
ratio–Young’s modulus space. The red boundary is defined by 𝑐11 > 2𝑐44 for non-auxetic materials, and itis then back-propagated into other spaces.

Figure 4.13: Film 155 (11nm thick) elastic constants. The same process is carried out here as for Film
145, shown in Fig. 4.12. The points and lines shown have the same meanings as above. Note how crossing
the 𝑐11 = 2𝑐44 boundary in (c) brings the lower bound on Poisson’s ratio to 0 in (d). The same plot windows
are used in (b), (c), and (d) here as are used in Fig. 4.12. Note Film 148 behaves similarly to Film 155.

95
Following the process described above, we extract the elastic properties of all three SiOC:H films,
presented in both 𝑐11 –𝑐44 space and Young’s modulus – Poisson’s ratio space in Fig. 4.14. We observe noevidence of thickness-dependence as all three films’ elastic constants overlap within uncertainty. Thisagrees with the lack of thickness-dependence (aside from differences in fabrication) for measurements ofporous films in [124]. Note, however, that the films in our study are not porous.

Figure 4.14: Elastic properties of varying-thickness SiOC:H films. Results are reported as 𝑐11 and 𝑐44
on the left, and as Young’s modulus and Poisson’s ratio on the right. The range of possible elastic constants
is consistent between films, within experimental uncertainty, indicating a lack of thickness-dependence.
Note that these films lie close to the 𝑐11 = 2𝑐44 boundary (red line) separating conventional materials fromauxetic materials (negative Poisson’s ratio). Calculations spanning to the limits of experimental uncertainty(solid points) cross this boundary for the thinner two films, pulling the lower bound on Poisson’s ratio to 0.
The 11nm film (blue) is fit using its 20nm linewidth grating, the 19nm film (green) is fit using its 20nm
linewidth grating, and the 44nm film (pink) is fit using its 30nm linewidth grating. Shaded regions represent
the full range of the elastic constants within uncertainty, approximated from the calculated points.
Considering the body of work showing thickness-dependent elastic properties in other materials, itis worth asking why there is no thickness dependence for these SiOC:H films. While it is possible that 11nm
is simply not thin enough to see thickness-dependent changes, depending on what model prediction is96
correct, these films are also at the rigidity percolation threshold, which we showed above significantly
affects the behavior of a material. With this in mind, we hypothesize that the reduced network connectivity
in the bulk of the film, caused by hydrogenation, reduces the impact of under-coordinated atoms at thesurface. In other words, there is only a minimal difference for these films between the average coordination
numbers of the surface and the bulk, producing only minimal surface effects. For most materials, there
would be a much larger difference between bulk and surface coordination numbers, in general. In [125],
the authors develop a simple effective coordination number model based on the differences between surface
and bulk coordination numbers in nanocrystals, and they successfully predict size-dependent properties
including the surface energies of nanodroplets. They show that this model agrees with the bond contraction
model mentioned above [98]. With this in mind, the best test of our hypothesis is to study an additional setof varying-thickness films with much more interconnected networks in their bulk.
While we do not currently have a set of SiOC:H films at higher network connectivity, we do havea pair of SiC:H films that have network connectivity well above the rigidity percolation threshold. As these
two films have nominal thicknesses of 5nm and 50nm, they are a good initial test of our hypothesis.
However, these films (numbers 503 and 335, see Table 4.2) were accidentally double deposited during
fabrication. This means the process of ammonia plasma treatment to improve film adhesion followed by
SiC:H film deposition was performed twice on the silicon substrates. This makes for a stack of two films
on top of the native oxide of the wafer, separated by a nitrogen-containing layer from the plasma treatment.
We need very precise characterization of these layers to correctly model the system and extract elastic
constants from our measurements, especially for a film only 5nm in thickness. For this reason, we
performed cross-sectional transmission electron microscopy (TEM) on a section of film in the center of oneof our 500nm linewidth gratings. Representative images are shown in Fig. 4.15, along with the measured
thicknesses of each layer in the stack. In addition to the film thicknesses, we observe a difference in contrast
between the two nominally identical films in the stack. To explore this further, we also performed energydispersive x-ray spectroscopy (EDS) with the electron beam for high resolution elemental composition.

97

Figure 4.15: TEM measurements of film 503 – double deposited 5nm SiC:H films. This image wastaken of the full stack of films in a region between two 500nm nickel nanolines. The bright field (BF) image(a) exhibits the best contrast between the two SiC:H films and the nitrogen-rich layers coming from theplasma treatment in the deposition process. The high angle annular dark field (HAADF) image (b) exhibits
better contrast between the top SiC:H film and the EUV-deposited carbon coating the surface of the sample.
Thicknesses and uncertainties listed are the averages and standard deviations, respectively, of several
measurements spanning the width of the image taking both the widest and narrowest extents of the gradients
between different layers. In this way, the uncertainty reflects both the diffusiveness of the boundaries andthe variation across the imaged region. Note that (b) was used to help identify the top surface of the upper
SiC:H film in the measurements of (a).
We performed EDS measurements to determine the elemental composition of film 503 beyond its
nominal values. Results are shown in Fig. 4.16(b). Referencing Table 4.2, the nominal fractions of Si andC should be roughly equal in the film, but EDS shows roughly seven times more Si than C. This could
indicate a fabrication-dependent change in the film, though it should be noted EDS cannot detect hydrogen,which is a significant fraction of this material, and it is difficult to detect C, N, and O. Aside from the nonnominal relative concentrations of Si and C, EDS shows that there are only few-percent differences between
the compositions of the top and bottom films. This could indicate that the difference in HAADF contrast
comes from different amounts of hydrogen in the two films. It has been observed that plasma treatments
can remove terminal CH3 groups and soften SiOC:H films [126]. However, for SiC:H films similar to ours,
N2 plasma treatments did not significantly change the Young’s modulus of multilayers down to 1nm layer

98thickness [111]. It is thus reasonable to approximate the top and bottom films as having identical elastic
properties for preliminary calculations. Note that the EDS spatial map indicates a surface oxide layer on
the top film with a similar thickness as the N-rich layers. This could be the cause of the poor contrast
between the top film and the EUV-deposited C in Fig. 4.15(a). We are currently performing TEM and EDS
on film 335 to confirm its layer thicknesses and verify its composition matches that of 503.

Figure 4.16: HAADF and EDS measurements of film 503. a) Integrated HAADF intensity as a functionof depth for a region topped by a 500nm nickel nanoline. Note that the carbon deposited by the e-beam andthat deposited by EUV is resolvable by HAADF contrast. The large amount of deposited carbon (12nm) isdue to the fact that we used this sample for calibration measurements (additional EUV exposure) between
the completion of its elastic characterization, and the cross-sectional TEM measurement. This measurementshows the strain in the silicon substrate that we account for when fitting the acoustic waves of large
linewidth gratings b) EDS measurement of Si, O, N, and C for a region with no nickel nanoline, both
spatially resolved (top) and integrated (bottom). Though the bottom SiC:H film shows only slightly more
Si and less C than the top film, its HAADF contrast is significantly different. Note that EDS did not resolve
the N-rich and native oxide layers resolvable by TEM in Fig. 4.15.
With these more detailed measurements of the films, we can now report preliminary elastic
characterizations of films 503 and 335. First, the fLAW reflects from the N-rich interface between the two
SiC:H films, in both cases. This means the 𝑐11 we calculate is that of the top film. As shown in Fig. 4.17(a),𝑐11 stays constant between the 5nm and 50nm films. This calculation used Eqn. 3, the nominal film density,

the top film thickness measured by TEM for film 503, and for film 335 I used half of the XRR measurementof the full stack thickness (95nm). Second however, the SAW dispersion exhibits a strong thickness-

99
dependence, as shown in Fig. 4.17(b). For the thicker film 335, the SAW fully confined to the top film hasa velocity of 4600±200m/s. For the thinner film 503, the most confined SAW (15nm linewidth grating) has
a velocity of 2400±200m/s. This indicates a significant softening of the 5nm film, compared to the 50nm
film. FEA calculations for precise determination of the two films’ elastic constants are ongoing to accountfor the surface oxide, potential differences between the top and bottom films in each stack, and whether theisotropic assumption is less valid than using a transversely isotropic elastic tensor.

Figure 4.17: Elastic characterizations of SiC:H films 335 and 503. a) The calculated 𝑐11 for both the
5nm (503) and 50nm (335) film is nearly identical. This elastic constant is calculated from the measured
fLAW, which reflects from the N-rich interface between the two SiC:H films in the double-deposited stack.
Thus, the reported 𝑐11 is that of the top SiC:H film, even if the two films in the stack are different. b) The
SAW dispersions for the two films indicate a significant softening in the 5nm film. Even when confined tothree times the film thickness, the SAW velocity in 503 is already significantly lower than that of the SAW
fully confined to the top film in 335. FEA calculations are being performed to extract the films’ elastic
constants from these measurements.
The observed differences between the two thicknesses of films could indicate that high network
connectivity SiC:H exhibits thickness dependence while low connectivity SiOC:H does not. This would
support our hypothesis that the more interconnected bulk of the film is more affected by the undercoordinated atoms at the surface. However, we cannot yet conclude that this hypothesis is correct. We stillneed to verify that films 335 and 503 have the same composition as one another to confirm that thedifferences in their elastic constants come from thickness dependence. We additionally cannot rule out thatthe existence of thickness dependence in the SiC:H pair but not in the SiOC:H series is due to some othermaterial difference, or simply because the SiC:H pair includes an even thinner film than the SiOC:H series.

100
Future work systematically studying both film thickness and network connectivity will clarify thesequestions.
The measurements of SiC:H films at 50nm and 5nm thicknesses show the importance of having
access to nanoscale SAWs with high surface confinement by using an EUV probe. Deviations in elastic
properties between these two films is only observable in the SAW dispersion, and not in 𝑐11 , which can bemeasured by visible picosecond ultrasonics. This makes our EUV nanometrology technique an important
addition to Brillouin light scattering (BLS) for the full elastic characterization of ultrathin films. BLS has
been used to characterize films down to 2.2nm thickness, though this measurement had to assume aPoisson’s ratio between 0-0.5 to obtain fits [89, 90]. Moving forward, both techniques will be valuable forultrathin film characterization.
4.4 Conclusions and Future WorkIn this chapter, I have presented measurements of the elastic properties of low dielectric constantultrathin films. These films are directly applicable to improving the efficiency and speed of integrated
circuits, but they are also a great platform to explore both materials engineering and fundamental thicknessdependence coming from surface effects. I first presented measurements of SiC:H films with varying levelsof hydrogenation, which lowered their network connectivities below the rigidity percolation threshold. At
this point, Poisson’s ratio diverged towards incompressible behavior, which contradicts the constant-value
assumption frequently used when characterizing such thin films. I then presented measurements of two setsof films with varying thickness, one set of SiOC:H films, and one set of SiC:H films. The SiOC:H set offilms exhibited no thickness-dependent changes in elastic properties down to 11nm thickness. The SiC:H
pair of films showed no change in 𝑐11 for 50nm and 5nm thicknesses, but their SAW velocities did show asignificant softening of the thinner film, indicating thickness-dependent changes to 𝑐44 .
In future work, we will confirm the thickness dependence of the SiC:H films and conduct a more
systematic study of thickness dependent changes to elastic properties for different material compositions.
We will further use intermediate grating sizes to individually characterize the elastic properties of a series

101of buried layers in a multilayer system. We will also continue to develop dynamic coherent diffractive
imaging, which measures local thermal and elastic dynamics by reconstructing full field images ofnanostructured systems [127]. In parallel to this effort to obtain local information, we have continued todevelop our established technique, which measures average information from periodic samples. To
characterize still more complicated nanosystems with this technique, we must extend it beyond what has
been described in Chapters 3 and 4. In Chapter 5, I will show the different ways we are extending ourtechnique to characterize more novel nanosytems.

Chapter 5
Extending Dynamic EUV Metrology

In the previous two chapters, I have explored the fundamental thermal and elastic behavior ofmaterials either at a semi-infinite surface, or with one dimension of nanoscale confinement as ultrathin
films. There are, however, many exciting materials systems that have two and three dimensions of
nanoscale confinement, with complex geometries and engineered structure. These nanoengineered and selfassembled materials have unprecedented thermal and elastic properties, with promising applications in
waste heat recovery, advanced electronics, ultralight materials, and more. For many of these materials
systems, the fabrication of grating transducers is either impossible, or could fundamentally change thephysics at play.
In this chapter, I present efforts to extend from our established, grating-based approach, to measure
more general materials systems. First, I present nanoscale thermal and acoustic measurements of 3D silicon
metalattices using grating transducers, as described previously. This represents the most complex sample
characterized to date using our grating-based approach. Second, I present measurements of long-range
thermal transport in novel nanosystems where grating fabrication is impossible. These samples are instead
excited using visible transient gratings (TG), and still probed by a coherent EUV beam. Finally, I present
measurements of nanoscale thermal and elastic dynamics launched without the need for transducers byusing an EUV transient grating. These measurements were made using both a visible and EUV probe while
collaborating at the free electron laser, FERMI, in Italy. I further present the work to date in implementing
EUV TG on a tabletop using HHG at JILA.

103
5.1 Introduction and Motivation
Metamaterials can achieve naturally-unattainable material properties, resulting from theirhierarchical structuring. Examples range from photonic crystals [128] to phononic bandgap materials [49]to structural metamaterials that are ultralight yet stiff [129]. For most of these applications, periodic
structuring must be done at the nanoscale, yet have a well-defined periodicity over orders of magnitudelarger length scales. One solution to this nanofabrication challenge is to use self-assembly, for instance byforming colloidal crystals of nanoparticles.
Colloidal crystals are a powerful and versatile platform for materials engineering. Nanoparticles
can be organized into a variety of extended crystal structures, and they can optionally be functionalizedwith ligands before assembly [130]. The nanoparticle cores can range from inert to plasmonic to magnetic
materials, and the ligands coating them can range from relatively short conductive molecules to longpolymer chains optimized for specific applications [131, 132]. Colloidal crystals have already been
demonstrated to perform a number of functions, including ultrathin membrane filtration [133], chemicaland biological sensing [134], economic hypersonic phononic crystal fabrication [135], and molecularelectronics and switching [136]. Furthermore, 2D and 3D colloidal crystals of nanospheres can be used as
scaffolds or templates for the fabrication of complex nanostructured materials [137, 138]. Both metals andsemiconductors can be infiltrated into the interstitial spaces between spheres, resulting in novel material
properties coming from both the material’s confinement to nanoscale “meta-atoms” and their extended
periodic ordering into “metalattices” [139, 140].
In this chapter, I present a series of works that extend our established thermal and elastic metrology
technique to colloidal-crystal-based systems, with potential for measurement of other more general
samples. I first present thermal and acoustic characterizations of 3D silicon metalattices using our gratingbased technique. I then present transducer-free measurements based on transient grating (TG) excitations.
Measurements are obtained on 2D ligand-coated gold nanoparticle arrays and on 2D gold nanomeshes
fabricated from a nanosphere template. Finally, I present work at the FERMI free electron laser and at JILA

104
to implement an EUV transient grating. EUV TG represents a significant advance in materials
characterization, as it directly excites deep nanoscale and ultrafast dynamics without the need fornanofabrication of metallic transducers. This enables the measurement of a much wider variety of samples,
as well as other applications such as super resolution imaging and magnetic characterization via nanoscale
spin gratings.
5.2 Silicon Metalattices
First, I test the limits of our established nanograting-based approach by characterizing complex,
three dimensional materials systems. Specifically, I characterize 3D, hierarchical nanosystems known asmetalattices, fabricated by the Badding group at Penn State University. I measure both the acoustic andthermal properties of three different silicon metalattices, using the same technique as described in Chapters
3 and 4. These metalattices have periodic structure on scales from the single nanometer to several microns,
making them interesting for a variety of applications, from thermoelectrics to ultralight materials.
The structure of a metalattice is best described by considering its fabrication process. First, silica
nanospheres are assembled into a colloidal crystal with hexagonal close-packed order from monolayer
thickness up to micron thickness [138]. This colloidal crystal serves as a template for the metalattice, as
shown in Fig. 5.1(a). Next, the interstitial space between nanospheres is infiltrated with the metalatticematerial. This material can be metallic, such as Ni, Pd, or Pt, or it can be semiconducting, such as Ge, or
Si. We measure Si metalattices, as shown in Fig. 5.1(b). Next, the silica spheres may be selectively etchedout, leaving only the highly-porous metalattice material, as shown in Fig. 5.1(c). In this case, confinedvolumes of silicon called “meta-atoms” are connected by thin necks, or “meta-bonds”, that are only a few
nanometers wide. Finally, the metalattice may then be re-infiltrated to produce a less porous system, withstronger interactions between meta-atoms. This is shown in Fig. 5.1(d).

105Figure 5.1: Fabrication of silicon metalattices from silica templates. a) A colloidal crystal of silica(SiO2) nanospheres (30nm diameter pictured) is first assembled. b) Silicon is then infiltrated into theinterstitial regions between nanopheres. Two of the metalattices we measure (at 30nm diameter and 14nm
diameter templates) are at this stage of fabrication. c) The silica spheres may then be selectively etched
away, leaving a highly porous silicon metalattice. d) The metalattice may then be re-infiltrated with silicon
to reach a lower porosity. One of the metalattices we measure (starting from a 30nm template) is at thisstage of fabrication. SEM images courtesy of John Badding.
Previous work on thermal transport in porous 3D metalattices shows many interesting phenomena.
Measurements of silicon metalattices at 100s of nanometer periods show evidence of coherent phonon
scattering at grain boundaries and very low thermal conductivity [141]. Copper metalattices at similar scales
exhibit quasiballistic electron transport, due to increased scattering at surfaces [142]. Molecular dynamics(MD) calculations of few-nanometer silicon metalattices show up to a 10,000-fold reduction in thermal
conductivity compared to bulk, due to phonon localization [143]. Our work focuses on silicon metalattices
obtained from 14nm – 30nm nanosphere templates. These systems are an order of magnitude smaller thanexperimental works to date, and come closer to the small scales investigated by MD.

106
For our measurements, we characterize three different silicon metalattices fabricated on top ofsilicon substrates. Two of the metalattices have the silica spheres remaining inside. One starts from a 14nm
diameter nanosphere template, and the other starts from a 30nm diameter nanosphere template. The third
metalattice is porous, where the 30nm template has been etched out, and the metalattice re-infiltrated to alower porosity. After fabricating these three silicon metalattices, collaborators at Penn State place nickel
nanogratings on top as transducers for our experiments. The gratings range from 1000nm linewidth down
to 30nm linewidth, and periods range from 4000nm to 120nm, respectively. As shown in Fig. 5.2(a), the ebeam lithographically deposited nanolines conform to the surface of the metalattice. Even for 30nm gratinglines at a period of 400nm, we are successfully able to diffract our EUV probe and measure transient change
in diffraction signal (Fig. 5.2(b-d)).

Figure 5.2: Measurements of 30nm gratings on a 30nm metalattice. a) AFM height map of a 30nm
linewidth, 120nm period nickel grating deposited on a silicon metalattice with the 30nm silica sphere
template remaining inside. b) Cross-section SEM of a 30nm linewidth, 400nm period nickel grating
deposited on the same metalattice. Note the silica spheres are visible in the cross section of the metalatticelayer (~800nm thickness). c) Vertically binned EUV diffraction pattern obtained from the grating in (b).
Note the low diffracted orders at approximate pixel numbers 100, 300, 600, and 750. The second orderdiffracted beams are partially clipped by a filter before the CCD. d) Change in diffraction signal as a fractionof integrated intensity on the camera is visible above noise for this grating (b) and diffraction pattern (c).
SEM image courtesy of Nabila Nova.

107
For larger grating sizes, namely 50nm linewidth to 1000nm linewidth, the signal is quite strong. As
shown in Fig. 5.3(a), the diffraction efficiency changes by up to 1.5% of the total integrated intensity of theprobe on the camera. This is stronger than the signals we have previously measured by about an order ofmagnitude. One possible explanation for the signal strength comes from the broadened diffraction of our
EUV probe due to the highly uneven surface of the metalattice, demonstrated in Fig. 5.4. By having a morediffuse beam on the camera, we are less likely to saturate pixels in our CCD. Though we ensure the fully
vertically binned image recorded by our camera does not saturate, it is possible that some of the individual
pixels of the CCD do in general saturate before being summed. Thus, because the diffracted beams fromthe metalattice are more diffuse, the individual CCD pixels do not saturate, leading to an enhanced signal.

Figure 5.3: Measured signals on larger gratings for the two 30nm metalattices. a) The diffraction signal
is shown as a fraction of total intensity on the camera, without further scaling. b) Signals are arbitrarily
normalized to separate the signals for clarity. Insets: silica spheres remain inside (a) but are removed in (b).
Actual sample porosity in (b) is lower than that shown for illustrative purposes in the inset.

108

Figure 5.4: Comparing diffraction from a 100nm linewidth, 400nm period grating. The verticallybinned diffraction pattern for the metalattice sample (blue line) is much broader than the diffraction pattern
for a flat fused silica substrate (purple line). Note that the vertical axis is on a log scale. The vertical offset
between the two diffraction images comes from differences in IR pump intensity, camera exposure time,
and the number of camera exposures averaged together. The two images were also collected at slightly
different angles, and different camera-sample distances, producing the slight differences in diffracted orderspacing, and where the CCD filter edge cuts off the diffraction pattern. I have manually shifted themetalattice image horizontally to align the DC beam, for clarity.
To analyze the measured signals from each grating, we first consider the acoustic waves that areclearly visible in each trace. As seen in Fig. 5.3, multiple surface acoustic wave (SAW) frequencies arepresent, which define the range of the SAW dispersion we can access. As described in Chapter 4, SAWs
are confined to the surface by about 1/π of their wavelength. Thus, as shorter wavelength SAWs aremeasured, the SAW velocity changes from that of the silicon substrate, to that of the metalattice layer. For
the smallest gratings, the mass loading effect of the nanolines becomes dominant, and begins to suppress
the SAW velocity below the metalattice’s intrinsic value. This is shown in Fig. 5.5. The SAW velocities ofthe middle grating sizes agree well with finite element analysis (FEA) calculations for an ideal FCC
silica/silicon structure. This indicates that the elastic properties of these metalattices follow continuum
mechanics, even for surface acoustic waves down to a wavelength of 4 nanospheres, and penetration depthsof ~1.5 layers. The measurements on the re-infiltrated metalattice also validate our technique on a porous

109
material. We can launch and probe SAWs, even with large pores of vacuum throughout the material. Notethat all the frequencies we excite with our transducers are below the predicted phononic bandgap for these
periodic structures.

Figure 5.5: Surface acoustic wave dispersion in silicon metalattices. At large penetration depths, theSAW velocity approaches that of the silicon substrate. At intermediate penetrations, results agree with the
FEA prediction. At the smallest penetration depths, mass loading from the transducer nanolines becomes adominant effect, depressing the SAW frequency and lowering the SAW velocity. SAW penetration depth
is the SAW wavelength (defined by the grating period) divided by pi. The SAW velocity is calculated fromthe measured SAW frequency and the grating period. Each point corresponds to the fundamental SAW
launched by each grating on the 30nm template metalattice (red points) and the 14nm template metalattice(green points). Note that both metalattices have the same porosity, and so are predicted to have the same
SAW velocity.
Though the elastic properties of these metalattices can be designed using continuum mechanics
assumptions, their thermal properties are very non-classical. As shown in Fig. 5.3, nanolines on these
metalattices have a much slower thermal decay than that observed on bulk silicon, as shown in Chapter 3.
The fitting of our data with an effective Fourier model is still underway, as we need an independentmeasurement of the porosity of the re-infiltrated sample. However, the slow decays in the raw data alreadysupport molecular dynamics (MD) calculations by Ismaila Dabo at Penn State. In these calculations, our110collaborators show an enhanced scattering factor beyond what continuum mechanics would predict forporous metalattices made from smaller templates (few nanometer spheres). This leads to a greatly reduced
thermal conductivity, as shown in Fig. 5.6. The Dabo group is currently pursuing non-equilibrium MD
simulations of a nanoline-on-metalattice system analogous to ours to inform the interpretation and fittingof our experimentally observed thermal decays.

Figure 5.6: MD calculations of thermal transport in porous silicon metalattices. a) Equilibrium MD
calculations of varying porosity metalattices (largest pore size is 2.2nm) show a reduced thermal
conductivity compared to FEA predictions. Our collaborators attribute this to an enhanced scattering factor
not captured in continuum mechanics b) Work is currently underway to understand the mechanisms behind
our thermal transport measurements in these systems, using nonequilibrium MD with transient heating.
Note the nanoline is made of copper in this simulation, rather than nickel, due to constraints on the interatomic potentials available to build MD models. Courtesy of Ismaila Dabo.
The measurement of complex hierarchical structures by our EUV nanometrology technique showsthat it is not limited to bulk and thin film systems. However, to extend to more general materials systems,
and to in-situ environments where nanograting fabrication is impossible, the technique must be broadened
to a fully non-contact, truly non-destructive modality. Moreover, relying on nanoline transducers both limits
the smallest excitations that we can produce, and the presence of the nanolines themselves can affect thesystem being measured. To overcome all of these challenges, I have pursued integrating transient grating
excitations with our established EUV nanometrology technique.

111
5.3 Optical Transient Gratings
As a first step towards fully non-contact measurements, I have implemented a visible transient
grating pump to launch dynamics at the micron scale, as described in Chapter 2. While this setup currently
only reaches down to 2μm TG periods, it is sufficient for characterizing the effective thermal conductivityof novel materials systems. While an EUV probe is not necessary at these large TG periods, by using one
we avoid the hot electron signal at early times that is measured by visible probes. The shorter wavelengthof our EUV probe also has higher sensitivity to small displacements than a visible probe. This visible TG
setup broadens our capabilities to more general samples, and serves as an important stepping stone to EUV
TG, where direct, non-contact excitation of nanoscale dynamics is possible. Additionally, the simplicity of
TG excitation permits analytical and semi-analytical solutions to the diffusion equation and/or thethermoelastic equation, depending on the system being studied. Thus, I first derive a fitting procedure forour measured signal from TG excitations that avoids the need for computationally intensive FEA modeling.
5.3.1 Data Processing and Fitting
A standard and successful method for fitting TG signals is to use the Fourier transform approach,as in [144]. However, when I implemented this method to fit our signals, I found it was not a good approachfor our data. The samples we have measured to date with TG excitations have very low thermal
conductivity, and they do not fully relax within the time window we can probe. Note, however, that the
sample does fully relax before the next pump pulse arrives. Because our time window is short compared tothe thermal dynamics we measure, we must calculate too large of a vector in frequency space to fit it. To
have enough points in our time window to fit the experimental data, the calculated signal must have asufficiently large maximum frequency. However, to properly sample the calculated signal in the frequency
domain, it must also have a sufficiently short frequency step. These two requirements together define acalculated data set in the frequency domain that is cumbersome to try to calculate and Fourier transformback into the time domain. Thus, I followed a different approach to calculate theoretical signals to fit to ourexperimental data.

112
To accommodate our relatively short time window for these low conductivity samples, I follow the
eigenfunction expansion method of Ju, et al. [145]. I modify the derivation the authors present to be for asingle layer on a thick substrate and to explicitly introduce a thermal boundary resistivity at the interface.
The calculation geometry is shown in Fig. 5.7, where the index i=1 corresponds to the film, and i=2
corresponds to the substrate.

Figure 5.7: Geometry for eigenfunction expansion calculations. Note that the positive z-axis is definedas going down into the depth of the film-substrate system. The film is l1 thick, while the substrate is l2 thick.
The thermal diffusivity, Di, and optical absorption coefficient, αi, are different for the film and substratematerials, in general. The transient grating period is λ. Relevant points on the z-axis are labeled with thedefinitions d0 = 0, d1 = l1, and d2 = l1 + l2.
The derivation begins with the 2D thermal diffusivity equation,

where Di is the thermal diffusivity of the film or substrate. I take the heat source, Qi, to bewhere I0 is the fluence of the incident pump beam (J/m2), and λ is the period of the transient grating. Notethat the instantaneous excitation at t = 0 is a reasonable approximation in this case. The ultrafast laser pulse
heats the electrons on the order of femtoseconds, which in turn thermalize with the lattice in a few
picoseconds. This is essentially instantaneous excitation for thermal decays of several nanoseconds.

113
Mathematically, this means the heat source serves as an initial condition on the temperature field. The
quantity Ei is given by

where α is the optical absorption coefficient (1/m), ρ is the density, and cp is the specific heat capacity [146].
Using separation of variables and eigenfunction expansion in the z-direction, the solution iswhere ε is the eigenvalue. To solve for the four A and B coefficients, as well as the eigenvalues, I apply
boundary conditions in the z-direction. The top and bottom surfaces are adiabatic (heat flux is zero):where κ is the thermal conductivity. At the interface, the heat flux must be continuous, but the temperature
has a discontinuity according to the thermal boundary resistivity, R,

Noting that d2 – d1 = l2 and d1 – d0 = l1, I apply the boundary conditions to obtain the four equations

Eqn. 9 immediately determines B1. The other coefficients and the eigenvalues are obtained byexpressing the remaining equations in matrix form,

114

By setting the determinant of this matrix equal to zero, I obtain the transcendental equation

which can be solved by graphical intersection for the infinite series of eigenvalues, εj. To numerically searchfor each intersection between the left- and right-hand sides of Eqn. 14, it is convenient to partition the ε axis
into intervals containing one intersection each. This condition should be met by searching over the intervalsof ε defined by the zeros of the left-hand side. This is shown in Fig. 5.8.

Figure 5.8: Graphical intersection method for the determination of eigenvalues. The left- and righthand sides of Eqn. 14 are plotted as a function of ε (blue and orange, respectively). Each intersection point,marked with a red circle, defines an eigenvalue, εj. This plot shows the 49th – 53rd eigenvalues for a gold
film on BK7 glass. Note there is exactly one intersection per interval between zeros of the left-hand side.

115
To determine the remaining coefficients, explicitly solve the system of equations in terms of A1:

The bracketed expression in Eqn. 15 is exactly zero at each eigenvalue, according to Eqn. 14. Thus, A1 is afree parameter for each εj. Plugging back into Eqn. 5, we obtain expressions for Z(z) in the film andsubstrate,

where j indexes the eigenvalues, εj. To determine the Aj, I use the initial condition defined by the heat source
applied at t = 0. First, I write the expression for Ti with the more explicit eigenfunction expansion as,

where i still indexes the layers, and j indexes the eigenvalues. Applying the initial condition,

Now, doing a “Fourier’s Trick” integral on both sides,

where 𝑍𝑖𝑗𝐴 (𝑧) is defined by the relationship

116
Pulling the sum outside the integral, I obtain

where the final step used the fact that the integral on the right-hand side of Eqn. 23 is zero unless j = m.
Relabeling m → j, I finally obtain

Defining

I obtain the final expression for the temperature field,

Assuming linear thermal expansion, the surface displacement measured by our EUV probe iswhere ai is the linear thermal expansion coefficient of each layer. To obtain a change in diffraction signal
to fit to our data, I numerically diffract an ideal Gaussian beam from this surface using Fresnel propagation,
as described in Chapter 2. Note that in general, the coupled thermoelastic equations should be solved, but
this equation may be sufficient when acoustic waves are not important.
Using this approach, I calculate theoretical change in diffraction efficiency signals to fit to ourexperimental data. I calculate signals for a sweep of film thermal conductivities and interpolate between
calculated signal curves to fit the best effective thermal conductivity to the data using least squares. The

117
other film material properties are defined as inputs to the calculation depending on the material system
under consideration.
5.3.2 Thermal Transport in Novel Nanosystems
We successfully measured two novel nanosystems using EUV-probed optical TG, both of which
are promising thermoelectric materials, as described in Chapter 1. The first sample is a 2D hybrid
nanoparticle array, fabricated by the Hihath group at UC Davis. Arrays like these have been shown to havevery low thermal conductivities, below 0.5W/mK, which is promising for a high thermoelectric figure of
merit, ZT [147]. Our arrays consist of one to four layers of oleylamine-coated, 10nm-diameter gold
nanoparticles in a hexagonal close packed (HCP) arrangement. The oleylamine molecules inter-digitate,such that the molecule length (1.5nm) is also the inter-particle spacing, as described in [131]. Part of thevalue of these systems is that the ligands can be exchanged for more electrically conductive molecules,
further enhancing ZT. The arrays are assembled as single monolayers, which are then stamp transferred
onto substrates, in our case BK7 glass and a silicon wafer. We characterized both monolayers and fourlayer assemblies on the two substrates. See Fig. 5.9 for an example of these nanoparticle arrays.

Figure 5.9: Nanoparticle molecular hybrid arrays. a) TEM image of a ligand coated gold nanoparticle
array with 2D HCP order. b) Comparison of average nanoparticle separation (black squares) versus
molecular ligand length (blue circles) when exchanging a variety of different length organic molecules onto
the nanoparticle surfaces. The good agreement between the two lengths indicates the ligands stronglyinterdigitate. For our measurements, the nanoparticles are coated in oleylamine, which should result in a
nanoparticle spacing of 1.5nm. Figure from [131].

118
The second material we characterize is a gold nanomesh, consisting of periodic holes in an HCP
arrangement in a thin gold film. We measure three different nanomeshes on a BK7 substrate, with holediameters and periods of 758nm/919nm, 1245nm/1500nm, and 2105nm/2560nm, respectively. Each gold
film is 13nm thick and is prepared as described in [137]. Unlike the nanoparticle arrays, the nanomeshes
have a large enough period to diffract our probe within the numerical aperture of our camera. Fig. 5.10
shows both an example of nanomeshes, and the 2D EUV diffraction that we measure. The hexagonal
symmetry of our diffraction indicates the nanomesh is ordered on the scale of at least 50-100μm. Whenscanning the beam across the sample, we observe regions where two, slightly rotated hexagonal diffraction
patterns are visible. This indicates the presence of grains and grain boundaries in the array of nanomeshholes.

Figure 5.10: Example of nanomesh geometry and diffraction. a) Experimentally measured EUV
diffraction from a 2560nm period gold nanomesh on BK7 glass. Note the six-fold symmetry expected for
the HCP order of the nanomesh. b) Example of nanomesh geometry, taken from [137]. Pictured is an SEM
image of a stack of nanomeshes. Our samples consist of a single gold nanomesh, 13nm thick, on a BK7
glass substrate.
To fit the conductivity of these materials using the approach described in Section 5.3.1, we must
input effective densities, specific heats, thermal expansion coefficients, and optical absorption coefficientsfor the film materials. I use a volume-weighted average for density, expansion coefficient, and absorption
coefficient, where the ratios of gold to vacuum to oleylamine (in the case of the nanoparticle arrays) aredetermined by geometry, and the constituent values are assumed to be bulk. For the heat capacity, a massweighted average must be used, so density and volume ratios are both used, according to119𝑐𝑝,𝑎𝑣𝑔 =𝑓𝐴𝑢 𝜌𝐴𝑢 𝑐𝑝,𝐴𝑢 + 𝑓𝑂𝑙 𝜌𝑂𝑙 𝑐𝑝,𝑂𝑙 + 𝑓𝑉𝑐 𝜌𝑉𝑐 𝑐𝑝,𝑉𝑐
,
𝜌𝑎𝑣𝑔

(29)

where fi is the volume fraction of each constituent, and ρavg is the volume-weighted-average density of thematerial. To calculate the volume ratios from geometry, I approximate both the nanomeshes and eachnanoparticle array layer as having ideal HCP order. For the 4 layer nanoparticle arrays, however, the orderin the stacking direction is not controlled. Thus, I fit the data using two different cross plane orders to
capture the bounds of maximum and minimum packing. For maximum packing, I assume 3D HCP orderof the four layers, while for minimum packing I assume each nanoparticle in a given layer is stacked directly
above a nanoparticle in the layer below. The volume ratios of the different constituent materials in thenanoparticle arrays and nanomeshes is given in Table 5.1. Table 5.2 lists the bulk material properties used.
Sample1 layer NP array
4 layer NP array (HCP)
4 layer NP array (stack)919nm nanomesh
1500nm nanomesh
2560nm nanomesh

Fraction gold
35.17%
44.588%38.50%
38.303%37.524%38.683%Fraction oleylamine
37.64%
43.436%39.99%-

Fraction vacuum
27.19%
11.976%21.51%
61.697%62.476%61.317%Height (nm)
13
42.88
47.5
13
13
13

Table 5.1: Volume fraction of constituent elements in nanoparticle arrays and nanomeshes. Eachligand-coated nanoparticle in the 2D array (top three rows) has a full diameter of 13nm. However, theligands interdigitate by 1.5nm, resuling in the volume fractions shown. For the four layer samples, the order
is not controlled in the cross-plane direction, so two limiting cases are considered. Minimum volume of
vacuum is achieved by a full 3D HCP order (FCC used in calculation) in the bulk of the layer. Maximum
volume of vacuum is achieved by stacking the nanoparticles in each layer directly on top of one another.
Note that the height of each sample is also listed, as the assumption chosen for the cross-plane order changes
what the total height is. Fractions for the nanomeshes are calculated assuming perfect HCP order in-plane.
Propertyκ [W/mK]ρ [kg/m3]
cp [J/kgK]
Thermal expansion [1/K]
Absorption coeff. [1/m]Gold(in effective fit)
19300 [148]
129 [148]
14.2×10-6 [151]7.562×107 [153]

Oleylamine(in effective fit)
813 [149]
2349 [150]2.87×10-4 [152]*(neglected)

BK7 glass
1.2 [63]*2650 [70]*750 [70]*
0.54×10-6 [65]*0.1482 [154]

Silicon149 [62]
2330 [62]710 [61]
3×10-6 [69]98820 [155]

Table 5.2: Bulk material properties used in eigenfunction calculations. Thermal conductivities of gold
and oleylamine in the samples are not input, as the effective conductivity is the fitting parameter used. The
specific heat for oleylamine is a calculated value, based on the group additivity approach. I have neglected
the optical absorption coefficient of oleylamine in the eigenfunction calculations as the absorption is very
high for gold, and small percent changes to this value do not affect the resulting signal. Note the (*) marks
properties of SiO2 used as approximations for BK7 borosilicate glass, and octadecane used as an
approximation for oleylamine.

120
To fit the data collected on the nanoparticle arrays, we first had to exclude all the measurements
where the sample was burned by our pump beam, as shown in Fig. 5.11. Because the pump fluence used islarge due to the low signal from these systems, and because the material is easy to burn, it is vital to checkevery measurement for permanent change or damage that could skew results. To do this, I calculate thechange in the un-pumped diffraction pattern from the beginning of a measurement to the end of ameasurement, and then I compare that result to the change in diffraction upon pumping at the beginning ofa measurement. The pump-dependent change shows where the diffracted orders from TG excitation are.
Any change in the unpumped diffraction over the course of the measurement at those camera pixels
indicates that some permanent change has taken place at that spatial frequency, i.e. the TG pattern has been
burned into the sample. An example of this process is given in Fig. 5.12(a). While measurements that burned
the sample are unsuitable to learn about the repeatable dynamics of these materials, subsequentcharacterization of the burns allows for confirmation of the TG period. By analyzing the Fourier transformsof optical microscope images of burned regions, as shown in Fig. 5.12(b), I confirm that the TG periods
achieved are 2.48±0.02μm and 3.18±0.02μm. These values are extremely close to the nominal 2.5μm and3.2μm phase masks used to generate TG, indicating good alignment of the 4f imaging system.

121

Figure 5.11: Optical microscope images of burned-in TG patterns. a) One layer of nanoparticle
molecular array on a BK7 substrate. Defects, gaps, and edges are visible in this image, as well as faintperiodic lines. These lines have a period that matches the expected period the TG excitation. b) Four layersof nanoparticle molecular array on a silicon substrate. At the most intense area of the burn, the dark silicon
substrate is visible. Note the sample area is otherwise uniform. The goal is always to measure in uniform
sample regions like (b), rather than in defect-heavy regions like (a), but this is not perfectly enforced.

122

Figure 5.12: Identification of sample burning and subsequent calculation of TG period. a) Each scanof our experiment consists of several sweeps through the time window, which are averaged together. The
difference between the pumped and unpumped diffraction patterns for the first sweep (red dotted line)
shows the appearance of TG diffracted orders. The difference between unpumped diffraction patterns
between the first sweep and the last sweep (blue line) indicates whether permanent change has taken place.
Note the random changes in the DC order come from mode fluctuations and/or carbon deposition, notnecessarily just burning. Data shown here is for 3.2μm TG on a hole-free region of the 919nm nanomesh(uniform gold film). b) After taking optical microscope images of burned-in gratings (left), I Fouriertransform the image to find the spatial frequency of the periodic grating (right). The blue arrows indicate
the burned-in grating area and the resulting peak in the Fourier transform. Data shown is for 3.2μm TG on
one layer of nanoparticle molecular array with a silicon substrate.
After I exclude the damaging measurements, I average together all traces for a given substrate,
layer thickness, and TG period configuration. I obtain fits for the effective thermal conductivity of each of

123
these average traces, as shown in Table 5.3 and Fig. 5.13. The best fit thermal conductivities are in the samerange from 0.1-0.5W/mK, as obtained by thermoreflectance measurements on similar, thicker stacks ofarrays in [147]. This is a reduction of three orders of magnitude from the thermal conductivity of bulk gold.
The best fit thermal conductivity in each of our configurations varies only slightly, indicating this is indeed
the effective thermal conductivity of the nanoparticle arrays. Note that one exception is the sample of four
layers on BK7 glass, which has a notably lower thermal conductivity than the others. Due to the agreement
between the other fits, I attribute this discrepancy to sample deviations. Similar to Fig. 5.11(a), the sample
of four layers on BK7 has several cracks, gaps, and other large scale defects. Because each measurement
location was not tightly controlled, it is possible that the only remaining measurements after excluding
burning correspond to locations with large defects, resulting in a spuriously slow decay.
Sample1 layer, 3.2μm TG, Si
1 layer, 3.2μm TG, BK7
4 layers, 3.2μm TG, BK7
4 layers, 2.5μm TG, Si
4 layers, 2.5μm TG, BK7

Effective κ [W/mK]
0.27±0.01
0.27±0.02
0.37±0.03
0.36±0.02
0.13±0.01

Table 5.3 Best fit conductivities for nanoparticle molecular arrays. These are the five samples that haveuseable data on them. Note that not all combinations of 1 vs 4 layers, 2.5μm vs 3.2μm periods, and silicon
vs BK7 substrates have useable data, after excluding burning measurements. The error bars on the effective
thermal conductivity are preliminary and reflect only variance in model substrate thickness and 4-layercross-plane order. The error bars will be larger once all sources of uncertainty are accounted for.
The next step is to verify the numerical model fits by also fitting the data using the FEA approachdescribed in Chapter 3. There are a number of assumptions in the numerical model derivation that need tobe confirmed as valid, most notably the uncoupled treatment of temperature and strain. Once the model isconfirmed, it may be applied to analyze the data taken on the gold nanomesh samples as well.

124

Figure 5.13: Best fits for the nanoparticle molecular arrays. Experimental data traces (dark blue lines)
are obtained by averaging together all non-burning measurements for a given layer number, TG period, andsubstrate combination. In the numerical model, the substrate thickness is specified. The full experimental
thicknesses of ~1mm for BK7 and ~400μm for silicon, however, are not required for the transport that
happens within our experimental time window. Additionally, using too thick of a substrate, sampledproperly, becomes computationally expensive. The above fits show the results for a variety of substrate
thicknesses, once the substrate is thick enough to avoid finite-thickness effects. The results indicate our best
fits are not spuriously dominated by model substrate thickness. For the four-layer samples, the light blue
curves correspond to HCP cross-plane order, while the purple curves correspond to nanoparticle-overnanoparticle stacking cross-plane order.
5.4 Extreme Ultraviolet Transient Gratings
Transient gratings achieved by the interference of two EUV beams has long been a goal fornanosystem metrology. The approach is truly non-contact and non-destructive—EUV TG excitations reachdeep nanoscale lengths and ultrafast times without the limits or unintended influence of nanostructured
transducers. It is furthermore a very general approach to studying surface excitations, as all materials
strongly absorb EUV. In addition to the thermal and elastic characterizations described above, the EUV TG
geometry is suitable for super-resolution imaging via structured illumination microscopy [156, 157],

125
nanoscale spin gratings by interfering crossed polarization beams [158], and even nonlinear optics in the
EUV wavelength range [159]. To achieve the full utility of EUV TG at the deep nanoscale, however, there
are some fundamental challenges that must be overcome.
Transient gratings generated by EUV can be tuned from micron scale periods at low crossing
angles, to nanoscale periods at very large crossing angles. This is defined by the familiar transient grating
relationshipΛ=𝜆,2 sin 𝜃(30)

where Λ is the grating period, λ is the wavelength of the excitation beams, and θ is the half crossing angle.
Eqn. 30 shows that reaching a TG period that matches the wavelength of the EUV beam used requires ahalf crossing angle of 30°. This is a substantial angle to reach, given the requirement of designing EUV
optical systems with no refractive elements, and with high losses at each reflective element. This is the firstchallenge in reaching the deep nanoscale with EUV TG – achieving large crossing angles with sufficient
EUV flux at the sample.
The next challenge in reaching deep nanoscale TG periods is in exciting measurable dynamics inthe sample itself. Materials reflect EUV more when approaching a glancing angle of incidence. This limits
the energy that is absorbed by the sample, once going to very large crossing angles. The ultrafast nature of
the EUV beams also makes it difficult to excite a measurable number of TG periods. The EUV bursts
generated during HHG lie within an ~10fs envelope, and free electron laser (FEL) sources can reachfemtosecond scale pulse durations as well. This means that pulse front tilt severely reduces the interaction
region between the two TG arms at high crossing angle, as shown in Fig. 5.14. While pulse front tilt is
mitigated at visible wavelengths through a 4f imaging system [44], the lack of refractive optics in the EUV
eliminates the same solution. It remains an open question how to design an EUV optical system with
optimized pulse front tilt to produce interference across the full width of the pump arms.

126

Figure 5.14: Pulse front tilt and the 4f optical TG setup. a) Ultrafast lasers produce very narrow pulses
in the longitudinal direction (short temporal duration). When crossing two such pulses at large angles, thisseverely reduces the region where the two pulses can interfere to produce a TG pattern. b) A 4f imaging
system with a transmission phase mask to generate the two TG beams produces a favorable pulse front tilt
to achieve interference across the full width of the crossing beams at the sample plane. Panels (c) and (d)are charge coupled device (CCD) images of crossing two near infrared 30fs pulses in the (a) and (b)geometries, respectively. Images taken from [44].
Though these challenges are fundamental, we have already demonstrated EUV TG at the seeded
FEL, FERMI, at Elettra, near Trieste, Italy. In the following section, I will present optical measurements ofthermal and acoustic dynamics launched by EUV TG, obtained in collaboration with the Nelson group at
MIT and the team at FERMI. I will further present FERMI’s first demonstration of EUV TG down to 28nm
period, probed by a third EUV beam. I will close this section with a description of the work so far to
implement EUV TG on a tabletop, using HHG beams at JILA. To reduce the complexity of the experiment,

127
and to optimize one component at a time, both the experiments at FERMI and the designs at JILA havebegun with an optical probe, and correspondingly large EUV TG periods.
5.4.1 EUV TG with an Optical Probe
The first work I collaborated on with FERMI used the mini-TIMER setup in the DiProI end station.
In this experiment, the beam from the FEL is slowly focused, then split in two by a half mirror. Two otherflat mirrors then recombine the two beams at shallow crossing angles to launch thermal and acoustic
dynamics at large enough EUV TG periods to be probed optically. Fig. 5.15 shows the setup used. In thepresent work, we generate EUV TG with 12.7nm FEL radiation, with 70fs pulse duration, 3μJ total pulse
energy, 50Hz repetition rate, and a 100μm diameter spot size on the sample. The full crossing angle is 2.62°,
producing a 277nm period TG pattern. This period is smaller than the visible TG periods commonly used,
but it is still large enough to be probed by an optically-generated 400nm wavelength probe.

Figure 5.15: The mini-TIMER setup for optically-probed EUV TG. a) Model of the experimental setup
and beam paths. The two TG arms are split from the FEL beam by clipping half of the beam at the edge ofa sharp half mirror. The two arms are then recombined in a parallelogram geometry, with their relative
timing adjustable by the M2 mirror position. b) Top view of the beam paths. The probe beam is incident at
the Bragg condition, producing both a transmitted and reflected first diffracted order. The reflected, or backdiffracted, beam is refocused to a separate camera from the transmitted diffracted beam, allowing
simultaneous signal measurement. c) Side view of the beam paths. The sample is tilted downward by asmall angle, φ, allowing the reflected diffraction beam to be collected by a mirror just below the incident
optical probe. Figures from [160].

128
To probe the EUV transient gratings, we use the second harmonic of the FEL seed laser. It has
400nm wavelength, 100fs pulse duration, and good timing stability with the FEL radiation. The probe wasdelayed by up to 600ps after the pump via a mechanical delay stage, defining the maximum time window
for this experiment. Because we are probing with visible wavelengths, we simultaneously measure signals
both in reflection and transmission geometries, for transparent samples. We probe at the Bragg condition,
meaning that the reflected diffracted order is collinear and counterpropagating to the incident probe beam.
To separate it for measurement, we slightly tilt the sample downward, as shown in Fig. 5.15(c). This angles
the reflected beam below the incident probe beam, where it is collected by an additional optical system.
Measurement of this reflected signal is vital when characterizing opaque samples.
We measured a variety of samples during this experiment. First, we measured thick, transparent
samples, where we characterized acoustic waves both on the surface (Rayleigh waves) and in the bulk
(longitudinal waves) of the sample. These samples included single-crystal diamond, BK7 glass, and thescintillator Bi4Ge3O12. We then measured thermal transport in the same diamond sample, a 349nm silicon
membrane, and a bulk silicon germanium sample (6% Ge). For the SiGe measurement, the sample was
opaque to 400nm light, meaning that only the reflected signal could be used.
The signal obtained by the optical probe is proportional to the refractive index modulation squared.
This must be taken into account to understand the dynamics observed. Consider the simplified equation
2

𝐼 = [𝐴𝑒 𝑒 −𝑡⁄𝜏𝑒 + 𝐴𝑡ℎ 𝑒 −𝑡⁄𝜏𝑡ℎ + 𝐴𝐿 cos(𝜔𝐿 𝑡) + 𝐴𝑅 cos(𝜔𝑅 𝑡)] ,(31)

where the Ai refer to the amplitudes of the electronic, thermal, longitudinal wave, and Rayleigh wave
signals, in order, and I is the measured intensity of the diffracted beam. When the square is carried out, thedirect terms will appear to have half the decay times, τ, and double the frequencies, ω, while the cross terms
will have the intrinsic values of the sample. The thermal-acoustic cross terms will be single-frequency
oscillations on top of a thermal decay. The acoustic-acoustic cross terms, however, will produce beatfrequencies in the measured signal. Each of these terms has a different starting amplitude and decay time,
meaning that they will be dominant at different points in the measured total signal. In general, the electronic

129
decay will dominate at the earliest times, followed by the thermal decay and acoustic waves. If the thermal
signal decays before the acoustic waves, then there is an apparent “frequency doubling” as the thermalacoustic cross terms die out.
With this simple model in mind, we can now discuss the results of the optically-probed EUV TG
measurements. First, I present the acoustic results. For diamond, both the electronic and thermal signals
decay within the first 50ps, leaving just the frequency-doubled and beat frequency acoustic signals, as
shown in Fig. 5.16. Acoustic wave velocities calculated from the measured frequencies are close to theliterature values for diamond [23]. For BK7, we compare the reflection signal to the transmission signal, as
shown in Fig. 5.17. Because of the low thermal conductivity of BK7, the thermal signal persists for the fullexperimental time window, and the measured frequencies are those of the intrinsic Rayleigh and bulk
longitudinal waves. In reflection, the Rayleigh wave signal is stronger, as it is confined to the surface. We
also observe a second longitudinal frequency that is back diffracted from the bulk of the sample. Thisdiffraction can be phase matched and observed because of the very short penetration depth of the EUV
pump, which produces a broad spatial frequency distribution in the depth-direction. See [23] for moredetails. Measured frequencies for BK7 also agree with expectations from literature. Finally, in Bi 4Ge3O12,
we directly excite a 2.8THz optical phonon with the EUV TG beyond just the Rayleigh and longitudinal
waves, as shown in Fig. 5.18. This optical phonon mode has previously been observed, though with aslightly lower frequency.
In addition to demonstrating acoustic characterizations, the mini-TIMER experiment demonstrates
that thermal characterizations are indeed possible using EUV TG. Though limited to shallow crossing
angles by the visible probe, the 277nm period achieved is sufficient to launch non-diffusive thermaltransport in silicon and diamond. Both of these materials have long phonon mean free paths at room
temperature, as discussed in Chapter 3. Though work is ongoing to analyze the thermal measurements,
preliminary results show quasiballistic thermal transport in silicon, as expected at this TG period, and nearly
fully ballistic transport in diamond. Together, all of these results show that EUV TG is a versatile andpowerful metrology tool. All that is needed to go to the deep nanoscale is the addition of an EUV probe.

130

Figure 5.16: Optically probed EUV TG signal on single crystal diamond. As is common for opticalprobes, the first few picoseconds are dominated by the hot electron signal (a). Afterward, thermal andacoustic dynamics are visible, as shown in the composite trace in (b). This signal is amplified 200x to makeit comparable to the electronic signal in (a). The inset shows the long timescale acoustic dynamics amplified
by 6000x compared to (a). These inset data are then Fourier transformed in (c) to show the doubled andbeat frequencies of the Rayleigh (R) and bulk longitudinal (L) waves. All data presented are obtained intransmission geometry. Figure from [23].

131

Figure 5.17: Optically probed EUV TG signal on BK7 glass. Panels (a) and (b) show the experimental
signal and its Fourier transform, respectively, for the transmitted diffracted beam. Panels (c) and (d) show
the signal and Fourier transform, respectively, for the reflected diffracted beam. The additional longitudinalfrequency in reflection mode comes from a back diffracted beam that is phase matched in the bulk. Figure
from [23].

Figure 5.18: Optically probed EUV TG signal on Bi4Ge3O12. a) Long timescale data show the Rayleighand bulk longitudinal wave frequencies of Bi4Ge3O12. The Fourier transform of these data are shown in (b).
c) At short timescales, a 2.8THz optical phonon is observed, as seen in the Fourier transform (d). Figure
from [23].

132
5.4.2 EUV TG with an EUV Probe
After the successful demonstration of optically probing EUV TG at mini-TIMER, we returned to
FERMI, along with members of the Keith Nelson and Gang Chen groups from MIT, to attempt an all-EUV
experiment at the TIMER end station, where both the pump and probe are FEL radiation. In this setup, theEUV beam, which contains multiple harmonics, is first split between pump and probe arms using a half
mirror, as for mini-TIMER. The half that serves as probe is delayed via a mechanical stage, using multilayer
mirrors that select a single EUV wavelength to serve as probe. The half that serves as pump is split once
again by a half mirror, and then filtered to the desired pump wavelength by thin solid state filters. Each of
the three beams is focused onto the sample using a separate toroidal mirror. A schematic of the TIMER
setup is given in Fig. 5.19. To change TG period, the pump toroids are removed, allowing the beams topropagate closer to the sample, where shorter object distance toroids are set up to recombine the beams at
a larger crossing angle. The TG period is further tuned by changing which FEL harmonic is used as thepump wavelength. For this work, the full crossing angle is maintained at 27.6°, while the pump wavelength
is changed from 53.4nm to 39.9nm, to 13.3nm. This produces the record TG periods 110nm, 85nm, and28nm, respectively.
Excited dynamics are measured in a transmission geometry, requiring that the samples be very thin,
as the EUV probe will be strongly absorbed by any material. Probe wavelengths are chosen to be 17.8nm,
13.3nm, and 13.3nm, respectively for each pump wavelength. The probe is incident at 4.5°, achieving the
Bragg condition for the first two geometries. The smallest TG period is not probed at the Bragg condition,
but the grating is sufficiently thin that appreciable signal is still obtained. The FERMI team successfullymeasured thermal and acoustic dynamics in thin Si3N4 and Si membranes, marking the first demonstrationof all-EUV TG with three distinct EUV beams. The work we did during our collaboration with FERMI
contributed to obtaining this result a short time later.

133

Figure 5.19: Schematic of the TIMER experiment with EUV-probed EUV TG. A) Schematic of the
splitting, delay, filter, and focusing optics. Additional toroidal mirrors for larger crossing angle TG are notshown. B) The FEL beam is split with half going to the probe, and the remaining two quarters being split
into the two pump beams. The lossy EUV mirrors in the delay line reduce the probe intensity to be belowthat of the pumps. C) The three beams are recombined on the sample. The delay optics filter the probe to
the desired wavelength, while separate solid state filters select the pump wavelength. D) Focusing after
splitting achieves a better mode at the sample, as indicated by the signal beam measured on the CCD
camera. Note the signal is “background free” as the diffracted beam is not colinear with any other beam.
Figure from [24].
The signals obtained on Si3N4 and Si are shown in Figs. 5.20 and 5.21, respectively. These signals
contain both thermal decays, and acoustic oscillations. On Si3N4, the calculated acoustic wave velocities
and thermal diffusivities agree with literature values [24]. This indicates that the TG excitation is behaving
as expected, even in the EUV regime, and it also indicates that Si3N4 membranes exhibit diffusive thermaltransport even down to 28nm periods. For silicon, however, there is strong non-diffusive thermal transport,
as expected for a long mean free path material. The observed decay is much slower than the diffusive
prediction, as shown in Fig. 5.21. Note that for both samples the earliest times are not dominated by theelectronic response present in optically-probed measurements. As expected from our work using EUV toprobe optically excited metals, the probe is largely insensitive to hot electron dynamics, as long as the EUV
energy is tuned away from core-level resonances. The signal is instead dominated by the lattice response,
e.g. thermal expansion and acoustic waves.

134

Figure 5.20: EUV TG signal on a Si3N4 membrane. TG periods are 110nm, 85nm, and 28nm in panels(A), (B), and (C), respectively. Panel (D) compares the short timescale signal from the 85nm TG to
optically-probed 280nm EUV TG on a thicker Si3N4 membrane. The optically-probed signal (in blue) has
been scaled down to match the EUV signal. The small bump circled in red is the electronic response
measured by the EUV probe. Red lines are exponential fits thermal signals. Figure from [24].

Figure 5.21: EUV TG signal on a Si membrane. Data (black) follow a much slower thermal decay (fit inred) than the diffusive prediction (blue) at 110nm TG period. Inset: there is no dominant electronic signal
at early times. Figure from [24].

135
5.4.3 EUV TG on a Tabletop
Shortly before getting the opportunity to collaborate with the FERMI team, we began designing an
EUV TG experiment using HHG beams at JILA. We considered many approaches to producing an EUV
transient grating using the unique properties of HHG. The ideas ranged from recombining the distinct beams
from non-collinear HHG in a gas jet [161], to doing all of the beam steering in the visible, and finallygenerating EUV with a separate waveguide for each TG arm. However, the final design we chose, to
optimize both stability and flux, is to split the EUV beam with a half mirror, similar to the design used bymini-TIMER. This design can be rather compact and monolithic, and it limits the number of opticalelements, which are always lossy in the EUV wavelength range. This approach also allows for spatial andtemporal overlap to be found by adjusting the final flat mirrors in situ. Fig. 5.22 shows a schematic of ourexperimental design. While our design shares many similarities with the FEL experiments, it is worth noting
one difference that takes advantage of our HHG sources. We push the delay line on the probe from the EUV
to the optical portion of the beam path. This allows us to reduce the losses encountered at FERMI in the
EUV delay line.

Figure 5.22: Design for initial demonstration of EUV TG at JILA. We will follow a focus-then-split
design to generate EUV TG. The blue rectangles represent a silicon rejector mirror, a 3:1 toroidal mirror,
the half mirror, the two recombining mirrors, and finally the sample. The 3:1 toroid should compensate forthe slope error that expands the focal spot by ~3x for standard toroidal mirrors. Using a 150μm fiber, thisshould result in a 30μm diameter pump spot on the sample. The probe will first be 780nm wavelength for
optical probing, and it will then be up-converted to 30nm by HHG for all-EUV TG. Note the probe here is
not at the Bragg condition, so the reflected signal beam is not colinear with the input probe. The Bragg
condition here is 46.8°. Probing at the Bragg angle would require only one access port to both input theprobe and output the signal beam. Distances are listed in mm. The full 700mm to the fiber is not shown.

136
To develop EUV TG capabilities, we will first use shallow crossing angles and a visible probe.
This allows us to simplify the problem and troubleshoot the TG setup without the added complexity of an
HHG probe. Conveniently, once the system is working with an optical probe, we can add a fiber to theexisting probe line, turn up the power, and generate an HHG probe with relatively small changes to thesetup. One difference between the two probes, however, is that the Bragg angle would be very different
between 780nm and 30nm wavelengths, given the same grating being probed.
The Bragg condition is a phase matching angle where the strongest diffracted signal will beobtained. This phase matching can be understood by matching the wavevectors of the probe and signal
beams to the wavevector of the TG, as shown in Fig. 5.23(b). Note, however, that this condition is only
required when the grating is optically thick. For a thin grating, the Bragg requirement is relaxed, and several
diffracted orders may be observed at any incident angle [29], as shown in Fig. 5.23(a). Due to the highabsorption of EUV photons in materials, EUV TG should always be relatively thin, allowing for non-Bragg
probing. There is still a reduction in the diffracted beam intensity when not meeting the Bragg condition,
however, given by the factor𝐿

𝛼 = sinc 2 (Δ𝑘2),(32)

where L is the grating thickness, determined by either the EUV penetration depth, or the sample thickness,
whichever is smaller. Δk is the longitudinal wavevector mismatch between the probe and signal beams,
given byΔ𝑘 = 𝑘𝑝𝑟𝑜𝑏𝑒 (cos 𝜃𝑠𝑖𝑔𝑛𝑎𝑙 − cos 𝜃𝑝𝑟𝑜𝑏𝑒 ),(33)

where kprobe is the magnitude of the probe wavevector, θprobe is the incident angle of the probe (measured
from the sample normal), and the output angle of the diffracted signal is defined by𝜃𝑠𝑖𝑔𝑛𝑎𝑙 = arcsin (𝜆𝑝𝑟𝑜𝑏𝑒− sin 𝜃𝑝𝑟𝑜𝑏𝑒 ),Λ

(34)

where λprobe is the wavelength of the probe beam, and Λ is the TG grating period [24, 159]. Though nonBragg probing is possible for a thin grating, the losses can still be substantial, as shown in Table 5.4. Thisis a significant concern for EUV TG using HHG, where photon flux is the main constraint.

137

Figure 5.23: Thin and thick grating diffraction conditions. a) For thin gratings, the probe may be set toany angle, and several diffracted orders will be produced. b) For thick gratings, the incident probe beam,“C”, must meet the Bragg condition to phase match the diffracted beam, “D”, and emit it from the sample
with appreciable intensity. Right: the phase matching condition is met by matching the probe and diffracted
beams’ wavevectors to the TG wavevector, 𝑞⃗, which is itself determined by the two pump wavevectors “A”
and “B”. Figures from [29].

Grating thickness
L=10nm
L=20nm
L=50nm
L=100nm𝜽𝒑𝒓𝒐𝒃𝒆 = 𝜽𝒑𝒖𝒎𝒑
99%
97%
83%
46%

𝜽𝒑𝒓𝒐𝒃𝒆 = 𝜽𝒑𝒖𝒎𝒑 + 𝟏°
99%
97%
85%
52%

𝜽𝒑𝒓𝒐𝒃𝒆 = 𝜽𝒑𝒖𝒎𝒑 − 𝟏°
99%
96%
81%
41%

Table 5.4 Example signal strength reductions for non-Bragg probing. In this example, I assume a 30nm
EUV beam is probing a 56nm period TG pattern, generated by crossing two 56nm beams at 60° full crossing
angle. The probe is incident very close to the incident pump angle, which would be the case for steering theprobe to the same recombining mirrors as the pump (to avoid geometric constraints). Though most of thesignal is preserved for a 10nm thick grating, the diffracted intensity is reduced by half for a 100nm thickgrating. Reductions in signal strength for other configurations may be calculated by Eqns. 32-34.

138
The main challenge to implementing EUV TG on a tabletop is generating enough flux to excite thesample. Whereas the FEL has more than enough flux to burn the sample with EUV [162], HHG in generalproduces much lower flux. Thus, it is essential to test the maximum HHG flux achievable, in a configuration
suitable for TG excitation, before construction of the experiment can begin.
To obtain a high-flux, single-frequency source for EUV TG, we test “blue-driven” harmonics. As
described in [163], high harmonics driven by the second harmonic of the Ti:sapphire laser (390nm
wavelength, blue-UV regime) are more widely separated in frequency space because each photon being
added is of a higher energy. This makes the harmonic comb easier to separate into single frequencies
suitable for the generation of a pure TG pattern. When driving high harmonics in krypton, the cutoff energy
for the high harmonic process lands at about the 9th harmonic, and the absorption of a thin aluminum filter
lands at about the 5th harmonic. This leaves nearly single-frequency HHG at the 7th harmonic, 56nm
wavelength, without the need for additional filtering by multilayer mirrors, which can be lossy. Moreover,
HHG driven by UV pulses can have enhanced brightness and efficiency from enhanced single-atom yield,
and phase matching of multiply ionized atoms [164]. However, second harmonic generation, for instancein a thin BBO crystal, is at best only 40% efficient. This more than doubles the energy required from theTi:sapphire amplifier, even before high harmonics are generated. The 2mJ laser system described in Chapter
2 is insufficient for this high-flux, single harmonic HHG approach, especially once a second HHG beam isrequired to be generated for the probe. Instead, we collaborate with another project team within our groupthat has a higher-powered driving laser and do flux tests with them. Their laser system produces 8-10mJ of
Ti:sapphire light at a 1kHz repetition rate. Using a 400nm wavelength driving beam, a 150μm innerdiameter fiber, a silicon rejector mirror (to help filter the driving laser), an aluminum filter, and a toroidal
mirror, we measure a per-pulse fluence of 13.7μJ/cm2, with a 72μm diameter spot at the focus. This is very
promising, as the mini-TIMER experiment only needed 2μJ/cm2 (with a 100μm diameter spot) to makethermal and acoustic EUV TG measurements [23]. This lower-energy regime of HHG is known to havehigher flux, comparable to synchrotron radiation [165], which is another reason we will start at 56nm
wavelength rather than 30nm wavelength. Note, however, that the 13.7μJ/cm2 quoted above is measured

139
for HHG in argon, which could have different spectral content. We must measure the spectrum of this HHG
beam to ensure it is still single-frequency for EUV TG.
Single frequency beams are required to generate a purely sinusoidal TG pattern. Even relatively
small intensities of other wavelengths from neighboring harmonics will create beat frequencies thatmodulate the TG pattern. See Fig. 5.24 for an example. Because our measurement is primarily sensitive tothe phase grating defined by thermal expansion, the multiple sine waves of the multiple interfering
harmonics will be added in phase to define the diffraction grating. Adding in phase is equivalent tomultiplying complex exponentials together. According to the rules of Fourier transforms, these terms arethen convolved to define the peaks in diffraction space. Thus, the presence of low-intensity additional
wavelengths in the pump arms will produce “side bands” around both the DC beam and the diffracted ordersof a pure, single-frequency TG diffraction pattern. This is shown in Fig. 5.24. This would complicate theanalysis of the measured data, though perhaps with appropriate modelling it would allow us to measuredynamics at multiple spatial frequencies simultaneously by selecting different diffracted orders.

Figure 5.24: Calculated TG patterns and diffraction for single- and multi-frequency beams. The
imperfect filtering of the HHG comb to a single wavelength leads to beat frequencies in the TG patterngenerated. This effect is significant even for relatively low (5%) intensities of undesired harmonics. Whendiffracting from the phase grating produced by thermal expansion, side-bands are observed from theconvolution of each sine wave from each interfering harmonic.

140
Now that we have a promising flux measurement, and a plan for single-frequency excitation, weare simply waiting for components to arrive to start building the EUV TG setup. In the meantime, however,
we have been designing an optimization procedure for the experiment. First, to optimize the mode and fluxof the HHG beam, it must be directed to a CCD camera. For the optically-probed initial demonstration,
there is no need for an EUV-sensitive CCD camera to measure the diffracted signal, so we may place ourcamera directly behind the sample. Then, we simply move the splitting mirror and sample out of the way,so the full beam is visible on the CCD. We may then insert the splitting mirror until half of the beam isclipped, and the other TG arm may also be directed to the CCD. As long as the crossing angle is sufficiently
shallow, and the CCD sufficiently close to the sample, both beams should be simultaneously observable.
Even with only the first TG arm visible on the camera, we will still be able to observe the point when halfof the beam has been clipped by the splitting mirror. For the all-EUV TG experiment, the probe and pump
HHG beams may be directed by sample rotation to a flange-mounted CCD camera placed in a reflection
measurement geometry. This would require either a highly reflective sample, or a suitably reflective surface
to be mounted on the sample holder, near the sample of interest.
To achieve spatial and temporal overlap, we have a number of tools at our disposal. Phosphorousand cerium YAG scintillators both fluoresce upon EUV irradiation, allowing the use of simple visible
cameras to spatially overlap the beams using the final flat mirrors. Phosphorous is particularly useful as it
may be painted on the corner of a sample to ensure the overlap plane matches the sample plane precisely.
At high crossing angles, even micron scale differences between planes can rapidly take the sample out of
the interference region of the beams. We will also have adjustability of the sample in the z-direction (along
the bisector of the TG arms) to ensure the interference and sample planes match. To overlap the two TG
arms and the probe temporally, we can get close by the careful measurement of optical path lengths, and/or
the use of a fast oscilloscope. EUV radiation can generate electrons in the exposed core of an SMA cable,
which produces a measurable signal on an oscilloscope. The two TG arms may then be coarsely
synchronized by moving one of the mirrors to change the separation between the mirrors, and adjusting themirror angles as needed. This is similar to what is described in [160, 162]. Once the TG arms are roughly

141
overlapped in time, they will be more precisely overlapped by using EUV-induced change in opticalreflectivity. When pumped by EUV radiation, the intensity of reflected or scattered optical light istransiently changed [162]. Using this effect, each TG arm may be separately synchronized to the opticalprobe, which in turn synchronizes the two TG arms to one another as well. During our experiment using
TIMER at FERMI, we found that CoFeB has a particularly strong optical reflectivity signal. We have threesamples of CoFeB films at JILA now, to aid in finding temporal overlap using this method. As describedin Chapter 2, the HHG pulses consist of attosecond bursts under a 10fs envelope. Trying to align thesebursts would greatly increase the difficulty in finding temporal overlap between the two pump arms, and it
would make them more susceptible to jitter than if only the 10fs envelopes needed to be synchronized. As
no one has produced EUV TG using HHG, it is not known how much of a concern this would be. Accordingto [164], UV driven harmonics have shorter attosecond bursts due to reduced intrinsic chirp, indicating theindividual bursts may have to be synchronized. However, even occasional overlap due to jitter would still
produce signal, just requiring more pulses to be averaged over in the measurement.
The initial demonstration of EUV TG at JILA using a visible probe will be designed for a single
crossing angle. We can vary the angle somewhat in situ by increasing the recombining mirror separationand translating the sample, but large changes would require a large displacement of the mirrors, and likely
a repositioning of the camera. However, it is still possible, even at a fixed angle, to produce different TG
periods by changing from 56nm wavelength harmonics to 30nm, and finally to 13nm. The flux constraints
become more dire at shorter wavelengths, as the HHG efficiency is reduced and additional filtering would
be needed to select a single frequency, so we will likely try all-EUV TG before changing HHG wavelengths.
For all-EUV TG, we will add additional mirrors on translation stages to allow for in situ tuning of
the TG period, and to reach very large crossing angles, as shown in Fig. 5.25. By translating the final two
additional recombining mirrors along the arcs of two ellipses whose foci are at the first pair of recombining
mirrors and the sample, respectively, the total path length of each arm will be preserved, maintaining
temporal overlap, while still tuning the TG crossing angle. This can be achieved by placing the two final
mirrors on programmable x-y stages. Though this design adds optical elements, generally sources of loss

142
in the EUV, it also makes each reflection more glancing, which increases reflectivity. These two effects
cancel each other out at 56nm and 30nm wavelengths, producing a similar final flux at the sample. At 13nm,
the four-mirror design actually transmits a greater flux as the reflectivity is more angle-dependent. Notethat this design has a very short working distance, making it difficult to probe the TG pattern. This constraint
can be overcome by probing slightly out of plane, or by reflecting the probe off of the same recombining
mirrors to impinge on the sample at a similar angle to one of the pump arms. For a 30nm probe wavelength,
the resulting diffracted beam is emitted nearly normal to the sample surface, which may then be collectedby an in-vacuum photodiode. This arrangement does, however, depart from the Bragg condition, withresulting signal strength losses shown in Table 5.4. In the end, however, these are engineering constraints
that can be addressed with more insight once we have done the initial demonstration of EUV TG with an
optical probe. The flux numbers are promising that we can match mini-TIMER, where EUV TG has alreadybeen demonstrated, and the outlook is good for demonstrating tabletop EUV TG metrology using HHG.

Figure 5.25: Schematic of four-mirror design for all-EUV TG at JILA. a) The design is similar to thatshown in Fig. 5.22, but with the addition of two more recombining mirrors. Also shown are two dotted
mirrors that represent an alternative position for these new mirrors as they are moveable to change the TG
crossing angle. Note the path lengths have also been scaled up compared to Fig. 5.22 to counteract the
reduction in working distance caused by the additional mirrors. b) By using x-y translation and rotation
stages, the final two mirrors may be moved along the arcs of two ellipses whose foci are at the firstrecombining mirrors and the sample. This preserves the path length, and thus the temporal overlap.

143
5.5 Conclusions and Future WorkIn this chapter, I have presented works that extend our established EUV metrology technique tomore complex samples. First, I presented thermal and acoustic characterizations of silicon metalattices, a
3D hierarchical material promising for thermoelectrics. I then presented the reinstitution of optical TG
pumping capabilities in our lab for the characterization of novel samples that preclude transducer
fabrication. I used this capability to measure ultra-low thermal conductivity in nanoparticle moleculararrays and gold nanomeshes. Finally, I presented work using EUV transient gratings to directly excite
dynamics in the deep nanoscale without the need for separate nanostructure fabrication. The first results
were obtained in collaboration with the FERMI free electron laser in Italy, and work is underway todemonstrate EUV TG on a tabletop at JILA.
The future work will primarily consist of the full development of a robust EUV TG metrology
technique using HHG beams. This will allow for nanoscale characterization of a general set of novel
samples, such as metamaterials, magnetic systems, and even liquids. The TG geometry moreover enables
the extension of powerful visible techniques into the EUV regime, such as super-resolution imaging andfour wave mixing. Now that the EUV TG approach has been proven at the FERMI FEL, and we haveachieved comparable fluxes using HHG, all that remains is to build the experiment on a tabletop.

Chapter 6
Conclusions and Future WorkIn this thesis, I have used coherent, ultrafast EUV beams to measure how nanoscale thermal andelastic properties deviate from bulk predictions. As our technique is one of the few that accesses the deepnanoscale at its intrinsic timescales, the results we have obtained are critically important to inform both
functional nanosystems design, and theoretical and computational approaches to nanosystem modeling. To
measure a more general set of samples and move towards in-situ metrology, I have additionally extended
our EUV measurements to a fully non-contact modality using transient gratings (TG). The main conclusionsof this thesis are as follows:•

Heat no longer travels diffusively when the relevant length scales of heat transfer arecomparable to the mean free paths (MFP) of the heat carriers. Instead, some of the heat carriers
travel ballistically across the relevant length scales and deposit their heat non-locally,
producing a thermal conductivity much lower than what Fourier’s law of heat conduction
would predict. For silicon and other semiconductors, the MFP’s of heat-carrying phonons reachfrom nanometers to several microns—sizes extremely relevant to nanoelectronics, where heat
management is of major concern. We have found that bringing such quasiballistic heat sources
closer together, so that their spacing is also comparable to the phonon MFP’s, counteracts thequasiballistic effect and produces faster thermal transport, closer to the diffusive prediction. By
separately tuning the size and spacing of laser-heated nanostructures on a silicon substrate, I
have mapped the transitions between the diffusive, the quasiballistic, and the collectively

145
diffusive regimes of thermal transport. I have also performed a control study on fused silica
glass, where no non-diffusive behavior is expected or observed.
•

Acoustic waves are an excellent tool to characterize the elastic properties of materials. Our
EUV probe allows us to measure surface acoustic waves far below the diffraction limit ofvisible light. By confining acoustic waves to nanoscale dimensions using nanoline transducers,
I have successfully characterized the full elastic tensor of ultrathin films, down to 11nm
thickness. In my first study, we found that hydrogenation, which lowers a film’s dielectric
constant for nanoelectronics applications, causes a divergence towards incompressible
behavior, once the hydrogenation has broken a critical number of bonds. In my second study,
we found that two different materials, with two different levels of hydrogenation, exhibited
different thickness dependence of their elastic properties. For highly-hydrogenated SiOC:H,
we observed no thickness dependence down to 11nm thickness, while for less-hydrogenated
SiC:H we observed evidence of thickness-dependent softening at 5nm thickness. We
hypothesize that the under-coordinated atoms at a film’s free surface have less of a softening
effect once hydrogenation has already terminated many network bonds in the bulk of the film.

•

To go beyond bulk substrates and 1D-confined thin films, I have worked to extend ourtechnique to general samples, including those with 3D complex structures. First, we usednanoline transducers to characterize the thermal and elastic properties of silicon metalattices,formed by infiltrating silicon into the interstitial regions between self-assembled silica
nanospheres. We found that these metalattices had very low thermal conductivity, but
predictable elastic properties according to continuum mechanics. Next, I used optical TG tocharacterize the effective thermal conductivity of 2D nanoparticle-molecular arrays, without
disrupting the system with nanoline transducers. We found the thermal conductivity of thesearrays was extremely low, which is potentially useful for thermoelectric applications. Finally,
I collaborated with the FERMI free electron laser to directly excite nanoscale thermal andelastic dynamics using EUV transient gratings. We demonstrated this technique on a number

146of transparent and opaque materials, and our work later contributed to the first demonstrationof all-EUV TG measurements by the FERMI team. By both pumping and probing samples with
EUV beams, this technique directly accesses deep nanoscale dynamics without the need fortransducers, and with the promise of compatibility with any sample that has a flat surface.
The next step for our experiment is to build an EUV TG setup here at JILA, using HHG beams. So
far, we have designed our approach, conducted flux tests, and gathered most of our equipment. It is very
promising to have a first demonstration of tabletop EUV TG within a few months’ time. This will greatly
broaden the number of samples we can measure with our nanometrology technique and remove thecomplexity and time it takes to perform nanofabrication prior to our measurements. Additionally, the EUV
TG geometry enables other techniques, such as super-resolution imaging using structured illumination
microscopy, and nonlinear four wave mixing.
In addition to the new EUV TG setup, there is still much opportunity for study using our establishednanograting-based technique. We already have samples of long MFP materials with nanostructure sizes thatreach into the fully ballistic transport regime. At these scales, the ultrafast dynamics we measure can be
dominated by highly non-equilibrium transport, which depends greatly on the specific excitation pathways
accessed in the material. By using laser-heated nanostructures, our established grating-based technique uses
a fundamentally different mechanism for depositing energy in the substrate, compared to direct excitation
by EUV TG. Comparing the two excitation approaches on the same substrate, at the same grating periods,
would be a new way to study ultrafast couplings and light-matter interactions in the EUV regime. HHG
sources would be particularly well-suited for these studies, given their exquisite time resolution. Similarly,
by changing between optical and EUV probing of our measurements, we can cleanly distinguish between
ultrafast phonon and electron dynamics—non-resonant EUV probes are almost exclusively sensitive to theformer, while optical probes are dominated by the latter at early times. A study such as this would enable
the precise study of electron-phonon coupling in metal nitrides and other optically-excitable materials.
Both the nanograting and transient grating techniques are complementary to our dynamic imaging
technique, which accesses local nanostructure information. The former techniques rapidly characterize the147
overall material properties of a sample, while the latter technique informs what is happening near a single
nanostructure. All of these approaches will be needed to characterize advanced, hierarchical nanosystems
that have critical dynamics at sizes from the single nanometer, up to microns.
Given the ability of our nanograting-based technique to study technologically relevant materials,
and the promising ability of TG to measure even more novel samples, coherent EUV nanometrology basedon HHG sources is well-positioned to continue informing the next generation of materials development.

Bibliography[1] L. R. Hirsch, R. J. Stafford, J. A. Bankson, S. R. Sershen, B. Rivera, R. E. Price, J. D. Hazle, N. J. Halas,
and J. L. West, "Nanoshell-mediated near-infrared thermal therapy of tumors under magnetic resonanceguidance," Proc. Natl. Acad. Sci. U. S. A. 100(23), 13549–13554 (2003).
[2] K. Riehemann, S. W. Schneider, T. A. Luger, B. Godin, M. Ferrari, and H. Fuchs, "Nanomedicine—
Challenge and Perspectives," Angew. Chem. Int. Ed. 48(5), 872–897 (2009).
[3] G. Mie, "Contributions to the Optics of Turbid Media, Especially Colloidal Metal Solutions," Ann.
Physik 25(3), 377–445 (1908).
[4] S. Link and M. A. El-Sayed, "Size and Temperature Dependence of the Plasmon Absorption of Colloidal
Gold Nanoparticles," J. Phys. Chem. B 103(21), 4212–4217 (1999).
[5] I. Freestone, N. Meeks, M. Sax, and C. Higgitt, "The Lycurgus Cup — A Roman nanotechnology,"
Gold Bull. 40(4), 270–277 (2007).
[6] M. M. Waldrop, "More than Moore," Nature 530(7589), 144 (2016).
[7] G. Chen, "Nonlocal and Nonequilibrium Heat Conduction in the Vicinity of Nanoparticles," J. Heat
Transfer 118(3), 539–545 (1996).
[8] Y. K. Koh and D. G. Cahill, "Frequency dependence of the thermal conductivity of semiconductor
alloys," Phys. Rev. B 76(7), 075207 (2007).
[9] M. E. Siemens, Q. Li, R. Yang, K. A. Nelson, E. H. Anderson, M. M. Murnane, and H. C. Kapteyn,"Quasi-ballistic thermal transport from nanoscale interfaces observed using ultrafast coherent soft X-ray
beams," Nat. Mater. 9(1), 26–30 (2010).
[10] A. J. Minnich, J. A. Johnson, A. J. Schmidt, K. Esfarjani, M. S. Dresselhaus, K. A. Nelson, and G.
Chen, "Thermal Conductivity Spectroscopy Technique to Measure Phonon Mean Free Paths," Phys. Rev.
Lett. 107(9), 095901 (2011).
[11] K. Esfarjani, G. Chen, and H. T. Stokes, "Heat transport in silicon from first-principles calculations,"
Phys. Rev. B 84(8), 085204 (2011).
[12] D. Shamiryan, T. Abell, F. Iacopi, and K. Maex, "Low-k dielectric materials," Mater. Today 7(1), 34–
39 (2004).
[13] L. G. Zhou and H. Huang, "Are surfaces elastically softer or stiffer?," Appl. Phys. Lett. 84(11), 1940–
1942 (2004).

149[14] N. Nakamura, H. Ogi, T. Yasui, M. Fujii, and M. Hirao, "Mechanism of Elastic Softening Behavior ina Superlattice," Phys. Rev. Lett. 99(3), 035502 (2007).
[15] M. Sigalas and E. N. Economou, "Band structure of elastic waves in two dimensional systems," Solid
State Commun. 86(3), 141–143 (1993).
[16] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, "Acoustic band structure ofperiodic elastic composites," Phys. Rev. Lett. 71(13), 2022–2025 (1993).
[17] R. Liontas and J. R. Greer, "3D nano-architected metallic glass: Size effect suppresses catastrophic
failure," Acta Mater. 133, 393–407 (2017).
[18] B. L. Davis and M. I. Hussein, "Nanophononic Metamaterial: Thermal Conductivity Reduction by
Local Resonance," Phys. Rev. Lett. 112(5), 055505 (2014).
[19] M. Martín-González, O. Caballero-Calero, and P. Díaz-Chao, "Nanoengineering thermoelectrics for21st century: Energy harvesting and other trends in the field," Renew. Sust. Energ. Rev. 24, 288–305(2013).
[20] K. M. Hoogeboom-Pot, J. N. Hernandez-Charpak, X. Gu, T. D. Frazer, E. H. Anderson, W. Chao, R.
W. Falcone, R. Yang, M. M. Murnane, H. C. Kapteyn, and D. Nardi, "A new regime of nanoscale thermaltransport: Collective diffusion increases dissipation efficiency," Proc. Natl. Acad. Sci. U. S. A. 112(16),201503449 (2015).
[21] T. D. Frazer, J. L. Knobloch, K. M. Hoogeboom-Pot, D. Nardi, W. Chao, R. W. Falcone, M. M.
Murnane, H. C. Kapteyn, and J. N. Hernandez-Charpak, "Engineering Nanoscale Thermal Transport: Sizeand Spacing-Dependent Cooling of Nanostructures," Phys. Rev. Appl. 11(2), 024042 (2019).
[22] J. N. Hernandez-Charpak, K. M. Hoogeboom-Pot, Q. Li, T. D. Frazer, J. L. Knobloch, M. Tripp, S.
W. King, E. H. Anderson, W. Chao, M. M. Murnane, H. C. Kapteyn, and D. Nardi, "Full Characterizationof the Mechanical Properties of 11–50 nm Ultrathin Films: Influence of Network Connectivity on the
Poisson’s Ratio," Nano Lett. 17(4), 2178–2183 (2017).
[23] A. A. Maznev, F. Bencivenga, A. Cannizzo, F. Capotondi, R. Cucini, R. A. Duncan, T. Feurer, T. D.
Frazer, L. Foglia, H.-M. Frey, H. Kapteyn, J. Knobloch, G. Knopp, C. Masciovecchio, R. Mincigrucci, G.
Monaco, M. Murnane, I. Nikolov, E. Pedersoli, A. Simoncig, A. Vega-Flick, and K. A. Nelson, "Generationof coherent phonons by coherent extreme ultraviolet radiation in a transient grating experiment," Appl.
Phys. Lett. 113(22), 221905 (2018).
[24] F. Bencivenga, R. Mincigrucci, F. Capotondi, L. Foglia, D. Naumenko, A. A. Maznev, E. Pedersoli,
A. Simoncig, F. Caporaletti, V. Chiloyan, R. Cucini, F. Dallari, R. A. Duncan, T. D. Frazer, G. Gaio, A.
Gessini, L. Giannessi, S. Huberman, H. Kapteyn, J. Knobloch, G. Kurdi, N. Mahne, M. Manfredda, A.
Martinelli, M. Murnane, E. Principi, L. Raimondi, S. Spampinati, C. Spezzani, M. Trovò, M. Zangrando,
G. Chen, G. Monaco, K. A. Nelson, and C. Masciovecchio, “Nanoscale transient gratings excited andprobed by extreme ultraviolet femtosecond pulses,” Sci. Adv. (Submitted).
[25] R. I. Tobey, E. H. Gershgoren, M. E. Siemens, M. M. Murnane, H. C. Kapteyn, T. Feurer, and K. A.
Nelson, "Nanoscale photothermal and photoacoustic transients probed with extreme ultraviolet radiation,"
Appl. Phys. Lett. 85(4), 564–566 (2004).
[26] D. H. Peregrine, G. Shoker, and A. Symon, "The bifurcation of liquid bridges," J. Fluid Mech. 212,
25–39 (1990).

150[27] J. A. Rogers, A. A. Maznev, M. J. Banet, and K. A. Nelson, "Optical Generation and Characterizationof Acoustic Waves in Thin Films: Fundamentals and Applications," Annu. Rev. Mater. Sci. 30(1), 117–
157 (2000).
[28] G. Cachier, "Optical excitation of high‐amplitude surface waves," Appl. Phys. Lett. 17(10), 419–421(1970).
[29] H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings. New York, NY: SpringerVerlag Berlin Heidelberg (1986).
[30] C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, "Surface generation and detection of phonons by
picosecond light pulses," Phys. Rev. B 34(6), 4129–4138 (1986).
[31] A. Neubrand and P. Hess, "Laser generation and detection of surface acoustic waves: Elastic properties
of surface layers," J. Appl. Phys. 71(1), 227–238 (1992).
[32] K. M. Hoogeboom-Pot, E. Turgut, J. N. Hernandez-Charpak, J. M. Shaw, H. C. Kapteyn, M. M.
Murnane, and D. Nardi, "Nondestructive Measurement of the Evolution of Layer-Specific Mechanical
Properties in Sub-10 nm Bilayer Films," Nano Lett. 16(8), 4773–4778 (2016).
[33] J. L. Krause, K. J. Schafer, and K. C. Kulander, "High-order harmonic generation from atoms and ionsin the high intensity regime," Phys. Rev. Lett. 68(24), 3535–3538 (1992).
[34] T. Popmintchev, M.-C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, "The attosecond nonlinear
optics of bright coherent X-ray generation," Nat. Photonics 4(12), 822–832 (2010).
[35] M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, "Theory of highharmonic generation by low-frequency laser fields," Phys. Rev. A 49(3), 2117–2132 (1994).
[36] A. Rundquist, C. G. Durfee, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn,"Phase-Matched Generation of Coherent Soft X-rays," Science 280(5368), 1412–1415 (1998).
[37] K. J. Schafer, B. Yang, L. F. DiMauro, and K. C. Kulander, "Above threshold ionization beyond thehigh harmonic cutoff," Phys. Rev. Lett. 70(11), 1599–1602 (1993).
[38] P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71(13),1994–1997 (1993).
[39] M.-C. Chen, C. Mancuso, C. Hernández-García, F. Dollar, B. Galloway, D. Popmintchev, P.-C. Huang,
B. Walker, L. Plaja, A. A. Jaroń-Becker, A. Becker, M. M. Murnane, H. C. Kapteyn, and T. Popmintchev,"Generation of bright isolated attosecond soft X-ray pulses driven by multicycle midinfrared lasers," Proc.
Natl. Acad. Sci. U. S. A. 111(23), E2361–E2367 (2014).
[40] J. Chen, E. Louis, C. J. Lee, H. Wormeester, R. Kunze, H. Schmidt, D. Schneider, R. Moors, W. van
Schaik, M. Lubomska, and F. Bijkerk, "Detection and characterization of carbon contamination on EUV
multilayer mirrors," Opt. Express 17(19), 16969–16979 (2009).
[41] R. I. Tobey, M. E. Siemens, O. Cohen, M. M. Murnane, H. C. Kapteyn, and K. A. Nelson, "Ultrafast
extreme ultraviolet holography: dynamic monitoring of surface deformation," Opt. Lett. 32(3), 286–288(2007).
[42] Q. Li, K. Hoogeboom-Pot, D. Nardi, M. M. Murnane, H. C. Kapteyn, M. E. Siemens, E. H. Anderson,
O. Hellwig, E. Dobisz, B. Gurney, R. Yang, and K. A. Nelson, "Generation and control of ultrashort-151
wavelength two-dimensional surface acoustic waves at nanoscale interfaces," Phys. Rev. B 85(19), 195431(2012).
[43] A. R. Duggal and K. A. Nelson, "Picosecond–microsecond structural relaxation dynamics inpolypropylene glycol: Impulsive stimulated light‐scattering experiments," J. Chem. Phys. 94(12), 7677–
7688 (1991).
[44] A. A. Maznev, T. F. Crimmins, and K. A. Nelson, "How to make femtosecond pulses overlap," Opt.
Lett. 23(17), 1378–1380 (1998).
[45] R. I. Tobey, M. E. Siemens, M. M. Murnane, H. C. Kapteyn, D. H. Torchinsky, and K. A. Nelson,"Transient grating measurement of surface acoustic waves in thin metal films with extreme ultraviolet
radiation," Appl. Phys. Lett. 89(9), 091108 (2006).
[46] Z. Qin and J. C. Bischof, "Thermophysical and biological responses of gold nanoparticle laser heating,"
Chem. Soc. Rev. 41(3), 1191–1217 (2012).
[47] Y. Lan, A. J. Minnich, G. Chen, and Z. Ren, "Enhancement of Thermoelectric Figure-of-Merit by a
Bulk Nanostructuring Approach," Adv. Funct. Mater. 20(3), 357–376 (2010).
[48] P. Pichanusakorn and P. Bandaru, "Nanostructured thermoelectrics," Mater. Sci. Eng. R Rep. 67(2),19–63 (2010).
[49] M. Maldovan, "Sound and heat revolutions in phononics," Nature 503(7475), 209–217 (2013).
[50] H. Honarvar, L. Yang, and M. I. Hussein, "Thermal transport size effects in silicon membranes
featuring nanopillars as local resonators," Appl. Phys. Lett. 108(26), 263101 (2016).
[51] J. A. Johnson, A. A. Maznev, J. Cuffe, J. K. Eliason, A. J. Minnich, T. Kehoe, C. M. S. Torres, G.
Chen, and K. A. Nelson, "Direct Measurement of Room-Temperature Nondiffusive Thermal Transport
Over Micron Distances in a Silicon Membrane," Phys. Rev. Lett. 110(2), 025901 (2013).
[52] J. Cuffe, J. K. Eliason, A. A. Maznev, K. C. Collins, J. A. Johnson, A. Shchepetov, M. Prunnila, J.
Ahopelto, C. M. Sotomayor Torres, G. Chen, and K. A. Nelson, "Reconstructing phonon mean-free-path
contributions to thermal conductivity using nanoscale membranes," Phys. Rev. B 91(24), 245423 (2015).
[53] K. T. Regner, D. P. Sellan, Z. Su, C. H. Amon, A. J. H. McGaughey, and J. A. Malen, "Broadband
phonon mean free path contributions to thermal conductivity measured using frequency domain
thermoreflectance," Nat. Commun. 4, 1640 (2013).
[54] T. Oyake, M. Sakata, and J. Shiomi, "Nanoscale thermal conductivity spectroscopy by using gold
nano-islands heat absorbers," Appl. Phys. Lett. 106(7), 073102 (2015).
[55] Y. Hu, L. Zeng, A. J. Minnich, M. S. Dresselhaus, and G. Chen, "Spectral mapping of thermal
conductivity through nanoscale ballistic transport," Nat. Nanotechnol. 10(8), 701–706 (2015).
[56] L. Zeng, K. C. Collins, Y. Hu, M. N. Luckyanova, A. A. Maznev, S. Huberman, V. Chiloyan, J. Zhou,
X. Huang, K. A. Nelson, and G. Chen, "Measuring Phonon Mean Free Path Distributions by Probing
Quasiballistic Phonon Transport in Grating Nanostructures," Sci. Rep. 5, 17131 (2015).
[57] F. Yang and C. Dames, "Mean free path spectra as a tool to understand thermal conductivity in bulk
and nanostructures," Phys. Rev. B 87(3), 035437 (2013).

152[58] I. Horcas, R. Fernández, J. M. Gómez-Rodríguez, J. Colchero, J. Gómez-Herrero, and A. M. Baro,"WSXM: A software for scanning probe microscopy and a tool for nanotechnology," Rev. Sci. Instrum.
78(1), 013705 (2007).
[59] F. Banfi, F. Pressacco, B. Revaz, C. Giannetti, D. Nardi, G. Ferrini, and F. Parmigiani, "Ab initio
thermodynamics calculation of all-optical time-resolved calorimetry of nanosize systems: Evidence ofnanosecond decoupling of electron and phonon temperatures," Phys. Rev. B 81(15), 155426 (2010).
[60] COMSOL, Inc. (2013) COMSOL Multiphysics, Version 4.3b (COMSOL, Inc., Burlington, MA).
[61] P. D. Desai, “Thermodynamic Properties of Iron and Silicon”, J. Phys. Chem. Ref. Data 15(3), 967–
983 (1986).
[62] J. A. Dean and N. A. Lange, Lange’s Handbook of Chemistry, 15. ed, New York, NY: McGraw-Hill(1999).
[63] Y. S. Touloukian, Thermophysical Properties of Matter: Vol 2. Thermal Conductivity – Nonmetallic
Solids. New York, NY: Plenum Publishing Corporation (1970).
[64] D. Nardi, M. Travagliati, M. E. Siemens, Q. Li, M. M. Murnane, H. C. Kapteyn, G. Ferrini, F.
Parmigiani, and F. Banfi, "Probing Thermomechanics at the Nanoscale: Impulsively Excited Pseudosurface
Acoustic Waves in Hypersonic Phononic Crystals," Nano Lett. 11(10), 4126–4133 (2011).
[65] M. J. Weber, Handbook of Optical Materials: Vol 3. Optical Materials, Part 1. Boca Raton, FL: CRC
Press, LLC (2003).
[66] J. Zacharias, "The Temperature Dependence of Young’s Modulus for Nickel," Phys. Rev. 44(2), 116–
122 (1933).
[67] J. J. Wortman and R. A. Evans, "Young’s Modulus, Shear Modulus, and Poisson’s Ratio in Silicon
and Germanium," J. Appl. Phys. 36(1), 153–156 (1965).
[68] T. G. Kollie, "Measurement of the thermal-expansion coefficient of nickel from 300 to 1000 K anddetermination of the power-law constants near the Curie temperature," Phys. Rev. B 16(11), 4872–4881(1977).
[69] Y. Okada and Y. Tokumaru, "Precise determination of lattice parameter and thermal expansion
coefficient of silicon between 300 and 1500 K," J. Appl. Phys. 56(2), 314–320 (1984).
[70] Y. S. Touloukian, Thermophysical Properties of High Temperature Solid Materials: Vol. 4. Oxides
and their solutions and mixtures. New York, NY: The Macmillan Company (1967).
[71] A. A. Maznev, J. A. Johnson, and K. A. Nelson, "Onset of nondiffusive phonon transport in transient
thermal grating decay," Phys. Rev. B 84(19), 195206 (2011).
[72] R. Cheaito, J. T. Gaskins, M. E. Caplan, B. F. Donovan, B. M. Foley, A. Giri, J. C. Duda, C. J.
Szwejkowski, C. Constantin, H. J. Brown-Shaklee, J. F. Ihlefeld, and P. E. Hopkins, "Thermal boundary
conductance accumulation and interfacial phonon transmission: Measurements and theory," Phys. Rev. B
91(3), 035432 (2015).
[73] P. B. Allen and J. L. Feldman, "Thermal conductivity of disordered harmonic solids," Phys. Rev. B
48(17), 12581–12588 (1993).

153[74] C. Hua and A. J. Minnich, "Heat dissipation in the quasiballistic regime studied using the Boltzmann
equation in the spatial frequency domain," Phys. Rev. B 97(1), 014307 (2018).
[75] P. Torres, A. Torelló, J. Bafaluy, J. Camacho, X. Cartoixà, and F. X. Alvarez, "First principles kineticcollective thermal conductivity of semiconductors," Phys. Rev. B 95(16), 165407 (2017).
[76] A. Ziabari, P. Torres, B. Vermeersch, Y. Xuan, X. Cartoixà, A. Torelló, J.-H. Bahk, Y. R. Koh, M.
Parsa, P. D. Ye, F. X. Alvarez, and A. Shakouri, "Full-field thermal imaging of quasiballistic crosstalk
reduction in nanoscale devices," Nat. Commun. 9(1), 255 (2018).
[77] B. Vermeersch and N. Mingo, "Quasiballistic heat removal from small sources studied from firstprinciples," Phys. Rev. B 97(4), 045205 (2018).
[78] P. Torres, A. Ziabari, A. Torelló, J. Bafaluy, J. Camacho, X. Cartoixà, A. Shakouri, and F. X. Alvarez,"Emergence of hydrodynamic heat transport in semiconductors at the nanoscale," Phys. Rev. Mater. 2(7),076001 (2018).
[79] A. Beardo, M. Calvo-Schwarzwälder, J. Camacho, T. G. Myers, P. Torres, L. Sendra, F. X. Alvarez,
and J. Bafaluy, "Hydrodynamic Heat Transport in Compact and Holey Silicon Thin Films," Phys. Rev.
Appl. 11(3), 034003 (2019).
[80] N. K. Ravichandran and A. J. Minnich, "Coherent and incoherent thermal transport in nanomeshes,"
Phys. Rev. B 89(20), 205432 (2014).
[81] J. Lee, W. Lee, G. Wehmeyer, S. Dhuey, D. L. Olynick, S. Cabrini, C. Dames, J. J. Urban, and P.
Yang, "Investigation of phonon coherence and backscattering using silicon nanomeshes," Nat. Commun.
8, 14054 (2017).
[82] S. W. King, J. Bielefeld, G. Xu, W. A. Lanford, Y. Matsuda, R. H. Dauskardt, N. Kim, D. Hondongwa,
L. Olasov, B. Daly, G. Stan, M. Liu, D. Dutta, and D. Gidley, "Influence of network bond percolation on
the thermal, mechanical, electrical and optical properties of high and low-k a-SiC:H thin films," J. NonCryst. Solids 379, 67–79 (2013).
[83] C. A. Yuan, O. van der Sluis, G. Q. Zhang, L. J. Ernst, W. D. van Driel, R. B. R. van Silfhout, and B.
J. Thijsse, "Chemical–mechanical relationship of amorphous/porous low-dielectric film materials,"Comput. Mater. Sci. 42(4), 606–613 (2008).
[84] H. Li, J. M. Knaup, E. Kaxiras, and J. J. Vlassak, "Stiffening of organosilicate glasses by organic crosslinking," Acta Mater. 59(1), 44–52 (2011).
[85] R. J. Wang, C. Y. Wang, and Y. T. Feng, "Effective geometric size and bond-loss effect innanoelasticity of GaN nanowires," Int. J. Mech. Sci. 130, 267–273 (2017).
[86] H. Ogi, M. Fujii, N. Nakamura, T. Yasui, and M. Hirao, "Stiffened Ultrathin Pt Films Confirmed byAcoustic-Phonon Resonances," Phys. Rev. Lett. 98(19), 195503 (2007).
[87] P. Villain, P. Goudeau, P.-O. Renault, and K. F. Badawi, "Size effect on intragranular elastic constantsin thin tungsten films," Appl. Phys. Lett. 81(23), 4365–4367 (2002).
[88] N. Nakamura, A. Uranishi, M. Wakita, H. Ogi, M. Hirao, and M. Nishiyama, "Elastic stiffness of L10
FePt thin film studied by picosecond ultrasonics," Appl. Phys. Lett. 98(10), 101911 (2011).

154[89] M. G. Beghi, A. C. Ferrari, C. E. Bottani, A. Libassi, B. K. Tanner, K. B. K. Teo, and J. Robertson,"Elastic constants and structural properties of nanometre-thick diamond-like carbon films," Diam. Relat.
Mater. 11(3), 1062–1067 (2002).
[90] M. G. Beghi, C. S. Casari, A. Li Bassi, C. E. Bottani, A. C. Ferrari, J. Robertson, and P. Milani,"Acoustic phonon propagation and elastic properties of nano-sized carbon films investigated by Brillouinlight scattering," Thin Solid Films 420–421, 300–305 (2002).
[91] C. S. Ewing, S. Bhavsar, G. Veser, J. J. McCarthy, and J. K. Johnson, "Accurate Amorphous Silica
Surface Models from First-Principles Thermodynamics of Surface Dehydroxylation," Langmuir 30(18),5133–5141 (2014).
[92] J. Weissenrieder, S. Kaya, J.-L. Lu, H.-J. Gao, S. Shaikhutdinov, H.-J. Freund, M. Sierka, T. K.
Todorova, and J. Sauer, "Atomic Structure of a Thin Silica Film on a Mo(112) Substrate: A TwoDimensional Network of SiO2 Tetrahedra," Phys. Rev. Lett. 95(7), 076103 (2005).
[93] C. Lee, X. Wei, J. W. Kysar, and J. Hone, "Measurement of the Elastic Properties and Intrinsic Strengthof Monolayer Graphene," Science 321(5887), 385–388 (2008).
[94] D. C. Hurley, V. K. Tewary, and A. J. Richards, "Thin-film elastic-property measurements with laserultrasonic SAW spectrometry," Thin Solid Films 398–399, 326–330 (2001).
[95] X. Li, T. Ono, Y. Wang, and M. Esashi, "Ultrathin single-crystalline-silicon cantilever resonators:
Fabrication technology and significant specimen size effect on Young’s modulus," Appl. Phys. Lett. 83(15),3081–3083 (2003).
[96] A. I. Fedorchenko, A.-B. Wang, and H. H. Cheng, "Thickness dependence of nanofilm elastic
modulus," Appl. Phys. Lett. 94(15), 152111 (2009).
[97] C. M. Stafford, B. D. Vogt, C. Harrison, D. Julthongpiput, and R. Huang, "Elastic Moduli of Ultrathin
Amorphous Polymer Films," Macromolecules 39(15), 5095–5099 (2006).
[98] C. Q. Sun, "Size dependence of nanostructures: Impact of bond order deficiency," Prog. Solid State
Chem. 35(1), 1–159 (2007).
[99] P.-O. Renault, E. Le Bourhis, P. Villain, P. Goudeau, K. F. Badawi, and D. Faurie, "Measurement of
the elastic constants of textured anisotropic thin films from x-ray diffraction data," Appl. Phys. Lett. 83(3),473–475 (2003).
[100] H. Ogi, M. Fujii, N. Nakamura, T. Shagawa, and M. Hirao, "Resonance acoustic-phonon
spectroscopy for studying elasticity of ultrathin films," Appl. Phys. Lett. 90(19), 191906 (2007).
[101] J. Luo, J. Wang, E. Bitzek, J. Y. Huang, H. Zheng, L. Tong, Q. Yang, J. Li, and S. X. Mao, "SizeDependent Brittle-to-Ductile Transition in Silica Glass Nanofibers," Nano Lett. 16(1), 105–113 (2016).
[102] R. E. Miller and V. B. Shenoy, "Size-dependent elastic properties of nanosized structural elements,"
Nanotechnology 11(3), 139–147 (2000).
[103] R. Dingreville, J. Qu, and Mohammed Cherkaoui, "Surface free energy and its effect on the elastic
behavior of nano-sized particles, wires and films," J. Mech. Phys. Solids 53(8), 1827–1854 (2005).
[104] K. Geng, F. Yang, and E. A. Grulke, "Nanoindentation of submicron polymeric coating systems,"
Mater. Sci. Eng. A 479(1), 157–163 (2008).

155[105] J. Hay and B. Crawford, "Measuring substrate-independent modulus of thin films," J. Mater. Res.
26(6), 727–738 (2011).
[106] B. Gong, Q. Chen, and D. Wang, "Molecular dynamics study on size-dependent elastic properties ofsilicon nanoplates," Mater. Lett. 67(1), 165–168 (2012).
[107] C. M. Flannery, T. Wittkowski, K. Jung, B. Hillebrands, and M. R. Baklanov, "Critical properties ofnanoporous low dielectric constant films revealed by Brillouin light scattering and surface acoustic wave
spectroscopy," Appl. Phys. Lett. 80(24), 4594–4596 (2002).
[108] P. A. Mante, J. F. Robillard, and A. Devos, "Complete thin film mechanical characterization using
picosecond ultrasonics and nanostructured transducers: experimental demonstration on SiO2," Appl. Phys.
Lett. 93(7), 071909 (2008).
[109] M. Colletta, W. Gachuhi, S. A. Gartenstein, M. M. James, E. A. Szwed, B. C. Daly, W. Cui, and G.
A. Antonelli, "Picosecond ultrasonic study of surface acoustic waves on periodically patterned layered
nanostructures," Ultrasonics 87, 126–132 (2018).
[110] S. W. King, M. M. Paquette, J. W. Otto, A. N. Caruso, J. Brockman, J. Bielefeld, M. French, M.
Kuhn, and B. French, "Valence and conduction band offsets at amorphous hexagonal boron nitride
interfaces with silicon network dielectrics," Appl. Phys. Lett. 104(10), 102901 (2014).
[111] A. Giri, S. W. King, W. A. Lanford, A. B. Mei, D. Merrill, L. Li, R. Oviedo, J. Richards, D. H. Olson,
J. L. Braun, J. T. Gaskins, F. Deangelis, A. Henry, and P. E. Hopkins, "Interfacial Defect Vibrations
Enhance Thermal Transport in Amorphous Multilayers with Ultrahigh Thermal Boundary Conductance,"
Adv. Mater. 30(44), 1804097 (2018).
[112] T. Denneulin, D. Cooper, J.-M. Hartmann, and J.-L. Rouviere, "The addition of strain in uniaxiallystrained transistors by both SiN contact etch stop layers and recessed SiGe sources and drains," J. Appl.
Phy. 112(9), 094314 (2012).
[113] L. Bluestein, "A linear filtering approach to the computation of discrete Fourier transform," IEEE
Trans. Audio Electroacoust. 18(4), 451–455 (1970).
[114] A. A. Maznev, "Band gaps and Brekhovskikh attenuation of laser-generated surface acoustic waves
in a patterned thin film structure on silicon," Phys. Rev. B 78(15), 155323 (2008).
[115] H. T. Grahn, H. J. Maris, J. Tauc, and B. Abeles, "Time-resolved study of vibrations of a-Ge:H/aSi:H multilayers," Phys. Rev. B 38(9), 6066–6074 (1988).
[116] C. Rossignol, B. Perrin, B. Bonello, P. Djemia, P. Moch, and H. Hurdequint, "Elastic properties ofultrathin permalloy/alumina multilayer films using picosecond ultrasonics and Brillouin light scattering,"
Phys. Rev. B 70(9), 094102 (2004).
[117] J. L. Arlein, S. E. M. Palaich, B. C. Daly, P. Subramonium, and G. A. Antonelli, "Optical pumpprobe measurements of sound velocity and thermal conductivity of hydrogenated amorphous carbon films,"
J. Appl. Phys. 104(3), 033508 (2008).
[118] B. A. Auld, Acoustic Fields and Waves in Solids, 2nd ed. (Krieger Publishing Company, Inc., 1990),
1.
[119] T. Atanackovic and A. Guran, Theory of Elasticity for Scientists and Engineers (Birkhauser Boston,2000).

156[120] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 2nd ed., Course of Theoretical Physics No. 7(Pergamon Press Ltd., 1970).
[121] D. Nardi, F. Banfi, C. Giannetti, B. Revaz, G. Ferrini, and F. Parmigiani, "Pseudosurface acoustic
waves in hypersonic surface phononic crystals," Phys. Rev. B 80(10), 104119 (2009).
[122] J. C. Phillips and M. F. Thorpe, "Constraint theory, vector percolation and glass formation," Solid
State Commun. 53(8), 699–702 (1985).
[123] J. C. Mauro, "Topological constraint theory of glass," Am. Ceram. Soc. Bull. 90(4), 31–37 (2011).
[124] J. Zizka, S. King, A. Every, and R. Sooryakumar, "Acoustic Phonons and Mechanical Properties of
Ultra-Thin Porous Low-k Films: A Surface Brillouin Scattering Study," J. Electron. Mater. 47(7), 3942–
3950 (2018).
[125] M. A. Shandiz, "Effective coordination number model for the size dependency of physical propertiesof nanocrystals," J. Phys. Condens. Matter 20(32), 325237 (2008).
[126] J. Zizka, S. King, A. G. Every, and R. Sooryakumar, "Mechanical properties of low- and high-k
dielectric thin films: A surface Brillouin light scattering study," J. Appl. Phys. 119(14), 144102 (2016).
[127] R. M. Karl, G. F. Mancini, J. L. Knobloch, T. D. Frazer, J. N. Hernandez-Charpak, B. Abad, D. F.
Gardner, E. R. Shanblatt, M. Tanksalvala, C. L. Porter, C. S. Bevis, D. E. Adams, H. C. Kapteyn, and M.
M. Murnane, "Full-field imaging of thermal and acoustic dynamics in an individual nanostructure using
tabletop high harmonic beams," Sci. Adv. 4(10), eaau4295 (2018).
[128] A. D. Sinelnik, M. V. Rybin, S. Y. Lukashenko, M. F. Limonov, and K. B. Samusev, "Opticalproperties of honeycomb photonic structures," Phys. Rev. A 95(6), 063837 (2017).
[129] L. R. Meza, G. P. Phlipot, C. M. Portela, A. Maggi, L. C. Montemayor, A. Comella, D. M. Kochmann,
and J. R. Greer, "Reexamining the mechanical property space of three-dimensional lattice architectures,"
Acta Mater. 140, 424–432 (2017).
[130] M. A. Boles, M. Engel, and D. V. Talapin, "Self-Assembly of Colloidal Nanocrystals: From Intricate
Structures to Functional Materials," Chem. Rev. 116(18), 11220–11289 (2016).
[131] C. E. McCold, Q. Fu, S. Hihath, J.-M. Han, Y. Halfon, R. Faller, K. van Benthem, L. Zang, and J.
Hihath, "Ligand exchange based molecular doping in 2D hybrid molecule-nanoparticle arrays: length
determines exchange efficiency and conductance," Mol. Syst. Des. Eng. 2(4), 440–448 (2017).
[132] W. R. Lenart and M. J. A. Hore, "Structure–property relationships of polymer-grafted nanospheresfor designing advanced nanocomposites," Nano-Struct. Nano-Objects 16, 428–440 (2018).
[133] J. He, X.-M. Lin, H. Chan, L. Vuković, P. Král, and H. M. Jaeger, "Diffusion and Filtration Propertiesof Self-Assembled Gold Nanocrystal Membranes," Nano Lett. 11(6), 2430–2435 (2011).
[134] F. J. Ibañez and F. P. Zamborini, "Chemiresistive Sensing with Chemically Modified Metal and Alloy
Nanoparticles," Small 8(2), 174–202 (2012).
[135] E. Alonso-Redondo, M. Schmitt, Z. Urbach, C. M. Hui, R. Sainidou, P. Rembert, K. Matyjaszewski,
M. R. Bockstaller, and G. Fytas, "A new class of tunable hypersonic phononic crystals based on polymertethered colloids," Nat. Commun. 6, 8309 (2015).

157[136] S. J. van der Molen, J. Liao, T. Kudernac, J. S. Agustsson, L. Bernard, M. Calame, B. J. van Wees,
B. L. Feringa, and C. Schönenberger, "Light-Controlled Conductance Switching of Ordered
Metal−Molecule−Metal Devices," Nano Lett. 9(1), 76–80 (2009).
[137] C. Stelling and M. Retsch, "Nanomeshes at Liquid Interfaces: From Free-Standing Hole Arrays
toward Metal–Insulator–Metal Architectures," Adv. Mater. Interfaces 5(10), 1800154 (2018).
[138] G. F. Mancini, R. M. Karl, E. R. Shanblatt, C. S. Bevis, D. F. Gardner, M. D. Tanksalvala, J. L.
Russell, D. E. Adams, H. C. Kapteyn, J. V. Badding, T. E. Mallouk, and M. M. Murnane, "Colloidal crystal
order and structure revealed by tabletop extreme ultraviolet scattering and coherent diffractive imaging,"
Opt. Express 26(9), 11393–11406 (2018).
[139] J. E. Han and V. H. Crespi, "Abrupt Topological Transitions in the Hysteresis Curves of
Ferromagnetic Metalattices," Phys. Rev. Lett. 89(19), 197203 (2002).
[140] Y. Liu, S. Kempinger, R. He, T. D. Day, P. Moradifar, S.-Y. Yu, J. L. Russell, V. M. Torres, P. Xu,
T. E. Mallouk, S. E. Mohney, N. Alem, N. Samarth, and J. V. Badding, "Confined Chemical Fluid
Deposition of Ferromagnetic Metalattices," Nano Lett. 18(1), 546–552 (2018).
[141] J. Ma, B. R. Parajuli, M. G. Ghossoub, A. Mihi, J. Sadhu, P. V. Braun, and S. Sinha, "Coherent
Phonon-Grain Boundary Scattering in Silicon Inverse Opals," Nano Lett. 13(2), 618–624 (2013).
[142] M. T. Barako, A. Sood, C. Zhang, J. Wang, T. Kodama, M. Asheghi, X. Zheng, P. V. Braun, and K.
E. Goodson, "Quasi-ballistic Electronic Thermal Conduction in Metal Inverse Opals," Nano Lett. 16(4),2754–2761 (2016).
[143] L. Yang, N. Yang, and B. Li, "Extreme Low Thermal Conductivity in Nanoscale 3D Si Phononic
Crystal with Spherical Pores," Nano Lett. 14(4), 1734–1738 (2014).
[144] O. W. Käding, H. Skurk, A. A. Maznev, and E. Matthias, "Transient thermal gratings at surfaces forthermal characterization of bulk materials and thin films," Appl. Phys. A 61(3), 253–261 (1995).
[145] H. Ju, S. Zhang, Z. Li, X. Shui, and P. Kuo, "Size-dependent thermal diffusivity of multilayered thinfilms deposited on substrates characterized by transient grating technique," Appl. Phys. Lett. 94(3), 031902(2009).
[146] R. M. Slayton, K. A. Nelson, and A. A. Maznev, "Transient grating measurements of film thickness
in multilayer metal films," J. Appl. Phys. 90(9), 4392–4402 (2001).
[147] W.-L. Ong, S. M. Rupich, D. V. Talapin, A. J. H. McGaughey, and J. A. Malen, "Surface chemistry
mediates thermal transport in three-dimensional nanocrystal arrays," Nat. Mater. 12(5), 410–415 (2013).
[148] Y. S. Touloukian, Thermophysical Properties of High Temperature Solid Materials: Vol. 1. Elements.
New York, NY: The Macmillan Company (1967).
[149] Oleylamine, ≥98% (primary amine), Sigma-Aldrich. Accessed April 4, 2019.
[150] X. Yin, J. Wu, P. Li, M. Shi, and H. Yang, "Self-Heating Approach to the Fast Production of Uniform
Metal Nanostructures," ChemNanoMat 2(1), 37–41 (2016).
[151] Y.S. Touloukian, Thermophysical Properties of Matter: Vol 12. Thermal Expansion – Metallic
Elements and Alloys. New York, NY: Plenum Publishing Corporation (1975).

158[152] R. A. Orwoll and P. J. Flory, "Equation- of-state parameters for normal alkanes. Correlation withchain length," J. Am. Chem. Soc. 89(26), 6814–6822 (1967).
[153] R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke,"Optical dielectric function of gold," Phys. Rev. B 86(23), 235147 (2012).
[154] Optical Glass Data Sheet, Schott, 2017. Accessed via refractiveindex.info April 4, 2019.
[155] C. Schinke, P. Christian Peest, J. Schmidt, R. Brendel, K. Bothe, M. R. Vogt, I. Kröger, S. Winter,
A. Schirmacher, S. Lim, H. T. Nguyen, and D. MacDonald, "Uncertainty analysis for the coefficient ofband-to-band absorption of crystalline silicon," AIP Adv. 5(6), 067168 (2015).
[156] M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A.
Agard, and J. W. Sedat, "Three-Dimensional Resolution Doubling in Wide-Field Fluorescence Microscopy
by Structured Illumination," Biophys. J. 94(12), 4957–4970 (2008).
[157] E. S. Massaro, A. H. Hill, and E. M. Grumstrup, "Super-Resolution Structured Pump–Probe
Microscopy," ACS Photonics 3(4), 501–506 (2016).
[158] A. R. Cameron, P. Riblet, and A. Miller, "Spin Gratings and the Measurement of Electron Drift
Mobility in Multiple Quantum Well Semiconductors," Phys. Rev. Lett. 76(25), 4793–4796 (1996).
[159] L. Foglia, F. Capotondi, R. Mincigrucci, D. Naumenko, E. Pedersoli, A. Simoncig, G. Kurdi, A.
Calvi, M. Manfredda, L. Raimondi, N. Mahne, M. Zangrando, C. Masciovecchio, and F. Bencivenga, "First
Evidence of Purely Extreme-Ultraviolet Four-Wave Mixing," Phys. Rev. Lett. 120(26), 263901 (2018).
[160] R. Mincigrucci, L. Foglia, D. Naumenko, E. Pedersoli, A. Simoncig, R. Cucini, A. Gessini, M.
Kiskinova, G. Kurdi, N. Mahne, M. Manfredda, I. P. Nikolov, E. Principi, L. Raimondi, M. Zangrando, C.
Masciovecchio, F. Capotondi, and F. Bencivenga, "Advances in instrumentation for FEL-based four-wavemixing experiments," Nucl. Instrum. Methods Phys. Res. A 907, 132–148 (2018).
[161] D. D. Hickstein, F. J. Dollar, P. Grychtol, J. L. Ellis, R. Knut, C. Hernández-García, D. Zusin, C.
Gentry, J. M. Shaw, T. Fan, K. M. Dorney, A. Becker, A. Jaroń-Becker, H. C. Kapteyn, M. M. Murnane,
and C. G. Durfee, "Non-collinear generation of angularly isolated circularly polarized high harmonics,"
Nat. Photonics 9(11), 743–750 (2015).
[162] R. Mincigrucci, F. Bencivenga, E. Principi, F. Capotondi, L. Foglia, D. Naumenko, A. Simoncig, S.
Dal Zilio, A. Gessini, G. Kurdi, N. Mahne, M. Manfredda, A. Matruglio, I. Nikolov, E. Pedersoli, L.
Raimondi, R. Sergo, M. Zangrando, and C. Masciovecchio, "Timing methodologies and studies at the
FERMI free-electron laser," J. Synchrotron Rad. 25, 44–51 (2018).
[163] S. Eich, A. Stange, A. V. Carr, J. Urbancic, T. Popmintchev, M. Wiesenmayer, K. Jansen, A. Ruffing,
S. Jakobs, T. Rohwer, S. Hellmann, C. Chen, P. Matyba, L. Kipp, K. Rossnagel, M. Bauer, M. M. Murnane,
H. C. Kapteyn, S. Mathias, and M. Aeschlimann, "Time- and angle-resolved photoemission spectroscopy
with optimized high-harmonic pulses using frequency-doubled Ti:Sapphire lasers," J. Electron Spectros.
Relat. Phenomena 195, 231–236 (2014).
[164] D. Popmintchev, C. Hernández-García, F. Dollar, C. Mancuso, J. A. Pérez-Hernández, M.-C. Chen,
A. Hankla, X. Gao, B. Shim, A. L. Gaeta, M. Tarazkar, D. A. Romanov, R. J. Levis, J. A. Gaffney, M.
Foord, S. B. Libby, A. Jaron-Becker, A. Becker, L. Plaja, M. M. Murnane, H. C. Kapteyn, and T.
Popmintchev, "Ultraviolet surprise: Efficient soft x-ray high-harmonic generation in multiply ionizedplasmas," Science 350(6265), 1225–1231 (2015).

159[165] H. Wang, Y. Xu, S. Ulonska, J. S. Robinson, P. Ranitovic, and R. A. Kaindl, "Bright high-repetitionrate source of narrowband extreme-ultraviolet harmonics beyond 22 eV," Nat. Commun. 6, 7459 (2015).

160

Appendix
COMSOL Finite Element Analysis

Throughout the work of this thesis, I have relied on the commercial software COMSOL to analyzeboth thermal and elastic measurements of nanoscale systems. COMSOL uses the finite element method tosolve differential equations that model physical systems, a process known as finite element analysis (FEA).
While the differential equations used assume classical, continuum mechanics, this approach has great power
and utility in that it can model our real experimental geometries, and it does not require prohibitive amountsof computing power. For the thermal transport problem, the use of an effective Fourier model means thatthe bulk diffusion equation may be directly implemented in COMSOL, as long as either conductivity or
resistivity is adjusted to account for nanoscale effects. For the elastic properties problem, we have observed
good agreement between COMSOL and diffraction measurements, indicating that continuum mechanics
remains a reasonable approximation for acoustic propagation, even at deep nanoscale sizes. In the following
appendix, I will discuss in more detail the COMSOL modelling performed for each of these measurements.
A.1 Finite Element Analysis
To solve partial differential equations, the finite element method first discretizes a system into a
series of points. These are the discrete locations where variables such as temperature and displacement arecalculated as a function of time. These points together make a mesh that may be explicitly defined or, morecommonly, “grown” from the defined boundaries according to established algorithms and user inputs.
Generally, a higher density mesh is needed in locations with more confined dimensions. For our nanoline
systems, the mesh must be high density at the nanoline-substrate interface, and it can grow to a much larger

161
spacing between mesh points as it extends into the depth of the substrate. Fig. A.1 gives examples ofstandard meshes used in our analysis.

Figure A.1: Finite element analysis meshes. These meshes are grown using one of COMSOL’s built-in
algorithms, which is based off of a triangular lattice. a) 3D simulation cell used for 2D nanocubes. b) 2D
simulation cell used for 1D nanolines. c) Calculated displacements on mesh in (b) after thermal expansion.
Displacements are represented both by moving the point positions in cartesian space and by color.
Displacements are exaggerated for visualization in (c). Panels (b) and (c) taken from [20].
One important practical aspect of FEA is that an improper mesh can give inaccurate results orinhibit the convergence of the calculation. As an example, if the mesh point spacing is similarly sized tothe acoustic wavelength being studied, the wave can unphysically diffract or disperse due to the mesh itself
and produce dynamics not present in the experiment. It is thus important to give great care to the meshgeneration when using FEA to interpret experimental results. We confirm there is no spurious meshdependence in our calculations by changing the number of mesh points, rerunning the simulation, andverifying there is no change in our results.
Similar to requiring proper sampling in space with an appropriate mesh, FEA requires a smallenough step size in time to properly sample the dynamics that are occurring. If too large of a time step isused, the simulation will not be properly constrained, and it will produce unrealistic values or in the worst
cases diverge. As mentioned in Chapter 3, the fastest dynamic we simulate in COMSOL is electron-phonon
coupling according to the two-temperature model. For these simulations, the step size must start at 2fs until
the ultrafast transients have decayed. For both time step size and number of mesh points used, there is a

162
trade-off between accuracy and computational time. The more steps and mesh points used, the longer ittakes for a simulation to finish. Generally, we try to use the fewest time steps and mesh points we can,without losing accuracy.
A.2 Thermal modelling and fitting
To analyze our thermal experiments, we first model a single unit cell of the periodic grating, aspresented in [20], and described below. As all of the COMSOL-based thermal analysis presented in thisthesis uses nanolines, I will limit discussion here to the 2D cross section case (approximates nanolines as
infinite), as shown in Fig. A.1(b). The sides of the simulation cell use periodic boundary conditions tocapture the dynamics of an infinite nanograting. This is a reasonable approximation as we probe the centerof our gratings. The nanoline-substrate interface has continuous displacements (no-slip condition), but
discontinuous temperature due to the thermal boundary resistivity. The top surface has unconstrained
displacement, but the heat flux is set to zero as losses due to radiation are minimal, and convection is not aconcern for our in-vacuum experiments. The substrate is modeled as several microns thick – enough so thatno heat will reach the bottom boundary within the simulated time window. The bottom surface has bothzero displacement and heat flux. We solve for the temperature, T, and displacement, u, at every point as afunction of time using the coupled thermoelastic equations,∇ ⋅ (c : ∇(u − αΔT)) = ρ
ρ𝑐𝑝

∂2 u
,
∂t 2

∂𝑇
+ ρ𝑐𝑝 𝑢 ⋅ ∇𝑇 = ∇ ⋅ (κ∇𝑇) + 𝑄,∂𝑡where c is the elastic tensor, α is the linear thermal expansion coefficient, ρ is the density, cp is the specificheat capacity, κ is the thermal conductivity, and Q is the heat source (absorbed laser pulse in our case). The
effective resistivity, reff, at the nanoline-substrate interface is introduced using the relationship𝑛 ⋅ (κ∇𝑇) = −

Δ𝑇
,
𝑟𝑒𝑓𝑓

163
where n is the unit vector normal to the surface, and ΔT is the temperature discontinuity across the interface
introduced by the boundary resistivity.
Using this approach for each COMSOL calculation, we perform a suit of simulations to fit eachgrating geometry (linewidth and period). We first calculate the surface displacement for a variety ofeffective resistivities, with all other parameters held constant. We then numerically diffract from eachsimulated surface displacement, using the Fresnel approximation. Building change in diffraction signals inthe same way as for experimental measurements, we obtain a single change in diffraction efficiency signal
versus time for each COMSOL simulation. This is shown for a single grating on silicon in Fig. A.2.

Figure A.2: Sweep of effective resistivity simulations for a nickel grating on silicon. As the effective
resistivity is increased, the thermal decay slows down, as expected. A separate set of simulations is
performed for each grating geometry on each sample being investigated. Note that these diffraction signals
are calculated under the quasistatic approximation, which removes acoustic oscillations.
When performing thermal simulations, we use the quasistatic approximation rather than calculating
the full elastic dynamics of the system. This retains the necessary information for thermal analysis but
greatly reduces computation times. The quasistatic approximation discards the second derivatives ofdisplacement in the thermoelastic equations, meaning that acoustic waves are eliminated from thecalculated displacements, but the thermal expansion is still properly influenced by the elastic tensor of the164material. As shown in Fig. A.3, the quasistatic diffraction signal goes through the average of the acoustic
dynamics, enabling least squares fitting of the quasistatic signals to experimental signals.

Figure A.3: Comparison of full inertial and quasistatic simulations. The blue line is measured data for
a 100nm linewidth, 400nm period nickel nanograting on a fused silica substrate. The green dotted line is afull inertial COMSOL calculation that includes all acoustic dynamics in this system. The red dashed line isa quasistatic calculation for this system, which passes through the moving average of the acoustic waves.
note that the full inertial model is calculated at a very fine time step, but it is down-sampled here to provide
a better comparison with the measured data.
After calculating a series of diffraction signals as a function of r eff, we must find the value of reffthat produces the best fit curve for our measured data. For improved resolution in our fit, we require someform of continuous interpolation between the series of simulated curves, such as those shown in Fig. A.2.
For the work presented in this thesis, we do this by first fitting a double exponential to each simulated curve.
We then look at how each of the four parameters (amplitude and time constant for both exponentials) change
from one resistivity to the next, and fit a polynomial to each parameter as a function of resistivity. This
produces a final analytical function that closely approximates our simulated curves, but allows forcontinuous tuning of resistivity between the simulated bounds. Mathematically then, the function we fit toour experimental data trace is𝑓(𝑡) = 𝑎(𝑟)𝑒 −𝑡/𝑏(𝑟) + 𝑐(𝑟)𝑒 −𝑡/𝑑(𝑟) ,165
where the parameters a(r), b(r), c(r), and d(r) are polynomial fits to how the double exponential parameterschange as a function of resistivity. Because the polynomial and double exponential fits are simply used toanalytically reproduce our COMSOL-calculated signals, we do not ascribe any physical meaning to theseparameters. The polynomial fits to the double exponential parameters are usually of degree three, as that is
sufficient to reproduce our COMSOL results, though some simulations require a fourth order polynomial
for the polynomial fit to reliably reproduce COMSOL results.
In addition to fitting the effective resistivity, we include three other fit parameters that aid in theregistration of the simulated curve to the experimental data. These parameters are horizontal offset, verticaloffset, and normalization, as shown in Fig. A.4. Of course, unconstrained offsets and normalizations cangive spurious and non-unique fits, so we limit these parameters according to the experimental data trace
itself. The horizontal offset is limited to sweep the points along the rise in signal from time 0 to the peak ofthermal expansion. This range is chosen because the appropriate start point lies somewhere within thisrange for the quasistatic curve that excludes inertial effects at early times. The vertical offset is bound by
the noise level before time 0, which can obscure where the real zero signal line lies. The normalization is
the most difficult parameter to bound because the inertial effects excluded by the quasistatic approximation
can cause significant changes in signal amplitude at early times. The most reliable method for setting therange of normalization is thus to compare the initial heights of full inertial and quasistatic simulations of
the same grating geometry. As the full inertial simulation captures all the acoustic dynamics present in theexperiment, this relative height bounds the relative height between the quasistatic simulation and theexperimental data.
Once all of these bounds are determined, we perform a least squares fit to the data for eachcombination of reff, horizontal offset, vertical offset, and normalization. The combination that has the lowest
residual error is then taken to be the best fit. In most cases, there is a clear minimum in the residual that is
not at the edges of the simulated parameter space, indicating a true fit. In some cases, we do have to limit
the offsets or normalization at the edge of their allowed values, according to the physical constraints listed
above. For reff, however, we always ensure the best fit lies somewhere in the middle of the calculated range

166
so that the best fit value is not artificially limited by the scope of our calculated range. This process isrepeated for each independent measurement on a given grating, with the final quoted value and uncertainty
of reff for that geometry being the average and standard deviation, respectively, of each best fit.
A.3 Acoustic wave modelling for elastic properties
To extract the elastic properties of ultrathin films using our experimentally measured surface
acoustic waves (SAW), we follow the procedure described in [22] and outlined in Chapter 4. This process
fully accounts for the mass loading of the nanostructures, which depresses the SAW velocities comparedto the values that can be derived for a pure film-on-substrate sample without any nanostructures. Similar to
the thermal simulations, our procedure minimizes the amount of full inertial calculations needed, as they
are the most computationally expensive. The quasistatic approximation is not an option in this case, since
it removes all acoustic waves. Instead, we leverage the power of eigenfrequency analysis in COMSOL,
which only requires the static geometry of the nanoline – thin film – substrate unit cell. In this analysis,
COMSOL solves for the eigenfrequencies, f, of the static structure using the equation
ρ(𝑓2π)2 𝑢 − ∇ ⋅ σ = 𝐹𝑉 ,where ρ is the density, u is the displacement, σ is the stress, and FV is the load on the material. These
calculations proceed very fast, and give all of the supported eigenfrequencies of the structure, within a
specified frequency range. The surface acoustic waves we measure must match one of these frequencies.
To determine which of these eigenmodes are excited in our experiment, we now turn to the fullinertial COMSOL simulations. Using the same thermoelastic equations described in the previous section,
we calculate the complete dynamics of the system from laser heating to maximum thermal expansion. This
only requires a simulation time window of ~100-200ps, which is well below the ~8ns time window required
to reproduce the full experimental measurement. Once we have calculated the point of maximum thermal
expansion, we have the excitation that launches all the acoustic waves we observe. Thus, we can project
the maximum displacement field onto the basis of eigenmodes we previously calculated to determine which
eigenfrequency corresponds to our measured SAW frequency. As shown in Fig. A.4, the projected

167
amplitude of the maximum displacement field, decomposed onto the eigenmode basis, will reach amaximum at the eigenfrequency corresponding to the excited SAW frequency. This provides a readyidentification of the calculated SAW frequency that can be directly compared to the measured SAW
frequency, without having to simulate a full inertial signal that is fit to the raw data.

Figure A.4: Determination of SAW frequency using eigenfrequency analysis. When the maximum
displacement field is projected onto the basis of eigenmodes for the nanostructure-film-substrate system,
each eigenfrequency has a different weight contributing to the maximum displacement field. As the laserheated nanoline grating predominantly excites the fundamental SAW, the large amplitude projection
corresponds to this wave. Note that flips in sign are just a 180 degree phase shift when adding that mode toform the total displacement field, and this does not always line up with the maximum projected amplitude.
We tune the elastic properties of the film-substrate system until the calculated SAW frequency
matches the measured SAW frequency. As described in Chapter 4, we first simulate a large grating, where
the SAW has a long penetration depth and travels mostly in the substrate. The wave speed is thus primarily
sensitive to the substrate elastic properties, and we tune the elastic constants of the substrate a small
percentage from their nominal values to fit the measured SAW frequency. This accounts for manufacturing
imperfections and film-induced strain in the substrate. Next, we simulate a small grating where the wave isfully confined to the film. By tuning the Young’s Modulus of the film in COMSOL, we match the calculated
and measured SAW velocities, and thus extract the elastic constants of the film accounting for mass loading.