Nanoscale thermal, acoustic, and magnetic
dynamics probed with soft x-ray lightbyMark E. Siemens
B.S., Engineering Physics, Colorado School of Mines, 2003

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
2010This thesis entitled:
Nanoscale thermal, acoustic, and magnetic dynamics probed with soft x-ray light
written by Mark E. Siemens
has been approved for the Department of Physics

Prof. Henry C. Kapteyn

Prof. Margaret M. Murnane

Date

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

iii
Siemens, Mark E. (Ph.D., Physics)
Nanoscale thermal, acoustic, and magnetic dynamics probed with soft x-ray light
Thesis directed by Prof. Margaret M. Murnane and Prof. Henry C. Kapteyn

This thesis discusses the application of coherent, ultrafast beams of soft x-ray lightfrom high-order harmonic generation (HHG) to study thermal, acoustic, and magnetic
processes in nanostructures. This short-wavelength light is a uniquely powerful probeof surface dynamics since it has both a very short wavelength and duration.
First, this thesis reports the first observation and quantitative measurements ofthe transition from diffusive to ballistic thermal transport for the case of heat flow awayfrom a heated nanostructure into a bulk substrate. This measurement provides insight
into the fundamentals of thermal energy transport away from nanoscale hot spots, anddemonstrates a fundamental limit to the energy dissipation capability of nanostructures.
Further, we propose a straightforward correction to the Fourier law for heat diffusion,necessary for thermal management in nanoelectronics, nano-enabled energy systems,
nanomanufacturing, and nanomedicine.
Second, this work discusses dynamic measurements of ultra-high frequency surface
acoustic waves (SAW) and the first SAW dispersion measurement in a nanostructured
system. These results are directly applicable to adhesion and thickness diagnostics ofvery thin films.
Finally, this thesis reports the first use of light from HHG to study magnetic
orientation. Using the transverse magneto-optic Kerr effect and soft x-ray light near
the M-absorption edges of Fe, Co, and Ni, magnetic asymmetries up to 8% are observed
from thin Permalloy (Ni80Fe20) films. This signal is 1-2 orders of magnitude higher
than that observed using optical methods, showing great promise for dynamic imaging
of domain flipping at the 100 nm level.

iv

Acknowledgements

My experience as a graduate student at CU/JILA was awesome- mostly becauseof the people around me. Margaret and Henry were fantastic advisors, giving me great
resources and freedom to accomplish this work. Ronggui Yang was as an extra advisor,
giving up many afternoons to patiently explain heat transfer to me. In the KM group, I
worked with many people who often gave up their time to help me fix my laser or figureout my experiment, including (in order of Margaret’s email list): David Gaudiosi, Tim
Lei, Ariel Paul, Erez Gershgoren, Nick Wagner, Etienne Gagnon, Amy Lytle, Xiaoshi
Zhang, Isabell Thomann, Daisy Raymondson, Tenio Popmintchev, Xibin Zhou, Andrea
Wuest, Arvinder Sandhu, Guido Saathoff, Robynne Hooper, Wen Li, Richard Sandberg,
Ming-Chang Chen, Alon Bahabad, Michael Gerrity, Craig Hogle, Matt Seaberg, Ethan
Townsend, and Adrienne Pon. I worked more closely with a few others: Jing Yin mademe laugh a lot, Oren Cohen inspired me to think big scientifically and always got thesame thing for lunch as me at the Thai place, and Luis Miaja-Avila was a great friend
from the time we took classes together until we both finally graduated. Finally, a special
thanks to those I worked with daily: Ra’anan Tobey for “teaching me all he knows”,
Qing Li for pushing me to understand more with her questions, and Chan La-O-Vorakiat
for his enthusiasm for life and fun-to-say last name. It was a pleasure working with allof you, and I will miss you greatly.
Of course, the person I have been most encouraged, challenged, and loved by is my
beautiful wife Katie. I can’t imagine the last four years without her (or Abbie!).

Contents

Chapter
1 Introduction1

1.1

Pump-probe measurement technique . . . . . . . . . . . . . . . . . . . .

2

1.2

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32 Ultrafast laser-initiated transport: photons, electrons, and phonons

6

2.1

Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.3

Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.4

Modeling thermal transport in real systems: circuit analogy . . . . . . .

16

2.4.1

Energy transport across an interface . . . . . . . . . . . . . . . .

16

2.4.2

Energy transport in the bulk . . . . . . . . . . . . . . . . . . . .

18

2.4.3

Putting the pieces together . . . . . . . . . . . . . . . . . . . . .

19

2.5

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 High-order harmonic generation

20
21

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.2

Advantages of EUV light. . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.2.1

Shorter wavelength . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.2.2

Elemental selectivity . . . . . . . . . . . . . . . . . . . . . . . . .

23

High-order harmonic generation . . . . . . . . . . . . . . . . . . . . . . .

24

3.3

vi

3.4

3.3.1

Step 1: Tunneling ionization. . . . . . . . . . . . . . . . . . . .

26

3.3.2

Step 2: Free electron propagation in a laser field . . . . . . . . .

28

3.3.3

Step 3: Recombination and radiation . . . . . . . . . . . . . . . .

30

3.3.4

HHG in hollow waveguides . . . . . . . . . . . . . . . . . . . . .

32

3.3.5

Characteristics of HHG . . . . . . . . . . . . . . . . . . . . . . .

33

Conclusion: Unique advantages of HHG . . . . . . . . . . . . . . . . . .

35

4 Time-resolved EUV holography

37

4.1

Holography with EUV light from HHG . . . . . . . . . . . . . . . . . . .

38

4.2

Fresnel propagation model . . . . . . . . . . . . . . . . . . . . . . . . . .

42

4.3

Longitudinal acoustic propagation measurement . . . . . . . . . . . . . .

46

4.4

Advantages of probing surface displacement with EUV light . . . . . . .

48

4.5

Conclusion

51. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Experimental setup for EUV probing of dynamics in nanostructures
5.1

54

EUV detection and noise management . . . . . . . . . . . . . . . . . . .

54

5.1.1

Laser system and beampath stability . . . . . . . . . . . . . . . .

55

5.1.2

Acquisition technique: chopping faster . . . . . . . . . . . . . . .

56

5.1.3

The danger of normalization . . . . . . . . . . . . . . . . . . . .

58

5.2

Sample design and fabrication . . . . . . . . . . . . . . . . . . . . . . . .

59

5.3

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

5.3.1

Pump beam setup . . . . . . . . . . . . . . . . . . . . . . . . . .

61

5.3.2

Glancing-incidence toroidal focusing mirror for EUV light . . . .

63

5.3.3

Aluminum filter holder and CCD adapter . . . . . . . . . . . . .

65

5.3.4

Mechanical setup . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Data acquisition and analysis . . . . . . . . . . . . . . . . . . . . . . . .

70

5.4.1

Data acquisition and time-dependent signal extraction . . . . . .

71

5.4.2

Signal linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.4

vii
5.5

Time-resolved EUV acquisition outlook . . . . . . . . . . . . . . . . . .

6 High-Frequency Surface Acoustic Wave Propagation in Nanostructures

74
77

6.1

Surface acoustic wave propagation for thin film measurements . . . . . .

78

6.2

EUV measurements of SAW dynamics . . . . . . . . . . . . . . . . . . .

81

6.2.1

Are we really seeing surface acoustic wave propagation? . . . . .

81

6.2.2

Surface acoustic wave dispersion measurements . . . . . . . . . .

84

SAW dispersion calculation: Modified thin film theory . . . . . . . . . .

86

6.3.1

SAW dispersion due to a thin film: Perturbative Auld model . .

87

6.3.2

SAW dispersion due to a thin film: Nelson model . . . . . . . . .

90

6.3.3

Comparing models to the data . . . . . . . . . . . . . . . . . . .

93

6.3

6.4

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Quasi-ballistic heat transport from nanoscale interfaces
7.1

7.2

Ballistic thermal transfer in and through a nanostructure . . . . . . . .

94
96
97

7.1.1

The Boltzmann transport equation for transport in a nanostructure 98

7.1.2

Basic theory of quasi-ballistic thermal transfer in a nanostructure

7.1.3

Experimental observations of ballistic transport in nanostructures 101

99

Quasi-ballistic thermal transport at an interface . . . . . . . . . . . . . . 103
7.2.1

Thermal boundary resistance across a bulk-constricted interface

104

7.2.2

Simple analytical prediction: ballistic-diffusive model . . . . . . . 106

7.2.3

Propagating vs reservoir phonons . . . . . . . . . . . . . . . . . . 110

7.2.4

Previous experiment and open questions . . . . . . . . . . . . . . 114

7.3

Measurement of ballistic heat transport at a nanoscale interface . . . . . 114

7.4

Multiphysics model for data analysis . . . . . . . . . . . . . . . . . . . . 116
7.4.1

Thermal transport across the interface and diffusion . . . . . . . 116

7.4.2

Thermal expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.4.3

Fresnel optical propagation . . . . . . . . . . . . . . . . . . . . . 128

viii
7.4.4

Discussion of model results . . . . . . . . . . . . . . . . . . . . . 128

7.4.5

Fitting the model to our experimental data . . . . . . . . . . . . 130

7.5

Implications of the measurement . . . . . . . . . . . . . . . . . . . . . . 133

7.6

Discussion of the time-resolved measurement . . . . . . . . . . . . . . . 136

7.7

EUV probing of nanoscale thermal transport outlook . . . . . . . . . . . 137

7.8

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8 Magneto-optics with EUV light

140

8.1

Historical development of the magneto-optic Kerr effect (MOKE) . . . . 141

8.2

Derivation of the transverse magneto-optic Kerr effect . . . . . . . . . . 142

8.3

8.2.1

Macroscopic (phenomenological) description . . . . . . . . . . . . 142

8.2.2

Microscopic (fundamental) description . . . . . . . . . . . . . . . 145

Experimental Magneto-Optic Kerr effect using EUV from HHG . . . . . 148
8.3.1

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8.3.2

Magnetic contrast measured with EUV light from HHG . . . . . 152

8.3.3

Hysteresis measurement . . . . . . . . . . . . . . . . . . . . . . . 153

8.3.4

Surface sensitivity measurement . . . . . . . . . . . . . . . . . . 154

8.4

Outlook: EUV measurement of demagnetization dynamics . . . . . . . . 155

8.5

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

9 Conclusion

158

Bibliography

161

Appendix
A Fresnel propagation code

178

B Multiphysics thermal and optical model

180

Chapter 1

Introduction

The remarkable advances in nanofabrication technology in the past few decades
have opened up unprecedented opportunities in the engineering of optical, electronic,
magnetic, and thermal properties of materials. These advances have been in large partby the relentless microelectronics industry (which is, of course, driven by a desire to takeyour money), so it is natural that electronic properties at the nanoscale have received atremendous amount of attention. Similarly, there are many scientifically interesting anduseful applications of optical phenomena at the nanoscale, including field enhancement.
Advances in magnetism have been driven in part by the information storage industry,
and individual bits stored on a hard disk, spintronic devices and magnetic random accessmemory are already exploiting sub-micron magnteic structures. Finally, thermal management in nanostructures is becoming increasingly “hot”, with applications requiring
both optimized transport (e.g. heat-sinking a transistor) and minimized transport (e.g.
thermoelectric energy conversion).
Nanostructures not only provide extremely fine engineering control useful for
applications, but also offer opportunities to observe the fundamental length scales ofphysical phenomena. The confined walls of a nanostructure can be smaller than optical
wavelengths, magnetic domains in the bulk, or phonon scattering distances (or evenphonon wavelengths). By looking at the system properties as a function of nanostructure
size and temperature, insight into physical fundamentals and the interaction of energy

2carriers is possible.
This thesis focuses on the study of physical phenomena in nanostructures on withlength scales near the fundamental limits of acoustic, thermal, and magnetic effects.

1.1

Pump-probe measurement technique
Most of the experiments discussed in this thesis make use of the pump-probe

technique, in which a laser pulse is split into a stronger pulse that excites a system anda weaker, time-delayed pulse probing the change caused by the pump. Using ultrafast
lasers at optical wavelengths (λ ∼ 400 nm - 800 nm), it has been possible to measure
optical excitation of electrons in metals [1] and semiconductors, phonon transport in
the bulk and in thin films, ultrafast phase transformations and fundamental electron
and phonon transport in exotic systems such as carbon nanotubes and gold nanorods.
However, optical light is not always useful for exciting or observing the phenomena we want to see. Observing acoustic, thermal, and magnetic effects in materials isparticularly difficult because the reflection of optical light couples very weakly to theseprocesses. Another problem is that imaging resolution is limited by the wavelength ofthe probe light by the diffraction limit, so optical light is limited to images with ∼ 500
nm resolution. Very sensitive detection of surface displacement can be obtained using an
optical interferometer, but there is potential for even better interferometric sensitivity
if shorter-wavelength light is used. Finally, direct excitation of inner-shell electrons also
requires higher energy (shorter wavelength) light. Direct excitation is useful for studies
such as element-specific imaging [2], Auger decay [3], or x-ray absorption fine structure
measurements [4].
There is clearly a strong case to use shorter wavelength (ultraviolet to x-ray) lightin pump-probe experiments. High-order harmonic generation (HHG) is an ideal source
for such experiments because it produces light that is

3• coherent, allowing for interferometric measurement and detection,• tunable and short-wavelength, with light generated in the Extreme Ultraviolet(or soft x-ray) region of the spectrum from 10 - 1 keV (120 - 1 nm),
• ultrafast, with pulses less than 10 fs in duration,• bright – A pulse energy of greater than 10 nJ is possible when phase-matched.
We will describe experiments implementing HHG light as a highly-sensitive probe ofnanoscale surface dynamics in thermal, acoustic, and magnetic systems.

1.2

OutlineWe will begin with an introduction to the physics of ultrafast laser-solid interac-tions in Chapter 2. The chapter describes the physical processes that occur when an
ultrafast laser pulse hits a material: photon absorption by an electron, electron-electron
and then electron-phonon coupling. A discussion of phonon diffusion follows, including
a derivation of the Fourier law for thermal transport. The chapter concludes with asimple method for accounting for competing transport mechanisms in a system, bothin the bulk and at an interface.
In Chapter 3, the high-order harmonic generation process is introduced. The
semiclassical three step model is reviewed, along with the benefits of performing highharmonic generation in a hollow waveguide. The unique advantages of light from HHG
over alternative sources of EUV and soft x-ray light are considered.
Chapter 4 presents a proof-of-principle experiment showing thermal and acoustic
effects measured in a thin film system. This experiment demonstrates that EUV lightfrom HHG is a sensitive tool for measuring very small thermal and acoustic dynamics
that can be determined from surface deformations. Furthermore, because the index ofrefraction of most materials at EUV wavelengths is very insensitive to density changes,

4
we make a pure surface phase measurement, unpolluted by a change in surface reflectivity. This is in contrast to experiments at optical wavelengths, where thermal densitychanges lead to a large change in reflectivity and complicates a determination of the
dynamic surface profile.
The experimental setup for measuring very small thermal, acoustic, and magnetic
signals is explained in detail in Chapter 5. Techniques for enhancing the stabilityof the light from HHG and filtering other sources of noise are discussed. The basic
optical pump, EUV probe experimental geometry is introduced, along with design andfabrication details for a typical nanostructured sample. Representative data is shown
along with an explanation of the process of extracting useful information from raw data.
In Chapter 6, experiments studying surface acoustic wave (SAW) propagation innanostructures are discussed. As described in Chapter 5, an ultrafast optical laser pulse
hits a nanostructured sample, impulsively stressing the surface with a periodicity set by
the nanostructure. This periodic and impulsive stress launches SAWs that propagate
along the surface with a penetration depth depending on their wavelength: deep penetration from long wavelengths, and shallow penetration for short wavelengths. SAW
wavelengths as short as 125 nm are considered, with a frequency of nearly 50 GHz anda penetration depth of ∼ 20 nm, almost an order of magnitude better than what is possible using optical excitation methods to generate the SAW. This technique therefore
has applications in studying the properties of and interfaces beneath very thin films.
At the same time as the SAW propagation is occurring, the heated nanostructures are also cooling, as energy is transported into the substrate. Chapter 7 discusses
this thermal transport, emphasizing the effect of reducing the size of a nanoscale heatedregion below the phonon mean free path of the heat sink. In this domain, heat transport near the interface can no longer be considered diffusive, as there are not enoughscattering events to establish local thermal equilibrium. However, the Fourier law forthermal transport assumes that diffusion is valid over any length scale. This assumes

5
additional scattering events, which causes a significant over-prediction of the heat flux
away from nanoscale heat sources and may have catastrophic effects in thermal engineering of nanoscale devices. This chapter discusses the first experimental observationand quantitative measurement of this effect, as well as a proposed method for correcting
the Fourier law at small length scales: addition of a size-dependent interfacial resistivity
that accounts for the over-prediction caused by assumed diffusion. Excellent agreement
between this model and the experimental data are observed.
Chapter 8 introduces the first measurements of static magnetic flipping asymmetry with soft x-ray light from HHG. These experiments make use of the transverse
magneto-optic Kerr effect (T-MOKE). The signal observed is commensurate with thatobtained from synchrotron radiation. Near the absorption edges of the magnetic elements under study (nickel and iron in a thin film Permalloy Ni80 Fe20 sample), the signal
is more than two orders of magnitude higher than that measured optically. This strong
contrast is very promising for future experiments exploring ultrafast demagnetizationand dynamic microscopy of domain flipping.
The thesis is concluded in Chapter 9 with a summary of the thermal, acoustic,
and magnetic phenomena observed with soft x-ray light. It closes with a few remarks
about the future outlook of HHG light as a source for studying surface processes. The
appendices provide two MATLAB programs. The program in Appendix A performs
Fresnel propagation, and is useful for determining how light diffracts from a surface withnanoscale features (either static nanstructure of laser thermal deformation). Appendix
B provides a multiphysics model for laser heating, heat diffusion, thermal expansion,and optical diffraction from a thermally-deformed surface and subsequent propagation.
The model was used in the data analysis for Chapter 7.

Chapter 2

Ultrafast laser-initiated transport: photons, electrons, and phonons

The interaction of photons, electrons, and phonons has been studied extensively
and is covered in a number of textbooks [5, 6]. In this introduction, I will focus on
the issue of energy transport driven by short-pulse laser excitation, and heat transfer
mediated by interfaces at short length scales. Characteristic time scales for the processes
discussed in this chapter are given in Table 2.1.

2.1

Photons
When considering the energy-carrying ability of electromagnetic radiation (pho-tons), it is usually easiest to think in terms of the particle nature, where each photon
has an energy given by
E = hν,(2.1)

where h is Planck’s constant (h = 6.626 × 10−34 m2 kg/s) and ν = c/λ is the frequencyof the photon inversely proportional to the wavelength. 800 nm (red- on the edge ofinfrared) light has a frequency of 3.74 × 1014 Hz, so each photon has an energy of
E = 2.48 × 10−19 J. The unwieldiness of such numbers can be alleviated by a changeof units to electron volts (1 eV = 1.6 × 10−19 J, the energy gained by a free electron
after accelerating through a 1 V electrostatic potential), where a 600 nm photon has anenergy of 1.55 eV.

7Fundamental physical processes
Optical electronic excitation
Electron thermalization
Electron cooling
Phonon relaxation
Thermal diffusion
Thermal melting
Thermal ablation

Interaction
photon-ee-ee-phononphonon-phonon
phonon-bulk
bulk-bulk
bulk-freeTime scale (s)
10−14
10−13
−13
10
− 10−1210−12
10−12 − 10−11
10−11
10−10

Table 2.1: Time scale of ultrafast interactions (Adapted from D. von der Linde, et.
al. [7]). The line separates thermal (below) from nonthermal (above) processes. Melting
and ablation will occur only for sufficiently high incident intensities.

A typical natural light source emits its radiation over 4π steradians and is incoherent, that is, light is emitted in all directions and there is no relation between
one photon and another in direction or phase. The transport of energy by photons hastaken a tremendous step forward with the invention and development of the laser, whichallowed for the creation of coherent beams of photons that have(1) directionality, useful for the efficient transport of energy or information, and(2) both spatial coherence (photons move “in step” across the laser beam) andtemporal coherence (photons move “in step” over time past a single point),
allowing for high-precision interference measurements and holography.
**** The primary fundamental laser-matter interaction is the absorption of photons
by electrons, which drives electrons from their equilibrium states to excited states. In
the case of CW (continuous wave, i.e. unpulsed) laser illumination, the time between
excitation events is longer than the relaxation of the energized electrons by electronelectron and electron-phonon collisions [8]. This means that there is equilibrium between
the electrons and phonons in the system during the laser-heating process.
In contrast to the CW case, ultrafast lasers have pulses ∼ 20 fs (1 femtosecond =1 × 10−15 s), a time almost inconceivably short: a pulse traveling at the speed of lightin air is only 3 µm long, about 1/30th the diameter of a human hair. Modern ultrafast

8
lasers are based on Kerr lens modelocking, in which thermal lensing in the laser crystal
creates a condition of stable operation for pulsed, rather than CW, operation [9, 10].
Titanium-doped sapphire is an excellent lasing medium for these ultrafast lasers because
it has very good thermal conductivity (making it easier to remove excess heat) andvery broadband absorption and emission. That this is important can be seen from theHeisenberg uncertainty relations for energy time:∆E∆t ≥ h̄/2.

(2.2)

From this expression, it is clear that to get a pulse that is very short in time, a broad
range of energies is required. Ti:Sapphire has an emission bandwidth of ∼ 250 nm,
centered at approximately 800 nm [11], so this is where most ultrafast lasers operate.
Of course, just having a broadband source does not guarantee ultrafast pulses, so dispersion compensation (correcting small phase drift between wavelengths as the bluer
wavelengths travel slightly slower than the red in air) is also critical, and can be accomplished in an oscillator by the use of a prism pair. Laser oscillators give fairly weakpulses, < 20 nJ/pulse [10], but these can be amplified using a stretcher-compressor
chirped pulse amplifier also based on lasing in a Ti:Sapphire crystal [12].
Relaxation

532 nmPump

800 nmLasingRelaxation

Figure 2.1: Laser energy level diagram for Ti:Sapphire. Note the wide bands for absorption and emission, providing a broad frequency base for ultrafast pulse generation.

The applications of ultrafast laser pulses are numerous: they can measure dy-

9namic processes with unprecedented temporal resolution, and initiate nonlinear andnonequilibrium energy transfer. Femtosecond-level resolution is important because thefundamental time scales of many carrier interactions and transport processes are on
this level. For example, molecules vibrate on a time scale of 10-100 fs and electrons are
ionized (ripped off of atoms) and excited between levels in less than 1 fs. Another key
feature of ultrafast lasers is their high peak power. For example, given the same totalenergy, a 100 fs pulse has a peak energy density 13 orders of magnitude higher than a
CW laser. Because the peak intensity of ultrafast lasers is so high, a single pulse caninitiate highly-nonlinear physical processes, such as the high-order harmonic generation
process discussed in Chapter 3.

2.2

Electrons
The concentrated energy delivered by an ultrashort laser pulse can generate highly

excited, nonequilibrium electron distributions. The electrons are excited by direct photon absorption (see Figure 2.2) on a timescale limited only by the duration of the laser
pulse (therefore the electron response time is less than 10 fs and is negligible). The spatial depth profile is determined by the optical penetration depth in the metal (ζo ∼ 10
nm at an optical wavelength of 800 nm- see Table 2.2), as shown in the top row in
Figure 2.3.
Electrondensityof states

Optical excitation
OccupiedFermi Energy

Available
EnergyFigure 2.2: Electron density of states near the Fermi energy for a typical metal. If an
optical pulse is incident on the material, a photon of energy E = hν can be absorbed
by an electron, creating the nonequilibrium electron distribution shown in the first stepof Figure 2.3. In the case of very short pulses with high peak intensities, mulitphoton
ionization can excite electrons from deeper in the valence band or from lower-energy
bands, but it is still less likely than direct photon-electron absorption.

Nonequilibrium electrons

t=0

ζo
Depth

DOS

Te=?

te-e~10-13 s
tth~10-14 s

Hot electron diffusionzEFte-ph~10-12 s

Ballistic electrons

Electrons in equilibrium
Te>Tlattice

DOS

10

z

EF
DOS

Excited electron density

Electrons and lattice inthermal equilibrium
Thermal diffusion

Te=TlatticeEFz

Figure 2.3: Non-equilibrium relaxation dynamics of electrons in a bulk metal excitedby an ultrafast laser pulse. Time since the arrival of the pulse increases from top tobottom (labeled on the far left). The left column shows the electron density of states
as a function of energy (EF is the Fermi energy), while the right column shows thedepth distribution of the electrons excited above the Fermi energy by the pulse. At
times earlier than te−e , the excited electrons are out of equilibrium with the rest of
the electrons, and only at times later than te−ph are phonons in equilibrium with theelectrons. Figure modified from [13].

Immediately following excitation, the excited electrons travel ballistically deeper
into the material at velocities close to the Fermi velocity (determined by the electronenergy at the Fermi level, vF ∼ 106 m/s). The average distance of travel (penetration
depth ζe ) of these ballistic electrons is also called the mean free path Λe , and is dependent
on the strength of electron-electron scattering in the metal (proportional to the electricalconductivity). Values range from over 100 nm for noble metals to 10-20 nm for d-band
metals; ballistic electron penetration depths are shown for a few common metals in Table
2.2. As the excited ballistic electrons travel out among unheated electrons and scatterwith them, eventually (50-100 fs after initial laser excitation) an electron equilibrium
is reached (sometime around 50-100 fs after initial laser excitation), as shown in thesecond row in Figure 2.3. At this point, electron transport is no longer ballistic, and11diffusive electron transport carries energy still deeper, though much more slowly thanin the ballistic case. However, these thermalized electrons are still out of equilibriumwith the lattice phonons.
Metal
Gold
SilverAluminum
NickelChromiumζo (nm)
13
12
7.5
14
18

ζe (nm)
105
142
46
11
14

Table 2.2: Optical penetration (ζo = λ/4πni , where ni is the imaginary component of
the index of refraction) at 800 nm [14] and hot electron transport (ζe ) depths in selected
metals. ζe,N i is an estimate, but others are from experiment [13].

Electrons will finally interact with the lattice on an even longer time scale givenby the electron-phonon coupling time (te−ph ∼ 10−12 s). This can be described by a twotemperature model (the caption of Figure 2.4 refers to a “Three temperature model”,
with the only difference being that it includes an additional temperature for the spins inthe system) proposed by Anisimov [15,16], in which the electrons and lattice (phonons)
have separate temperatures (Te and Tl ) and heat capacities (Ce and Cl ). The system
can then be modeled as two coupled differential equations
∂Te
∂t
∂Tl
Cl
∂t

Ce

= k∇2 Te − α(Te − Tl ) + A(r, t),(2.3)= G(Te − Tl ),(2.4)

where k is the thermal conductivity, A(r, t) is the pulsed heating of the driving laser,
and G is the electron-phonon coupling constant [17]. As the electron and phonon temperatures equilibrate, the electron temperature typically drops by one to two orders ofmagnitude because the heat capacity of the electrons is so much smaller than that ofthe phonons (e.g. in Ni, Ce = 3.74 × 105 and Cl = 39.52 × 105 J/m3 K), as shown in
Figure 2.4.
Further transport of heat is by thermal diffusion, with the phonons and electrons

12
in equilibrium, as shown in the bottom row of Figure 2.3. It should be noted that
this picture of electron transport through diffusion and before eventual coupling to thelattice is valid only for metals. For semiconductors and insulators, the electron mobility

ARTICLES

NATURE MATERIALS DOI: 10.1038/NMAT2341

is much lower and the transport will be dominated by phonons.
abElectrons
LatticeSpin

3.0τel¬lat∆T/∆T∞

e¬Electrons (Te)τ el¬spτ lat¬sp

2.0

1.0

Lattice (Tl)0

Spins (Ts)

c

0

d

2.5
τ (ps)

5.0

7.5

2.0

∆T/∆T∞

Figure 2.4: Three temperature model, in which the electrons, lattice, and spins in asystem are allowed
to havedifferent temperatures.1.5 An ultrafast laser is absorbed bye¬τel¬latelectrons, which couple to the lattice with a characteristic time τe−ph ∼ 1 ps. The
Electrons (Te)spintemperature is important in magnetic systems,1.0and the electron-spin coupling timeτe−sp has not been measured with sufficient resolution, as discussed in Chapter 8. Figure
Lattice (Tl)
τ el¬sp0.5
τ lat¬spfrom [18].
Spins (Ts)

0
¬5

2.3

Phonons

0

5

10

15

20

τ (ps)

Figure 2 | Three-temperature model as determined by rate equations. a–d, Schematic representation and numerical calculations for the case of aferromagnetic metal (a,b) and for the case of the half-metal (c,d). In the case of the half-metal, the direct pathway for electron–spin interaction is blocked(c), resulting in a slow increaseof thetemperature(d). is carried and stored by vibrations of atoms about theirHeat
in spina crystallatticemanganite La0.66 Sr
spin sub-bandhas aaremuch
larger where$%↑(↓) is defined
the theirsum overthe corresponding quantity
0.33 MnO3 . One
equilibrium
positions.
Atoms
connected
by chemicalbondsas toneighboring
effective mass that barely contributes to the transport and Pi for majority (minority) spin n↑(↓) (EF ) weighted by their velocities
reaches a value ofatoms,
more than
IIIA ). Aas
generalextensivev↑(↓)(Esprings,
to shown
the powerof i.left
Aspanelthe transmission
T 2 (TF ) raised asand90%can (type
be treatedmassesconnected
byin theofoverview of the different families of half-metals in addition to the tunnelling matrix element) for an amorphous barrier such as Al2 O3materials mentioned
above—zinc-blenderuthenates
decreases
with theFermi velocity,
predominantly
Figure2.5. The right compounds,
panel of Figure
2.5 showsa typical
interatomic
potential,
which s, p-like statesand double-perovskites, chalcospinels, pyrites and organic half- contribute to the tunnelling current for the Meservey–Tedrow
metals—is given by Katsnelson et al.11 .
technique15 (Fig. 1b). Thus, values obtained by the Meservey–
demonstrates the reason that this mass-spring model works so well: near equilibrium,
To characterize the value of the spin polarization, different Tedrow technique PT and by Andreev reflection in the diffusivetechniques are available12 . They can be classified into two main limit Pnv 2 are almost equal12 . A less common but direct method
the potential
is verythenearlyapproximatedharmonic.
energiesof a har- resolved is spinsubgroups: (1) thosethat determine
densityof states n↓ as toprobe the The
bulk allowed
density-of-states
momentum16
and n↑ at the Fermi level using a photoemission experiment or resolved positron annihilation . However, using the ‘classical’oscillator
system
are given(2) experiments monic
determining
the spincurrentsi↓ andbyi↑ in a methods, the interpretation is not always clear, and sample
spin-transport device (Fig. 1a–c). In a photoemission experiment preparation can be tedious, expensive or demanding. The finding(Fig. 1a), the density of states is mapped. The photoelectrons are by Zhang et al.17 of a very slow demagnetization of the spin system
En = hν(nfor+ 1/2),(2.5)
attributed to their energy and spin. In Fig. 1b, the Meservey–
the half-metal CrO2 in all-optical pump–probe
experiments
Tedrow technique is shown. By applying a magnetic field to gave rise to the idea to implement a novel contact-free technique
the superconducting electrode, the quasiparticle density shows to characterize half-metallic materials for spin-electronic devices:
Zeeman split resonances and the current through the barrier can be if the demagnetization time τm of the spin system probed byseparated for both spin directions. In Fig. 1c, the superconductor the time-resolved Kerr rotation after femtosecond excitation isis a probe for the diffusive transport (Andreev reflection). The characteristically different (Fig. 1d), then it should be possible toelectron probes the superconductor for a certain time before a use it as a measure of the degree of half-metallicity of a material.
Cooper pair is formed and a hole of opposite spin polarization is This opens up the way towards an alternative technique thatreflected. On the basis of these experimental techniques, Mazin and overcomes the drawbacks of the ‘classical’ methods; it is fast, does
co-workers defined a more general spin polarization13,14 including not involve the preparation of a transport device and is suited for13
where n = 0, 1, 2, 3, ... is an energy level counter and ν is a fundamental frequency ofvibration, describing a single phonon. In gases and other weakly-coupled systems, theRepulsion

Potential

harmonic energy levels are discrete, but in a solid, the levels merge together.

Harmonic
approximation
Attraction
Interatomic
distancer0: Equilibrium
Figure 2.5: Basic model for energy transport and storage in a lattice. Left: the lattice
can be treated as a collection of masses (atoms) connected by springs (interatomic
potential). Right: a schematic interatomic potential.

A crystal with more than one atom per unit cell (e.g. Al2 O3 is hexaganol and Ni
is cubic face-centered) will have two major phonon branches : the acoustic branch that
corresponds to in-phase oscillations of all the atoms in a unit cell, and an optical branchfor out-of-phase oscillations of atoms making up a unit cell, as shown in Figure 2.6. Eachof these branches can be further split into three more, corresponding to longitudinal,
and both transverse directions of phonon propagation.
The coherence length of room-temperature phonons is ∼ 1 nm [19], so in mostcases, interference effects can be neglected and phonons can be treated and thought ofas particles with velocity, density, and collision properties. It is these collisions thattransport energy throughout the crystal, and given this phonon particle picture, itis easy to derive the Fourier law for thermal transport by phonons (or other kinetic
particles) in a 1D case.
Consider a collection of phonons (particles) in a box where one side is heated
and the other remains cold. We are interested in the flux along the x-direction qx , at a

ω(k)

14

LO
TO

LA
TA
-π/a

0

k

π/a

Figure 2.6: Phonon dispersion for a typical three dimensional crystal with lattice constant a. There are longitudinal (L) and transverse (T) modes for both acoustic (A) andoptical (O) phonons. There are two transverse modes for each pictured because of thetwo transverse directions.

qx

CooledHeatedx0vxτ vxτFigure 2.7: Simple calculation of the heat flux for a system of heated kinetic particles(e.g. phonons). Modified from [19].

position x, which is determined by the difference between the energy flux of the phonons
flowing in the positive and negative directions. In each case, phonons within a distancevx τ can reach the flux surface before scattering with another carrier. vx is the averagecomponent of the phonon velocity that is in the x-direction, and τ is the average time

15
a phonon travels before being scattered, called the relaxation time. Given a phonon
density per unit volume n and a particle energy E, the net flux at the surface at x is
q(x) =

1
2(nE(x)vx )x−vx τ

−1
2(nE(x)vx )x+vx τ

.

(2.6)

The factors of 1/2 take into account the fact that only half of the phonons will be
traveling in the required direction, since their motion is random. If we now apply a
Taylor expansion to the two components we findqx =1d(nE(x)vx )  1 d(nE(x)vx ) 
v 2 τ dU dT,nE(x)vx − vx τ−nE(x)vx + vx τ
=−
2
dx
2
dx
3 dT dx(2.7)

where U (x) = nE(x) is the position-dependent energy density and v is the averagevelocity given by vx2 = v 2 /3, assumed to be position-independent. Now we recognizethat dU/dT is the volumetric heat capacity C, and we have obtained the Fourier law
for heat conduction:
qx = −Cv 2 τ dT= −k∇T.
3 dx

(2.8)

Here we have assumed that the thermal conductivity k can be given by
k = Cv 2 τ /3 = CvΛ/3,(2.9)

where Λ = vτ is the average distance a phonon travels before scattering, the phononmean free path.
This understanding of thermal transport relies on scattering occurring at a distance Λ determined by the material. If the scattering length is limited by the presenceof interfaces or if the flux needs to be determined at shorter length scales, the Fourier
law for heat diffusion will obviously fail. Chapter 7 discusses the first observation of the
breakdown of Fourier description near nanostructured interfaces, as well as a discussionof how the Fourier law can be corrected to account for this error.
In addition to the phonon diffusion discussed here, it is also possible to launchand propagate coherent phonon wavepackets. In general, diffusive phonons propagation

16
is excited by thermal flows and coherent phonon wavepackets are excited by impulsive
heating (e.g. from an ultrafast laser) causing stresses and strains that relax with thephonons.

2.4

Modeling thermal transport in real systems: circuit analogy
An understanding of the fundamental energy carriers and their interactions does

us little good if we cannot apply that knowledge to real systems. In this section, a
simple way to understand energy flow in terms of an analogy to circuit diagrams [20] is
discussed, with obvious applications to practical engineering and design problems.
We will treat each transport interaction as having a resistivity to energy flow. In
the literature, the terms resistance and resistivity are often used interchangeably, but
we will take the term resistance (R) to be an extensive parameter dependent on the sizeof the system with units K/W, while resistivity (r) is an intensive, area-independent
parameter with units m2 K/W. They are related by R = r × A, where A is the crosssectional area of thermal flow.

2.4.1

Energy transport across an interface

The concept that an interface could have a significant effect on the transport ofenergy was considered in the late 1930’s, but not experimentally demonstrated until
Kaptiza’s measurement of a temperature drop at a helium-solid interface when heat
flowed across the boundary in 1941 [21]. The resistance to heat flow at the boundary(sometimes called Kaptiza resistance or thermal boundary resistivity, rT BR ) is definedas the ratio of the temperature drop at the interface to the energy flowing across theinterface per unit area:rT BR =

∆TBoundary.
q(2.10)

In practice, the boundary resistivity is so dependent on device preparation and fabrication, the presence defects and voids, and slight material disordering, that an accurate

17
theoretical prediction is impossible. However, it is instructive to consider the potential transport mechanisms to develop an understanding of the fundamental interactions
governing interfacial energy transport.
At an interface between different materials, both phonon-phonon and electronphonon interactions may play a role in the energy flow across the boundary. For example,
an electron reflecting off of the boundary can cause the boundary to recoil and emit
phonons. This possibility was considered in numerous calculations, and was determined
to be negligible in the case of solid/solid interfaces. We therefore neglect the contribution
from electron-phonon interactions at an interface, and consider only phonon-phonontransport.
Phonon-phonon coupling across an interface can be understood by two basic models. In the acoustic mismatch model (AMM), phonon transport at an interface is handledin much the same way as a Snell’s law calculation for optical light: a phonon incident on
the surface at an angle θ can be transmitted across the boundary at a refraction angle
related to the ratio of the acoustic impedances of the two sides, or reflected back at
an angle θ. This model assumes that no scattering takes place at the boundary, so thephonon spectrum emitted into the second material matches that of the phonons in thefirst material. Additionally, whether a given phonon is transmitted or reflected is determined by the ratio of the acoustic impedances of the materials making up the interface:a large mismatch will give a high reflectivity and vice versa. In contrast to the AMM,
the diffuse mismatch model assumes that all of the phonons incident on the surface arescattered completely. Since each incident phonon is scattered, the phonon emission isdetermined only by the available phonon states on each side of the interface (at a typicalsolid-solid interface, the emission is ∼ 50% into each side [22]). Both of these models
for phonon transport at an interface involve major simplifications, so their relevanceto real-life situations is limited, although both are valid at temperatures below ∼ 50
K [22]. Other effects, such as inelastic phonon scattering [23] and disorder-induced

18
multiple-phonon scattering [Hopkins 18,19], may play a significant role, especially at
non-cryogenic temperatures.

2.4.2

Energy transport in the bulk

Using the Fourier law for carrier transport derived in Equation 2.8, the flux (q)through a bulk material of thickness L is given by
q = −k∇T = −k

∆T
∆T
= −k
.
∆x
L

(2.11)

Using the electrical analogy, we can recast this equation in terms of a resistivity tothermal transport, which is defined as the ratio between the material thickness and thethermal conductivity (r = L/k),
q=−

∆T
.
r

(2.12)As phonons and electrons carry heat simultaneously, the thermal resistivity due tothese two channels must be considered in parallel. Electron transport can be describedin terms of separate resistivites to electron-electron (e − e), electron-phonon (e − ph)and electron-dislocation/interface (e − d) events. Similarly, phonon resistivities include
phonon-phonon (ph − ph), phonon-electron (ph − e), and phonon-dislocation/interface(ph − d) components. Each type of interaction creates a resistivity to heat transfer and,for one carrier, all of these resistances must be added:
1
1
1
=
+ ,
rT otalrph re(2.13)

rph = rph−ph + rph−e + rph−d ,(2.14)

re = re−e + re−ph + re−d .

(2.15)In metals, both electrons and phonons can carry heat, so they both contribute to thetotal resistivity (Figure 2.8, blue box). However, in semiconductors and dielectrics, thethermal transport is dominated by phonons, so we consider only phonon interactions(Figure 2.8, gray box).

re-e

Metal
re-ph

Interface

re-d

T1

T2A
rph-ph

rph-e

rTBR

T2B

19

Dielectric
rph-ph

rph-d

T3

rph-d

Figure 2.8: Resistivity model for determining thermal flow rates in the case of theexample considered in section 2.4.3. The example layout and temperature distributions
are shown in Figure 2.9.

Temperature

q

LA

LB

A

B

T1

q

T2A

T=-q/kA

T2B
LA
PositionT=-q/kB

T3
LA+LB

Figure 2.9: Heat flow between two solids in contact illustrating the effect of boundary
resistance. Modified from [24].

2.4.3

Putting the pieces together

Let us consider a simple example, illustrated in Figure 2.9, of a metal (A) in
contact with a dielectric (B). We know the resistivity of each of the constituent parts(A, B, and interface), and by applying energy conservation, the same flux must flow
through each. The total resistivity can then be determined by summing the resistivitiesof each component:q=∆T
T1 − T3=,
rtotal
LA /kA + rT BR + LB /kB(2.16)

20
where rtotal = rA + tT BR + rB .
This method of treating flow as channels in series or parallel is a helpful simplification that we will refer to in Chapter 7. Particularly important for our later use
are the boundary resistivity and the concept that energy flow in a bulk system canbe summarized by a resistivity to flow that is connected in series with the interfacial
resistivity.

2.5

Conclusion
In conclusion, ultrafast laser pulses are a powerful tool for exciting systems

through subsequent photon → electron → phonon excitations and subsequent equilibrations. Applications of this excitation system to study acoustic, thermal, and magnetic
phenomena in nanostructures are discussed in the following chapters.

Chapter 3

High-order harmonic generation

3.1

Introduction
The human eye will respond to photons of wavelength between about 350 to 750

nm in air (commonly referred to as ”optical” light), but this represents only a minisculeportion of the entire electromagnetic spectrum. At longer wavelengths, infrared (∼10µm), microwave (∼10 mm), and radio (∼1 km) radiation are useful for communications
and measurement. Because the energy of each photon is so small, these types of radiation
can pass through many materials without being absorbed. On the other end of the scale,
hard x-rays with wavelengths shorter than 0.1 nm can also pass through many materials,
and by tuning the wavelength (related to the photon energy by Ephoton = hc/λ, where
h is Planck’s constant and c is the speed of light) to a region where one material istransmissive and another absorptive, element-specific imaging can be performed. In acrude sense, this is what is taking place in a medical ”x-ray”, where the radiation used
is of high enough energy to pass through the soft tissue like skin and clothing, butinteracts strongly with bones composed of heavier elements.

3.2

Advantages of EUV light
Between hard x-rays and ultraviolet light (such as the UV radiation we get fromthe sun) lies a vast region of the electromagnetic spectrum that was largely unexploredfor many years [25]. However, the very reasons that made it challenging to explore

22

Figure 3.1: The electromagnetic spectrum(high absorption of light due to multiple atomic resonances) make it very interesting forstudying light-matter interactions.

3.2.1

Shorter wavelength

There are many reasons to generate and use light in the EUV region of thespectrum, and the most obvious of these is that EUV light has a much shorter wavelengththan optical radiation. This allows for tighter focusing and better resolution in imaging.
Similarly, short-wavelength light can also diffract from smaller structures. Consider thediffraction equation
mλ = d sin(θ),(3.1)

where m is the diffraction order, d is the grating constant (center-to-center line spacing),and θ is the angle of diffraction. If we limit ourselves to the case of first-order diffraction,
m = 1, andθ = arcsin(λ/d).

(3.2)

Taking the arcsin of a number greater than one is geometrically impossible, so thisexpression for the diffraction angle is only valid for λ < d and there is no diffraction for
gratings with spacings smaller than the probing wavelength (”subwavelength” diffraction
gratings). This diffraction signal has numerous experimental applications, including: asa spectrometer in element or photon-energy specific studies, and in the detection andimaging of nanostructures. However, Eq. 3.2 states that optical probe beams will not
diffract from structures smaller than ∼ 400 nm. Structures down to 10 nm can be

23

Figure 3.2: Illustration of the increased sensitivity to small phase shifts in interferometricmeasurements when the wavelength of light is short. This is one advantage of using EUV
light to study surfaces.

manufactured with plasma beam etching, and such small structures require EUV probe
light for sensitive detection, analysis, and imaging.
The shorter wavelength also provides additional contrast in interferometric measurements because a smaller absolute phase shift is required to get the same relative
phase shift. This concept is illustrated in Figure 3.2, where we can see that shortwavelength light EUV light provides better phase contrast than much longer-wavelength
optical light for the same phase difference. Applied to surface metrology, this meansthat EUV light has higher spatial resolution not only in the transverse (horizontal, perpendicular to the beam propagation) directions, but also in the vertical (displacement)
direction.

3.2.2

Elemental selectivity

Because of the relatively high energies accessible in the EUV region of the spectrum, many core level electron resonances can be accessed. Light near resonance edges istransmitted or reflected determined by the exact wavelength. For example, the energydependent photon transmission for carbon and water is shown in Figure 3.3. The ability
to selectively study or image the properties of a specific element in a molecule or lattice
opens tremendous opportunities for fundamental chemistry and physics experiments.
One particular example of the utility of this elemental selectivity is the amplified mag-Transmission (through 1 µm)

24
0.8

0.6

0.4

0.2

0
100

200

300

400

500

600

Photon energy (eV)Figure 3.3: Transmission of soft x-ray light through 1 µm of water (blue) and carbon(black). The sharp edges are useful for composition determination of materials andelement-specific imaging and dynamics probing. The photon energy range between 290540 eV is referred to as the ”water window”, where water is transparent and carbon isabsorbing, providing contrast useful for biological imaging. Data from [26].

netic contrast near absorption edges, discussed and experimentally probed in Chapter
8.
Another advantage of working near absorption edges is that at nearby energies,
the absorption can be so strong that the penetration depth is very shallow. This is thecase for most materials at EUV energies below ∼100 eV, as shown in Figure 3.4. These
shallow penetration depths enables high experimental sensitivity to surface properties.

3.3

High-order harmonic generation
The most common source of experimentally-useful soft x-ray light is a synchrotron,

which is a bright (high flux) and tunable source, but requires facilities the size of a football field and is therefore expensive and has limited access. An alternative to synchrotron
sources is the high-order harmonic generation light source (HHG, a comparison to synchrotrons is shown in Table 3.1), a highly-nonlinear process in which intense optical or
infrared radiation can be upconverted to the EUV region of the spectrum. The non-

25

1000

Penetration depth (nm)

Aluminum
Silicon Nitride

100

Sapphire
Ni

10

Pd

1

40

50

60

70

80

90

Photon energy (eV)Figure 3.4: Penetration depths at 45o incidence for a few materials mentioned in thisthesis. Penetration depth is defined as the point of 1/e transmission measured along
the surface normal. Data from [26].

linear nature of the HHG process requires very intense, very short laser pulses focused
to a very small spot with peak intensities of 1014 W/cm2 or higher, well within thecapabilities of modern ultrafast laser amplifier systems. At such intensities, the laser
electric field has a strength comparable to the Coulomb field binding an electron to an
atom, so the perturbative models of the light-atom interaction break down and newtheory is required.
While quantum models are required to rigorously explain HHG [28] [29,30], many
basic characteristics of HHG can be intuitively understood using the semi-classical model
proposed by Corkum [31, 32]. The semi-classical model breaks down the high harmonic
Category
Photon energy
Pulse duration
Flux (photons/second)
Spatial coherence
Physical size

Synchrotron
10 eV - 2 GeV
100 fs - 100 ps
1016
Poor
Football field

HHG
10 eV - 1 keV< 10 fs1012
High
Tabletop

Table 3.1: Synchrotron vs high harmonic generation sources of EUV light. Data from[25] and [27].

12Unperturbed atom,bound electron

03

Laser field lowers barrier,electron tunnels out

026
Laser field oscillates,
electron accelerates away0

Step 1
direction, 5 Electron reverses direction 6
Electron arrives at ion,
4Laser field switches
electron slows
can rescatter or recombine
000Step 2
Laser field
1

2

Step 33

6
4

5

Figure 3.5: Schematic explanation of high-order harmonic generation using the semiclassical three step model. The upper panels illustrate the instantaneous potentials,
while the bottom panel shows where in the laser cycle the snapshots are taken.

generation process into three steps summarized in Figure 3.5:
(1) Tunneling ionization of an electron from a laser-modified atomic potential(2) Free electron propagation in the oscillating electric field of the laser(3) Recombination of the electron with the ion, emitting a high-energy photon.

3.3.1

Step 1: Tunneling ionization

The photoionization of an electron from an atom can be grouped into three
regimes, depending on the strength of the exciting laser field [33]:• Multiphoton ionization is the dominant mechanism for photoionization in thepresence of a small electric field. In this case, an electron is ionized by the27
absorption of enough photons to get to an energy higher than the ionization
potential of the atom.
• For high intensities, ionization occurs by tunneling through a distorted Coulomb
barrier. Here, the laser is strong enough to significantly modify the Coulomb
potential of the parent atom, enough so that an electron can tunnel out into
the continuum.
• Even higher laser fields lead to above-barrier ionization.

In this case, theCoulomb potential is suppressed below the ionization potential, allowing electrons to freely leave the atom.
Tunneling photoionization rates were first calculated by Ammosov, Delone, andKrainov [34], and later, the generalized Coulomb potential of the ion was added toextend the results to more complex atom cores [35]. The result is that the ionization
rate is:ω(t) = ωp |Cn∗ |24ωp 2n∗ −1
4ωp 
exp −
,
ωt
3ωt

(3.3)

where
ωp = Ip /h̄ pqe E(t) / 2me Ipq= Z IH /Ip

ωt =
n∗


∗
|Cn∗ |2 = 22n / n∗ Γ(n∗ + 1)Γ(n∗ ) .
In these definitions, Ip is the ionization potential of the atom, h̄ is Planck’s constant hdivided by 2π, E(t) is the time-dependent laser electric field, me and qe are the electronmass and charge, Z is the ion charge after ionization, IH is the ionization potential ofatomic hydrogen, and Γ(x) is the Gamma function.

28
The fraction of electrons that have escaped an atom is given by the fraction of
atoms that have been ionized, determined by:Z t
η(t) = exp −ω(t0 )dt0 .

(3.4)

−∞

Figure 3.6 shows a calculation of the time dependence of the ionization of argon within
an ionizing laser pulse. Also shown is the electric field of the ionizing laser pulse – thesteps in the ionization fraction occur at times that match peaks in the electric field,
demonstrating that the ionization is very sensitive to the time-dependent field. The
ionization also depends strongly on spatial variations in the intensity, so an accurate
understanding of laser profile and propagation is important for determination of theionization process.

Ionized fraction η

1.0

Ionization fraction η

0.5

Pulse electric field(shifted and normalized)

0.0
-40

-20

0

20

40

Time (fs)Figure 3.6: Fraction of ionization of Argon from a 25 fs, 5×1014 W/cm2 peak intensity
pulse.

3.3.2

Step 2: Free electron propagation in a laser field

After an electron is ionized, the three-step model assumes that it is accelerated
in the laser field, starting with zero velocity. The basic idea is as follows: the laser field
first accelerates the electron away from the ion core, but the field is oscillating, so it29soon changes direction. This accelerates the electron back toward the ion, slowing it
until eventually it is heads back towards the ion it came from, where it can recombine
and emit radiation.
The propagation of the now-free electron can be calculated using classical equations of motion. If we take the laser field to be E(t) = E0 cos(ωt) (In the case ofultrashort laser pulses, this is an approximation because the sinusoidally-oscillating
field is modulated by the pulse envelope, as shown in Figure 3.6), then the electron
velocity is determined by F = ma and F = qE. The acceleration of the electron is thena(t) = (qe /me )E(t), and the velocity can be determined by integrationqev(t, t0 ) =mev(t, t0 ) = −Z t0

E(t0 )dt0

t0


qe E 0sin(ωt) − sin(ωt0 ) .
me ω

(3.5)

The position can be determined by integrating further to getx(t, t0 ) =qe E0cos(ωt0 ) − cos(ωt) − sin(ωt0 )(ωt − ωt0 ) .
2
me ω

(3.6)

The resulting electron propagation is highly-dependent on the phase of ionization withrespect to the laser field. One might expect the optimal results to occur when theelectron is excited at the peak of the laser field, but since the electron started withan initial velocity of zero and experiences a symmetric potential, it would arrive backat the atom with zero velocity, as shown by the black curve at laser phases of 2π
and 4π. Electron paths for various phases of ionization with respect to the laser field
are shown in Figure 3.7. Zero crossings for the position represent opportunities forthe electron to collide with the parent ion with a velocity (recall that kinetic energy
KE ∝ v 2 ) given by the slope of the curve at that point. While some paths do notever return to the ion (e.g. π/2), others encounter the ion multiple times (e.g. π/32).
The semiclassical model only considers the first interaction event because the electron
distribution spreads transversely as it propagates at a rate of ∼ 1.5Å/fs [31], meaning

30
that the electron distribution is broader at later times and therefore much less likely toElectron travel from atom (qeE0/meω2)

recombine with the parent ion.
0 radians

2
0

-2

π/32

Drivinglaser field

.09π

-4

π/8

-6

π/4

-8

-10

π/2

-12
0

1

2

3

4

Laser phase (rad/π)Figure 3.7: Free electron propagation in a sinusoidal laser field (red dotted line). Electron displacement from the atom is shown for electrons excited at several delays (marked
at right) relative to the peak laser field. Every zero-crossing for a particular trajectory
is a potential collision even with the ion core.

3.3.3

Step 3: Recombination and radiation

When an electron that has been ionized (Step 1) and accelerated in the laser’s
electric field (Step 2) returns to the atom, it can recombine with the atom, falling backinto the potential energy well it originated in. The excess energy the electron had, bothkinetic and potential, is given out as a radiated photon. The potential energy component
is given by the ionization potential of the atom (Ip ). In order to get the highest-energy
harmonics, we use Noble gases that have filled (and therefore tightly bound with a highionization potential) valence electron shells. For example, the ionization potential ofargon is 15.7 eV, and that of neon is 21.6 eV. The kinetic energy component is given
by me v 2 /2, where v is the electron velocity when it arrives back at the atom. This
return velocity can be determined by finding the first zero-crossing of Equation 3.6 and31plugging that value in to Equation 3.5. Figure 3.8 shows the electron’s return kinetic
energy as a function of laser phase when the electron was ionized. The peak occurs fora phase of ∼ 0.1π radians, and has a value of ∼ 3.2 × Up , where Up is the ponderomotiveenergy, which is the maximum kinetic energy gained by the electron due to accelerationin one cycle of the laser field. The maximum photon energy that can be radiated is thengiven byEph,max = P Ee + KEe,max = Ip + 3.2Up .

(3.7)

Electron return KE/Up

3

2

1

0
0.0

0.1

0.2

0.3

0.4

0.5

Laser phase at electron ionization (rad/pi)

Figure 3.8: Kinetic energy of electron returning to the atom as a function of laser phasewhen the electron was ionized.

An understanding of the kinetic energy of the returning electron (Figure 3.8)
yields additional insights into the HHG process. First, the peak return kinetic energy
is proportional to the ponderomotive potential, which depends on the properties of thedriving laser: Up ∝ Iλ2 , where I and λ are the intensity and wavelength of the laser.
So to get higher-energy harmonic light, we can increase the laser peak intensity (shorter
pulses or tighter focii) or increase the wavelength (of course, using a longer wavelength
will increase the time an electron spends in the driving laser field, so the distribution
spread will decrease the recollision probability [31]). Also, there are two laser phases

32which give the same kinetic energy, known as the short (highest initial phase value)and long(smallest phase) trajectories. These electrons each spend a different amountof time in the driving laser field, so comparing their amplitude as a function of laser
intensity and wavelength gives insight into intensity-dependent induced phase [36, 37].
In general, both harmonics from both short and long trajectories are generated, but by
using phase matching techniques, a single trajectory can be selected , which is importantfor experiments in which a highly monochromatic spectrum is desired [38].

3.3.4

HHG in hollow waveguides

High-order harmonic generation can be performed by focusing an ultrafast laser
pulse into a collection of gas atoms (gas jet), but this technique offers a very limited
interaction region because of the tight pump focus and limited jet size. An alternativetechnique pioneered by the Kapteyn-Murnane group uses hollow glass waveguides tohold the gas, guide the laser, dramatically extend the interaction region, and provide
more control over the harmonic generation process [39]. Like any parametric conversion
process, efficient harmonic conversion requires that the process be phase-matched. As
the driving laser beam propagates, harmonics are generated coherently and build up,
adding constructively with past generation events only if the driving laser light andgenerated harmonic light travel with the same phase velocity in the medium. For
conversion that is not phase-matched, generation is limited to a distance known as thecoherence length Lc = π/∆k, the distance over which the driving and generated waves
slip in phase by half a cycle, leading to constructive interference between new and past
generated light. The phase mismatch ∆k = qklaser − kEU V , where q is the harmonicorder, and has contributions from vacuum, neutral gas dispersion, non-linear refractive
index, plasma dispersion, and waveguide dispersion [39]. Fortunately, these components
have different sign: the first three add positively to the dispersion, while the last two
are negative. This means that by balancing the atomic and waveguide dispersion by33
pressure tuning, we can obtain phase-matching (∆k = 0) and see coherent buildup of
the generated harmonic beam along the propagation direction, with an increase of up
to two orders of magnitude more flux for a given pulse energy [40].
Unfortunately, this phase-matching technique works only at relatively low photon
energies (EEU V < 50 eV for argon and EEU V < 130 eV for helium), because above thesephoton energies, high levels of laser-induced ionization cause the plasma dispersion term
to dominate beyond what can be compensated by simple pressure tuning. Instead, quasiphasematching techniques must be used, which include the use of modulated waveguides[41] and counterpropagating pulses [42].
At the single-atom level, the HHG process is governed by the (usually excellent) coherence of the driving laser, but the highly-nonlinear process also gives rise toa dynamically-changing index of refraction and complex multi-mode generation that
leads to only partially-coherent light being generated [43–45]. However, when the HHG
process is phase-matched in a follow waveguide, a fully spatially-coherent beam with
an improved mode quality results [46]. Thus, high-order harmonic generation in a hollow waveguide gives high-flux, coherent beams of extreme ultraviolet light useful forsensitive detection of chemical and physical phenomena.

3.3.5

Characteristics of HHG

Regardless of whether the harmonics are generated in a gas jet or a hollow waveguide, the experimentally-observed harmonics have common features. The most obviousof these is given by the name: spectrally, a series of harmonics of the driving laser energy
is observed that extend up to the cutoff energy in Equation 3.7, as shown on the rightside of Figure 3.9. Experimentally, we typically observe a HHG spectrum from argon
gas as shown in Figure 3.10: a comb of ∼ 5 harmonics strongly peaked near 42 eV (27thharmonic). There are two reasons we don’t observe a plateau of even-amplitude harmonics: at the lower energies, the generated harmonics are absorbed by the Ar gas [39, 47],

34
and at the higher energies, we see the ionization limit to phase-matching [39]. Usingneon as the harmonic medium, the spectrum is a plateau comb of harmonics from 35
eV to above 70 eV. A typical spectrum from neon is shown in Figure 8.7.

Temporal domain

Spectral domain

Broad frequency range UV - keV

Figure 3.9: Characteristic light from HHG in the temporal (left) and spectral (right)
domains.

Amplitude (a.u.)

0.4

0.3

0.2

0.1

0
35

40

45

50

Photon energy (eV)Figure 3.10: Typical harmonic spectrum from Argon.

The temporal structure of a harmonic beam is a series of attosecond bursts spacedby half an optical cycle (as shown in Figure 3.9, left) because harmonics are generated
near each peak in the electric field. Because HHG is such a highly-nonlinear process, it
is localized to the peak of the driving laser pulse: a driving laser with a duration of 25
fs generates a HHG pulse shorter than 10 fs [48].

35

3.4

Conclusion: Unique advantages of HHG
As the usable range of harmonic light that can be generated is pushed to higher

and higher energies [49, 50], access to new absorption edges and shorter wavelengths forfiner-resolution in imaging enables experiments that previously could only be done atsynchrotrons. However, the use of HHG has a few particular advantages over the generalcharacteristics of EUV light mentioned in Section 3.2. The most obvious advantage of
EUV light from HHG is the accessibility of having an EUV light source in the labrather than at a large user facility. The strict time schedules and beamline sharingthat is the norm for most synchrotron-based projects pose a significant experimental
challenge. Additionally, many synchrotron users live and work far away from the source
itself, requiring expensive trips and dangerous equipment transport. Another obvious
advantage of HHG is the financial freedom of a source that costs ∼ 4000x less to build
and maintain.
The primary scientific advantage of EUV light from HHG is that the light issimultaneously short wavelength (high energy) and ultrafast, combining the photon
energies of a synchrotron, with a temporal resolution even better than that possible using
an ultrafast visible laser. The use of HHG as a probe has been recently demonstrated inobservations of molecular rotation [51,52] and dissociation dynamics [53,54], observationof acoustic dynamics in damascene [55], and photoelectron spectroscopy of solid-statesystems [48, 56].
The short wavelength light from HHG makes it an obvious candidate for imaging. Most synchrotron imaging is currently performed using incoherent light, but thesestudies are limited to a resolution of ∼ 15 nm, defined by the optics (smallest minimum width on a zone plate) [57]. Coherent imaging is one way to circumvent thisroadblock since the resolution is theoretically limited only by the wavelength of light.
Pioneering work on applying the lensless imaging technique previously demonstrated at

36
a synchrotron [58] to use a high harmonic source was performed by Sandberg and Paul,and they observed images with 90 nm resolution using 29 nm light from HHG [59, 60].
More recently, using a specialized sample with reference holes, they implemented Fouriertransform holography capable of fast image reconstruction and 70 nm resolution [61],and by combining this with lensless imaging reconstruction algorithms, the resolution
improved to 53 nm. This technique is continuing to mature and ongoing work is pursuing the use of shorter wavelengths for improved resolution, time-resolved imaging of
biological samples, and the reflective geometry.

Chapter 4

Time-resolved EUV holography

Time resolved interferometric imaging techniques have attracted much attention
recently because of their ability to precisely monitor dynamic changes in both space andtime [62–64]. These experiments make use of a pump-probe geometry, where an opticalpump distorts the sample, while a second, time-delayed, probe captures the induced
dynamics. Interferometric information is obtained by placing the sample in one arm of
a Mach-Zender or Michelson interferometer. In some experiments [62, 63], a dynamic
interference pattern is then captured on a CCD camera, and a reconstruction algorithm
is used to obtain the sample evolution. Specifically, femtosecond electron dynamics
at metal and semiconductor surfaces can be studied with lateral spatial resolution approaching 1 µm, and with phase sensitivities of λ/1000 at 800nm. Interferometric detection has also been employed to study surface deformation on ultrafast time scales [64],where a Sagnac interferometer can be used to obtain phase resolutions on the order of
1 pm.
The use of extreme ultraviolet (EUV) radiation for interferometric detection offers the potential for simultaneous sub-Angstrom displacement sensitivity and increased
lateral spatial resolution, because of the shorter probe wavelength. However, the useof EUV light in interferometric measurements has proven difficult because of the poor
reflectivity of optical components (∼ 1%) at these wavelengths, and because even poorer
transmission makes an EUV beamsplitter impractical. Typical sample reflectivities are38around 1%, making a Michelson interferometer impractical. Nevertheless, a few experiments to date have employed interferometric detection with EUV light on relatively
long timescales and large spatial dimensions. For example, the electron density profile of laser-produced plasmas can be measured using EUV interferometry [65] because
the high frequency of the EUV light is above the plasma frequency of the expanding
plasma. Other interferometric experiments have directly measured the complex index
of refraction of sample materials at EUV wavelengths [66]. However, to date EUV interferometry or holography with simultaneously high spatial and temporal resolution
has not been experimetally realized.

4.1

Holography with EUV light from HHG
Among EUV sources, high-order harmonic particularly useful for dynamics ex-

periments because the very short pulse duration (¡10 fs) allows for excellent temporal
resolution. In this experiment, we implement a dynamic, time-resolved extension ofprevious work in our group that used EUV light from HHG for static Gabor holography [46]. Here, we study the dynamics of optically-initiated thermoelastic processes
in a thin metal film with simultaneously high time resolution and spatial resolution in
the vertical dimension. A femtosecond near-infrared laser pulse is used to launch theacoustical processes. A fully-coherent EUV beam [46] generated using the process ofhigh-order harmonic generation in a gas-filled waveguide serves as the probe of the subsequent acoustic oscillations at the surface of the thin film. We use a novel excitationgeometry shown in Figure 4.1: a area is excited by a line-focued infrared-pump laser,which is then probed by a much larger EUV beam. The unperturbed sample reflects
the EUV (reference) beam, while the pumped region diffracts the object beam to form adynamic hologram. This leads to a simple, robust, single-reflection geometry for studying transient dynamics on surfaces with femtosecond time resolution. We demonstrate
excellent sensitivity to surface displacements and we find that EUV probing of surface

39
deformation is largely free of ambiguity associated with electronic effects. This allows
investigation of surface deformation immediately following impulsive heating - a regime
in which the surface deformation signal is generally screened by the coherent electronic
signal if probed by visible wavelengths [67].
The experimental geometry is shown in Figure 4.2. Light from a titanium-doped
sapphire laser-amplifier system [10, 12]operating at a repetition rate of 2 kHz, with a
25 fs pulse duration, is split into pump and probe beams. Light in the probe arm is
upconverted into the EUV using the process of high harmonic generation (HHG) in an
argon-filled hollow waveguide [40]. The waveguide allows the conversion process to bewell phase-matched over an extended propagation length, leading to the generation ofa fully coherent EUV output beam.6 The EUV pulses generated in this manner yields
a spectrum of 5 (odd) harmonic orders, peaked at the 27th harmonic order with awavelength of 29 nm, as shown in Figure 3.10. The EUV light then passes throughthree 200 nm thick aluminum thin film filters to eliminate the residual laser light while

IR pumpObject beam

EU

V

pr
ob
e

ef
R

ce
n
e
er

am

be

80nm Ni film

Fused Silica substrate

Figure 4.1: Scheme for dynamic EUV Gabor holography. The infrared laser pump is
focused to a small line on the nickel thin film sample, which causes a thermal bump.
The EUV probe spot is larger than the bump, so most of the EUV beam is simply
reflected, forming the reference beam. The EUV that is diffracted from the thermalbump is the object beam. The interference between these beams is detected on a CCD.

40transmitting approximately 3% of the EUV radiation. The pump beam is sent into
a computer-controlled delay line. A single cylindrical lens is used to generate a 1.5
cm×77 µm line focus on the sample. Experiments were performed on thin metal films
that were thermally evaporated onto fused silica substrates. The probe spot is 1 mm indiameter and is spatially and temporally overlapped with the pump beam on the sample
surface, as shown in Figure 4.1. In contrast with conventional pump-probe techniques,
the excitation region of the sample resides wholly within the region being probed. Thisgeometry allows the EUV reflection from the unperturbed region of the sample to serveas the reference beam for holographic detection. When the sample is perturbed by theline-focus pump excitation, the EUV intensity pattern on the camera corresponds tothe interference of light reflected from the flat surfaces of the sample and light diffracted
from a dynamically varying, line-pumped, region. We observe a dynamically changing
single-slit diffraction pattern that is shown in Figure 4.3. This representative image wastaken by reflecting EUV light from a 2 µm thick aluminum film, at a time delay of 150
ps after the pump pulse.
7ns delay

Ti:Sapph Amplifier
2mJ, 2KHz, 25fs

PUMP IR beam (300mW)

CCD camera

Shutter!/2

Coherent EUV
generation

Al Filter
Al Filter

IR beam 2.5W

Figure 4.2: Experimental setup for time-resolved EUV Gabor holography. The infraredlaser pump is absorbed by the thin nickel film, and the thermal and acoustic transientsthat follow are dynamically probed with EUV pulses.

The simplicity and ease of alignment of this method for holographic imaging of41

Pump onPump off

Difference

Figure 4.3: Recorded holograms with the pump on, pump off, and pump-probe difference
hologram. It is impossible to see a change between the pump on and off images, but thedifference image is clear. Figure 4.4 shows the image reduced to 1D by vertical binning.

surfaces is not without its drawbacks. We would like to extend this technique to ahigher lateral resolution, but we are limited by the finite spread of the reference beam.
Since the image and reference beams will only interfere where they overlap and thefinest-detail information is located furthest from beam center, the signal we observe
contains only low-resolution information. In the experiment described here, we imaged
an optically-pumped spot with a width of approximately 100 µm, which is much larger
than the limit of our optical focusing system – we were unable to see a holographic
signal for smaller surface features. It may be this is a signal to noise problem: as thepump spot is made smaller and smaller, the relative strength between the object andreference beams is made weaker and weaker. This could be remedied by using a smaller
probe spot, but it is desirable to have a large reference beam at the interference planeand necessary to have a pump spot much smaller than the probe spot on the sample,
so these factors must be balanced.

42

Difference intensity (a.u.)

Fit
Data

CCD pixel (26µm/pixel)
Figure 4.4: Difference in reflected EUV intensity from a 2 µm thick aluminum film, withand without the pump pulse present, at a time delay of 150 picoseconds. The solid line
shows the dynamic hologram exhibiting the characteristic far field diffraction pattern ofa slit. The calculated signal is shown as a dashed line. By analyzing many such images
as a function of time delay, the amplitude of the diffraction pattern oscillates as abulk longitudinal acoustic wavepacket propagates into the bulk and undergoes multiple
reflections at the film/substrate interface..

4.2

Fresnel propagation model
In determining how to model the optical propagation, an important parameter isthe F -number F = D2 /(zλ), where D is the diffraction feature size on the sample andz is the sample to CCD distance. In this experiment, D is equal to the pump beam
spot on the sample, D ∼ 100 µm. z was measured to be 25 cm so F = 1.4 > 1, andtherefore the propagation is in the Fresnel regime.
Given an electric field distribution U (x0 ) at z = 0, the field S(x) at a distance z
away (as shown in Figure 4.5) can be calculated by
Z Z
i
eikr
S(x) = −U (x0 )
cos(θ)dx0 ,
(4.1)
λ
r
p
where k = 2π/λ, r = (x − x0 )2 + z 2 , and cos(θ)=z/r. The Fresnel approximation

43
assumes that the scale of the entire image planes U (x0 ) and S(x) is less than z, which
is valid here (x0max ∼ 1 mm, xmax ∼ 2 mm, z = 250 mm). In this case, calculationof r can be simplified by the use of a Taylor series expansion and the angle θ can beassumed to be very close to zero, so cos(θ) ∼ 1. Under these assumptions, Equation 4.1
reduces toi eikzS(x) = −
λ z

Z Z

0 2

U (x0 )eik(x−x ) /2z dx0 ,(4.2)

which is the Fresnel diffraction equation [68, 69].

y

y’
r
x’

x

z

Figure 4.5: Object and image plane definitions for the discussion of the Fresnel diffraction integral. In the Fresnel approximation, r ∼ z and S(x, y) can be calculated by aconvolution with a transfer function [70].

One way to proceed with Fresnel propagation is to use a transfer function h(x)and its Fourier transform H(fx ) defined as [70]eikz −ikπx2 /2z
h(x) =and H(fx ) =ezr

2 ikz 2πifx2 z/k
e e
ikπz

(4.3)

where the first term is a constant phase and is neglected. The Fresnel integral in
Equation 4.2 can then be expressed in terms of a convolution between the initial field
U (x0 ) and the transfer function h(x):S(x) = U (x0 ) ∗ h(x).

(4.4)

44
We can then make use of the convolution theorem, which says that the Fourier transformof the convolution of two functions f and g is the product of their Fourier transforms:
F [f ∗ g] = F [f ] × F [g]. Then the propagated field distribution ishiS(x) = U (x0 ) ∗ h(x) = F −1 F [U (x0 )] × H(fx ) .

(4.5)

This propagation can be performed for both Upumped (x) and U0 (x) to determine theimage fields in the pumped and unpumped cases. Subtracting the intensities (I(x) =|S(x)|2 ) gives a calculation for what was experimentally measured. The full matlab
code for the propagation calculation is shown in Appendix A.
The initial field distribution in the case of no laser pump is determined by agaussian EUV input reflecting from a pristine surface with reflectivity r0 :2
U0 (x0 , t) = r0 exp(−x02 /wEU
V ).

(4.6)

In the case of the laser-heated surface, the 29 nm EUV light (a single harmonic is used in
the calculations) experiences a complex reflectivity coefficient of r0 +∆r(x0 ) exp iφ(x0 ) ,where ∆r(x0 ) represents the pump-induced change in reflectivity due to photo-elastic
and electronic effects, and φ(x0 ) is the phase change as a result of the thermal surface
deformation.11 However, as we will show below, amplitude effects can be neglected
because the EUV probe is only sensitive to surface displacements. Assuming that the
profile of the surface deformation is proportional to the Gaussian profile of the pumpintensity, the pump-induced phase change at the surface is given by:φ(x0 , t) =4π∆`(t)2)exp(−x02 cos(θ)2 /wIR
λ cos(θ)(4.7)

where λ is the probing wavelength, ∆`(t) is the peak surface expansion dependent on
the pump-probe delay time at a given a pumping fluence, θ = 45◦ is the angle between
the probe and the surface normal, and w is the width of the pumped region, which in
our experiment is 77 µm. The complex field at the sample is then the product of the45gaussian EUV spot and the pump-induced complex reflectivity:2
Up (x0 , t) = r0 + ∆r(x0 , t) exp(iφ(x0 , t)) exp(−x02 /wEU
V ),

(4.8)

which can be used in Equation 4.5 to determine the field distribution propagated a
distance z away.
The only free parameter in fitting the calculated to the experimental holograms isthe peak surface expansion. Figure 4.4 shows one such fit to holographic data collected
150 ps after time zero, yielding a surface deformation of ∆` = 0.022 ± 0.003 nm. In
order to determine if this is a reasonable value, we determine the thermal expansion of
the bump byZ∆`(x) = α∆T (x, y)dy,(4.9)

where α and ∆T (x, y) are the coefficient of thermal expansion (αN i = 12.4 × 10−6 )and the temperature distribution in the film. As will be discussed in section 7.4.2,
this is a poor assumption that overestimates the thermal expansion by a factor of two
to three [71], but it will serve an upper limit. The data was collected at a pump
power of 300 mW and a repetition rate of 2 kHz, so with a nickel reflectivity at normal
incidence of 69%, the energy absorbed per pulse is Eabsorbed = 48 µJ. Then the change
in temperature is given by∆T =Eabsorbed,
Cp ρVabsorption(4.10)

where all the properties are for nickel: Cp = .44 J/gK, ρ = 8.9 g/cm3 , and Vabsorption isthe absorption volume, equal to the product of the beam dimensions with the absorption
depth ∼ 15 nm. This calculation yields a peak temperature change in the nickel filmof ∆Tmax = 260 K, so the heated nickel is still well below the melting temperature of∼ 1730 K. Plugging in the value for ∆T in Equation 4.9, we determine an upper boundon the surface deformation of 0.07 nm, indicating that our measurement of ∆` = 0.022
nm is reasonable. Differences between the signal and simulation in Figure 4.4 are likely
due to beam pointing instabilities in the laser system. Based on our fit results and a

46
0.004

ΔSignal (deformation)

Thick Nickel sample (2000nm)
Thin Nickel sample (80nm)

800nm probe

-0.05

0.003

!R/R (%)
ΔR/R
(%)

-0.20

-0.15

0.002

-0.20

EUV probe

0.001

-0.25

-0.3

0.000
4

06

8
2

10
4

12
6

P ump-probe delay time (ps)

14

-0.2

0.0

0.2

0.4

0.6

Pump-probe delay time (ps)
P ump-probe
delay time (ps)Figure 4.6: Comparison of ultrafast pump-probe data from nickel films. At left is thesignal from 80 nm (blue) and 2000 nm (red) thick Ni films using an EUV probe, with arise time ∼ 2 ps. At right, an 800 nm probe beam is used and a much faster rise time∼ .2 ps is observed. The difference is attributed to the EUV sensitivity to the surface
deformation which takes longer to occur. 800 nm data reproduced from reference [73].

peak signal-to-noise of 35 (obtained from Figure 4.7), we concluded that our sensitivity
to surface deformation is 0.7 pm. This sensitivity is comparable to that obtained using
the best visible interferometric techniques [72], and shows great promise for further
improvements of dynamic measurement and imaging of thermal processes.

4.3

Longitudinal acoustic propagation measurement
By analyzing the time evolution of a diffraction pattern such as that in Figure

4.4, the transient thermo-acoustic dynamics excited by the ultrafast pump pulse canbe extracted. Data are accumulated by averaging 25-75 CCD exposures, where eachexposure is 20-100 ms (40-200 laser pulses) long. Scans with a good signal-to-noise
ratio take about 1 hour to complete. A single data point is comprised of the integrated
difference between CCD images taken with and without excitation of the sample, forexample, the integrated absolute deviation from zero in Figure 4.4. Figure 4.6 shows
the fast signal rise at time zeroas a function of time delay between the pump and probe
for an 80 nm nickel film on fused silica. The onset of transient dynamics is marked

47
by a sharp rise in signal level immediately after time zero, with a risetime of τ20−80 =2
ps. This initial rise is due to the delayed responses of electron thermalization, electronphonon coupling and thermal expansion of the lattice surface after impulsive excitation,
as discussed in Chapter 2.
The laser-induced thermal stress near the surface relaxes by launching a broadband, longitudinally-polarized strain pulse into the metal film. This acoustic wavepacket
propagates through the thin film until it reaches the film-substrate interface, where a
portion of the wavepacket is transmitted and the rest is reflected back towards the sample surface. The arrival of the wavepacket back at the probed surface is marked by an

C

Optical Pump

B
C

D

Thin film (Ni)Substrate (FS)Surface displacement (nm)

A

B
D

A
Pump-probe delay time (ps)Figure 4.7: Temporal evolution of the holographic signal, measuring both the initial
thermal expansion of the surface and the repeated arrival of an acoustic wavepacket atthe probed surface. The acoustic dynamics are summarized in a series of steps on the left:(A) Optical absorption and thermal expansion, (B) longitudinal acoustic wavepacket islaunched (C) wavepacket reaches the film-substrate interface and is partially reflected(D) the reflected portion of the acoustic pulse returns to the film surface, modulates
the surface, and is reflected back into the film where steps B-D are repeated. The step
locations in time are marked on the data at right. An oscillation frequency of ∼46 GHz
indicates a nickel film thickness of 66 nm.

48
additional modulation of the surface, and detected in the experiment as a change inthe holographic signal. Partial reflection of the acoustic wavepacket at the free surface
leads to successive echoes as the wavepacket bounces between the two interfaces [74–76]
By Fourier transforming the temporal evolution of the diffraction patterns as shown in
Figure 4.7, we extract the oscillatory surface deformation due to multiple reflection of
the acoustic wavepacket with a frequency of 45.7 ± 1.1 GHz. Using the accepted value
for the longitudinal acoustic velocity in bulk nickel (v` = 6040 m/s), we retrieve a film
thickness of 66.1 ± 1.6 nm. This value is in qualitative agreement with the nominal
film thickness of 80 ± 20 nm measured by a crystal microbalance during the thin film
evaporation process.

4.4

Advantages of probing surface displacement with EUV light
Of particular interest in Figure 4.7 is the absence of signal arising from electroninteractions at time zero, in contrast with optical probe measurements that can experience two electron effects [77]. The first is a coherent electron bleaching when the pumpand probe wavelengths are the same, as shown on the left side of Figure 4.8. Whenelectrons are excited by the pump beam to energies above the Fermi level, then probe
light of the same frequency will experience a decrease in absorption and a commensurateincrease in transmissio n and reflection [78]. This effect is very short-lived, lasting only
slightly longer than the pump and probe beam overlap time since electron decoherence
times are on the order of 10 fs [7]. The second electron effect is an incoherent change
in reflectivity when the energy of the probe photons is near an interband transition
threshold (ITT) [79]. The Fermi edge is distorted by non-equilibrium ”hot” electrons
that have absorbed the pump laser energy, as shown in Figure 2.3. Probe photons withenergy slightly below the ITT energy see less electrons than are normally near the Fermi
level, so their absorption is increased. The opposite is true for probe photons with energy above the ITT energy: there are more occupied electron states in the final probing

ΔR
R

d

Density of States

Density of States

49

Probe
Pump: hω0s,pEnergyProbe
EnergyCoherent artifact

ΔR
R

d
ITT
s,p
EF

EnergyProbe
EnergyHot electron interaction

Figure 4.8: Schematic illustration of electronic effects near time zero common to opticalpump-probe experiments. In the coherent artifact (left), the pump and probe energy
are very similar and the probe beam interacts with electrons excited by the pump,artificially increasing the probe reflectivity. In the case of hot electrons (right), theprobe light with energy near the interband transition threshold (ITT) sees a modified
Fermi edge (dashed line) due to the non-thermal electrons, which changes the probereflectivity. EUV light (shown by the light purple arrows) is insensitive to both ofthese effects because the photon energy is so high, it does not interact with the excitedelectrons.

state, so the absorption decreases. Although slightly longer than the coherent electron
effect, the hot electron effect is also short-lived because the electrons thermalize withthe lattice in less than a picosecond (electron-electron and electron-phonon thermalization are discussed in section 2.2). However, both the coherent and hot electron effects
effectively obscure any contribution to the signal from the thermal turn-on processes in
the lattice. On the other hand, the reflectivity of EUV probe light is not changed by
the hot or coherent electrons, as shown in Figure 4.8 by the long purple arrows (notethat the actual arrows should be much larger if on a linear energy scale). The photon
energies in the EUV probe are so much higher than the optically-excited electrons, they
do not interact and the EUV is sensitive only to the thermal surface displacement.
Another unique feature of using EUV light to probe surface displacement dy-

50
namics is that the EUV light doesn’t experience the reflectivity change normally used
to measure a signal. Standard transient thermoreflectance (TTR) measurements use
the sample reflectivity to detect thermal density change and the subsequent stresses
due to the propagating wavepacket [72, 74–77, 80–82]. While these measurements givelarge signals, they do not give surface profile information directly, but rather a complicated convolution of dynamic density and strain information [76, 83]. A temperature
increase of 260 K (as determined in section 4.2) results in a maximum density decreaseof .3%, which in turn affects the index of refraction. At EUV wavelengths, the index
of refraction can be written as n = (1-δ)-iβ, where δ and β are proportional to thedensity [25]. For light at 42 eV incident on nickel, δ0 = 0.1467 and β0 = 0.1158, so
the density-modulated values are δ∆ = 0.1465 and δ∆ = 0.1156. The Fresnel equations
can be used to relate this small change in the index of refraction to reflectivity. For
reflection of s-polarized light incident at angle θ (variables are defined in Figure 4.9),
the field reflection coefficient is given byrs =Er
n2 cos(θ) − n1 cos(θ0 )=,Ein2 cos(θ) + n1 cos(θ0 )(4.11)

whereθ0 = sin−1

n

1

n2


sin(θ) ,(4.12)and the intensity coefficient Rs is |rs |2 . Using the values for the pumped and unpumped
indices of refraction and the incident angle θ = 45◦ , we calculate a density-inducedchange in reflectivity of 0.00016. This is 20× lower than the change in reflectivity
for an optical probe, as determined from temperature-dependent absorption data inreference [84]. The modeling propagation results for holographic signal from surface
displacement and reflectivity change for both 30 nm and 800 nm probe light are in
Figure 4.10. According to our model, the strength of the holographic signal from thisEUV reflectivity change is less than 2% of the total signal, compared to 98% for probing
at 800 nm.

Ei
θi

θr

θr = θi = θn2Interfacen151

θt = θ’
Et

Er

Figure 4.9: Definition of variables for the reflection of s-polarized light from an interface as calculated by the Fresnel equation (Equation 4.11) and the refraction angle inEquation 4.12.

The surface sensitivity is manifest in the difference in the shape of the EUV measurement of acoustic pulses (as shown in Figure 4.7) compared to the acoustic pulses
normally measured using optical reflectivity measurements [82]. This is because reflectivity measurements couple to the stress in the surface, detecting the temporal profileof the acoustic wavepacket, while the EUV measurement is sensitive to the simple compression and expansion of the surface (the two are related by a derivative). The acoustic
signature we measured matches the results of experiments in Wright’s group [76, 77],
in which they extracted a signal dominated by surface modulation of Ni. The samegroup later proposed an experimental method for direct measurement of surface deformation [83], but it has not been realized to date. This straightforward method for
determination of surface deformation by EUV probing is largely free of ambiguities associated with electronic and photoelastic effects, a significant advantage over alternative
photoacoustic measurement techniques.

4.5

Conclusion
We have demonstrated a simple and robust holographic geometry for dynamic

phase-sensitive detection of surface deformation using coherent EUV radiation. Theseexperiments extend the scope of previously demonstrated Gabor holography to thestudy of transient acoustic dynamics. With phase sensitivities of greater than λ/43000

Normalized signal

52

800 nm illumination

30 nm illumination
CCD pixel
Figure 4.10: Modeled ”pump on” - ”pump off” image results for illumination with 800
nm near-IR light (top) and 29 nm EUV light (bottom). The total signal in each caseis marked in blue while the contributions from the phase and amplitude change on thesurface are in red and green, respectively. Signal from 800 nm illumination is dominatedby a strong reflectivity contribution, but the signal in the case of EUV illumination isdominated by the phase bump. EUV light is therefore the logical choice for a probe ofsurface displacement.

53
at 30 nm, we detect ultrafast surface deformations and subsequent acoustic echoes inthin metal films. This surface measurement is unobscured by hot electron dynamics orstress-induced density changes in the material.

Chapter 5

Experimental setup for EUV probing of dynamics in nanostructures

After completing the dynamic EUV holography work discussed in the previous
chapter, we decided to use EUV light to time-resolve the acoustic and thermal dynamics
that occur in a nanostructure after impulsive heating by an ultrafast laser pulse. The
driving motivation in pursuing these experiments was to experimentally measure theeffect of a nanostructured interface on energy transport, both along and across theinterface. The analysis and discussion of the interfacial effects we measured can be foundin Chapters 6 and 7; here we discuss the prerequisite work in extracting small timeresolved signals from noisy backgrounds, and how the data presented in the following
chapters was collected and processed.

5.1

EUV detection and noise management
The HHG process is very inefficient (less than one IR photon in 105 is converted)and highly sensitive to small changes in the parameters: gas pressure, and especiallythe peak power and pointing of the driving laser are all critical. In order to perform anexperiment, it must be carefully constructed so as to minimize noise. In his thesis three
years ago [55], Ra’anan Tobey discussed a number of techniques for managing EUV
noise in a system and many of his suggestions are still employed in the experiments
described in this thesis. However, it is better to reduce the source noise than it is to
manage it, and since Ron’s noise measurements four years ago, a number of steps have55
been made to improve the stability of the harmonic source.

5.1.1

Laser system and beampath stability

The harmonic generation process is only as stable as the laser source driving it, so
perhaps the most important improvements made in the past few years are those made to
the laser amplifier. The amplifier itself is now a multipass churped-pulse amplification(CPA) ”Red Dragon” system from KM labs. Along with an improvement in the multipass ring design that minimizes beam astigmatism, the most important improvement
is the use of a new DM-40 pump laser from Photonics Industries. This laser reliably
produces above 30 W at a repetition rate of 2 kHZ with a shot-to-shot timing jitterof < 5 ns and power jitter of < 1%. These noise specifications are much better than
Quantronix Falcon system used previously that also required flashlamp replacement andfrequent service, and this dramatically improved the shot-to-shot power stability of thelaser amplifier. Since HHG is a very high-order nonlinear process, the power fluctuation
improvement has a significant effect on the EUV source stability.
Another important improvement in the stability of the driving laser involved theelimination of air currents around the beampath, both in the amplifier and on the way
to the HHG setup. The amplifier housing was upgraded from a sim4 foot tall fiberboard
box to a ∼8 inch tall metal enclosure. Even more useful was the construction of a plastic
box surrounding the beampath from the laser amplifier to the HHG setup. The noise
reduction after construction of such a system is shown in Figure 5.1 which compares
results with a covered vs uncovered box- without a box at all, the signal is even noisier.
The signal itself is a thermal excitation and decay and will be discussed more in thefollowing sections.
A final component of the optical setup that saw significant improvement was in
the design of the mounting system for the hollow-core fibers for HHG. The old systemof gluing a fiber inside of a glass tube caused very slight bending in the inner fibers,

56
1.0

Uncovered beampath

Amplitude (a.u.)

0.8

0.6

0.4

Covered beampath
0.2

0.0
-100

0

100

200

300

400

Time (ps)Figure 5.1: Chopped, averaged, time-resolved thermal signal for a boxed and covered
beampath (blue) and a boxed but uncovered beampath (red), demonstrating the noisereduction from enclosing the beampath before the HHG fiber.

which significantly reduced the HHG output. The new design uses a v-groove to keep
the fiber supported and confined along a straight line and the resulting harmonics arebrighter, more stable, and much more consistent from fiber to fiber [60].

5.1.2

Acquisition technique: chopping fasterIn Tobey’s analysis of noise management of EUV light from HHG for measurements, he found that the noise was dominated by low frequency components. In order
to filter this low frequency noise, he implemented a chopping method of toggling thepump beam on and off between consecutive exposures. The pump-induced signal could
then be observed by considering the difference of the averages of the pump on and pumpoff exposures.
Three significant improvements were made in this technique since Tobey’s experiments, which are summarized in Figure 5.2:
• We changed the data collection scheme from computer control to be triggered offof the CCD exposure timing, eliminating the need to wait time in the computer
control. In order to toggle the pump beam on and off with each CCD exposure,
we designed and built the circuit shown in Figure 5.3. This circuit was an

57
CCD
Computer
CCD

CCD

Shutter

110ms dead time per exposure

Computer
CCD

ComputerComputer

Shutter

Shutter

Computer control and single CCD acquisition

Shutter

CCD control and kinetic series acquisition

Figure 5.2: Illustration of the difference between two methods of data acquisition: computer control of a series of single CCD acquisitions (left) and CCD control with a kinetic
series of acquisitions.

integral part of all of the data collected for the next three chapters.
• We realized that the biggest source of lag in the experiment was the CCD
readout time: for each exposure, an additional 100 ms was used in downloading
the data to the computer. This meant that in a 20 ms exposure, the system
was collecting data only 16% of the time. This problems was solved by using akinetic series acquisition: instead of repeated I/O commands from the computer
to the CCD, a single command is sent to the CCD instructing the CCD to takethe total number (say 1000) of exposures, and then download all of them to thecomputer when finished. This eliminated the 100 ms data transfer delay (except
for one at the end of the series), and greatly improved the data acquisition speed
and efficiency.
• We switched to use a faster LS 200 shutter from NM laser capable of continuous
operation at 200 Hz, much faster than the 20 Hz limit of the previous shutter.
With all of these improvements, we were able to take data nearly seven times
as fast as before, which not only gave better statistics faster (since signal-to-noise is√
proportional to 1/ N , where N is the number of measurements), but to eliminate lowfrequency noise up to frequencies 7× higher than before. This dramatically improved

58
Divide by two with delay

BNC out+5V

Circuit schematic
November 9, 2007

Mastertoggle+5V.1µF

Updated January 30, 2009

.1µF

BNC in16, Power

Signal In

A, 114, Power

RX

14, CEXT

D, 2

5, Q

CX

15, REXT/CEXT

74LS123(16 pin delay)

+5V

B, 2

4, Q
8, GND

74ALS74(14 pin ÷2)
CLK, 3

6, Q

7, GND

1, CLR 1
Pushbutton
toggleInput waveform(from CCD FIRE)
Transition waveform(from delay chip)T

Output waveform(from ÷2)
T = delay from rising edge (ms) = 0.37 x RX x CX, RX =1k potentiometer, and CX=100µF for T (0->37ms)
Note: +5V regulated by a voltage regulator and additional 1F capacitor to groundFigure 5.3: Schematic for divide by two with delay to enable fast, synchronized excitation synchronization with the EUV CCD detection.

the signal-to-noise of our measurements, and enabled the detection of small signals with
1 photon in 105 sensitivity.

5.1.3

The danger of normalization

Normalization is another technique Tobey suggested to deal with EUV noise.
However, normalization is dangerous because it discards information that was present
in the data (total counts) for normalized data. It is also a big problem when there ispump light leakthrough because pump light hitting the CCD away from the area ofdetection can still influence the total signal. This caused a strong signal to show up,
even before time zero, that masked the real probed signal. We found that it is better toleave the data unnormalized- in this case, the pump-only background can be subtracted

Normalized change in counts (10 -4)

59
0

-2

-4

-6

-8
350

400

450

500

550

600

CCD pixel

Figure 5.4: Signal before time zero caused by scattered pump light hitting the CCD
and skewing the normalization. This illustrates the danger in normalizing the data.
Compare with Figure 5.12 (bottom), which shows real signal before and after time zero.

as a baseline from the pumped data.

5.2

Sample design and fabrication
Sapphire was selected as the substrate material because it has a very high acoustic

velocity (Rayleigh velocity vSAW,Sa ∼6300 m/s) and is transparent to the 800 nm pumplight used for exciting the nanostructure. Nickel lines were manufactured using electron
beam lithography and liftoff in 120 µm×120 µm square regions with 240 µm between
them. Within each region, the lines were 120 µm long with a fixed width L between 65
and 2000 nm (period confirmed using a SEM, as shown in Figure 5.5). Figure 6.4 shows
the two grating geometries used: one with nickel strips of height h = 20 nm and period
p = 4L, the other with h = 10 nm and p = 2L. The periodicities were selected such thatthe product of the height and the duty cycle η = L/p was constant, useful for comparing
the acoustic dispersion in the perturbative approximation discussed in section 6.3.1. To
ensure uniform heating in the vertical (surface normal) direction, the height of thenanostructures was chosen to be < 20 nm — between the optical penetration depth
and ballistic electron transport, the nanostructure should be uniformly heated over thisdimension.

60

L=50 nm

L=70 nm

L=100 nm

L=140 nm

L=200 nm

L=70 nm

L=100 nm

L=200 nm

L=500 nm

L=1000 nm

Figure 5.5: Scanning electron microscope images of typical nickel on sapphire nanograting structures. p = 4L (η = .25, left column) and p = 2L (η = .5, right column)
structures are shown.

The samples were fabricated by our collaborator Erik Anderson at Lawrence
Berkeley National Laboratory. The fabrication process is summarized in Figure 5.6.

61(1) Spin PMMA, Bake(2) Evaporate Al

Sapphire(3) Expose
Sapphire(4) Remove Al in TMAH
developer (LDD26w)(5) Develop PMMA

Sapphire(6) Evaporate 20nm Ni

Sapphire(7) Liftoff in NMP

Figure 5.6: Nanograting sample fabrication process.

5.3

Experimental setup
As in the experiment discussed in Chapter 4, the output of an ultrafast Ti:Sapphirelaser-amplifier system is split into pump and probe beams, as shown in Figure 5.7. A
few notable details of the experimental setup are discussed in this section.
Ti:Sapph amplifier
2kHz, 2mJ, 25fs

IR pump beam

Nanostructured
Ni/Sapphire sample

Beamsplitter

Time-delay stage
Al filter
Hollow waveguidefor HHG

EUV probe beam

Toroidal mirror

Figure 5.7: Setup for nanothermal and nanoacoustic experiments.

5.3.1

Pump beam setup

The 800 nm pump beam is sent through a computer-controlled time-delay stage
and is then loosely focused onto the sample. A relatively large pump spot (∼ 700µm)
is used so that the area probed will see a uniform laser fluence. The sapphire sub-62
strate is transparent to the 800 nm pump light, so the pump pulse only heats the Ni
nanostructure.
In order to determine the actual thermal excitation of the pump laser, it is important to account for the power losses of p-polarized light between the location in the
beampath of the power measurement (30 mW) and the thermal excitation of the nickel:• Beam clipping in the shutter, pellicle steering mirror reflections, and front andback reflections from a vacuum window at near-normal incidence (85% measured),• Front and back reflections from the sapphire substrate at 45o incidence, as well
as reflection from and transmission through the nickel (52% calculated [14,69]).
Accounting for these losses reduces the measured power of 30 mW to 13 mW thermally
exciting the sample, which is an intensity of 1.7 mJ/cm2 for a laser repetition rate
at 2 kHz and a pump spot size of 700 µm. Using this absorbed energy density inEquation 4.10 along with the density and specific heat listed immediately below that
equation and a nickel thickness of 20 nm, we obtain a peak temperature rise of thenickel nanostructure of 220 K. Of course, the nonequilibrium electrons that absorb theincident pump pulses will get to much higher temperatures than this, but because our
EUV detection is sensitive to the lattice response and is insensitive to electron dynamics(see section 4.4), we are not concerned with this.
In the lab, final pump alignment is performed by looking at an image of the pumpbeam on the sample. The pump beam covers approximately 8 gratings, so pump-probe
overlap isn’t critical, but the imaging setup allows for easy location of the pump beam
to the grating being probed.

63
5.3.2

Glancing-incidence toroidal focusing mirror for EUV light

The probe light is converted to EUV by focusing the 800 nm light into a gas-filled
hollow waveguide to generate high-order harmonics. The emitted harmonics can betreated as a gaussian beam with a 1/e2 diameter of 50 nm and a wavefront curvatureof 0.02 m [85]. Therefore, in order to have a small, intense EUV spot useful for probing
the 100 µm × 100 µm sample patches, we need to focus the EUV light.
Since materials are absorptive in the EUV region of the spectrum, a transmission
optic (lenses) is impossible to use. Diffractive (zone plates) enable near diffractionlimited spot sizes [25] and high-resolution imaging [57, 86, 87] at higher photon energies(250 eV and above), but they are very chromatic (the different harmonics will focus
to different distances from the optic, giving a very smeared focus) and at for photons
with energies of <100 eV, they are very inefficient. Reflective (mirrors) optics can beused at any wavelength, but again the absorption is a significant challenge: for example,
reflection from a gold-coated mirror is only 3.5% at normal incidence, which is where
curved mirrors should be used in order to avoid astigmatism in the focused beam.
The reflectivity of normal-incidence mirrors can be improved by using multilayer
coatings [25, 88–90]. Like zone plates, these mirrors are also more reflective for higherenergy photons, with performance optimized at 13.3 nm: using alternating layers of Mo
and Si with a periodicity of λ/2, a narrowband reflectivity over 70% can be obtained,
but if the same materials are used in a multilayer optimized for 29 nm light, the peakreflectivity is only 25%, and even using multilayer materials optimized for this wavelength (MgC and Si) gives a peak reflectivity of 38%. These values are for narrowband
reflectivity (determined by coating many ∼ 40 layer pairs), so all neighboring harmonic
are rejected, which is useful when a single harmonic is desired for energy resolution, forexample in lensless imaging or photoelectron experiments [cite Richard, Luis]. Broadband mirrors can also be manufactured (by coating fewer ∼ 10 layer pairs), but the64peak reflectivity is usually lower by ∼ 50% in these cases.
Another useful option when broadband reflectivity is desired is to use a glancing
incidence mirror because the reflectivity at any wavelength improves dramatically as theangle of incidence increases from normal. For example, the reflectivity of 42 eV lightfrom a gold-coated mirror is 71% at 10o from glancing, a dramatic improvement from
3.5% at normal incidence. As alluded to previously, the problem with this configurationis the strong astigmatism that is introduced by an off-normal reflection from a sphericalmirror. For light incident at an angle θ, as shown in Figure 5.8, the effective focal
lengths in the tangential (in the plane of the angle of incidence) and sagittal (out of theplane) directions are modified as follows:
ft = R cos(θ)/2,

(5.1)


fs = R/ 2 cos(θ) ,where the radius and normal-incidence focal length of a curved mirror are related by
f = R/2. In practice, this modification of the effective focal lengths in the two directions means (for light reflecting from a curved mirror with an incidence off-normal
horizontally) that there will be two line foci: first a sharp vertical (tangential) focus,
and then a blurry horizontal (sagittal) focus. This line focusing may be useful in someexperiments, as in Chapter 8, but in this case we wanted a stigmatic, circular focused
spot to get the EUV light efficiently on the sample.
We used a toroidal mirror to correct the astigmatism. A toroid is a donut-likesurface which is a curved mirror with different radii of curvature in two perpendicular
directions. The idea is to correct the two different focal lengths by using two radii ofcurvature, i.e. setft = fs in Equations 5.1 and solve for Rs and Rt :Rs
= cos2 (θ).
Rt

(5.2)

If the ratio of the sagittal and tangential radii of curvature follow this equation for agiven angle, then light incident on the toroid at θ will be focused stigmatically. Note

65

Object

θ

Rs
Rt

Image
Figure 5.8: Geometry for a toroidal mirror with radii of curvature Rs and Rt in thesagittal and tangential directions, for astigmatism-corrected reflection of light incident
at an angle θ satisfying Equation 5.2. If Rs = Rt , the mirror is spherical and lightincident at an angle θ will be astigmatically reflected. Figure modified from [91].

that a toroid must be designed with a particular angle of incidence in mind, and slightdeviation from that angle will introduce astigmatism.
The toroid we used was designed for use at φ = 90o − θ = 10o from glancing anda focal length of 50 cm. It was manufactured by ARW Optical, and had the following
parameters: 50 mm × 25 mm Pyrex substrate, Rt = 3220 mm, Rs = 90.5 mm, surface
roughness 0.8-1.2 nm. We gold-coated the toroid and then mounted it with springloaded screws to avoid thermal deformations. It was then placed in the beamline withindependent fine control over the sagittal and tangential angles, as well as rotation toalign the incident beam with the tangential toroid axis. With the toroid 65 cm fromthe end of the HHG capillary, we expected a EUV spot size of ∼ 100 µm at the focus,
where we place the nanograting sample.

5.3.3

Aluminum filter holder and CCD adapter

In pump-probe experiments, rejection of the pump light before it reaches theprobe detector is essential to avoid signal contamination. This is a significant challenge

66
in optical pump, EUV probe experiments because there are so many more pump photons. For example, at ∼30 mW and 2 kHz used for the experiments discussed here,
there are ∼ 6 × 1013 1.58 eV photons incident on the sample per pulse, which is nearlyfour orders of magnitude more than the number of harmonic photons generated. Since
there are so many more pump photons, a very efficient filtering method is required.
Additionally, we must consider the residual laser light from the harmonic generation process: nearly 2 W of power is coupled into the waveguide, and only a small
fraction of it is converted to harmonic light. This is a problem not only for detection of
the harmonic photons by a CCD, but also for the integrity of the sample: the nanostructured samples sit at the focus of the beam (from the toroid), so if the laser pulses wereallowed to hit the sample, the intensity of ∼ 109 W/cm2 would generate a temperature∼ 1014 K (well above the melting point of nickel at 1728 K), and would certainly destroy
any structure. So here too efficient filtering of the 800 nm light is required, while still
keeping as much of the harmonic light as possible.
Although there are other methods for filtering the driving laser light from thegenerated EUV light (Brewster’s angle reflection from silicon mirrors is one intriguing
possibility [92]), transmission through thin metal foils is by far the most robust and lighttight system. The concept was developed in the 1960’s in the space program [93], andthe techniques for manufacturing unsupported metal films of a few hundred nanometersin thickness was developed for use in EUV spectroscopy a decade later [94, 95].
Typically in our group, we eliminate the pump and residual laser amplifier light bytransmission through two aluminum films with a thickness of 200 nm. The transparencyof 42 eV photons through 200 nm of pure aluminum is 62%, but aluminum oxidizes
readily in atmosphere, so a few nm of more-absorptive aluminum oxide (Al2 O3 ) grows
on the surface, reducing the effective transmission of each filter to ∼ 40%. Ideally,
we would minimize the number of aluminum surfaces that can be oxidized by using asingle, thicker filter, but unfortunately, an intrinsic property of thin film filters is the67
presence of pinholes that allow a small amount of unwanted light through [96]. For thisreason, at least two filters must be used, but in pump-probe experiments, we use threealuminum filters: two are needed after the sample to effectively reject the pump light,
and an additional one is placed in the probe line before the sample to prevent sample
damage. The total harmonic transmission through the filters is 6%, which seems low
until we compare it with the the transmission of red light, which is 10−34 smaller.
A significant challenge of using thin metal film filters is that they are extremely
delicate. EUV experiments must be conducted in a vacuum because of the high atmospheric absorption, so thin film filters placed in the beamline (usually in a vacuum-tightseal to establish a light-tight seal and prevent scattered light leakthrough) must be able
to withstand viscous forces applied during the evacuation and venting of the vacuum
chamber. Ariel Paul developed an EUV filter wheel dealing with this problem thatallows for the removal of the filters during venting and pumpdown while still giving alight-tight seal when desired [60].
Unfortunately, this filter wheel requires 6 cm of beampath, which is unreasonable
in experiments looking at small structures, where the CCD must be placed very close to
the sample in order to capture the diffraction. How close the CCD needs to be can be
determined by calculating the diffraction angle (parameters are defined in Figure 5.9:mλ = d sin(a) ± sin(b) ,(5.3)

where m = 1 for first-order diffraction and the wavelength in these experiments lambda =29 nm. If we consider the grating with the smallest period that we sill be studying,d = 125 nm, then given an angle of incidence a = 45o , the diffraction angle a−b = 28.4o .
We can then determine the deviation of this first-order diffracted beam from the zeroorder beam bytan(a − b) = A/B.

(5.4)

The CCD we used is an Andor DO-420BN, which has a sensor that is 1 inch across, so

68
if the reflected beam hit one edge of the sensor, the diffracted beam would have to be
A ∼ 2 cm away. This sets an upper bound on the sample-to-CCD distance of B < 6.7
cm. The sample was held in a cryostat to allow for temperature control of the substrate
while under vacuum, and from the sample to the outer wall of the cryostat was ∼ 5 cm,
so only 1.7 cm can separate the chamber and the CCD, ruling out the use of a thin film
filter wheel.
A

B

Figure 5.9: Schematic for calculation of diffraction with an angle of incidence off ofnormal in Equation 5.3. The deviation of the first-order diffraction from the reflectedbeam (A) can then be calculated as a function of distance from the grating (B).

In order to minimize the sample-to-CCD distance while still allowing for two thinfilm filters to be placed between the sample and the CCD, I designed and machined athin CCD adapter with the following features:• Vacuum is retained by o-ring seals on both sides.
• The adapter is only 1.25 cm thick so that the CCD can be placed 6 cm fromthe sample (the distance from the sample to the edge of the cryostat chamberis <5 cm).
• One or two Al filters can be held in the center by aluminum retaining rings.

69

Figure 5.10: Cryostat-CCD adapter plate designed to hold two and provide vacuum
bypass for two aluminum filters while minimizing the sample-to-CCD distance.

• Small holes with many right angle turns to allow air bypass around both filters
but reject light scatter.

5.3.4

Mechanical setup

Each nanostructured grating is 120 µm×120 µm in area, so they present a verysmall target to be hit by the 100 µm EUV probe beam. In order to get good probesample overlap, a means of fine-tuning alignment while the system is under vacuum isnecessary. We considered steering the beam to the sample by turning the toroid, but
this caused astigmatism in the toroid’s focusing, as well as being hard to work withbecause of the distance between the sample and the toroid. Eventually, we settled on
an alignment system that takes the opposite approach: aligning the sample to the beam.
More accurately, we move the whole vacuum chamber to the beam since the sample ison a cold finger and cannot be moved inside the cryostat vacuum chamber.
The final chamber positioning system is shown in Figure 5.11. It has the following
features:• Independent control over the sample position along three axis, as shown by thered, blue, and green arrows in Figure 5.11. This allows for sample selection and70final sample-beam overlap optimization as well as sample movement along thedirection of the beam to position the sample at the focus of the probe beam.
• Stable chamber support by using flange-mounted bearings to hold the chamber
steady despite the vacuum pressure pulling the bellows is ∼ 40 pounds =(650torr)×(.0194 psi/torr)×(3 in2 ). Sample stability in case of a bump or small
adjustment is also very good.

Figure 5.11: Sample positioning system that allows for vertical (red arrows) and horizontal translation of the sample along the direction of the incident beam (blue arrows)
and a direction parallel to the surface of the sample (green arrows), as well as rotation,
all while holding the chamber very stable.

5.4

Data acquisition and analysis
There are two responses to the impulsive, localized, and periodic heating from a

800 nm pump pulse:(1) The heat-induced periodic stress launches SAWs that travel along the surface
perpendicular to the orientation of the nickel lines, as shown in Figure 6.3.
These SAWs interact with the nanostructured nickel strips as they travel, slow
down (since the SAW velocity in nickel is slower than in sapphire), and displace
the nanostructure surface. This displacement is periodic in time as the SAWs
travel on the surface, so they introduce a sinusoidal oscillation of the signal with71
a frequency given by the acoustic velocity divided by the nanostructure period.
A detailed analysis of the SAW propagation is conducted in Chapter 6.
(2) Thermal expansion of the nanostructure as it is heated, and a slow shrinking
as it cools and heat is transported across the interface into the substrate. Thiseffect is the subject of Chapter 7, and can be seen from the same data setsconsidered here as the quasi-exponential thermal decay time under the SAW
oscillation.

5.4.1

Data acquisition and time-dependent signal extraction

Both of these effects introduce very small (∼picometer) displacements of thenanostructure surface, which the EUV light is capable of detecting them because of
the short wavelength. The displacements can be detected in changes in the diffraction
efficiency of the probe EUV beam from the nanograting for a series of (pump on) (pump off) CCD exposures, as shown in Figure 5.12. As discussed in section 5.1.2, thefaster theseIn order to extract a time-resolved signal, we add up the difference intensities at
each time. To gain further insight into the origin of the signal, we can separately consider
the contributions to the signal from the first and second-order diffraction as well as thezero-order reflection. When adding these components together, we flip the sign whenappropriate (for example, the total signal in Figure 5.12 from the ±1, 0 orders is given
by inverting the diffraction components and adding them to the reflected. Repeating
this process at each time delay, we obtain the dynamic signal.

5.4.2

Signal linearity

In order to experimentally determine whether both the thermal and acoustic
signals are linear, we can observe the dynamic signal as a function of pump power. The
results of this test for 90 nm lines are shown in Figure 5.13, and clearly both the thermal

72

Figure 5.12: Top: raw data obtained from averaged pump-on (blue, shifted) and pumpoff (red) exposures of EUV diffraction from a L = 1 µm nanograting. Inset shows ablow-up of the first diffracted order, demonstrating the imperceptible difference between
exposures with the pump on and off. Bottom: Pump-on, pump-off difference from top
figure, both before (orange, no signal) and after (green, change in diffraction efficiency
signal) time zero.

73
decay and acoustic oscillations scale linearly with pump power in this range. Similar
tests on other samples showed the same linear signal dependence. All data presented in
the following chapters was taken with pump intensity at or below these values.
40mW pump
30mW pump
20mW pump
10mW pump
5mW pump
2.5mW pump

0.010
0.008

Normalized ∆ counts signal

0.006
0.004
0.002
0.000

40mW, 1x30mW, 4/3x
20mW, 2x10mW, 4x5mW, 8x2.5mW, 16x

0.012
0.010
0.008
0.006
0.004
0.002
0.000
0

1000

2000

3000

4000

Time (ps)Figure 5.13: Signal from L = 90 nm nanogratings as a function of pump power. The
top graph shows the unscaled signals at each power, while the bottom graph scales allof the data by the appropriate power ratio in order to determine the signal linearity as
a function of pump power.

Modeling of the expected signal predicts linear signal dependence on temperature
as well. The thermal surface displacement is directly proportional to the change intemperature ∆T by ∆h = αh∆T , where α is the coefficient of thermal expansion of
the heated nanostructure (for nickel, α = 12.7 × 10−6 /K). In the experiments that
follow, we will excite temperature changes of ∼ 100 K, so for a h = 20 nm structure,
the expected peak displacement is ∼ 25 pm. The calculated change in signal for a 1
pm displacement is shown in Figure 5.14, and although the signal is not linear over a
few-nm scale, over a 25 pm scale, the change in the signal is very small, so the signal

74
can be assumed to be linear.
This brings up another point: the baseline diffraction signal in these experiments
shifts with a change in the initial temperature because of thermal expansion/shrinkageof the unperturbed grating. However, whatever the initial temperature, a thermal
excitation of ∼ 100 K can still be treated as a linear effect, as long as the baseline

Change in signal with Δh = 1pm

diffraction of the initial temperature is measured.
.0001

0

-.0001
16

20

24

28

Nanostructure height h (nm)32

Figure 5.14: Signal from changing line height by 1pm as a function of nanostructure line
height h from reflection (top, green) and diffraction (bottom, black). The total change
in signal is given by to sum of the absolute value (total separation) of the two curves.

5.5

Time-resolved EUV acquisition outlook
There is still significant room for improvement beyond what has already been

done in refining the optical, mechanical and acquisition components in the detectionof EUV probes. The 10× increase in acquisition rate is a useful step, but ideally, we
would do single-shot measurements. The laser amplifier runs at 2 kHz, so this would
require a detector with a sub-500 µs acquisition time, 40× faster than our current best.
Such shot-to-shot measurements are possible using fast CCD cameras (Roper Scientific)
using binning to run even faster or photodiode detection. Photodiodes are capable of> 1 GHz acquisition rates, so they can be used with a lock-in amplifier, which would

75
greatly increase the signal-to-noise for a given acquisition time. However, the practical
efficiency of photodiodes is much lower than CCD detectors and they provide much
much less spatial information than a CCD (a photodiode is a single-pixel detector),so we have used CCDs to this point. If enough flux could be obtained, it would be
useful to use a hybrid CCD/photodiode system: using the CCD for alignment and thephotodiode for sensitive signal detection, as shown in Figure 5.5
EUV transmission
grating

Normalization
photodiodeslit
photodiode

Sample for dynamic
measurements
slitphotodiode

CCD
Alignmentwith CCD

CCD
Data collection
with photodiodeFigure 5.15: Hybrid CCD-photodiode setup. The HHG and sample alignment can be
optimized by looking at the CCD (left), and then fast data collection can be conductedwith a photodiode and an adjustable slit for selecting a single diffracted order (right).
Also shown in this figure is a proposed technique for shot-to-shot signal normalization
using a weakly diffracting grating and a photodiode to detect the HHG intensity before
hitting the sample.

Despite our efforts to improve stability, the biggest source of noise in the systemis the harmonic generation process. Sending a small portion of the beam to a photodiode for shot-to-shot intensity normalization would significantly reduce the noise on
the signal. This is also shown in Figure 5.5. Another way to improve the signal is toimprove the total HHG flux, which could be done by using a shorter hollow capillary
or reducing the number of EUV filters or moving them to be before the sample (using
infrared pump light would eliminate the need for filters to be between the sample andthe CCD since the CCD is insensitive to infrared light).
A final source of noise that must be considered for high rep-rate CCDs with

76
finite exposure times is pulse-exposure offset, as illustrated in Figure 5.16. This effectgets stronger as the exposure time gets shorter, and the solution is to synchronize theexposures with he pulses by triggering the exposures at a fixed delay to the probe
pulses. However, triggering a CCD exposure usually adds considerable dead time to thetotal acquisition, so the advantages of short exposure times must be balanced by thisconsideration.
2

1

2

1

1

2

1 CCD exposure
Probe pulse

2

2

2

2

2

Synchronized

Figure 5.16: Illustration of the exposure time/probe pulse walk-off that can cause anirregular number of probe pulses to be recorded on each exposure, which becomes abigger problem as shorter exposures are taken. The solution is to trigger the CCD
exposure on the probe pulses, but this adds considerable dead time to the acquisition.

Chapter 6

High-Frequency Surface Acoustic Wave Propagation in
Nanostructures

The previous chapter discussed the application of EUV light from HHG in probing
longitudinal acoustic propagation in thin films, which can be used for measurements ofthin films. Longitudinal acoustic pulses inherently propagate into a surface, and the
back-reflections observed at the surface can be utilized as a diagnostic of thickness andproperties of thick films. Here, ”thick” means thicker than the width of the acoustic
wavepacket, set by the excitation depth. In laser ultrasonics, the excitation depth isdetermined by the optical penetration depth ζo of the pump laser and (for metals) the”hot” ballistic electron range ζe . For most metals (See Table 2.2), optical penetration
depths are on the order of 10 nm and ballistic electrons can carry this heat another10-150 nm before thermalizing, so this technique is useful only for films of thicknessh ≥ 100 nm. Thin film technology has advanced to the point that atomic-layer films
can be deposited, so much better depth sensitivity is required.
Additionally, interfacial sensitivity is very low for longitudinal pulses: their propagation is dominated by the film properties because their relative interaction with interfaces is so little. A precise understanding of interface properties is of great interest to anumber of engineering communities, because as thin film stacks are grown with thinner
and thinner layers, the interface properties become increasingly important. Diffusion
between layers is also a major concern, and the ability to perform quick, noncontact

78
testing of buried interfaces is important for device diagnostics and coating methodology
improvements. For these reasons, an optical probe of properties of interfaces beneathvery thin films is desired.

6.1

Surface acoustic wave propagation for thin film measurements
One technique that holds promise for studying interfaces beneath very thin films

is surface acoustic wave (SAW) spectroscopy. SAW modes are confined to the surface with a very shallow surface penetration, and thus their propagation is sensitive
to surface structure and layer composition. Also, because they propagate parallel tothe surface, they interact strongly with buried interfaces and defects [97], so they are
better-suited to detecting interfacial anomalies than longitudinal acoustic pulses. To
study films of sub-micron thickness, however, it is necessary to use SAWs with comparably short wavelengths, since the penetration depth ζSAW of the SAW is proportional
to the acoustic wavelength ΛSAW by [98]
ζSAW =

ΛSAW
.
2π

(6.1)

Thermally exciting the film using a transient grating geometry is a common technique for generating narrow-band surface acoustic waves [99–101]. The basic idea is
illustrated in Figure 6.1 – when two pulsed laser beams are crossed at a surface, thelaser intensitiy distribution at the surface forms an interference pattern given byI(x, t) = I0 1 + cos 4π sin(θ)x/λlaser exp − t2 /τ 2 exp − x2 /a2 ,(6.2)

where θ is half of the crossing angle and I0 , a, and tau denote the peak intensity, thebeam radius, and the pulse duration of the laser. If the driving laser is short comparedto the SAW propagation time (τ  ΛSAW /vSAW < 1 ps), this intensity distribution
impulsively excites the surface in a spatially-periodic way, which stresses the surface
and launches a narrow-band surface acoustic wave with a wavelength given by the79interference period,
ΛSAW = λlaser /(2 sin(θ)).

(6.3)

However, this method is limited to SAW wavelengths on the order of or higher than
optical wavelengths because the generated grating can be determined by using the crossing angle of the two pump beams in Equation 3.1. As a result, the shortest acoustic wavelength generated to-date using the transient grating method is approximately
750nm [102,103]. This presents a challenge for conventional optical methods of creating
and detecting SAWs, and would seem to limit this technique to film thicknesses h ≥ 100
nm, the same order as longitudinal methods.

θFigure 6.1: Schematic depiction of the transient grating scheme for generating surface
acoustic waves. Two pulsed laser beams (red in the figure) are crossed on a surface
at an angle θ, which sets up an interference pattern and spatially-periodic heating.
This generates a periodic stress that launches a SAW with wavelength set by ΛSAW =λlaser /(2 sin(θ)). The SAW dynamics can then be probed by observing the transient
diffraction from the acoustic ripple of a third, probe pulse (shown in blue).

An alternative approach that overcomes this limitation is to lithographically pattern a nanostructure on the surface. This pattern can then be optically excited, locallystressing a surface and thus exciting a SAW with a wavelength limited only by the resolution of the lithographic pattern [104, 105]. However, this approach raises the questionof how much the presence of the nanostructure will affect the propagation of the SAW.
This question is critical for analysis of the underlying sample characteristics. Past mea-

80
surements employing thin Al patterned absorbers suggested that these issues can beneglected [105–108]. However, these conclusions were based on data taken at a single
SAW wavelength.Very recent work has shown 2D SAW propagation in arrays of nanodisks and suggested that there may be dispersion from the nanostructure loading [109].
To date, no work has sampled the full dispersion curve of a patterned thin film in order
to come to a definitive conclusion on this issue.
The Maris group at Brown University made the first to experimental attempt
to generate high-frequency SAWs in a nanostructure in 1992. They used a λ = 596
nm, 3 ps laser pulse to excite and probe h = 40 nm high gold stripes on a fused silica
substrate. However, they only saw vibrational modes of the wires themselves and not
SAW propagation [110]. Since that time, work on similar structures has focused onwhether the observed signal is dominated by SAW propagation in the substrate or
normal mode resonance of individual wires [105, 107, 110].
All previous of surface acoustic wave dynamics in nanostructures used optical lightto probe dynamical acoustic processes in a nanostructure, so detection is also an issue.
Visible light will not diffract from a sub-wavelength structure, so current techniques
measure very small (< 1 part in 105 [110]) changes in surface reflectivity to observe
SAW propagation [105,107–109,111,112]. However, these reflectivity changes arise froma complex mix of thermal density changes and local and interfacial stresses and strains.
Direct transient grating diffraction from the surface would give more reliable, easierto-interpret, and higher signal-to-noise data — however, such an experiment requires
light with a wavelength shorter than the SAW wavelength. EUV light from HHG is an
ideal source to fill this need because the < 10 fs pulse duration provides excellent time
resolution and the short wavelength gives strong diffraction and increased sensitivity tosurface displacements.

81

6.2

EUV measurements of SAW dynamics
By measuring the change in diffraction as a function of pump-probe delay time,

we can observe dynamic surface acoustic wave propagation. Four scans with p = 3 µm,
1.5 µm, 300 nm, and 160 nm are shown in Figure 6.5. The relative amplitude of theoscillatory SAW component of the signal is 10× larger than in a similar experimentusing an optical probe [108]. For a given grating period p, we extract the SAW frequency
ν by performing a Fourier transform, as shown in the insets of Fig. 6.5.
800nm laser pump
To CCD

EUV probe

Nickel lines

Diffracted EUV
Surface acoustic wave
Sapphire substrate

Figure 6.2: Experimental setup for ultrafast measurements of SAW propagation with
EUV light. An 800nm laser pulse (red) heats the nickel nanostructure, which createsa periodic stress and excites a SAW. The propagation of this SAW is then observed byobserving dynamically changing diffraction of an ultrafast EUV probe pulse (blue) fromthe structure.

6.2.1

Are we really seeing surface acoustic wave propagation?

Past work on generating and detecting acoustic waves from nanostructures was
focused on whether the observed signals are due to SAW propagation or vibrations ofisolated structures, so a natural question is: Are we really seeing surface acoustic wave
propagation? To determine the relative contribution of these modes, we compared theresults from p = 4L samples to those from p = 2L samples, as shown in Figure 6.5.
A Fourier transform of the data reveals a narrow-band oscillation in all cases, and the

t=0

82

t=p/8vt=p/4vt=3p/8v

t=p/2vt=5p/8v

t=3p/4v
t=7p/8v

Figure 6.3: Illustration of SAW propagation: the sum of two equal-amplitude sinusoidal
waves traveling in opposite directions (blue traveling left and red right, arrows markpropagation of a specific crest) is a standing wave (black). The propagation and surface
displacement (black) are shown over one full cycle, with the times marked in the upperleft of each panel.

results of such an analysis are summarized in Table 6.1. In the case of the L = 800
nm structures, the SAW propagation in the 2L (p = 1.5µm) sample has a frequency
two times greater than the frequency in the 4L(p = 3µm) samples. The frequency ofoscillation is proportional to the nickel strip period rather than the line width. If weassume that the oscillation is due to SAW propagation, both the 4L and the 2L data setsyield a SAW velocity ∼ 6000 m/s, which is the expected Rayleigh velocity for sapphire.
We therefore conclude that the oscillation is due to SAW propagation in the substrate,with a SAW wavelength set by the strip period (ΛSAW = p).

A

L

p=4L

83

h=20 nm Nickel
Sapphire

B

p=2L
L

h=10 nm Nickel
Sapphire

Figure 6.4: Schematic showing two sample configurations for nickel nanolines of width
L on sapphire substrates: (A) h=20 nm and nanostructure periodicity p = 4L (25%
duty cycle), (B) h=10 nm and nanostructure periodicity p = 2L (50% duty cycle).

Figure 6.5: Dynamic EUV diffraction from surface acoustic waves on samples of sapphire with periodically spaced lines of nickel. Nickel line periods of p = 1.5 and 3 µm
with width of L = 800 nm (top) and p = 300 and 160 nm with L = 80 nm (bottom)
were heated and probed. Impulsive heating and rapid thermal expansion launch surface
acoustic waves whose oscillations are superimposed upon a nonoscillatory signal component that decays due to thermal diffusion. The SAW frequencies are shown in theinsets.

84
Period p
1.56 µm
3.12 µm
145 nm295 nmLinewidth L
800 nm
800 nm80 nm
80 nm

SAW frequency ν
4.01 ± .01 GHz
2.01 ± .01 GHz
40.5 ± .1 GHz
20.6 ± .1 GHz

SAW velocity vSAW = pν6280 ± 15 m/s
6270 ± 30 m/s
5880 ± 25 m/s
6077 ± 30 m/s

Table 6.1: Extracted surface acoustic wave frequency ν for samples shown in Figure 6.5.
The calculated velocity is dependent on the period p rather than the linewidth L,

6.2.2

Surface acoustic wave dispersion measurements

While the analysis for the large-linewidth values of L = 800 nm in the previous
section revealed a constant velocity, the small-linewidth velocities from L = 80 nm data
are different, both from the L = 800 nm values and for different periods. However, thechange is relatively small (∼ 7%), so the same mechanism of SAW propagation in thesubstrate is dominant, but it is being perturbed by a factor dependent on the periodicity(or acoustic wavelength ΛSAW ). The cause of this change in measured SAW velocity isa penetration depth shift-induced dispersion: as the SAW periodicity is decreased, thepenetration depth is equally decreased (recall ζ = ΛSAW /2π), so the SAW propagation isincreasingly confined in the nickel nanostructure. Therefore at small periodicities on theorder of hη, where eta is the nanostructure duty cycle eta = L/p, the SAW propagation
is no longer simply determined by substrate properties – the effect of the nickel must
be taken into account. This effect is understood and has been clearly experimentallyobserved in thin films [103], but had not been observed in nanostructures.
We repeat the measurement of SAW frequency on structures with intermediate
linewidths to determine the dispersion of the SAW propagation in the Ni/sapphire
structure. The resulting dispersion in SAW frequency is shown in Figure 6.6 plottedagainst the SAW wavevector k = 2π/ΛSAW = 2π/p up to k = .05, corresponding toa periodicity of 125 nm. This period excites a SAW wavelength of 125 nm, where wemeasured a SAW frequency of ν = 47 GHz, which is the shortest wavelength and highestfrequency SAW observed (and even time-resolved!) by optical means. For long acoustic

85
wavelengths (large p, small k), the penetration depth is long, so the nanostructure canbe ignored and the SAW velocity is equivalent to the Rayleigh velocity in the sapphire
substrate, as shown in Table 6.1. Numerically, this means that measured frequency
is in good agreement with ν = vSa /p, shown as the dashed line in Figure 6.6. For
shorter acoustic wavelengths (small p, large k), the penetration depth (see top x-axis
in Figure 6.6) is decreased and the SAW is increasingly localized in the nickel. The
Rayleigh velocity in nickel is approximately half the velocity in sapphire, so confined
SAW propagation in the nickel slows the SAW propagation, so the frequency deviatesfrom the sapphire Rayleigh prediction.
50

1000

100

Penetration depth = p/2 (nm)50

33

SAW velocity (m/s)/2 : SAW frequency (GHz)

40

lo
ve
h
ig

yle

30

Ra

y
cit

25

in

s

ph
ap

ire

20

ate

tr
bs
u
s

Rayleigh velocity of substrate

20

Data from Sample B:
h=10nm, p=2L

6300
6200
10
6100

Data from Sample A:
h=20nm,Modiﬁed p=4L
thin-ﬁlm

6000
5900

5800
0
00

dispersion calculation

0.01
0.05

0.02
0.1

0.03
0.15

h k k = 2 /ph L(1/nm)
/ p2

0.04
0.2

0.05
0.25

Figure 6.6: Measured surface acoustic wave frequency for propagation in two sets ofsamples with nanopatterned nickel on sapphire. As the periodicity p decreases withincreasing wavevector (k = 2π/p), the penetration depth ζ decreases and the SAW is
increasingly confined in the nickel nanostructure and deviates from substrate-dominatedpropagation (dashed line). Error bars are plotted for each point, but are only visible
for the largest values of k.

86This observation of surface acoustic wave dispersion in a nanostructure is in contrast to previous studies of Al/quartz [107] and Al/Si [108] structures at isolated acoustic
wavelengths around ΛSAW ∼200 nm, which didn’t observe any deviation from substratedominated propagation and suggested that the SAW dispersion is minimal. Since they
were dealing with only one or two data points with large error bars in properties of thenanostructure, substrate, and interfaces, it is easy to see how they came to this conclusion, but a comprehensive wavelength-dependence study such as this one is necessary
to observe the SAW dispersion.
This effect is seen even more clearly when the data are converted from the measured frequency to SAW velocity by vSAW = νp. The results of this conversion, presented in Figure 6.7, show that the velocity dispersion introduced by the nickel nanostructures is nearly linear over this range of parameters. We present the data in termsof hkη = 2πhL/p2 because in terms of this dimensionless parameter, data from the two
samples should yield the same dispersive behavior. This is the case except for a small
offset between the data from the two different samples, which we attribute to a slightdifference in the actual duty cycles or line heights.

6.3

SAW dispersion calculation: Modified thin film theory
A quantitative calculation of SAW propagation is complicated by the modified

Brillouin zone, stopbands, and multi-mode propagation introduced by the nanostructure[113]. However, to first approximation, an effective mass model from thin film theory
can be used. We will discuss two models here: Auld’s model based on a perturbative
expansion of the stress at the surface [114], and Nelson’s model based on the propertiesof the materials themselves [102, 115]. In order to connect dispersion in a thin film todispersion in a nanostructure, the thin film theory in each case is slightly modified andcompared with the measured dispersion data.

87
Rayleigh velocity of sapphire substrate

6300

SAW velocity (m/s)

6200

Data from Sample B: h=10nm, p=2L
6100

6000

Data from Sample A: h=20nm, p=4L

5900

5800
0

0.05

0.1

0.15

0.2

0.25

h k η = 2π h L/p2

Figure 6.7: Measured surface acoustic wave velocity for propagation in two sets ofsamples with nanopatterned Ni on sapphire. Significant deviation from the Rayleighvelocity in sapphire is observed for the smaller grating periods, as the SAW propagation
is increasingly confined in the nickel nanostructure (vN i ∼ 3000 m/s). The measured
velocity dispersion agrees with an effective mass-loading calculation.

6.3.1

SAW dispersion due to a thin film: Perturbative Auld model

For very thin films or long acoustic wavelengths, the surface acoustic wave dispersion can be calculated by adding a small perturbation to the substrate value of the SAW
velocity [116]. The velocity shift can be determined by calculating the first-order effectof the film on the particle velocity components at the surface. Assuming an isotropic
substrate and film, the normalized particle velocity components are [114]ss(νRy )y=0
fy(νRz )y=0
fz
=
ω and
=
ω,
2
1/2
1/2
ρ s vsρs vs2
P
P
R

R

(6.4)

88
vR,Sa
6312

vt,Sa
6639

v`,Sa
11250

vt,N i
3150

v`,N i
6040

ρSa
3900

ρN i
8900

µN i
76

λN i
130

Table 6.2: Parameter values used in model calculations of the acoustic dispersion fromthe nickel nanostructures. Units for velocity v are m/s, densities ρ are given in kg/m3 ,and shear modulus µ and elastic modulus λ are in GPa.

where
PR = Power flow per unit width along x
3/2
 2
4χ2 1 − (vR /vs )2
vs
fy =
vR
3χ − 2χ(vR /vs )2 − 1
1
fz = fyχ
1 − (vR /vs )2 (vs /v` )2χ2 =.
1 − (vR /vs )2

These parameters can then be used in an expression for the frequency-dependent SAW
dispersion:


 
!
4µf λf + µf
|νRy |2
∆vR
∆βR
vR h
|νRz |2=−=−ρf+ ρf − 2
vR
βR
4
PR
PR
vR λf + 2µf. (6.5)y=0

Substituting ω = vR k = 2πvR /ΛSAW and inserting values for material properties
recorded in Table 6.2, this reduces to a simple expression for the SAW dispersion ina thin film.
One final step is necessary because the case of a nanostructure is quite different
from a thin film from even a simple mass loading point of view. Datta and Hunsinger extended thin film theory to nanostructures by introducing a duty cycle factor η (η = L/p)

vAB
A

vBC
B

C

Figure 6.8: Geometry for a simple net velocity calculation. The total velocity from A
to C is not given by a weighted sum of the velocites, but must be determined using
Equation 6.6

89
by which the dispersion effect is multiplied [117]. This has been used in all work using amodified thin film theory for dispersion in a nanostructure [105, 107–109, 111, 112], and
it is a very intuitive approach since we expect the dispersion to be dependent on theamount of a whole film the nanostructure covers. However, this approach fails to properly explain velocity dispersion: by saying that the nanostructure-induced dispersion is
proportional to the area of coverage, the model is saying that velocities in series can be
added. We can think of each period of the nanostructure as having two regions withdifferent velocites: a perturbed velocity under the nanostructure, and pure substratevelocity in the gaps, as shown in Figure 6.8. The net surface acoustic velocity vAC isdetermined by dividing the total distance travelled from A to C by the time it takes togo from A to B and B to C:vAC =dAB + dBC
dAC
dAB vAB + dBC vBC=
6=.
tAB + tBCdAB /vAB + dBC /vBC
dAC

(6.6)

The expression on the right in Equation 6.6 is the result predicted by Datta and Hunsinger, and it is wrong for all values of η except for zero (nothing on the surface at all)and one (complete film). The dispersion curves using the correct determination of net
velocity in the nanostructure from Equation 6.6 are shown in Figure 6.9 along with thelinear dispersion predicted by the simple Datta and Hunsinger correction. The correct
velocity expressions predict a much stronger dispersion effect that increases with decreasing duty cycle η, as the error of the simple velocity prediction increases. However,this is the opposite effect from the actual, nonlinear dispersion we expect to see — as
we will see from the calculation in the next section that goes beyond the perturbation
method. The flawed logic of the Datta and Hunsinger method that underpredicts thedispersion is compensated by the overprediction of the perturbation method. Both effects are only seen at values of hkη > .25, putting them outside the reach of current
experiments.
In making comparisons to our measured data, we use the (admittedly flawed)

90
6400

Datta's L/p correction
η=.5 with correct velocity expression
η=.25 with correct velocity expression
Expected dispersion

SAW velocity (m/s)

6200
6000
5800
5600
5400
5200
5000
4800
0

0.2

0.4

h k η = 2π h L / p

0.6

0.8

Figure 6.9: Surface acoustic wave velocity dispersion calculated using Auld’s perturbation method and (green) a simple duty-cycle scaling and (red and black) geometricallyhonest velocity mapping for conversion from a thin film to a nanostructured case. The
expected dispersion is shown as the dashed green line; even though the simple method
is incorrect, it is closer to the expected results at large values of hkη, and is used here.

Datta and Hunsinger method for converting a thin film dispersion to the case of ananostructure by a simple duty cycle factor η. Significantly for this experiment, thelinear dependence on hη means that it predicts identical dispersion for our two samples
since the ratio hkη is constant between (10 nm×.5 k) and (20 nm×.25 k). In thesenondimensional units, the dispersive change in velocity is given by ∆vR = −1887(hkη),so the total velocity isvR = vR,Sa − 1887(hkη),(6.7)

and this linear dispersion is shown as the solid green line in Figures 6.9 and 6.11.

6.3.2

SAW dispersion due to a thin film: Nelson model

In reference [115], Duggal et. al laid out the framework for a rigorous calculationof the acoustic mode dispersion introduced by a thin film overlay on a substrate. They
were concerned with the transient grating excitation geometry, which they referred to

91as ”ISTS” (Impulsive stimulated thermal scattering). While their detailed calculationsof displacements and stresses were geometry-specific, the mode dispersion analysis was
general: true regardless of the excitation scheme. This allows us to apply their conclusions to our nanostructure excitation geometry. In his thesis, Tobey goes through asimilar analysis without assuming a particular excitation geometry and arrives at thesame equations [55].










C=









(1 + s2 )e−nkh

2ise−skh(1 + s2 )enkh−2iseskh2ine−nkh−2inenkh−(1 + s2 )e−skh −(1 + s2 )eskh

1 + s2

1 + s2

2is

−2is

2in

−2in

−(1 + s2 )−(1 + s2 )

−1

−1

−is

is

in

−in

−1

−1

0

0





0
0



−(1 + r2 )g
2irg



2
2img
(1 + r )g 




1
−ir


im
1

with the following definitions:
n=

q

2 ),
1 − (v 2 /vLfq
1 − (v 2 /vT2 f ),q
2 ),m = 1 − (v 2 /vLsqr = 1 − (v 2 /vT2 s ),s=g = ρs vT2 s /ρf vT2 f ,where vLs and vLf are the longitudinal acoustic velocities in the substrate and film, andvT s and vT f are the transverse acoustic velocities in the substrate and film (values used
for a nickel film and a sapphire substrate are given in Table 6.2). The dispersed mode
velocity v for given wavenumber k can be determined by setting the determinant of thematrix C to zero:det(C) = 0.

(6.8)

92
The complete dispersion curve can be calculated by repeatedly solving Eqn. 6.8 overthe desired range of k values. For a given value of k, there can be many solutions,and each corresponds to a SAW mode. Numerical solutions are dangerous because they
may fall into false solutions or switch between nodes without notifying the user. A safer
method that shows all of the modes is to graphically solve the determinant: plot thevalue of det(C) as a function of v and look for zeros. This method is demonstrated in
Figure 6.10 for a 10 nm nickel film on sapphire with k=0.025/nm, so ΛSAW = 250 nm.
The zero-crossings give the solutions to Equation 6.8, and so are viable SAW modes:
the mode at ∼5680 m/s is the Rayleigh mode altered by the presence of the nickel film,

Det(C) with k=.025/nm

while the faster mode ∼6030 m/s is a higher-order (Sezawa) mode [115].

20

10

0
5000

5500

6000

Velocity (m/s)

Figure 6.10: Plot of det(C) for determining SAW modes: locations where the curve
touches zero are valid SAW modes by Equation 6.8. Parameters used are k=0.025/nm(ΛSAW = 250 nm), h=10 nm thick nickel film on sapphire.

Modifying the results of this calculation to account for the nanostructure is notas simple as in the previous model because this model gives the predicted velocity
rather than the change in the velocity. This can be dealt with by subtracting theRayleigh velocity for sapphire (vSAW for k = 0), then applying the correction factor
η, and finally adding back the subtracted baseline velocity. Numerically, this means

93
that vSAW,f ilm (k) obtained from the model for thin film dispersion can be converted tovSAW,nanostructure (k) for the velocity dispersion due to the nanostructure byvSAW,nanostructure (k) = vSAW,f ilm (k) − vSAW (k = 0) η + vSAW (k = 0).

6.3.3

(6.9)

Comparing models to the data

The results of each model are plotted in Figure 6.11 along with the measured
velocity dispersion data, and the match to the data seems to be quite good overall. The
rms error values for the Auld perturbative model and the Nelson model for h = 20 nm
are very similar, indicating that in this range of hkη, either model can be used. However,at larger values of hkη (further along the x-axis, as shown in the inset to Figure 6.11),
the models strongly diverge:• The Nelson model predicts a dispersion limit determined by the velocities ofthe substrate and the nanostructure and the duty cycle η, which is reasonablebecause the dispersion of the nickel is only in effect where there is nickel, andnot in the gaps:
vdispersive limit =vN i vSa,
η vSa + (1 − η)vN i(6.10)

where each velocity is the surface acoustic wave velocity. This leads to dramatically different dispersive limits for the η = 25% and η = 50% samples.
• The linear dependence on hkη in the Auld calculation is insensitive to velocity
limits, even predicting a SAW velocity lower than vSAW,N i when hkη > 1.65.
The model works well for values of hkη up to 0.25 studied here, but for larger
values or smaller duty cycles (where the minimum velocity limit is higher), themodel gives nonphysical predictions.

94

6.4

Conclusion
We used ultrafast, coherent, EUV beams to measure high frequency surface acous-

tic wave propagation in nanostructures at frequencies up to ∼50 GHz, the highest observed using optical detection. Such a high frequency corresponds to SAWs with apenetration depth ∼20 nm, 6 × lower than is possible using the state-of-the-art in optical transient grating methods. The shorter penetration depth realized in our techniqueallows new high-selectivity studies of the properties of interfaces beneath thin films with
10-100 nm thickness.
By repeating the measurements on nanostructures with different periods, we measured the SAW dispersion in the nanostructured system and confirmed the validity of
two models modified from their original use in thin film theory. These dispersion measurements are a critical step in the advancement of this technique: by understanding theinfluence of the nanostructure on the SAW propagation, we can isolate it in and remove
it from future measurements in order to precisely measure the properties of layers andinterfaces below the nanostructure. This technique of visible laser excitation and EUV
probing can easily be extended to even shorter wavelengths (higher frequencies), limited
only by nanopatterning capabilities. Thus, it holds promise for studying materials andinterface properties of extremely thin films.

95ARayleigh velocity of sapphire

6300

SAW velocity (m/s)

6200

6100

6000

Auld calculations

Nelson calculation:
10 nm, p=2L

5900

Nelson calculation:
20 nm, p=4L

5800
0.0

0.05

0.1

0.15

h k = 2 h L / p2

0.2

0.25

vSa= 6300 m/s

B

6000

SAW velocity (m/s)

5500

5000

vlimit, =.25 = 4840 m/s

Nelson calculation:
20 nm, p=4L
Nelson calculation:
10 nm, p=2L

4500

vlimit, =.5 = 3930 m/s

4000

Auld calculations

3500

vNi = 3150 m/s
3000

0

0.5

1

h k = 2 L/p2

1.5

2

Figure 6.11: Measured veloctiy dispersion along with predictions from two models of
SAW dispersion: a perturbation model discussed in section 6.3.1 that was proposed by
Auld [114], and a detailed stress/strain model discussed in section 6.3.2 proposed by
Nelson [115]. (A) shows that Auld’s perturbation method fits the data better over therange of hkη measured in this experiment. (B) shows the calculated dispersion overa larger range of hkη and demonstrates the expected breakdown of the perturbative(Auld) model for hkη > .5. The blue box indicates the range in viewed in part (A).

Chapter 7

Quasi-ballistic heat transport from nanoscale interfaces

As nanofabrication techniques continue to improve and smaller structures can be
manufactured for finer control of electronic, optical, mechanical and thermal properties,
there is an ever-greater need to understand the fundamentals of heat transfer in suchnanostructures. Of particular importance is the rapid progress in the microelectronics
industry, where transistors with 45 nm gate widths are now in production and 32 nm
gates are right around the corner [118]. A fundamental and quantitatively-accurate
understanding of how heat flows away from a nanoscale heat source – the nanoscale”heat sink” problem – is critical for future advances in nanotechnology because thethermal energy generated in the operation of these devices must be dissipated.
For time scales shorter than the energy relaxation time and length scales smaller

Figure 7.1: Schematic thermal images illustrating the difference between diffusive andquasi-ballistic transport for heat dissipation from a nanostructure into a substrate.

97
than the mean free path of energy carriers (for example, the phonon mean free path Λ is
∼ 250 nm in silicon), heat flow can deviate significantly from the Fourier conduction law
q = −k∇T , the governing equation of diffusive thermal transport that predicts smoothtemperature gradients [19, 119]. When the device size is much larger than the phononmean free path Λ, heat flow is diffusive and at length scales significantly smaller than Λ,
the transport away from the interface is ballistic (Figure 7.1). When the length scale ofa device is on the same order as Λ, we will describe the heat transfer as quasi-ballistic.
In this chapter, we discuss the theoretical and experimental background of thistransition from diffusive to ballistic thermal transport within nanostructures. Then we
consider the important case of transport away from a confined heat source defined bya nanostructure coupled to a bulk substrate. This includes background, basic theory,and the first experimental observation and quantitative, time-resolved measurement ofquasi-ballistic thermal transport away from a nanostructure [120].

7.1

Ballistic thermal transfer in and through a nanostructure
Before discussing the distinctives of heat transport away from a heated nanos-

tructure hot spot, we will first consider the better-understood case of heat transport
in or through a nanostructure. The different transport regimes in a nanostructure areillustrated in Figure 7.2, which schematically depicts temperature distributions in a
T2

T2
h

T1

DiffusiveL >> Λ

T2
hT1

Quasi-ballistic
L~Λ

h
T1

Ballistic
L << Λ

Figure 7.2: Schematic illustration of steady-state temperature distributions in the diffusive, quasi-ballistic, and ballistic transport regimes in a 1-dimensional slab of thickness
h. Modified from [121].

98
1-dimensional slab in the diffusive, quasi-ballistic, and ballistic cases. The thermal conductivity will be affected by the nanostructure, as has been demonstrated by both basic
and advanced theory, numerical simulation, and experiment.

7.1.1

The Boltzmann transport equation for transport in a nanostructure

When the length scale of a system is much larger than the mean free path of theenergy carriers, the transport can be well-described by the Fourier law derived in section
2.3 and the transport is diffusive, i.e. dominated by multiple scattering events between
energy carriers and characterized by local thermal equilibria existing throughout thesystem. In the opposite limit, when the length scale is much smaller than the mean freepath, the energy flux can be described by the Stefan-Boltzmann law q = σT 4 where σ is
the Stefan-Boltzmann constant defined in Equation 7.11, and the transport is radiative,
i.e. dominated by the interfaces since carriers travel directly from interface to interface
without scattering. In the transition region where the length scale of the system is on
the order of the mean free path, a determination of the flux requires the solution of thetime-dependent Boltzmann transport equation (BTE):∂f~ r f + F~ · ∇~ p f = ∂f
+ ~v · ∇+ s,∂t∂t collision(7.1)

where f is the time and space-dependent carrier distribution, p~ is the momentum coordinate, ~r is the position coordinate, F~ is an applied force, and s is a source term. The
collision term is given by the matrix element
hf (p~0 )|S(p~0 , p~)|f (~
p)i,

(7.2)

where S(p~0 , p~) is the scattering probability. The collision term can be converted to aset of integrals over each momentum dimension. So all together, the BTE is a sevendimensional integro-differential equation. Approximations can be applied to reduce thecomplexity of the BTE, but a full analytical solution is impractical for physical systems.

99
Numerous numerical methods of determining energy transport in nanostructured
systems have been developed [122], including Monte Carlo [123] and molecular dynamics simulations [124], and a numerical solution of the BTE using a discretize lattice [125–127]. Chen proposed a simplification in which the BTE is split into ballistic anddiffusive terms (called “ballistic diffusive equations”, or BDE) describing the thermaltransport [128, 129]. An alternative simplification (called the “equation of phonon radiative transfer”, or EPRT) was proposed by Majumdar [121,130], which yields slightly
different results than the BDE [131].

7.1.2

Basic theory of quasi-ballistic thermal transfer in a nanostructure

By considering a basic model for thermal conductivity based on the Fourier heat
conduction law (Equation 2.8) derived in Chapter 2, we can gain insight into a few basic
questions regarding ballistic phonon transport effects in nanostructures:• Why do we expect to see ballistic thermal thermal transport in a nanostructure?
• At what length scale do we expect to see ballistic effects?
• How will ballistic transport affect the thermal conductivity?
In the Fourier law for diffusive transport, the constant of proportionality connecting the temperature gradient to the flux is the conductivity k = CvΛ/3 (Equation 2.9).
This expression uses the Debye approximation, which assumes that all of the phonons
can be lumped together with a single velocity (given by the k → 0 limit of the acoustic
velocity) and specific heat. The thermal conductivity of a bulk material can then be
determined if the average phonon mean free path is known, or, as is more commonly thecase, a mean free path can be determined from a known thermal conductivity. This was
the standard method for determining the phonon mean free path until the recent observation that the phonon dispersion can have a significant effect on the results [132]. The

100
example of sapphire at room temperature will demonstrate just how bad this approximation is for real materials. Using textbook parameters CSa = .7788 J/gK, ρSa = 4
g/cm3 , vl,Sa = 11250 m/s, and kSa = 46 W/mK, the mean free path from k = CvΛ/3 isΛSa = 3.9 nm. Because this model assumes that all phonons have the acoustic velocity,which is the highest phonon velocity in the material, the mean free path of the phonons
that actually carry heat is significantly under-estimated. Numerical calculations taking the phonon dispersion into account predict ΛSa ∼ 100 − 150 nm for heat-carrying(acoustic) phonons.
In a crystalline material, the phonon mean free path is dominated by different
effects depending on the temperature, with a typical temperature dependence shown in
Figure 7.3. For temperatures within an order of magnitude of the Debye temperature(TDebye,Sa ∼ 900 K), the phonon mean free path is dominated by phonon-phonon scattering outside of the Brillouin zone. This is known as Umklapp scattering and gives
rise to a phonon with opposite momentum [5]. Umklapp scattering increases as thephonon temperature increases (nearer TDebye ) because higher-energy phonons are thermally excited, making outside-the-Brillouin-zone scattering more likely, so the mean
free path decreases in this temperature region (Λ ∝ 1/T ). As the temperature is decreased, phonons travel much further between scattering events and begin to scatteroff of defects and impurities in the sample. If the temperature is further decreased andapproaches absolute zero, the phonon mean free path is determined by the dimensions of
the material: phonons travel ballistically from interface to interface without scattering
in-between. The mean free path is limited by the interface scattering, so this sample
dimension sets an absolute limit: Λ < L, where L is the structure size.
Combining this information about the phonon mean free path with the fact thatthe thermal conductivity is directly proportional to Λ, it is then easy to see how thethermal conductivity is determined by the nanostructure size. As the nanostructure
dimension is decreased below the phonon mean free path, the thermal conductivity

101

Boundary
scattering

Defectscattering

Phonon-phonon(Umklapp) scattering

Figure 7.3: Temperature dependence of the phonon mean free path Λ for sapphire(from k and C and the Debye model). Λ is determined by Umklapp phonon-phonon
scattering at high temperature (orange), defect and impurity scattering at somewhat
lower temperature (blue), and sample size near zero temperature (pink). The mean freepath is shown here for a 10 µm-thick slab of sapphire.

should decrease in direct proportion to the dimension of the structure. This is illustrated
in Figure 7.4, where the low-temperature thermal conductivity is decreased by twoorders of magnitude when the sample size is decreased from 10 µm to 100 nm. The
phonon mean free path is constant at low temperature, so the thermal conductivity is
directly proportional to the specific heat C in the ballistic case. Also, the temperature atwhich the thermal conductivity peaks is shifted higher for a smaller structure – compare
dark and light green curves on Figure 7.4).

7.1.3

Experimental observations of ballistic transport in nanostructures

The transition from diffusive to ballistic transport regimes has been studied aggressively for the past ten years, with numerous experimental observations of nonFourier heat transport across and through nanostructures in different geometries. Earlyexperiments observed ballistic thermal transport in a thin film (see Figure 7.5) and aseries of thin films (superlattice) [19, 119, 133–135]. Recent experiments have demonstrated reduced thermal conductivity of nanowires [136,137], nanocomposites [138,139],

102
10

10 µm layer

k=

CvΛ
3

1

100 nm layer

0.1

k reduced
0.01

BufferCalc(0*8)
k, Sapphire (W/cmK)C, Sapphire (J/gK)
Mean free path (µm)
Mean free path (µm)

0.001

10

100

Guesses marked by

1000

Temperature (K)Figure 7.4: Thermal conductivity k, specific heat C, and phonon mean free path Λof sapphire, related by k = CvΛ/3. As the sapphire thickness is reduced, Λ is also
decreased, which causes a reduction in the thermal conductivity.

molecular chains [140] and multi-walled carbon nanotubes [?, 141]. Theoretical work
has also predicted quantized thermal transport in single-walled carbon nanotubes [142].
Current experiments are driven in large part by potential application to thermoelectric energy conversion (TEC) [145–147]. For TEC devices to function efficiently, atemperature difference must be maintained while a current is generated, so a material
with high electrical conductivity and low thermal conductivity is required. Introducing
multiple interfaces to reduce the thermal conductivity is one route that shows promise,
although interfaces inhibiting thermal flow also affect electrical transport, so engineered
geometries and surfaces are required [139]. Silicon nanowires are particularly interesting
because they can be made so narrow as to force the phonon transport beyond ballistic into a quantum regime (restricted not by phonon mean free path, but by phonon
wavelength), while retaining a relatively high electrical conductivity [136, 137, 148].

103

Figure 7.5: Thermal conductivity of silicon as a function of film thickness or wire
diameter. Figure from [19], and data in the figure are from [134, 136, 143, 144].

7.2

Quasi-ballistic thermal transport at an interface
As described in the previous section, a number of past experiments have studied

non-diffusive thermal transport in nanostructured systems. However, despite its direct
relevance to the practical issue of thermal management in nanoelectronics [149], nanoenabled energy systems [10, 11], nanomanufacturing [150], and nanomedicine [13], thenanoscale heat sink problem (i.e. heat transfer from a nanoscale heat source into its
bulk surroundings) has received much less attention. Here, we are concerned not withenergy flow within a nanostructure, but the efficient coupling of thermal energy out of
the structure and into the substrate. When heat flows from a hotspot with a dimension
L that is smaller than the mean free path of phonons in the heat sink Λ, energycarrying phonons travel ballistically away from the source for a significant distance
before experiencing collisions. In the Fourier picture, heat flow is assumed to be diffusive,with the thermal flux q determined by the temperature gradient (q = −k∇T , where k isthermal conductivity). When there is a significant energy density change over a distance104
less than the phonon mean free path, as is the case for heat flow away from a small
source, there is no local phonon equilibrium near the source and therefore temperaturecannot be defined, as illustrated in Figure 7.1. Under these circumstances, the Fourierlaw, which assumes sufficient scattering events to obtain local thermal equilibrium,
overestimates the thermal transport [151–153].
Numerical simulations using the BDE [154], EPRT [155], and discrete BTE methods [156] have demonstrated the error in the Fourier law for dissipation of thermal energy
from a heat source smaller than the phonon mean free path in the heat-sinking material.
However, there is no agreement as the the strength or exact dependence of this effecton temperature and structure size.

7.2.1

Thermal boundary resistance across a bulk-constricted interface

In practice, there often exists an interface between a small heated region andthe heat-sink substrate, and this interface thermal resistance must also be taken into
account. Measurements of the interface thermal resistance can be performed for structures at bulk length scales and in thin films using time-resolved optical pump-probe
methods such as transient thermoreflectance (TTR) measurements [133, 157]. The interface resistance between two materials in contact is called thermal boundary resistance(TBR).
There is some confusion in the literature regarding how to describe ballistic thermal transport away from an interface. It is important to distinguish this ballistic interface effect from the thermal boundary resistance and spreading resistance that areused to characterize interfaces in the bulk approximation, valid when the interface contact size in much larger than the phonon mean free path of the heat sink. In thatbulk/Fourier case, the total effective resistance is given by a sum of these two and theFourier resistance of the substrate slab:
RInterf ace = RT BR + RSpreading + RF ourier .

(7.3)

105
As described in section 2.4.1, RT BR describes the interfacial phonon resistance [22].
It has been (misleadingly) called the ballistic component of the boundary resistance.
However, it has no dependence on the phonon mean free path and instead scales with
the Biot number [158] (see Equation 7.16), which measures the prevalence of surface vsbulk resistance.
Spreading resistance is also a bulk concept, which accounts for thermal resistancedue to heat diffusion into the substrate (along with RF ourier from section 2.4.2 that
accounts for diffusion in a slab). It is a means to express the geometrical thermal
spreading in the substrate in terms of a single resistance that can be applied in series
with other resistivities to heat flow, as discussed in section 2.4.3. It is commonly used inthermal engineering as a way to estimate the resistance to thermal flows after passing
through a constriction, e.g. heat flow from a transistor, through a film, and into asubstrate. It has been analytically calculated for a number of different geometries [159],
and for the infinite strip we are concerned with here, it is related to the ratio  of thecontact size L to the structure repeat distance by [160]∞

1 X sin2 (nπ)Rspreading = 3 2.
π  ks
n3

(7.4)

n=1

In the case of our samples with nickel nanostructures of width L and fixed periodicity
4L on sapphire,  = .25 and Rspreading = .00202 Km/W, which is in units of resistanceper length since it assumes an infinite contact in one dimension. Spreading resistancewill be neglected in our final analysis because we calculate the thermal diffusion in
the substrate directly from the heat equation. However, it is important to distinguishbetween diffusive spreading effects and ballistic effects, and the concept of spreading
resistance does motivate the assumption we will make later: the ballistic effect due totransport surrounding a nanoscale heat source can be accounted for by including an
additional resistance to energy flow at the boundary between the heat source and its
surroundings (heat sink).

106
Some models claim to show that the boundary resistance goes up as the structure
size is decreased, but in these models [158], the effect is due only to geometric scalingof the system (for example, from Equation 7.4, lim→0 Rspreading = ∞) — similar to
the electrical resistance of a wire increasing with decreasing diameter. This is obviously
not a truly ballistic effect because it has no dependence on the phonon mean free pathand further, it can be predicted by the Fourier law! In order to avoid this confusion,
we discuss non-diffusive heat transport in terms of the intensive parameter resistivityr, which is independent of linewidth in the bulk approximation for a single line (thecase of  → 0). In the case studied here where  is constant for all nanostructured
samples, Rspreading is a constant and so rspreading is proportional to L, as is rF ourier .
The ballistic effect would then be manifest as an additional component rBC in theboundary resistivity, so the total resistivity lumped into an interface effect is given by
rInterf ace = rT BR + rSpreading + rF ourier + rBC .

7.2.2

(7.5)

Simple analytical prediction: ballistic-diffusive model

Gang Chen made the first prediction of this ballistic effect near a nanoscale heatsource in 1996 [161]. He considered the heat transfer between concentric spheres (withthe outer sphere having an infinite radius) in the limits of purely radiative/ballistic andpurely diffusive/bulk transport. He assumed that the component predicting a lower
heat flux would constrain the transport to that value. Chen found that when theheated inner sphere was larger than the phonon mean free path of the material between
the spheres, many scattering events occured and thermal diffusion was the dominanttransport mechanism. However, when the inner sphere was small compared with Λ,
scattering events were greatly reduced and the heat transport could be considered to
be radiative, so the ballistic component dominated.
The net heat flux is easier thought of in terms of resistivity (as discussed above,
where r = ∆T /q), where the total resistivity is simply the sum of the Fourier and107

M >> L

L/2

T1

Λ
2L

T2

Figure 7.6: Geometry for simple analytic model to demonstrate the transition fromdiffusive to ballistic thermal transport away from a nanoscale heated strip of width
L. For simplicity in cylindrical coordinates, the nickel strip is modeled as a cylinderof diameter L/2, and the outer bounding cylinder has a diameter of 2L, 4× greaterthan the inner diameter as in the experiment. The transport is determined in both theballistic and diffusive regimes.

ballistic components, which will automatically provide the correct limits to the heat
flux. In this case, the Fourier law under-predicts the resistivity at length scales shorter
than the mean free path, and ballistic transport over-predicts the resistivity in the bulk.
Of course, our system of nanoscale lines on a substrate is not obviously connectedto the worked example of concentric spheres. Our system does have something close
to cylindrical geometry, so we modified the calculation to get results directly related
to our situation. The simplified geometry used in this calculation is summarized in
Figure 7.6, where the line contact of width L is treated as an embedded half-cylinder ofdiameter L. This is the heated surface, which is analogous to the inner sphere in Chen’scalculation. The outer surface is a half-cylinder in the substrate with a diameter of
4L, which is determined by the line separation. The entire sample can be considered bylining up a series of these half-cylinder shells edge-to-edge, and we will assume that they
are non-interacting so that the heat transport of the entire sample can be understoodby considering a single unit half-cylinder. Of course, neighboring half-cylinder units

108
do interact, but periodic boundary conditions are equivalent to insulating boundary
conditions.
We will first consider the diffusive transport between the half-cylinders. In the~ .
Fourier equation, the heat flux is determined by the temperature gradient: ~q = −k ∇TIn this cylindrical geometry, the gradient operator is~ = ∂ r̂ + 1 ∂ θ̂ + ∂ ẑ,
∇
∂r
r ∂θ
∂z

(7.6)

where θ is the angle about the axis of the cylinder and z is along the direction of M , thelength of the nano-lines. We assume that the heat will diffuse radially from the innercylinder and the length of these lines is much greater than the diameter (M >> L), so
we can neglect the θ̂ and ẑ components of the derivative. The Fourier expression thenreduces to ~q = −kdT /drr̂. The heat flux q is given by the total heat Q divided by thearea over which the heat flows, which is A = πrM for a half-cylinder. Plugging this into the radial Fourier law and regrouping terms, we obtainQ

dr= −πM kdT.
r

(7.7)

Integrating both sides from r1 = L/2 to r2 = 2L, we get an expression for the total heatflow between the half-cylindersQ=

2πM k(T1 − T2 ).
log(r2 /r1 )(7.8)

Now we can plug in the high-temperature Debye law approximation, k = CvΛ/3 anddivide the area back out to get the flux between the half-cylinders
2CvΛ(T1 − T2 ).
3 log(4)L/2(7.9)

3 log(4)L∆T=.
qF ourier
4CvΛ

(7.10)

qF ourier =Finally, we convert to resistivity
rF ourier =109
In calculating the ballistic thermal transport, we start with the Stefan-Boltzmann
law q = σT 4 . σ is the Stefan-Boltzmann constant, derived from other fundamental
constants including the Boltzmann constant kB :σ=4
2π 5 kB= 5.67 × 10−8 J s−1 m−2 K−4 .
15c2 h3(7.11)The radiation flux between concentric cylinders is given by [162, 163]q=

σ(T14 − T24 ),1/1 + (1/2 − 1)A1 /A2(7.12)

where 1,2 are the spectral emissivities of the inside and outside cylinder, respectively. If
we assume that both surfaces are blackbody radiators, 1,2 = 1, then the denominator of
Equation 7.12 is 1. The numerator can be simplified by applying σT 4 = CvT . Finally,
the temperature rise at the interface is also changed because we assume that there are
no reflections at the interface, therefore the inner surface temperature T1 is the averageof the emitted phonon temperature from A1 and the phonons arriving from the outersurface A2 : T1 → (T1 + T2 )/2 [151]. This results in an extra factor of two, giving us anexpression for flux in the ballistic case, which can be simply converted to resistivity,
rBallistic =∆T
qBallistic

=

1
.
2Cv

(7.13)The ballistic resistivity is independent of L, so we normalize both the ballistic
and the Fourier resistivities to the ballistic value, and plot the results in Figure 7.7.
When the linewidth L is much larger than the phonon mean free path Λ, the resistivity
is dominated by the Fourier component (blue), but for L < Λ, the ballistic resistivity(red) is more significant. The total linewidth-dependent resistivity is given by the sum of
the Fourier and ballistic parts, and the ballistic correction resistivity rBC that accounts
for the error of using the Fourier law alone is determined byrBC ∝rBallistic
2Λ
Λ
=
∝ .
rF ourier
3 log(4)LL(7.14)

Chen determined the same Λ/L proportionality in the spherical case, so this is characteristic of the ratio between diffusive and ballistic transport away from a nanoscale hot

110spot. Therefore, we will be looking for a deviation from the Fourier law proportional toΛ/L in our experimental data.

Figure 7.7: Analytical prediction of the Fourier (blue) and ballistic (red) components of
the resistivity for thermal transport away from a half-cylinder of diameter L as shownin Figure 7.6, based on the spherical case discussed in [161]. We measure the correction
term (green) that can be used to modify the Fourier law.

7.2.3

Propagating vs reservoir phonons

Phonon dispersion is another factor that becomes increasingly important in thecase of non-local transport where there are not enough scattering events to establish
phonon equilibrium near the a hot spot. In Fourier transport, phonons are assumed to be
in local equilibrium, so for a given phonon temperature, the phonon distribution can becomputed directly from the density of states and the Bose-Einstein occupation function
at that temperature (as shown in Figure 7.8). Therefore the energy and velocity of thephonons can be determined if the dispersion is known – see Figure 7.9 for the phonon
density of states and dispersion in sapphire. Even in a bulk system, the dispersion must
be taken into account because phonons of different energy have different velocity, as seen

111
0.04

4
0.03

Density of states
3
0.02

500 K

2

300 K
0.01
1

0

Phonon distribution function (thin lines)

Bose-Einstein distribution function (thick lines)

5

0
0

20

40

60

80

100

Phonon energy (meV)

Figure 7.8: Phonon distribution in sapphire at 300 K (green) and 500 K (red). The total
distribution is given by the product of the phonon density of states (black, from [165])and the Bose-Einstein occupation function at that temperature (thick curves, left axis).
This method of determining the phonon modes at a given location is valid only if there
is local phonon equilibrium and a temperature can be defined.

by the slope of the dispersion in the right panel of Figure 7.9. The dashed purple line
shows at approximate division between fast phonons (efficient energy carriers because
they move quickly) at lower energy and slow phonons at higher energies. We will referto these two sets of phonons as propagating (fast) and reservoir (slow) [164].
In the case of many scattering events, equilibrium between fast and slow phonons
is maintained and the thermal energy in the slow phonons can be efficiently transported
by first transferring the energy to the fast phonons by scattering. However, as thenumber of scattering events is reduced at short length scales, the slow phonons nolonger have opportunities to “offload” their energy onto more efficient carriers, but aretrapped as “reservoir” phonons. Meanwhile, the faster “propagating” phonons travel
quickly and freely out of the heated area. This issue has been examined before in theOptical
500K (heated)
Room temperature

Acoustic

Phonon density of states

112

Sapphire phonon dispersion

Figure 7.9: Phonon density of states (left) and dispersion (right) in sapphire. The
green and red lines represent the energy in meV of kB T at room temperature and 500
K, respectively. The dashed purple line is an approximate delimiter between ”slow”
optical (higher energy) and ”fast” acoustic (lower energy) phonons. Both parts of thisfigure are taken from [165].

case of thermal transport away from a nanoscale source, and was determined to causea significant reduction in flux from that predicted by the Fourier law with a change inresistance of RReservoir /RF ourier = Λ2 /L2 [166, 167]. A ratio of resistances is the sameas a ratio of resistivities because the width dependence is present in both numerator anddenominator. However, this experiment considered heat generated by direct electron
excitations, and since electrons readily excite optical phonons, they assumed that only
reservoir phonons were excited.
Since we are assuming that the electrons and phonons in the heated nickel nanostructures are in equilibrium (> 1 ps after initial laser heating) and the energy transport
across the nickel-sapphire interface is driven by phonon-phonon processes (see section
2.4.1), we expect to excite acoustic phonons. However, the question remains: how muchof an effect will we see from the inefficient transport of reservoir phonons? The answer
depends on exactly what is happening as phonon energy is transported across the interface. We will consider results as determined by the two models for phonon-phonontransport at an interface discussed briefly in section 2.4.1: The acoustic mismatch model

113

0.8

0.01

0.6

0.008
0.006

0.4

0.004

0.2

0.002
0

0

20

40

60

80

Phonon energy (meV)

100

0

1

AMM

0.14
0.12

0.8

0.1

0.6

0.08
0.06

0.4

0.04

0.2

0.02
0

Population below given energy

0.012

Normalized phonon population

1

DMM

0.014

Population below given energy

Normalized phonon population(AMM) and the diffusive mismatch model (DMM).

0

20

40

60

80

Phonon energy (meV)100

0

Figure 7.10: Excited phonon population (red)as a function of energy for thermal flow
between nickel and sapphire. The left figure uses the diffuse mismatch model (DMM),
in which the excited phonons match the available density of states in the sapphire.
The figure at right is derived from the acoustic mismatch model (AMM), in which
phonon states are excited directly by same-energy phonons in nickel (orange dotted line,from [168]). In each figure, the dashed purple line represents an approximate boundary
between optical (higher energy) and acoustic (lower energy) phonons, as shown in Figure
7.9.

In the DMM, the excited phonon spectrum (shown in the left panel of Figure7.10) is determined solely by the available density of states. In this case, ∼ 50% of
the excited phonons are the slower, reservoir phonons. In the other limiting case, theexcited phonons are determined by both the available density of states and the exciting
phonon energies, determined by the phonon density of states in nickel. This results in the
excitation of much lower-energy phonons (see the right panel of Figure 7.10), with nearly
all of them being the highly-efficient propagating phonons. As discussed in section 2.4.1,
These models for phonon interactions at a boundary break down at high temperatures
such as room temperature, so the real excited phonon spectrum is probably somewhere
in between the predictions of the diffuse and acoustic models. However, since > 50% of
the excited phonons are in propagating modes, we do not expect this to be a dominant
effect, as it was in the case of electrical excitation [166, 167]. The strength of this effectat real surfaces and high temperatures is another open experimental question. We will

114
keep this effect in mind if we see Λ2 /L2 deviation from the Fourier law in our data.

7.2.4

Previous experiment and open questions

Although there have been many models predicting a non-diffusive effect at theboundary of a nanoscale heat source, only one experiment has claimed to observe it.
In that experiment, Sverdrup et al. used electrical resistance thermometry to observe
steady-state temperature distributions in a thin 5 µm silicon membrane near a 300 nmwide doped resistor and saw slight deviation from the Fourier prediction below 200 K,
but no size-dependent effect at room temperature [167]. However, this experiment did
not allow for any systematic and quantitative comparison between experiment and theory due to the limitations posed by the sample geometry and steady-state measurement,leaving several open experimental questions:• How does the ballistic effect show up at an interface?
• Can it be measured at room temperature?
• Is the effect of ballistic transport time-dependent?
• Will it still be present in a bulk substrate, or is a thin membrane required?
• How does the strength of the ballistic transport vary with linewidth?
• What is the quantified effect of ballistic transport- can heatsink calculations
that use Fourier transport be easily corrected?

7.3

Measurement of ballistic heat transport at a nanoscale interface
Here, we use a beam of ultrafast coherent soft x-rays, at a wavelength of ∼ 29 nm,

to directly observe cooling of a nanoscale heat source into its bulk surroundings. By
interferometrically monitoring displacement in a heated nanostructure using diffractionof short wavelength light, we can detect dynamic temperature changes more sensitively

115
than would be possible using optical probing. This short-wavelength transient diffraction methodology, illustrated in Figure 7.11 and described in detail in Chapter 5, represents a newly-developed experimental technique that allows us to map out nanoscale
thermal transport over a range of nanostructure sizes. In this experiment, an infraredlaser pulse heats nanoscale nickel lines deposited on a transparent substrate (sapphireor fused silica), making it possible to heat a confined area, as illustrated schematicallyin Figure 7.1. The response (i.e. thermal expansion and acoustic response) is observedby monitoring the diffraction of the coherent soft x-rays from the nanostructure. This
allows us to observe heat propagation through the interface and into the substrate.

Sof
pr t x-r
ob ay
e

ed
ct ys
lf e -raD
Re ft x CC
o
s to

d
Diffracte
s
y
a
r
x
soft

12
0µ

m

120µm
IR pump

Figure 7.11: Geometry of the sample showing infrared laser illumination and EUV
detection scheme.

In this experiment, we used a sapphire substrate for two reasons: it is transparent
to the 800 nm laser used for heating the nanostructure, and the average mean free pathof heat-carrying phonons in sapphire is long (phonon mean free path in sapphire is∼ 100 − 150 nm at room temperature). An identical sample of nanostructured nickel
lines was prepared on a fused silica substrate (phonon mean free path is ∼ 2 nm atroom temperature) in order to compare the thermal transport between substrates withvery different values of Λ. The pump-probe setup, and data optimization, collection,and processing are described in detail in Chapter 5. Results from four characteristic

116
scans at varying nanostructure linewidth L are shown in Figure 7.12, along with two
fitting results at each linewidth. The red curve is the best fit using the thermal modelthat will be discussed in the following section, and the blue curve results from using thebulk value of rInterf ace in the model. As the linewidth is decreased, the bulk resistivity
increasingly under-estimates the best fit resistivity. This error is due to size-dependentballistic effects that cannot be accounted for in a Fourier model.
We approach the analysis in a way similar to standard TTR measurements ofthermal boundary resistance at thin film interfaces [23, 133, 157, 169–171] . In theseexperiments, the Fourier model is used to interpret experimentally measured data; weattribute deviation of our experiments from the Fourier model to a size-dependent ballistic correction resistivity rBC ∝ rBallistic /rF ourier , given by Equation 7.14. Recall thatthe effective total resistivity including ballistic effects (lumped into the total resistivity
which will be applied at the interface) is then given by
rInterf ace = rT BR + rBC .

(7.15)

By fitting our data to a multiphysics model described in the next section, we will retrieve the linewidth-dependent effective interface resistivity rInterf ace , and by subtracting rT BR , determine the ballistic correction resistivity.

7.4

Multiphysics model for data analysis
Data analysis is complicated by thermal expansion of the substrate as heat isdissipated from the nickel lines. In order to extract the cooling dynamics of the nickel
lines, we used the parameters in Table 7.1 in a multiphysics model.

7.4.1

Thermal transport across the interface and diffusion

In order to understand how the heat flowing from the nanostructure into the substrate effects the signal, we must be able to accurately calculate the thermal transport

1
1.0

L = 810L =nm
810 nma.
a

117

Measured data
Measured datarRBD
2.1(Best
BD==2.1
resistivity (bestﬁt)= 2.1ﬁt)rRBD
2.0
2.0(Bulk=value)
BD==value)
resistivity (bulk
2.0

0.8

0.6

0.5
0.4

0.2
0
0.0
11

00

2000
2000

4000
4000

8000
8000

L = 350L =nm
350 nmb.
b.
b

Measured data
Measured data
rRBD
2.9(Best
BD==2.9
resistivity (bestﬁt)= 2.9ﬁt)rRBD
2.0
2.0(Bulk=value)
BD==value)
resistivity (bulk
2.0

0.8

0.6

0.5

Normalized ΔDiffraction Signal

6000
6000

0.4

0.2
00

1
1.0

00

2000
2000

1000

c.
c

4000
4000

3000

6000
6000

5000

L = 190
L =nm
190 nmMeasured data
Measured data
rRBD
3.1(best
BD==3.1
resistivity (bestﬁt)= 3.1ﬁt)rRBD
2.0
2.0(bulk=value)
BD==value)
resistivity (bulk
2.0

0.8
0.6

0.5
0.4
0.2
0
0.0

1
1.0

00

1000
1000

d.
d

2000
2000

3000
3000

L = 80Lnm= 80 nm

4000
4000

Measured data
Measured data
rRBD
4.4(best
BD==4.4
resistivity (bestﬁt)= 4.4ﬁt)rRBD
2.0
2.0(bulk=value)
BD==value)
resistivity (bulk
2.0

0.8

0.6

0.5
0.4

0.2
0
0.0

00

500
500

1000
1000

Time (ps)

1500
1500

2000
2000

Figure 7.12: Normalized dynamic diffraction signal demonstrating the observed deviation from bulk theory for thermal transport from small nanostructures. Experimental
data (black) from Ni linewidths of a. 810 nm, b. 350 nm, c. 190 nm, and d. 80
nm are shown. The signal in each case consists of a sharp rise due to impulsive laser
heating, a thermal decay due to interfacial thermal transport, and an oscillation fromsurface acoustic wave propagation. The best fit for rInterf ace for each linewidth isshown as the red line, while the blue line shows the result of plugging in the bulk valueof rInterf ace = rT BR .

Parameterh
CN i , Cs
rInterf aceαsσsαN i , αρ

118
Value in Model (Ni, Sa, FS)
20 nm
Ni: 4.07, Sa: 2.63, FS: 2.2 [106 W/m3 K]
Determined from data, [W/m2K]
Sa: 15.64, FS: .54 [10−6 m2 /s]
Sa: .25, FS: .17
Ni: 12.77, Sa: 5.31, FS: .54 [10−6 /K]
Ni: 8.91, Sa: 4.00, FS: 2.65 [g/cm3 ]Physical meaning
Nanostructure thickness
Volumetric specific heat
Boundary conductivity
Thermal diffusivity= k/C
Substrate Poisson ratio
Linear coef of thermal expansion
Density

Table 7.1: Parameters used in the multiphysics model described in this section.

across the boundary, the diffusion into the substrate, and the effect of the boundary
conditions. The fixed periodicity of the nanostructure means that the entire system canbe studied by simply looking at one cycle of the repeated structure: one nickel line andthe substrate beneath it, as shown in Figure 7.13. In the case of the 25% duty cycle
samples studied here, that means a nickel line of width L and a substrate beneath it ofwidth 4L.

4L
C
1µm

E

L

Equation 7.17
D

Equation 7.18

A

B
F

Figure 7.13: Setup for model “cell” for thermal transport from a nanoscale contact ofwidth L and periodicity 4L into a bulk heat sink. The two equations governing transport
across the interface and within the substrate are indicated, as well as the interfaces for
identification of boundary conditions.

We assume that the nickel nanoline can be treated as a “lumped capacitance”,
which means that it is thin enough that we can neglect transport inside the nickel. This
assumption can be checked by calculation of the Biot number Bi for the interface, which119
is a measure of the relative importance of bulk and boundary effects:
Bi =hrT BR kN i,

(7.16)

where kN i is the thermal conductivity of the nanostructure. If Bi < 1, it is valid to use
the lumped capacitance model to ignore diffusion in the nickel. The room-temperaturethermal conductivity of nickel is kN i = 90.62 W/mK and later measurements will showthat rT BR ∼ 2 × 10−9 m2 K/W. So for a h = 20 nm nanostructure, Bi = .1 < 1 — thelumped capacitance model is valid. This means that the nanostructure can be modeled
as having no temperature dependence in the vertical direction. Temperature dependence
across the surface is still considered, however, as the edges of the nanostructure coolmore rapidly than the center due to thermal spreading.
Since we can neglect diffusion inside the nanostructure, the thermal model iscomposed of only two parts:(1) Coupling of the nanostructure and substrate boundary temperatures, accomplished by conservation of energy across the interface. This is determined by
TN i (x, t) − Ts (x, y = 0, t)
dTN i (x, t)
ρN i hCN i=−,dt
rInterf ace(7.17)

where TN i and Ts are the time and position-dependent nickel nanostructureand substrate temperatures, and x and y are the coordinates along the surface
and depth directions, as shown in Figure 7.16. Equation 7.17 essentially saysthat the amount of energy lost by the nanostructure (left side) in a differential
amount of time is equal to the energy flowing across the interface, into thesubstrate (right side).
(2) Diffusion of heat in the substrate beneath each nickel line, which is modeled bynumerically solving the heat equation∂Ts (x, y, t)= αs ∇2 Ts (x, y, t).
∂t(7.18)

120
This equation is solved numerically in two spatial dimensions with time-stepping
using the Crank-Nicolson and Alternating-Direction Implicit (ADI) methods.
The following boundary conditions are used (see Figure 7.13 for boundary labels):
A,B: vertical sides of modeling cell have periodic boundary conditions, whichis equivalent to saying that there is no flux across the boundaries –∂Ts (x = 0 or 4L, y, t)=0∂x(7.19)

C,D: twin regions on top of the substrate flanking the nanostructure havevacuum on the other side of the interface and therefore cannot have heatflow across them (other than radiation, which is neglected), so they also
have no flux boundary conditions –∂Ts (x, y = 0, t)= 0.
∂y(7.20)

E: specified flux boundary condition –Tf (x, t) − Ts (x, 0, t)∂Ts (x, y = 0, t)=−.
∂yrInterf ace ks(7.21)

Numerical implementation of this boundary condition is challenging because of the finite grid size. The usual numerical derivative at an interface∂f (y0 )f (y0 + dy) − f (y0 )'+ O(dy)∂ydy(7.22)

is valid only when the grid spacing dy is much smaller than the distanceover which the derivative acts. In the case of thermal propagation, this√distance is the thermal diffusion length ` = αs t. For sapphire, αs =1.5748×10−5 m2 /s, so for a time one picosecond after excitation, ` ∼ 4 nm.
Computing the thermal propagation on such a fine grid is unreasonable for121the larger nanostructures, so instead we chose to implement a multi-point
numerical derivative, still centered at y0 (with y1 = y0 + dy...) [172]:∂f (y0 )−3f (y4 ) + 16f (y3 ) − 36f (y2 ) + 48f (y1 ) − 25f (y0 )'+ O(dy 4 ),∂y
12dy
(7.23)
which greatly increases the accuracy of the derivative calculation.
F: back of the substrate has a no flux boundary condition assuming that thesemi-infinite approximation is valid –∂Ts (x, y = 1µm, t)= 0.
∂x(7.24)The semi-infinite approximation requires that the substrate is thick enoughthat no heat will be able to diffuse to the back surface in the given amountof time t. Our pump pulse delay stage is limited to ∼ 8 ns, and the
maximum extent of thermal diffusion in sapphire at this timescale is ` =√αs t ∼ .35 µm. This means that the semi-infinite approximation is valid
both in the modeling, where we used a 1 µm depth, and the experiment,
where the substrate was 3 mm thick.
The initial conditions describe uniform laser-induced heating in the nickel andan unheated sapphire substrate. Final modeling was run using a nickel heatedtemperature of 500 K (determined from deposited laser energy and nickel properties), but actual-temperature results deviate from normalized temperature(heated nickel temperature initially set to 1 K, sapphire set to 0 K) by lessthan 2% (error due to temperature-dependent changes in the materials properties, shown in Figure 7.15), so we present normalized results here. The initialconditions, then are:
TN i (x, t = 0) = 1,

Ts (x, y, t = 0) = 0.

(7.25)

122
Figure 7.14 presents modeling results for L = 1 µm (top) and L = 50 nm (bottom)
at selected times after the excitation of the nickel nanostructure. The half-width of themodeling region in the first case is 2 µm, much farther than heat diffusion travels in
8 ns (longest time available for measurement). This leads to a thermal distributionthat is primarily confined beneath the heated L = 1 µm, even at 8 ns. This result isin contrast to the modeling results using a nanostructure linewidth of L = 50 nm. In
this case, the modeling region half-width is only 100 nm, and there is nearly-uniform
thermal distribution across the surface only 1 ns after excitation. Therefore, thermal
expansion of the substrate, is expected to be localized beneath the nanostructure for
L = 1 µm and evenly-distributed across the surface for L = 50 nm. These results
represent extreme but indicative examples of the sample sizes we used in experiments.

7.4.2

Thermal expansion

Linear thermal expansion is described by a linear coefficient of thermal expansion
α in the equationZ∆hx = α∆T (x)dx,(7.26)

where ∆T (x) is the temperature difference from equilibrium. We will refer to this as
the “vertical integration technique”, and it has implicit assumptions:• The temperature change is low; the coefficient of thermal expansion is actuallytemperature-dependent (see Figure 7.15), so very large changes in temperature
can cause errors. This can be corrected by including a temperature-dependent
coefficient of thermal expansion and integrating over the change in temperature.
• The object is free to expand; otherwise, pressure and stress will build up withoutcausing displacement.
This expression is valid for the vertical displacement of the nickel lines because
they are bound on the bottom only and are free to expand away from the substrate.

123

100ps 500ps

1ns

5ns

8ns

.3
Temperature with TNi(t0) = 1

50ps

4µm

1µm
10ps

200nm
10ps

50ps

100ps

500ps

1ns

5ns

8ns

1000nm

L = 50nm.2Temperature with TNi(t0) = 1

200nm

L=
1µm 0

0

Figure 7.14: Time snapshots of thermal diffusion modeling for a single sapphire cell from10 ps to 8 ns. Results are shown with a 4 µm wide region with a 1 µm nickel heater (top)and 200 nm wide region with a 50 nm nickel heater (bottom). The different temperaturescales are the result of different peak temperatures at these times- the nanostructured
nickel heaters started at the same temperature. These thermal images are rotated 90◦
counter-clockwise from the modeling setup shown in Figure 7.13 for ease of presentation.

Their horizontal displacement is neglected in the model because it will be much smaller
since the nickel is rigidly bound to the substrate in that direction. Also, in the EUV

Thermal conductivity k (W/mK)

124
1600
1400
1200
1000
800
600
400
200
0

Volumetric specific heat
capacity (1E6 J/m3 K)

6

Nickel5
4Sapphire3210Coefficient of thermal
expansion (1E-6 /K)

20

200

400

600

800

1000

200

400

600

800

1000

15

10

5

0

Temperature (K)Figure 7.15: Temperature-dependence of thermal conductivity k, specific heat C, andcoefficient of thermal expansion for nickel [173] and sapphire [174–177].

125
diffraction detection method used, horizontal displacement has a much smaller effectthan a vertical one. Because there is no depth dependence of the nickel lines in the thermal model, the position-dependent displacement is directly-proportional to the change
in temperature:∆hN i,y (x) = αN i hN i ∆T (x).

(7.27)

In contrast to the simple treatment for thermal expansion of the nickel lines, thethermal expansion of the surface of the substrate can not be determined by integrating
the thermal profile vertically and multiplying by the coefficient of thermal expansion(see Figure 7.17). A simple integration of the thermal profile implies that a hotspot deep
within the substrate has the same effect as the same hotspot on the surface, which can’tbe true because there are forces binding each lattice point to its neighbors. The thermal
expansion should actually be calculated starting from the time-dependent Navier-Stokes
equation [178]2~ ∇~ · ~u) + F~ − α∇T~ = ρ ∂ ~u ,µ∇2 ~u + (ν + µ)∇(∂t2(7.28)

where µ is the shear modulus, ν is Poisson’s ratio, F~ is an external force term, and ~u isthe displacement at a particular location. If we assume that there are no external forces
acting on the system, then F~ = 0. It is also assumed that the material is isotropic,
which is true for fused silica and is > 97% true for sapphire (based on ratio of thermal
conductivities along and orthogonal to the axis).
Further simplification can be made under the quasi-static assumption ∂ 2 ~u/∂t2 =0, which implies that the acceleration of the surface can be neglected. This will give
us a stationary, time-independent partial differential equation that assumes that thesurface displacement at a given time is coupled to the temperature distribution atthat time only. That this is valid is not obvious for the picosecond time scale weare dealing with here, but we can make a rough estimate of the time scale of thesesurface acceleration effects. Assuming that the maximum surface velocity is given by

126
the longitudinal acoustic velocity, and the maximum surface expansion is 1 nm, thenwe can neglect surface acceleration effects for time step sizes > 100 fs (this is much
longer than the typical time step ∼ 10 ps used in these experiments). Implementing
these three assumptions, Equation 7.28 simplifies to the thermoelastic equation [179]3(1 − ν) ~ ~3(1 − 2ν) ~~ × ~u = α∇T.
~∇(∇ · ~u) −∇×∇
1+ν
2(1 + ν)(7.29)In order to obtain a useful solution to the thermoelastic equation, we make a final
assumption that the surface is stress-free. This is true at the unbound (free) surface ofa thin film or substrate, but not technically true at the surface of our substrate, where thecenter is coupled to a nickel strip. However, the stress-induced change to the deformation
is a second order effect and therefore much smaller than the thermal displacement
itself [180]. The thermoelastic equation can then be solved using Green’s functions todetermine the vertical displacement at the surface as a function of position [71](1 + ν)α∆hy (x, z) =
3π

Z

Z

Z

dx0

dy0

dz0

z0
(x − x0 )2 + (z − z0 )2 + y02

3/2 ,(7.30)

which involves a weighted integration over the three-dimensional temperature distribution at each point on a two-dimensional surface (see Figure 7.16, 2D).
z

Δhy(x,z)Δhy(x)xxy

2D
surfacey

1Dsurface

Figure 7.16: Coordinate systems for surface displacement calculation for a twodimensional (left, Equation 7.30) and one-dimensional (right, Equation 7.31) surface.

We convert this to a two-dimensional distribution by integrating over the z-axis
from −∞ → ∞. Finally, the remaining integrals are converted to discrete sums and127
nearest neighbor temperature profiles are included in the calculation (since the neighboring cells can have a significant effect near the edges – neighbor properties determinedby wrapping original cell), leading to the final expression for the thermal surface displacement of the substrate:ix,max jy,max
1jy(1 + νs )αs dx X X X∆hy (x) = 2
T (ix , jy ).
3π(x/dx + N × ix,max − ix )2 + (jy )2
N =−1 ix =1

jy =1(7.31)
Here, ix and jy are the summing steps in x and y respectively, dx is the step size in x,
and N is an index accounting for nearest neighbors. The results of the calculation are
shown as the red curve in Figure 7.17A for a typical temperature distribution shownin 7.17B. There is significant deviation in both amplitude and shape from the vertical

Vertical displacement (10s of picometers)

integration technique.

B

A

Surface position (10s of nm)

Figure 7.17: (A) Calculated surface displacement using three different techniques: vertical integration (blue), and depth and distance weighting and integration with (red, asin Equation 7.31) and without (green) including contributions from neighboring lines.
The most realistic results are given by the weighted calculation including neighbors(red), and are used in this model. (B) Illustration of the difference between the vertical
integration method (long white rectangle) and weighted integration method (concentric
diminishing circles) for determination of the vertical displacement of a point on thesurface.

128
7.4.3

Fresnel optical propagation

To complete the connection of the thermal model to the measured data, we needto calculate how the thermally-induced surface profile affects the diffraction of the EUV
probe, which we measure experimentally. We use the same Fresnel optical propagation model discussed in section 4.2 and Appendix A to calculate the diffraction fromthe thermal surface deformation, which can then be subtracted from the diffraction of
the undeformed structure to yield the expected signal. A summary of the steps in thecalculation is shown in Figure 7.18, with the final change in diffraction shown in thebottom panel. This calculated change in diffraction compares well with our raw experimental data, as seen by the green curve in Figure 5.12. Integrating the absolute valueof the change in diffraction gives the total signal, which can be compared directly tothe experimentally measured values when both are normalized to the peak signal.

7.4.4

Discussion of model results

In order to provide insight into the significance of the contributions to the measured signal, the Fresnel propagation portion of the calculation is performed three times
to determine the signal from: the nickel lines alone, the sapphire substrate alone, andthe combined effect of the Ni and Sa together. These results are summarized in the top
panel of Figure 7.19 for the L = 1000 nm nanostructures. The signal from the nickel
lines is close to a double-exponential decay, as expected for thermal transport across asurface and into a substrate. Because of the quasi-static approximations discussed in
section 7.4.2, the decay of the Ni and the rise time of the sapphire signal are directly
related to the rate of thermal energy flow between them, and by conservation of energy,
are equal.
The signal from the substrate logically starts at zero since it is initially unheated,gradually increases as heat flows from the Ni lines into the substrate, and eventually

129

Figure 7.18: Results of Fresnel porpagation calculation of diffraction signal change. Top:
sample profile and EUV beam on sample. Middle: Reflected beam (as if there were no
grating present) and diffracted beam. Bottom: pump-on, pump-off difference signal,
showing a change in diffraction efficiency.

decreases. The EUV probe makes an interferometric measurement, so it is the relative
displacement between the substrate and the nickel lines that matters. This fall timeof the signal from the sapphire substrate is determined by the diffusion time for heatto flow in the substrate from the edge of a nickel line to the center between two lines.
The fixed duty cycle of the nanostructured samples means that this diffusion distance is
proportional to L itself, so the time for heat to diffuse to the edge of the modeling cellis strongly linewidth-dependent, as shown in Figure 7.14. In fact, this diffusion time isquadratic in L [19]–τ ∼ L2 /αs .

(7.32)

This dramatic decrease of the decay time as the linewidth is reduced means that for130very small structures (which are very close together), the contribution to the total signal
from the sapphire substrate is much smaller than their large-L counterparts which aremuch further apart, as alluded to at the end of section 7.4.1. This is borne out in thecalculations of the contributions to the signal from the nickel and sapphire for different
linewidths, as shown in the bottom panel of Figure 7.19.
The total signal measured in the experiment is equal to the sum of the contributions from thermal expansion in the nanostructure and the substrate, and therefore
cannot be fit by an exponential, as we tried to do in our original data analysis. In thiserroneous analysis, we observed decay times dominated by L2 at large linewidths forthe sapphire, and no linewidth dependence for the data on fused silica substrates. This
L2 dependence is due to the diffusion time for the heat to flow to between the lines(Equation 7.32), and can be corrected by thorough numerical modeling, as describedabove. The effect of the substrate in the case of the fused silica substrate is much smaller
because of the smaller coefficient of thermal expansion.

7.4.5

Fitting the model to our experimental data

In order to fit the modeling results to the data, we ran the model several times fora given nickel linewidth, varying the input interface resistivity between runs. Resultsfrom a typical series of modeling runs for L = 1 µm linewidths are shown in Figure 7.20,
where each curve corresponds to modeling results using a different resistivity (values ofconductivity h are shown, h = 1/r). Each curve is fit to a quasi-exponential function,with terms optimized for best fitting. For example, for small linewidths, the modeled
total signal is very close to an exponential, but for large linewidths, additional terms are
required to get good fits. The fitting parameters are then fit to a quadratic function,with the result being a parameterization of the modeled time-dependent “signal” as afunction of interface resistivity. This result for the thermal component is then added
to a decaying sinusoid to account for the surface acoustic wave component discussed

1311

Calculated normalized signal

0.8

0.6

Net signal (measured)

0.4

0.2

0

Signal from thermal expansionof the substrate0

2

1

Signal from nanostructure(what we want to determine)
4Time (ns)6

2000nm1000nm
500nm
200nm
100nm

Calculated normalized Signal

0.8

k=45
rTotal=1E-8

0.6

Ni+Sa Signal

Ni Signal

8

0.4

0.2

Sa Signal
0

0

2

4

Time (ns)68Figure 7.19: Calculated change in diffraction signal from thermal expansion. The top
panel shows the signal from L = 1000 nm lines from the Ni nanostructure itself (lightblue), the sapphire substrate (purple), and total (dark blue). The bottom panel shows
the linewidth dependence of these signals: as the linewidth is decreased, the signal fromthe sapphire substrate becomes smaller. rInterf ace = 10−8 m2 K/W in these calculations.

132
in Chapter 6 and finally fit to the data by minimizing the RMS error between theexperimental data and the fitting function.

Calculated normalized signal

1.0
h=600 W/m2 K
h=500 W/m2 K
h=400 W/m2 K
h=300 W/m2 K
h=200 W/m2 K

0.8

0.6

0.4

0.2

0.0

0

2000

4000

6000

8000

Time (ps)Figure 7.20: Normalized dynamic total signal from full thermal diffusion-thermal
expansion-optical propagation model for L = 1 µm linewidth as a function of conductivity (conductivity h = 1/r, and units of given conductivities are in 106 ).

For each data set at a given linewidth, we retrieve the total boundary resistivity
rInterf ace , which includes both conventional thermal boundary resistivity and ballistic
effects in the substrate, and fit to the time-resolved thermal decay data. The best
fits to the data shown in Figure 7.22 (red lines) demonstrate that the measured interface resistivity increases as the heated interface gets smaller. The measured values ofrInterf ace , rT BR , and rBC for both sapphire and fused silica substrates are shown in
Figure 7.22. The Fourier resistivity, rF ourier (including spreading resistance discussed
in section 7.2.1), is automatically accounted for in our numerical model for heat propagation in the substrate (Equation 7.18). By denition, large linewidth measurements on
samples with both samples (sapphire and fused silica substrates) yield the bulk values ofthermal boundary resistivity rT BR (equivalent to TTR measurements of thermal boundary resistance on thin lms), as shown by the horizontal dotted lines in Figure 7.22. We
measured a very small (zero within error bars) deviation from rT BR for the fused silica

133
substrate, because the phonon mean free path in fused silica (ΛF S ∼ 2 nm) is much
shorter than the smallest nickel linewidths. However, in the case of samples on sapphire
substrates, the extracted resistivity at the smallest linewidths is more than three times
higher than the bulk value. The blue dotted and red dashed lines in Figure 7.22 show
the predicted resistivity values rInterf ace (with rBC = rInterf ace − rT BR proportional toΛ/L) determined from the analytical ballistic-diffusive model (Figure 7.6 and Equation

Normalized Thermal Signal7.14) for phonon mean free paths of ΛF S = 2 nm and ΛSa = 120 nm.

Measured data

1.0

0.5

Ni+Sa Signal
Sa Signal

Ni Signal

0.0
0

2

4

6

Time (ns)Figure 7.21: Results of the multiphysics model for L = 1000 nm linewidth structureswith the best fit value of rInterf ace = 2.2 × 10−9 m2 K/W (purple). Also included
is a sample measurement (green). At this linewidth, the sapphire substrate makes asignificant contribution to the total signal.

7.5

Implications of the measurement
One way that we can understand these results is by considering what we willcall a “minimum heated region” model, in which the thermal conductivity surrounding
a nanoscale heat source is reduced in an area including one phonon mean free path
length [161]. We can see the effect of this idea by considering a simplified calculation,
starting with the Fourier law for heat flow: q = −k∇T . If we consider only a single

134

Effective Interface Resistivity rInterface (m2 K/W)

7x10-9

Fused silica substrate

6x10-95x10-94x10-9Sapphire substrate
3x10-9rBC

2x10-9rTBR

1x10-9

0

100

rTBR

1000

Interface contact size L (nm)Figure 7.22: Measured effective interface resistivity for nickel nanostructures of width
L on fused silica (blue hollow dots) and sapphire (red solid dots) substrates. The
measured resistivity for the samples with fused silica substrates (ΛF S ∼2 nm) is nearly
independent of linewidth, while the boundary resistivity for the sapphire substrates(ΛSa ∼120 nm) increases rapidly as the linewidth is decreased due to the ballistic
correction of the over-prediction of the flux by the Fourier law.

dimension, ∇T is ∂T /∂x ∼ ∆T /∆x. The normal dimension of the system is ∆x ∼ L,
but in this case, we use the phonon mean free path Λ as a minimum to the length scale:∆x ∼ L + Λ. So for L  Λ, ∆x ∼ L, and for L < Λ, ∆x ∼ Λ. The modified Fourier
law for the minimum heated region model is thenqM inimum heated region model = −

k∆T
,
L+Λ

(7.33)

L

135

Λ

Figure 7.23: Minimum heated region model: If the size of a heated area is smaller thanthe phonon mean free path Λ, heat flows as if coming from a spot of size Λ.

and if we consider the ratio of this modified flux to the traditional Fourier flux, we findqM inimum heated region modelL=.
qF ourierL+Λ

(7.34)In terms of resistivity (r = ∆T /q),rM inimum heated region modelΛ=1+ .
rF ourierL(7.35)

This is the same result as the calculation in section 7.2.2, and agrees with our experimental results. The model effectively says that the system can only respond to a heatedspot larger than a phonon mean free path; if the heated area is smaller than Λ, the heatflows as if it were coming from a spot of size Λ.
It is important then to distinguish the case of heat transfer away from a nanoscaleheat source and transport inside a nanostructure.
• For heat transfer inside a nanostructure, illustrated in Figure 7.2, ballistic transport means that the phonon mean free path is limited by the size of the structure(for L  Λ, Λ → L).
• In the case considered in this experiment of heat transfer away from a nanoscaleheat source, illustrated in Figure 7.23, ballistic transport means that the effective source size is limited by the phonon mean free path (for L  Λ, L → Λ).

136
Therefore, our measurement of ballistic thermal transport away from a nanoscale heatsource is fundamentally different from previous experiments which observed ballistictransport in or through a nanostructure.

7.6

Discussion of the time-resolved measurement
One important aspect of this experiment that we haven’t yet emphasized is the

dynamic time-resolved nature of the measurement. This is in contrast to most experimental techniques for studying thermal transport in a nanostructure, which use
steady-state [141] or oscillatory (alternating current) [181] measurements. Without emphasizing its importance, we have already seen the importance of the time-resolved
nature of the experiment in Figure 7.12d. In this figure, the difference between the bulk
and ballistic-sensitive fits is only seen at short times (< 1 ns). The importance of a dynamic measurement is also illustrated in Figure 7.24, which shows normalized modeling
results for the temperature decay in the nickel lines for L = 1000 nm (red) and 65 nm(blue lines). The dark blue line shows the thermal decay in the nanostructure using thebest-fit value of rInterf ace , and the light blue curve shows the thermal decay when thebulk interface resistivity is used. The difference between the two curves, which depict achange in rInterf ace > 3×, is very interesting: the bulk-resistivity curve predicts much
faster thermal decay in the first 500 ps, but by 3 ns, the predicted temperature profiles
are the same. This long-time convergence is due to the thermal conductivity in thebulk portion of the substrate, which is independent of the interface resistivity. Therefore, a dynamic measurement is necessary to observe ballistic effects from an impulsive
excitation.
This result, determined from the modeling fit to our time-resolved experimental
data, gives insight into the time-dependent temperature in the nanostructure that isn’t
possible using static probes. This short-time information is also quite useful: the green
line at 500 ps corresponds to a frequency of 2 GHz, which is the operating frequency of

137
the processor in my computer.

7.7

EUV probing of nanoscale thermal transport outlook
The ballistic effect seen near a nanostructured interface should be even larger formaterials with longer phonon mean free paths. The obvious example is silicon, which
has a phonon mean free path Λ ∼ 250 nm at room temperature, and is commonlyused in the IC industry as a substrate. We have already obtained a sample with asilicon substrate, and are now in the process of modifying the experimental setup toaccommodate the sample. One problem is that Si is not transparent to the 800 nm
pump light, so we cannot pump from the backside. The absorption of Si also presents
a problem in that we will no longer be able to assume that the substrate not directly
heated by the laser pump pulse. However, the penetration depth for 800 nm light insilicon is rather long (7.5 µm [14]), so this may not change the calculation critically.
500 ps
(2 GHz)1.0

L = 1000 nm, rBD = 2.0 m K/W
L = 65 nm, rBD = 6.2 m K/W
L = 65 nm, rBD = 2.0 m K/W

Normalized average TNi

0.8

0.6

0.4

0.2

0.0
0

1000

2000

Time delay (ps)3000

Figure 7.24: Modeling results for best fits to data showing the extracted average temperature of the heated nickel line as a function of time. The red curve shows the best
fit to the data from the L = 1000 nm linewidth samples, which gives the bulk valueof boundary resistivity. The light blue and dark blue curves are the results for a 65
nm linewidth, using the bulk and best fit values of resistivity respectively. Resistivity
values should be ×10−9 .

138Figure 7.25: Quantum efficiency of Andor CCD as a function of wavelength. Note thezero efficiency above 1100 nm. Figure from [183].

On the other hand, the thermal expansion of silicon is about half as large as that insapphire, so the signal from the substrate will be smaller.
The concerns about the transparency of the silicon substrate bring up an interesting consideration: using laser light with a longer wavelength to pump the sample (suchinfrared light can be obtained by downconversion of the 800 nm light from the laser
amplifier to a wavelength of 1100-3000 nm using an optical parametric amplifier [182]),
would allow for the same backside pumping we used before, and because of the reduced
efficiency of silicon-based CCDs in the infrared (see Figure 7.25), may allow for theremoval of one or both of the aluminum filters between the sample and the CCD. This
would allow for 3× higher flux on the CCD, could enable faster vacuum pumpdown
and venting, and would reduce or eliminate beam deformities caused by transmission
through the filters after the sample. In this setup, silicon windows would be used on thecryostat containing the sample to allow the infrared pump light to enter while effectively
blocking the optical light in the room.
Another advantage of using a silicon substrate is that as a semiconductor, silicon
has a much lower electrical resistivity than sapphire and fused silica. This higher elec-

139tron mobility will hopefully allow experiments at low temperature, where the sapphire
substrates became quickly contaminated because of space-charge buildup.

7.8

Conclusion
In summary, we have observed the transition from diffusive to quasi-ballistic heattransport at nanoscale interfaces smaller than the phonon mean free path. We saw
the breakdown of the Fourier law at such short length scales, and demonstrated thatheat transfer calculations can be corrected by adding a ballistic correction resistivity
proportional to the Knudesn number Λ/L. We also demonstrated that at a basic level,
the heat flow from a nanoscale source smaller than the phonon mean free path canbe thought of as emanating from a spot the size of the mean free path, which sets
a fundamental limit on the ability of the system to handle small hot spots. Theseexperiments advance understanding of heat transport fundamentals, and are important
for thermal management and design for integrated circuits.

Chapter 8

Magneto-optics with EUV light

Ultrafast magnetism at nanometer scale lengths is a topic directly relevant toindustry, since individual bits stored on a hard disk are already of sub-micron dimension. Advances in storage capacity depend on further increases in magnetic storage
density. A thorough understanding of switching dynamics is also critical for improving writing speeds commensurate with capacity. Continued technological progress depends on developing an understanding of the dynamics of magnetic systems at a fundamental level, which ultimately requires studies that combine nanometer spatial resolution with femtosecond time resolution. This is a challenging proposition, but one
that can be addressed using newly-developed experimental tools. Magneto-optical measurements [184–186] for magnetic properties currently use either x-rays from large-scale
synchrotron x-ray facilities or visible-wavelength light from ultrafast lasers. X-rays gives
high spatial resolution and high contrast at the elemental absorption edges of the ferromagnetic materials; for example, magnetic soft x-ray microscopy takes advantage ofcircular dichroism (XMCD - a difference in the absorption of left and right-circularly
polarized light near a core-level absorption edge) as a contrast mechanism for the microscope. Using soft x-ray photons 700 eV and zone plate imaging, domain structures withelemental selectivity have been imaged with a resolution of 15 nm [86, 187]. However,with photons from synchrotron radiation, the time resolution is too low to see the fastest
dynamics involved in domain switching. Visible ultrafast lasers, on the other hand, give

141
short pulses (∼ 30 fs), enabling studies in the ultrafast regime using the magneto-optical
Kerr effect (MOKE). But magneto-optical signals are usually very small and the longwavelengths used limit these studies to low spatial resolution.

8.1

Historical development of the magneto-optic Kerr effect (MOKE)
The field of magneto-optics (MO) has a long history, dating back to 1845, when

Michael Faraday observed polarization rotation of light passed through lead-boroilicate
glass in a magnetic field [188]. This first observation of the interaction of light withmagnetism was followed in 1876 by John Kerr’s discovery of a similar effect in reflection[189]. Kerr observed a that linearly polarized light reflecting off of the surface of amagnet experienced an induced polarization ellipticity. He observed a change both whenthe magnetic field was oriented orthogonal (polar geometry, P-MOKE) and parallel tothe surface and the plane of incidence [190] (longitudinal geometry, L-MOKE). The final
possible geometry for the Kerr effect (transverse, T-MOKE, where the magnetizationof the sample is parallel to the surface but perpendicular to the plane of incidence) was
first demonstrated by Zeeman in 1896 [191]. These observations encouraged thinking
that light and magnetism were coupled, and played a significant role in the developmentof the theory of electromagnetism.

M

Polar

M

Longitudinal

M

Transverse

Figure 8.1: The polar, longitudinal, and transverse geometries of the magneto-optic
Kerr effect.

More than 100 years later, the Kerr MOKE is still used to make optical measure-142
ments of the magnetic properties of materials. At optical wavelengths, the P-MOKE
and L-MOKE geometries are commonly used with laser probes with very precise polarization properties, enabling very sensitive measurements of MO polarization rotation.
The T-MOKE geometry is less-commonly used however, because although it does not
require polarization analysis, T-MOKE signals from 3p transition metals (e.g. Fe, Co,Ni) are small ( 1% upon full magnetization rotation [192]) when using visible laser
probing. However, the strength of the T-MOKE signal is increased at soft x-ray wavelengths where the energy of the incident light is just above or just below absorption
edges. Most experiments use the stronger T-MOKE effect at the L edge (700-800 eV),but recent studies have found a change in reflectivity greater than 50% [192] at the Medge (50-70 eV) [192–194]. All past work was performed using a synchrotron source. In
this chapter, we describe the first observation of the T-MOKE at the M-edge using atabletop source of soft x-ray light.

8.2

Derivation of the transverse magneto-optic Kerr effect
In the case of EUV probing discussed here, polarization analysis is challenging

because of the low transmission of most materials. It can be done using multiple reflections [186, 195, 196], but since each reflection is < 1% efficient, the reduction in flux issignificant and makes hard experiments even more challenging. For this reason, the factthat T-MOKE can be measured without polarizers greatly simplifies the experiment,
and of the magneto-optic effects, we will focus our attention on the specifics of the
T-MOKE geometry.

8.2.1

Macroscopic (phenomenological) description

The response of isotropic materials to light at a given wavelength can be de~ to thescribed by the dielectric constant . This constant connects the electric field E~ by D~ = E.
~ However, the high-symmetry conditions are broken whendisplacement D

143→~ is present, and in this case,  must now be expressed as a tensor ←a magnetization M ,given by←→
~ × E/|~ E|,~
 = 0 + iQM(8.1)

where Q is a complex material parameter that is roughly proportional to the saturation
magnetization of the sample and that describes the strength of the Kerr effect [197].
In the case of a cubic ferromagnet, the dielectric tensor can be expressed in Cartesian
coordinates asxy xz 
 xx




←
→
 =  −xy xx yz  .
−xz −yz xx
The off-diagonal components of the dielectric tensor arise from the occurrence of magnetism, and indicate magneto-optical anisotropy. The Onsager relations mn = −nmuse symmetry to simplify the off-diagonal components further [198].
In the transverse geometry with the coordinates as defined in Figure 8.2, themagnetic field is in the ±x̂ direction, so the dielectric tensor reduces to [199]


0
0 
 1




←
→
 =  0 1  2  ,
0 −2 1

s

p

M=Mx

Transverse MOKE
Figure 8.2: Detailed geometry for the transverse magneto-optic Kerr effect. The magnetization is in the x-direction and the incident light can be s-polarized (x-direction, nomagnetic signal) of p-polarized (y + z-direction, maximum signal).

144
where 1 and 2 are the on and off-diagonal elements respectively.
Now that we have an expression for the dielectric permeability, we must step backand consider the fundamental connection between magnetism and the optical response:
Maxwell’s equations.
~ ·D~ = 4πρ
∇

(8.2)

~ ·B
~ =0
∇

(8.3)

~
∂H
,
∂t
~
∂E
→
~ = 0 ←
∇×H
 ·
.
∂t
~ = −µ0
∇×E

(8.4)
(8.5)

A laser incident on the surface can be described by a plane electromagnetic wave with~ and H~ given byE~ = E~0 exp − i(ωt − ~k · ~r
E



~ =H
~0 exp − i(ωt − ~k · ~r
H



(8.6)

where ω is the frequency and k is the wavevector of the incoming laser light. By taking~ = i~k × E~ and ∇ × H~ = i~k × H.
~ Timethe curl of Equations 8.6, we see that ∇ × E~~ and ∂ H/∂t~
~ Thesederivation of the same equations yields ∂ E/∂t
= −iω E= −iω H.
~ = ωµ0 H~ and ~k × H~ =expression can be used in Equations and 8.5 to get ~k × E→~ We can eliminate H~ from these equations and use the vector relationship−ω0 ←
 · E.
~ × (A~ × B)~ = A(~ A~ · B)~ − A2 B~ to arrive atA→~k(~k · E)
~ − k2 E~ + 0 µ 0 ω 2 ←~ = 0.
 ·E

(8.7)

√
If we now use the relation for the speed of light c = 1/ µ0 0 and define ~n ≡ c~k/ω, thenthe Fresnel equation results [200]→~ = 0,
n2~1 − ← − ~n : ~n · E(8.8)

with ~1 the identity matrix and ~n · ~n = ni nj the dyadic product. Because the dielectric

145
tensor is a given, fixed quantity, the normal modes follow from the solutions of→
det n2~1 − ← − ~n : ~n .

(8.9)

One interesting feature of this expression is that it gives at most two independent vector
solutions for ~n [197]. The eigenmodes for the electric field are also required, and these
also follow directly from the solution of Equation 8.9.
In the transverse geometry, the solutions of Equation 8.9 are
n2s = 1 , n2p = 1 +

22
,
1

(8.10)

and we immediately see that there is no magnetic coupling for s-polarized light because
it depends only on 1 . The electric field k-vectors corresponding to these modes are asshown in Figure 8.2:
~ s = E0 x̂,• s-polarized, Ez = Ey = 0, so E√
~ p = E0 (ŷ + ẑ)/ 2.
• p-polarized, Ez = Ey , Ex = 0, so E
Unique to the transverse geometry, the normal modes of the system are the two directions of linearly-polarized light, unlike the P-MOKE and L-MOKE geometries wherecircular polarizations comprise the normal modes. This is significant because in thetransverse geometry, linearly-polarized light reflected from a magnetic material will
experience a magnetism-dependent change in intensity, rather than a polarization rotation. Remarkably, the index of refraction is directly proportional to the off-diagonalcomponent (which is in turn proportional to the magnetization in Equation 8.1), so
the change in reflectivity is directlyproportionaltothemagnetization in the T-MOKEgeometry [201].

8.2.2

Microscopic (fundamental) description

To this point, we have considered magneto-optic effects in terms of macroscopic
explanations and symmetry arguments. The fundamental origins of the magneto-optical

146
effect are electron transition selection rules and spin-orbit coupling. In the case of
linear, p-polarized light with components in the y and z directions, the selection rules~ toare ∆l = ±1 and ∆ml = 0, ±1. Spin-orbit coupling links the orbital momentum L~ allowing optical probing of the spin state as shown below.
the spin momentum S,
The fundamental quantity of the magneto-optic interaction is not the dielectric
tensor, but the conductivity, which is related to the dielectric tensor by
←
→
←
→
→
 = 1 + 4πi←
σ.

(8.11)

Making use of Equation 8.1, which says that the off-diagonal components of the dielectric~ , we see that the conductivity is directly
tensor are proportional to the magnetization M
proportional to the magnetization, and by Equation 8.10, directly proportional to theindex of refraction (and therefore the detected signal) for p-polarized light.
The conductivity (and the related off-diagonal components of the dielectric tensor) can be calculated by [197]
β

σαβ =

1
iq 2 X f (n ) − f (n0 ) Παn0 n Πnn0
= 2e
,
4πi
me h̄Vωnn0
ω + − ωnn0
0

(8.12)

nn

where qe and me are the electron charge and mass, En is the energy of state n, hωnn0 =En − En0 , f (E) is the Fermi-Dirac function, Παnn0 = hn | pα | n0 i the matrix element of
the momentum operator, and τ , the lifetime broadening of the transition [78].
The matrix elements Παnn0 determine the transition strengths and are proportional
to the density of available electron states. In an unmagnetized medium, the ↑ and ↓ spinstates have the same valence levels, but in a magnetized system, the valence levels of the~ where the magnetic field B~ is proportional↑ and ↓ spin states shift according to −µ · B,
~ . This shift of the valence levels, illustrated in the left panel
to the magnetization M
of Figure 8.3, causes an energy shift in the density of electron states dependent on themagnetic spin, illustrated in the top-right of Figure 8.3. The full-contrast magnetic
signal( probed by excitation of an inner-shell electron), given by the difference between

147
the conductivity of the ↑ and ↓ spin states has an up-down shape centered at the Fermi
level, as shown in the bottom right panel of Figure 8.3 [202,203]. If optical probe light isused, the situation is much less clear because the matrix elements are a convoluted messof smeared contributions from initial and final electron energies that may be anywherein the continuum, rather than starting with a defined, bound energy. This removes the
direct proportionality between magnetization and total signal strength, and gives muchsmaller and more complex signals [202].

ValenceEF
3dEnergyEnergyErskine et. al, PRB 8 1239 (1973)

Theory
Core
2p3/2
2p1/2

L3
L2

Figure 8.3: Microscopic origin of the magneto-optic Kerr effect at an absorption edge.
The applied magnetic field shifts the valence levels for up and down spins relative to eachother, which affects the relative density of states. The difference between the available
spin-up and spin-down density of states goes into Equation 8.12 for the conductivity,
which is proportional to the magneto-optic signal.

148

8.3

Experimental Magneto-Optic Kerr effect using EUV from HHG
Previous experimental studies of magneto-optics in the EUV region of the spec-

trum have been limited to synchrotron sources. Particularly relevant work is summarized in Figure 8.4, which shows the measured Faraday rotation signal as a functionof energy for a permalloy (Ni80 Fe20 ) sample [194] and the T-MOKE signal near 55 eVfrom an iron sample [192]. There is great interest in using HHG light for these same
sorts of measurements since the source is more accessible and also offers much better
temporal resolution.

ab

Figure 8.4: Results of previous magneto-optic experiments at the M-edge, demonstrating the expected signal. a) shows results of the Faraday effect on Peralloy, and b) shows
angle-resolved measurements on iron.

8.3.1

Experimental setup

In the experiment, we use light from a Ti:sapphire laser amplifier system operated
at 3 kHz with an average power of 4.3 W, which we couple into a hollow fused-silica
capillary (Figure 8.5a). The capillary is filled with neon gas to phase-match a broad
range of harmonics from 42 eV to 72 eV covering both Fe and Ni M-edges, as shown in
Figure 8.7. The comb of harmonics emerging from the capillary is separated from theresidual driving laser light by transmission through two 200 nm thick aluminum filters.

149
The harmonics are then focused by a near-glancing reflection from a 2 m focal lengthgold-coated curved mirror. Finally, they are spectrally resolved by diffraction from a 1µm line (2 µm period) ferromagnetic permalloy grating, Figure 8.8), and the resulting
diffraction is recorded on an x-ray CCD. The Permalloy grating was fabricated by alift-off process whereby photolithography was used to define the grating pattern over a
1 cm ×1 cm area on a thermally oxidized Si wafer. Ion beam deposition was used todeposit a 3 nm Ti seed layer and a 10 nm Permalloy layer which was lifted-off in an
ultrasonic acetone bath.

abFigure 8.5: Typical CCD image, showing reflected beam containing all harmonics (onleft) and diffracted monochromatic stripes (first-order in middle, second-order on right).

!
Figure 8.6: Typical CCD image, showing reflected beam containing all harmonics (onleft) and diffracted monochromatic stripes (first-order in middle, second-order on right).

A typical energy-calibrated harmonic spectrum is displayed in Figure 8.7, whichshows the discrete harmonics diffracting from the permalloy grating. Due to the spherical symmetry of the non-linear medium (noble gas), each harmonic is separated by 3.16

Intensity normalized to reflected peak

150

0.4

0.3

0.2

0.1

0.0
40

45

50

55

60

65

70

75

Photon energy (eV)Figure 8.7: Typical harmonic spectrum in neon.

eV, which is twice the photon energy of the fundamental laser light of 1.58 eV. The
sudden edge above which no harmonic light is seen corresponds to the strong absorption edge of the aluminum filters at 72 eV. It is then straightforward to calibrate theharmonic spectrum: we identify the photon energy immediately below the Al edge (72.5
eV/1.58 eV = 45.8 harmonic order at cutoff, so last harmonic before absorption edge
is given by 1.58 eV ×45 = 71.1 eV) and determine the energies of the lower harmonics
accordingly. The angle of incidence and the position of higher diffraction orders can also
be related to this information by an oblique incidence diffraction formula for a grating.
The system is set up in a transverse geometry (T-MOKE), i.e. the magnetic
field is applied perpendicular to the plane of incidence, as shown in Figure 8.5a. Since
only p-polarized light contributes to the T-MOKE signal, we rotate the polarizationof the fundamental laser light, which minimizes the reflected intensity of harmonics
from the sample. The sample magnetization is oriented by a Helmholtz coil capable ofproducing magnetic fields as large as ±60 Gauss, well above the coercivity field (fieldrequired to reverse the magnetization) of the permalloy of ∼ 10 Gauss Gauss. The coilmust be water-cooled since it is inside a vacuum chamber with a pressure of 10−5 torr.

151

Figure 8.8: Scanning electron microscope images of a permalloy grating fabricated by
Justin Shaw at NIST Boulder, shown at 1250× (top) and 16000× (bottom) magnification. The ferromagnetic permalloy (Ni80 Fe20 ) stripes are 1 µm wide and separated by
2 µm (center-to-center), with a height of 10 nm.

The diffracted harmonic intensities are measured at magnetic fields of ±60 Gauss andexpressed in terms of asymmetry, defined in Equation 8.13. In order to improve thesignal-to-noise level of the measurement, we alternate the field between +60 and -60
Gauss (to flip the magnetization completely, as shown in Figure 8.5b) between each152
exposure of the harmonic spectrum, and average these two sets of exposures before
calculating the asymmetry. This removes low frequency noise due to slow drifts in theharmonic intensity and pointing.

8.3.2

Magnetic contrast measured with EUV light from HHG

This geometry enables straightforward detection of the magnetic state with soft xray light. The strength of the T-MOKE is traditionally reported in terms of asymmetry
between two magnetization states, defined as
A=I↑ − I↓I↑ + I↓(8.13)

where I↑ and I↓ denote the reflected intensities recorded for the two magnetizations.
The shape of the resulting asymmetry for different incidence angles of the probe lightis shown in Figure 8.10 as the colored lines. Large changes in sign and magnitude ofasymmetry are found at M absorption edges of Ni (67 eV) and Fe (55 eV). Because the
T-MOKE signal is so strong at the absorption edges, the signal is clearly visible after
only a second of acquisition, and a signal-to-noise ratio of more than 30 is evident in
100 seconds. The shape of the asymmetry, shown along with raw spectral data from the
CCD in Figure 8.9, is in agreement with both theoretical arguments based on spin-orbit
coupling in the valence band discussed in section 8.2.2, and with the measurement of
Faraday rotation in a permalloy film using synchrotron radiation [194]. We also show
the dependence of the T-MOKE signal on polarization by repeating the measurementwith s-polarizated light, where we see no asymmetry, as shown in Figure 8.10. We
measure asymmetries up to 6% near the Ni M-edge, when the angle of incidence of theprobe light is nearest 45 degrees [185], and observe a decrease as the angle of incidence
is increased.

153
200

Harmonics diffracted
from permalloy grating:energy labeled in eV

0.05

53

0.04

62

71

100

47

65
44

0.02

0

Zero orderreflection

Al edge(72eV)

Ni edge at 67eVCountsAsymmetry

0.03

50
59 56

68

0.01

Fe edge at 55eV
-100
0

-0.01
200

300

400

500

600

700

-200
800

Pixel

Figure 8.9: Spectral data from CCD. The blue curve (right axis) shows the total counts
measured in a single 100 ms exposure (data collected by full vertical binning an image,as shown in Figure 8.6). The red curve (left axis) shows the asymmetry after a seriesof exposures with the magnetization alternating between up and down.

8.3.3

Hysteresis measurement

In order to demonstrate the use of the HHG source as a useful analytical probeof magnetization, we measured the hysteresis of the asymmetry, which is proportional
to magnetization. Figure 8.11 shows three hysteresis loops: the first one (a) monitoring
the harmonic close to the Ni edge, the second (b) close to the Fe edge, and the third (c)is data from an element-insensitive optical measurement of the T-MOKE signal [204].
As expected, both edges yield identical hysteresis loops, with a coercivity field of around
10 Gauss.

154s polarization
55 deg62 deg72 deg0.06

Asymmetry

0.04
0.02
0
-0.02
-0.04
-0.06
40

45

50

55

60

65

70

Photon Energy (eV)Figure 8.10: Magnetic asymmetry from the permalloy grating sample at various angles
of incidence from normal (p-polarized light). The asymmetry increases as the angle of~ ×E~ is maximized there (see Equation 8.1). Also
incidence approaches 45o , since M
shown is data from s-polarized light, which shows no asymmetry, as expected from
Equation 8.10.

8.3.4

Surface sensitivity measurement

As with soft x-ray light from other sources, light from HHG interacts strongly
with most materials, and the short penetration depth can be utilized to examine surfacesensitive processes. Furthermore, light just below an absorption edge has a penetration
depth ∼ 4× longer than light above the edge [26], so depth-dependent properties ofmagnetic materials can be measured with elemental specificity.
In order to demonstrate the surface sensitivity of this technique, we measure
the asymmetry of permalloy gratings with nonmagnetic palladium films of different
thickness coated on top, as shown in Figure 8.12. Here a series of identical Permalloy
gratings had varying thickness (2 - 15 nm) of sputtered Pd deposited on top. Pd was usedsince it doesnt oxidize or have an M-edge signature near that of Ni or Fe. This allows us
to systematically bury the Permalloy and measure how quickly the signal decays for Fe
and Ni strictly from adsorption by the Pd. We perform the measurements at an angle of

155

a

c

b
d

Figure 8.11: EUV measurement of element-specific asymmetry as a function of magnetic
field (hysteresis) near the nickel (a) and iron (b) edges, with the selected regions for (a)and (b) highlighted in (d). Part (c) shows an optical hysteresis measurement performedin the Silva lab at NIST, Boulder [204].

incidence of 63 degrees, where we had observed good reflectivity of p-polarized light aswell as a high asymmetry value (see Figure 8.10). The measured asymmetries near theNi and Fe absorption edges are plotted in Figure 8.13 as a function of permalloy capping
layer thickness. The measurements show an exponential decay of the asymmetry as afunction of the capping layer thickness. Exponential fits give decay constants of 3.01 nm(Ni edge) and 2.67 nm (Fe edge). This shallow penetration depth shows the potential
for using this technique as a surface-sensitive probe for characterization of magnetic
structure, and also for surface magnetization imaging.

8.4

Outlook: EUV measurement of demagnetization dynamics
As discussed in Chapter 3, HHG offers significant advantages over other sourcesof EUV light because the sub-10 fs pulse duration [48] provides excellent temporal resolution for dynamics measurements. This makes HHG an ideal light source for studying

Palladium capping layer

156

Figure 8.12: Schematic demonstrating the presence of the palladium capping layer,
present over the whole surface (not just the nanostructure).

Figure 8.13: Measurement of element-specific asymmetry as a function of capping layer
thickness. The red and blue data correspond to data taken at the harmonics labelled in
the inset. In each case, the penetration depth was measured to be 3 nm, demonstrating
the highly surface-sensitive nature of the measurement.

demagnetization dynamics following thermal excitation by an ultrashort intense laser
pulse [205–207]. The pulse duration advances the currently available soft x-ray source
resolution of the fs slicing of synchrotron radiation by more than an order of magnitude [208].
Because the soft x-ray light from HHG is fully spatially coherent [46], coherent

157
diffractive imaging techniques can be applied to magnetic systems in order to determine
the domain sizes and patterning which can scale from a fewµm down to few nm. The
lensless imaging techniques demonstrated by Sandberg et. al [61] could be adapted tomagnetic systems as a new microscope for observing magnetic domain patterns similar
to the Fresnel-zone plate technique [86] or a scanning Kerr microscope [209]. This
technique could be applied to image magnetic orientation in nanostructures, where
exotic effects are predicted [210, 211].

8.5

Conclusion
We measure the T-MOKE effect on permalloy at the M-edges of Fe and Ni with

soft x-ray light from a HHG source. By using photon energies near absorption edgesof the elements of interest, the T-MOKE signal is much higher than at optical wavelengths and the information about a magnetic material can be measured with elemental
specificity, high signal-to-noise, and short acquisition time. We find high asymmetries
at 55 eV (4%) and 65 eV (6%), corresponding to the absorption edges of Fe and Ni, respectively. We further demonstrate that light from HHG is a useful source for studying
magnetic materials by the measurement of hysteresis and the surface sensitivity of theasymmetry. This work enables many further possibilities for a tabletop ultrafast lightsource to study magnetic systems.

Chapter 9

Conclusion

One exciting aspect about applying EUV light to surface science is the broad
range of available phenomena that EUV light lends itself to studying. In this thesis, I
have focused on the application of Extreme Ultraviolet/soft x-ray light from high-orderharmonic generation to study acoustic, thermal, and magnetic processes in nanostructures. In the previous chapters, we:• Performed dynamic EUV holograpy and surface displacement measurement with
sub-picometer sensitivity [212]. In this experiment, we demonstrated the factthat in the EUV region of the electromagnetic spectrum, small density changes
do not change the index of refraction, so our surface displacement measurement
was unpolluted by changes in reflectivity that dominate optical measurements.
• Made a dynamic measurement of surface acoustic wave propagation, observing
the highest-freqency narrowband SAWs ever detected optically and the dispersive effect of acoustic propagation in a nanostructure [213]. Here, we took advantage of the full spatial coherence of phase-matched harmonic generation [46].
• Observed quasi-ballistic thermal transport away from a nanostructured heatsource, and quantified the size-dependent error in the Fourier law [120]. This
experiment utilized the short wavelength of the EUV light to not only diffract
from sub-100 nm structures that would be undetectable to optical light, but also

159
provide a sensitive, interferometric measurement of displacement that could beconnected to temperature.
• Probed the magnetic orientation using the T-MOKE for the first time using
EUV light from HHG [214]. Here, we took advantage of the elemental selectivityof the EUV light: by looking at photon energies near an atomic absorption edge,
we saw a significant increase in magnetic signal and were able to measure themagnetic orientation of an alloy with elemental specificity.
Each of the last three chapters detailing experimental observation of surface
acoustic wave (Chapter 6), thermal (Chapter 7) and magnetic (Chapter 8) effects innanostructures included a brief description of the immediate future of that experiment.
Indeed, the future is bright for application of EUV light from high-order harmonic generation to study surface science. Of course, the real power of these experiments lies inthe short wavelength of the EUV probe light. We are currently taking advantage of
surface sensitivity (and elemental selectivity in the magnetic studies), but the potential
for small spot sizes and diffraction from very small objects has not yet been explored in
the context of thermally-sensitive measurements. Probing of a small spot on an isolated
nanostructure would enable other experimental possibilities studying a broad range ofphenomena, such as:• In acoustics experiments, high-resolution surface profile imaging would give insight into the acoustic mode shaping as a surface acoustic wave passes througha nanostructure.
• Time-resolved thermal microscopy of the surface near a sub-100 nm line would
give direct insight into the dynamic evolution of the non-local temperature
distribution shown schematically in Figure 7.1.
• Dynamic microscopy of magnetic systems offers the potential to time-resolve

160
the magnetic orientation of sub-100 nm domains, something no other tabletop
source can do.
The simultaneously short pulse duration and short photon wavelength of EUV
light from HHG makes it a uniquely useful tool in studying fundamental length andtime scales of processes. This thesis has discussed some of the first applications of thissource to study acoustic, thermal, and magnetic effects in nanostructures, and the future
prospects for dynamic imaging these and other effects with sub-10 fs time resolution
and sub-100 nm spatial resolution is very exciting.

Bibliography[1] C. A. Paddock and G. L. Eesley, “Transient thermoreflectance from thin metal
films,” Journal of Applied Physics, vol. 60, no. 1, pp. 285–290, Jul 1986.
[2] M. Berglund, L. Rymell, M. Peuker, T. Wilhein, and H. Hertz, “Compact waterwindow transmission X-ray microscopy,” Journal of Microscopy- Oxford, vol. 197,
no. Part 3, pp. 268–273, Mar 2000.
[3] J. M. Schins, P. Breger, P. Agostini, R. C. Constantinescu, H. G. Muller, G. Grillon, A. Antonetti, and A. Mysyrowicz, “Observation of laser-assisted auger decay
in argon,” Phys. Rev. Lett., vol. 73, no. 16, pp. 2180–2183, Oct 1994.
[4] D. E. Sayers, E. A. Stern, and F. W. Lytle, “New technique for investigating
noncrystalline structures: Fourier analysis of the extended x-ray absorption finestructure,” Physical Review Letters, vol. 27, p. 1204, 1971.
[5] N. W. Ashcroft and N. D. Mermin, Solid State Physics.
Winston, 1976.
[6] M. Born and E. Wolf, Principles of Optics.

Holt, Rinehart, andPergamon Press, Oxford, 1980.

[7] D. vonderLinde, K. Sokolowski-Tinten, and J. Bialkowski, “Laser-solid interactionin the femtosecond time regime,” Applied Surface Science, vol. 110, pp. 1–10, Feb
1997.
[8] R. W. Schoenlein, W. Z. Lin, J. G. Fujimoto, and G. L. Eesley, “Femtosecond
studies of nonequilibrium electronic processes in metals,” Phys. Rev. Lett., vol. 58,
no. 16, pp. 1680–1683, Apr 1987.
[9] D. E. Spence, P. N. Kean, and W. Sibbett, “60-fsec pulse generation from aself-mode-locked ti:sapphire laser,” Opt. Lett., vol. 16, no. 1, pp. 42–44, 1991.
[Online]. Available: http://ol.osa.org/abstract.cfm?URI=ol-16-1-42[10] M. T. Asaki, C.-P. Huang, D. Garvey, J. Zhou, H. C. Kapteyn, and M. M.
Murnane, “Generation of 11-fs pulses from a self-mode-locked ti:sapphirelaser,” Opt. Lett., vol. 18, no. 12, pp. 977–979, 1993. [Online]. Available:
http://ol.osa.org/abstract.cfm?URI=ol-18-12-977[11] W. R. Rapoport and C. P. Khattak, “Titanium sapphire laser characteristics,”
Appl. Opt., vol. 27, no. 13, pp. 2677–2684, 1988. [Online]. Available:
http://ao.osa.org/abstract.cfm?URI=ao-27-13-2677

162[12] S. Backus, R. Bartels, S. Thompson, R. Dollinger, H. C. Kapteyn, and M. M.
Murnane, “High-efficiency, single-stage 7-khz high-average-power ultrafast laser
system,” Opt. Lett., vol. 26, no. 7, pp. 465–467, 2001.
[13] J. Hohlfeld, S. S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke, and E. Matthias,“Electron and lattice dynamics following optical excitation of metals,” Chemical
Physics, vol. 251, no. 1-3, pp. 237–258, Jan 2000.
[14] Luxpop online collection of optical properties of materials. [Online]. Available:
www.luxpop.com[15] S. I. Anisimov, “Effect of very short laser pulses on absorbing substances,” Soviet
Physics JETP, vol. 31, no. 1, July 1970.
[16] S. Anisimov, B. Kapeliovich, and T. Perel’man, “Electron emission from metal
surfaces exposed to ultrashort laser pulses,” Soviet Physics JETP, vol. 39, no. 2,
August 1974.
[17] Z. B. Lin and L. V. Zhigilei, “Temperature dependences of the electron-phonon
coupling, electron heat capacity and thermal conductivity in ni under femtosecond
laser irradiation,” Applied Surface Science, vol. 253, no. 15, pp. 6295–6300, May
2007.
[18] G. M. Mueller, J. Walowski, M. Djordjevic, G.-X. Miao, A. Gupta, A. V. Ramos,
K. Gehrke, V. Moshnyaga, K. Samwer, J. Schmalhorst, A. Thomas, A. Huetten,
G. Reiss, J. S. Moodera, and M. Muenzenberg, “Spin polarization in half-metals
probed by femtosecond spin excitation,” Nature Materials, vol. 8, no. 1, pp. 56–61,
Jan 2009.
[19] G. Chen, Nanoscale Energy Transport and Conversion. Oxford University Press,
2005.
[20] P. Chantrenne, M. Raynaud, D. Baillis, and J. L. Barrat, “Study of phonon
heat transfer in metallic solids from molecular dynamic simulations,” Microscale
Thermophysical Engineering, vol. 7, no. 2, pp. 117–136, Apr-Jun 2003.
[21] P. Kapitza, “The study of heat transfer in helium ii,” J. Phys. (USSR), vol. 4, p.
181, 1941.
[22] E. T. Swartz and R. O. Pohl, “Thermal-boundary resistance,” Reviews of Modern
Physics, vol. 61, no. 3, pp. 605–668, Jul 1989.
[23] P. E. Hopkins, P. M. Norris, and R. J. Stevens, “Influence of inelastic scattering
at metal-dielectric interfaces,” Journal of Heat Transfer, vol. 130, no. 2, p. 022401,
2008.
[24] Wikipedia, “Thermal contact conductance.” [Online]. Available://en.wikipedia.org/wiki/T hermalc ontactc onductance[25] D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation.
versity Press, 1999.

http :

Cambridge Uni-

163[26] E. Gullikson, “X-ray interactions with matter, center for x-ray optics,” 2009.
[Online]. Available: http : //henke.lbl.gov/opticalc onstants/[27] H. Kapteyn, M. Murnane, and I. Christov, “Extreme Nonlinear Optics: Coherent
X Rays from Lasers,” Physics Today, vol. 58, no. 3, pp. 030 000–44, Mar. 2005.
[28] J. Krause, K. Schafer, and K. Kulander, “High-order Hormonic Genration from
Atoms and Ions in the High-Intensity Regime,” Physical Review Letters, vol. 68,
no. 24, pp. 3535–3538, Jun 15 1992.
[29] T. Auguste, B. Carre, and P. Salieres, “Quasi-phase-matching of high-order harmonics using a modulated atomic density,” Physical Review A, vol. 76, no. 1, JUL
2007.
[30] T. Pfeifer and M. C. Downer, “Direct experimental observation of periodic intensity modulation along a straight hollow-core optical waveguide,” Journal of the
Optical Society of America B-Optical Physics, vol. 24, no. 5, pp. 1025–1029, May
2007.
[31] P. Corkum, “Plasma perspective on Strong-field Multiphoton Ionization,”
Physical Review Letters, vol. 71, no. 13, pp. 1994–1997, SEP 27 1993.
[32] M. Lewenstein, P. Balcou, M. Ivanov, A. LHuillier, and P. Corkum, “Theory ofHigh-Harmonic-Generation by Lof-Frequency Laser Fields,” Physical Review A,
vol. 49, no. 3, pp. 2117–2132, Mar 1994.
[33] L. Keldysh, “Ionization in field of a strong electromagnetic wave,” Soviet Physics
JETP-USSR, vol. 20, no. 5, p. 1307, 1965.
[34] M. Ammosov, N. Delone, and V. Krainov, “Tunnel ionization of complex atoms
and of atomic ions in an alternating electromagnetic field,” Soviet Physics
JETP-USSR, vol. 64, no. 6, p. 1191, 1986.
[35] A. Scrinzi, M. Geissler, and T. Brabec, “Ionization above the coulomb barrier,”
Physical Review Letters, vol. 83, no. 4, p. 706, 1999.
[36] P. Balcou, P. Salieres, A. LHuillier, and M. Lewenstein, “Generalized phasematching conditions for high harmonics: The role of field-gradient forces,”
Physical Review A, vol. 55, no. 4, pp. 3204–3210, Apr 1997.
[37] M. Lewenstein, P. Salieres, and A. LHuillier, “Phase of the Atomic Polarization in
High-order-Harmonic Generation,” Physical Review A, vol. 52, no. 6, pp. 4747–
4754, Dec 1995.
[38] X. Zhang, A. L. Lytle, O. Cohen, M. M. Murnane, and H. C. Kapteyn,“Quantum-path control in high-order harmonic generation at high photon energies,” New Journal of Physics, vol. 10, Feb 29 2008.
[39] C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C.
Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys.
Rev. Lett., vol. 83, no. 11, pp. 2187–2190, Sep 1999.

164[40] A. Rundquist, C. Durfee, Z. Chang, C. Herne, S. Backus, M. Murnane, and
H. Kapteyn, “Phase-matched generation of coherent soft X-rays,” Science, vol.
280, no. 5368, pp. 1412–1415, May 29 1998.
[41] A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M.
Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation ofcoherent extreme-ultraviolet light,” Nature, vol. 421, no. 6918, pp. 51–54, 01
2003/01/02/print. [Online]. Available: http://dx.doi.org/10.1038/nature01222[42] X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn,
M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path
control of high-harmonic generation using counterpropagating light,” Nat
Phys, vol. 3, no. 4, pp. 270–275, 04 2007/04//print. [Online]. Available:
http://dx.doi.org/10.1038/nphys541[43] P. Salieres, A. L’Huillier, and M. Lewenstein, “Coherence control of high-orderharmonics,” Physical Review Letters, vol. 74, no. 19, pp. 3776–3779, May 1995.
[44] T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. G. Tisch, D. D. Meyerhofer,and M. H. R. Hutchinson, “Spatial coherence measurement of soft x-ray radiation
produced by high order harmonic generation,” Phys. Rev. Lett., vol. 77, no. 23,
pp. 4756–4759, Dec 1996.
[45] P. Salieres, T. Ditmire, M. Perry, A. LHuillier, and M. Lewenstein, “Angular distributions of high-order harmonics generated by a femtosecond laser,”
Journal of Physics B-Atomic Molecular and Optical Physics, vol. 29, no. 20, pp.
4771–4786, Oct 28 1996.
[46] R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus,
I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatiallycoherent light at extreme ultraviolet wavelengths,” Science, vol. 297, no. 5580, pp.
376–378, Jul 2002.
[47] E. Constant, D. Garzella, P. Breger, E. Mével, C. Dorrer, C. Le Blanc, F. Salin,
and P. Agostini, “Optimizing high harmonic generation in absorbing gases: Modeland experiment,” Physical Review Letters, vol. 82, no. 8, pp. 1668–1671, Feb 1999.
[48] L. Miaja-Avila, C. Lei, M. Aeschlimann, J. L. Gland, M. M. Murnane, H. C.
Kapteyn, and G. Saathoff, “Laser-assisted photoelectric effect from surfaces,”
Physical Review Letters, vol. 97, no. 11, p. 113604, Sep 15 2006.
[49] A. L. Lytle, X. Zhang, P. Arpin, O. Cohen, M. M. Murnane, and H. C. Kapteyn,“Quasi-phase matching of high-order harmonic generation at high photon
energies using counterpropagating pulses,” Opt. Lett., vol. 33, no. 2, pp. 174–176,
2008. [Online]. Available: http://ol.osa.org/abstract.cfm?URI=ol-33-2-174[50] T. Popmintchev, M.-C. Chen, O. Cohen, M. E. Grisham, J. J. Rocca, M. M.
Murnane, and H. C. Kapteyn, “Extended phase matching of high harmonics
driven by mid-infrared light,” Opt. Lett., vol. 33, no. 18, pp. 2128–2130, 2008.
[Online]. Available: http://ol.osa.org/abstract.cfm?URI=ol-33-18-2128

165[51] X. Zhou, R. Lock, W. Li, N. Wagner, M. M. Murnane, and H. C.
Kapteyn, “Molecular recollision interferometry in high harmonic generation,”
Physical Review Letters, vol. 100, no. 7, Feb 22 2008.
[52] W. Li, X. Zhou, R. Lock, S. Patchkovskii, A. Stolow, H. C. Kapteyn, and M. M.
Murnane, “Time-Resolved Dynamics in N2O4 Probed Using High Harmonic Generation,” Science, vol. 322, no. 5905, pp. 1207–1211, Nov 21 2008.
[53] E. Gagnon, P. Ranitovic, X.-M. Tong, C. L. Cocke, M. M. Murnane, H. C.
Kapteyn, and A. S. Sandhu, “Soft X-ray-driven femtosecond molecular dynamics,”
Science, vol. 317, no. 5843, pp. 1374–1378, SEP 7 2007.
[54] A. S. Sandhu, E. Gagnon, R. Santra, V. Sharma, W. Li, P. Ho, P. Ranitovic,
C. L. Cocke, M. M. Murnane, and H. C. Kapteyn, “Observing the Creation ofElectronic Feshbach Resonances in Soft X-ray-Induced O-2 Dissociation,” Science,
vol. 322, no. 5904, pp. 1081–1085, Nov 14 2008.
[55] R. I. Tobey, “Linear and nonlinear photo-acoustic spectroscopy with extreme ultraviolet radiation,” Ph.D. dissertation, University of Colorado, 2006.
[56] L. Miaja-Avila, G. Saathoff, S. Mathias, J. Yin, C. La-o vorakiat, M. Bauer,
M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Direct measurement of core-level relaxation dynamics on a surface-adsorbate system,”Physical Review Letters, vol. 101, no. 4, p. 046101, Jul 25 2008.
[57] W. Chao, B. Harteneck, J. Liddle, E. Anderson, and D. Attwood, “Soft X-ray
microscopy at a spatial resolution better than 15nm,” Nature, vol. 435, no. 7046,
pp. 1210–1213, Jun 30 2005.
[58] J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodologyof x-ray crystallography to allow imaging of micrometre-sized non-crystalline
specimens,” Nature, vol. 400, no. 6742, pp. 342–344, 07 1999. [Online]. Available:
http://dx.doi.org/10.1038/22498[59] R. L. Sandberg, A. Paul, D. A. Raymondson, S. Haedrich, D. M. Gaudiosi,
J. Holtsnider, R. I. Tobey, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Lensless diffractive imaging using tabletop coherent high-harmonic soft-x-ray beams,”
Physical Review Letters, vol. 99, no. 9, p. 098103, Aug 31 2007.
[60] A. J. Paul, “Coherent euv light from high-order harmonic generation: Enhancement and applications to coherent euv light from high-order harmonic generation:
Enhancement and applicaitons to lensless diffractive imaging,” Ph.D. dissertation,
University of Colorado, 2007.
[61] R. Sandberg, D. Raymondson, W. Slaughter, M. Murnane, and H. Kapteyn,“Closing the gap to the diffraction limit: Tabletop soft x-ray fourier transform
holography,” Optics Letters, 2009 (In press).
[62] D. J. Funk, D. S. Moore, S. D. McGrane, J. H. Reho, and R. L. Rabie, “Ultrafast spatial interferometry: a tool for characterizing material phase and hydrodynamic motion in laser-excited metals,” Applied Physics A-Materials Science &Processing, vol. 81, no. 2, pp. 295–302, Jul 2005.

166[63] V. Temnov,
K. Sokolowski-Tinten,
P. Zhou,and D. von der
Linde,“Femtosecond
time-resolved
interferometricmicroscopy,”
Applied Physics A-Materials Science & Processing, vol. 78, no. 4, pp. 483–
489, Mar 2004.
[64] D. H. Hurley and O. B. Wright, “Detection of ultrafast phenomena by use of amodified sagnac interferometer,” Opt. Lett., vol. 24, no. 18, pp. 1305–1307, 1999.
[Online]. Available: http://ol.osa.org/abstract.cfm?URI=ol-24-18-1305[65] J. Filevich, J. J. Rocca, M. C. Marconi, R. F. Smith, J. Dunn, R. Keenan,
J. R. Hunter, S. J. Moon, J. Nilsen, A. Ng, and V. N. Shlyaptsev, “Picosecondresolution soft-x-ray laser plasma interferometry,” Appl. Opt., vol. 43, no. 19, pp.
3938–3946, 2004. [Online]. Available: http://ao.osa.org/abstract.cfm?URI=ao43-19-3938[66] C. Chang, E. Anderson, P. Naulleau, E. Gullikson, K. Goldberg, and D. Attwood,“Direct measurement of index of refraction in the extreme-ultraviolet wavelength
region with a novel interferometer,” Opt. Lett., vol. 27, no. 12, pp. 1028–1030,
2002. [Online]. Available: http://ol.osa.org/abstract.cfm?URI=ol-27-12-1028[67] G. L. Eesley, “Generation of nonequilibrium electron and lattice temperatures incopper by picosecond laser-pulses,” Physical Reivew B, vol. 33, no. 4, pp. 2144–
2151, Feb 1986.
[68] E. Hecht, Optics, 3rd ed.

Addison Wesley Publishing Company, 1997.

[69] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of
Propagation, Interference and Diffraction of Light, 7th ed. Cambridge University
Press, 1999.
[70] J. W. Goodman, Introduction to Fourier Optics.

McGraw-Hill, Boston, 1996.

[71] M. Vicanek,
A. Rosch,
F. Piron,and G. Simon,“Thermaldeformationofasolid-surface
under
laser
irradiation,”
Applied Physics A-Materials Science & Processing, vol. 59, no. 4, pp. 407–
412, Oct 1994.
[72] C. J. K. Richardson, M. J. Ehrlich, and J. W. Wagner, “Interferometric detectionof ultrafast thermoelastic transients in thin films: theory with supporting experiment,” Journal of the Optical Society of America B-Optical Physics, vol. 16, no. 6,
pp. 1007–1015, Jun 1999.
[73] M. van Kampen, J. Kohlhepp, W. de Jonge, B. Koopmans, and R. Coehoorn, “Sub-picosecond electron and phonon dynamics in nickel,” Journal of
Physics-Condensed Matter, vol. 17, no. 43, pp. 6823–6834, Nov 2 2005.
[74] N. W. Pu and T. C. Li, “Pulse-echo measurement of longitudinal sound velocity
in nanometer thin films,” Appled Physics B-Lasers and Optics, vol. 82, no. 3, pp.
449–453, Mar 2006.

167[75] C. Thomsen, J. Strait, Z. Vardeny, H. J. Maris, J. Tauc, and J. J. Hauser, “Coherent phonon generation and detection by picosecond light pulses,” Physical Review
Letters, vol. 53, no. 10, pp. 989–992, 1984.
[76] O. B. Wright and K. Kawashima, “Coherent phonon detection from ultrafast
surface vibrations,” Phys. Rev. Lett., vol. 69, no. 11, pp. 1668–1671, Sep 1992.
[77] T. Saito, O. Matsuda, and O. B. Wright, “Picosecond acoustic phonon pulse
generation in nickel and chromium,” Physical Reivew B, vol. 67, no. 20, p. 205421,
May 2003.
[78] M. van Kampen, “Ultrafast spin dynamics in ferromagnetic metals,” Ph.D. dissertation, Eindhoven University of Technology, 2003.
[79] J. G. M. S. S. W. J. Hohlfield, U. Conrad and E. Matthias, “Femtosecond timeresolved linear and second-order reflectivity of metals,” in Nonlinear Optics inMetals, ser. Oxford Science Publications, K. H. Bennemann, Ed. Clarendon
Press, 1998, ch. 4, pp. 219–267.
[80] X. Wang and X. Xu, “Thermoelastic wave induced by pulsed laser heating,”
Applied Physics A-Materials Science & Processing, vol. 73, no. 1, pp. 107–114,
Jul 2001.
[81] O. Matsuda and O. B. Wright, “Reflection and transmission of light in multilayers
perturbed by picosecond strain pulse propagation,” Journal of the Optical Societyof America B-Optical Physics, vol. 19, no. 12, pp. 3028–3041, Dec 2002.
[82] C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation anddetection of phonons by picosecond light pulses,” Physical Reivew B, vol. 34,
no. 6, pp. 4129–4138, Sep 1986.
[83] O. Matsuda, K. Aoki, T. Tachizaki, and O. B. Wright, “Direct measurement of
ultrafast surface displacement in laser picosecond acoustics,” Journal de PhysiqueIV, vol. 125, pp. 361–363, Jun 2005.
[84] M. Shiga and G. P. Pells, “The optical properties of nickel above and below thecurie temperature,” Journal of Physics C (Solid State Physics), vol. 2, no. 2, pp.
1847–1856, 1969.
[85] A. Paul and R. Sandberg, personal communication regarding gaussian parameters
used to model the coherent HHG mode emerging from the capillary.
[86] P. Fischer, T. Eimuller, G. Schutz, G. Denbeaux, A. Pearson, L. Johnson, D. Attwood, S. Tsunashima, M. Kumazawa, N. Takagi, M. Kohler, and
G. Bayreuther, “Element-specific imaging of magnetic domains at 25 nm spatial
resolution using soft x-ray microscopy,” Review of Scientific Instruments, vol. 72,
no. 5, pp. 2322–2324, May 2001.
[87] P. Fenter, C. Park, Z. Zhang, and S. Wang, “Observation of subnanometrehigh surface topography with x-ray reflection phase-contrast microscopy,” Nature
Physics, vol. 2, no. 10, pp. 700–704, Oct 2006.

168[88] T. Barbee and D. Keith, “Synthesis of metastable materials by sputter deposition
techniques,” in Synthesis and Properties of Metastable Phases, E. Machlin and
T. Rowland, Eds. Metalurgical Society, Amer. Inst. Mech. Eng., 1980, p. 93.
[89] J. Underwood and T. Barbee, “Soft-x-ray imaging with a normal incidence mirror,” Nature, vol. 294, no. 5840, pp. 429–431, 1981.
[90] R. Haelbich and C. Kunz, “Multilayer interference mirrors for the xuv range
around 100 ev photon energy,” Optics Communications, vol. 17, no. 3, pp. 287–
292, 1976.
[91] Optometrics
llc
opticalwww.optometrics.com

components

catalog.

[Online].

Available:[92] E. Takahashi, H. Hasegawa, Y. Nabekawa, and K. Midorikawa, “High-throughput,
high-damage-threshold broadband beam splitter for high-order harmonics in theextreme-ultraviolet region,” Optics Letters, vol. 29, no. 5, pp. 507–509, Mar 1
2004.
[93] W. R. Hunter, “Thin films and their uses for the extreme ultraviolet,” Applied
Optics, vol. 4, p. 891898, 1965.
[94] G. N. Steele, “Development and fabrication of large-area extreme ultraviolet filters
for the applo telescope month,” Space Optics Proceedings, Ninth International
Congress of the International Commission for Optics, 1974.
[95] W. R. Hunter, “The preparation and use of unbacked metal films as filters in theextreme ultraviolet,” Phys. Thin Films, vol. 7, pp. 43–114, 1973.
[96] W. R. Hunter, J. D. Purcell, and G. N. Steele, “Evaluation of pinholes in unbacked
metal film filters to be used in rocket- and satellite-borne spectrographs,” Applied
Optics, vol. 12, no. 8, p. 1874, 1973.
[97] C. Glorieux, K. Van de Rostyne, J. Beers, W. Gao, S. Petillion, N. Van Riet,
K. Nelson, J. Allard, V. Gusev, W. Lauriks, and J. Thoen, “Acoustic waves atinterfaces studied by laser ultrasonics (invited),” Review of Scientific Instruments,
vol. 74, no. 1, Part 2, pp. 465–469, Jan 2003.
[98] A. Every, “Measurement of the near-surface elastic properties of solids and thin
supported films,” Measuremend Science & Technology, vol. 13, no. 5, pp. R21–
R39, May 2002.
[99] A. A. Maznev, A. Mazurenko, L. Zhuoyun, and M. Gostein, “Laser-based surface
acoustic wave spectrometer for industrial applications,” vol. 74, no. 1. AIP,
2003, pp. 667–669. [Online]. Available: http://link.aip.org/link/?RSI/74/667/1[100] A. P. M. A. Lomonosov and P. Hess, “Laser-based surface acoustic waves in materials science,” in Modern acoustical techniques for the measurement of mechanicalproperties, ser. Experimental Methods in the Physical Sciences, H. E. B. M. Levy
and R. Stern, Eds. Academic Press, 2001, vol. 39, ch. 3, pp. 65–134.

169[101] J. A. Rogers, M. Fuchs, M. J. Banet, J. B. Hanselman, R. Logan, and K. A.
Nelson, “Optical system for rapid materials characterization with the transient
grating technique: Application to nondestructive evaluation of thin films used in
microelectronics,” Applied Physics Letters, vol. 71, no. 2, pp. 225–227, 1997.
[Online]. Available: http://link.aip.org/link/?APL/71/225/1[102] J. A. Rogers, L. Dhar, and K. A. Nelson, “Noncontact determination of transverse
isotropic elastic moduli in polyimide thin films using a laser based ultrasonicmethod,” Applied Physics Letters, vol. 65, no. 3, pp. 312–314, 1994.
[103] R. I. Tobey, M. E. Siemens, M. M. Murnane, H. C. Kapteyn, D. H. Torchinsky, andK. A. Nelson, “Transient grating measurement of surface acoustic waves in thinmetal films with extreme ultraviolet radiation,” Applied Physics Letters, vol. 89,
no. 9, p. 091108, Aug 2006.
[104] 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,” Applied Physics Letters, vol. 85,
no. 4, pp. 564–566, Jul 2004.
[105] G. A. Antonelli, P. Zannitto, and H. J. Maris, “New method for the generationof surface acoustic waves of high frequency,” Physica B: Condensed Matter, vol.
316, pp. 377–379, May 2002.
[106] G. A. Antonelli, H. J. Maris, S. G. Malhotra, and J. M. E. Harper, “Picosecond
ultrasonics study of the vibrational modes of a nanostructure,” Journal ofApplied Physics, vol. 91, no. 5, pp. 3261–3267, 2002. [Online]. Available:
http://link.aip.org/link/?JAP/91/3261/1[107] B. Bonello, A. Ajinou, V. Richard, P. Djemia, and S. M. Cherif, “Surface acoustic
waves in the ghz range generated by periodically patterned metallic stripes illuminated by an ultrashort laser pulse,” The Journal of the Acoustical Society ofAmerica, vol. 110, no. 4, pp. 1943–1949, 2001.
[108] D. Hurley and K. Telschow, “Picosecond surface acoustic waves using a suboptical
wavelength absorption grating,” Physical Reivew B, vol. 66, no. 15, Oct 15 2002.
[109] C. Giannetti, B. Revaz, F. Banfi, M. Montagnese, G. Ferrini, F. Cilento, S. Maccalli, P. Vavassori, G. Oliviero, E. Bontempi, L. E. Depero, V. Metlushko, and
F. Parmigiani, “Thermomechanical behavior of surface acoustic waves in ordered
arrays of nanodisks studied by near-infrared pump-probe diffraction experiments,”
Physical Reivew B (Condensed Matter and Materials Physics), vol. 76, no. 12, p.
125413, 2007.
[110] H.-N. Lin, H. J. Maris, L. B. Freund, K. Y. Lee, H. Luhn, and D. P. Kern, “Studyof vibrational modes of gold nanostructures by picosecond ultrasonics,” Journalof Applied Physics, vol. 73, no. 1, pp. 37–45, 1993.
[111] D. H. Hurley, R. Lewis, O. B. Wright, and O. Matsuda, “Coherent controlof gigahertz surface acoustic and bulk phonons using ultrafast optical pulses,”170
Applied Physics Letters, vol. 93, no. 11, p. 113101, 2008. [Online]. Available:
http://link.aip.org/link/?APL/93/113101/1[112] D. H. Hurley, “Optical generation and spatially distinct interferometric detectionof ultrahigh frequency surface acoustic waves,” Applied Physics Letters, vol. 88,
no. 19, p. 191106, 2006.
[113] A. A. Maznev, “Band gaps and brekhovskikh attenuation of laser-generated
surface acoustic waves in a patterned thin film structure on silicon,” Physical
Reivew B (Condensed Matter and Materials Physics), vol. 78, no. 15, p. 155323,
2008. [Online]. Available: http://link.aps.org/abstract/PRB/v78/e155323[114] B. A. Auld, Acoustic Fields and Waves in Solids.
Company, 1990, vol. II.

Robert E. Krieger Publishing[115] A. R. Duggal, J. A. Rogers, and K. A. Nelson, “Real-time optical characterizationof surface acoustic modes of polyimide thin-film coatings,” Journal of Applied
Physics, vol. 72, no. 7, pp. 2823–2839, 1992.
[116] H. Robinson, Y. Hahn, and J. N. Gau, “A comprehensive analysis of surface
acoustic wave reflections,” Journal of Applied Physics, vol. 65, no. 12, pp.
4573–4586, 1989. [Online]. Available: http://link.aip.org/link/?JAP/65/4573/1[117] S. Datta and B. J. Hunsinger, “First-order reflection coefficient of surface acoustic
waves from thin-strip overlays,” Journal of Applied Physics, vol. 50, no. 9, pp.
5661–5665, 1979.
[118] International technology roadmap for semiconductors. [Online]. Available:
http://www.itrs.net[119] D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J.
Maris, R. Merlin, and S. R. Phillpot, “Nanoscale thermal transport,” Journal ofApplied Physics, vol. 93, no. 2, pp. 793–818, Jan 2003.
[120] M. Siemens, Q. Li, R. Yang, K. Nelson, E. Anderson, M. Murnane, and
H. Kapteyn, “Quasi-ballistic thermal transport from nanoscale hotspots observed
using ultrafast coherent soft x-ray beams,” Nature (submitted), 2009.
[121] A. A. Joshi and A. Majumdar, A., “Transient ballistic and diffusive phonon heattransport in thin-films,” Journal of Applied Physics, vol. 74, no. 1, pp. 31–39, Jul
1993.
[122] S. Sinha and K. Goodson, “Review: Multiscale thermal modeling in nanoelectronics,” International Journal for Multiscale Computational Engineering, vol. 3,
no. 1, pp. 107–133, 2005.
[123] D. Lacroix, K. Joulain, D. Terris, and D. Lemonnier, “Monte carlo
simulation of phonon confinement in silicon nanostructures: Application to
the determination of the thermal conductivity of silicon nanowires,” Applied
Physics Letters, vol. 89, no. 10, p. 103104, 2006. [Online]. Available:
http://link.aip.org/link/?APL/89/103104/1

171[124] S. K. Saha and L. Shi, “Molecular dynamics simulation of thermal transport at
a nanometer scale constriction in silicon,” Journal of Applied Physics, vol. 101,
no. 7, p. 074304, Apr 2007.
[125] R. A. Escobar, S. S. Ghai, M. S. Jhon, and C. H. Amon, “Multilength and time scale thermal transport using the lattice boltzmann
method with application to electronics cooling,” International Journal of
Heat and Mass Transfer, vol. 49, no. 1-2, pp. 97 – 107, 2006. [Online].
Available:
http://www.sciencedirect.com/science/article/B6V3H-4HVDNBP1/2/263fbe183aac5224ee972f7a63e2c799[126] S. V. J. Narumanchi, J. Y. Murthy, and C. H. Amon, “Boltzmann transport
equation-based thermal modeling approaches for hotspots in microelectronics,”
Heat and Mass Transfer, vol. 42, no. 6, pp. 478–491, Apr 2006.
[127] R. Yang and G. Chen, “Thermal conductivity modeling of periodic twodimensional nanocomposites,” Physical Review B, vol. 69, no. 19, p. 195316, May
2004.
[128] G.
Chen,
“Ballistic-diffusive
heat-conduction
equations,”
Physical Review Letters, vol. 86, no. 11, pp. 2297–2300, Mar 12 2001.
[129] ——, “Ballistic-diffusive equations for transient heat conduction from nano tomacroscales,” Journal of Heat Transfer, vol. 124, no. 2, pp. 320–328, Apr 2002.
[130] A. Majumdar, “Microscale heat-conduction in dielectric thin-films,” Journal of
Heat Transfer, vol. 115, no. 1, pp. 7–16, Feb 1993.
[131] F. X. Alvarez and D. Jou, “Memory and nonlocal effects in heat transport: From
diffusive to ballistic regimes,” Applied Physics Letters, vol. 90, no. 8, p. 083109,
Feb 2007.
[132] G. Chen, “Thermal conductivity and ballistic-phonon transport in the cross-plane
direction of superlattices,” Physical Reivew B, vol. 57, no. 23, pp. 14 958–14 973,
Jun 1998.
[133] W. S. Capinski, H. J. Maris, T. Ruf, M. Cardona, K. Ploog, and D. S. Katzer,“Thermal-conductivity measurements of gaas/alas superlattices using a picosecond optical pump-and-probe technique,” Physical Reivew B, vol. 59, no. 12, pp.
8105–8113, Mar 1999.
[134] Y. S. Ju and K. E. Goodson, “Phonon scattering in silicon films with thickness oforder 100 nm,” Applied Physics Letters, vol. 74, no. 20, pp. 3005–3007, 1999.
[135] R. Costescu, D. Cahill, F. Fabreguette, Z. Sechrist, and S. George, “Ultra-low
thermal conductivity in W/Al2O3 nanolaminates,” Science, vol. 303, no. 5660,
pp. 989–990, Feb 13 2004.
[136] D. Li, Y. Wu, P. Kim, L. Shi, P. Yang, and A. Majumdar, “Thermal conductivityof individual silicon nanowires,” Applied Physics Letters, vol. 83, no. 14, pp.
2934–2936, 2003.

172[137] R. Chen, A. I. Hochbaum, P. Murphy, J. Moore, P. Yang, and A. Majumdar,“Thermal conductance of thin silicon nanowires,” Physical Review Letters, vol.
101, no. 10, p. 105501, Jan 2008.
[138] C. Chiritescu, D. G. Cahill, N. Nguyen, D. Johnson, A. Bodapati, P. Keblinski, and P. Zschack, “Ultralow thermal conductivity in disordered, layered WSe2
crystals,” Science, vol. 315, no. 5810, pp. 351–353, Jan 19 2007.
[139] B. Poudel, Q. Hao, Y. Ma, Y. Lan, A. Minnich, B. Yu, X. Yan, D. Wang, A. Muto,
D. Vashaee, X. Chen, J. Liu, M. S. Dresselhaus, G. Chen, and Z. Ren, “Highthermoelectric performance of nanostructured bismuth antimony telluride bulk
alloys,” Science, vol. 320, no. 5876, pp. 634–638, Jan 2008.
[140] Z. Wang, J. A. Carter, A. Lagutchev, Y. K. Koh, N.-H. Seong, D. G. Cahill, andD. D. Dlott, “Ultrafast flash thermal conductance of molecular chains,” Science,
vol. 317, no. 5839, pp. 787–790, Aug 10 2007.
[141] P. Kim, L. Shi, A. Majumdar, and P. L. McEuen, “Thermal transport measurements of individual multiwalled nanotubes,” Phys. Rev. Lett., vol. 87, no. 21, p.
215502, Oct 2001.
[142] S. Berber, Y.-K. Kwon, and D. Tománek, “Unusually high thermal conductivityof carbon nanotubes,” Phys. Rev. Lett., vol. 84, no. 20, pp. 4613–4616, May 2000.
[143] M. Asheghi, M. Touzelbaev, K. Goodson, Y. Leung, and S. Wong, “Temperaturedependent thermal conductivity of single-crystal silicon layers in SOI substrates,”
Journal of Heat Transfer- Transactions of the ASME, vol. 120, no. 1, pp. 30–36,
Feb 1998.
[144] W. Liu and M. Asheghi, “Phonon-boundary scattering in ultrathin single-crystal
silicon layers,” Applied Physics Letters, vol. 84, no. 19, pp. 3819–3821, 2004.
[Online]. Available: http://link.aip.org/link/?APL/84/3819/1[145] G. Chen, “Nanoscale heat transfer and nanostructured thermoelectrics,” IEEE
Transactions on Components and Packaging Technologies, vol. 29, no. 2, pp. 238–
246, Jun 2006.
[146] F. DiSalvo, “Thermoelectric cooling and power generation,” pp. 703–706, Jan
1999.
[147] C. Wood, “High-temperature thermoelectric energy conversion. 2. materials survey,” Energy Conversoin Management, vol. 24, no. 4, pp. 331–343, Jan 1984.
[148] A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang, “Enhanced thermoelectric performance of roughsilicon nanowires,” Nature, vol. 451, no. 7175, pp. 163–U5, Jan 2008.
[149] E. Pop, S. Sinha, and K. E. Goodson, “Heat generation and transport innanometer-scale transistors,” Proceedings of the IEEE, vol. 94, no. 8, pp. 1587–
1601, Aug 2006.

173[150] J. Lee, T. Beechem, T. L. Wright, B. A. Nelson, S. Graham, and W. P. King,“Electrical, thermal, and mechanical characterization of silicon microcantileverheaters,” Journal of Microelectromechanical Systems, vol. 15, no. 6, pp. 1644–
1655, Dec 2006.
[151] G. Chen, D. Borsa-Tasciuc, and R. Yang, “Nanoscale heat transfer,” in
Encyclopedia of Nanoscience and Nanotechnology, H. Nalwa, Ed. American
Scientific Publishers, 2004, vol. 7, pp. 429–459.
[152] F. Claro and G. D. Mahan, “Transient heat transport in solids,” Journal ofApplied Physics, vol. 66, no. 9, pp. 4213–4217, Nov 1989.
[153] G. D. Mahan and F. Claro, “Nonlocal theory of thermal conductivity,” PhysicalReview B, vol. 38, no. 3, pp. 1963–1969, Jul 1988.
[154] R. G. Yang, G. Chen, M. Laroche, and Y. Taur, “Simulation of nanoscale multidimensional transient heat conduction problems using ballistic-diffusive equations
and phonon boltzmann equation,” Journal of Heat Transfer, vol. 127, no. 3, pp.
298–306, Mar 2005.
[155] S. V. J. Narumanchi, J. Y. Murthy, and C. H. Amon, “Simulation of
unsteady small heat source effects in sub-micron heat conduction,” Journalof Heat Transfer, vol. 125, no. 5, pp. 896–903, 2003. [Online]. Available:
http://link.aip.org/link/?JHR/125/896/1[156] P.
Heino,“Multiscale
Lattice-Boltzmann
finitedifference
model
for
thermalconduction
fromnanoscalehot
spots,”
International Journal for Multiscale Computational Engineering, vol. 6, no. 2,
pp. 169–178, 2008.
[157] D. G. Cahill, “Analysis of heat flow in layered structures for time-domain thermoreflectance,” Review of Scientific Instruments, vol. 75, no. 12, pp. 5119–5122,
2004.
[158] R. Prasher, “Predicting the thermal resistance of nanosized constrictions,” Nano
Letters, vol. 5, no. 11, pp. 2155–2159, 2005.
[159] M. Yovanovich, Y. Muzychka, and J. Culham, “Spreading resistance of isoflux
rectangles and strips on compound flux channels,” Journal of Thermophysics andHeat Transfer, vol. 13, pp. 495–500, 1999.
[160] Mikic and Rohsenow, “Thermal contact resistance,” Rohsenow Heat and Mass
Transfer LaboratoryTechnical Reports, vol. 4542, 1966.
[161] G. Chen, “Nonlocal and nonequilibrium heat conduction in the vicinity ofnanoparticles,” Journal of Heat Transfer, vol. 118, no. 3, pp. 539–545, Aug 1996.
[162] R. Siegel and J. Howell, Thermal Radiation Heat Transfer, 4th ed.
Francis, 2001.

Taylor and[163] C. H. Christiansen, “Absolute bestimmung des emissions-und absorptionsvermogens fur warmes,” Ann. Phys. Wied., vol. 19, p. 267, 1883.

174[164] B. H. Armstrong, “Two-fluid theory of thermal conductivity of dielectric crystals,”
Physical Review B, vol. 23, no. 2, pp. 883–899, Jan 1981.
[165] R. Heid, D. Strauch, and K. P. Bohnen, “Ab initio lattice dynamics of sapphire,”
Physical Reivew B, vol. 61, no. 13, pp. 8625–8627, Apr 2000.
[166] P. G. Sverdrup, “Simulation and thermometry of sub-continuun heat transport insemiconductors,” Ph.D. dissertation, Stanford University, Dec 2000.
[167] P. G. Sverdrup, S. Sinha, M. Asheghi, S. Uma, and K. E. Goodson, “Measurementof ballistic phonon conduction near hotspots in silicon,” Applied Physics Letters,
vol. 78, no. 21, pp. 3331–3333, May 2001.
[168] M. Kresch, O. Delaire, R. Stevens, J. Y. Y. Lin, and B. Fultz, “Neutron
scattering measurements of phonons in nickel at elevated temperatures,” PhysicalReview B (Condensed Matter and Materials Physics), vol. 75, no. 10, p. 104301,
2007. [Online]. Available: http://link.aps.org/abstract/PRB/v75/e104301[169] R. J. Stoner and H. J. Maris, “Kapitza conductance and heat flow between solids
at temperatures from 50 to 300 k,” Physical Reivew B, vol. 48, no. 22, pp. 16 373–
16 387, Dec 1993.
[170] K. E. Goodson, O. W. Kading, M. Rosler, and R. Zachai, “Experimental investigation of thermal conduction normal to diamond-silicon boundaries,” Journal ofApplied Physics, vol. 77, no. 4, pp. 1385–1392, 1995.
[171] A. P. Caffrey, P. E. Hopkins, J. M. Klopf, and P. M. Norris, “Thin film nonnoble transition metal thermophysical properties,” Microscale ThermophysicalEngineering, vol. 9, no. 4, pp. 365–377, Oct-Dec 2005.
[172] sitmo, “Resources for financial engineers, equations archive.” [Online]. Available:
http://www.sitmo.com/eq/262[173] T. G. Kollie, “Measurement of the thermal-expansion coefficient of nickel from
300 to 1000 k and determination of the power-law constants near the curie temperature,” Physical Reivew B, vol. 16, no. 11, pp. 4872–4881, Dec 1977.
[174] Tabulated
data
for
sapphire.
[Online].
http://global.kyocera.com/prdct/fc/product/pdf/s c sapphire.pdf

Available:[175] S. Burghartz and B. Schulz, “Thermophysical properties of sapphire, aln andmgal2o4 down to 70 k,” Journal of Nuclear Materials, vol. 215, pp. 1065–1068,
Sep 1994.
[176] Y. S. Touloukian, Ed., Thermophysical Properties of High Temperature
Solid Materials. The Macmillan Company, 1967, vol. 4. [Online]. Available:
TA418.26.P8[177] Y. S. Touloukian, R. W. Powell, C. Y. Ho, and P. G. Klemens, Eds.,
Thermal conductivity: Nonmetallic Solids, Thermophysical Properties of Matter.
IFO/Plenum, NY, 1970, vol. 4. [Online]. Available: QC171.P83

175[178] W. Nowacki, Thermoelasticity. Addison-Wesley Publishing Company, Inc., 1962.
[179] Landau and Lifshitz, Theory of Elasticity.

Pergamon Press, 1986.

[180] Albagli, D and Dark, M and Vonrosenberg, C and Perelman, L and Itzhan, I and
Feld, MS, “Laser-induced Thermoelastic Deformation - a 3-dimensional Solution
and its Application to the Ablation of Biological Tissue,” Medical Physics, vol. 21,
no. 8, pp. 1323–1331, Aug 1994.
[181] D. G. Cahill, “Thermal conductivity measurement from 30 to 750 k: the 3 omega
method,” Review of Scientific Instruments, vol. 61, no. 2, pp. 802–808, 1990.
[Online]. Available: http://link.aip.org/link/?RSI/61/802/1[182] G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Review
of Scientific Instruments, vol. 74, no. 1, pp. 1–18, Jan 2003.
[183] A. Technology, “Ccd spectral response (quantum efficiency).” [Online]. Available:
http : //www.andor.com/learn/digitalc ameras/?docID = 315[184] P. Pershan, “Magneto-optical effects,” Journal of Applied Physics, vol. 38, no. 3,
pp. 1482–&, 1967.
[185] M. Freiser, “A survey of magnetooptic effects,” IEEE Transactions on Magnetics,
vol. MAG4, no. 2, pp. 152–&, 1968.
[186] H. Mertins, S. Valencia, A. Gaupp, W. Gudat, P. Oppeneer, and
C. Schneider, “Magneto-optical polarization spectroscopy with soft X-rays,”
Applied Physics A-Materials Science & Processing, vol. 80, no. 5, pp. 1011–1020,
Feb 2005.
[187] D. Kim, P. Fischer, W. Chao, E. Anderson, M. Im, S. Shin, and S. Choe, “Magnetic soft x-ray microscopy at 15 nm resolution probing nanoscale local magnetic
hysteresis (invited),” Journal of Applied Physics, vol. 99, no. 8, APR 15 2006.
[188] M. Faraday, Philisophical Transactions of the Royal Society, vol. 136, p. 1, 1846.
[189] J. Kerr, Report of the British Association for the Advancement of Science, p. 40,
1876.
[190] ——, Philisophical Magazine, vol. 5, p. 161, 1878.
[191] P. Zeeman, Leiden Communications, vol. 29, 1896.
[192] M. Hecker, P. Oppeneer, S. Valencia, H. Mertins, and C. Schneider, “Soft X-ray
magnetic reflection spectroscopy at the 3p absorption edges of thin Fe films,”
Journal of Electron Spectroscopy and Related Phenomena, vol. 144, no. Sp. Iss.
SI, pp. 881–884, Jun 2005.
[193] M. Pretorius, J. Friedrich, A. Ranck, M. Schroeder, J. Voss, V. Wedemeier,D. Spanke, D. Knabben, I. Rozhko, H. Ohldag, F. U. Hillebrecht, and E. Kisker,“Transverse magneto-optical kerr effect of fe at the fe 3p threshold,” Physical
Reivew B, vol. 55, no. 21, pp. 14 133–14 135, Jun 1997.

176[194] S. Valencia, A. Gaupp, W. Gudat, H.-C. Mertins, P. M. Oppeneer, D. Abramsohn,and C. M. Schneider, “Faraday rotation spectra at shallow core levels: 3p edgesof Fe, Co, and Ni,” New Journal of Physics, vol. 8, Oct 25 2006.
[195] P. Antoine, B. Carre, A. LHuillier, and M. Lewenstein, “Polarization of high-orderharmonics,” Physical Review A, vol. 55, no. 2, pp. 1314–1324, Feb 1997.
[196] X. Zhou, R. Lock, N. Wagner, W. Li, H. C. Kapteyn, and M. M. Murnane,“Elliptically polarized high-order harmonic emission from molecules in linearlypolarized laser fields,” Physical Review Letters, vol. 102, no. 7, p. 073902, 2009.
[Online]. Available: http://link.aps.org/abstract/PRL/v102/e073902[197] P. M. Oppeneer, “Magneto-optical kerr spectra,” in Handbook of Magnetic
Materials, K. H. J. Buschow, Ed. Elsevier, 2001, vol. 13, ch. 3, pp. 229–422.
[198] I. Boswarva, R. Howard, and A. Lidiard, “Faraday effect in semiconductors,”
Proceedings of the Physical Society of London Series A- Mathematical andPhysical Sciences, vol. 269, no. 1336, pp. 125–&, 1962.
[199] Birss, Symmetry and Magnetism. North-Holland Publishing Company, Amsterdam, 1964.
[200] K. Shinagawa, “Faraday and kerr effects in ferromagnets,” in Magneto-Optics,
ser. Solid State Sciences, S. Sugano and H. Kojima, Eds. Springer, 2000, vol.
128, pp. 137–178.
[201] Z. J. Yang and M. R. Scheinfein, “Combined three-axis surface magneto-opticalkerr effects in the study of surface and ultrathin-film magnetism,” Journalof Applied Physics, vol. 74, no. 11, pp. 6810–6823, 1993. [Online]. Available:
http://link.aip.org/link/?JAP/74/6810/1[202] J. L. Erskine and E. A. Stern, “Magneto-optic kerr effects in gadolinium,” Phys.
Rev. B, vol. 8, no. 3, pp. 1239–1255, Aug 1973.
[203] J. Erskine and E. Stern, “Calculation of the m23 magneto-optical absorption
spectrum of ferromagnetic nickel,” Phys. Rev. B, vol. 12, no. 11, pp. 5016–5024,
Dec 1975.
[204] T. Silva, J. Shaw, and H. Nembach, “Optical measurement of t-moke signal,”
measurement performed at National Institute of Standards and Technology, Electromagnetics Division.
[205] A. Laraoui, V. Halte, M. Vomir, J. Venuat, M. Albrecht, E. Beaurepaire, andJ.-Y. Bigot, “Ultrafast spin dynamics of an individual CoPt3 ferromagnetic dot,”
European Physical Journal D, vol. 43, no. 1-3, pp. 251–253, Jul 2007.
[206] G. Malinowski, F. D. Longa, J. H. H. Rietjens, P. V. Paluskar, R. Huijink, H. J. M.
Swagten, and B. Koopmans, “Control of speed and efficiency of ultrafast demagnetization by direct transfer of spin angular momentum,” Nature Physics, vol. 4,
no. 11, pp. 855–858, Nov 2008.

177[207] M. Cinchetti, M. S. Albaneda, D. Hoffmann, T. Roth, J.-P. Wustenberg,
M. Krauss, O. Andreyev, H. C. Schneider, M. Bauer, and M. Aeschlimann,“Spin-flip processes and ultrafast magnetization dynamics in co: Unifying
the microscopic and macroscopic view of femtosecond magnetism,” PhysicalReview Letters, vol. 97, no. 17, p. 177201, 2006. [Online]. Available:
http://link.aps.org/abstract/PRL/v97/e177201[208] H. A. Duerr, C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast, K. Holldack,
S. Khan, C. Lupulescu, E. F. Aziz, M. Wietstruk, and W. Eberhardt, “Ultrafast
electron and spin dynamics in nickel probed with femtosecond X-ray pulses,”IEEE Transactions on Magnetics, vol. 44, no. 7, Part 2, pp. 1957–1961, JUL 2008.
[209] A. Hubert and R. Scäfer, Magnetic Domains- The Analysis of Magnetic
Microstructure. Springer, 1998.
[210] R. D. McMichael and M. D. Stiles, “Magnetic normal modes of nanoelements,” vol. 97, no. 10. AIP, 2005, p. 10J901. [Online]. Available:
http://link.aip.org/link/?JAP/97/10J901/1[211] J. M. Shaw, T. J. Silva, M. L. Schneider, and R. D. McMichael, “Spin dynamics and mode structure in nanomagnet arrays: Effects of size and thickness onlinewidth and damping,” Physical Reivew B (in press), 2009.
[212] 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,” Optics Letters, vol. 32, no. 3, pp. 286–288, Feb 2007.
[213] M. E. Siemens, Q. Li, M. M. Murnane, H. C. Kapteyn, R. Yang, E. H.
Anderson, and K. A. Nelson, “High-frequency surface acoustic wave propagationin nanostructures characterized by coherent extreme ultraviolet beams,”
Applied Physics Letters, vol. 94, no. 9, p. 093103, 2009. [Online]. Available:
http://link.aip.org/link/?APL/94/093103/1[214] C. La-O-Vorakiat, M. Siemens, J. Shaw, H. Nembach, S. Matthias, R. Adam,
C. M. Schneider, M. A. T. Silva, M. Murnane, and H. Kapteyn, “Magneto-opticalkerr effect probed using ultrafast high-order harmonic euv light,” Physical Review
Letters, in preparation, 2009.

Appendix A

Fresnel propagation code

% fresnel.m Use the fresnel approximation to calculate the interference of gaussian pump% induced height and reflectivity changes. All distances and constants are given in meters

lambda = 30E-9;
z = 25E-2;
w = 50E-6;
spot = 2E-3;
h = .13E-9;
dx0 = 6.5E-6;
r0 = sqrt(.0535);
dr = sqrt(.0533)-sqrt(.0535);
N = 1024;
k = 2*pi/lambda;
I=sqrt(-1);

zcrit = N * dx0^2/lambda;
x = -N/2: 1 :N/2-1; % Initialize x-arrayx0 = dx0 * x; % Set x-array to pixel size of camera

bump = h * exp(-x0.^2 / w^2); % Phase term of reflectivity amplitude coefficientrefl = r0 + dr * exp(-x0.^2 / w^2); % Amplitude term of reflectivity amplitude coefficientgauss = exp(-x0.^2 / spot^2); % Probe spot of reflectivity amplitude coefficient
iB = spot^-2*z/k;
Bfactor = (1+I*iB)^-1
spotz = sqrt(real(Bfactor))^-1 * spot

179

f1 = gauss .* exp(I*4*pi/lambda * bump) * r0; % Probe spot modulated by phase term
fa = gauss .* refl; % Probe spot modulated by amplitude reflectivity term
ff = gauss .* exp(I*4*pi/lambda * bump) .* refl;% Probe spot modulated by amplitude and phase therms

du = 1./(N*dx0);
u = [0:N/2-1 -N/2:-1]*du;% Note order of points- for convolution

H = exp(-i*2*pi^2*u.^2*z/k);% Fourier transform of kernel

f2 = ifft( fft(f1) .* H );% Convolution with phase term

fb = ifft( fft(fa) .* H );% Convolution with amplitude term

fF = ifft( fft(ff) .* H );% Convolution with both terms

f0 = ifft( fft(gauss*r0) .* H );dx1 = lambda*z*du/26E-6;
x1 = [-N/2:N/2-1]*dx1;% Convolution with no modulation% dx0 / 26E-6% Baseline for output in pixels

pti = abs(f2).^2 - abs(f1).^2; % Phase term intensity
ati = abs(fb).^2 - abs(fa).^2; % Amplitude term intensity
bti = abs(fF).^2 - abs(ff).^2; % Both terms intensity

plot(x1, pti, x1, ati, x1, bti)axis([-10,10]);
title(’Holographic signal including amplitude effects for a 30nm probe’);xlabel(’CCD pixel’)
ylabel(’signal level’);%label the x-axis%label the y-axis

bump_max = max(pti);
refl_max = max(ati);
both_max = max(bti);
gauss_max = max(abs(gauss).^2);
ratio_separate = refl_max / bump_max;
ratio_amp = refl_max / both_max;
ratio_both = (both_max-bump_max) / both_max
signaltoprobe = both_max/gauss_max

Appendix B

Multiphysics thermal and optical model% New in version 8: take T-dependence of k, C, CTE (with heating) into
% accountclear;

dt=1*10^-12; tf=8*10^-9; dt2=dt;

Tsave=[0:.1d-10:1d-10 2d-10:1d-10:1d-9 1.25d-9:.25d-9:2d-9 2.5d-9:.5d-9:5d-9 6d-9:1d-9:tf 10d-9];
TempVary=[310 300];
Lvary=[1000]*1d-3;
hBDvary=[150 200 350 500 700]*1d6;
Tf0=500;

for xVary=Lvary% Define grid parameters
width=xVary; xscale=2; dx=width/xscale*10^-8; xs=width*4*10^-6; ys=xs; xf=width*10^-6;
Nt=round(tf/dt)+1; Nxs=round(xs/dx); Nys=round(ys/dx); Nxf=round(xf/dx);
Nx1=round(3*Nxs/8); Nx2=round(5*Nxs/8);

df=2*10^-8;% film thickness in m%--------------------- Set up diffraction model ---------------------%
I = sqrt(-1);% lambda = Wavelength of the probe lightlambda = 29E-9;

181
kk = 2*pi/lambda;% dx0 = Original mesh resolution, N = # of points in meshdx0 = dx*xscale;
N = 1024*1024;
x = -N/2: 1 :N/2-1; % Initialize x-arrayx0 = dx0 * x; % Set x-array% z = Distance from the sample to the cameraz = 6E-2;% w = Full linewidth of the Ni line on the sample, dw = pump-induced change, off = pump offset from centerw = xf;dw = 0E-12;
off = 0*dx0;% spot = EUV 1/2, 1/e width at the focus from the toroid
spot = 50E-6;% h = height of the Ni line on the sample, dw = pump-induced change% h = 18.25E-9;
h = df;% dh = 18.7E-12;% r0 = reflectivity of the sapphire substrate (45), dr = reflectivity% difference between sapphire and Ni
r0 = sqrt(.0535);
dr = sqrt(.0529)-sqrt(.0535);% zfoc = distance from EUV focus to the sample, prop = propagator for that distancezfoc = 0E-3;
prop = (1+I*2*zfoc/(kk*spot^2))^-1;% wsample/wcamera = unperturbed probe size at sample and camera, respectively
wsample = sqrt(real(prop/spot^2))^-1;
wcamera = sqrt(real(1/spot^2*(1+I*2*(z+zfoc)/(spot^2*kk))^-1))^-1;

182
zcrit = N * dx0^2/lambda;% Normalization of probe counts: say 1E4 photons in EUV beam at sample (including% toroid reflectivity (71%), quantum efficiency of the CCD (20%), and the 3 Al
% filters). This is distributed over the entire gaussian, integrated to sqrt(pi/a),% where a=1/spot^2. This leads us to a normalization constant ’C’ for the amplitude% gaussian: C = sqrt(1E4/(spot*sqrt(pi/2)));
C=1;

nmax = floor(120/(xs*2d6));
dutycycle = 1;bump = grating((x0)/w, nmax, dutycycle); % Profile of Ni line on sample

amp = r0 + dr*grating((x0-off)/w, nmax, dutycycle); % Reflectivity on sample
amp2 = r0 + dr*grating((x0-off)/(w+dw), nmax, dutycycle);gauss0 = C*exp(-x0.^2/spot^2); % Probe spot at focus
gauss = C*prop * exp(-x0.^2/spot^2*prop);f = amp.* gauss.* exp(I*4*pi/lambda * h*bump);
f2 = FProp(f,x0,dx0,N,z);

xmax2=Nxs; ymax2=Nys;xmax1=Nxf; ymax1=1; dy=dx; dy0=dx0;% Initialize CN calculation matrices
a=zeros(Nxs,1); b=a; c=a; d=a;

% Substrate matrix coefficients for x-implicitb(1)=-1; c(1)=1;% Top boundarya(Nxs)=1; b(Nxs)=-1;

a2=zeros(Nys,1); b2=a2; c2=a2; d2=a2;
b2(1)=-1; c2(1)=1;
a2(Nys)=1; b2(Nys)=-1;

fa=zeros(Nys,1); fb=fa; fc=fa; fd=fa;
fa(Nys)=1; fb(Nys)=-1;% bottom boundary% Substrate matrix coefficients for y-implicit% Top boundary% bottom boundary% substrate matrix coefficients beneath film
% bottom boundary

183

for Temp=TempVary
Tsh=Temp-69;% Define thermal parameters of the film (nickel) and substrate (sapphire)load kKCTENiSaT5.txt;
ksT= kKCTENiSaT5(:,4); kfT= kKCTENiSaT5(:,2);
asT= kKCTENiSaT5(:,5); afT= kKCTENiSaT5(:,3);% W/m K
% J/m^3 K

axsT=kKCTENiSaT5(:,6)*1d-6; axfT=kKCTENiSaT5(:,7)*1d-6;
CfT=kKCTENiSaT5(:,8)*8.9d6;
sig=.25;% Poisson ratio for sapphire% Cf = 2.44*10^6; kf = 237;%

% Aluminum film

ks = 1.38; Cs = .740*2.2E6; axs=.54d-6; sig=.17; % FS substrateasxT=asT;for hBD=hBDvary
C2=1; C1=100;%Define initial conditions in the film and substrate
Ts1=zeros(Nxs,Nys); Ts2=Ts1; Tft=Tf0*ones(Nxf,1); Tft2=Tft;

FileName2 = sprintf(’SigNi%02d T%03d h%02d.txt’,[xf*1d9 Temp round(hBD/1e6)]);FileName3 = sprintf(’SigNiSa%02d T%03d h%02d.txt’,[xf*1d9 Temp round(hBD/1e6)]);%FileName4 = sprintf(’TNiMid%02d T%03d k%02d h%02d.txt’,[xf*1d9 Temp round(ks) round(hBD/1e6)]);%FileName5 = sprintf(’TNiEdge%02d T%03d k%02d h%02d.txt’,[xf*1d9 Temp round(ks) round(hBD/1e6)]);%FileName6 = sprintf(’TSaMid%02d T%03d k%02d h%02d.txt’,[xf*1d9 Temp round(ks) round(hBD/1e6)]);%FileName7 = sprintf(’TSaEdge%02d T%03d k%02d h%02d.txt’,[xf*1d9 Temp round(ks) round(hBD/1e6)]);%FileName8 = sprintf(’TSaCorner%02d T%03d k%02d h%02d.txt’,[xf*1d9 Temp round(ks) round(hBD/1e6)]);for t=2:Nt,% Time stepping

if(t.*dt>=Tsave(C2));
for k=1:Nxs/xscalehSa(k)=0;
end

184

% Determine thermal deformation at the surface of the substrate
for Neighbors=-1:1% Sum over nearest neighbors

for Ci=(1:xscale:Nxs)+floor(xscale/2)
for Cj=(1:xscale:Nys)+floor(xscale/2)
for Ck=1:round(Nxs/xscale)% Sum over x (across suface)% Sum over z (into subtrate)% Sum over new index (across surface)hSa(Ck)=hSa(Ck)+2*(1+sig)*axsT(round(Ts1(Ci,Cj))+Tsh)*dx0*dy0/(3*pi)*...
(Ts1(Ci,Cj))*Cj*dy/((Ck*xscale+Neighbors*Nxs-Ci)^2*dx^2+Cj^2*dy^2);
end
end
end
end

% Determine thermal expansion of the Ni lines
for Ci=1:xmax1/xscalehNi(Ci)=sum(axfT(round(Tft(Ci*xscale-floor(xscale/2),:))+Temp-69)*...
Tft(Ci*xscale-floor(xscale/2),:))*h/ymax1;
end

pump = transpose(bumpNi2(hNi,x0,nmax)); % Profile of pumped Ni line on sample
hump = transpose(gauss3(hSa,x0,nmax)); % Profile of pumped Ni with Sa deformationp = amp.* gauss.* exp(I*4*pi/lambda * (h*bump + pump));q = amp.* gauss.* exp(I*4*pi/lambda * (h*bump + pump + hump));p2 = FProp(p,x0,dx0,N,z);q2 = FProp(q,x0,dx0,N,z);sigbump=sum(abs((abs(p2).^2-abs(f2).^2)));sighump=sum(abs((abs(q2).^2-abs(f2).^2)));
if C2==1, SigNorm=sighump; end;

SigNi(C2,1:2)=[round(Tsave(C2)*1e12),sigbump];
SigSa(C2,1:2)=[round(Tsave(C2)*1e12),sighump/SigNorm];%

TNiMid(C2,1:2)=[round(Tsave(C2)*1e12),Tft(100)];%

TNiEdge(C2,1:2)=[round(Tsave(C2)*1e12),Tft(1)];%TSaMid(C2,1:2)=[round(Tsave(C2)*1e12),Ts1(400,1)];%

TSaEdge(C2,1:2)=[round(Tsave(C2)*1e12),Ts1(300,1)];

185%

TSaCorner(C2,1:2)=[round(Tsave(C2)*1e12),Ts1(1,1)];%

PercentRealSignal(C2+1)=sigbump/sighump;%

PercentRealSignal(C2+1)%

FileName = sprintf(’PercentRealSignal%04d.dat’,xf*1d9);%

dlmwrite(FileName,PercentRealSignal(:), ’ ’);
dlmwrite(FileName2,SigNi(:,:), ’ ’);
dlmwrite(FileName3,SigSa(:,:), ’ ’);%

dlmwrite(FileName4,TNiMid(:,:), ’ ’);%

dlmwrite(FileName5,TNiEdge(:,:), ’ ’);%

dlmwrite(FileName6,TSaMid(:,:), ’ ’);%

dlmwrite(FileName7,TSaEdge(:,:), ’ ’);%

dlmwrite(FileName8,TSaCorner(:,:), ’ ’);fprintf(1,’%03d ’, t);fprintf(1,’%s ’,’/’);fprintf(1,’%03d\n’,Nt);C2=C2+1;
end;%

Tf0 = dt*hBD/(df*Cf)*(0-Tf0)+Tf0;% implicit in y direction

for i=1:Nxf% In and under the film

k=i+Nx1;%d(1)=-dx*hBD/ks*(Tft(i)-Ts1(k,1));% Set u_y at film-substrate interface

DeltaTf=dt*hBD/(df*CfT(round(Tft(i))+Tsh))*(Ts1(k,1)-Tft(i));fb(1)=-25; fc(1)=48;% Left boundary

for j = 2:Nys-1
as=asT(round(Ts1(k,j))+Tsh);fa(j) = -2*as*dt; fb(j) = 4*dx^2+4*as*dt; fc(j) = -2*as*dt;
fd(j) = 2*as*dt*Ts1(k,j+1)+(4*dx^2-4*dt*as)*Ts1(k,j)+2*dt*as*Ts1(k,j-1);
end
Z=16+3*fb(4)/fc(4); Y=36-3*fa(4)/fc(4)+Z*fb(3)/fc(3);fb(1)=fb(1)+Y*fa(2)/fc(2); fc(1)=fc(1)-Z*fa(3)/fc(3)+Y*fb(2)/fc(2);fd(1)=-12*dx*hBD/ksT(round(Ts1(k,1))+Tsh)*(Tft(i)-Ts1(k,1))+...

186
3*fd(4)/fc(4)-Z*fd(3)/fc(3)+Y*fd(2)/fc(2);Tft(i) = DeltaTf+Tft(i);[Ts2(k,:)]=TDMAsolver(fa,fb,fc,fd);% Solve the system%

DeltaEfi=DeltaTf*Cf*df*dx;%

DeltaEsi=(sum(Ts2(k,:))-sum(Ts1(k,:)))*Cs*dx^2;%

Ts2(k,:)=Ts1(k,:)+abs(DeltaEfi/DeltaEsi)*(Ts2(k,:)-Ts1(k,:));%

uy0=(Ts2(k,1)-Ts2(k,2))/dx;%

fprintf(1,’uy0 = %03d, d(1) = %03d \n’, uy0/abs(DeltaEfi/DeltaEsi), d(1)/dx);%

if i==1, fd(i)=0;% Pure implicit in x- 1D in the

%

film

%

elseif i==Nxf, fd(i)=0;%

else fd(i) = 2*af*dt*Tft(i+1)+(4*dx^2-4*dt*af)*Tft(i)+2*dt*af*Tft(i-1); end
end%[Tft(:)]=TDMAsolver(fa,fb,fc,fd);

for i=1:Nx1
d2(1)=0; d2(Nys)=0;
m=i+Nx2;% Solve the system- film temperature in 1D% Flanking film
% u_x = 0 at both edges% index for on the other side of the film

for j = 2:Nys-1
as=asT(round(Ts1(i,j))+Tsh);a2(j) = -2*as*dt2;

b2(j) = 4*dx^2+4*as*dt2;

c2(j) = -2*as*dt2;

d2(j) = 2*as*dt*Ts1(i,j+1)+(4*dx^2-4*dt*as)*Ts1(i,j)+2*dt*as*Ts1(i,j-1);
end[Ts2(i,:)]=TDMAsolver(a2,b2,c2,d2);% Solve the system

for j = 2:Nys-1
as=asT(round(Ts1(m,j))+Tsh);a2(j) = -2*as*dt2;

b2(j) = 4*dx^2+4*as*dt2;

c2(j) = -2*as*dt2;

d2(j) = 2*as*dt*Ts1(m,j+1)+(4*dx^2-4*dt*as)*Ts1(m,j)+2*dt*as*Ts1(m,j-1);
end[Ts2(m,:)]=TDMAsolver(a2,b2,c2,d2);% Solve the system

187
end

% implicit in x direction
for(j=1:Nys)% In the substratefor(i=2:Nxs-1)
asx=asxT(round(Ts2(i,j))+Tsh);a(i) = -2*asx*dt2;

b(i) = 4*dx^2+4*asx*dt2;

c(i) = -2*asx*dt2;

d(i) = 2*asx*dt*Ts2(i+1,j)+(4*dx^2-4*dt*asx)*Ts2(i,j)+2*dt*asx*Ts2(i-1,j);end[Ts1(:,j)]=TDMAsolver(a,b,c,d);% Solve the systemend%

DeltaEs=(sum(sum(Ts2(:,:)))-sum(sum(Ts1(:,:))))*Cs*dx^2;%

EnergyConservation=-DeltaEf/DeltaEs;%

Ts1(:,:)=Ts1(:,:)+EnergyConservation*(Ts2(:,:)-Ts1(:,:));%

C1=C1+1;%

if C1>=100 fprintf(1,’Energy Conservation = %03d \n’, -DeltaEfi/DeltaEsi); C1=0; end
end
end

end
end

% Calculate DeltaEnergy substrate