Ultrafast Carrier and Lattice Dynamics in Highly

Transcription

Ultrafast Carrier and Lattice Dynamics
in Highly Photo-Excited Solids
A thesis presented
by
Albert Mjong-Tschol Kim
to
The Division of Engineering & Applied Sciences
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the subject of
Applied Physics
Harvard University
Cambridge, Massachusetts
May 2001
c
2001
by Albert Mjong-Tschol Kim
All rights reserved.
Advisor: Prof. Eric Mazur
Albert M.-T. Kim
Ultrafast Carrier and Lattice Dynamics in Highly Photo-Excited Solids
Abstract
In this dissertation we report femtosecond time-resolved measurements of the spectral dielectric function of amourphous GaAs, GeSb thin films and single-crystalline Te. In all
materials we measured the evolution of ε(ω) over a broad energy range (1.7 – 3.4 eV),
following an impulsive excitation by an ultrashort laser pulse.
The ε(ω) data on a-GaAs show evidence of a nonthermal, structurally driven
semiconductor-to-metal transition. A comparison to previously taken ε(ω) data on c-GaAs
is especially illuminating in terms of the influence of the initial structure on the phase
transition.
Our results on GeSb thin films reveal a new nonthermal, metallic phase. The
ε(ω) data provide a detailed picture of the transition from the amorphous phase to the
crystalline phase of these thin films. Furthermore we refute a previous claim on an ultrafast
disorder-to-order transformation in these materials.
The time resolved ε(ω) data on Te reveal a great wealth of new information on
impulsively driven coherent phonons, including their influence on the electronic bandstructure. We find evidence indicative of a new nonthermal phase of matter which we call a
frustrated metal.
Table of Contents
Abstract
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Table of Contents
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List of Figures
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List of Tables
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Acknowledgements
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Citations to Published Work
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1 Introduction
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2 Linear and Nonlinear Optical Properties of Solids
2.1 The Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The Classical Picture of Absorption — The Complex ε(ω) . . . . . .
2.1.2 Relation of the Dielectric Function to Material Properties . . . . . .
2.2 The Fresnel Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Single Vacuum-Material Interface . . . . . . . . . . . . . . . . . . . .
2.2.2 Multilayer Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Uniaxial Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Optical Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Non-Resonant Nonlinearities — Wave-Mixing Phenomena and Self
Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Resonant Nonlinearities — Two-Photon Absorption . . . . . . . . .
2.3.3 White-Light Generation . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Tools of Ultrafast Spectroscopy
3.1 Ultrashort Pulse Generation . . . . . . . . . . . .
3.1.1 Mode Locking in Laser Resonators . . . .
3.1.2 AM Mode Locking . . . . . . . . . . . . .
3.1.3 Analytical Treatment of Mode Locking by
3.1.4 Kerr Lens Mode Locking . . . . . . . . .
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Fast Saturable Absorbers
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Table of Contents
3.2
3.1.5 The KML Oscillator . . . .
Chirped Pulse Amplification . . . .
3.2.1 Compression and Stretching
3.2.2 Multipass Amplifier Design
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of Femtosecond Optical Pulses
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4 Techniques in Ultrafast Spectroscopy
4.1 Overview of Ultrafast Spectroscopic Techniques . . . . . . . . . . . . . . . .
4.1.1 The Pump-Probe Scheme — Obtaining Femtosecond Time Resolution
4.1.2 Different Detection Geometries for Different Phenomena . . . . . . .
4.2 Characterization of Ultrashort Pulses . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Autocorrelation Measurements . . . . . . . . . . . . . . . . . . . . .
4.2.2 Frequency Resolved Optical Gating . . . . . . . . . . . . . . . . . . .
4.3 Femtosecond Time-Resolved Ellipsometry . . . . . . . . . . . . . . . . . . .
4.3.1 Multi-Angle Ellipsometry for Isotropic, Bulk Materials . . . . . . . .
4.3.2 Accounting for Oxide Layers and Extension to Uniaxial Materials and
Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 The Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Calibration and Error Estimate . . . . . . . . . . . . . . . . . . . . .
4.3.5 Temporal Resolution: Chirp Correction etc. . . . . . . . . . . . . . .
4.3.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Ultrafast Processes in Semiconductors — an Overview
5.1 Carrier Relaxation . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Thermalization and Cooling of Carriers . . . . . .
5.1.2 Intervalley Scattering . . . . . . . . . . . . . . . .
5.1.3 Carrier Recombination . . . . . . . . . . . . . . . .
5.2 Coherent Dynamics in Semiconductors . . . . . . . . . . .
5.2.1 Coherent Carrier Dynamics: Quantum Beats . . .
5.2.2 Coherent Lattice Dynamics . . . . . . . . . . . . .
5.3 Ultrafast Phase Transitions in Semiconductors . . . . . .
5.3.1 Nonthermal Melting . . . . . . . . . . . . . . . . .
5.3.2 Dielectric Function Measurements in c-GaAs . . .
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6 Ultrafast Phase Transitions in Amorphous GaAs
6.1 Experimental Results . . . . . . . . . . . . . . . . .
6.1.1 Low Fluence Regime (F < 1.7 Fth-a ) . . . .
6.1.2 Medium Fluence Regime (F ≈ 1.7 Fth-a ) . .
6.1.3 High Fluence Regime (F > 3.2 Fth-a ) . . . .
6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Low and Medium Fluence Regimes . . . . .
6.2.2 High Fluence Regime . . . . . . . . . . . .
6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . .
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Table of Contents
vi
7 Ultrafast Phase Transitions in GeSb Thin Films
7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . .
7.2 Experimental Results . . . . . . . . . . . . . . . . . .
7.2.1 Dielectric Functions of Unexcited a-GeSb and
7.2.2 Below Fcr . . . . . . . . . . . . . . . . . . . .
7.2.3 Above Fcr . . . . . . . . . . . . . . . . . . . .
7.2.4 Above 3.5 Fcr . . . . . . . . . . . . . . . . . .
7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Phase Dynamics Below Fcr . . . . . . . . . .
7.3.2 Phase Dynamics Above Fcr . . . . . . . . . .
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
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c-GeSb
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8 Coherent Phonons in Tellurium
8.1 Fundamental Properties of Te . . . . . . . . . . .
8.1.1 Structural and Electronic Properties of Te
8.1.2 Sample Preparation . . . . . . . . . . . .
8.2 Experimental Results . . . . . . . . . . . . . . . .
8.2.1 The Ordinary ε(ω) . . . . . . . . . . . . .
8.2.2 The Extra-Ordinary ε(ω) . . . . . . . . .
8.3 Discussion . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Detailed Phonon Dynamics . . . . . . . .
8.3.2 The Effects on the Bandstructure . . . . .
8.4 Conclusions . . . . . . . . . . . . . . . . . . . . .
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9 Summary and Outlook
180
References
183
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
Brillouin zone, bandstructure, and dielectric function of c-GaAs. . . . . . .
Electronic bandstructure of Cu. . . . . . . . . . . . . . . . . . . . . . . . . .
Dielectric function of a free electron gas, described by the Drude model. . .
Schematic illustration of free carrier absorption. . . . . . . . . . . . . . . . .
Dielectric functions of Te, a-GaAs, l-Si and c-GaAs at various temperatures.
Reflection and refraction at the interface between a uniaxial material and an
isotropic material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Reflection configurations at an interface between a uniaxial crystal and an
isotropic crystal where the interface contains the c-axis. . . . . . . . . . . .
2.8 Schematic illustration of self focusing. . . . . . . . . . . . . . . . . . . . . .
2.9 Schematic illustration of self-phase modulation. . . . . . . . . . . . . . . . .
2.10 White-light spectrum generated in a 2 mm thick piece of CaF2 . . . . . . . .
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3.1
3.2
3.3
3.4
3.5
Schematic representation of active mode-locking. . . . . . . . . . . . . . . .
Schematic representation of saturable absorber action. . . . . . . . . . . . .
Three mirror configuration of a Kerr-lens mode-locked laser cavity. . . . . .
Four mirror configuration of a Kerr-lens mode-locked laser cavity. . . . . . .
Schematic illustration of the Kapteyn-Murnane-Laboratories Ti:sapphire oscillator cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Illustration of a so-called soft, or self-induced aperture. . . . . . . . . . . . .
3.7 Principle of Chirped Pulse Amplification. . . . . . . . . . . . . . . . . . . .
3.8 Compressor design based on diffraction gratings. . . . . . . . . . . . . . . .
3.9 Compressor design based on prisms. . . . . . . . . . . . . . . . . . . . . . .
3.10 Stretcher based on diffractive gratings. . . . . . . . . . . . . . . . . . . . . .
3.11 Stretcher using just one grating at Littrow-angle. . . . . . . . . . . . . . . .
3.12 Multipass amplifier design. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.1
4.2
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4.3
4.4
Schematic illutration of a pump-probe setup. . . . . . . . . . . . . . . . . .
Detection geometries for Four Wave Mixing and Transmissive Electrooptic
Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Detection geometry for autocorrelation measurement using SHG. . . . . . .
SHG FROG trace of ultrashort laser pulse from an amplified Ti:sapphire
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
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List of Figures
TPA FROG trace of ultrashort laser pulse from an amplified Ti:sapphire
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Temporal chirp of white-light continuum pulse generated in CaF2 . . . . . .
4.7 Mappings from dielectric function space to reflectivity space via the Fresnel
formulae. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Dependence of the ratio of the differential change in reflectivity with real and
imaginary part of ε on angle of incidence. . . . . . . . . . . . . . . . . . . .
4.9 Schematic representation of femtosecond time resolved ellipsometry setup. .
4.10 Correction factors for various materials for poor choice of parameters. . . .
4.11 Correction factors for various materials for good choice of parameters. . . .
viii
4.5
Carrier excitation followed by thermalization and cooling of the carrier distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Schematic illustration of intervalley scattering. . . . . . . . . . . . . . . . .
5.3 Schematic illustration of recombination. . . . . . . . . . . . . . . . . . . . .
5.4 Dependence of recombination times on excited carrier density. . . . . . . . .
5.5 Schematic illustration of a quantum beat. . . . . . . . . . . . . . . . . . . .
5.6 Schematic illustration of Displacive Excitation of Coherent Phonons. . . . .
5.7 Schematic illustration of the A1 -mode in Te. . . . . . . . . . . . . . . . . . .
5.8 Dielectric function response of c-GaAs to low fluence laser excitation. . . . .
5.9 Lattice heating in c-GaAs following excitation with a low fluence ultrashort
laser pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.10 Dielectric function response of c-GaAs to medium fluence laser excitation. .
5.11 Dielectric function response of c-GaAs to high fluence laser excitation. . . .
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5.1
6.1
6.2
6.3
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Dielectric function of unexcited a-GaAs. . . . . . . . . . . . . . . . . . . . .
Evolution of the dielectric function of a-GaAs after excitation at 0.9 Fth-a . .
Laser-induced lattice heating of a-GaAs for 0.9 Fth-a , tracked using the shift
of ε(ω) to lower photon energies. . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Evolution of the dielectric function of a-GaAs after excitation at 1.7 Fth-a . .
6.5 Evolution of the dielectric function of a-GaAs after excitation at 5.7 Fth-a . .
6.6 Evolution of the measured dielectric function of a-GaAs for excitation at
5.7 Fth-a , for time delays between −667 and 667 fs. . . . . . . . . . . . . . .
6.7 Comparison of the dielectric function of c-GaAs in the medium fluence regime
with that of a-GaAs measured following excitation with a low-fluence pulse.
6.8 Comparison of the dielectric function of c-GaAs at the lower end of the high
fluence regime (0.8 Fth-c ) with that of a-GaAs measured following excitation
with a pulse of fluence on the border between its low and high fluence regimes
(1.7 Fth-a ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Plasma frequency as a function of time delay in the high fluence regime for
c-GaAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Plasma frequency as a function of time delay in the high fluence regime for
a-GaAs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.1
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Schematic illustration of physical mechanisms in rewritable DVDs. . . . . .
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List of Figures
7.2
7.3
7.4
7.5
7.6
Femtosecond time resolved reflectivity transients of GeSb thin films. . . . .
Dielectric function of unexcited a-GeSb thin film. . . . . . . . . . . . . . . .
Dielectric function of unexcited c-GeSb thin film. . . . . . . . . . . . . . . .
Microscope images of irradiated spots of GeSb thin films. . . . . . . . . . .
Evolution of the dielectric function of a-GeSb thin film following excitation
with a fluence of 0.6 Fcr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7 Evolution of the dielectric function of a-GeSb thin film following excitation
7.8 Evolution of the dielectric function of a-GeSb thin film following excitation
7.9 Liquid layer model for the evolution of the reflectivity of a-GeSb thin films
following excitation at fluences below Fcr . . . . . . . . . . . . . . . . . . . .
7.10 Dielectric function of a-GeSb thin film 200 fs after excitation at F = 1.6 Fcr.
7.11 Calculated reflectivity transients at various angles and energies for three different excitation fluences. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.12 Reflectivity spectra of a-GeSb thin film 200 fs after excitation at three different fluences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
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Crystal structure of Te and motion of A1 phonon mode. . . . . . . . . . . . 159
Derivation of the Te crystal structure from a distortion of the primitive cubic
lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.3 Optical phonon modes of Te lattice. . . . . . . . . . . . . . . . . . . . . . . 161
8.4 Electronic bandstructure of Te at normal conditions. . . . . . . . . . . . . . 163
8.5 Electronic bandstructure of Te for lattice distortion along A1 -mode. . . . . 164
8.6 Laue diffraction pattern of Te. . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.7 Evolution of the ordinary part of the dielectric function of Te following excitation with a fluence of 5.6 mJ/cm3 . . . . . . . . . . . . . . . . . . . . . . . 168
8.8 Zero-crossing of real part of ordinary ε(ω) vs. time for different fluences. . . 169
8.9 Evolution of the extra-ordinary part of the dielectric function of Te following
excitation with a fluence of 5.6 mJ/cm3 . . . . . . . . . . . . . . . . . . . . . 171
8.10 Zero-crossing of real part of extra-ordinary ε(ω) vs. time for different fluences.172
8.11 Frequency dynamics of the coherent phonon modes in Te. . . . . . . . . . . 174
8.12 Maximum value of Fourier transform coefficients for transforms of time resolved zero-crossing traces of ε(ω). . . . . . . . . . . . . . . . . . . . . . . . 176
8.1
8.2
List of Tables
8.1
Table of optical phonon modes in Te. . . . . . . . . . . . . . . . . . . . . . .
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Acknowledgements
Completing this Ph.D. thesis was a long, but very rewarding experience for me. As with
most things in life, this is not a single person’s achievement, but many people have contributed to this work. My special thanks go to my advisor Eric Mazur. His amazing personal
skills paired with a sixth sense for picking the “right” graduate students enable him to create an extremely collegial and friendly environment in his research group. Throughout the
years in Eric’s group I was always happy to go to “lab” in the morning — even if it was not
for the work sometimes, it was always worth it for the people. I appreciate Eric’s respectful
attitude towards his student, behaving as a primus inter pares.
I would like to thank the members of my committee Profs. Mike Aziz, Henry
Ehrenreich, and Peter Pershan for their expertise and guidance throughout the development
of the work described in this thesis. My special thanks go to Prof. Henry Ehrenreich for
his deep insights and extremely useful discussions on optical properties of semiconductors.
I am hoping that the continuation of the work described in this thesis will involve further
stimulation by Prof. Ehrenreich and a collaboration with Mike Aziz’s group.
The laser and data acquisition system used for the experiments, as well as the
experiments on a-GaAs and a-GeSb thin films were completed in collaboration by Paul
Callan and myself. Paul and I spent endless hours side by side in the lab struggling with
laser, electronics, and of course physics problems. Paul’s inquisitive nature and razor sharp
mind always kept me honest and on my toes when discussing physics and working in the lab.
From the moment I joined the lab in 1996 up until Paul’s graduation in 2000 we developed
a frienship that hopefully will last a lifetime.
xi
Acknowledgements
xii
In the summer of 1999, Chris Roeser joined our lab. The experiments on coherent
phonons in Te would not have been possible without Chris’ help. Chris is one of those
people which get twice as much work done in half the time as anyone else — how do you
do it Chris?
Chris Schaffer joined Harvard at the same time I did. It was easy to “bond” with
Chris since we share a passion for aquatic activities such as surfing, windsurfing, sailing,
and scuba diving. Chris is not only one of the most impressive experimentalists I know, but
he has also taught me the American way of borrowing against the future (right Chris?). I
hope that I picked up more of his experimental skills than the latter one.
I would also like to thank the other members of the Mazur group. Jon Ashcom’s
sharp wit and his calm and well-balanced character always make it a pleasure to interact with
him. By the way Jon: you are an extremely bright individual and should stop underselling
yourself. Rebecca Younkin, apart from being an outstanding scientist, was always the
conscience of our group. Her efforts to hold up the “moral flag” were severely under attack
sometimes, but she always succeeded in the end. The group would not be the same without
her. Jim Carey joined our lab in the summer of 2000. He definitely brought some fresh
impulses into the group. His cheerful nature spreads a good mood around. Also, his
extensive programming knowledge keep our website running smoothly. I also want to thank
all the other members of the group I interacted with during my time here: Andre Brodeur,
Nan Shen, Tsing-Hua Her, Claudia Wu, Rich Finlay, Li Huang, Adam Fagen, Catherine
Crouch, Rafael Gattass, and Aryeh Feder — all of you made my stay at Harvard extremely
pleasureable and it would not have been the same without any one of you.
I would like to thank Prof. Roland Allen and Dr. Lorin Benedict for insightful
discussions on the ultrafast dynamics in highly excited GaAs. I am indebted to Prof. Peter
Grosse for providing the excellent quality single-crystalline Tellurium sample for the coherent phonon experiments. I thank Prof. Stephen Fahy and Dr. Paul Tangney for fruitful
discussions on the theoretical aspects of coherent phonon dynamics in Tellurium. My special
thanks go to my advisor for my Diplom-thesis at the RWTH-Aachen (Germany) Prof. Hein-
Acknowledgements
xiii
rich Kurz and my co-workers Dr. Thomas Dekorsy, Dr. Stefan Hunsche, and Dr. Gyu-Cheon
Cho for their excellent introduction into the field of ultrafast optics. I would like to thank
Prof. Howard Stone for sharing his dedication to teaching with me. I was lucky enough to
be a Teaching Fellow for Howard twice, assisting him in his justifiably extremely popular
course on differential equations.
One of the most important ingredients to happiness in life are good friends. This
thesis would not have been completed without the support from my friends. I want to
thank Kyung Wha Byun for her love and support during the last half year of the thesis
work. Francesca Chang, Gregg Favalora, Julie Frantsve, David Hong, Ted Miguel, Boris
Müller, Tim Nosper, Giovanna Oddo, Swantje Rietfort, Claire Ryan, Oliver Rychter, Heike
Schoof, Andrea Stürmer, Denis Yu, and Gary Zabow are all wonderful friends who gave me
soul support during my five year stay in graduate school and I thank all of you. Juliana
Josef was a great friend for the first three years of my graduate school career and I owe her
many of my views on life.
My special thanks go to my Judo coach Roland Schiffler. Judo and Roland were
central parts of my life for a period of 15 years and I benefited from both in a great way. Even
though it might seem very remote, Roland has contributed to this thesis a lot by drilling me
to and beyond my physical limits and at the same time providing a great friendship. Along
similar lines, Jacob Goldfield is a coach for me in the game of life. Jacob’s argumentative
nature and sharp mind have kept me on the ball during my long decision process in search
of a career path after graduate school. Thank you for your advise.
Last, but definitely not least, I want to thank my family. My father Bjong-Ro is
not only a brilliant scientist but also the most admirable human being on the face of this
planet. His character continues to give me an ideal to strive for. My mother Suh-Young and
my brother Philip Mjong-Hyon both give me the unconditonal love and limitless support
only family can give each other and I would not be at this point in my life without them.
Acknowledgements
xiv
Acknowledgements of Financial Support
I gratefully acknowledge support through a fellowship from the Deutscher Akademischer Auslandsdienst (DAAD). The experiments described in this thesis were supported by
the Joint Services Electronics Program under contract ONR N0014–89–J–1023, by the Army
Research Office under contract DAAG55-98-1-0077 and by the National Science Foundation
under contract DMR-9807144.
Citations to Published Work
Parts of the data, analysis and text in this dissertation can be found in the following papers:
A. M.-T. Kim, C. A. D. Roeser, and E. Mazur, Direct observation of displacive excitation
of coherent phonons in Te, manuscript in preparation.
A. M.-T. Kim, C. A. D. Roeser, J. P. Callan, L. Huang, E. N. Glezer, Y. Siegal, and E.
Mazur, Femtosecond Time-Resolved Ellipsometry, manuscript in preparation for Journal of
the Optical Society of America A.
A. M.-T. Kim, J. P. Callan, C. A. D. Roeser, and E. Mazur, Ultrafast phase transitions in
crystalline and amorphous GaAs, manuscript in preparation for Physical Review B.
J. P. Callan, A. M.-T. Kim, C. A. D. Roeser, and E. Mazur, Universal dynamics during
and after ultrafast laser-induced semiconductor-to-metal transitions, submitted to Physical
Review B
J. P. Callan, A. M.-T. Kim, C. A. D. Roeser, and E. Mazur, Ultrafast laser-induced phase
transitions in amorphous GeSb films, Physical Review Letters 86, 3650 (2001).
J. P. Callan, A. M.-T. Kim, C. A. D. Roeser, and E. Mazur, Ultrafast dynamics and phase
changes in highly excited GaAs, in Ultrafast Processes in Semiconductors 67, 151, edited
by K.-T. Tsen, (Academic Press, San Diego, 2001).
J. P. Callan, A. M.-T. Kim, L. Huang, and E. Mazur, Ultrafast electron and lattice dynamics
in semiconductors at high excited carrier density, Chemical Physics 251, 167 (2000).
xi
For my parents Drs. Bjong-Ro and Suh-Yong Kim who gave and continue to give me the
love and opportunities needed for achievements like the completion of this thesis.
By three methods we may learn wisdom: First, by reflection which is the noblest; second,
by imitation, which is the easiest; and third, by experience, which is the bitterest.1
Confucius (551 - 479 BC)
1
For this thesis, as in all research, all three methods were used.
Chapter 1
Introduction
The invention of the laser by Maiman in 1960 enabled access to a source producing light
with extreme intensity and monochromaticity [1]. Soon, the laser would revolutionize optical
physics and successively many other areas of science as well as engineering. The demand
for ever higher intensities led to the development of pulsed laser sources. Various methods
of pulsed laser operation, such as Q-switching and Mode-Locking, continously drove down
the temporal pulse width and increased the peak power. It took 16 years, before passive
mode-locking techniques led to the first sub-picosecond pulse [2]. The current world record
of short pulse generation is at less than 5 femtoseconds [3]. Since a single cycle of an
electromagnetic wave in the visible frequency range corresponds to about 3 femtoseconds,
the regime of tens or hundreds of femtoseconds is the ultimate lower limit to the duration
of light pulses in the visible range. Researchers therefore call these sub-picosecond pulses
ultrashort pulses.
Ultrafast optics has since blossomed into a mature but still exciting and rapidly
developing field of science. The variety of ultrafast spectroscopy techniques and their applications has grown rapidly along with the development of ultrafast laser sources [4]. Researchers in academia and industry use ultrafast spectroscopic techniques to probe dynamics
in semiconductors on time scales never attained before [5] and to time resolve chemical reactions [6].
1
Chapter 1: Introduction
2
Femtosecond laser pulses are not only very short, but also have extreme peak
powers — peak powers up to 60 Terawatt have been reported [7]. Focusing such a powerful
laser pulse to a standard spot size of hundreds of square micrometers creates intensities of
Exawatts (1018 ) per square centimeter. The relatively easy access to such power levels has
led to a large amount of research on femtosecond laser-induced damage in all kinds of matter
such as metals [8], transparent materials [9, 10, 11, 12], and semiconductors [13, 14, 15].
These studies focus on excitation conditions far beyond the threshold for damage. Material
is ablated off of the irradiated surface or voids are formed within the bulk of the sample.
There is an interesting intermediate regime, where the material is excited with fluences on
the order of the damage threshold. This is the fluence regime we are mostly concerned with
in the experiments described in this thesis.
Highly excited solid state materials, especially semiconductors, are of prime interest not only for basic scientific reasons but also for industrial applications. For example,
(1) high carrier densities drive phase transitions between structurally different phases which
are important in optical storage devices and micromachining; (2) the injection currents in
modern high power laser diodes generate very high carrier densities necessitating a detailed
understanding of the dynamics at these densities [16]. It turns out that the spectral dielectric function is an excellent candidate to provide detailed information on the exact state of
a material at a given time after an intense photo-excitation. In this dissertation we report
femtosecond time-resolved measurements of the spectral dielectic function of amourphous
GaAs, GeSb thin films and single-crystalline Te.
Content and Organization of this Dissertation
The thesis begins with a review of the linear and nonlinear response of matter to electromagnetic radiation in Chapter 2. Special attention is given to the dielectric function of
solids and the Fresnel reflectivity formulae. We describe in detail the relation between ε(ω)
and microscopic properties of materials and derive extensions to the Fresnel formulae for
multi-layered materials and also non-isotropic materials.
Chapter 1: Introduction
3
Chapters 3 and 4 describe the fundamental tools and techniques of an ultrafast
spectroscopy laboratory. In Chapter 3 we focus our discussion on the generation and amplification of femtosecond optical pulses. In Chapter 4 we briefly review the pump-probe
setup (which enables femtosecond time resolution) and several variations of it, but mainly
focus on femtosecond time-resolved ellipsometry.
Chapter 5 gives an overview of ultrafast processes in semiconductors. We mention
a few specific scattering events which are of importance to the experiments described in
this dissertation, and also discuss ultrafast phase transitions in solids.
In Chapters 6, 7, and 8 we describe and analyze experimental results on highly
photo-excited a-GaAs, GeSb thin films, and Te. In all materials we measured the evolution
of ε(ω) over a broad energy range (1.7 – 3.4 eV), with femtosecond time resolution following
an intense excitation by an ultrashort laser pulse.
The ε(ω) data on a-GaAs show evidence of a nonthermal, structurally driven
semiconductor-to-metal transition. A comparison to previously taken ε(ω) data on c-GaAs
is especially illuminating in terms of the influence of the initial structure on the phase
transition.
GeSb thin films are of prime interest to engineers in the optical storage industry
because they are stable in an amorphous and a crystalline phase which have largely different
reflectivities. Our results on the transition between these two phases yield a much improved
understanding of the transition mechanisms, aside from refuting a previous claim of an
ultrafast disorder-to-order transformation in these materials.
Last but not least, in Chapter 8 we revisit the phenomenon of coherent phonons
in Te. The time resolved ε(ω) data reveal a great wealth of new information on coherent
phonons in Te, including their influence on the electronic bandstructure. We find indicative
evidence of a new non-thermal phase of matter which we call frustrated metal.
Chapter 2
Linear and Nonlinear Optical
Properties of Solids
The fundamental goal of any optical experiment is to deduce information on a sample from
its optical properties. It is therefore crucial to develop a good understanding of the link
between material properties and their influence on the optical response. In this chapter,
we review the linear and nonlinear optical properties of solids which are crucial for the
understanding of the experiments described in this thesis. In the discussion of the linear
response we focus on the dielectric function and its link to microscopic material properties.
We briefly review the Fresnel reflectivity formulae and derive extensions for multilayer
samples and non-isotropic materials. Among the multitude of nonlinear optical effects we
restrict ourselves to a brief description of wave-mixing processes, self-focusing, and twophoton-absorption. These are the processes which are important for the understanding of
the apparatus used for the experiments described in this thesis.
2.1
The Dielectric Function
The interaction of electro-magnetic radiation with matter is governed by Maxwell’s equations in conjunction with the material equations [17]. In the field of optics it is sufficient to
4
Chapter 2: Linear and Nonlinear Optical Properties of Solids
5
consider the response of the material to the electric field only, because the interaction with
the magnetic field component is negligible in comparison. In the case of moderate electric
fields, the response of a material can be very well described by linear response theory. The
linear relation between the electric field and the resulting polarization is given by the linear
susceptibility χ or equivalently by the dielectric function ε(ω). In this section we discuss the
relation between specific material properties (such as lattice and electronic structure) and
ε(ω), as well as the way in which ε(ω) governs the linear optical properties of a material.
2.1.1
The Classical Picture of Absorption — The Complex ε(ω)
Let us begin by briefly reviewing the role of ε(ω) in the propagation of an electro-magnetic
wave. The evolution of such a wave in an arbitrary medium is governed by Maxwell’s
equations [17]:
4π
1
j
curlH − Ḋ =
c
c
1
curlE − Ḃ = 0
c
(2.1)
(2.2)
divD = 4πρ
(2.3)
divB = 0
(2.4)
The influence of the medium on the light field is dictated by the material equations:
j = σE
(2.5)
D = εE
(2.6)
B = µH
(2.7)
In vacuum, there are no free carriers making the RHS of Eq. 2.3 equal to zero. Furthermore,
the conductivity σ is equal to zero and both the dielectric function ε and the magnetic
susceptibility µ are equal to one (for optical fields the latter is always a good approximation
and we will take µ = 1 from now on, in any material). We can easily derive the wave
equation for an E-field by eliminating H from Eqs. 2.1 and 2.2:
∇2 E =
1
Ë
c2
(2.8)
6
Therefore in vacuum the phase velocity of a light wave is equal to c.
In the case of an insulator (which is defined as a material with bandgap much
higher in energy than optical photon energies) the conductivity can be neglected, but the
dielectric function ε(ω) is not equal to one anymore. The phase velocity is then given by
√
v = nc , where n = ε denotes the refractive index of the material [17].
In a semiconductor or a metal we have to address absorption of the light as it
propagates through the material. Classically, this can be done using Eq. 2.5. Having a
non-zero conductivity σ leads to a phase velocity of the form [17]:
v=
c
ε + i 4πσ
ω
(2.9)
Usually, the conductivity σ and the dielectric function ε are absorbed into one complex
quantity ε̂:
ε̂(ω) = ε + i
4πσ
ω
(2.10)
For ease of notation we drop the “hat” from now on and refer to this new complex quantity
as the dielectric function which now has a real and an imaginary part: ε = εr + iεi .
Following the convention in the non-absorbing case, we define a complex index of refraction
√
as n̂ = ε = n + ik.1 A complex index of refraction causes the amplitude of an electromagnetic field to die off exponentially (as can be easily verified by looking at the plane-wave
solution to the wave equation in an absorbing material) [17]. It is more common to consider
the drop-off of the intensity. The exponential coefficient is then given as α = 2 ωc k and also
referred to as the absorption coefficient. The intensity is governed by a differential equation
of the form:
dI
= −αI.
dx
2.1.2
(2.11)
Relation of the Dielectric Function to Material Properties
The classical treatment of absorption formally defines the macroscopic material parameter
governing absorption — the absorption coefficient α. In this section we discuss the most
1
We will drop the “hat” over n from now on as well.
7
common models which provide a link between microscopic material properties and the
absorption coeffients and therefore the dielectric function. A comprehensive treatment
of optical properties in solids can be found in Yu and Cardona’s book Fundamentals of
Semiconductors [18].
Direct absorption — A Two-body Process
The simplest case of absorption is a two-body process — one electron absorbs one photon. This case is usually treated using a semi-classical approach. That is, the electron is
treated quantum-mechanically whereas the electric field is taken into account as a classical
perturbation to the electron Hamiltonian (which is given by the structure of the respective
solid). Since the wave-vector of the electro-magnetic field is small compared to the extent
of a crystal Brillouin zone it is valid to make the electric-dipole approximation.2 Following
Fermi’s Golden rule the probability per unit time for the absorption of a photon is then
given by [18]:
2π e 2 E(ω) 2 |Pvc |2 δ(Ec (k) − Ev (k) − ω),
R=
mω 2 (2.12)
k
where Pvc denotes the electric dipole matrix element and Ec (k) and Ec (k) denote the
dispersion relations for the conduction band and the valence band of the crystal respectively.
To link this probability to the absorption coefficient it is instructive to consider the power
loss due to absorption which is given by the transition probability per unit time multiplied
by the energy for each photon: Power loss = Rω.
On the other hand, the power loss can also be expressed as:
dI dx
c
εi ωI
dI
=
= − αI = − 2 .
dt
dx dt
n
n
The intensity I is related to the field amplitude by: I =
n2
2
8π |E(ω)| .
(2.13)
Using Eqs. 2.12 and
2.13 we can therefore obtain a microscopic expression for the imaginary part of the dielectric
2
The electric-dipole approximation only considers “vertical” transitions, i.e. it takes the wave-vector of
a photon to be equal to zero [18].
8
function:
εi (ω) =
2πe
mω
2 |Pvc |2 δ(Ec(k) − Ev (k) − ω).
(2.14)
k
Since the real and imaginary part of ε(ω) are related by the Kramers-Kronig relations [17],
one obtains the real part εr “for free”:
4πe2 2
|Pvc |2
,
εr (ω) = 1 +
2 − ω2
m
mωcv ωcv
(2.15)
k
where ωcv = Ec (k) − Ev (k).
As Eq. 2.14 shows, the imaginary part of ε(ω) is directly determined by the dispersion of the conduction and valence band. In analogy to a simple Lorentzian oscillator
model, Im[ε(ω)] peaks close to energy values where electronic transitions in the material
are present and the magnitude of those peaks scales with the oscillator strength associated with each transition. The real part is correlated to the imaginary part through the
Kramers-Kronig relations and exhibits a dispersive “wiggle” at each absorptive peak in the
imaginary part [17]. Figure 2.1 shows the Brillouin zone (a), the bandstructure (b) and the
dielectric function (c) of c-GaAs. The characteristic absorption peaks in the imaginary part
of the dielectric function at 3.1 eV (E1 ) and 4.7 eV (E2 ) are due to a large joint densities
of states around the L-valley and X-valley in the bandstructure as indicated by the shaded
regions in the graph. The real part shows the characteristic dispersive structure near the
peaks in Im[ε(ω)].
Higher Order Absorption Processes
The semiclassical model described above only considers “vertical” absorption processes,
i.e. processes where the momentum change from the initial state to the final state of the
electron is taken as equal to zero. This is a very good approximation if only two-body
processes are important because the momentum of photons is negligible compared to typical
electron momenta. In insulators these vertical absorption processes do in fact dominate the
optical properties and the dielectric functions obtained using the semiclassical model are
9
K
L
W
U
Γ
K
X
U
(a )
+4
40
GaAs
GaAs
E1
dielectric function
energy (eV)
+2
E2
0
−2
20
E2
E1
Re ε
Im ε
0
Eo
(c )
(b )
−4
L
Γ
−20
X
0
2
4
6
photon energy (eV)
Figure 2.1: (a) First Brillouin zone [19], (b) bandstructure [20], (c) and dielectric function
of c-GaAs [21].
very accurate. For metals or semiconductors, however, there are other, higher order effects,
which are important in determining the absorption or dielectric function of a specific sample
material.
Let us first consider metals. Figure 2.2 shows the bandstructure of Cu. The
electrons involved in conduction or absorption are close to the Fermi level indicated by
EF in the figure. In describing the dielectric function of metals it is very common to
approximate the electron dispersion around the Fermi level as parabolic.3 This seems
like a very rough approximation, but it proves to work very well for most metals. Once a
parabolic approximation is made, it is possible to treat the electron system as a free electron
3
2 2
k
As a quick reminder, a free electron has a parabolic dispersion relation: E(k) = ~2m
.
10
+10
Cu
energy (eV)
+5
EF
−5
−10
Γ
X
W
L
Γ
K
Figure 2.2: Electronic bandstructure of Cu, EF denotes the Fermi energy [22].
gas (corrected for the effective mass of the electrons4 ). Thus, the dielectric function for a
metal can be approximated by a Drude model [22]:
ωp2
ε∞ ωp2 γ
−
i
ε(ω) = ε∞ 1 − 2
ω + γ2
ω(ω 2 + γ 2 )
(2.16)
where
ωp2 =
4πN e2
m∗ ε∞
(2.17)
is the plasma frequency, N is the electron density, m∗ is the effective mass of electrons near
the Fermi level, τ = 1/γ is the phenomenological average time between two electron scattering events, and ε∞ is the dielectric constant of the material at infinity frequency. Figure
2.3(a) shows the dielectric function according to the Drude model for a plasma frequency
of ωp = 12 eV and a scattering time of τ = 0.18 fs.5 The most striking characteristics
4
The effective mass of an electron in a solid is proportional to the inverse of the curvature of the bandstructure at the wavevector of the respective electron [22].
5
A plasma frequency of 12 eV corresponds to an electron density of 1022 cm−3 according to Eq. 2.17
which is typical for metals. The value of τ = 0.18 fs is also very typical for metals.
11
40
40
20
Cu
dielectric function
dielectric function
Drude model
Im ε
0
20
Im ε
0
Re ε
ωp
Re ε
(b )
(a)
−20
−20
0
5
10
15
0
photon energy (eV)
2
4
6
photon energy (eV)
Figure 2.3: (a) Dielectric function (solid curve = Re[ε(ω)]; dashed curve = Im[ε(ω)]) of
a free electron gas, described by a Drude model with plasma frequency ωp = 12 eV and
relaxation time τ = 0.18 fs. (b) Dielectric function of Cu [21].7
of a Drude shape are a monotonically decreasing imaginary part and a real part which is
negative for energies below the plasma frequency and small but positive for energies above
ωp . For comparison, Fig. 2.3(b) shows the dielectric function of Cu. In spite of the very
rough approximation of assuming parabolic electron dispersion, the Drude model produces
an ε(ω) which is very close to the one of a “real” metal like Cu.
The Drude model is a simple but powerful technique to describe free carrier absorption which is illustrated in Fig. 2.4. To absorb the energy of an incident photon the electron
has to gain the momentum ∆k. Because a photon carries virtually no momentum at all, free
carrier absorption is inherently a three-body process, where the momentum is provided (or
taken up) by a third particle, such as a phonon or an impurity [18]. A thorough quantum
mechanical treatment is quite difficult for such a three body process. Instead, the Drude
model incorporates all potential scattering events into the phenomenological scattering time
leading to a simple but surprisingly accurate description of free carrier absorption.
Another prominent three-body absorption process in semiconductors is absorption
into so-called excitonic states [18]. An exciton is a bound state formed by an electron in the
conduction band and a hole in the valence band. Their mutual Coulomb attraction gives
rise to the formation of a Hydrogen-atom-like state. The absorption resonance of such an
12
E
EF
∆k
k
Figure 2.4: Schematic illustration of free carrier absorption. ∆k denotes the momentum
change the electron undergoes due to the absorption.
exciton is lowered below the bandgap of the semiconductor by the binding energy between
hole and electron. It turns out that the absorption into these excitonic states is very strong
and needs to be taken into account in order to accurately model the absorptive properties
of semiconductors.
Of course the story of absorption in solids does not end here, but there is a whole
myriad of other many-body processes involved. A thorough treatment would exceed the
frame of this thesis, however. We refer the interested reader to the excellent overview given
in Yu and Cardona’s book Fundamentals of Semiconductors [18].
The Dielectric Function for Different Phases of Materials
Let us now turn to actual dielectric functions of various materials, as well as the ε(ω)
of different phases of materials. Figure 2.5(a) shows ε(ω) for the extra-ordinary part of
Te. Te is a semiconductor with a small bandgap of about 0.3 eV. The zero-crossing of
Re[ε(ω)]denotes the energy of the main transition in the material — also referred to as the
bonding-antibonding split of the material using the nomenclature of molecular quantum
13
40
90
60
30
a-GaAs
Im ε
dielectric function
dielectric function
Te (ext)
Re ε
20
Re ε
Im ε
0
0
(a)
−30
1.5
2.0
2.5
3.0
(b)
−20
1.5
3.5
2.0
3.0
3.5
4.0
liquid Si
Im ε
dielectric function
30
20
0
GaAs
723 K
423 K
293 K
20
10
Re ε
Im ε
0
Re ε
(d)
(c)
−20
1.5
4.5
40
40
dielectric function
2.5
photon energy (eV)
photon energy (eV)
2.0
2.5
3.0
3.5
photon energy (eV)
4.0
4.5
−10
1.5
2.0
2.5
3.0
3.5
4.0
4.5
photon energy (eV)
Figure 2.5: Dielectric functions of (a) crystalline Te (the extra-ordinary part), (b) amorphous GaAs, (c) liquid Si, and (d) crystalline GaAs at different temperatures.
mechanics [24] — and is relatively low at 2.0 eV.8 Fig. 2.5(b) shows ε(ω) for amorphous
GaAs. The distinct features of ε(ω) for the crystalline material are washed out because
there is no requirement for the conservation of crystal momentum in an amorphous material.
Fig. 2.5(c) shows ε(ω) for liquid Si. The liquid phase of Si is metallic as the Drude-like shape
of the dielectric function indicates (see Fig. 2.3 for comparison). Lastly, Fig. 2.5(d) shows
ε(ω) of c-GaAs at different temperatures. The dielectric function takes on cleary distinct
shapes for different temperatures.
In summary, it is possible to extract information on the material itself and the specific phase of a material from an accurate measurement of the spectrally resolved dielectric
function. We make use of this fact in all experiments decribed in this thesis.
8
For comparison, Fig. 2.1 shows that the bonding-antibonding split for crystalline GaAs (bandgap of 1.55
eV at room temperature) is about 4.7 eV.
2.2
14
The Fresnel Formulae
Given the dielectric function, one can deduce all linear optical properties, such as reflectivity,
absorption, or transmissivity of a material. For the purpose of the experiments discussed
in this thesis, we are most interested in the relation between the reflectivity of a material
and ε(ω). The reflectivity is dictated by ε(ω) through the Fresnel formulae. Given the real
and imaginary part of ε(ω) at a particular photon energy, the Fresnel formulae predict the
reflectivity for any angle and for any polarization of the incident light beam. In this section
we discuss the Fresnel formulae for an interface between vacuum and an isotropic material
as well as extensions to multilayer stacks and uniaxial materials.
2.2.1
Single Vacuum-Material Interface
Let us first consider the easiest case — the interface between an isotropic material and
vacuum. In this case the Fresnel formulae are [17]:
(ε1 + i ε2 ) cos θi − (ε1 + i ε2 ) − sin θi 2
rp =
(ε1 + i ε2 ) cos θi + (ε1 + i ε2 ) − sin θi 2
√
ε1 + i ε2 cos θi − (ε1 + i ε2 ) − sin θi 2
,
rs = √
ε1 + i ε2 cos θi + (ε1 + i ε2 ) − sin θi 2
(2.18)
(2.19)
where rp and rs are the ratios of the incident and reflected wave amplitudes for p- and spolarization respectively, θi is the angle of incidence, and ε1/2 denote the real and imaginary
part of the dielectric function — ε(ω) = ε1 + iε2 . The refractive index of air is taken to
be unity for simplicity. The power reflectivity R is given by the absolute square of the
2
Fresnel factors rp/s: Rp/s = rp/s . The derivation of this most simple case of the Fresnel
equations is quite straightforward and can be found in any reasonably advanced textbook
about electro-magnetism. Unfortunately, the cases where it is possible to use this simple
version of Fresnel’s formulae are very rare in the real world. In most cases, the sample has
one or all of the following complications: (1) there is at least one interface between layers
that both have refractive indices different from one; (2) there are multiple layers of different
15
media; (3) the sample is not isotropic. We will address these issues in the following two
sections.
2.2.2
Multilayer Stacks
Most solid materials develop a native oxide layer when exposed to air. In fact, the oxide
layer on Si is absolutely crucial to the design of current microchips. For optical experiments, especially at higher angles of incidence, the native oxide layer on the surface of a
semiconductor or a semimetal contributes significantly to the reflectivity of the surface. It
is therefore not sufficient to assume the special case of a vacuum-material interface. Typically, native oxide layers have thicknesses of about 2-10 nm with an ε(ω) that can be well
approximated by a constant ranging from roughly 4 to 10. Thus, there are now three media
involved in the reflectivity of the material: air, oxide layer and the bulk material (following
the direction of the incident light). We account for native oxide layers with a multilayer
reflectivity formula that is derived in Priniciples of Optics by Born and Wolf [17]:
r12 + r23 ei2β 2
2
R = |rtotal| = 1 + r12 r23 ei2β where
β=
(2.20)
ω d ε2 − ε1 sin2 θi
c
is related to the thickness d of the oxide layer, and r12 and r23 are Fresnel factors for the
air-oxide and oxide-sample interfaces respectively, as calculated from a simple extension of
Eq. 2.18 or Eq. 2.19. The Fresnel equations above have to be slightly modified to account
for the fact that in the case of the oxide-sample interface the first medium is not vacuum:
n22 cos θi − n1 n22 − n21 sin θi 2
,
(2.21)
r12p =
n22 cos θi + n1 n22 − n21 sin θi 2
n1 cos θi − n22 − n21 sin θi 2
,
(2.22)
r12s =
2
2
2
n1 cos θi + n2 − n1 sin θi
where n1 is the refractive index of the first material and n2 is the refractive index of the
second material (again, following the direction of the incident light).
16
When there are more than three layers, closed-form formulae for the reflectivity
are difficult to calculate. This situation arises if one wants to measure ε(ω) of a thin film
deposited on a substrate which happens to develop a native oxide layer in air. Now, there
are four media — air, oxide, thin film and substrate. In this case, it is best to use the
characteristic matrix method, described by Born and Wolf [17]. We define a characteristic
matrix for each layer l,

Ml = 
where
− pil
cos βl
−ipl sin βl

sin βl
,
(2.23)
cos βl
pl = nl cos θl = .l − .0 sin2 θ0
and
βl =
2π
ω
nl hl cos θl = hl pl .
λ
c
θl is the angle the beam makes with the normal, and we relate it to the angle of incidence θ0
using Snell’s Law. Variables nl , .l and hl denote the refractive index, dielectric constant and
thickness of layer l, respectively. If there are N layers between the air or vacuum (labelled
by “0”) and the substrate (which we label “N + 1”), the characteristic matrix for the entire
stack of layers is the matrix product
M=
N
Ml .
(2.24)
l=1
The power reflecitivity is related to M via [17]:
(M11 + M12 pN +1 ) p0 − (M21 + M22 pN +1 ) 2
R = (M11 + M12 pN +1 ) p0 + (M21 + M22 pN +1 ) (2.25)
where we calculate p0 and pN +1 just the same way as for the other layers. As this reflectivity
formalism shows, the reflectivity of a multi-layer stack is determined by the thickness and
the dielectric function of each layer in the stack.
2.2.3
17
Uniaxial Materials
From Section 2.2.1 and 2.2.2 we know how to model the reflectivity of samples which consist
of multiple layers of different isotropic media. The experiments described in Section 8 were
performed on Te, which is a uniaxial crystal. For the discussion of anisotropic materials
let us turn to the special case of an interface between an isotropic medium and a strongly
absorbing, uniaxial medium. The Fresnel formulae for the slightly simpler case of the
interface between vacuum and a strongly absorbing uniaxial has been treated by Mosteller
and Wooten in [25, 26]. In the following we will extend the Fresnel formula to the more
general case, closely following the work by Mosteller and Wooten from 1968.
Uniaxial materials have two independent entries in the dielectric tensor. It is
therefore not possible to treat the tensor ε as a scalar. The relationship between the
refractive index and ε(ω) becomes much more involved in this case. To derive this new
relation, consider the Maxwell equations 2.1 and 2.2 in their Fourier representations:
where k =
ω
cn
n × H = −D
(2.26)
n × E = H,
(2.27)
and D̂ = 4πi
ω j + D = εE. Here, D̂ denotes the D-field including the current
j. For ease of notation we will drop the hat in the following. It is understood that from now
on D includes the current j. Equivalently, ε includes the imaginary part in the dielectric
function induced by the conductivity tensor in Eq. 2.5, ε̂ = ε + 4πiσ/ω.
Eliminating H from Eqs. 2.26 and 2.27 we obtain:
D = n E − (n · E)n.
(2.28)
Now, because we know that D = εE, we can extract the relation between ε and n. After a
lot of messy algebra we arrive at:
n2 (εx n2x + εy n2y + εz n2z ) −
[n2x εx (εy + εz ) + n2y εy (εx + εz ) + n2z εz (εx + εy )] + εx εy εz ] = 0.
(2.29)
18
optic axis
z
k(t)
θt
uniaxial medium
interface P
x
isotropic medium
k(i)
θi
θr
k(r)
Figure 2.6: Reflection and refraction at the interface between a uniaxial material and an
isotropic material. For s-polarized light, the electro-magnetic wave only “sees” the ordinary
index. For p-polarized light the wave “sees” an effective index consisting of a mixture of
ordinary and extra-ordinary index depending on the angle of incidence.
This simplifies to the simple, standard relation ε = n2 in the isotropic case, as can be easily
verfied.
For the uniaxial case, we assume the configuration depicted in Fig. 2.6, that is, a
reflection off the basal plane of a crystal. We consider an ordinary wave incident on the
plane P (polarized along the y-axis) and an extra-ordinary wave (polarized in the plane of
incidence). We set εz = ε and εx,y = ε⊥ . Now, the expression in Eq. 2.30 becomes:
(n2 − ε⊥ )[ε n2z + ε⊥ (n2x + n2y ) − ε⊥ ε ] = 0.
(2.30)
We thus obtain two solutions for the complex refractive index in a uniaxial material, as
19
expected:
n2 = ε⊥ , and
(n2x + n2y )
n2z
+
= 1.
ε⊥
ε
(2.31)
The first solution determines the refractive index in the ordinary direction no which, as
expected, is simply the square-root of ε⊥ . The second solution determines the effective
refractive index nef f of an extra-ordinary wave incident on the plane P. Dividing by nef f
we obtain:
1
nef f
sin2 (θ) cos2 (θ)
+
.
ε
ε⊥
=
(2.32)
For the following derivation of the reflectivity formulae it is useful to express the
transmitted H-fields in terms of E-fields. Using Eq. 2.27 we obtain for the ordinary case:
Ho =
√
ε⊥
ko × Eo
.
|ko |
(2.33)
In the extraordinary case, the complicated relation between n and ε does not allow
such a simple step. Instead we start from Eq. 2.26. Since He is pointing in the y-direction
by definition, the magnitude of the extra-ordinary H-field He is:9
He = −
1
1
Dx = − εx Ee cos(θt ),
nz
nz
(2.35)
where θt is the angle between the z-axis and the Poynting vector.
The last piece we need before we can go ahead and derive the extended Fresnel
formulae is the well known Snell’s law [17]. Snell’s law can be readily derived from the
boundary conditions on the E and H-fields which come out of Maxwell’s equations [17]:
H(1) = H(2)
(1)
Et
9
Remember that
0
n×H=@
(2.36)
(2)
= Et
ny Hz − nz Hy
nx Hz − nz Hx
nx Hy − ny Hx
(2.37)
1 0D 1
A = −@ D A.
x
y
Dz
(2.34)
20
The H-field and the tangential components of the E-field are always continous across an
interface. Using these boundary conditions, Snell’s law is easily obtained [17]:
sin(θi ) = sin(θr )
n(i) sin(θi ) = n(t) sin(θt ).
(2.38)
We are now ready to derive the Fresnel equations for an interface between an isotropic and
a uniaxial material. As a first step we write down all participating E and H-fields:
(i) ·r−ωt)
Ei = E(i) ei(k
Hi = k(i) × Ei /k(i)
(r) ·r−ωt)
Er = E(r) ei(k
Hr = k(r) × Er /k(r)
(t)
i(ko
Eot = E(t)
o e
·r−ωt)
Hot = n⊥ (k((t))
× Eot /ko(t) )
o
(t)
i(ke
Eet = E(t)
e e
Het = −
·r−ωt
1
εx Ee cos(θt )
nz
(2.39)
Here, we use refractive index rather than dielectric function notation — n2⊥ = ε⊥ and
n2 = ε .
Let us first consider the ordinary case in the basal plane configuration which
corresponds to s-polarized light. In this case, the E-field has only a component in the ydirection in the framework of Fig. 2.6. Matching the fields at the interface according to the
boundary conditions from Eqs. 2.36 and 2.37 leads to:
Ey(i) + Ey(r) = Ey(t)
(2.40)
Hx(i) + Hx(r) = Hx(t)
(2.41)
Plugging in the corresponding expressions for the H-field from Eqs. 2.39 into Eq. 2.41
21
gives:10
(t)
ni Ey(i) cos(θi ) − ni Ey(r) cos(θi ) = n⊥ Eoy
cos(θot )
(2.43)
We can now eliminate cos(θot ) using Snell’s law and combine Eqs. 2.43 and 2.40 to solve
for the Fresnel factor rs :
rs =
(r)
Ey
(i)
Ey
=
ni cos(θi ) −
n2⊥ − n2i sin2 (θi )
ni cos(θi ) + n2⊥ − sin2 (θi )
(2.44)
This is, as expected in equivalent to the standard Fresnel formula in Eq. 2.22 because an
ordinary wave only “sees” the ordinary part of the index of refraction — the material is
“isotropic” as far as the ordinary wave is concerned.
The more interesting case is the one for an extra-ordinary wave, which in the basal
plane case corresponds to p-polarized light. Matching the fields at the interface according
to the boundary conditions leads to:
Ex(i) + Ex(r) = Ex(t)
(2.45)
Hy(i) + Hy(r) = Hy(t).
(2.46)
Plugging in the respective expressions for the fields from Eqs. 2.39 we obtain:
)
E (i) cos(θi ) − E (r) cos(θr ) = Ee(t) cos(θet
)Ee(t)
ni E (i) + ni E (r) = (n2⊥ /nz ) cos(θet
(2.47)
(2.48)
(t)
Again, eliminating cos(θet ) and Ee gives:
nz ni E (i) + nz ni E (r) = n2⊥ (E (i) cos(θi ) − E (r) cos(θi ))
E (r)(nz ni + n⊥ cos(θi )) = E (i)(n2⊥ cos(θi ) − nz ni )
10
k
H=n ×E=n
k
0
@
ky Ez − kz Ey
kx Ez − kz Ex
kx Ey − ky Ex
1
A,
(2.49)
(2.50)
(2.42)
where Ey is the only non-zero component, since we are considering the ordinary case here. So, Hx = −kz Ey
(i,r,t)
— where kz
= ±ni,⊥ cos(θi,ot ), according to Fig. 2.6.
22
So, we arrive at the following expression for the Fresnel factor rp:
rp = E (r)/E (i) =
n2⊥ cos(θi ) − nz ni
n2⊥ cos(θi ) + nz ni
(2.51)
We can now express nz in terms of n and n⊥ using Eq. 2.31:
n2x + n2y
n2z
+
=1
ε⊥
ε
(2.52)
Further simplification leads to:
n2z
ε⊥
n2x + n2y
= 1−
,
ε
(2.53)
where ny = 0 because the wavevector of the extra-ordinary wave lies in the plane P. Hence,
ε⊥
(ε − n2x ),
ε n⊥ 2
ε⊥
=
ε − n2i sin2 (θi ) =
n − n2i sin2 (θi ) using nx = ni sin(θi ) (2.54)
ε
n
n2z =
nz
Replacing nz in Eq. 2.51 we obtain:
rp =
n⊥ cos(θi ) − ni nn⊥
n⊥ cos(θi ) − ni nn⊥
n2 − n2i sin2 (θi )
n2 − n2i sin2 (θi )
.
(2.55)
We have derived the Fresnel reflectivity formulae for an interface between an
isotropic medium and an absorbing, uniaxial medium. Only the special case of a reflection off the Basal plane was considered. It is reasonably straightforward to extend the
treatment to the case of other crystal faces. In the experiments in Chapter 8 we measure
reflectivities off a surface in Te which contains the c-axis. The corresponding reflectivity
formulae are:
rp =
n2⊥ cos(θi ) −
n2⊥ cos(θi ) +
n⊥ n − n2i sin2 (θi )
n⊥ n − n2i sin2 (θi )
,
(2.56)
for the case where the c-axis is perpendicular to the plane of incidence as shown in Fig. 2.7(a)
and,
n2i n2⊥ − n4i sin2 (θi )
,
rp =
n n⊥ cos(θi ) + n2i n2⊥ − n4i sin2 (θi )
n n⊥ cos(θi ) −
(2.57)
23
c-axis
E-field
E-field
c-axis
(a)
(b)
Figure 2.7: Reflection configurations at an interface between a uniaxial crystal and an
isotropic crystal where the interface contains the c-axis. In (a) the plane of incidence
contains the c-axis and in (b) the c-axis is perpendicular to the plane of incidence. For both
configurations the E-field is p-polarized.
for the configuration shown in Fig. 2.7(b) where the c-axis is contained in the plane of
incidence.
As a “sanity” check, it is instructive to let n = n⊥ in all of the Eqs. 2.44, 2.55,
2.56 and 2.57. It is easy to show that all equations go over to the standard form of Fresnel’s
equations for isotropic materials (Eq. 2.22 and 2.21), as expected.
2.3
Optical Nonlinearities
So far, we have assumed that the response of the material to an incident electro-magnetic
wave is purely linear. If the intensity of the light propagating through a medium increases
to magnitudes where the electric field strength becomes comparable to the field strengths
between the valence electrons and their host ions, the response of the material becomes
nonlinear. Typical atomic field strengths are on the order of 3 · 108 V/cm. In a laser beam,
an intensity of 1 W/cm2 corresponds to a field strength of about 12 V/cm. So, if a laser
24
reaches intensities of about 106 W/cm2 , nonlinear effects start to kick in. The nonlinear
interaction causes a multitude of intruiging phenomena such as wave-mixing, self-focusing
and white-light generation. The field of nonlinear optics is largely based on the seminal
work by Bloembergen et al. who published a comprehensive treatment of the nonlinear
wave dynamics in Ref. [27]. In this section we discuss a selection of nonlinear optical effects
that are of importance within the framework of our experiments.
2.3.1
Non-Resonant Nonlinearities — Wave-Mixing Phenomena and Self
Focusing
Once the nonlinear components of the susceptibility of a material start to play a role, a
whole world of intriguing phenomena is unleashed. The simplest case are second-order
processes where the polarization is generated not only by the linear susceptibility χ(1) but
there is also a nonlinear polarization due to the second order susceptibility χ(2) :
P(total) = χ(1) · E + χ(2) : EE
(2.58)
This is the most general expression for the nonlinear polarization P. To describe a specific
nonlinear process it is useful to change to a real electric field description: E(ω) = 12 (Eeiωt +
Ee−iωt ), with an equivalent Ansatz for the polarization. For instance the process of sumfrequency generation (SFG) can be described by picking the polarization which oscillates
at the sum of the two input field frequencies:
P(2)(ω1 + ω2 : ω1 , ω2 ) = χ(2) : E(ω1 )E(ω2 )
(2.59)
This oscillating polarization re-radiates an electro-magnetic wave at the sum-frequency of
the two generating waves. In the most general case, χ(1) is a second rank tensor and χ(2)
is a third rank tensor. Usually, crystal symmetries reduce the number of entries in these
tensors significantly. In the degenerate case, where the two input fields are of the same
frequency, this process is called second-harmonic-generation (SHG), because the frequency
of the input fields is doubled.
25
There is, of course, also the corresponding process — called difference frequency
generation (DFG). The polarization at the difference-frequency between the two input fields
is generated according to the same formalism used in Eq. 2.59. Except that in the case of
DFG the polarization is generated by the product of one field and the complex conjugate
of the second field:
P(2) (ω1 − ω2 : ω1 , ω2 ) = χ(2) : E(ω1 )E∗ (−ω2 )
(2.60)
There is a degenerate case for DFG just as there is one for SFG. For DFG the generated
field in the degenerate case is a DC-field and the process is called optical rectification [28].
Both processes DFG and SFG are, in principle, always present and of equal importance. There is a restriction, however, called the phase-matching [28] requirement that
dictates whether one or the other effect dominates. For DFG or SFG to be efficient, the
generated nonlinear wave has to propagate through the nonlinear medium at the same
phase velocity as the two generating waves. This cannot be achieved in isotropic materials because the phase velocity of light at the fundamental wavelengths is usually different
from the one at the sum or the difference, respectively. Phase-matching can be achieved
in uniaxial materials which have different refractive indices for light polarized along the
c-axis (ne , the extra-ordinary index) and light polarized perpendicular to it (no , the ordinary index). Let us consider the specific case of SHG in a so-called Type I phase-matching
configuration [29]. In Type I phase-matching the fundamental waves are both polarized
along the ordinary plane in the uniaxial crystal. Let us further consider the SHG wave
generated perpendicular to the polarization of the input waves. Given a negative, uniaxial
crystal where ne (ω) < no (ω), it is now possible to find a propagation direction at a certain
angle to the c-axis where the effective index of refraction for the SHG wave nef f (2ω) is
equal to no (ω) (where nef f (2ω) can be calculated using Eq. 2.32). It is along this direction
where SHG will be most efficient and where the generation of the doubled frequency will be
optimal. This nonlinear process of SHG or more generally SFG, is very common and has
many applications in science as well as engineering. We will make use of SHG in Section
26
4.2 to measure the temporal width of an ultrashort pulse.
Let us now turn to third order nonlinearities. For materials with vanishing χ(2)
(i.e. centrosymmetric materials [30]) we can write:
P(total) = χ(1) · E + χ(3) :: EEE
(2.61)
Again, there are a number of different processes induced by χ(3). For example, wave mixing
produces light at different frequencies including at the third harmonic of the fundamental
— a process called third harmonic generation. The approach to describe these wave mixing
phenomena is very similar to that of SFG and DFG — a new field at a new frequency
is generated due to χ(3) . There is a completely different phenomenon, however, which is
also triggered by χ(3) . This phenomenon is called the optical Kerr effect and results in self
focusing of the incident beam if its intensity is sufficiently high. To understand the Kerr
effect, it is instructive to rewrite Eq. 2.61 as:
P=
3
χ(1) + χ(3) |E|2 E
4
(2.62)
We know from Section 2.1.2 that D = E + 4πP and D = εE. Thus,
ε = 1 + 4π(χ(1) + χ(3) |E|2 )E
Since the linear εlin is given by εlin = 1 + 4πχ(1) and
(2.63)
√
ε = n this can be written as:
εlin + 3πχ(3) |E|2
3π (3)
χ |E|2
= n 1+
ε
3π (3)
χ |E|2 ),
≈ n(1 +
2n
n =
for small
2π (3)
|E|2 .
n χ
(2.64)
So, the χ(3) nonlinearity induces an intensity dependent refractive
index. In general, the refractive index is a complex quantity. For the purpose of this
discussion we consider only the real part of the index. This is a reasonable approximation if
there are no resonant transitions in the nonlinear medium at any of the frequencies present,
gaussian
wavefront
27
nonlinear medium
Figure 2.8: Schematic illustration of self focusing. The incoming wavefront gets distorted
due to the nonlinearity of the refractive index: the center portion of the beam is retarded
due to a higher effective index.
i.e. including the input waves and the generated waves [31]. The intensity dependent change
is then given by:
3π (3)
2
χ |E| ≡ n2 |E|2
∆n = Re
2n
(2.65)
If a Gaussian beam propagates through such a nonlinear medium with positive n2 , the
central part of the beam experiences a larger refractive index than the outer regions of the
profile due to the higher intensity. Thus the central part of the beam travels at a slower
velocity than the edge distorting the wavefront of the beam, in a similar fashion to a focusing
lens, as depicted in Fig. 2.8.
There is a competing process, however. Any beam of finite width will suffer from
diffraction. Depending on which of the two processes dominates, a high intensity beam
will either spread out, focus itself further and further, or remain unchanged in width. The
critical power for this transition can be derived quite straightforwardly. The formulae for
the angular spread due to diffraction and the maximal angle for total internal reflection
inside a region of higher refractive index are [32]:
θDif f
λ/n
, θtot−ref ≈
≈
w
2∆n
n
(2.66)
setting them equal and solving for the power (implicit in ∆n) gives:
Pc =
π.o cλ2
8n2
(2.67)
28
A beam with a power higher than Pc will catastrophically focus until other nonlinear effects
come into play or until ∆n saturates [28]. The breakdown of optical beams into narrow
filaments due to this effect is a well established experimental fact. If the power is lower
than the critical value, the beam will spread out due to the dominance of diffraction. In the
special case where the intensity of the beam exactly matches the critical power, the beam
will propagate for a long distance without changing its width. This phenomenon is called
self trapping [28]. As one can imagine, the general analytical treatment of the propagation of
a spatially nonuniform beam through a nonlinear medium is highly complex. In the special
(but widely used and quite useful) case of Gaussian beams, this problem is analytically
solvable in an elegant manner. We return to this problem in Section 3.1.4 when describing
the generation of ultrashort optical pulses using the Kerr-effect.
2.3.2
Resonant Nonlinearities — Two-Photon Absorption
In the previous section we discussed non-resonant nonlinear processes, also called parametric
processes. Let us now consider the case of resonant processes, i.e. we include resonant
transitions at the optical frequencies involved. More specifically, we consider a degenerate
third order process — there are three input waves at the same frequency ω — with a resonant
transition in the nonlinear medium at 2ω. At high enough intensities, there will be energy
transfer from the electro-magnetic fields into the material by simultaneous absorption of
two photons to bridge the energy gap of 2ω. This process is called Two-Photon-Absorption
(TPA). TPA can be described in a very similar framework to the Kerr effect. Except that
in this case the dominant part of the nonlinear refractive index is the imaginary one due to
the resonance. From Eq. 2.65 we know the dependence of the nonlinear index on the input
fields. For the imaginary part we have:
3π (3)
2
χ |E| = k2 |E|2 ,
∆k = Im
2n
(2.68)
where k denotes the imaginary part of the refractive index — ntot = n + ik. We therefore
obtain an intensity dependent imaginary part of the refractive index.
29
The absorption coefficient of a material in linear response theory is given by [17]
(see also Section 2.1.2):
ω
α = 2 nk(ω)
c
(2.69)
The linear absorption coefficient at frequency ω for a medium where the first absorptive
transition is at 2ω is practically equal to zero, because the linear k is equal to zero. Instead,
at high enough intensities for TPA to be important the absorption is dominated by nonlinear
absorption. We can define a nonlinear absorption coefficient using Eqs. 2.68 and 2.69:
3π (3)
ω
ω
2
χ |E| ≡ 2 nk2 |E|2 .
α = 2 n Im
c
2n
c
(2.70)
The intensity evolution of a light wave as it propagates through a linearly absorbing
medium is given by Beer’s law [17]:
dI
= −αI
dz
(2.71)
In the case of nonlinear absorption the α is given by Eq. 2.70 leading to a nonlinear Beer’s
law:
ω
ω
dInonlin
= −2 nk2 |E|2 I = −2 nk2 I 2
dz
c
c
(2.72)
We will return to the discussion of TPA in Section 4.2.2 where we discuss the
characterization of ultrashort optical pulses using TPA.
2.3.3
White-Light Generation
One of the most striking nonlinear optical phenomena is white-light generation. Whitelight generation is caused by an intricate interplay between a number of different nonlinear
effects. The most basic picture of white-light generation is based on an effect called selfphase modulation (SPM). Figure 2.9 schematically illustrates this process. Assuming an
intensity dependent refractive index there is not only a self-induced spatial variation of the
index (as described in Section 2.3.1 in the context of self-focusing), but also a temporal
n0+∆n
n0
n0
blue
shift
30
red
shift
Figure 2.9: Schematic illustration of self-phase modulation. The high index of refraction
at the temporal center of the pulse causes slow phase velocity. The back end of the pulse
“runs into the center” whereas the front end “runs away” from the center causing a temporal
modulation of the phase and thereby creates new frequency componennts broadening the
spectrum of the pulse.
modulation due to the intensity peak in time of an ultrashort optical pulse. The refractive
index is thus largest for the central portion of the pulse. This causes the phase velocity to
be slower in the middle of the pulse than at the front end or back end of the pulse. The
light wave at the front end therefore “runs away” from the center whereas the wave at the
back end “runs into” the center of the pulse. As indicated in the figure that leads to a
temporal modulation of the phase of the electro-magnetic wave resulting in the generation
of new frequency components and ultimately a broader spectral bandwidth of the pulse.
SPM is one mechanism of spectral broadening. The magnitude of the broadening is
determined by the peak intensity and the steepness of the seed pulse. As depicted in Fig. 2.9,
the broadening due to SPM is completely symmetric for a symmetric seed pulse which does
not agree with experimentally observed spectra which are much more broadened towards
higher frequencies than lower frequencies. The typical, assymetrical white-light spectrum
can only be understood if other nonlinear effects are taken into account. The nonlinear
31
107
intensity (a.u.)
10
6
105
10
4
103
102
1.5
2.0
2.5
3.0
3.5
energy (eV)
Figure 2.10: White-light spectrum generated in a 2 mm thick piece of CaF2 . The solid curve
indicates the original spectrum and the dashed curve indicates the flattened spectrum after
the light passes through a blue glass filter.
effects which play a role in white-light generation are SPM, self-focusing, space-time focusing, self-steepening, avalanche ionization and multiphoton ionization, and plasma coupling
[33]. Self-focusing (see Section 2.3.1) enhances all other nonlinear effects by increasing the
intensity in the material. The ionization effects put a limit to the intensity increase because the light scatteres off of the generated free carriers inhibiting further focusing. Most
important for the understanding of the asymmetrical broadening are self-steepening and
space-time focusing. Self-steepening occurs due to the fact that not only phase but also
group-velocity is intensity dependent. That leads to a steepening of the back and of the
pulse shifting the peak towards the back. The steep back and of the pulse enhances SPM
producing a large blue broadening of the spectrum. This process gets further enhanced by
space-time focusing. Space-time focusing denotes the change in diffraction strength due to
the temporal steepness of the pulse. The steeper back end diffracts less strongly then the
front end increasing the relative intensity at the back end which further enhances the blue
32
broadening.
Figure 2.10 shows the white-light spectrum generated by focusing a 800 nm, 50
fs laser pulse of about 1 µJ energy into a 2 mm thick piece of CaF2 . The solid curve
indicates the strongly blue-broadened spectrum. As indicated by this curve, the major
part of the pulse energy remains at the wavelength of the incident seed pulse (800 nm =
1.55 eV). For spectroscopic experiments it is strongly advisable to flatten the spectrum to
make full use of the dynamic range of the used photodetectors. In our setup we choose to
flatten the spectrum using a blue glass filter which has a sharp absorption peak at 800 nm
(Schott, BG − 40). The dashed curve in Fig. 2.10 indicates the resulting spectrum. In all of
the experiments described in this thesis the broadband probe light was generated in CaF2
and flattened using a BG − 40 glass filter.
Chapter 3
Tools of Ultrafast Spectroscopy
The fundamental tool of any ultrafast spectroscopy laboratory is the laser system which
generates femtosecond pulses. In the early part of this chapter we describe in detail how
modern femtosecond laser oscillators utilize a non-resonant optical nonlinearity — Kerr
Lensing — to achieve pulse duration well below 100 fs. For the experiments described in
this thesis it is necessary to amplify the pulses produced by the femtosecond oscillator. We
discuss the method used in our experiments — Chirped Pulse Amplification — in the latter
part of this chapter.
3.1
Ultrashort Pulse Generation
The coupling of longitudinal modes in a laser resonator by locking their phases, usually
referred to as mode locking, is by far the most widely used technique for generating short
light pulses. Ever since its discovery in 1964 by Hargrove et al. [34], there has been a
tremendous effort to develop sources for short and ultrashort pulses motivated by demand
in physics, engineering, chemistry and biology. The first generation of pulsed lasers were
solid state lasers, which were either actively modulated or used a fast response saturable
absorber [35]. The pulse durations of these systems were around 100 ps as a result of
shortcomings in the active modulation process or the finite response time of the absorbers.
33
Chapter 3: Tools of Ultrafast Spectroscopy
34
The discovery of organic dye lasers in 1970 [35], led to the rise of the second generation of
pulsed lasers, which persisted for about 15 years. Those lasers, some of which are still in
active use, used a saturable absorber in conjunction with a saturable gain to achieve efficient
pulse shaping down to sub-100-fs pulses [35]. In spite of these excellent results, the demand
for stable, easy to use and to maintain, ultrashort pulse generating lasers motivated research
on mode-locked solid-state lasers. A number of new broadband gain media were discovered,
one of which was the Titanium doped sapphire crystal which is the most prominent one
today [36]. Apart from bandwidth there is the problem of ultrafast modulation to achieve
short optical pulses. This problem was solved by the use of the nonresonant Kerr effect
(see Section 2.3.1). The exact implementations of the mode locking techniques using this
effect varied [35], but commonly used the quasi-instantaneous response of that nonlinearity.
The scheme which has proven to be most stable and easy to handle is the so called Kerr
lens mode locking scheme. It was first discovered purely experimentally [37], lacking any
theoretical explanation. But soon afterwards numerical and analytical investigations were
able to explain the responsible physical effects, which will be the main topic of this section.
3.1.1
Mode Locking in Laser Resonators
There are two basic schemes for phase locking longitudinal modes in a laser resonator. One
method modulates the amplitude of the electric field in the laser cavity and is referred to
as amplitude modulation (AM), borrowing terminology from electrical engineering. This is
by far the most commonly used one. The second technique modulates the phase of the field
rather than the amplitude and is referred to as frequency modulation (FM) [38]. Since all
the modern sources of short light pulses use AM mode locking, this paper will focus on that
technique. Furthermore, there are two fundamentally different approaches to achieve either
of the two modulations of the electric field. One can either externally ”switch” the field
on and off by actively modulating the loss or the gain. This is referred to as active mode
locking and was and still is widely used to generate light pulses. Obviously however, the
time duration of these pulses is limited by the available switching mechanisms, which usually
35
lie above 1 ns (for example, the recombination time of direct bandgap semiconductors). In
order to achieve pulse durations below this time regime one has to resort to so-called passive
mode locking where the laser beam modulates itself due to nonlinear effects either directly
inside the lasing medium or in nonlinear elements which are purposefully added to the cavity.
One important fact to note is that the number of longitudinal cavity modes which can be
phase locked to each other is limited by the gain bandwidth of the laser medium used. So
one basic requirement for achieving ultrashort pulses is to have a broad bandwidth material.
For a certain bandwidth δω of the lasing transition, a lower bound for the achievable pulse
duration can be derived directly from Fourier transform relation as 1/δω.
3.1.2
AM Mode Locking
The actual process of forcing mode locking upon a laser is nonlinear and its details depend
heavily on the specific system in question [39]. The following description is a general
semiquantitative approach to give some basic physical insight into the fundamental process
of AM mode locking. Let us consider the electric field inside a resonator. For simplicity
but without loss of generality, we assume all fields to be scalars:
Em(z, t) = Em sin(km z) sin(ωm t + φm )
(3.1)
Now suppose the amplitude is not constant, but is modulated at a certain frequency Ω and
with a modulation depth .:
Em = Eo (1 + . cos(Ωt))
(3.2)
This leads to a modulated electric field of the form:
Em (z, t) = Eo(1 + cos(Ωt)) sin(kmz) sin(ωm t + φm )
(3.3)
Using trigonometric identities this can be written as:
.
Em (z, t) = Eo sin(ωm t + φm ) + sin[(ωm + Ω)t + φm )]
2 .
+ sin[(ωm − Ω)t + φm )] sin(km z)
2
(3.4)
36
Thus the modulation of the amplitude generates sidebands at frequencies ω ± Ω:
∆ = ωm+1 − ωm = π
c
L
(3.5)
Now, if the modulation frequency matches exactly the roundtrip frequency of the
laser cavity, (in other words the longitudinal mode spacing in frequency space) the sidebands
associated with each mode will correspond exactly to the two modes adjacent to it. In this
case, each mode becomes strongly coupled to its nearest neighbors. One can understand
why this locking of the phases of different longitudinal modes leads to the generation of
pulses in different ways. Purely mathematically, it is possible to show that as a direct
consequence of the above formalism, the summation over N different longitudinal modes
leads to pulses in the time domain which have a width of 2L/cN , where L is the length
of the laser cavity [38]. This is analogous to the decomposition of a wave packet into its
plane wave Fourier components. One can also think of it more physically, by assuming that
laser action is only possible when the modulated gain (loss) is bigger (smaller) than the
natural loss (gain), which will happen in time intervals of T = 2π/∆ = 2L/c. Of course
it is necessary to match the modulation frequency to the round-trip frequency of the laser
for this purpose, so that a pulse suffers from a equal gain (loss) on each round-trip in the
cavity.
Active and Passive Mode Locking
As mentioned above, there exist two basic schemes to achieve mode locking in a laser. The
obvious and straightforward one is referred to as active mode locking and can be described
fairly easily by the following differential equation [39]:
1 d2
g 1 + 2 2 − (Lo + Lm (1 − cos(ωm t))) E(t) = 0
ωg dt
(3.6)
which describes the balance between gain and loss. The gain is assumed to have Lorentzian
profile with a bandwidth g. For small gain, the change in the electric field amplitude
due to that gain can be expressed as above. For short pulse durations, the harmonic
2.8
37
2.8
4.0
2.6
2.6
3.0
2.4
2.4
2.2
2.2
2.0
2.0
0.0
1.8
−1.0
(b)
1.8
0
1
2
3
4
5
intennsity (a.u.)
loss (a.u.)
gain (a.u.)
(a)
2.0
1.0
0
1
time (a.u.)
2
3
4
5
time (a.u.)
Figure 3.1: Schematic representation of active mode-locking. The loss (solid curve) is
actively modulated to dip below the gain (dotted curve) to achieve pulsed laser action.
modulation of the loss can be approximated as parabolic, which leads to a Gaussian solution
E(t) = Eoe−t/τ , where the pulse width τ is given by:
1
1
g 4
8
τ=
ωg ωm
Lm
(3.7)
The principle of active mode locking is illustrated in Fig. 3.1.
This kind of mode locking is still widely used, but is obviously limited by the
external switching time constants. For ultrashort pulse generation it is necessary to achieve
mode locking without having to modulate gain or loss externally. This concept is referred
to as passive mode locking [40]. The fundamental idea is the same for all passively mode
locked systems. A lossy element with an intensity dependent absorbance is inserted into
the cavity. The absorbance sharply decreases above a certain threshold intensity, favoring
pulsed lasing above continous wave (cw) operation. This kind of element is referred to
as a saturable absorber. The rate at which the laser beam now modulates itself is only
limited by the material properties of the saturable absorber, and can be much higher than
the highest possible active switching rates. In fact, the essential time constant of these
saturable absorbers — namely the time for recovery from saturation — divides them into
two main classes: slow and fast saturable absorbers [40].
In the case of slow saturable absorbers, pulse formation is due not only to saturation of the absorber (decrease in loss), but also due to saturation of the gain medium
gain / loss
38
loss
gain
(a)
intensity
gain / loss
loss
gain
(b)
(c)
time
Figure 3.2: Schematic representation of saturable absorber action. (a) shows the principle
of a slow saturable absorber: the lasing window is defined by the fast onset of absorption
i.e. fast onset of loss (solid curve) combined with fast onset of gain saturation (dotted curve).
(b) shows the principle of a fast saturable absorber: the pulse duration is only determined
by the speed of the absorber. (c) shows the resulting pulse.
(decrease in gain) as depicted in Fig. 3.2(a) [40]. If the saturation intensities of the absorber
and gain media are comparable, pulses with durations much shorter than the individual recovery times can be achieved. The pulse shaping comes about since the absorber will absorb
the leading edge of a pulse before going into saturation and the gain medium will cut off
the trailing edge of the pulse by going into saturation itself. In this way a pulse will suffer
from shortening every roundtrip. This scheme was the favored one during about 15 years
following the discovery of organic dye lasers in the 70’s [35]. Pulses with durations shorter
than 100 fs were achieved [35].
39
In spite of the fact that extremely short pulses can be achieved using the aforementioned technique, most of these systems have been replaced by more convenient solid
state sources. Solid state lasers in general stand out, from a technical point of view, by their
reliability, their compactness, and the high degree of reproducibility of their performance.
In general, broadband tunable solid state lasers can be achieved when the electron-phonon
coupling between the lasing ion and the host lattice is strong [36]. After the first realization of such a vibronic transition laser in 1963 [36], many different materials were studied
and successfully used as lasing media. By far the most widely used material is Ti:sapphire
(Ti:Al2O3 ), whose use as a laser medium was first demonstrated by Moulton in 1982 [35].
Ti:sapphire has an exceptionally wide tuning range (from 680 nm to 1100 nm) covering
most of the interesting wavelengths for physics, engineering or biochemical applications.
This characteristic along with other reasons such as high optical quality, make it the most
popular material to date. For a long time, one remaining problem with solid state lasing
media was that the gain saturation threshold lies far above the usual cavity intensities
[35, 36], ruling out passive mode locking by slow saturable absorber techniques. Therefore
one has to resort to fast saturable absorbers, where the pulses are shaped only by the lasing
window, which is created by the saturation and recovery of the absorber. The principle of
such a fast saturable absorber is depicted in Fig. 3.2(b).
Now, the straightforward way of implementing such an absorber would be to find
the material with the fastest decaying transition at the desired wavelength. Unfortunately
typical decay time constants for transitions at optical wavelengths are above about 1 ns. So
the task is to think of non-trivial fast saturable absorbers. There has been a tremendous
research effort in this field and there have been several clever solutions to this problem [40].
All the solutions involve nonresonant optical nonlinearities to achieve so-called artificial
fast saturable absorbers. The most successful one is called Kerr lens mode locking which
will be discussed in detail in Section 3.1.4. But to understand the very specific case of Kerr
lens mode locking one first has to understand mode locking with a fast saturable absorber,
which will be described in detail in the following section.
3.1.3
40
Analytical Treatment of Mode Locking by Fast Saturable Absorbers
In the terminology of nonlinear optics, a saturable absorber is basically a medium with
significant third order optical susceptibility χ(3). In this case, the optical properties, including absorption of that medium, will be dependent on intensity, since the dielectric function
must now be written as [28]:
. = 1 + χ + χ(3) |E|2 E
(3.8)
Thus the nonlinear contribution to the dielectric function is given by
∆.N L = χ(3) |E|2 ,
(3.9)
which leads to an intensity dependent absorption coefficient of the form (see Section 2.3.1):
3π (3)
2
2
2
(3)
(N L)
χ |E| ,
= Im
(3.10)
∆n = χ |E| ⇒ ∆k
2n
where ∆k stands for the absorption coefficient. That will in turn cause a change in the
amplitude of a beam going through this nonlinear medium:
∆E =
ω
∆k(N L)E = γ |E|2 E
c
(3.11)
So the absorption will also depend on the intensity, opening the possibility of using these
kinds of materials as saturable absorbers.
The problem of describing mode locking by artificial fast saturable absorbers is
complicated by the fact that the use of a specific optical nonlinearity requires intensities that
will always invoke other nonlinear effects. These effects have to be taken into account in a
thorough analytical model. Haus et al. proposed a model which predicts a sech(t) shape of
the generated pulse, which was experimentally verified [41]. The model is essentially based
on the same type of differential equation as the treatment of active mode locking described
above. However, the loss is not harmonically modulated but dependent on the intensity of
the laser beam, leading to a change in the electric field amplitude of:
∆E = γ |E|2 E
(3.12)
41
This instantaneous treatment implies, however that the relaxation time of the
respective material is considerably shorter than the pulse durations: τa << τ . Otherwise
one would have to take into account the time evolution of the absorption after the pulse has
passed. Furthermore, one has to account for the phase shift due to self phase modulation
by:
∆E = −iδ |E|2 E,
(3.13)
There will also be group velocity dispersion:
∆E = iD
d2
E,
dt2
(3.14)
where D is the paramater indicating the magnitude of group velocity dispersion and linear
phase shift:
∆E = −iψE
(3.15)
resulting in a differential equation of the form:
d2
1 d2
g 1 + 2 2 − l − γ |E|2 + i D 2 − iψ − δ |E|2 E = 0
ωg dt
dt
(3.16)
This differential equation is solved by the Ansatz:
E = Eosech(t/τ )eiβln(sech(t/τ )
(3.17)
Inserting this back into the differential equation leads to an equation with multipliers in sech(t) and sech2 (t). Setting the coefficients of each of them to zero and separating
into real and imaginary parts leads to four equations, which can be solved for the three
parameters β (chirp), τ (pulse duration) and Eo (pulse amplitude). This quite elaborate
algebra gives conditions on the parameters from which design optimizations can be inferred.
Physically, it is crucial to see that a nonlinear absorption coefficient, denoted here by γ can
lead to passive mode locking generating pulses down to durations of the time scale of the
relaxation time constant of the absorber material.
3.1.4
42
Kerr Lens Mode Locking
We have seen in Section 2.3.1 that intense laser beams are subject to self focusing in nonlinear optical media. It turns out that one can design an artificial ultrafast absorber based on
self focusing. If an aperture is placed into a laser cavity and self focusing is present, there
will be an intensity-dependent loss due to the varying beam waist. The smaller beam waist
at higher intensities will guarantee that the beam can pass through the aperture without
being attenuated. At lower intensities, the edge of the beam will be cut off by the aperture.
Thus a change in the electric field amplitude of the form ∆E = γ |E|2 E is obtained. Owing
to the quasi-instantaneous response of the nonresonant Kerr effect, ultrafast saturable absorber action can be simulated. In properly designed resonators this amplitude modulation
mechanism favors pulsed operation over cw laser oscillation. This scheme of mode locking
is referred to as Kerr lens mode locking, so-named by Spinelli et al., and has been successfully used by many groups to produce ultrashort optical pulses [40]. The first experimental
observation of Kerr lens mode locking (KLM) occured in 1990 using a Ti:sapphire crystal
as the laser medium and was published under the title of “... self-mode-locked Ti:sapphire
laser”, already suggesting that people were very unsure about the physical origins of the
mode locking process [37]. At that time the mechanism was not very well understood and
even the term “magic mode locking” was commonly used. What happened in that experiment was that the beam apertured itself due to the spatial gain profile in the laser. So the
nonlinearity of the gain medium can be responsible for both self-focusing and aperturing. In
subsequent experiments KLM with “hard” apertures (as opposed to the “soft” self-induced
aperture) was also achieved [42], and led to improved results. In the following sections we
will focus on KLM with hard apertures, because it gives a more intuitive understanding.
There were also a number of analytical treatments, one of which will be discussed in detail
in the following section.
43
Propagation of Gaussian Beams
Before we can go ahead and tackle the analytical model of KLM, we need to develop a
formalism to treat the propagation of a Gaussian beam through a nonlinear medium when
self-focusing occurs. This is a highly non-trivial problem. An elegant treatment of the
propagation of Gaussian beams through a nonlinear medium was given by Belanger et
al. [43] and has been widely adapted. The basic idea is that, under certain approximations
(which are usually well fulfilled), the propagation of a Gaussian beam in the nonlinear
medium can be described as free space propagation if the generalized radius of curvature of
the Gaussian beam q is renormalized appropriately.
A Gaussian beam can be represented as [43]:
2
ikr
− 2q(z)
ΨG (ρ, z, ω) = AG e−iP (z) e
,
(3.18)
where r represents the lateral coordinate and q(z) is the generalized radius of curvature,
which has the form:
1
−2i
1
=
+
,
q(z)
R(z) kw 2 (z)
(3.19)
where R(z) is the curvature of the wavefront and w(z) is the beam waist.
Consider first the propagation of the beam through a short distance dz, containing
a weak lens of differential focusing strength d(1/f ). The ABCD matrix [44] for such a
system is:



A B
C D

=

1
dz
−d(1/f )
1

(3.20)
The transformation of the q parameter leads after some straightforward algebra to the
following differential equation for the q parameter:
1
d 1
d 1
= 2+
−
dz q
q
dz f
Since the intensity of the Gaussian beam is given by A2G e(−2r
(3.21)
2 /w2 )
, the phase delay caused
by the Kerr medium after propagating through the slab dz can be expressed as
φ=
2π
2
2
2P
2π
n2 A2G e−2r /w dz ≈
n2
(1 − 2r 2 /w 2 ),
λ
λ
πw 2
(3.22)
44
where the beam profile is approximated as parabolic (which is well fulfilled, if appropriate
filtering mechanisms in the laser prevent modes other than the fundamental Gaussian mode
from oscillating) and the photon flux P is introduced as P =
πw2 2
2 AG .
The relative phase
shift between different lateral regions of the beam is:
∆φ =
2π 4P
2π 2P 2r 2
n2
n2
(π/λ2)Im2 [1/q]r 2 dz
dz =
4
λ
π w
λ
π
(3.23)
As one can directly read off from the form of the Gaussian beam above, the phase shift is
related to the curvature of the phase front 1/R by
kr 2
2R
= ∆φ. Using this relation one can
find the focal strength of the Kerr medium:
8P
1
= d(1/f ) =
(π/λ2)n2 Im2 [1/f ]dz
R
π
(3.24)
Substituting this q dependence of the focal strength into the equation obtained by the
ABCD formalism above, yields:
1
1
2 1
= 2 + 2KIm
,
(3.25)
q
q
q
π 2
n2 has been introduced. Separating
where the dimensionless Kerr parameter 2K = 8P
π
λ
d
−
dz
this equation into real and imaginary parts, leads to two equations, which can be simplified
into one by renormalizating the q parameter in the following manner:
1
1
1
+ iIm
(1 − 2K)1/2
= Re
q
q
q
In terms of this new parameter the differential equation above simplifies to:
1
d 1
= ,
−
dz q
2q
(3.26)
(3.27)
which corresponds to the simple differential equation describing the propagation of a Gaussian beam in free space. That means that the propagation of the fundamental mode of a
Gaussian beam in a nonlinear medium can be described by the ABCD formalism in free
space, with the only correction being the renormalization of input and output q parameters. All ABCD matrix elements stay the same. Thus the treatment of Gaussian beams in
nonlinear media is fairly straightforward, since all the nonlinear effects such as the change
45
l2
l1
gain / kerr
medium
(1)
(2)
(3)
Figure 3.3: Three mirror configuration of a Kerr-lens mode-locked laser cavity.
of beam waist and of wavefront curvature are taken care of by the above renormalization,
which is mainly dependent on the intensity of the beam and the n2 value of the material.
One major point to note with respect to the main topic of this section is that this Kerr
nonlinearity is nonresonant and therefore almost instantaneous. It has a relaxation time
constant below 10 fs [35] and can therefore be exploited as the fundamental basis for an
artificial fast saturable absorber as will be described in the following section.
Analytical Solution
As mentioned before, there have been a number of different analytical studies addressing
KLM [42, 45, 46]. The complexity of the analysis of KLM systems increases rapidly with
the number of optical components in the respective resonator. Earlier analytical treatments
therefore studied simple two mirror cavities extending the analysis to three mirror cavities
[46]. In more recent studies the full four mirror cavity as commonly used in experiments
is described [42, 45]. For the sake of clarity we describe the treatment of the three mirror
cavity given by Haus et al. [46] in this section, since it already provides enough insight into
the physical process and all the mathematical tools necessary to understand the full four
mirror configuration. Consider the three mirror cavity configuration depicted in Fig. 3.3.
The main goal of the following calculation is to obtain an expression for the γ
46
parameter, which, as discussed above governs the intensity dependent loss and therefore
the possibility of Kerr lens mode locking. A power-dependent loss can be described mathematically as:
γ |E|2 = −P
dl
dl dy1
= −P
,
dP
dy1 dP
(3.28)
where l denotes the loss and y1 is a the imaginary part of the Gaussian parameter q and
thus denotes the beam waist size. The differential loss per change in waist size is easily
obtained from the radius of the aperture, since everything outside that aperture will be lost.
Including a factor of two (due to the two passes per round trip) one obtains:
∞
2
2
2
2
2
2lP =
2πrdrA2G e−2r /wa = e−2Ra /wa P = P e−2πRa /λy1 ,
(3.29)
Ra
where Ra denotes the radius of the aperture and wa the waist of the beam at the aperture.
The differentiation by y1 can be carried out straightforwardly.
The actual challenge is to describe the transformation of the Gaussian beam due
to the Kerr medium, i.e. from plane (1) to plane (2) and thus obtain an expression for
the intensity-dependent waist size y1 (P ). The rest of the propagation from plane (2) to
plane (3) can be described by the standard ABCD formalism. As worked out above, the
propagation through the Kerr medium can be described easily when the Gaussian beam
parameters are properly renormalized. In this case, the transformation of the renormalized
parameters is simply the one for free space:
q2 = q1 +
L
n
(3.30)
Since the mirror at plane (1) is flat, the Gaussian profile has infinite radius of curvature, so
q1 must be purely imaginary. Assuming small K values, the renormalized q1 is then:
q1 = √
where y1 =
πw12
λ .
iy1
≈ iy1 (1 + K),
1 − 2K
(3.31)
q2 at reference plane (2) can be easily determined using the free space
propagation property above. Inverting the normalization leads to q2 :
L/n − iy1
2K(L/n)y12
2Kiy1 (L/n)2
1
1
=
−
−
= (0) + ∆,
2
2
2
2
2
2
2
2
q2
(L/n) + y1
[(L/n) + y1 ]
[(L/n) + y1 ]
q2
(3.32)
47
where terms were already ordered for their real and imaginary parts and for their intensitydependent and -independent ones. From this form of 1/q2 , one can directly read off the
intensity-dependent changes of curvature and beam waist. From the above equation q2 can
(0)
(0)
(0)
(0)
be written as q2 = q2 − [q2 ]2 ∆ = q2 + δa2 . But since q2
(0)
is simply q2
(0)
= q1 + L/n,
the change in q2 can also be expressed as a correction to q1 :
(0)2
δq1K = − q2 ∆
(3.33)
It has therefore been shown that the Kerr medium has induced a focusing wavefront and
decreased the beam waist just as expected.
Let’s disregard the Kerr effect for a moment. In the absence of the Kerr medium,
the transformation from plane (1) to plane (3) is described by the ABCD matrix:


 
1−l2
l1 l2
l
+
l
−
A B
1
2
f 

= f
,
l1
1
−f
1− f
C D
(3.34)
where f denotes the focal length of the lens. The q-parameter is subject to the usual
transformation. With the constraint that the real part of q3 has to be zero, since the mirror
at plane (3) is flat, one can derive an equation for y1 :
y12
1−
BD
= f2
=
AC
1−
l1
f
l2
f
l1
1−
f
l2
1−
f
(3.35)
In the presence of the Kerr effect, the real part of q3 has has to remain zero. That
is, a change δq1K induced by the Kerr effect, has to be cancelled by an appropriate change
in the initial y1 by δy1 :
Re
dq3
dq1
(iδy1 + δq1K ) = 0
(3.36)
The derivative dq3 /dq1 can be easily obtained from the equation obtained by the ABCD
transformation carrying q1 over into q3 . Substituting this and δq1K from above leads to an
equation for δy1K :
2K L
δy1
n
=
2
2
y1
y1 + L
n2
L
AD + BC
−
=M
2AC
n
(3.37)
48
d = f 1 + f2 + δ
l1
l2
gain / kerr
medium
f1
f2
Figure 3.4: Four mirror configuration of a Kerr-lens mode-locked laser cavity.
Then the derivative of y1 by P is simply
dy1
dP
=
δy1
P
=
y1 M
P
for small enough P . Therefore
one now has everything to calculate γ:
γ=−
L
πR2a 2πR2a/λy1 K nf
dl dy1
=−
e
dy1 dP
λ
P
AD+BC
2ACf
y12
f2
+
L
− nf
L2
n2 f 2
(3.38)
In order to achieve KLM, γ must be positive, which implies that the term in brackets must
be negative. Thus it has been shown that the self focusing effect, in conjunction with an
appropriate aperture inside a laser cavity ,can lead to the effect of an ultrafast saturable
absorber (in this notation, a positive γ).
It is worthwhile mentioning that the positive dispersion inside the Ti:sapphire
laser medium severely limits the pulse width [37]. One has to compensate for this effect by
an intracavity negative dispersive element, e.g. a prism combination, to achieve the actual
pulse limit given by KLM. We will discuss dispersion control later, in Section 3.2.1. The
dispersion at the output coupler will further stretch the pulse, so that the pulse coming
out is actually longer than the intracavity pulse. This again can be compensated by an
external compression element, i.e. a dispersive element [37]. For sake of completeness, the
most commonly used four mirror configuration is shown in Fig. 3.4. The formalism from
above can be extended to that configuration leading to a more complicated ABCD matrix
[42, 45]. Again, the result is a power-dependent loss, leading to ultrafast saturable absorber
action.
49
Numerical Results and Corrections
Based upon the knowledge of the latest numerical studies of KLM, it is only fair to say that
the analytical approach described above reveals the basic physical idea of the process, but is
not necessarily accurate. To predict the exact lasing characteristics, especially at the limit
of transform limited pulse widths, additional nonlinear effects inside the cavity must be
taken into account [47, 48]. We give a brief overview of these effects and their consequences
for the pulse characteristics.
The main improvements in the latest theoretical investigations [47] are that: (1)
a full three dimensional calculation of Maxwell’s equations is carried out; (2) instead of the
simple ABCD model, which assumes parabolic beam profiles, an accurate treatment of the
evolution of the pulse through the nonlinear crystal is provided; (3) dispersion in the lasing
material — especially higher order dispersion (up to fourth order) — is taken into account;
(4) the slowly varying envelope approximation is abandoned; (5) the finite response time
of the Kerr nonlinearity is taken into account; (6) the restriction that the power of the
beam has to be below the threshold for self-filamentation is lifted. Of course, under these
conditions, solutions can only be obtained numerically. There are several major points of
correction to the view of the situation provided by the analytical approach. First of all,
the presence of dispersion results in different pulse widths at different locations within the
resonator, since the pulse will be chirped after passing through the crystal. This means that,
in addition to the spatial focusing, there is also a focusing in time referred to as space-time
focusing in the literature. The calculations predict that the pulse characteristics depend
sensitively on the relative position between the spatial and temporal focus, being optimal
when they are matched within the crystal. Furthermore, dispersion will cause the pulse to
broaden significantly after reaching the space-time focus (at which the intensity can be well
above self-filamentation threshold), preventing any further self-focusing and catastrophic
filamentation. Lastly it was found that the effect of space-time focusing also makes the
mode-locking process less sensitive to the response time of the Kerr effect, predicting pulse
durations down to 5 fs. All of the aforementioned predictions are verified experimentally
50
[49] supporting the need for enhanced models, including heavy numerical calculations for
high end ultrashort pulse lasers, operating at the transform pulse width limit.
Summary
In summary, a guided tour from the principles of mode locking to state of the art ultrafast
pulse generation was given. The tour started out with the concept that amplitude modulation can lead to the locking of the phases of several longitudinal modes in a laser cavity,
which in turn produces pulsed laser radiation rather then continuous waves. We pointed
out that actively modulating the amplitude (active mode locking) can only achieve a certain
pulse duration, which is the reason why resarch efforts focus on passive mode locking. The
basic physical ideas behind the two main techniques - slow and fast saturable absorbers were discussed. It turns out that in solid state lasers one has to apply fast saturable absorber techniques. An analytical treatment of mode locking with fast saturable absorbers
was given. To achieve pulse durations in the sub-100 fs regime one has to resort to so-called
artificial fast saturable absorbers, which make use of nonresonant optical nonlinearites. One
special kind, which is the most successful one to date is the Kerr effect. An analytical theory of Kerr lens mode locking was presented, which leads to an analytical expression for
the intensity dependent loss. The guided tour concluded with a brief report of the latest
numerical investigations carried out on that subject, which predict pulse durations of 5 fs.
The current world record for ultrashort pulse generation is held by Krausz et al. and is a
pulse duration of 5 fs [3].
3.1.5
The KML Oscillator
In our experiments, we use a commercially available KLM oscillator from Kapteyn Murnane
Laboratories (KML). Like almost all KLM oscillators, it uses a Ti:sapphire crystal as the
lasing medium. Figure 3.5 shows a schematic of the laser cavity. The KML oscillator is
a standard KLM laser cavity following the four mirror configuration discussed in Section
3.1.4, where the lenses are replaced with focusing mirrors to reduce dispersive stretching
51
prisms for
dispersion compensation
output
coupler
end
mirror
PM
L
PM
Ti:sapphire
crystal
pump beam
Figure 3.5: Schematic illustration of the Kapteyn-Murnane-Laboratories Ti:sapphire oscillator cavity. PM and L denote steering mirrors and focusing lens for the pump beam.
in the cavity. The Ti:sapphire crystal is cut at the Brewster angle for the center wavelength (800nm) of the gain spectrum to minimize losses due to reflection at the crystal face.
Furthermore, the KML oscillator uses a so-called soft, or self-induced aperture to achieve
KLM. The aperture is induced by the spatial gain profile inside the Ti:sapphire crystal,
which in turn is determined by the Gaussian intensity profile of the pump beam. Figure
3.6 illustrates the principle of a pump-induced aperture. The highest gain is achieved towards the center of the pump beam. The outer regions are pumped at a lower intensity and
therefore provide less gain. This Gaussian gain profile is effectively acting as a soft aperture
because it amplifies smaller beams more than wider beams. As already mentioned in Section 3.1.4, the prism combination introduces negative dispersion to cancel the dispersion in
the Ti:sapphire crystal.
Let us turn to the alignment and day-to-day use of the KML oscillator. It is
absolutely crucial in the alignment of the laser to perfectly overlap the pump and the probe
beam in the Ti:sapphire crystal to achieve KLM action. It turns out that the astigmatism
introduced due to the necessary non-zero angle orientation of the curved inner cavity mirrors
actually helps achieving a clean focus inside the crystal. It compensates for the astigmatism
introduced by the Brewster-cut crystal face. The Brewster-cut induced astigmatism of the
pump beam is cancelled by slightly rotating the focusing lens L, by about 12o . For the
52
Gain
Profile
High (H)
Intensity
Low (L)
Intensity
Figure 3.6: Illustration of a so-called soft, or self-induced aperture. The Gaussian gain
profile, as induced by the profile of the pump beam acts as a soft aperture by amplifying
smaller diameter beams better than wider diameter beams.
first alignment, following the instruction in the manual is mostly sucessful. It is crucial
to exactly level all beams (pump and infrared beam) inside the cavity at the exact correct
height (center of all optics) to successfully achieve mode locking. Mode locking is triggered
by rocking the outer prism to introduce a seed for KLM. When the cavity is well aligned
a simple knock on the optical table should be enough to start pulsed laser action. Even
self-starting mode locking was achieved in our labs with this KML oscillator.
The KML oscillator produces pulses of down to 18 fs duration (full width at half
maximum — FHWM) after compression to cancel the dispersion of the output coupler.
The compression is usually done using a prism compressor which we describe in Section
3.2.1. The spectral bandwidth of laser pulses is about 80nm (780 – 860 nm, FWHM). The
repetition rate is about 90 MHz and the pulse energy is about 3 nJ per pulse. There is a
vast variety of experiments which have been performed using such a Ti:sapphire oscillator
from
oscillator
pulse
stretcher
53
amplifier
pulse
compressor
pump
laser
Figure 3.7: Schematic representation of Chirped Pulse Amplication.
and the research using this tool is still in full motion. For the experiments described in
this thesis, however, much higher pulse energies are needed. The amplification of ultrashort
optical pulses presents quite an experimental challenge. We use an approach called Chirped
Pulse Amplification (CPA) to overcome this challenge.
3.2
Chirped Pulse Amplification
The main problem in amplifying a femtosecond optical pulse lies in the fact that, at energy
levels of several micro-Joules per pulse, the peak power becomes so high that amplified
pulses damage the lasing medium or optical components in the amplifier. Even if damage
does not occur there are many nonlinear optical effect, such as self focusing, which either
cause dispersion that is very hard to compensate for, or affect the propagation of the beam
in a way that makes it hard to properly collimate or focus the beam. A solution to these
problems is an approach called Chirped Pulse Amplification (CPA). The main idea is to
stretch the seed pulse in time to durations where even if the pulse energy is raised to mJ
levels, damage and nonlinear effects do not play a major role. If the stretching is done in
a controlled fashion it is then possible to reverse the stretching process and recompress the
pulse to its orginal duration. The general principle of CPA is indicated in Fig. 3.7. CPA
was first successfully demonstrated by Mourou et al. [50, 51]. In this section we discuss the
main steps in CPA and describe the specifics of the system used in our experiments.
3.2.1
54
Compression and Stretching of Femtosecond Optical Pulses
Grating Compressor
The first step in CPA is the stretching of the femtosecond seed pulse to tens of picoseconds
or sometimes even nanosecond durations. To understand the details of pulse stretching it is
instructive to first consider pulse compression. The notion of optical pulse compression in
the time domain is usually equivalent to inducing negative dispersion to cancel the positive
dispersion in most materials.1
The most easily understood optical device that achieves negative dispersion is the
grating-based pulse compressor. Figure 3.8 shows the first design which was proposed by
Treacy in 1969 [52]. It is still the basis for all grating-based dispersion control devices in
use.
If a monochromatic beam at wavelength λ hits the first grating it gets deflected
according to the standard grating formula:
sin θr = sin θi +
mλ
d
(3.39)
where the integer m is the order of the diffraction (we consider m = −1 for our purposes), λ
is the free space wavelength and d is the groove spacing on the gratings. Different wavelength
components follow different paths as shown in Fig. 3.8, the path being shorter for higher
frequency components.
Following the nomenclature introduced in Fig. 3.8(b), the path length for a single
frequency is given by:
p(ω) = l1 + l2 = constant + 2LG (sec θr + cos θi + sin θi tan θr )
(3.40)
where LG is the perpendicular distance between the gratings. The important quantity in
determining the temporal shape of an optical pulse is the phase delay which is related to
1
Broadband optical pulses widen temporally as they propagate through materials, which generally exhibit
positive dispersion.
55
blue
red
(a )
l2
i- r
LG
l1
r
i- r
i
(b )
Figure 3.8: (a) Compressor based on diffraction gratings. The labels red and blue denote the
paths of the long wavelength and the short wavelength components of the light respectively.
(b) Shows the detailed path of monochromatic light through the first grating pair.
the path length by
∆φ = kp =
ω p(ω)
.
c
(3.41)
The influence of dispersion on the phase is most commonly tackled in ascending
order of the Taylor expansion of the functional relation between the phase and the frequency
ω. A very instructive treatment of the effect of these linear and higher order dispersion terms
on the temporal shape of the seed pulse is given by Glezer in Ref. [53]. It turns out that the
linear term in ∆Φ(ω) does not effect the pulse shape at all but merely changes the pulses
56
overall velocity. The first important term is the second order derivative of ∆φ(ω) — also
called second order dispersion (SOD).
In our example we can easily calculate all derivatives of Eq. 3.41 given Eqs. 3.40
and 3.39. The SOD is
2 −3/2
2πc
8π 2 c
∂ 2 ∆φ(ω)
− sin θi
= 3 2 LG 1 −
,
∂ω 2
ω d
ωd
where
2πc
ωd
(3.42)
− sin θi = sin θr according to Eq. 3.39. Therefore, the SOD of this grating
compressor is negative. The exact magnitude of SOD induced by this optical device depends
on the values of the grating separation LG and the angle of incidence θi . We can use such
a grating compressor to compensate for the positive SOD of materials by finetuning these
values. It is perfectly sufficient to just use one grating pair if all one is concerned about
is temporal shape of the output pulse. In laboratory applications, however, it is necessary
to properly collimate and steer optical beams. The output of a single grating pair is still
spectrally dispersed as indicated in Fig. 3.8. This can be fixed by passing through an
equivalent grating pair in the opposite direction. Thereby the pulse experiences twice the
SOD and is spectrally refocused and recollimated. In our grating compressor (which we
discuss in Section 3.2.2) we double pass one grating pair by retro-reflecting the beam back
into the grating pair, thus passing the pair twice. The retro-reflecting mirror is slightly tilted
horizontally to ensure that the beams travel through the grating pair at slightly different
heights each time. That allows clean injection and ejection of the beam. This method of
double passing is often used to “save” optical components, and is referred to it as “folding”.
Prism Compressor
One of the major disadvantages of compressing a pulse using a grating based design, as
decribed above, is loss. There is quite a significant amount of loss (up to 50%) due to
specular reflections and diffractions into different orders. An alternative approach for pulse
compression which does not suffer from diffractive losses is the prism based compressor [54].
Figure 3.9 shows a schematic drawing of such a device. The details of the prism design are
57
blue
red
(a )
B′
C′
C
B
translation
for fine adjustment
LP
A′
A
(b )
Figure 3.9: (a) Compressor based on prisms. The labels red and blue denote the paths
of the long wavelength and the short wavelength components of the light respectively. (b)
Shows the detailed path of monochromatic light through the first prism.
given in Refs. [54, 55]. We will just outline the basic scheme here. The SOD induced by a
four-prism compressor is given by [55]:
∂ 2 ∆φ(ω)
∂ω 2
=
=
8π 2 c d2 p
ω 3 dλ2 2 dθ
8π 2 c
d2 θ
− cos θ(λ)
LP − sin θ(λ)
,
3
ω
dλ
dλ
(3.43)
58
where the first term in the square brackets describes the SOD due to the material of the
prisms (which is positive as for most materials) and the second term describes the SOD due
to the geometry of the setup. If the separation of the prisms is large enough the second
term wins out and the total SOD is negative.
We use a prism compressor to compress the pulses from the Ti:sapphire oscillator
in some of the experiments described in Chapter 8. Prism compressors are the preferred
devices for pulse compression if there is no need to amplify pulses for the experiment,
because they are cheaper, easier to build and have less loss than grating compressors.2 In
our experiments we use a double-passed prism compressor, where the configuration shown
in Fig. 3.9(a) is folded along the dashed line. That is done by putting a retroflector in place
of the dashed line. Of course the second prism pair is now superfluous. The amount of SOD
can be changed by varying the distance between the two prisms.
A very nice summary of SOD and higher order dispersion effects of grating pairs,
prism pairs, and materials is given in Ref. [56].
Grating Stretcher
The previous discussion was centered around the topic of compressing optical pulses in the
time domain. As we mentioned earlier, it is necessary to stretch a pulse before amplification.
Stretching of an optical pulse is equivalent to inducing positive dispersion — consistent with
the stretching of a pulse as it propagates through material. Of course, the easiest way of
imposing positive dispersion on an optical pulse is to send it through a slab of material.
The problem with this straightforward solution is that one has to stretch a femtosecond
pulse by about 4–6 orders of magnitude to be able to amplify it to mJ levels. The slab of
material required for this kind of stretching would be very thick (on the order of meters for a
standard fs pulse). Also, it is desirable to have a controlled way of imposing the dispersion.
That makes it possible to cancel the artificially imposed positive dispersion in a compressor
2
In the case of amplified pulses on the other hand it is necessary to prestretch and then recompress
ultrashort pulses as we describe in Section 3.2. One therefore has to exactly cancel the dispersion caused by
the stretcher and the compressor respectively. In that case the only viable choice is using gratings for the
dispersion control.
59
FM
blue
G1
L1
f
G2
red
f
L2
f
D
f −D
Figure 3.10: Stretcher based on diffractive gratings. The two gratings G1 and G2 have
negative separation due to their position relative to the telescope consisting of the two
lenses L1 and L2 .
by inducing the equivalent negative dispersion.
Martinez proposed a clever scheme to create a negatively dispersive optical device
in 1986 [57]. As we know from Eq. 3.42, the SOD of a grating pair is linearly dependent
on the separation of the two gratings and the total expression is negative. If one could
achieve negative distances, the effective SOD of the grating pair would be positive. Martinez
achieved an effective negative grating separation by using a telescope as shown in Fig. 3.10.
The grating G1 is at the focus of lens L1 . The two identical lenses are separated by exactly
twice their focal length f . That makes a 1-to-1 telescope which images the focal plane of
lens L1 exactly to the focal plane of lens L2 . So placing the second grating at the focus of L2
would be equivalent to having the two gratings stuck together with zero distance between
them. Instead, grating G2 is placed at a distance D after the second lens L2 which is smaller
than the focal length f . That has the effect of having a negative distance of D − f between
the gratings. Therefore, according to Eq. 3.42, a positive SOD is introduced which stretches
the incident pulse significantly. In the setup shown in the figure, the grating pair is double
passed — F M indicates the folding mirror used to retroreflect the beam back through the
grating pair. This makes the second grating pair superfluous. Since the induced dispersion
can be exactly controlled, it is easy to cancel it using a grating compressor as discussed in
60
1
4
3
2
1
3
1
4
3
2
2
4
1
Figure 3.11: Stretcher design only using one grating. The grating is used at Littrow-angle
so that all optics are in line with one common optic axis. The beam path is such that it
starts at the point denoted 1 and exits the point denoted 2.
Section 3.2.1.
In our experiments we use a very special design of a grating compressor. The
schematic setup is shown in Fig. 3.11. We use only one grating for the entire stretcher.
This can be achieved by folding the two grating configuration again at the plane indicated
by the dashed line in Fig. 3.10. Before folding, one has to make sure that the setup is
completely symmetric about the folding plane. Moving G1 towards L1 creates symmetry
while preserving the effective negative distance between the gratings. After folding we
replace the focusing lens by a parabolic mirror to minimize achromatic aberrations. Lastly,
we use the grating at Littrow angle3 which allows aligning all optics along one common
optic axis. We use a grating that is blazed at Littrow angle which increases the energy
diffracted along the Littrow angle quite signifcantly to about 90% — the remaining 10%
is lost to specular reflection and diffraction into different orders). The last folding mirror
which “replaced” the dashed line in Fig. 3.10 has to be positioned exactly one focal length
f apart from the parabolic mirror just as the dashed line is exactly f between both lenses.
The grating on the other hand is only at a distance D from the mirror to achieve negative
effective distance which puts it between the folding mirror and the parabolic mirror as
indicated in Fig. 3.11. This specific setup stretches the pulses from the KML oscillator (see
Section 3.1.5) by 3 orders of magnitude to about 50 ps.
3
The Littrow-angle is the angle at which the first negative order diffraction is diffracted directly back
into the direction the incident beam is coming from.
3.2.2
61
Multipass Amplifier Design
Now that we know how to stretch and compress ultrashort optical pulses, we can turn to
the discussion of the amplification of these pulses using CPA. In our experiments, we use a
scheme called multi-pass amplification.4 The full amplifier system is illustrated in Fig. 3.12.
This design follows the design by Backus et al. from 1995 [59]. Let’s follow the path of the
light coming from the KML oscillator and see what happens at each stage in the multi-pass
amplifier.
First, the pulse is stretched to about 50 ps in the stretcher (described in Section
3.2.1). It then passes through a pulse picker which consists of two crossed polarizers with
a Pockels cell (Quantum Electronics Model QC-10) in between. This arrangement allows
the selection of single pulses out of the 90 MHz pulse train from the Ti:sapphire oscillator.
We divide the 90 MHz train down to a 1 kHz train. The reason for doing this is that it
is necessary to pump the gain medium at a very high power levels to achieve the wanted
amplification factors. Commercially available laser sources obviously have certain power
limitations. Lasers which can provide the necessary pulse energies of about 15 mJ per pulse
in the green (optimal pump wavelength for the Ti:sapphire crystal is around 530 nm) have
a repetition rate of just around 1kHz — hence the division of the oscillator pulsetrain to
1 kHz. This 1 kHz train of 50 ps pulses is then sent to the heart of the amplifier — the
Ti:sapphire crystal.
We use a Ti:sapphire crystal from Crystal Systems with the following specifications:
10 mm diameter, 4.75 mm path length, 0.25% Ti doping, Brewster cut, Figure of Merit
(FOM) > 150. The crystal is positioned between two spherical mirrors which each have
a focal length of 50 cm. Both are slightly tilted towards a very broad flat mirror which
allows a triangular path with a focus at the crystal. We pump the crystal using a doubled
Q-switched YLF laser from Quantronix which produces 150 ns pulses of 15 mJ energy at
a repetition rate of 1kHz. We focus the output of this laser into the crystal using a 40 cm
4
There is another very common scheme called regenerative amplification. We refer the interested reader
to Ref. [58] for detailed information these systems.
62
EM
multipass amplifier
IM
Ti:S
P
PM
PD
P
pulse
picker
PC
MK
shutter
PH
compressor
Nd:YLF pump laser
(1 kHz, 15 mJ, 527 nm, 150 ns)
Argon pump laser
(CW, 14 W, 514 nm)
PE
PE
Ti:sapphire oscillator
(90 MHz, 4 nJ, 805 nm, 80-nm wide)
stretcher
Figure 3.12: Detailed schematic of the Ti:sapphire amplifier. All major stages are labelled:
stretcher, pulse picker, multipass amplifier, mechanical shutter and compressor. Components other than flat mirrors, spherical mirrors, lenses, gratings and alignment irises are
labelled as follows: PM = parabolic mirror, PD = photodiode (receives signal from specular reflection off the stretcher grating and sends it to the pulse picking electronics), PE =
down periscope (set of two mirrors that points the beam down and rotates its polarisation),
P = polariser, PC = Pockels’ cell, IM = amplifier injection mirror, EM = amplifier ejection
mirror, MK = amplifier mask, PH = pinhole (an iris tightened down at the beam focus to
clean up the amplified beam mode).
lens. 70% of the pump energy is absorbed in the first pass through the crystal. We reflect
the remaining energy back into the crystal using a 40 cm spherical mirror on the opposite
63
side.
To synchronize the arrival of the infrared seed pulse and the green pump pulse in
the crystal we slave both the Pockels cell and the YLF laser to the pulse train produced by
the Ti:sapphire oscillator. We can do this quite easily by picking off a higher order diffraction
from the stretcher grating with a photodiode and using the so-produced electrical pulse train
as the master clock for the following electronics. First, the MHz train serves as an input
into a down counter (Quantum Electronics Model DD1 Divider Delay Unit) that generates
a 1 kHz train which is synchronized to the MHz train. This kHz train serves as a clock for a
delay generator box (Stanford Research Systems Model DG-535). This delay generator box
serves two purposes: (1) it allows the selection of a specific temporal window for opening
the pulse picker — that is, when to turn on the Pockels cell and when to turn it off5 ; (2)
it allows generation of a pulse train at a variable delay (with respect to the Pockels cell
trigger) which we use to trigger the YLF laser, allowing exact temporal overlap of the green
and the infrared pulse inside the Ti:sapphire crystal. If the spatial overlap of the two pulses
is equally good, a single pass gain of almost 10 can be achieved. Following the triangular
path described above the infrared pulse makes 8 round-trips. Great care has to be taken
to make sure that all 8 passes overlap in the crystal. Theoretically the seed pulse has now
accumulated a gain of 108 . It turns out that the gain saturates, however, at a level of about
106 . We opt for two more round trips to ensure that the infrared pulse “clears” out all
the excited electrons. At optimal alignment the output pulses out of the amplifier have
energies of about 1 mJ. A seemingly small but crucial element in the amplifier is the mask
M K. It serves two purposes: (1) it prevents amplified stimulated emission to build up in
the triangular cavity which would take some of the gain away from the stimulated emission;
(2) it reduces the effect of thermal lensing in the Ti:sapphire crystal — due to the large
amount of heat deposited into the laser rod the infrared beam diverges quite significantly
as it travels around the amplifier triangle; the mask clips the beam at each round trip to
its original diameter.
5
For the optimal window we achieve a contrast ratio of rejected pulses to admitted pulses of better than
1:1000.
64
The beam coming out of the amplifier triangle passes through a telescope which
compensates for the thermal lens in the laser rod and recollimates it. Just before the focus
of the telescope we place a shutter (nmLaserP roducts, Model LS200F N C) which we use
to pick individual pulses out of the 1 kHz train. Right at the focus of the telescope we place
an adjustable pinhole which is used to spatially filter the beam.6
The last remaining step is the recompression which we already described in Section
3.2.1. Unfortunately the throughput efficiency of grating compressors is inherently bad. In
our setup we achieve a throughput of about 60% so that the final pulse train we get consists
of pulses of 0.5 mJ energy at a repetition rate of 1 kHz. At the optimal position of the
compressor we measure pulse durations of about 35 fs. We will return to the characterization
of ultrashort optical pulses in Section 4.2.
6
Higher order modes and other imperfections in the spatial mode will not come to as small a focus as the
wanted TEM00 mode and can be clipped of at the focus with the appropriate diameter pinhole.
Chapter 4
Techniques in Ultrafast
Spectroscopy
There is a great variety of different detection techniques to observe ultrafast phenomena.
All of these techniques are based on the pump-probe scheme which we describe in the
beginning of this chapter. After giving a brief review of various ultrafast spectroscopic
techniques, we describe a new approach to characterize femtosecond optical pulses, which
is a variation of frequency resolved optical gating [60]. The main focus of this chapter is the
experimental technique used in all of the experiments described in this thesis: by combining
multi-angle ellipsometry with a white-light pump-probe setup we are able to measure the
spectral dielectric of a material with femtosecond time-resolution.
4.1
Overview of Ultrafast Spectroscopic Techniques
We have described the generation of ultrashort optical pulses in Chapter 3. Not only is
it possible to generate pulses as short as a few femtoseconds but we can also amplify and
these pulses to power levels up to 1011 Watts.1 As already mentioned in Chapter 1, there
are two main scientific applications of these short and powerful light pulses. One is to
1
Power levels of up to 1015 Watts have been reported by other research groups [61].
65
Chapter 4: Techniques in Ultrafast Spectroscopy
66
chopper
ultrafast
laser
BS
pump
sample
transmitted
probe
detector
probe
AF
time
delay
signal
BS
∆t
detector
delay
stage
0
time delay
Figure 4.1: Schematic illustration of a pump-probe setup. BS indicates a beam splitter.
use the extremely intense light field to access highly non-equilibrium states of matter —
that includes nonlinear optical effects as discussed in Chapter 2.3, optical breakdown, and
a variety of other extreme effects. The other is to use these ultrashort optical pulses as
temporal gates to achieve femtosecond time resolution. In this section we describe the basic
scheme which makes fs time resolution possible and a number of clever extensions to this
basic setup which enable the detection of many different phenomena.
4.1.1
The Pump-Probe Scheme — Obtaining Femtosecond Time Resolution
The basic scheme that is used to achieve fs time resolution is called pump-probe scheme. A
schematic pump-probe setup is shown in Fig. 4.1. A pulse coming from the ultrafast laser
source is split into two by a beamsplitter BS. Usually the beamsplitter is chosen such that
one of the pulses, say the transmitted one, is much more intense than the reflected pulse.
We refer to the intense pulse as pump and to the weaker pulse as probe pulse. Both pulses
are directed to a sample with flat mirrors. Usually a single lens focuses both beams such
that the pump and probe spot spatially overlap on the sample. In the probe-“arm” of the
setup there is a variable delay stage consisting of two mirrors as indicated in the figure. The
two mirrors basically function as a retroreflector that is mounted on a motorized stepper
67
stage. By controlling the position of the retroreflector we can adjust the time delay τ
between the pump and the probe pulse. For each micrometer of path length difference, the
relative time delay between pump and probe pulse changes by 3.3 fs. This value doubles for
the retroreflector because each micrometer of stage movement causes a path delay of two
micrometers. Commercially available motion controllers easily have accuracies of 0.1 µm
per step. It is therefore possible to control the time delay between pump and probe pulse
to a precision of single femtoseconds.
Let’s assume that we are interested in a pump-induced change in the absorptive
properties of the sample. A good indicator for changes in the absorption in a material is the
transmissivity (given that the reflectivity does not change too much at the same time). We
therefore measure the transmitted intensity of the probe pulse with a simple photodiode as
indicated in Fig. 4.1. By scanning the time delay τ from negative times (where the probe
pulse hits the sample before the pump pulse) to positive times we can track the evolution
of the transmissivity with time after the pump pulse excites the sample. A representative
trace is given in the figure. The fundamental limit of the time resolution of this setup is the
temporal pulse width from the ultrafast laser source. Equivalently one can also measure
reflectivity changes by measuring the reflected beam.
4.1.2
Different Detection Geometries for Different Phenomena
Differential Transmission/Reflection Spectroscopy
The simplest time-resolved measurements one can perform with a pump-probe setup are
measurements of pump-induced transmission and reflection changes as described in Section
4.1.1. In experiments which do not require amplification of the pulses from the Ti:sapphire
oscillator, i.e. experiments which can make use of the full repetition rate of the oscillator,
there are a few relatively simple improvements one can make to significantly enhance the
signal to noise ratio of such a setup. We use this kind of high-sensitivity setup in our
experiments described in Chapter 8. We illustrate the specifics of our setup in Fig. 4.1.
68
One source of noise is the laser source itself. There are random fluctuations in the output
power of every laser. To suppress laser noise, we split a portion of the probe beam off to a
detector using a second beam splitter. The signal read by this detector is then subtracted
from the signal measured by the detector measuring the actual transmission (or reflection)
in a differencing amplifier. To exactly cancel the two signals — which is done at negative
time delays — we use an adjustable neutral density filter wheel, as indicated by AF in
Fig. 4.1. Now, the output of the differencing amplifier will only be non-zero if there is
a pump-induced change in the transmissivity of the sample as the time delay is changed
from negative to positive values. To further suppress noise due to ambient roomlight and
any other potential source of noise we use a standard lock-in amplifier. The pump pulse
train is modulated by an optical chopper as indicated in the figure. The signal controlling
the chopper is then used as the reference input for the lock-in amplifier, ensuring that it
detects only pump-induced signal contributions. The use of this lock-in amplifier scheme is
only possible due to the fact that the repetition rate of the Ti:sapphire oscillator is much
faster than the response time of the photodiodes, therefore making the pulse train appear
as continous wave light. The slower modulation of the chopper is picked up without any
problems, however, thus creating a clean square wave form which the lock-in amplifier can
easily interpret.
Four Wave Mixing and Other Detection Geometries
There are many more sophisticated detection geometries which have been used very successfully to measure ultrafast phenomena. A nice overview of ultrafast spectroscopy techniques
can be found in Shah’s book Ultrafast Spectroscopy of Semiconductors and Semiconductor
Nanostructures [5].
Probably the most prominent detection geometry makes use of the χ(3) nonlinearity
which was discussed in Section 2.3. As Fig. 4.2(a) shows, the detected beam is the fourth
wave in a so-called four wave mixing process which propagates along the direction 2k1 −k2 as
indicated in the figure [28]. Intuitively this signal can be understood as being diffracted by a
69
(a)
(b)
2 k2 - k1
probe
pump
PBS
k2
k1
L
probe
pump
Figure 4.2: (a) Four Wave Mixing detection geometry. L = focusing lens for pump and
probe, ki denote the wavevectors of the incident light pulses; (b) Transmittive Electrooptic
Sampling geometry. PBS = polarizing beam splitter.
polarization grating set up by the two incident beams. The intensity of the diffracted beam
depends on the efficiency of the grating which in turn depends on the temporal evolution of
the polarization created by the pump pulse. For instance, if the polarization set up by the
pump pulse evolves over time, the diffracted beam carries the signature of this evolution.
This technique was used to observe Bloch oscillations and quantum beats in semiconductor
heterostructures [62, 63, 64].
Another clever geometry is shown in Fig. 4.2(b). By splitting a linearly polarized
probe beam into its vectorial polarization components ±45o degrees from the original polarization and subtracting these signals one can measure the anisotropic transmission and/or
reflection changes of the sample. This technique was dubbed transmissive/reflective electrooptic sampling. Kurz and coworkers used this detection geometry to measure Bloch oscillations and coherent lattice vibrations in bulk GaAs and GaAs heterostructures [65, 63, 66].
There is a wealth of other sophisticated detection schemes each of which are designed to observe a specific pump-induced process. To list and discuss all of those techniques
would exceed the limits of this thesis. The reader may have a look at Ref. [67] which is an
excellent bibliography of the entire field of ultrafast spectroscopy including detailed papers
70
describing the detection techniques.
4.2
Characterization of Ultrashort Pulses
In this section we describe the characterization of the ultrashort light pulses generated by,
e.g., Kerr lens mode locked cavities (see Section 3.1.5). That is, we describe methods to
determine the exact temporal duration and even the phase of ultrashort electro-magnetic
pulses. The short time duration of these pulses does not allow an electronic measurement
because even the fastest electronics have response times on the order of picoseconds. The
only way to measure an ultrashort optical pulse is to use the pulse to measure itself in
a pump-probe scheme as described in Section 4.1.1. Measuring one quantity with itself
is commonly referred to as autocorrelation. Below, we describe two specific methods of
autocorrelating optical pulses [68]. These autocorrelation measurements produce values for
the temporal intensity profile of the optical pulse, but do not contain any phase information.
For a measurement of the full electric field in a pulse, i.e. both the amplitude and the phase,
one has to resort to an extension to the simple autocorrelation techniques referred to as
frequency resolved optical gating (FROG). We conclude this section with the description of
different varieties of FROG.
4.2.1
Autocorrelation Measurements
SHG Autocorrelation
How is it possible to measure an optical pulse by using the pulse itself? The basic idea
is to use a pump-probe setup (see Section 4.1.1). The fundamental requirement for any
autocorrelation measurement (that provides information about the pulse duration) is to
have a nonlinearity that couples pump and probe pulses. The simplest form of such an
autocorrelation uses the χ(2) nonlinearity for SHG (see Section 2.3.1). The setup used for
such an SHG-based autocorrelation measurement is shown in Fig. 4.3. SHG is generated
probe
BBO
71
k2 , ω
k1 + k2 , 2ω
pump
detector
k1 , ω
L
2ω filter
Figure 4.3: Detection geometry for autocorrelation measurement using SHG. L = focusing
lens, BBO = nonlinear crystal. ki , ω denote the wavevectors and frequencies of the incident
light pulses.
by the nonlinear polarization described in Section 2.3.1. The filter in front of the detector
only transmits the light generated at the SHG frequency as indicated in the figure. The
electric field arriving at the detector is therefore:
E(2ω, t) ∝ χ(2) E(ω, t)E(ω, t + τ ),
(4.1)
where τ denotes the time delay between the pump and the probe beam. The photodetector
measures time-integrated intensity rather than electric field. The signal measured by the
detector is given by the time-integrated absolute square of the field in Eq. 4.1:
∞
I(ω, t)I(ω, τ + τ ).
S(2ω, τ ) ∝
(4.2)
−∞
Assuming a Gaussian pulse shape for the input pulse we can relate the width of
the signal trace S(2ω, τ ) to the width of the original pulse:
I(t) = Io e−(ln(2) Γ )2
(4.3)
SSHG (t) = So e−(ln(2) 2Γ ) ,
(4.4)
t
leads to an SHG trace of the form
ln2 2
where Γ denotes the FWHM of the original Gaussian pulse. The FWHM of the SHG trace
√
is therefore larger than the FWHM of the original pulse by a factor of 2.
As described in Section 2.3.1, the phasematching condition has to be satisfied for
SHG to be efficient. It is very common to use uniaxial crystals for this purpose. In our
autocorrelator we use BBO (Beta-Barium Borate — BaB2 O4 ) which has excellent conversion
efficiency values of up to 90% at optimal alignment.
72
TPA Autocorrelation
The SHG autocorrelation technique described in the previous section relies on the nonresonant second-order wave-mixing process. It is also possible to use a resonant χ(3) process to
perform autocorrelation measurements. In fact, the second most common way of autocorrelating ultrashort optical pulses after SHG autocorrelation is using TPA (see Section 2.3.2)
[69, 70]. TPA autocorrelations can be performed in a standard differential transmission
scheme as described in Section 4.1.2. The full electric field arriving at the detector is given
by:
TPA
(t) = E(t) + χ(3) E(t) |E(t − τ )|2
Etot
(4.5)
where τ denotes the time delay between pump and probe pulse. The detector will actually
measure intensity which is given as the absolute square of Eq. 4.5:
2
∗
TPA
(t) = I(t) + χ(3) I(t)I 2 (t − τ ) + χ(3) + χ(3) I(t)I(t − τ ).
Itot
(4.6)
As we discussed in Section 2.3.2, in the case of TPA we only consider the imaginary part
of χ(3) . The real part of χ(3) causes the Kerr effect (see Section 2.3.1) which does not
contribute to the TPA signal. Therefore, the last term in Eq. 4.6 vanishes. Furthermore,
the first term in Eq. 4.6 can be neglected as well, because the unperturbed intensity of
the probe pulse is subtracted in a differential transmission experiment. The remainder is a
slightly different relation than in the SHG case. Here, the signal generated by a Gaussian
pulse of the form given in Eq. 4.3 is of the form:
ST P A (τ ) = So e−(ln(2)
So, the FWHM of the original pulse is given by
2ln2 2
)
3Γ
.
(4.7)
2/3 times the FWHM of the TPA-trace.
This method of TPA-autocorrelation is convenient because there are no phase-matching
requirements. A disadvantage is that it is not background free, because the signal is propagating in the same direction as the probe beam as opposed to SHG, where the signal
propagates in a background free direction. Either technique has its place in certain appli-
73
cations where one or the other advantage/disadvantage is dominant. We will come back to
the debate of SHG vs. TPA in Section 4.2.2.
The fact that it is necessary to make initial assumptions about the original pulse
shape to calculate its FWHM indicates the shortcomings of simple autocorrelation techniques. They do contain information on the duration of the measured pulses but they
cannot determine the actual shape of the pulse. Strong assumptions on the pulse shapes
have to be made to find a value for the pulse duration. Fortunately, it is well known that
for standard Ti:sapphire KLM-oscillators, the output pulse shape is Gaussian. We therefore
use SHG- or TPA-autocorrelation techniques to determine the duration of the pulses used
in our experiment. In certain experiments it is necessary to know the actual shape of the
laser pulse, or even better, to know the amplitude and phase evolution in time of the laser
pulse. Trebino et al. demonstrated a clever technique to measure the full electric field of
an ultrashort laser pulse [60]. We describe this so-called FROG technique in the following
section.
4.2.2
Frequency Resolved Optical Gating
Frequency-resolved optical gating (FROG) is currently the most commonly used technique
to fully characterize an ultrashort optical pulse. Various species of FROG have been demonstrated which utilize different optical nonlinearities such as SHG (χ(2)), polarization gate,
transient grating, third harmonic generation and self diffraction (all χ(3)) [60]. The basic
idea of FROG is to spectrally resolve an autocorrelation signal. Let us consider SHG FROG
— the simplest species of FROG. Spectrally resolving the SHG signal gives a spectrogram
of the form [60]:
SHG
(2ω, τ )
SFROG
= ∞
−∞
−i2ωt
E(t)E(t − τ )e
2
dt
(4.8)
Thus, FROG measures the SHG signal at each frequency component in an ultrashort laser
pulse. Given this additional information, it is possible to numerically retrieve the amplitude
and phase of the original electric field E(t). An excellent review of FROG techniques is
74
given in Ref. [60]. We have built an SHG FROG system in our lab to characterize the pulses
from the KML oscillator as well as from the multipass amplifier.
SHG FROG and the other FROG species listed above are complimentary to each
other and work well for standard amplified and unamplified ultrashort pulses. All of these
conventional FROG techniques, however, have considerable difficulties in characterizing
white-light continuum pulses (see Section 2.3.3) due to limited phasematching bandwidth,
sensitivity, and/or upconversion to ultraviolet wavelengths where detection becomes an
issue. These white-light pulses, generated by focusing powerful ultrashort pulses into a
nonlinear medium, are commonly used in ultrafast spectroscopy as convenient sources of
a broadband probe [71] and we use broadband continua for all the experiments described
in this thesis. Only recently a new technique that uses spectral interferometry [72] with
a tunable reference pulse has been reported to measure ultrabroadband continuum pulses
[73].
We devised a new FROG technique based on two-photon absorption (TPA) which
is ideally suited for characterizing white-light continuum pulses. Two-photon absorption
in various materials is routinely used for autocorrelation measurements of ultrashort laser
pulses [69, 70]. Due to the resonant enhancement at the material’s band gap and the lack
of phasematching requirements for TPA, this χ(3) nonlinearity is an excellent candidate for
ultrashort pulse characterization. To produce a FROG trace based on TPA, two pulses
are spatially overlapped in the TPA crystal and the spectrum of one pulse is measured as
a function of the time delay between the two pulses. Spectrally resolving the TPA-signal
leads to a FROG trace of the form:
TPA
(ω, τ )
SFROG
= ∞
−∞
2
E(t) + E(t) |E(t − τ )|
−iωt
e
2
dt .
(4.9)
This is a slightly more complicated form than Eq. 4.8. A numerical retrieval should still be
possible, however.2
As a test of the new technique we have measured the SHG and TPA FROG traces
2
Efforts to retrieve the full wave form of a white light continuum pulse were ongoing at the writing of
this thesis.
75
370
wavelength (nm)
395
420
445
470
330
165
0
165
330
time delay (fs)
Figure 4.4: SHG FROG trace of ultrashort laser pulse from an amplified Ti:sapphire system.
The contours are equi-intensity lines which are equally spaced from the lowest to the highest
intensity region.
of pulses generated by a standard multipass Ti:sapphire amplifier. We use a BBO crystal
(see Section 4.2.1) for the SHG measurement and a GaP crystal for the TPA measurement.
GaP has a bandgap of 1.9 eV which makes it ideally suited for TPA of the 1.55 eV pulses
from the Ti:sapphire amplifier. The results are shown in Figs. 4.4 and 4.5. The SHG FROG
trace shows elliptical symmetry indicating at most linear chirp [60] whereas the TPA FROG
trace in Fig. 4.5 shows some signs of cross phase modulation due to the strong pump beam
and long interaction region in the GaP sample. An appropriate choice of sample thickness
and pump to probe intensity ratio should eliminate the cross phase modulation and make
the two traces agree.3
3
We are currently performing experiments on samples ranging from 50 µm to 0.5 mm in thickness.
76
900
wavelength (nm)
850
800
750
700
330
165
0
165
330
time delay (fs)
Figure 4.5: TPA FROG trace of ultrashort laser pulse from an amplified Ti:sapphire system.
The unmodulated laser spectrum is subtracted out for clarity of the graph. The nonlinear
medium used for the TPA FROG measurement is GaP (2.1 eV bandgap).
TPA FROG is especially well suited for characterizing broadband continuum pulses
because there are no phasematching bandwidth limitations. The TPA technique is limited
only by the bandgap of the nonlinear material. TPA starts at photon energies of half the
bandgap and stops at photon energies resonant with the bandgap because linear absorption
becomes dominant. An appropriately chosen set of materials allows pulses whose bandwidth
spans the entire visible wavelength range and beyond to be characterized. Figure 4.6 shows
preliminary results on the temporal chirp of the white light continuum obtained by a TPA
FROG measurement using GaP up to 2.1 eV and then ZnO (3.3 eV bandgap) up to 2.7 eV.
77
2.8
energy (eV)
2.6
2.4
2.2
2.0
1.8
−300
−150
0
150
300
time delay (fs)
Figure 4.6: Temporal chirp of white-light continuum pulse generated in CaF2 . The chirp
was measured using TPA in GaP and ZnO. The line represents a second order polynomial
fit.
4.3
Femtosecond Time-Resolved Ellipsometry
All the ultrafast spectroscopic techniques mentioned so far in this chapter have one common
characteristic. They all focus on measuring one particular optical quantity which is geared
towards the detection of a very particular process. At high excited carrier densities, however,
many different processes happen simultaneously and it becomes necessary to pick an optical
signature which can provide information on multiple processes simultanously (see Chapter
1). It turns out that the spectral dielectric function is an excellent candidate for providing
information on detailed carrier and lattice dynamics in solids upon excitation with an intense
ultrashort laser pulse.
We have developed a technique which allows the direct measurement of the real
and imaginary parts of the dielectric function of a material with fs time-resolution over a
broad energy range (1.8 eV to 3.4 eV). A detailed description of this technique will be the
78
topic of this section. Essentially, the technique combines multi-angle ellipsometry [74] with
a femtosecond pump-probe setup (see Section 4.1.1). Performing two standard white-light
time-resolved reflectivity experiments at two different angles of incidence under the exact
same excitation conditions produces two reflectivity values for each wavelength. If the two
angles are chosen appropriately one can uniquely invert the Fresnel reflectivity formulae to
obtain real and imaginary part of the spectral dielectric function ε(ω).
4.3.1
Multi-Angle Ellipsometry for Isotropic, Bulk Materials
The most widespread method of measuring ε(ω) is ellipsometry [74], where the reflectivity of
a sample is measured for many different polarization orientations and/or multiple angles of
incidence. The real and imaginary part of the dielectric function are extracted by inverting
the Fresnel reflectivity formulae using these reflectivity values. Let us look at this nontrivial step more closely. The reflectivity of the interface between vacuum and a material is
governed by the dielectric function of that material through the Fresnel reflectivity formulae
as described in Section 2.2.1. Thus, knowing both the real and imaginary part of ε(ω) allows
the exact determination of the reflectivity of a material at all angles of incidence and for all
polarization states of the incoming light. One can similarly use these equations to extract the
dielectric function from a reflectivity measurement. Numerically inverting those equations
(there is no analytical solution), provides a means to obtain exact values for the real and
imaginary part of ε(ω) from simple reflectivity data. Since there are two unknowns to be
solved for at each wavelength (Re[ε(ω)] and Im[ε(ω)]), either two knowns or an additional
relation between Re[ε(ω)] and Im[ε(ω)] are needed to invert the Fresnel formulae.
There is indeed a relation between the real and imaginary part of the dielectric
function, which follows from causality — the famous Kramers-Kronig relation. Given the
Kramers-Kronig relation, it is sufficient to know either Re[ε(ω)] or Im[ε(ω)] to obtain the
other. For an accurate calculation of the unknown quantity, it is necessary to fully characterize the known quantity over the entire bandwidth, from DC to infinite frequency.4
4
The Kramers-Kronig relation involves an integral from zero to infinity.
79
Any experimental restriction of the measured frequency range results in a decrease in accuracy of determining the full dielectric function. Previous experiments have utilized the
Kramers-Kronig relation and obtained excellent results by measuring the real part of the
refractive index5 over a very wide frequency range [23]. In fs time-resolved experiments
such a wide range of frequencies is impossible to obtain. One has to therefore resort to
measuring multiple optical quantities.
In standard ellipsometry there is a large number of different reflectivity values
since both the incident angle and the polarization of the incoming light can be altered to
obtain different reflectivity values. Typically the angle is kept fixed and the polarization
is rotated continuously from 0o to 360o, giving a potentially infinite number of reflectivity
values. This large number of values allows an accurate determination of both real and
imaginary part of ε(ω) or, if ε(ω) is known, of the exact structure of the examined surface
(measuring exact thicknesses of dielectric surface layers is in fact the most common use of
ellipsometry). For the purpose of a time-resolved measurement of ε(ω) this is not a practical
approach. Due to the inherent complexity of the experimental apparatus, it is desireable
to minimize the number of measurements to be taken. As mentioned above, the minimum
number of different reflectivity values needed to invert the Fresnel formulae is two. It turns
out that there is an optimum choice of parameters (angles and polarizations) for those two
reflectivities to achieve maximal accuracy in doing this inversion. To illustrate this fact
Fig. 4.7 (a) shows a grid of dielectric function values and the corresponding reflectivity
values for specific sets of parameters are shown in Fig. 4.7 (b) – (d). The reflectivity graphs
in Fig. 4.7 (b) – (d) are obtained by calculating the reflectivity values for two specific
parameter sets for each point in the ε(ω) grid. Thereby one generates a corresponding point
in the chosen reflectivity space (where the two axes represent the two chosen parameter sets).
Figure 4.7 (b) shows the reflectivity distribution for 60o angle of incidence and p-polarization
along the y-axis versus 45o angle of incidence and p-polarization along the x-axis. As
the graph shows, the two-dimensional grid in dielectric function space collapses onto a
5
The refractive index in absorptive materials is complex and the real and imaginary parts n and k are
are intimately related to the dielectric function via ε1 = n2 − k2 and ε2 = 2nk [17].
80
(a)
(b)
100
50
R(60 , p-pol)
30
20
10-1
o
Im[ε]
40
10
10-2
0
−25
0
25
10-3 -1
10
50
Re[ε]
R(45 , p-pol)
(d)
(c)
0
10
R(83o, p-pol)
R(75o, p-pol)
10
10-1
10-2
10-3 -1
10
100
o
R(45 , p-pol)
100
o
0
10-1
10-2
10-3 -1
10
100
o
R(45 , p-pol)
Figure 4.7: Mappings from dielectric function space to reflectivity space via the Fresnel
formulae. (a) shows an arbitrarily chosen grid of dielectric function values. (b) – (d) show
the respective points in reflectivity space where the axes correspond to the chosen angle
and polarization. The dielectric functions of a selection of materials (ranging from 1.5 - 3.5
eV are shown: ✷ c-GaAs, ◦ GeSb, ✸ Si, Sb. The values for GaAs are omitted in (c) and
(d) for clarity.
virtually 1-dimensional curve in the chosen reflectivity space. The error in mapping certain
points from this reflectivity space back into dielectric function space is very large, because
two very closely spaced reflectivity points can lead to dielectric function values which are
quite different. Thus, for a reliable inversion for this choice of angles/polarizations it is
81
necessary to have quasi infinite resolution in reflectivity space. Another way of looking at
this problem is to realize that the two reflectivity measurements are not linearly independent
and therefore cannot be used to uniquely invert the Fresnel formulae.
A clever choice of reflectivity parameters can provide a remedy to this problem. To
better understand the consequences of picking the right polarization and angle combination
it is instructive to look at the dependence of changes in reflectivity on changes in the
dielectric function [75]. If we choose two reflectivity parameter sets denoted by the subscript
i = 1, 2, the changes in reflectivity as the real and imaginary part of the dielectric function
change can be expressed as:
∆R1 =
∆R2 =
∂R1
∆Re[ε] +
∂Re[ε]
∂R2
∆Re[ε] +
∂Re[ε]
∂R1
∆Im[ε], and
∂Im[ε]
∂R2
∆Im[ε].
∂Im[ε]
(4.10)
(4.11)
These two equations can only be linearly independent — and therefore allow a unique
assignment of dielectric function values to a point in the chosen reflectivity space — if
∂R1
∂Re[ε]
∂R1
∂Im[ε]
=
∂R2
∂Re[ε]
.
∂R2
∂Im[ε]
(4.12)
The differentials of R1 and R2 can be calculated using Eq. 2.18 and 2.19. Figure 4.8
shows a plot of the ratio
∂Rp
∂Rp
/
∂Re[ε] ∂Im[ε]
for an arbitrary choice of dielectric function values
— Re[ε] = 12 and Im[ε] = 1 — as a function of angle of incidence. The solid curve shows
the results of the calculation for p-polarization, and the dashed curve shows the case of
s-polarization. The zero-crossing of the solid curve corresponds to the Brewster angle for
the chosen dielectric function values. The general shape of these curves is the same for
any choice of dielectric function values; only the value for the Brewster angle changes. The
graph clearly shows that, in the case of s-polarization, any pair of angles will lead to values
for the ratio in Eq. 4.10 which are very close to each other, thus making the reflectivity
measurements at those angles linearly dependent. It is therefore impossible to extract ε(ω)
from a set of measurements involving only s-polarized light. In the case of p-polarization,
however, choosing one angle sufficiently far below the Brewster angle and the other angle
82
∂R/∂Re[ε] / ∂R/∂Im[ε]
20
15
10
5
0
−5
0
30
60
90
angle of incidence
Figure 4.8: Dependence of the ratio of the differential change in reflectivity with Re[ε] and
Im[ε] on angle of incidence. The solid curve shows the dependence for p-polarized light; the
dashed curve shows the dependence for s-polarized light.
close to or sufficiently far above the Brewster angle leads to fairly different values for the
ratio mentioned above. Unfortunately this analysis is valid only for small changes in ε
around a certain point in dielectric function space. In fact the Brewster angle is itself
dependent on the exact values of the dielectric function [17]. And since every material
displays a certain dispersion ε(ω) it therefore has a whole range of Brewster angles in the
wavelength range used. It turns out, however, that this analysis provides a very good rule of
thumb for choosing a “good” set of angles and polarizations: choose one angle well below all
Brewster angles and the second angle higher than the maximal Brewster in the experiment.
To put the obtained conditions for those parameters to the test we plot the reflectivity values for p-polarized light for 45o angle of incidence vs. the values for 75o angle
of incidence in Fig. 4.7 (c). Now the grid of dielectric function points mostly maps onto a
fairly widespread 2-dimensional area. Even at this choice of angles there is still a sizeable
number of points in the dielectric function grid which end up on a 1-dimensional curve. The
83
reason for this remaining collapse of the 2-dimensional grid lies in the fact that for some
values in the dielectric function grid the Brewster angle is larger than 75o which results in
all of those points collapsing onto the 1-dimensional area in Fig. 4.7 (c). Fortunately it is
very unlikely that all dielectric function values in the space spanned by the grid in Fig. 4.7
(a) are involved in any given experiment. A given material’s dispersion is represented by a
curve through the 2-dimensional grid. The curves for c-GaAs, Si, Sb, and GeSb are shown
in Fig. 4.7 (a). Each point along these curves corresponds to the pair (Re[ε], Im[ε]) at a
particular wavelength within the experimental range (here, we illustrate ε(ω) in the range
from 1.5 eV to 3.5 eV). The corresponding points in reflectivity space at two different angles
is shown in Fig. 4.7 (c) and (d). The values for c-GaAs are omitted for clarity of the graph.
Figure 4.7 (c) shows that when choosing 75o vs. 45o the reflectivity points for Sb and GeSb
all lie in regions where the points are spaced out far enough to allow an accurate inversion
back to dielectric function space. In the case of Si, however, some of the points lie on a
virtually 1-dimensional curve. As discussed, it turns out that for Si the Brewster angles
for most of the relevant wavelength range lie above the chosen second angle of 75o , making
a reliable inversion of the Fresnel formulae impossible at those parameters. Figure 4.7 (d)
shows the same materials for the angles 83o vs. 45o . Now, even all the values of Si lie in
regions where the dielectric function points map onto a 2-dimensional area, because 83o is
above the Brewster angle in Si for all involved photon energies. Figure 4.7 clearly shows that
choosing the appropriate set of polarizations and angles is crucial for an accurate inversion
of the Fresnel formulae over the relevant wavelength range.
With the appropriate angle choice at hand we can now proceed to invert the
Fresnel formulae. The Fresnel formulae cannot be inverted analytically. We therefore use
a numerical inversion algorithm that is based on the simplex downhill method [76]. The
algorithm minimizes the difference between the measured reflectivity at each wavelength
and time delay and the reflectivity predicted by certain values for the real and imaginary
part of ε(ω) as Re[ε(ω)] and Im[ε(ω)] are varied according to the simplex downhill scheme.
The algorithm converges fast enough to complete an inversion in about 0.5 minutes on an
84
Apple Macintosh G-3 desktop computer. For typical values of about 40 wavelengths and
100 time delays the full inversion can therefore be completed within reasonable times, on
the order of a few hours.
4.3.2
Accounting for Oxide Layers and Extension to Uniaxial Materials
and Thin Films
In the previous section we demonstrated the general approach to extract the full dielectric
function from a multi-angle reflectivity measurement of a bulk, isotropic material. This
technique works very well for materials such as glasses, which are truly isotropic and have
no native oxide layers. As one moves to materials which have native oxide layers, as most
semiconductors and metals do, or are even uniaxial, a more sophisticated approach is necessary. Basically, one has to extend the reflectivity formulae used for the simple case and
follow the same procedure as described above. The only difference being that now the
inversion algorithm uses the new extended reflectivity formulae.
Accounting for Oxide Layers and Measuring Thin Films
Only a select group of materials which are very inert (e.g. noble metals or glasses) can
exist in air and not develop oxide layers. All the samples that were used in this thesis do
have native oxide layers and thus require a more sophisticated treatment to determine the
reflectivity of the exposed surface. We described these extended Fresnel formulae in Section
2.2.2. The Fresnel formulae for a three-layer system (in this case air-oxide-sample) are given
in Eq. 2.20. We also described the Fresnel formulae for materials consisting of more than
three layers. In this case the Fresnel formulae are given by Eq. 2.25. Assuming that all
thicknesses are known and the dielectric functions of all but one layer are known, there
are only two unknown quantities in determining the reflectivity of the multi-layer stack —
Re[ε(ω)] and Im[ε(ω)] of the one layer which we are interested in measuring. Hence, the
technique described in Section 4.3.1 will successfully extract the full dielectric function of
that layer. For example, we can extract the dielectric function of a thin film on a substrate
85
even if it has an oxide layer.
Therefore, the reflectivity formalism developed in Section 2.2.2 in conjunction with
the technique described in Section 4.3.1, allows us to extract the full dielectric function of
materials with oxide layers or even materials in such complex arrangements as thin films
on substrates with an oxide layer on the surface.
Uniaxial Materials
In the previous section it was assumed that all materials in the multi-layer stack are
isotropic. It is in fact possible to extract the dielectric function even of non-isotropic materials. We will discuss the easiest case of a uniaxial material in this section. A uniaxial
material has two independent entries in the susceptibility tensor. Quite intuitively, the
material has different refractive indices for light polarized along the c-axis (usually referred
to as the extra-ordinary index) and perpendicular to the c-axis (usually referred to as the
ordinary index). Thus there are four values to be extracted at each wavelength — the real
and imaginary part of each of the two refractive indices.
Let us discuss the special case of an interface between a uniaxial material and an
isotropic medium where the interface contains the c-axis. This case becomes important in
our experiments on Te in Chapter 8. We described the Fresnel reflectivity formulae for this
case in Section 2.2.3.
Given the technique described in Section 4.3.1, there is a straightforward way to
extract all four parts of the dielectric tensor in this configuration: perform reflectivity measurements for four different angles.6 However, there are significant drawbacks in choosing
this approach. First, the resolution for inverting the Fresnel formulae is significantly worse
than for the case of two angles since the four angles are inevitably closer together (see discussion in Section 4.3.1, especially Fig. 4.8). Second, it is necessary to run a four-dimensional
minimization algorithm. This leads to unreasonable inversion times of multiple days on
currently available desktop computers.
6
Choosing different polarizations is not an option, since one has to use p-polarized light to maintain a
reasonable resolution in inverting the Fresnel formula (as was shown in Section 4.3.1).
86
A better way to extract the ordinary and extraordinary dielectric function is to
perform two separate two-angle experiments at different orientations of the sample. The
first configuration is shown in Fig. 2.7(b). The surface contains the c-axis of the crystal,
and the plane of incidence is perpendicular to the c-axis. In the case of p-polarization, the
incident electric field only “sees” the ordinary part of the dielectric function. Therefore the
exact same technique as for the isotropic case (with an oxide layer if needed) can be applied
to extract the ordinary dielectric function values.
The second configuration is the one shown in Fig. 2.7 (a). In this case, the reflectivity is given by Eq. 2.57 if it is a pure bulk material. If there is an oxide layer on
the surface the reflectivity is given by Eq. 2.20, where now the Fresnel factors are given by
Eq. 2.18 for the air-oxide interface and Eq. 2.57 for the oxide-crystal interface. Now, we can
apply the two-angle technique because we know the values for the ordinary dielectric function from the measurement in the first configuration which reduces the number of variables
from four to two — the real and imaginary parts of the extraordinary dielectric function.
Hence, we can extract the full dielectric tensor of a uniaxial crystal with an oxide layer with
femtosecond time resolution.
We will present experimental details and examples of measured dielectric functions
for various materials in the next section.
4.3.3
The Experimental Setup
To measure the spectral dielectric function of a material with fs time resolution, we combine
the multi-angle reflectivity technique described in Section 2.2 with a standard white-light
pump-probe setup, i.e., we perform two time-resolved reflectivity measurements at carefully
chosen angles (see Section 4.3.1) under the exact same excitation conditions. Figure 4.9
shows a schematic representation of the experimental setup. The light source can be any
amplified femtosecond laser system. In our setup we use a KML oscillator (described in
Section 3.1.5) to seed a home-built 1 kHz repetition rate Ti:sapphire multipass amplifier
(described in Section 3.2.2). Amplification of the oscillator pulses is necessary to generate
M
87
CaF2
probe
λ/2
P
PM
L
CF
BS
M
M
pump
λ/2
P
M
sapphire
femtosecond
laser source
reference
L
sample
spectrometer
PM
M
L
L
M
filters
polariser
M
M
Figure 4.9: Schematic representation of FTRE setup. BS = polarizing beam splitter; M =
flat mirror; PM = parabolic mirror; L = lens; P = polarizer; λ/2 = half-wave plate; CF =
colored glass filter.
the broadband probe pulse and to achieve high enough pump fluence levels to induce the
phase transitions that are the subject of this thesis.
As Fig. 4.9 shows, the incoming fs pulse is split into a stronger pump and a
weaker probe pulse at beamsplitter BS. The pump pulse is directed to the sample via a
variable delay stage allowing for adjustable time delays between the pump and the probe
pulse. The pump pulse is then focussed onto the sample using a slowly focussing lens (20
cm focal length). The pump spot size can be adjusted according to specific experimental
requirements by varying the distance from the lens to the sample. It is important to keep it
at least 4 times larger than the probe spot to ensure probing of a (laterally) homogenously
excited region. The probe pulse still probes an inhomogenous excitation profile in the
direction perpedicular to the sample surface. This effect is minimized by two measures: (1)
we disregard data at photon energies lower than 1.7 eV. At 1.7 eV the absorption depth of
e.g. GaAs is reduced by almost a factor of two with respect to the pump photon energy
of 1.5 eV (450nm vs. 750nm); (2) immediately after the pump excitation, the reflectivity
88
of the sample is raised significantly due to the excited free carriers, which in turn lowers
the absorption depth abruptly. For high excitation densities, where the material becomes
metallic the absorption depth is on the order of tens of nanometers. We neglect the effect
of excitation inhomogeneity throughout this thesis which is a good approximation for the
bulk of the data presented here.
The handling of the probe beam is much more challenging. We discussed the
principles of white-light generation in Section 2.3.3, including the specifics of the whitelight setup in our experiment. As indicated in Fig. 4.9, we position the colored glass filter
to flatten the spectrum (see Section 2.3.3) between the CaF2 crystal and the sample to
prevent strong interaction (such as damage) of the sample with the probe pulse. In order to
prevent multiple beam filamentation CaF2 crystal, We use a half-wave plate and polarizer
to carefully tune the incident energy of the seed pulse to just barely exceed the threshold for
white-light generation.7 The single filament guarantees that the emitted white-light cone
has a perfectly Gaussian wavefront [77] allowing clean recollimation and focussing.
To prevent excessive dispersive stretching of the probe pulse due to its extremely
wide bandwidth we use reflective optics whereever possible between the white-light generation and the sample. We image the single filament of white-light on the exit face of the CaF2
crystal onto the sample using a one-to-one telescope consisting of two off-axis paraboloid
mirrors (10 cm focal length) as shown in Fig. 4.9. The 1/e2 focal spot diameter d is very
close to the size predicted by Gaussian optical theory because the wavefront evanescing
from the single filament is perfectly Gaussian: d = f λ/D [32], where f is the focal length
of the used lens, λ the wavelength, and D the beam waist of the incident beam. For our
setup, Gaussian optical theory predicts a focal spot of ∼ 6µm. We estimate the actual
spot size to be slightly larger (∼ 10µm) than the predicted value due to the broad bandwidth and experimental imperfections. We choose the pump spot size at least four times
larger to ensure the probing of a homogeneously excited region. Since alignment of off-axis
paraboloids is fairly tedious we recollimate the white-light with achromatic lenses because
7
cm.
About 1 µJ for our focussing conditions: 15 cm focal length lens and incident beam waist of about 1
89
pulse stretching after the sample does not inflence the time-resolution. After recollimation,
the probe beam gets focussed onto the entrance slit of an imaging spectrograph. We use a
home-built, prism-based spectrometer plus commercially available CCD (Jobin Yvon, CCD
3500) to capture the spectra for each time delay and excitation fluence.
Since we want to obtain the dielectric function by inverting the Fresnel formulae,
we need to measure absolute reflectivities — as opposed to other fs reflectivity experiments
where only differential reflectivity measurements are sufficient [65, 78]. To perform an absolute reflectivity measurement, we use a thin sapphire plate to split off a fraction of the
white-light probe as a reference (see Fig. 4.9). We steer the reference beam around the sample to the spectrograph and focus it onto the entrance slit slightly vertically displaced from
the reflected beam. Thus we simultaneously measure the “reflected” and the “reference”
pulses for each time delay and excitation fluence. The ratio between the two spectra is the
absolute reflectivity modulo some calibration. We will discuss the calibration of the setup in
Section 4.3.4. The simultaneous measurement of the reflected and reference spectra for each
laser pulse also protects the experiment from noise due to fluctuations in the white-light.
The white-light noise is quite significant because we operate close to white-light generation
threshold and the pulse energy out of the multipass amplifier fluctuates by values on the
order of 10 % and higher. But by computing the ratio between reflected and reference beam
we divide these fluctuations out.
In order to study highly excited materials at excitations close to and above the
threshold for permanent damage, the sample must be mounted on a translation stage to
allow translation to a fresh spot after each irradiation. Furthermore, since the laser system
generates pulses at a repetition rate of 1 kHz, we use a fast shutter (shutter model and
make) that is synchronized to the laser pulse train to enable the irradiation of the sample
with a single pulse. Fluctuations in the pulse energy not only induce noise in the probe
beam, but in the pump excitation as well. To account for fluctuations in the pump energy,
we measure the pump pulse energy using a thin glass plate to split off a small portion of
the beam The output of the photodiode used to measure this fraction of the pump pulse
90
is read by a track-and-hold, which is synchronized to the laser pulse train, thus allowing
us to track the fluence for each laser shot. We control time-delay and sample-translation
stages, shutter, track-and-hold and data acquisition using a central computer equipped
with data acquistion and controller cards (National Instruments PCI-1200 and 16-E4) and
LabViewTM software (National Instruments). Experiments above the damage threshold
of the examined materials necessarily have to be single shot measurements, which have
comparatively low signal-to-noise ratios. Typically, our setup is able to detect reflectivity
changes on the order of ∆R/R = 5−10%. However, the signal levels at these high excitation
fluence levels are typically quite high (e.g., ∆R/R ≈ 200% in GaAs).
On the other hand, if an experiment is carried out at fluence levels below the
threshold for permanent damage it is not necessary to move the sample between successive
shots. In this case, it is possible acquire data at the repetition rate of the laser source. It is
ideal to read off the CCD for each shot and average over multiple readings – thus making
full use of its dynamic range. Unfortunately there is no camera commercially available
which is able to read and clear its registers at a repetition rate of 1 kHz. The CCD system
in our setup is able to acquire images at about 2 Hz (much faster systems are available
but were not at hand at the time of building the setup). But rather than lowering the
data acquisition rate of the whole experiment to 2 Hz, it is much more advisable to allow
multiple shots to hit the CCD detector (in our case, ∼500 shots per CCD reading). In this
way the experiment still averages at the full repetition rate of 1 kHz. However, the dynamic
range is reduced by a factor of ∼500 because it is necessary to attenuate the incoming light
to prevent saturation of the CCD detector. Most modern CCD cameras, including the one
used in our setup, offer at least 16-bit resolution (i.e. 65000 counts per pixel) keeping the
remaining dynamic range above two orders of magnitude per shot, which is still more than
sufficient for all experiments described in this thesis. A separate benefit of below threshold
experiments is that sample translation is unnecessary. The elaborate electronic timing of
shutter, track-and-hold and data acquisition from the CCD camera are superfluous as well.
The reflectivity resolution of our setup in this fast mode is better than ∆R/R = 10−3 .
4.3.4
91
Calibration and Error Estimate
As mentioned above a careful calibration of the setup is necessary to measure absolute
reflectivities, which is vital to successfully extracting the dielectric function. Basically, a
correct calibration must account for all absorption losses and other optical imperfections in
the optics used in the setup. To do so, we measure the reflectivities of a number of standard
materials for which the dielectric functions are known. For each material, we compute the
ratio between the reflectivities predicted by the Fresnel formulae using literature values
of the dielectric functions [21] and the experimentally obtained reflectivities. This ratio is
wavelength-dependent and represents a wavelength-dependent correction factor (CF) to the
experimental data:
CF(ω) =
Reflectivitylit (ω)
Reflectivitycal(ω)
(4.13)
Multiplying the experimental data by this factor across the entire wavelength range gives
the correct reflectivity. This CF(ω) is entirely dependent on the experimental geometry, so
CFs of different materials should be identical given that there is no other artefact in the
system.
In calculating the reflectivities from the given dielectric function values for a material, the angle of incidence and the polarization are both input parameters. In addition,
for some materials the exact oxide layer thickness or ε(ω) of the oxide layer is unknown.
As discussed in Section 2.2 it is advisable to choose p-polarization for both angles. Once
the polarization is fixed, the angle and certain oxide values remain as free parameters when
comparing CFs of different materials. In the case of semiconductors, the oxides are insulators and their refractive indices are well approximated by a real constant across the
bandwidth of the white-light probe.
As long as there are more materials than free parameters, the problem is overdetermined and there is only one set of parameters where all correction factors match. The
accuracy with which the free parameters can be determined depends on how strongly the
CFs depend on those parameters or how different the dielectric functions of the chosen
92
0.6
correction factor
0.5
sapphire, 46.0o
a-GaAs, 46.0o, 4nm
Te, 46.0o, 15nm
0.4
0.3
0.2
0.1
1.5
2.0
2.5
3.0
3.5
energy (eV)
Figure 4.10: Correction factors taken in multi-shot mode for various materials: ✷ sapphire,
✸ a-GaAs, ◦ Te. The angle of incidence is 46.0o and the oxide layers on a-GaAs and Te are
4 nm and 15 nm respectively.
materials are. For a “good” choice of materials, matching the CFs determines the angle of
incidence to within a tenth of a degree and the oxide layer thicknesses to better than 1 nm.
Figure 4.10 shows the correction factors measured in our multi-shot setup for
sapphire, a-GaAs, and Te (only the ordinary part of ε(ω) is considered here). The free
parameters in this case are the angle of incidence and the thickness of the Te-oxide layer.
We determined the dielectric constant for TeO2 in other measurements to be ε = 5 which
is in agreement with other semiconductor-oxides8 . Figure 4.10 shows the correction factors
for a poor choice of parameters. Only slight changes in the two free parameters, namely
adjusting the angle of incidence from 46.0o to 49.7o and the Te-oxide layer thickness from
15 nm to 5.5 nm, leads to a very good match as shown in Fig. 4.11.
The typical relative standard deviation (i.e., standard deviation divided by the
mean) among the CF values shown in Fig. 4.11 is about 2%. This means that the accuracy
8
There is no literature data on the dielectric function of Tellurium oxides to our knowledge.
93
0.6
correction factor
0.5
sapphire, 49.7o
a-GaAs, 49.7o, 4nm
Te, 49.7o, 5.5nm
0.4
0.3
0.2
0.1
1.5
2.0
2.5
3.0
3.5
energy (eV)
Figure 4.11: Correction factors taken in multi-shot mode for various materials: ✷ sapphire,
✸ a-GaAs, ◦ Te. The angle of incidence is 49.7o and the oxide layers on a-GaAs and Te are
4 nm and 5.5 nm respectively.
for measuring absolute reflectivities is on the order of 2%. This error in absolute reflectivity
values is the same for the single-shot and the multi-shot setup since we average over multiple
shots even in the single-shot setup to obtain a CF to as high a precision as possible. The
dominant sources of this error are alignment issues and sample quality, as well as the quality
of the existing dielectric function literature values. There is another source of error, however,
which is the mismatch of reflectivity values between individual data runs (e.g. successive
time delays). This “relative” reflectivity error is much higher in the single-shot case than in
the multi-shot case because the multi-shot setup allows averaging over multiple acquisitions
which reduces the noise significantly. We find that for single-shot measurements the relative
error is about 5% and therefore dominates, whereas in the multi-shot case the relative error
is at least two orders of magnitude smaller allowing measurements of relative reflectivity
changes as small as ∆R/R = 10−3 . But even in the case of multi-shot experiments, the
error bars for the absolute values of the extracted dielectric function is determined by the
94
larger of the two errors which is 2%. The data shown in Chapter 8 indicate that this 2% is
perhaps too conservative.
4.3.5
Temporal Resolution: Chirp Correction etc.
As mentioned in Section 4.3.3 we minimize dispersive stretching of the white-light probe
pulse by using only reflective optics between the CaF2 crystal and the sample. However,
it is impossible to avoid the chirp induced as the white-light propagates through the CaF2
crystal itself. In principle there are two ways to account for this chirp. The brute force
approach is to recompress the white light to the original duration of the seed pulse to regain
the original time resolution of the single-color experiment [79]. However, recompression of
broadband pulses is a highly non-trivial task. Alternatively, one can measure the chirp
in an independent measurement and regain the time resolution by time-shifting the data
accordingly, i.e., the measured reflectivities at each wavelength are shifted with respect to
each other by time delays dictated by the chirp measurement. We choose the latter option
to account for the chirp in our probe. There are several ways to measure the chirp of a
stretched pulse [80, 69]. One of the more elegant ways is to use two-photon-absorption
[69]. This technique is convenient because of the lack of phase-matching requirements. A
slight extension of this technique even allows a comparatively simple FROG (frequency
resolved optical gating [60]) measurement of the chirped white-light pulse as described
in Section 4.2.2. Figure 4.6 shows the chirp of our white-light measured using the twophoton-absorption technique described in Ref. [69]. The white-light probe pulse stretches
to a duration of about 500 fs due to propagation in the CaF2 crystal itself and due to
propagation through the glass filter.
The temporal chirp of the probe pulse is not the only factor impacting the time
resolution of the setup. It is also necessary to think about the spectral resolution of the
spectrograph used to capture the reflectivity spectra. We know that the white-light probe
pulse stretches over a time period of 500 fs and has a bandwidth of about 500 nm. Therefore
the temporal resolution is also limited by the spectral resolution of the spectrograph in
95
that a resolution of say 10 nm limits the temporal resolution of the setup to 10 fs. The
spectrograph used in our setup has a spectral resolution of 1 nm, and thus is not a limiting
factor for the temporal resolution of the setup. The overall temporal resolution of the setup
is therefore given by the duration of the original pulse from the Ti:sapphire amplifier. We
measure a pulse width of 35 fs using second-harmonic generation in a BBO crystal [68].
4.3.6
Summary and Outlook
We have developed a multi-angle ellipsometric technique to measure the full spectral dielectric function of solid materials with femtosecond time resolution. Femtosecond time-resolved
ellipsometry is a powerful tool to study phase transitions and carrier and lattice dynamics
in highly excited solids on a femtosecond time scale. For single shot measurements, the
reflectivity sensitivity is on the order of 5 %. When excitation fluences are low enough to
prevent damage of the sample, multi-shot experiments are possible and reflectivity sensitivities are as good as ∆R/R = 10−3 . Here, the usage of a higher repetition rate laser
system in conjunction with a faster CCD camera could boost the sensitivity by at least
another order of magnitude simply by averaging over more shots. Image acquisition from
a CCD is inherently slow because of the large amount of data that has to be moved. The
fastest way of acquiring two spectra is to employ a pair of one-dimensional detectors, such
as photodiode arrays, each with a dedicated spectrometer. The cost of such a fast setup
would be quite high but sensitivities on the order of ∆R/R = 10−4 are certainly achievable.
Let us briefly mention two other techniques which are currently used to track
phase changes in solid materials on a femtosecond time scale. The first is the method
of femtosecond microscopy, in which the pumped region is imaged onto a CCD camera
using the probe pulse at different time delays after the excitation. This method has the
twin disadvantages of measuring only reflectivity and doing so only at a single frequency.
However, it is the only method which observes the variation of material response across
a pumped region. Sokolowski-Tinten, von der Linde and co-workers have employed the
technique in beautiful experiments to study ablation, melting and resolidification in various
96
materials [15, 81, 82].
A second, emerging technique — time-resolved X-ray diffraction — goes a step
beyond purely optical methods. Researchers can now generate femtosecond X-ray pulses
using femtosecond optical pulses. In the past two years, several groups have used such pulses
to carry out time-resolved X-ray diffraction on materials excited by an ultrashort laser pulse
[83, 84, 85, 86, 87, 88]. Time time-resolved X-ray techniques can provide definitive means of
studying ultrafast structural dynamics in solids and other materials. In combination with
optical measurements, which primarily detect electronic changes, X-ray data could provide a
complete picture of ultrafast electronic and lattice dynamics in solids. Currently, however,
femtosecond X-ray sources are still in its infancy and the experiments which have been
carried out are more calibrations of these new highly complex experimental setups rather
than sources of new physical results. Currently, femtosecond time-resolved ellipsometry
provides the most complete and accurate view of ultrafast dynamics in highly excited solids.
Chapter 5
Ultrafast Processes in
Semiconductors — an Overview
There is a myriad of different processes that occur after the excitation of carriers in a semiconductor by an ultrashort light pulse. Roughly speaking, these processes can be classified
into coherent and incoherent carrier/lattice dynamics. Following irradiation, incoherent
scattering processes drive the material from a highly non-equilibrium state back into its
equilibrium state. Coherent carrier and lattice dynamics on the other hand are only observable, when their dephasing times are longer than a few cycles of the coherent resonance.
In this chapter we discuss both, incoherent scattering processes as well as coherent carrier
and lattice dynamics in semiconductors. For more details the reader may refer to the excellent excellent review of ultrafast processes in semiconductors by Shah in his book Ultrafast
Spectroscopy of Semiconductors and Semiconductor Nanostructures.
5.1
Carrier Relaxation
When an ultrashort light pulse hits a semiconductor with a bandgap smaller than the
photon energy of the light, the light is absorbed and for each absorbed photon an electronhole pair is created. For the purpose of this section we restrict ourselves to the discussion of
97
Chapter 5: Ultrafast Processes in Semiconductors — an Overview
(a)
98
(b)
E
(c)
E
E
CCS
Phonon
Emission
k
Eexc
Egap
f(E)D(E)
f(E)D(E)
pump
Figure 5.1: (a) photoexcitation of an electron into the conduction band of a semiconductor,
Eexc denotes the excess energy above the bottom of the band; (b) carrier-carrier scattering
causes the initial nonthermal electron distribution to take on a Fermi-Dirac distribution,
f (E) denotes the distribution function and D(E) denotes the density of states in the band;
(c) the initally hot electron distribution equilibrates with the lattice by emission of phonons.
electron dynamics1 . The generated electron distribution first redistributes due to carriercarrier scattering. Then, it equilibrates with the lattice (i.e. it either cools down or heats
up) by emission or absorption of phonons. Redistribution of the excited carriers typically
happens within a few tens of femtoseconds whereas carrier-lattice equilibration takes a few
picoseconds. On a much longer time scale — nanoseconds — the electrons recombine with
holes under emission of photons. In this section we discuss these processes in more detail.
5.1.1
Thermalization and Cooling of Carriers
Thermalization
Figure 5.1(a) shows the excitation of electrons into the conduction band of a semiconductor.
The electrons are generated with an excess energy above the bottom of the conduction band,
1
The hole dynamics for the most part are equivalent to the electron dynamics and can be treated with
the same formalisms if certain parameters, such as effective masses etc., are changed
99
as indicated by Eexc in the figure. This excess energy is given by the difference of the photon
energy of the incident light and the energy gap of the material. Immediately after excitation
the electron distribution corresponds to the spectral shape of the pump pulse as indicated
by the dashed curve in Fig. 5.1(b).2 The abscissa of the graph represents the probability of
finding an electron at a certain energy as given by the product of the distribution function
and the density of states. This is a very peculiar state of matter which can only be achieved
with ultrashort laser pulses: we create electrons in a distribution that is different from
the Fermi-Dirac distribution! Carrier-carrier scattering (CCS) redistributes the energy of
the electrons such that a Fermi-Dirac distribution is assumed. The rate of this process is
dependent on the excited density because CCS is a two-body scattering event. For moderate
excitation densities on the order of 1018 carriers per cm3 , carrier thermalization3 takes
place in times of several hundreds of femtoseconds as experiments on bulk GaAs and GaAs
quantum wells indicate [89, 90]. For higher carrier densities, thermalization can happen
very rapidly — as quickly as a few femtoseconds.
Carrier Cooling
After the electron distribution has thermalized it is possible to assign a temperature to the
distribution. Even so, this thermalized state of the semiconductor is only obtainable using
femtosecond lasers. The temperature of the electrons and the temperature of the lattice are
different! With the appropriate wavelength of laser light it is possible to generate electron
distributions with temperatures of several thousand degrees while the lattice remains at
room temperature.4 This hot electron distribution equilibrates with the lattice via emission
of phonons, as indicated in Fig. 5.1(c). Both the macroscopic cooling of a hot electron
distribution and the microscopic emission of phonons has been observed experimentally
2
This is only true if the pump pulse is sufficiently short — i.e., sub-100 fs. Otherwise the electrons start
to redistribute before the pulse is over.
3
Thermalization is a misleading naming convention here since it denotes the process of taking on a thermal
(Fermi-Dirac) distribution. The following process where the electrons equilibrate with the lattice is usually
referred to as cooling/heating in the literature.
4
Room temperature corresponds to about 25 meV. A Ti:sapphire laser (Ephoton = 1.5 eV) excites electrons in c-GaAs (Egap = 1.4 eV) with roughly an excess energy of about 4 times room temperature which
corresponds to 1200 K.
E
100
Eexc
k
V1
V2
Figure 5.2: Schematic illustration of intervalley scattering.
[91]. In c-GaAs the scattering time for emission of a longitudinal optical phonon by a hot
electron is 170 fs [91]. Cooling of an entire electron distribution requires many of these
individual electron-phonon scattering process. The exact number depends on the number
of excited carriers and the initial temperature of the distribution. Typically, carrier cooling
times are on the order of a few picoseconds [5].
5.1.2
Intervalley Scattering
All of the scattering processes described above are so-called intraband scattering events. The
electrons scatter from one k-vector to another one, but stay in the same valley. There is
another mechanism that scatters an electron from one valley to another — called intervalley
scattering. This process is illustrated in Fig. 5.2. If electrons are excited into the valley
denoted by V1 with an excess energy that is large enough to be close to or above the bottom
of the valley V2 they can emit/absorb a large wave-vector phonon and scatter into the V2
valley. The reason why intervalley scattering only occurs for electrons with an excess energy
higher than the bottom of V2 as indicated in Fig. 5.2 is that the energy gained/lost by an
absorption/emission of a phonon is comparably small (on the order of 10 meV).
Since the mobility of the electrons in the valley into which they scatter can be
lower than in the valley they originally occupied, carriers can actually “slow down” after
being accelerated enough by an electric field to undergo intervalley scattering. Macroscopic
101
(a)
(b)
Ephoton = Egap
E
E
k
k
Figure 5.3: Schematic illustration of (a) radiative recombination and (b) Auger
recombination.
phenomenona such as negative differential resistance and also the Gunn effect [18] can
be understood in terms of intervalley scattering. The literature is still quite inconsistent
regarding the exact scattering times. The reported values for an intervalley scattering event
from the L to the Γ-valley in c-GaAs range from a few femtoseconds to several picoseconds
[92]. An extensive review of femtosecond time resolved studies of intervalley scattering can
be found in Ref. [92].
5.1.3
Carrier Recombination
All effects described in Sections 5.1.2 and 5.1.1 lower the total energy of the electron distribution but preserve the number of excited carriers. For the density of excited carrier to
decrease, an electron in the conduction band has to recombine with a hole in the valence
band, a process called recombination. We discuss the two most prominent recombination
pathways in this section.
Radiative Recombination
Figure 5.3(a) shows the lowest order process of recombination. As indicated, an electron
102
can directly recombine with a hole if the energy is conserved via emission of a photon
corresponding to the energy difference between the hole state and the electron state involved.
This process is called radiative recombination and is the dominant recombination pathway
in semiconductors. Since both an electron and a hole are involved in this process, its
characteristic scattering time is proportional to the square of the excited carrier density.
The typical time scale for radiative recombination in semiconductors is about a nanosecond
for an excitation density of 1018 cm−3 [93]. Since the maximal achieveable time delay in our
setup is 0.5 ns, we do not observe radiative recombination in our experiments.
Auger Recombination
The next most prominent recombination process is shown in Fig. 5.3(b). Here, an electron
recombines with a hole but in this case the energy is conserved through elevating a third
particle (the case of an electron is indicated in the figure, but it might as well be a hole) to a
higher energy state. This process is called Auger recombination. It is a three body process,
involving three carrier particles.5 Since energy as well as momentum must be conserved
(which is trivial in the radiative recombination case because the photon’s momentum is
negligible) only certain electron-hole pairs can recombine and simultaneously find a pair
of initial and final states for a second electron/hole which allows to conserve energy and
momentum at the same time. So, Auger recombination is very different for electrons at
different positions in the Brillouin zone. In general, however, the characteristic scattering
time is proportional to the cube of the excited carrier density because three carrier particles
are involved in Auger recombination.
Figure 5.4 shows the carrier density dependence of radiative vs. Auger recombination in GaAs. The values are based on experiments by Strauss [94]. For low carrier densities,
Auger recombination is much less likely to occur than radiative recombination because only
certain carrier triplets can undergo Auger recombination. Since Auger recombination depends strongly on the carrier density, it becomes dominant over radiative recombination for
5
Recall that radiative recombination involved a photon plus two carrier particles making it a three-body
process where two carriers and one photon interact.
103
log [recombination lifetime (s)]
−6
−9
radiative
−12
Auger
GaAs
−15
17
18
19
20
21
22
23
log [carrier density (cm−3 )]
Figure 5.4: Dependence of recombination times in GaAs on excited carrier density [94].
The dotted line indicates Auger recombination rates and the dashed line indicates radiative
recombination rates.
excited carrier densities above 1019 cm−3 . The simple power laws for the density dependence
of both recombination processes we have discussed are not valid for high carrier densites
because the dense electron-hole plasma screens the Coulomb interaction between individual
carriers. Theoretical work on Si predicts that Auger recombination rates saturate at about
0.15 ps−1 for carrier densities above 1020 cm−3 [95]. In spite of screening, it is still reasonable to expect Auger recombination to be the dominant recombination mechanism at high
excited carrier densities.
5.2
Coherent Dynamics in Semiconductors
All of the scattering events discussed so far have one thing in common: they involve incoherent processes. Here, “incoherent” refers to the fact that carrier-carrier scattering as
well as carrier-phonon scattering destroys the phase relation between excited carriers. It
104
is common to distinguish between interband coherence, which denotes the phase relation
between carriers and holes, and intraband coherence, which denotes the phase relation between carriers within a valence or conduction band. We discuss the intruiging phenomenon
of quantum beats in this section, which arise from interband coherence. A slightly different
type of coherence is found in lattice dynamics. Here, the quasi-instantaneous excitation of
the material due to the ultrashort laser pulse can excite coherent vibrational modes where
all lattice ions move in phase with each other. These peculiar phonon modes are referred
to as coherent phonons and are topic of the later part of this section.
5.2.1
Coherent Carrier Dynamics: Quantum Beats
Coherent carrier dynamics in semiconductors have attracted much attention over the past 10
years. Covering the whole field would far exceed the frame of this thesis. A very nice review
of the field to date can be found in Ref. [5]. To give the reader a flavor of these intruiging
phenomena we describe one of the simplest cases in this section — the quantum beat. The
expression quantum beat denotes the coherent superposition of two excited states sharing a
common ground state in a quantum mechanical system. This situation is illustrated in Fig.
5.5. If the bandwidth of the exciting laser pulse is wider than the energy separation between
the two excited states e1 and e2 , a coherent superposition of the two states is excited. Since
each state evolves in time like e− ~ ei t the superposition evolves in time as the beat note:
i
e− ~ (e2 −e1 )t . It turns out that in semiconductor quantum wells the confinement gives rise to
i
new electron and hole eigenstates with energy separations just in the right range.
If an electron is excited from the valence band into the conduction band there
is an interband polarization associated with this excitation. The polarization oscillates at
a frequency corresponding to the energy separation between electron and hole state. The
phase coherence between electron and hole is usually destroyed fairly quickly (depending
on the temperature of the sample and the excitation density within tens of femtoseconds to
a few picoseconds) by scattering processes. In the time when phase coherence is granted,
however, the polarizations of two excited electron states sharing one common hole state can
105
|e2
|e1
pump
|g
Figure 5.5: Schematic illustration of a quantum beat. If the bandwidth of the exciting
laser pulse is wider than the energy separation between the two excited states, a coherent
superposition of these states is excited.
interfere with each other producing a beat note at the difference frequency as depicted in
Fig. 5.5. If the sample is chilled and the carrier density is kept reasonably low, the phase
coherence is preserved for time periods exceeding several multiples of the beat period,
thus making an observation of quantum beats possible. The first successful observation of
quantum beats in semiconductor quantum wells was done by Leo et al. using Four Wave
Mixing (see Section 4.1.2) [62]. A more detailed study which showed that quantum beats
are not only due to interband coherence but also give rise to intraband coherence can be
found in Ref. [63].
5.2.2
Coherent Lattice Dynamics
In the previous section we have discussed coherent carrier dynamics in semiconductors. In
certain materials one can also generate (and detect) coherent lattice dynamics using ultrashort optical pulses. If the duration of the laser pulse used to excite these materials
is shorter than an oscillation period of the phonon modes it is possible to excite coherent
phonons, i.e. lattice vibrations where all ions move in phase with each other. Using various
106
ultrafast spectroscopic techniques (see Section 4.1.2) it is then possible to observe coherent
phonons in the time domain, i.e., to directly measure the relative amplitude and phase of
lattice vibrations. This technique is complementary to Raman scattering, which is traditionally used for the study of phonon modes in solids [96]. Reviews of coherent phonons in
solid materials can be found in Refs. [97, 98].
Generation Mechanisms
It is common to categorize phonon modes as Raman-active or IR-active. The latter are
optical phonon modes in lattices where the atoms are at least partly ionically bonded. The
distortion of the lattice due to an optical phonon generates a dipole moment to which an
electro-magnetic field can couple. Raman-active modes on the other hand can be present
in any lattice. In principle Raman scattering [19] can be thought of as a second-order wave
mixing process where one of the participating waves is given by the phonon mode. Following
the formalism in Section 2.3.1, the nonlinear polarization generated will have frequency
components at the sum and difference between the phonon mode and the fundamental light
frequency (the so-called Anti-Stokes and Stokes frequencies respectively).
In the case of IR-active modes, it is possible to drive a coherent phonon via so-called
resonant excitation. If two light waves of the proper frequencies are incident on a crystal, the
frequency of the wave generated by DFG can be exactly tuned to the resonance frequency of
an IR-active phonon. This process is called Impulsive Stimulated Raman Scattering (ISRS)
and has been successfully used to excite coherent phonons in a number of experiments[99].
ISRS is the only way to resonantly excite coherent phonons. Of course it is also possible
to directly irradiate the crystal with light at the phonon frequency. However, nonlinear
wave-mixing is the only way to date to efficiently generate electro-magnetic radiation at
optical phonon frequencies, which are typically on the order of several THz.
The other approach of exciting coherent phonons is referred to as impulsive excitation. In the case of IR-active phonons in c-GaAs the impulsive generation of coherent IR-active phonons has been achieved by the ultrafast screening of the strong surface
+
+
+
107
+
pump
Figure 5.6: Schematic illustration of Displacive Excitation of Coherent Phonons.
field, typical of III-IV semiconductors [65]. Specifically, the phonons observed in GaAs
are longitudinal optical (LO) phonons. In a mechanical analog, the springs between the
individual Ga and As atoms are stretched due to the surface field. As the surface field is
quasi-instantaneously “switched off” the springs relax to their new equilibrium position and
oscillate around it. This impulsive mechanism has been used to generate coherent phonons
in bulk c-GaAs as well as GaAs/AlGaAs heterostructures [98].
There is a second impulsive mechanism referred to as displacive excitation of coherent phonons (DECP) which was first observed by Cheng et al. [100, 101]. The researchers
performed fs time-resolved differential reflectivity experiments (see Section 4.1.2) on small
gap semiconductors (Te, Sb, Bi, and Ti2 O3 ). The reflectivity traces showed strong oscillatory features at frequencies corresponding to the the symmetry preserving A1 -mode [102]
in these materials. If the excitation mechanism was based on Raman scattering, one would
expect all Raman-active modes to be excited. The fact that only the A1 -mode was excited prompted the researchers to propose the DECP mechanism, which was subsequently
explained theoretically by Zeiger et al. [103]. Figure 5.6 shows a schematic illustration of
DECP. The left hand side of the figure shows a diatomic molecule in its equilibrium position. The electronic energy levels are indicated below the molecule. The strength of the
covalent bond, which in this case consists of four electrons (e.g., O2 ), is indicated by the
thickness of the spring. If a laser pulse excites an electron from the lower energy (bonding)
108
state into the higher energy (anti-bonding) state, the new electron configuration screens
the Coulomb repulsion of the ions less effectively, establishing a new equilibrium separation
for the ions. This effect is indicated in the right hand side of Fig. 5.6 where the spring is
weaker and the ions are further apart from each other. Note, that now the energy separation of the bonding and the antibonding states is less, due to the increased separation of
the host ions. This redshift of the bonding-antibonding split will become important in the
discussion of our experiments in Chapter 8. This simple molecular picture can be directly
appied to semiconductors. In solids, the electronic states are delocalized and form bands.
The valence band is made up of bonding states and the conduction band consists of antibonding states. Photoexcitation of semiconductors is equivalent to promoting electrons
from bonding to antibonding states [104]. An impulsive weakening of the bonds therefore
causes the lattice to rapidly relax to its new equilibrium position. If the excitation happens
on a much shorter time scale than a single phonon period, the lattice vibrates around the
new equilibrium position.
We now understand the basic mechanism for DECP. But why is it that only
the A1 -mode is excited in the experiments by Cheng et al.? The reason lies in the fully
symmetric nature of the excitation mechanism. As opposed to the resonant excitation or the
impulsive field screening mechanism discussed above, DECP does not impose any symmetry
breaking direction onto the crystal.6 Therefore, only phonon modes which fully preserve
the symmetry of the crystal structure are excited via DECP [103]. Of course only certain
lattice structures allow fully symmetry preserving phonon modes — so-called A1 -modes.
Figure 5.7 shows a schematic illustration of the A1 -mode in Te. The Te lattice consists of
three-fold helices which are positioned in a hexagonal pattern [102]. The figure shows a view
down the axis of the helices. The helices are all of the same screw direction, and Te with
both left and right handed helices exists. The magnification of an individual helix on the
right hand side of the figure indicates the atom motion in the A1 -mode. This “breathing”
of the lattice does not influence its symmetry properties at all. There is no other fully
6
In the resonant excitation case the polarization of the field breaks the symmetry, and in the field screening
case the surface field introduces a preferred direction.
conventional
unit cell of Te
109
A1 phonon mode
c-axis
a
Figure 5.7: Schematic illustration of the fully symmetric A1 -mode in Te.
symmetry preserving mode which is why Cheng and co-workers observed one only mode
in their experiments. The researchers found a frequency of 3.6 THz for the A1 -mode in
excellent agreement with previous Raman measurements [102].
There is an interesting implication involved in DECP. Since the lattice bonds are
weakened, the frequency of the A1 -mode can be expected to redshift from its original value.
In the early experiments by Cheng and co-workers, very low carrier densities of 1018 cm−3
were created. The observed phonon frequencies were indistinguishable from the values
previously obtained using Raman techniques because the weakening of the covalent bonds
is not appreciable at such low densities. At higher excitation levels, however, an observable
redshift should be expected. This “softening” of the A1 phonon mode was indeed observed
in beautiful experiments by Hunsche et al. in 1995 where carrier densities up to 1021 cm−3
were excited [78]. The phonon frequency redshifts linearly with increasing carrier density.
At the highest excitation density of about 1021 cm−3 , the phonon frequency shifts down to 3
THz. This experimental work was theoretically modelled by Tangney et al. a few years later
in 1999. The theory qualitatively agrees with the experimental data. Both, the experiment
and the theory further validate the DECP model of impulsive weakening of covalent bonds.
110
We will return to the experiments by Hunsche et al. and the theory by Tangney et al. in
the discussion of our data in Chapter 8.
Detection of Coherent Phonons
So far we have discussed different generation mechanisms for coherent phonons in solids.
Detecting these coherent modes is another challenge. Since typical optial phonon modes
in solids have resonance frequencies on the THz scale it is necessary to use ultrafast spectroscopic techniques, as described in Section 4.1.2, to resolve these lattice vibrations. As
already mentioned, it is crucial to pick the appropriate optical signature to observe a certain
phenomenon.
Choosing the appropriate optical signature requires knowledge of how the lattice
vibrations alter the optical properties of the material. In the case of LO-phonons in c-GaAs,
lattice vibrations induce an anisotropic reflectivity. A rapidly changing electric field along
the direction of vibration due to the polar binding between Ga and As atoms causes the
index of refraction to change anisotropically perpendicular to the field change [65]. This is
due to the electrooptic or Pockels effect based on the second order nonlinearity χ(2) [30].
In GaAs, the r43 element in the electrooptic tensor causes ellipsoidal breathing of the index
profile perpendicular to the field direction (the index is circular when no field is applied,
i.e., it is the same in all directions perpendicular to the field direction) [65]. Therefore
coherent LO-phonons in c-GaAs are ideal candidates for detection in a geometry as shown
in Fig. 4.2(b). Using this detection scheme paired with a sophisticated noise reduction
system, it is possible to detect the reflectivity changes caused by the coherent LO-phonon
mode in GaAs which are on the order of ∆R/R ≈ 10−5 .
In the case of DECP, the fully symmetric A1 -mode causes an isotropic change in
the reflectivity of the host material. The induced reflectivity changes are comparably large,
ranging up to 10% for the highest excitation densities [78]. Therefore, it was sufficient to use
a simple differential reflection technique in the experiments in Refs. [100, 101, 78]. However,
these simple reflectivity measurements cannot provide information beyond the phase and
111
relative amplitude of the coherent A1 -mode. The time resolved traces merely allow tracking
the exact frequency of the phonon for different excitation conditions. This proved to be
useful for comparison with certain aspects of theories [78, 105], but it would certainly be
extremely instructive to have a detection technique which can provide more information on
the exact processes in the material. As discussed in Section 4.3, FTRE is such a technique.
In fact, the detection of coherent phonons using FTRE is the main topic of Chapter 8.
5.3
Ultrafast Phase Transitions in Semiconductors
Sections 5.1 and 5.2 on low density excitations in semiconductors. Only at low carrier
densities is it possible to extract specific information on scattering processes, such as carrierphonon scattering rates, or observe coherent phenomena, such as quantum beats. In these
experiments the semiconductor is “tickled” a little bit by the laser pulse inducing certain
carrier or lattice effects, but the macroscopic phase of the material is not altered. This
sections deals with the physics of highly excited semiconductors, i.e., excitations on the
order of the threshold for permanent damage of the materal.
5.3.1
Nonthermal Melting
The term nonthermal melting was first introduced by van Vechten in the framework of
his pioneering work on short-pulsed laser annealing of Si [106]. Even though van Vechten
postulated nonthermal melting for the case of nanosecond pulses, which was proven to
be wrong later, he still deserves the credit for first publishing the idea of a nonthermal
disordering process.
This work triggered many experiments exploring nonthermal melting in Si and
GaAs [107, 13, 108, 14, 109, 15], and InSb [110]. In all of these experiments the researchers
found that semiconducting materials can undergo a structural change on time scales of
hundreds of fs, which is faster than the time required for a complete equilibration of the
generated hot carrier distribution with the host lattice (see Section 5.1.1). The experimental
112
findings were supported by the theoretical work of Stampfli and Bennemann [104, 111, 112].
The theory is based on impulsive weakening of the lattice as a light pulse promotes a large
number of carriers into the conduction band of a semiconductor.7 In more recent work,
Graves, Dumitrica and Allen use molecular dynamics simulations to calculate the dielectric
function of c-GaAs after intense excitation with an ultrashort laser pulse.[113, 114] All
calculations predict a critical density for lattice instability if 10% of all valence electrons
are excited into anti-bonding states.
There are two reasons why researchers refer to this optically induced disordering
process as “nonthermal melting”. First, as mentioned above, the lattice disorders on a time
scale too short for thermal equilibration to take place. Essentially, the lattice disorders while
it is still “cold”. The bonds between the atoms are weakend so much due to the optically
exicted, dense electron-hole plasma, that the crystalline order is lost. The intermediate
thermal step of energy transfer from the electronic system to the lattice through phonon
emission is skipped. Second, the energy required to disorder the lattice with a short optical
pulse is much less than the energy required for thermal melting [104, 111, 112]. For example,
the damage threshold for c-GaAs for femtosecond pulses is about 1 kJ/cm3 [115]. In thermal
terms, that corresponds to a temperature increase by about 600 K. So starting from room
temperature one would end up with a temperature of about 900 K, whereas the melting
point of c-GaAs is about 1500 K.
Most of the experiments were conducted using femtosecond microscopy [13, 109,
15], where images of the sample are taken at several time delays after the pump pulse
excites the sample. The image area is chosen to be larger than the pumped spot allowing
the simultaneous monitoring of areas exposed to different pump fluences. In all experiments,
a high-reflectivity phase is observed within 1 ps after the excitation which is shorter than
the time necessary for thermal equilibration between the hot carriers and the lattice to take
place (which is usually about an order of magnitude larger [5]). The researchers attribute
this high reflectivity to a molten phase because it is well known that liquid semiconductors
7
This process is analogous to DECP (described in Section 5.2.2).
113
are metallic. Other experiments used SHG to monitor this phase transition in c-GaAs
[14, 108, 116]. Since SHG is sensitive to the exact symmetry properties of the material (it
vanishes for centro-symmetric materials [28]), the researchers were able to conclude that
the samples lose long range order on length scales of the probe wavelength. The optical
experiments therefore provide evidence of a nonthermal transition of the material to a
disordered, metallic state.
A very different approach to the nonthermal melting problem was taken by Hunsche et al. As described in Section 5.2.2 Hunsche and co-workers observed an increasing
softening of the A1 phonon mode in Te with higher excitation densities. Extrapolating
the data to excitation densities where the phonon frequency approaches zero enabled the
researchers to extract a critical excitation density for nonthermal melting of about 10% of
all valence electrons. This is in agreement with the theoretical prediction by Stampfli et al.
[104, 111, 112].
The information gathered from the optical experiments to date do not reveal the
whole story of nonthermal melting, because it is still impossible to determine the exact
nature of the nonthermal phase semiconductors assume about 100 fs after excitation with a
strong femtosecond laser pulse. The most promising approach to reveal the exact structural
properties of this new state of matter is ultrafast x-ray spectroscopy as described in Section
4.3.6. Inspite of very intruiging first data, researchers are still struggling to optimize this
complex experimental tool and the results to date are almost calibrations of the new setups
to old results rather than revealing new physics. Presently the technique which can provide
the most detailed look at ultrafast phase dynamics in solids is FTRE (see Section 4.3).
FTRE measurements of phase transitions in a-GaAs and GeSb thin films are the topic of
Chapters 6 and 7 respectively. We present recent results on c-GaAs obtained in our group
in the next section.
40
40
0.32 F th
30
20
Re ε
10
Im ε
0.32 F th
30
500 fs
dielectric function
dielectric function
114
323 K
0
20
4 ps
Re ε
10
Im ε
473 K
0
(b)
(a)
−10
1.5
2.0
2.5
3.0
3.5
4.0
4.5
−10
1.5
2.0
photon energy (eV)
2.5
3.0
3.5
4.0
4.5
photon energy (eV)
Figure 5.8: Dielectric function of c-GaAs [• — Re[ε(ω)], ◦ — Im[ε(ω)]] for a pump fluence
of 0.32 Fth (a) 500 fs after excitation and (b) 4 ps after excitation. The curves show: (a)
the real and imaginary part of ε(ω) for GaAs at room temperature [21] (solid and dashed
curves respectively), (b) the real and imaginary part of ε(ω) for GaAs at 473 K [117] (solid
and dashed curves respectively).
5.3.2
Dielectric Function Measurements in c-GaAs
Our group significantly advanced the understanding of ultrafast phase changes in semiconductors by measuring the femtosecond time resolved dielectric function of c-GaAs [115].
We briefly review the findings of Huang and co-workers on c-GaAs in this section, which
will be helpful in our discussions of the experiments described in Chapters 6 and 7. The
response of c-GaAs to irradiation with intense femtosecond laser pulses can be categorized
into three fluence regimes — a low, a medium and a high fluence regime. It is useful to
parameterize the fluence levels in multiples of the threshold fluence of permanent damage.8
The damage threshold of c-GaAs for excitation with femtosecond laser pulses was found to
be about Fth = 1.0 kJ/m2 [115].
Low Fluence Regime
The dynamics of the dielectric function of c-GaAs are quite similar for fluences below 0.5 Fth .
Thus we define F < 0.5 Fth as the low fluence regime. Figure 5.8(a) shows the dielectric
function of c-GaAs 500 fs after excitation with a pump fluence of 0.32 Fth . The curves
8
Permanent damage was determined by post-mortem inspection under an optical microscope.
115
1000
c-GaAs
800
600
0.32 Fth
400
0.20 Fth
NONTHERMAL
lattice temperature (K)
0.45 Fth
200
0
0
5
10
15
20
time delay (ps)
Figure 5.9: Lattice heating in c-GaAs following excitation with ultrashort laser pulses of
varying fluence: • — 0.45 Fth , — 0.32 Fth , — 0.2 Fth . The curves are exponential fits
which share the same rise time of 7 ps.
represent the dielectric function of unexcited c-GaAs at room temperature as measured
using cw-ellipsometry. The dominant feature is the “bleaching” of the E1 -peak (see Section
2.1.2) in the imaginary part of ε(ω). Bleaching results from Pauli-blocking of transitions
due to the occupation of conduction band states by excited electrons. In these experiments,
the carriers are originally excited into the Γ valley, whereas the E1 peak corresponds to the
L valley of c-GaAs. It is therefore possible to extract intervalley scattering times from this
data (see Section 5.1.2). From the ε(ω) data is is possible to deduce a Γ − L scattering time
of a few hundred fs which is on the low end of reported values (see Section 5.1.2) — for
more details refer to Ref. [118].
Figure 5.8(b) shows the dielectric function of c-GaAs 4 ps after excitation with
a pump fluence of 0.32 Fth . The curves represent real and imaginary part of ε(ω) of cGaAs at 473 K as measured using cw-ellipsometry [117]. The data match the values for
heated c-GaAs almost perfectly. This indicates that after 4 ps, the electron distribution has
116
equilibrated with the lattice. Similar data sets for different time delays and different pump
fluences within the low fluence regime show equivalently nice fits for different temperatures
of c-GaAs. The low fluence regime is thus characterized by thermal lattice heating on
picosecond time scales.
Figure 5.9 shows the fitted lattice temperatures as a function of time delay for
three different fluences. The curves represent exponential fits with a same rise time of 7 ps.
It turns out that the observed lattice temperatures exceed the values predicted by a simple
model which only takes direct phonon emission into account (see Section 5.1.1). In such a
model, the final temperature depends on the excess energy of the excited electrons assuming that each electron equilibrates with the lattice by emitting phonons until the electron
distribution and the lattice have reached the same temperature [118]. A relaxation model
based on Auger recombination (see Section 5.1.3) provides a remedy to this discrepancy.
In Auger recombination an electron recombines with a hole giving its energy to another
electron. Thus, the total excess energy of the electrons above the bottom of the conduction
band is raised, explaining the high lattice temperatures reached [118].
Medium Fluence Regime
Huang et al. find a medium fluence regime from F = 0.5 − 0.8 Fth . Figure 5.10(a) shows
the dielectric function of c-GaAs 500 fs after excitation with a pump fluence of 0.70 Fth .
The curves represent the dielectric function of unexcited c-GaAs at room temperature, as
measured using cw-ellipsometry. The early electronic effects, such as bleaching of the E1
peak, are similar to the low fluence regime, but naturally more pronounced. The most
interesting behavior in this regime is shown in Fig. 5.10(b). At 4 ps after excitation, it
is not possible to fit the data with dielecric functions of heated c-GaAs at any temperature. Instead, the curves represent the real and imaginary part of ε(ω) of a-GaAs at room
temperature, as obtained by cw-ellipsometry [119]. The agreement between the data and
the cw values is quite good, indicating that the material has become disordered on time
scales of picoseconds after excitation with fluences in the medium fluence range. This find-
40
40
0.70 F th
30
0.70 F th
30
500 fs
dielectric function
dielectric function
117
20
Im ε
10
Re ε
0
4 ps
20
Im ε
10
Re ε
0
(b)
(a)
−10
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
4.0
4.5
−10
1.5
2.0
2.5
3.0
3.5
4.0
4.5
photon energy (eV)
curves respectively), (b) the real and imaginary part of a-GaAs at room temperature [119]
(solid and dashed curves respectively).
ing supports previous SHG measurements which also observe an ultrafast disordering of
c-GaAs due to intense photo-excitation [116] and furthermore provide an additional piece
of evidence for nonthermal melting as described in Section 5.3.1. It is not surprising that
the match between the data and the cw values is not perfect because the amorphous phase
of GaAs that is generated with the femtosecond pulse is certainly not at room temperature.
There are no cw measurements of heated a-GaAs reported in the literature to date. As
described in Chapter 6, we performed the first ε(ω) measurement of heated a-GaAs and use
these data to compare them to the ε(ω) of c-GaAs excited at medium fluences.
High Fluence Regime
Above fluences of F = 0.80 Fth , c-GaAs undergoes an ultrafast semiconductor-to-metal
transition. This phase transition is characteristic of the high fluence regime. Figure 5.11(a)
shows the dielectric function of c-GaAs 500 fs after excitation with a pump fluence of 1.60
Fth . The curves represent the dielectric function of unexcited c-GaAs at room temperature
as measured using cw-ellipsometry. The dielectric function takes on a completely different
shape even at this early time delay. The most telling feature in this graph is the position of
80
80
1.60 F th
40
Im ε
20
0
2.5
3.0
3.5
photon energy (eV)
4 ps
40
20
Im ε
0
Re ε
2.0
1.60 F th
60
500 fs
dielectric function
dielectric function
60
−20
1.5
118
Re ε
(a )
4.0
4.5
−20
1.5
2.0
2.5
3.0
3.5
(b)
4.0
4.5
photon energy (eV)
curves respectively), (b) the real and imaginary part of a Drude model dielectric function
with a plasma frequency of 12 eV and a relaxation time of 0.2 fs (solid and dashed curves
respectively).
the zero-crossing of Re[ε(ω)], which denotes the bonding-antibonding split of the material
(see Section 5.2.2). The zero-crossing has redshifted from its original position at 4.7 eV
(see Fig. 2.1(c)) to about 2.0 eV within 500 fs. This is an indicator of a progressively
closing bandgap. There is a greath wealth of theoretical and experimental studies of a
many body effect called bandgap renormalization (BGR) [120, 121, 122, 123, 124]. BGR
denotes the negative self-energy correction to the single electron energy due to the excited
electron plasma [125]. Usually predicted (and observed) as a small energy correction to the
bandgap of semiconductors for low density excitations, Kim et al. predicted that the direct
gap of c-GaAs would completely collapse at an excitation of 10% of all valence electrons.
BGR could therefore explain the bandstructure dynamics in the ε(ω) data purely due to
electronic effects. However, electronic effects should be largest immediately after excitation
when the carrier density is highest.9 Therefore, even though BGR certainly contributes
to the bandstructure dynamics observed by Huang and co-workers, the progression of the
zero-crossing of ε(ω) indicates that the transition is not a purely electronic effect but rather
9
BGR does depend on carrier temperature as well as density. However, the models by Zimmermann and
others [120, 121, 122, 123] indicate that the self-energy correction is only weakly dependend on temperature
making density the dominant factor.
119
structurally driven. Kim’s work does pose an interesting question for future experiments,
however: is it possible at even higher excitations to achieve complete BGR, i.e. can a
femtosecond pulse cause a semiconductor-to-metal transition at the time scale of electron
excitation?
Figure 5.11(b) shows the dielectric function of c-GaAs 4 ps after excitation with
a pump fluence of 1.60 Fth . The curves represent the real and imaginary part of a Drude
model dielectric function with a plasma frequency of 12 eV and a relaxation time of 0.2 fs.
The data fit the Drude model better than a typical metal does10 . The relaxation time of 0.2
fs is a typical value for metals [22]. The plasma frequency of 12 eV roughly corresponds to
the carrier density equivalent to all valence electrons in c-GaAs, indicating that the bandgap
has completely closed and all valence electrons are participating in conduction. Although
the Drude fit is excellent, a peak centered at 2.75 eV is visible in Fig. 5.11(b). This peak
implies a contribution due to interband transitions between 2.5 and 3 eV in photon energy,
even after the band gap has collapsed. The same phenomenon occurs in copper, whose
dielectric function is well described by a Drude model except for interband transitions
above 2 eV, which are responsible for the characteristic color of copper. The theoretical
results by Allen et al. [113, 114] are in qualitative agreement with these data showing a
residual interband contribution to Im[ε(ω)] around 2.5–3.0 eV after the semiconductor-tometal transition has occurred. The simulations suggest that this contribution comes from
some of the states in the valence and conduction bands which originally produced the E2 peak in unexcited c-GaAs, but which move closer together as a result of the excitation.11
Similar semiconductor-to-metal transitions in different materials are the main focus in all
experimental chapters in this thesis (6, 7, and 8).
10
As we have seen in Section 2.1.2, Cu for instance has residual peaks in Im[ε(ω)] due to strong interband
transitions.
11
Such residual transitions close to the bonding-antibonding split have been found before in (thermal)
liquid semiconductors [126].
Chapter 6
Ultrafast Phase Transitions in
Amorphous GaAs
In this chapter we present femtosecond time-resolved measurements of the dielectric function
of amorphous GaAs (a-GaAs) over a broad spectral energy range (1.7-3.4 eV), at carrier
densities below and above the threshold for permanent damage. A detailed analysis of the
data reveals many new insights about the dynamics of a-GaAs at high excitation fluences.
Comparing the ε(ω) data to previous results on c-GaAs (see Section 5.3.2) we develop a
deeper understanding of the non-thermal phase transition in GaAs.
6.1
Experimental Results
Since the experiments described in this chapter involve pump fluences above the damage
threshold, it was necessary to use the single-shot setup described in Section 4.3. The largest
Brewster angle for a-GaAs across the spectrum of the broadband probe is 77o . For this
experiment, we chose to measure reflectivities at 53o and 78o . Using the methods described
in Section 4.3.4, we determined the thickness of the native oxide layer on a-GaAs to be
1.7 nm. We assume that the oxide formed on a-GaAs is identical to that on c-GaAs and
use its value of ε = 4.0 for the dielectric constant of the oxide [127]. Figure 6.1 shows
120
Chapter 6: Ultrafast Phase Transitions in Amorphous GaAs
121
40
a-GaAs
dielectric function
30
20
no pump
Re ε
10
Im ε
0
−10
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
Figure 6.1: Dielectric function of unexcited a-GaAs (• = Re[ε(ω)]; ◦ = Im[ε(ω)]). The
curves show previous measurements of ε(ω)(solid lines = Re[ε(ω)]; dashed lines = Im[ε(ω)])
of a-GaAs at room temperature using ellipsometry with continuous wave light [119].
the dielectric function of our unpumped a-GaAs sample. The error bars are based on an
estimated 5% uncertainty in matching the correction factors (see Section 4.3.4). Assuming
a 5% deviation from the measured reflectivity value to higher and lower values produces
four additional reflectivity pairs since there are two angles. Running the inversion algorithm
on these four pairs produces four different imaginary and real parts of ε(ω). We assume
the lowest and highest of these values as the lower boundary and higher boundary for the
error bars shown in the graph. As indicated in the figure, the ε(ω) of our sample agrees well
with that of an amorphous GaAs sample generated by bombardment with 270-keV As+
ions [119]. This agreement confirms that the type of ions used to amorphize c-GaAs have
little effect on the optical properties of the final a-GaAs sample.
Our a-GaAs sample was prepared by Dr. C. W. White of Oak Ridge National
Laboratory using three stages of ion implantation with Kr+ ions into a crystalline wafer.
The sample was successively exposed to ions with an energy of 900 keV, 450-keV, and
122
225 keV. The dosing amounts were 4×1015 , 2×1015 and 1015 ions per cm2 , respectively. The
total dose of 7 × 1015 ions per cm2 constitutes less than 1% of the atoms in the amorphized
region of the material. After implantation the material is amorphous to a depth of 600 nm,
which is greater than the optical penetration depth of our probe pulse.
The response of a-GaAs to excitation by a femtosecond pulse varies strongly with
excitation fluence F . Throughout this chapter, we quote F as the peak fluence at the
center of the Gaussian beam profile in units of the threshold fluence for permanent damage
to the sample Fth-a = 0.1 kJ/m2 (as observed under an optical microscope). We observe the
formation of a crater on the a-GaAs surface (as opposed to simple discoloration) when the
fluence exceeds F = 4.5kJ/m2 . Recall that the threshold for permanent damage in c-GaAs
fluence is Fth-c = 1.0 kJ/m2 (see Section 5.3.2). This order of magnitude difference comes
from the fact that c-GaAs has to undergo a crystalline-to-amorphous transition whereas
when starting from a-GaAs it is an amorphous-to-amorphous transition. We have identified
three different regimes of behavior which we refer to as low, medium, and high fluence
regimes in the following presentation of the data.
6.1.1
Low Fluence Regime (F < 1.7 Fth-a)
Figure 6.2 shows the evolution of the dielectric function of a-GaAs following an excitation
of 0.9 Fth . We omit errors bars in this and the following graphs for sake of clarity of
the figures. Mostly, the size of the error bars is on the order of the marker size. Within
the first few hundred femtoseconds, Im[ε(ω)] rises and Re[ε(ω)] falls for frequencies at the
lower end of the spectral range. The dynamics at early times happen mostly at photon
energies close to the pump excitation of 1.5 eV. Hence, it is likely that the changes are
due to electronic effects, i.e. free electron absorption (described by the Drude model), Pauli
blocking of transitions, and changes in the allowed energy states that occur when electrons
screen the ions and each other. Since the Im[ε(ω)] rises, Pauli blocking does not seem to
play a major role in the electron dynamics at early times. Free carrier absorption and
screening effects on the other hand do cause the Im[ε(ω)] to rise and seem to be dominant
40
40
0.9 F th
a-GaAs
30
Im ε
20
10
0.9 F th
30
200 fs
dielectric function
dielectric function
123
Re ε
500 fs
Im ε
20
10
Re ε
0
0
(a )
−10
1.5
2.0
2.5
3.0
(b )
−10
1.5
3.5
2.0
40
3.0
0.9 F th
0.9 F th
30
dielectric function
2 ps
Im ε
20
10
Re ε
0
8 ps
Im ε
20
10
Re ε
0
(c )
−10
1.5
2.0
2.5
3.0
(d )
−10
1.5
3.5
2.0
photon energy (eV)
3.0
3.5
40
0.9 F th
30
Im ε
20
0.9 F th
30
16 ps
dielectric function
dielectric function
2.5
photon energy (eV)
40
10
Re ε
0
400 ps
Im ε
20
10
Re ε
0
(e )
−10
1.5
3.5
40
30
dielectric function
2.5
photon energy (eV)
photon energy (eV)
2.0
2.5
photon energy (eV)
3.0
(f )
3.5
−10
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
Figure 6.2: Evolution of the dielectric function of a-GaAs (• = Re[ε(ω)]; ◦ = Im[ε(ω)])
after excitation by pulses of about 0.9 Fth-a . The dielectric function of a-GaAs at room
temperature [119] is shown for comparison in all panels (solid curves = Re[ε(ω)]; dashed
curves = Im[ε(ω)]).
in a-GaAs at early times.
The peak in Im[ε(ω)] shifts below its initial position of 3.35 eV, reaching 2.8 eV
at 50 ps. Re[ε(ω)] also shifts to lower frequency with time. Such a downshift is expected
when a semiconductor heats [119, 128, 129, 130]. We cannot determine the temperature
124
0.8
downshift of ε (eV)
a-GaAs
0.6
0.4
Re ε = 10
0.2
Im ε = 15
0
1
10
100
1000
time delay (ps)
Figure 6.3: Laser-induced lattice heating of a-GaAs at 0.9 Fth-a , tracked using the shift of
ε(ω) to lower photon energies. We use two criteria to quantify this shift: the photon energy
ω where Re[ε(ω)] = 10 (•) and the lower value of ω where Im[ε(ω)] = 15 (◦). Lines join the
data points to guide the eye.
as a function of time delay because the linear optical properties of hot a-GaAs have never
been measured. However, the redshift of ε(ω) indicates that lattice heating starts after
a few picoseconds — a time delay corresponding to the electron-lattice coupling time in
c-GaAs. Figure 6.3 shows the extent to which Re[ε(ω)] and Im[ε(ω)] have moved to lower
photon energy as a function of time — an indirect measure of the lattice temperature. The
plot suggests that aGaAs heats for tens of picoseconds, then starts to cool again on a time
scale of several hundred picoseconds, with the dielectric function moving back to higher
photon energies. The cooling is due to thermal diffusion, and the time scale of several 100
ps agrees with theoretical estimates for heat diffusion over tens of nanometers (the optical
probe depth) [131].
40
40
1.7 F th
a-GaAs
1.7 F th
30
200 fs
dielectric function
dielectric function
30
Im ε
20
10
Re ε
0
Im ε
20
10
Re ε
(a )
(b )
2.0
2.5
3.0
−10
1.5
3.5
2.0
photon energy (eV)
3.0
3.5
40
1.7 F th
30
Im ε
20
10
1.7 F th
Im ε
30
2 ps
dielectric function
dielectric function
2.5
photon energy (eV)
40
Re ε
0
8 ps
20
10
Re ε
0
(c )
−10
1.5
(d )
2.0
2.5
3.0
−10
1.5
3.5
2.0
photon energy (eV)
2.5
3.0
3.5
photon energy (eV)
40
40
1.7 F th
Im ε
Im ε
30
16 ps
dielectric function
30
dielectric function
500 fs
0
−10
1.5
20
10
0
400 ps
10
0
2.0
Re ε
(f )
2.5
3.0
3.5
1.7 F th
20
Re ε
(e )
−10
1.5
125
−10
1.5
photon energy (eV)
2.0
2.5
3.0
3.5
photon energy (eV)
after excitation at 1.7 Fth-a . The dielectric function of a-GaAs at room temperature [119]
appears for comparison in all panels (solid curves = Re[ε(ω)]; dashed curves = Im[ε(ω)]).
6.1.2
Medium Fluence Regime (F ≈ 1.7 Fth-a)
For a-GaAs we don’t find a clearly distinct intermediate region. Instead we present data
obtained at F = 1.7 Fth-a , on the border between the low and high fluence regimes. Figure
6.4 shows the evolution of ε(ω) at F = 1.7 Fth-a . The initial electronic effects are more
126
pronounced than at 0.9 Fth-a . At 2 ps after excitation, the peak value of Im[ε(ω)] has
increased to a value of 30, and both the peak of Im[ε(ω)] and the zero-crossing of Re[ε(ω)]
have moved to lower photon energy. The zero-crossing of Re[ε(ω)] continues to drop in
frequency over the following picoseconds, while Im[ε(ω)] takes on a Drude-like shape [22].
At 16 ps, Re[ε(ω)] is negative across the entire spectral range. Although Im[ε(ω)] still has a
peak at around 2.4 eV, the dielectric function looks like one produced by the Drude model
for a free electron gas. This dielectric function indicates that a metallic layer forms.
6.1.3
High Fluence Regime (F > 3.2 Fth-a)
Figure 6.5 shows the evolution of ε(ω) for a pump fluence of 5.7 Fth-a ; the response of
the dielectric function is similar at other fluences above 3.2 Fth-a . At 500 fs the dielectric
function already exhibits Drude-like shape, indicating that the material has undergone a
non-thermal semiconductor-to-metal transition. Where appropriate, the measured ε(ω) is
fit by a Drude model ε(ω). The free parameters used in the Drude fit are the plasma
frequency and the relaxation time. Between 1 ps and 16 ps, the relaxation time remains
constant at 0.12 fs while the plasma frequency decreases — to 16.0 eV at 2 ps and 14.0 eV
at 8 ps. At 16 ps, the best-fit value of τ decreases to 0.10 fs, while ωp = 14.0 eV. By this
time, the material is in thermal equilibrium and the dielectric function is that of the liquid
phase. Between 50 ps (not shown) and 400 ps, enough heat diffuses from the surface layer
to cause resolidification, which is visible in the dielectric function at 400 ps.
For time delays less than a few picoseconds, the material cannot be in thermal equilibrium because there is insufficient time for the electrons to equalize in temperature with
the lattice. The dielectric function at 500 fs indicates that a non-thermal liquid/disordered
solid has been generated with a dielectric function close to, but not identical to, that of
the thermal liquid (i.e. the dielectric function after 16 ps). To accurately determine the
time required for the semiconductor-to-metal transition, we use the dielectric function data
without correcting for the chirp.
Figure 6.6 shows the uncorrected dielectric function for five the pump-probe delays.
40
40
5.7 F th
a-GaAs
Im ε
(b )
30
200 fs
dielectric function
dielectric function
30
20
10
Re ε
0
Im ε
2.0
2.5
3.0
Re ε
−10
1.5
3.5
2.0
photon energy (eV)
2.5
3.0
3.5
photon energy (eV)
40
40
5.7 F th
(c )
30
Im ε
20
10
0
2.0
8 ps
20
Im ε
10
0
Re ε
−10
1.5
5.7 F th
(d )
30
2 ps
dielectric function
dielectric function
500 fs
10
0
−10
1.5
5.7 F th
20
(a )
2.5
3.0
Re ε
−10
1.5
3.5
2.0
photon energy (eV)
2.5
3.0
3.5
photon energy (eV)
40
40
5.7 F th
(e )
30
(f )
Im ε
30
16 ps
dielectric function
dielectric function
127
20
Im ε
10
0
5.7 F th
400 ps
20
10
Re ε
0
Re ε
−10
1.5
2.0
2.5
photon energy (eV)
3.0
3.5
−10
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
after excitation at 5.7 Fth-a . The curves in (a) and (f ) show Re[ε(ω)] (solid lines) and
Im[ε(ω)] (dashed lines) for a-GaAs at room temperature.[119] The curves in (b), (c), (d)
and (e) show Drude-model dielectric functions with plasma frequencies of 19.0, 16.0, 14.0
and 14.0 eV, and relaxation times of 0.10, 0.12, 0.12 and 0.10 fs, respectively.
The data at negative time delays correspond to the semiconducting phase; those at positive
time delays to the metallic phase. At a nominal time delay of 0 fs, the data show a transition
from the semiconducting to the metallic phase because the red part of the probe arrives at
128
probe chirp (fs)
30
20
−150
+150
0
+300 +450
a-GaAs
5.7 F th
−667 fs
−333 fs
0 fs
333 fs
667 fs
Re ε
SC
10
0
metal
(a )
−10
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
probe chirp (fs)
30
−150
+150
0
+300 +450
Im ε
20
10
−667 fs
−333 fs
0 fs
333 fs
667 fs
0
(b )
−10
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
Figure 6.6: Evolution of (a) Re[ε(ω)] and (b) Im[ε(ω)] of a-GaAs for excitation at 5.7 Fth-a ,
for time delays between −667 and 667 fs. The nonlinear probe chirp is indicated at the top
of each plot. The stated time delay is only correct at 2.35 eV; for lower photon energies,
the data correspond to earlier times. The semiconductor-to-metal transition is indicated by
() and () for Re[ε(ω)] and Im[ε(ω)] respectively.
the sample before the blue part. From the (nonlinear) chirp data at the top of the graph
we see that the transition occurs between 50 fs and +170 fs, giving a transition time of 220
fs. At 14 Fth, we find that the transition occurs in 170 fs; at 3.2 Fth, we obtain a transition
time of 380 fs.
6.2
129
Discussion
The presentation of the dielectric function data in the previous section already revealed
some of the fundamental physics behind the dynamics occuring in amorphous GaAs after
excitation by an ultrashort laser pulse. In the following section we present a more detailed analysis of the data. Time resolved dielectric function data is an extremely useful
source of information of carrier dynamics for crystalline materials within the framework of
a bandstructure (i.e. interactions between carriers at specific points in the Brillouin zone or
scattering events from one point in the Brillouin zone to another) as we have seen in Section
5.3.2. However, a Brillouin zone cannot be defined for a-GaAs due to the lack of crystal
symmetry. The following discussion focuses on the structural dynamics that accompany
optically induced semiconductor-to-metal transitions.
6.2.1
Low and Medium Fluence Regimes
The dynamics at low fluences are quite straightforward. From the discussion of the ε(ω)
data of c-GaAs in Section 5.3.2 we know that the effects of low fluence excitation are
equivalent to thermal heating. The low fluence ε(ω) data in Fig. 6.2 present therefore the
first measurement of the dielectric function of heated a-GaAs. To our knowledge there are
no other experimental values reported in the literature to date. Approximating the density
and specific heat of a-GaAs with the corresponding values for c-GaAs (i.e., ρ ≈ 5.3 g cm−3
and cv ≈ 250 mJ g−1 K−1 ) we can predict that the temperature of a-GaAs should rise by
550 K at an excitation fluence of 0.9 Fth-a .1 The temperature rise is probably less than
550 K, as energy also goes to other channels such as luminescence.
We can now use the ε(ω) data of heated a-GaAs to test the hypothesis from Section
5.3.2 that c-GaAs transforms to a hot disordered solid after excitations at fluences above 0.5
Fth-c . Figure 6.7 shows ε(ω) for c-GaAs taken at 4 ps for F = 0.7 Fth-c and ε(ω) measured
for a-GaAs for F = 0.9 Fth-a at the same time delay. The match between the two data sets
is excellent, indicating that c-GaAs indeed undergoes a nonthermal transition to heated a1
That is assuming all of the absorbed energy is converted to heat within the excited region.
130
30
dielectric function
GaAs
Im ε
20
a-GaAs
10
0
1.5
c-GaAs: 0.7 F th-c, 4 ps
a-GaAs: 0.9 F th-a, 4 ps
Re ε
2.0
2.5
3.0
3.5
photon energy (eV)
Figure 6.7: Comparison of the dielectric function of c-GaAs in the medium fluence regime
with that of a-GaAs measured following excitation with a low-fluence pulse. The plot shows
the dielectric function (• = Re[ε(ω)]; ◦ = Im[ε(ω)]) for c-GaAs at 4 ps after excitation by
0.7 Fth-c together with the dielectric function ( = Re[ε(ω)]; = Im[ε(ω)]) of a-GaAs at
4 ps after excitation by 0.9 Fth-a . The curves show previous measurements of ε(ω) (solid
line = Re[ε(ω)]; dashed line = Im[ε(ω)]) for a-GaAs at room temperature.[119]
GaAs after excitation with fluences at 0.5 Fth-c < F < 0.8 Fth-c . Hence FTRE does not only
add another piece of evidence to the hypothesis of nonthermal melting, but also provides
detailed information on the final phase of the material.
This finding also allows an estimate of the crystallization energy of GaAs. The
threshold pump fluence of 1 kJ/m2 corresponds to a deposited energy of about 16.4 kJ per
mole of GaAs atoms (where we count both Gallium and Arsenic ions as individual atoms).
Since we know the pump fluence at which c-GaAs disorders into heated a-GaAs (around
0.7 Fth-c ) and we also know the fluence which heats a-GaAs to the same temperature
(around 0.9 Fth-a ), we can determine the crystallization energy of GaAs as the difference
between those energies. We find an energy of about 10.0 kJ/mol to disorder the lattice of
c-GaAs into a-GaAs. Since we do have an error of about 30% in estimating the exact pump
131
fluence, this error directly translates into the error for the crystallization energy. There have
been a number of experimental and theoretical studies on the energy of crystallization of
tetrahedrally coordinated semiconductors. Donovan et al. reported values of 11.6 kJ/mol
for Ge and 11.9 kJ/mol in Si [132]. The value for Si was later re-measured to be 13.4
kJ/mol [133]. To our knowledge, there is no experimental value for GaAs reported yet. We
were also unable to find a theoretical estimate for the heat of crystallization of GaAs. It
is to be expected, however, that the crystallization energy is close to the values of Si and
Ge because GaAs is also tetrahedrally coordinated. For comparison, the cohesive energy of
c-GaAs has been found to be about 135 kcal/mol of molecules which corresponds to 1080
kJ/mol of atoms [134].
We also find an excellent match between the ε(ω) data for a-GaAs about a picosecond after an excitation at 1.7 Fth-a and the ε(ω) data for c-GaAs several picoseconds
after an excitation at 0.8 Fth-c . Both fluences are at the lower boundary of the high fluence
regime where semiconductor-to-metal transitions occur. As Fig. 6.8 shows, the dielectric
functions match almost perfectly. At this fluence, however, a-GaAs is not merely heated.
The similarity between the data therefore offers a different perspective on the structural
dynamics in both crystalline and amorphous GaAs than the low fluence data discussed
above. According to all current models of non-thermal melting the electronic excitation
causes bonds to break and permits atoms to move around [106, 135, 104, 111, 112]. Since
the amorphous material is already disordered it takes some time before the atoms starting
in the crystalline phase “catch up” with the ones which start in the amorphous phase. The
agreement between the 1 ps data for a-GaAs and the 8 ps data for c-GaAs is consistent
with this picture.
6.2.2
High Fluence Regime
In the high fluence regime, the ε(ω) data provide evidence for a non-thermal, structurally
driven semiconductor-to-metal transition. We observe a metallic phase within a few picoseconds, just as in c-GaAs. For both materials, the rapid non-thermal transition happens
132
40
GaAs
dielectric function
30
c-GaAs: 0.8 F th-c, 8 ps
a-GaAs: 1.7 F th-a, 1 ps
20
Im ε
10
Re ε
0
−10
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
Figure 6.8: Dielectric function (• = Re[ε(ω)]; ◦ = Im[ε(ω)]) of c-GaAs at 8 ps after excitation
at 0.8 Fth-c together with the dielectric function ( = Re[ε(ω)]; = Im[ε(ω)]) of a-GaAs
at 1 ps after excitation at 1.7 Fth-a .
more quickly as the fluence increases. Amorphous GaAs undergoes a transition to the
metallic state within only 170 fs for 14 Fth-a . Crystalline GaAs undergoes its ultrafast
phase transition almost as quickly (260 fs) when the excitation fluence is about 17 times
the threshold for permanent change in c-GaAs [118]. Inspite of these very fast times, we
still observe a gradual drop in the zero-crossing of Re[ε(ω)] in both c-GaAs and a-GaAs.
This gradual drop adds additional credence to the claim that the semiconductor-to-metal
transition is driven by structural as well as electronic effects. As discussed in Section 5.3.2
a gradual transition cannot be solely electronically driven, because electronic effects are
largest immediately after excitation. Furthermore, the observed structural changes must
be non-thermal, since the semiconductor-to-metal transition occurs in less than 2 ps for
excitations in the high fluence regime, i.e. before the carriers have fully equilibrated with
the lattice.
The ε(ω) data does not only allow the observation of the ultrafast phase transition,
133
plasma frequency (eV)
20
0.9 Fth
15
1.2 Fth
1.3 Fth
10
1.6 Fth
2.1 Fth
relaxation time
τ = 0.15 fs
5
c-GaAs
0
0
5
10
15
20
time delay (ps)
Figure 6.9: Plasma frequency as a function of time delay in the high fluence regime for
c-GaAs: 0.9 Fth-c (◦), 1.2 Fth-c (•), 1.3 Fth-c (), 1.6 Fth-c () and 2.1 Fth-c (✷). The Drude
model fits that produce these values of ωp all take τ = 0.15 fs. The lines joining the data
on the plot are guides to the eye.
but also enables us to track the evolution of the metallic state. Figure 6.9 shows the plasma
frequency for c-GaAs as a function of time delay for data sets well-described by a Drude
model as derived from the ε(ω) data taken by Huang et al. [118]. Interestingly, the plasma
frequency is smaller for higher fluences and also decreases with time delay. The decrease
in plasma frequency with time is most likely caused by two effects. One is diffusion of
carriers into the material. Because the probe beam is only sensitive to the first 10 nm of the
metallic material, carriers can diffuse out of the probed region very rapidly. Alternatively,
rapid Auger recombination can lower the carrier density sufficiently to observe a lower
plasma frequency.
The second observation, namely that the plasma frequency decreases with increasing fluence, is more difficult to explain. The carrier density should be independent
of excitation fluence because the free carriers that generate the Drude dielectric function
134
plasma frequency (eV)
30
a-GaAs
25
relaxation time
τ = 0.10 fs
20
3.2 Fth
15
5.7 Fth
(a )
10
0.1
14 Fth
1
10
100
time delay (ps)
Figure 6.10: Plasma frequency ωp as a function of time delay for different excitations in
the high fluence regime for a-GaAs: 3.2 Fth-a (◦), 5.7 Fth-a (•) and 14 Fth-a (). The Drude
model fits to the dielectric function data that produce the values of ωp all take the relaxation
time to be τ = 0.10 fs. The lines on the plots are guides to the eye.
are produced mainly by the band-gap collapse and not the original excitation. Hence, the
electron density should always correspond to the total valence electron density of GaAs.
One possible explanation lies in the fact that the plasma frequency depends not only on the
carrier density but also on the effective masses of the free carriers. According to Eq. 2.17,
the plasma frequency decreases with increasing effective mass of the carriers.2 Future simulations might be able to calculate effective masses and plasma frequencies at different
fluences, in order to explain the dynamics in Fig. 6.9.
The dynamics of the metallic phase of a-GaAs is strikingly similar to the dynamics
observed in c-GaAs. Figure 6.10 shows the plasma frequency derived from Drude fits as
2
In current models for standard metals, the electron mass is always taken as the free electron mass which
leads to good results. The metallic state of GaAs and other highly excited semiconductors described here
is quite different from a normal metal, however, where potentially the electrons could have effective masses
different from the free electron mass.
135
a function of time and fluence. For consistency with Fig. 6.9 we keep the relaxation time
fixed at ωp = 0.10 fs.3 The plasma frequency decreases over time at a given fluence, and
decreases roughly with increasing fluence. Thus, this pattern of behaviour is independent
of the original structure of the material. In fact, we also found non-thermal transitions
to a metallic state, where the plasma frequency decreases with time and fluence, in our
experiments on GeSb thin films described in Chapter 7. This univeral pattern of behavior
for ultrafast semiconductor-to-metal transitions [136] invites new theories illuminating the
rich body of ε(ω) data.
6.3
Conclusions
The time-resolved dielectric function provides a wealth of new information on electron
dynamics and structural changes in a-GaAs following femtosecond laser excitation. At low
fluences, we see evidence of heating in a-GaAs, as previously observed for c-GaAs. To our
knowledge, the data represent the first measurement of the dielectric function of heated
a-GaAs. We find largely different values for the threshold fluences for permanent damage
in a-GaAs vs. c-GaAs: 0.1 kJ/m2 vs. 1.0 kJ/m2 . This order of magnitude difference can
be attributed to the fact that it is harder to drive a crystalline-to-amorphous transition
than an amorphous-to-amorphous transition. The boundary between the low and high
fluence regimes appears to be very different relative to the respective threshold fluences
for permanent damage — 0.5 – 0.8 Fth-c for c-GaAs and around 3.2 Fth-a for a-GaAs. In
absolute fluence terms, however, these values are close to each other indicating that the
fluence needed to drive the semiconductor-to-metal transition is independent of the original
structure of the material. Furthermore we estimate the crystallization energy of GaAs to
be about 3.1 kcal per mole of GaAs molecules. For fluences well above Fth-a , we confirm
that a-GaAs undergoes a semiconductor-to-metal transition to a metallic phase similar to
that reached by the crystalline material. Both cases show a decrease in plasma frequency
3
Note, that for some time delays slightly different relaxation times result in marginally better fits, as
shown in Fig. 6.5.
136
over time after the metallic phase appears. At the highest fluence, 14 Fth-a , the transition
to the metallic state in a-GaAs takes only 170 fs. This is still long enough to indicate that
the cause is an electron-induced, non-thermal structural change, similar to that observed in
c-GaAs. Even at these high fluences, we do not observe a collapse in the band gap due to
the free carriers themselves. Future experiments could investigate the possibility of a purely
electronically driven band gap collapse — a collapse within the duration of the pump pulse
— by probing even higher excitation densities.
Chapter 7
Ultrafast Phase Transitions in
GeSb Thin Films
In the past decade there has been considerable interest in laser-induced phase transitions in
Sb-rich GeSb thin films because of potential application of such films as media for optical
data storage devices [137, 138, 139]. The key features which favor GeSb films for this application are the large reflectivity difference between the amorphous and crystalline phase of
GeSb (up to 20 %) and the fact that it is possible to optically induce transitions between
those phases in less than a nanosecond [139]. In this chapter we discuss ε(ω) data illuminating the phase changes in a-GeSb thin films induced by femtosecond laser excitation. The
transition from the low reflectivity amorphous to the highly reflective crystalline phase of
a-GeSb films is studied with fs time resolution. The results reveal an ultrafast transition to
a new non-thermodynamic phase which is not c-GeSb, as previously believed [82].
7.1
Motivation
Re-writable optical data storage is an area of prime interest to today’s information technology industry. The current industry standard is the rewriteable DVD (digital versatile disc)
that is based on phase transformation technology. Basically, the technology relies on the
137
Chapter 7: Ultrafast Phase Transitions in GeSb Thin Films
138
fact that ε(ω) of the crystalline phase of a material is sufficiently different from ε(ω) of its
amorphous phase, permitting a detectable optical contrast.1 One can then use a focused
laser beam to drive the material from one phase into the other, thereby writing or rewriting
bits in the storage medium. The material of choice in today’s commercially available DVD
is an AgInSbTe-alloy. The melting temperature of this alloy is around 600o C. Therefore,
if the originally (poly-)crystalline material is heated above this temperature and consecutively cooled down rapidly enough, the material will solidify in the amorphous phase. On
the other hand, the critical temperature for annealing to be efficient is about 200o C. If an
amorphous sample is kept above 200o C for a sufficiently long time (longer than the crystallization time), the material undergoes a transformation to the crystalline phase. Figure 7.1
schematically shows the physical process of creating an amorphous and a crystalline region.2
Since the speed at which one can write and rewrite individual bits onto the medium is crucial in determining the overall speed of such an optical storage medium, researchers and
engineers have thought of many ways to minimize the phase transition times. The cooling
rate which determines the time for the amorphization step can be optimized by the disc
structure design. The crystallization time can be optimized by the chemical composition
of the alloy. However, these improvements cannot go beyond the fundamental times for
the structural phase transitions. AgInSbTe-alloy based DVD media are therefore limited to
writing speeds of about 25 Mbit/s.
To further enhance writing speeds, researchers in the late 1980’s started to consider metal and semi-metal-alloys because of their extremely fast crystallization speeds.
Especially Sb-rich alloys, such as InSb, GaSb and GeSb attracted great interest. In early
experiments, crystallization with pulses as short as 10 ns was observed [137, 138]. A breakthrough was achieved by Afonso et al. in 1992 where crystallization of Sb-rich a-GeSb thin
films was observed with pulses of only 500 fs duration [139]. Furthermore, the researchers
showed that it is possible to cycle the material back and forth between cyrstalline and
1
That of course implies that both, the crystalline and amorphous phase of the material are stable at
normal conditions.
2
The figure was published by Phillips, Inc. in a white-paper — http://www.dvdrw.org/whitepapers.html.
Creating amorphous regions
139
Creating polycrystalline regions
temp
temp
o
~600 C
melting point
o
~600 C
melting point
tcooling
o
crystallisation
temperature
~200 C
tpass
o
~200 C
time
tcrystal
crystallisation
temperature
time
tcrystal
Single sided disc
(Not to scale)
Label
Polycarbonate
2P resin
Reflective layer
Upper dielectric layer
Recording layer (Ag-In-Sb-Te)
Lower dielectric layer
Polycarbonated disc substrate
Laser beam
Groove
Figure 7.1:
Schematic illustration of the phase transformation
in rewritable DVDs.
From white-paper published by Phillips,
http://www.dvdrw.org/whitepapers.html.
mechanisms
Inc.
—
amorphous phase using different excitation fluences [140]. The reflectivity difference between crystalline and amorphous phase was determined to be 15–20% enabling clear optical
distinction between the two phases.
The process allowing such fast crystallization speeds is different from standard
annealing. To understand the process it is important to note that the a-GeSb films are
140
very thin — on the order of 50 nm. For fluences just above the melting threshold for aGeSb the material only melts within a depth shorter than the film thickness. Because the
thermal conductivity of the film is very high, the molten film is rapidly quenched, leading
to resolidification in the amorphous state. If, on the other hand, the excitation fluence
is much higher than the melting threshold, the entire film melts. The quenching rate is
then slowed down significantly due to the low thermal conductivity of the glass substrate,
allowing crystallization of the material. Heat takes about 200 ps to diffuse across the 50
nm layer, leading to heat loss to the subtrate for pulses exceeding this duration. A detailed
study by Solis et al. shows a gradual increase of crystallization threshold fluence for pulses
with duration above 200 ps [141]. Interestingly, in the same paper the researchers report
a constant crystallization threshold for pulses of durations between about 200 ps and 800
fs and a decrease of the threshold for durations below 800 fs. The authors conclude that
electronic effects are partly responsible for the crystallization at excitation with pulses
shorter than 800 fs.
This finding triggered a further experiment on the crystallization process of aGeSb thin films. Sokolowski-Tinten et al. used femtosecond microscopy (see Section 4.3.6)
to monitor the phase changes induced by irradiation with 100 fs laser pulses [82]. The
researchers report an ultrafast amorphous-to-crystalline transition (within about 200 fs) in
Sb-rich a-GeSb thin films based on single-angle and single-wavelength reflectivity measurements. Figure 7.2 shows the crucial results from the experiments in Ref. [82]. As indicated
in the graph, the reflectivity of the material approaches the reflectivity of the crystalline
phase on time scales on the order of 100 ns if the excitation fluence is above a threshold
value denoted by Fcr . Sokolowski et al. determined this crystallization threshold to Fcr =
0.19 kJ/m2 . For fluences below Fcr the material returns to its original amorphous state on
similar time scales.
These findings are completely in line with the abovementioned results where the
dependence of Fcr on pulse duration were studied [141]. The surprising result from Fig. 7.2
is the behavior of the reflectivity on the fs time scale. Within 200 fs, the reflectivity reaches
141
time zero
0.75
GeSb
crystalline
reflectivity
0.70
0.65
A: 2.4 Fcr
B: 1.1 Fcr
0.60
C: 0.6 Fcr
amorphous
0.55 −
1
10
1
10
2
10
3
10
4
10
5
10
time delay (ps)
Figure 7.2: Femtosecond time resolved reflectivity transients of GeSb thin films for three
different pump fluences from Ref. [82]: • = 2.4 Fcr ; ✸ = 1.1 Fcr ; = 0.6 Fcr.
exactly the value for c-GeSb for all fluences above Fcr.3 These findings led the authors
to conclude that a-GeSb thin films undergo an ultrafast disorder-to-order transition. This
hypothesis motivated us to ask a fundamental question: How can the atoms move fast
enough to assume an ordered structure within 200 fs? Having developed a powerful optical
tool to learn about material phase dynamics on the fs time scale (FTRE — see Section 4.3)
we decided to try to find answers to these questions. In the following sections we present
and analyze our fs time resolved ε(ω) data on a-GeSb thin films.
3
The reflectivity subsequently drops and then returns to the value of c-GeSb on much longer time scales
of nanoseconds indicative of melting and resolidification.
7.2
142
We used a sample from the same batch as Sokolowski-Tinten and co-workers [82]. The 50nm thick, amorphous Ge0.06 Sb0.94 film samples were prepared by Solis et al. at the Instituto
de Optica in Madrid. The films were deposited onto a fused silica subtrate in a multitarget
DC magnetron sputtering chamber. The experiments described in this chapter naturally
involve excitation fluences above the threshold for permanent change in the material. Hence,
we used the single-shot setup for FTRE as described in Section 4.3. Furthermore, we have
to account for multiple layers in determining the reflectivity of the thin film samples as
described in Section 4.3.2. The films consist of three layers: an oxide layer, the film itself
and the glass substrate. The dielectric constant of SbO2 was previously measured to be ε
= 4.2 [142]. The two angles used in this experiment were 52.85o and 79.40o. Matching the
correction factors for different materials as described in Section 4.3, we found the thickness
of the SbO2 -layer to be 1.25 nm.
7.2.1
Dielectric Functions of Unexcited a-GeSb and c-GeSb
Figure 7.3 shows the measured ε(ω) for unpumped a-GeSb. The error bars are based on an
estimated 5% uncertainty in matching the correction factors and obtained in the same way as
described in Section 6.1. As indicated in the figure, the match between the data obtained in
our setup and previously measured data obtained using continuous wave ellipsometry [143]
is excellent showing once more the accuracy of our technique. We are not only concerned
with the amorphous phase but also with the crystalline phase of GeSb. Figure 7.4 shows
unpumped ε(ω) data obtained from spots on the sample which were previously irradiated
with light pulses exceeding the crystallization threshold Fcr . These regions were shown to
be crystalline in separate SEM measurements [143]. The graph also shows previous results
obtained using continuous wave ellipsometry on polycrystalline [142] and crystalline Sb
[21]. The measured dielectric function of c-GeSb is similar but not identical, to that of
crystalline or polycrystaline Sb. The Ge concentration of 6% does have a non-negligible
143
40
GeSb
0.6 Fth
dielectric function
30
20
10
0
−10
(a)
−20
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
Figure 7.3: Dielectric function of unexcited a-GeSb thin film (• = Re[ε(ω)]; ◦ = Im[ε(ω)]).
The curves show previous measurements of ε(ω) (solid = Re[ε(ω)]; dashed = Im[ε(ω)]) of
a-GeSb using ellipsometry with continuous wave light [143].
effect on the optical properties of the alloy. Henceforth, we take our measurements of ε(ω)
in the crystalline region to be ε(ω) for c-Ge0.06 Sb0.94 .
Figure 7.5 shows three images of the Ge0.06 Sb0.94 thin film sample used for the
experiments described in this chapter. There is a clear contrast between the irradiated (and
thereby crystallized) regions and the originally amorphous material. For normal incidence
and at 2 eV photon energy (chosen representatively for the visible wavelength range), the
reflectivity of the amorphous phase is about 55%, vs. 70% for the crystalline phase, making
the difference about 15% and enabling clear optical distinction between the phases. The
black regions are spots where the laser fluence was high enough to ablate material. We
determine the crystallization threshold to Fcr = 0.22 kJ/m2 , in good agreement with the
previously measured value of 0.19 kJ/m2 [82]. The threshold for ablating material was
measured to be about 3.5 Fcr. In the following presentation, we categorized the data into
three regimes: (1) below Fcr; (2) above Fcr ; (3) above 3.5 Fcr .
144
50
c-Ge 0.06 Sb0.94
dielectric function
40
30
Im ε
20
10
0
Re ε
−10
−20
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
Figure 7.4: Dielectric function of crystallized spot on unexcited c-GeSb thin film sample (• =
Re[ε(ω)]; ◦ = Im[ε(ω)]). The curves show previous measurements of ε(ω)of polycrstyalline
Sb (solid = Re[ε(ω)]; dashed = Im[ε(ω)]) [142] and single-crystalline Sb (dash-dotted =
Re[ε(ω)]; dotted = Im[ε(ω)]) [21] at room temperature using ellipsometry with continuous
wave light.
7.2.2
Below Fcr
Figure 7.6 shows the temporal evoution of ε(ω) for a-GeSb excited at F = 0.6 Fcr . We
omit error bars for sake of clarity of the figure in this and the following graphs. Whereever
errors bars become an appreciable factor we discuss them separately. The shaded symbols
indicate the data for the previously shown time delay respectively. The solid and dashed
curves represent the cw ellipsometry results for a-GeSb [143]. The ε(ω) changes rapidly
(within 200 fs) as indicated in Fig. 7.6(a) − (c). Im[ε(ω)] falls rigidly across the observed
wavelength range for about 200 fs. Representatively taking the values at 2.5 eV, Im[ε(ω)]
falls from its original value of about 19 to 15 at 200 fs as indicated in Fig. 7.6(c). Re[ε(ω)]
also starts downshifting immediately after excitation. The low frequency end continuously
falls from originally about 10 at 1.7 eV to about 0 at 200 fs. Both, Im[ε(ω)] and Re[ε(ω)]
145
x2.5
500 µm
250 µm
x5
x40
30 µm
Figure 7.5: Microscope images of irradiated spots of GeSb thin films.
remain stable at these new values for more than 5 ps. After 5 ps Re[ε(ω)] starts falling
further to about -7 at 1.7 eV at 20 ps. It stabilizes there until the last measured time delay
of 475 ps as Fig. 7.6(f ) shows. As previously observed for c-GaAs and a-GaAs (see Sections
5.3.2 and 6.1.2), the zero-crossing of Re[ε(ω)] progressively falls for tens of ps resulting in a
Drude-like ε(ω) about 20 ps after excitation.
7.2.3
Above Fcr
For excitations at fluences above the crystallization threshold, the dynamics of ε(ω) become
more pronounced, as indicated in Fig. 7.7 for F = 1.6 Fcr . Again, the shaded symbols
indicate the data for the previously shown time delay respectively. The solid and dashed
curves represent the cw ellipsometry results for a-GeSb [143]. In addition we show the ε(ω)
146
100 fs
0 ps
40
40
Im ε
GeSb
0.6 Fcr
30
dielectric function
dielectric function
30
20
10
Re ε
0
Im ε
20
10
Re ε
0
−10
−10
(b)
(a)
−20
1.5
2.0
2.5
3.0
3.5
−20
1.5
2.0
200 fs
3.5
40
Im ε
GeSb
0.6 Fcr
30
dielectric function
30
dielectric function
3.0
5 ps
40
20
10
Re ε
0
−10
Im ε
GeSb
0.6 Fcr
20
10
Re ε
0
−10
(c)
−20
1.5
(d)
2.0
2.5
3.0
3.5
−20
1.5
2.0
photon energy (eV)
2.5
3.0
3.5
photon energy (eV)
20 ps
475 ps
40
40
Im ε
20
GeSb
0.6 Fcr
30
dielectric function
30
dielectric function
2.5
photon energy (eV)
photon energy (eV)
10
Re ε
0
−10
Im ε
20
GeSb
0.6 Fcr
10
Re ε
0
−10
(e)
−20
1.5
GeSb
0.6 Fcr
(f)
2.0
2.5
photon energy (eV)
3.0
3.5
−20
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
Figure 7.6: Dielectric function of a-GeSb thin film for an excitation with a fluence of 0.6
Fcr : • = Re[ε(ω)], ◦ = Im[ε(ω)]. The solid and dashes curves show real and imaginary part
of ε(ω) from previous cw measurements [143]. The gray-shaded symbols indicate the ε(ω)
for the previous time delay respectively.
data for c-GeSb (obtained from unexcited crystallized regions of our sample in a different
measurement — see Section 7.2.1) as dash-dotted (real part) and dotted (imaginary part)
curves. Im[ε(ω)] downshifts faster than for low fluence excitations. Within 100 fs Im[ε(ω)] at
2.5 eV falls from 19 to about 15 as indicated in Fig. 7.7(a) − (b). There is some oscillatory
147
0 ps
100 fs
40
40
Im ε
GeSb
1.6 Fcr
30
dielectric function
dielectric function
30
20
10
0
Re ε
−10
−20
1.5
2.5
3.0
3.5
10
0
−20
1.5
Re ε
(b)
2.0
photon energy (eV)
200 fs
Im ε
GeSb
1.6 Fcr
30
dielectric function
dielectric function
3.5
5 ps
20
10
0
Re ε
−10
Im ε
20
GeSb
1.6 Fcr
10
0
Re ε
−10
(c)
2.0
2.5
3.0
3.5
−20
1.5
(d)
2.0
photon energy (eV)
2.5
3.0
3.5
photon energy (eV)
20 ps
475 ps
40
40
Im ε
20
GeSb
1.6 Fcr
30
dielectric function
30
dielectric function
3.0
40
30
10
0
Re ε
−10
−20
1.5
2.5
photon energy (eV)
40
−20
1.5
GeSb
1.6 Fcr
20
−10
(a)
2.0
Im ε
Im ε
GeSb
1.6 Fcr
20
10
0
Re ε
−10
(e)
2.0
2.5
photon energy (eV)
3.0
3.5
−20
1.5
(f)
2.0
2.5
3.0
3.5
photon energy (eV)
of ε(ω) from previous cw measurements [143] and the dash-dotted and dotted curves show
the real and imaginary part of ε(ω) of c-GeSb as measured on the crystallized regions of our
sample. The gray-shaded symbols indicate the ε(ω) for the previous time delay respectively.
behavior at the low frequency end for a few picoseconds until Im[ε(ω)] stabilizes at the
values it had taken at 100 fs. The dynamics of Re[ε(ω)] are also more pronounced than
in the low fluence case. The zero-crossing downshifts much more rapidly and is out of our
148
wavelength range within 100 fs. The dielectric function looks Drude-like as early as 200 fs
after excitation. In fact, Re[ε(ω)] does not change after it has reached the Drude-like shape
at 200 fs. There are no further detectable dynamics over our entire detection range of 475
ps. It is important to note that ε(ω) after 200 fs is reasonably close, but not equivalent to
ε(ω) of c-GeSb. We will come back to this point in our analysis of the data in Section 7.3.
7.2.4
Above 3.5 Fcr
The evolution of ε(ω) for fluences above the threshold for cratering is actually quite similar
to the dynamics for F just above Fcr . Figure 7.8 shows the evolution of ε(ω) for a pump
excitation at 3.5 Fcr . Again, the shaded symbols indicate the data for the previously shown
time delay respectively. The solid and dashed curves represent the cw ellipsometry results
for a-GeSb [143]. In addition we show the ε(ω) data for c-GeSb (obtained from unexcited
crystallized regions of our sample in a different measurement — see Section 7.2.1) as dashdotted (real part) and dotted (imaginary part) curves. For F = 3.5 Fcr the dynamics of
Im[ε(ω)] are more pronounced than for F = 1.6 Fcr . Im[ε(ω)] continuous falling for several
ps — the value at 2.5 eV reaches values as low as 10 at 10 ps as indicated in Fig. 7.8(d).
For longer time delays of hundreds of ps Im[ε(ω)] recovers and rises again. In fact, Im[ε(ω)]
starts approaching the Im[ε(ω)] of the c-GeSb phase at 475 ps as Fig. 7.8(f ) shows. The
zero-crossing of Re[ε(ω)] rapidly redshifts out of our wavelength range. Just as in the case
of F = 1.6 Fcr the ε(ω) looks very Drude-like within 200 fs after excitation as indicated in
Fig. 7.8(b). Re[ε(ω)] does not undergo further dynamics for hundreds of ps. Only at the last
time delay of 475 ps does Re[ε(ω)] change as shown in Fig. 7.8(f ). It approaches Re[ε(ω)]
of the c-GeSb phase just as Im[ε(ω)], as described above. The real part, after having been
stable in the fairly straight Drude-shape it obtained within 200 fs, takes on a significant
curvature and actually crosses the zero-line again at about 1.9 eV. These features make
Re[ε(ω)] indeed similar to the Re[ε(ω)] of the crystalline phase.
149
0 ps
200 fs
40
40
Im ε
GeSb
4.0 Fcr
30
dielectric function
dielectric function
30
20
10
Re ε
0
−10
Im ε
20
10
Re ε
0
−10
(a)
−20
1.5
(b)
2.0
2.5
3.0
3.5
−20
1.5
2.0
photon energy (eV)
2.5
5 ps
20 ps
Im ε
20
GeSb
4.0 Fcr
30
dielectric function
dielectric function
3.5
40
30
10
0
Re ε
−10
Im ε
20
0
−10
−20
1.5
GeSb
4.0 Fcr
10
(c)
Re ε
(d)
2.0
2.5
3.0
3.5
−20
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
photon energy (eV)
475 ps
100 ps
40
40
Im ε
20
GeSb
4.0 Fcr
10
0
−10
Re ε
2.5
photon energy (eV)
20
Im ε
10
0
Re ε
−10
(f)
(e)
2.0
GeSb
4.0 Fcr
30
dielectric function
30
dielectric function
3.0
photon energy (eV)
40
−20
1.5
GeSb
4.0 Fcr
3.0
3.5
−20
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
of ε(ω) from previous cw measurements [143] and the dash-dotted and dotted curves show
the real and imaginary part of ε(ω) of c-GeSb as measured on the crystallized regions of our
sample. The gray-shaded symbols indicate the ε(ω) for the previous time delay respectively.
7.3
Discussion
Even though we excite carrier densities of about 1022cm−3 assuming linear absorption of
the 800 nm pump pulse, there is no evidence of electronic effects in our data. This lack
150
of electronic effects is mainly due to the fact that the original state of the material is
amorphous, washing out clear electronic transition features (recall the discussion of a-GaAs
in Section 6.2). In addition, the most pronounced electronic effects take place close to the
pump energy of 1.5 eV which is at the edge of our detection range.4 The following discussion
therefore focuses on structural dynamics.
7.3.1
Phase Dynamics Below Fcr
The rapid change in ε(ω) within the first 200 fs is most likely due to electronically induced
non-thermal structural effects along the lines of non-thermal melting, as described in Section
5.3.1. The dielectric function remains fairly constant over the period from 200 fs to about
5 ps. The reason for this is the fact that it takes several picoseconds for the hot electronic
system to equilibrate with the still cold lattice. 5 ps is a very reasonable time for this
equilibration to take place. After the electron-lattice system thermalizes, the ε(ω) is affected
by thermal effects.
We propose that the changes in ε(ω) starting about 5 ps after excitation are due to
an exponential temperature gradient in the material caused by the linear absorption of the
pump light. The material closest to the surface facing the laser is heated above the melting
temperature. The viscosity then exponentially increases into the film until it reaches practically the value for undisturbed a-GeSb at the back-end. It is very hard to model the optical
response of such a system, because it requires knowledge of the temperature dependence
of the full spectral ε(ω). Instead, we model the temporal evolution of the reflectivity by
assuming a liquid-solid interface that propagates into the material. In this model we only
need to know the ε(ω) of liquid GeSb (we take the ε(ω) 100 ps after excitation — when
thermal equilibrium is established) and a-GeSb. To gain a quantitative understanding of
this situation we compare the measured reflectivity spectra to a multilayer reflectivity model
(see Section 2.2.2). We assume a five layer system consisting of air, the oxide layer (1.25
nm of SbO2 ), a layer of l-GeSb (taken as the phase 100 ps after excitation) of thickness x
4
We actually completely omit them in the previous presentation of the ε(ω) data.
151
0.6
reflectivity
0.5
R 1 = R p (52.85ο)
0.4
R 2 = R p (79.4ο)
0.3
0.2
0.6 F cr
0.1
20 ps
(a )
0.0
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
liquid thickness (nm)
15
10
5
0.6 F cr
(b )
0
1
10
100
1000
time delay (ps)
Figure 7.9: Liquid layer model for the evolution of the reflectivity of a-GeSb thin films
following excitation at fluences below Fcr for 52.85o and 79.45o angle of incidence. (a) The
solid lines represent the reflectivity obtained using a five layer model: air, oxide, 12 nm
of l-GeSb (taken as phase 100 ps after excitation), 38 nm of a-GeSb and glass substrate.
The dash-dotted lines show the reflectivity spectra for c-GeSb. and represent the
reflectivity data obtained 20 ps after excitation at 0.6 Fcr. (b) Temporal evolution of liquid
layer thickness x. The curve is a guide to the eye.
nm, the remaining 50−x nm thick layer of a-GeSb, and the glass substrate. Figure 7.9(a)
shows the 20-ps data and reflectivity spectra according to the model, where x = 12 nm.
The two data sets agree very well indicating that at 20 ps the liquid layer is 12 nm thick.
Figure 7.9(b) shows the temporal evolution of x as a result of reflectivity fits to different
152
time delays. The melt front propagates into the material from 5 ps to 20 ps, increasing the
layer thickness from about 5 nm to 13 nm. This rate is consistent with a heat diffusion
rate of about 10 nm within tens of ps [143]. Cooling and resolidification starts at about 100
ps, which is extremely fast compared to typical resolidification times, but consistent with
previous observations by Siegel et al. [144].
7.3.2
Phase Dynamics Above Fcr
One of our main motivations to study a-GeSb thin films was the report by SokolowskiTinten et al. on ultrafast crystallization within 200 fs. Having the fs time resolved ε(ω) at
hand, we can make a more accurate comparison to the c-GeSb phase than simply comparing
single-angle, single-wavelength reflectivities. Figure 7.10 shows the ε(ω) data for the a-GeSb
thin film 200 fs after excitation at F = 1.6 Fcr and F = 4.0 Fcr. The data for both fluences
are almost identical to each other but clearly distinct from ε(ω) of c-GeSb. Hence, the
material evidently does NOT crystallize after 200 fs, but undergoes a non-thermal change
to a new phase. The justification for calling this non-thermal state of the material a phase
lies in the fact that the ε(ω) at 200 fs is fluence independent which is a very strong indicator
for a new phase. This new phase is metallic, as indicated in the figure by a Drude-fit with
a plasma frequency of 14.5 eV and a relaxation time of 0.18 fs. This plasma frequency
corresponds roughly to the valence electron density in a-GeSb indicating that the bandgap
has fully collapsed. Hence we conclude that a-GeSb thin films undergo an ultrafast phase
transition from the solid amorphous phase to a new nonthermal amorphous phase with
metallic properties upon excitation with fluences above Fcr .
There is, of course, the very distant possibility that this new phase is in fact a
new crystalline phase which is different from the c-GeSb obtained at large time delays after
excitation. Only a time resolved X-ray diffraction experiment can provide definite proof of
the exact structure of this nonthermal ultrafast phase. But it is extremely unlikely that the
material undergoes a disorder-to-order transition within such a short time.
153
200 fs
40
GeSb
1.6 Fcr
4.0 Fcr
dielectric function
30
20
10
0
−10
−20
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
Figure 7.10: Dielectric function of a-GeSb thin film 200 fs after excitation at F = 1.6 Fcr
and F = 4.0 Fcr (• = Re[ε(ω)]; ◦ = Im[ε(ω)]). The ε(ω) for F = 4.0 Fcr is indicated by
the shaded symbols. The solid and dashed curves show the real and imaginary parts of a
Drude fit (ωp = 14.5 eV; τ = 0.18 fs). The dash-dotted and dotted curves show the real
and imaginary parts of ε(ω) for the crystalline phase of the GeSb thin film.
This finding, even though comforting for one’s physical intuition,5 now poses the
question of compatibility between our experiments and the experiments by SokolowskiTinten et al. [82]. A direct comparison between the results is not immediately possible
because we measured the full ε(ω) as opposed to single-angle, single-color reflectivities.
Knowing the full ε(ω), however, allows us to calculate the transient reflectivity response of
the a-GeSb thin film for any angle, polarization, or wavelength. Figure 7.11 shows a variety
of calculated reflectivity transients. It is extremely instructive to compare Fig. 7.11(a)
with the data from Ref. [82] displayed in Fig. 7.2. The agreement is astonishing. For zero
degree angle of incidence, at a photon energy of 2.0 eV, the reflectivity of the material does
5
It is unlikely that materials undergo ultrafast disorder-to-order transformations upon impulsive laser
excitation.
time zero
time zero
2.01 eV, 0°
0.70
0.65
(d )
crystalline
crystalline
0.65
4.0 Fcr
1.6 Fcr
0.6 Fcr
0.60
reflectivity
reflectivity
3.00 eV, 0°
0.70
(a )
0.55
4.0 Fcr
1.6 Fcr
0.6 Fcr
0.60
0.55
amorphous
0.50 −2
10
−1
10
1
10
amorphous
GeSb
102
0.50 −2
10
103
−1
10
time delay (ps)
time zero
1
10
102
103
time delay (ps)
time zero
2.01 eV, 52.85°
0.55
3.00 eV, 52.85°
0.55
(b )
crystalline
0.50
0.50
4.0 Fcr
1.6 Fcr
0.6 Fcr
0.45
reflectivity
reflectivity
154
(e )
crystalline
4.0 Fcr
1.6 Fcr
0.6 Fcr
0.45
0.40
amorphous
0.40
0.35
amorphous
0.35 −2
10
−1
10
1
10
102
0.30 −2
10
103
−1
10
time delay (ps)
time zero
time zero
2.01 eV, 79.40°
0.40
(c )
reflectivity
reflectivity
102
103
3.00 eV, 79.40°
(f )
crystalline
0.45
crystalline
0.25
4.0 Fcr
1.6 Fcr
0.6 Fcr
0.20
0.40
4.0 Fcr
1.6 Fcr
0.6 Fcr
0.35
0.30
0.15
amorphous
amorphous
0.10 −2
10
10
0.50
0.35
0.30
1
time delay (ps)
−1
10
1
10
time delay (ps)
102
103
0.25 −2
10
−1
10
1
10
102
103
time delay (ps)
Figure 7.11: Calculated reflectivity transients at various angles and energies for three different excitation fluences: • = 4.0 Fcr ; ✸ = 1.6 Fcr ; = 0.6 Fcr.
in fact reach exactly the crystalline level within 200 fs, if and only if the pump fluence
exceeds Fcr . It then dips again before it reaches its final (crystalline) value at time scales of
155
nanoseconds. Unfortunately, our maximum time delay was insufficient to observe the final
crystalline state.6 However, the agreement proves that our data are consistent with the
results of Sokolowski-Tinten et al. The power of measuring the full ε(ω) is not only evident
from the ε(ω) data described above, but also from the Figs. 7.11(b) − (f ). For any other
parameter set (i.e. anything other than 2.01 eV, 0o ) the reflectivity transients show that the
material actually does not go through the crystalline phase at 200 fs after excitation. For
instance, Fig. 7.11(b) shows the reflectivity transient for 0o angle of incidence and a photon
energy of 3.0 eV. Clearly, the reflectivity does not approach the crystalline value. Similarly,
Fig. 7.11(c) shows the reflectivity transient a photon energy of 2.01 eV and an angle of
incidence of 79.4o . Again, the reflectivity does not peak at the value of the crystalline
phase.
The agreement between our measurements and the experiments by SokolowskiTinten et al. is further illustrated in Fig. 7.12 which shows reflectivity spectra at 200 fs
after excitation for (a) 0o and (b) 79.4o degree angle of incidence. The reflectivity spectra
for the amorphous and crystalline phase are indicated by the solid and dash-dotted curves.
Let us first consider Fig. 7.12(a). For excitations exceeding Fcr (indicated in the figure
by: • = 4.0 Fcr and 1.6 Fcr = ✸) the reflectivity spectra of the excited a-GeSb and the
unexcited c-GeSb cross at exactly 2 eV. For a different angle of incidence on the other hand,
as indicated in Fig. 7.12(b) the spectra cross at a different photon energy, namely around
2.4 eV.
7.4
Conclusions
Using FTRE we successfully disproved the claim of an ultrafast disorder-to-order transition
from Ref. [82]. The full ε(ω) data clearly show that the material 200 fs after excitation is
not in the crystalline state. We find that a-GeSb thin films rather undergo a transition to
a new non-thermal phase with metallic properties. For fluences below Fcr we successfully
6
At infinite time delays, however, (i.e. data taken several minutes after irradiation), we do find the
crystalline reflectivity.
156
200 fs, 0°
0.70
c-GeSb
reflectivity
0.65
0.60
4.0 Fcr
0.55
1.6 Fcr
0.6 Fcr
a-GeSb
(a )
0.50
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
200 fs, 79.40°
0.50
c-GeSb
reflectivity
0.40
0.30
4.0 Fcr
0.20
1.6 Fcr
0.6 Fcr
a-GeSb
(b )
0.10
1.5
2.0
2.5
3.0
3.5
photon energy (eV)
Figure 7.12: Reflectivity spectra of a-GeSb thin film 200 fs after excitation at three different
fluences: • = 4.0 Fcr ; ✸ = 1.6 Fcr ; = 0.6 Fcr. The solid curve indicates the spectrum for
the unexcited a-GeSb thin film and the dash-dotted curve the spectrum for the unexcited
crystalline phase of the thin film.
modelled the behavior of ε(ω) assuming a propagating liquid-solid interface. We were able
to extract heat diffusion speeds and resolidification rates for the material after excitation
with fs laser pulses. The rates turn out to be in agreement with previous measurements
which used longer pulse durations [144]. “Simulating” the experiment by Sokolowski-Tinten
et al. using simple Fresnel calculations shows that our experiment is in excellent agreement
157
with Ref. [82]. It turns out that the researchers in 1998 were extremely unlucky in choosing
exactly the one pair of incidence angle and probe wavelength, at wich the reflectivity of
a-GeSb reaches precisely the value of the crystalline phase.7
7
That is 200 fs after excitation with an intense pump pulse.
Chapter 8
Coherent Phonons in Tellurium
In this chapter we present fs time resolved dielectric function measurements of highly photoexcited single-crystalline Tellurium. The ε(ω) measurements have sufficient accuracy to
resolve the changes due to the impulsively excited coherent phonons in the lattice. Previous
differential reflectivity experiments were able to phase-sensitively detect coherent phonons
in Te (see Section 5.2.2). The data presented here provide a much more detailed picture
of the electron and lattice dynamics involved in the coherent A1 -mode of the Te lattice.
The ε(ω) data not only allow a very accurate comparison to existing theories on coherent
phonons, but also invite new theoretical work on this subject. As a striking example, we
directly observe DECP (see Section 5.2.2) for the first time. We track the dynamics of the
bonding-antibonding split (BAS) via the time resolved spectral ε(ω) and find an initial rapid
redshift followed by oscillations at the A1 -frequency, in excellent agreement with theories
for DECP [103, 105].
8.1
Fundamental Properties of Te
In this section we discuss the lattice structure and electronic properties of Te. Based on
the understanding of the Te lattice we then discuss the details of the A1 -phonon-mode
and its symmetry preserving property. Furthermore, we describe the bandstructure of Te
158
Chapter 8: Coherent Phonons in Tellurium
159
c-axis
d
x
Figure 8.1: Crystal structure of Tellurium. The Te atoms are positioned along right-handed
three-fold screws. The screws are arranged in a hexagonal lattice. The graph shows the
view down the screw axes. d denotes the interhelical distance and x denotes the radius of
each helix.
and its dependence on lattice parameters, which is important for our interpretation of the
experimental data.
8.1.1
Structural and Electronic Properties of Te
The Te lattice consists of three-fold helices which are hexagonally (almost) closed packed,
i.e. the helices are arranged on locations corresponding to points in the hexagonal Bravais
lattice. Tellurium with both right and left-handed helices exists. Figure 8.1 shows a view of
a Te lattice with right-handed helices down the helical axis. For future reference we define
d as the interhelical distance and x as the radius of each helix, as indicated in the figure.
Under normal conditions: x = 0.2686 d [105]. The standard unit cell of the lattice is the
(a)
160
(b)
Figure 8.2: Derivation of the Te crystal structure from a distortion of the primitive cubic
lattice. (a) shows part of a primitive cubic lattice. (b) shows the Te lattice as obtained
by shifting the center atom along the (110) direction. The screws are indicated by the fat
lines.
trigonal cell indicated by the dashed line in the figure and it contains three atoms. The
primitive unit cell is harder to visualize. The best way to think about it is to interpret the
Te lattice as a distortion of the primitive cubic lattice [102]. Figure 8.2(a) shows part of
a primitive cubic lattice. Shifting the center atom along the (110)-direction, one obtains
the Te lattice as indicated in Fig. 8.2(b). The fat lines indicate the screws. By looking at
the graph in a slightly different way, it is possible to think about this structure as a body
centered tetragonal (bct) lattice with a two atomic basis. The “footprint” of the bct lattice
√
consists of the four corner atoms making the bct unit cell a factor of 2 wider than the side
length of the cube we started from. The center atom is the top atom in Fig. 8.2(b), which
makes the height of the bct cell twice as high as the side length of the cube in Fig. 8.2(a).
The basis molecule, therefore, consists of an atoms of the bct lattice and an atom which was
displaced along the (110) directon (the previously centered atom on the LHS of the figure).
We have therefore shown that Te has a two atomic basis, or in other words the primitive
unit cell contains two atoms.
Given the lattice structure, one can calculate the vibrational eigenmodes of Te.
A1
A2
161
E’LO
E’TO
E’’LO
E’’TO
+
+
−
−
−
Figure 8.3: Optical phonon modes of Te lattice: A1 – Raman-active, A2 – only infra-red
active, ETO/LO – Raman and infra-red active, ETO/LO – Raman and infra-red active.
There are three atoms per unit cell, which implies that there are six optical phonon modes.
We are only concerned with optical modes for the purpose of the experiments described
in this thesis. Hence we neglect the discussion of acoustic modes. Figure 8.3 shows the
atomic motion for each of the optical eigenmodes within each helix of the lattice. The
A1 -mode fully preserves the symmetry of the crystal structure as Figs. 8.1 and 8.3 indicate.
It turns out that the A1 -mode is Raman-active but not infra-red active [102]. Raman
measurements reported the resonance frequency of this “breathing” mode to be f = 3.6
THz [145]. Furthermore, there is no internal polarization associated with the A1 -mode.
Table 8.1 summarizes the characteristics of the modes depicted in Fig. 8.3.
As we discussed in Section 5.2.2, it is possible to coherently excite Raman-active
modes through ISRS. For the symmetry preserving A1 -mode, an impulsive excitation via
DECP is also possible. The purely infra-red phonon modes cannot be excited by any of these
processes. Rather, an impulsive field effect has to be involved, e.g., the rapid screening of the
surface field in GaAs (see Section 5.2.2). We will return to the exact excitation mechanisms
162
Table 8.1: Table of optical phonon modes in Te [102, 145, 146].
mode
frequency
activity
internal polarization
A1
3.6 THz
Raman-active
none
A2
2.6 THz
infra-red active
Ec
2.94 THz
Raman and infra-red active
E⊥c
3.15 THz
E⊥c
4.22 THz
E⊥c
4.26 THz
E⊥c
ETO
ELO
ETO
ELO
in Section 8.3.
Let us turn to the electronic properties of Te. Figure 8.4 shows the bandstructure
for Te at normal conditions. As indicated, Te is a small bandgap semiconductor — it has
an indirect gap of about 0.3 eV. From the discussion of the structure we know that there
are two atoms in the primitive unit cell of Te. Since Te is a group VI element, each atom
contributes 6 valence electrons. Therefore, the total number of bands in Te is 2 × 6 = 12
[147]. The figure shows the top 6 bands. We omit the 6 lower lying bands since they are
not involved in the experiments. Because there are three atoms in the standard unit cell,
the lowest
3×6
2
= 9 bands are going to be filled [147]. Fig. 8.4 shows that the topmost three
bands are unoccupied conduction bands, in accordance with our discussion.
It is known that Te has a stable metallic phase under certain thermo-dynamic
conditions. The semiconductor-to-metal transition in Te has been the topic of previous
experimental and theoretical work. Specifically, a pressure-induced transition has been
found to occur at pressures of about 40 kbars [148, 149]. At such pressures, the crystal
structure of Te changes from the trigonal structure [102] to a closed packed structure with
metallic bonding.
Similarly, Tangney et al. have shown that the bandstructure of Te undergoes severe changes as the A1 -mode is strongly excited [150]. The researchers performed density
163
12
energy (eV)
Te
10
0.3 eV
8
6
Γ
A
H
K
Γ
Figure 8.4: Electronic bandstructure of Te at normal conditions. The shaded region indicates the indirect bandgap of 0.3 eV.
functional theory calculations of the bandstructure for different positions of the Te lattice
along the motion of the A1 -mode. According to their results the indirect gap closes as
x = 0.28 d which is equivalent to a 10% change in the helical radius. Figure 8.5 shows the
bandstructure for x = 0.3 d. Clearly, the indirect gap is closed for this lattice configuration.
In fact the top of the valence band (at the H point) and the bottom of the conduction band
(at the A point) overlap by about 0.4 eV.1
8.1.2
Sample Preparation
We use a single crystalline Te sample provided by P. Grosse at the University of Technology
Aachen (Germany). The crystal was grown using the Czrochalski method [102]. The first
1
If the A1 -mode is excited to even higher amplitude, i.e. x = 1/3 d, even the direct gap closes. The
lattice undergoes a Peierls distortion at this point. The crystal symmetry is raised and the lattice obtains
rhombohedral symmetry which includes inversion symmetry [150].
164
12
energy (eV)
Te
10
8
6
Γ
A
H
K
Γ
Figure 8.5: Electronic bandstructure of Te for lattice distortion along A1 -mode. The amplitude of the distortion is x = 0.3d. The indirect gap is closed for this lattice distortion.
challenge to overcome in this experiment was the preparation of the sample. As the crystal
is “pulled” out of the melt, it is indeed of single crystalline quality, but the surface is far
from optically flat. Furthermore, the direction of the c-axis is not unambiguous from the
raw material.
As a first step, we polished a surface which we estimated to be close to perpendicular to the c-axis. Tellurium is a very soft material, and polishing it is not a trivial matter.
The method that is most commonly recommended in the literature consists of a combined
chemical etch / mechanical polish [151]. A fine-woven cloth is soaked with the etching agent
consisting of 1 part CrO3 , 1 part concentrated HCl acid and 3 parts of H2 O (this is commonly called the Honeywell etch). The cloth is spanned over a flat glass plate mounted on a
rotating platform. The sample is then carefully lowered onto the cloth. The mild roughness
of the cloth in conjunction with the etch slowly removes material and a clean surface can
165
Figure 8.6: Laue diffraction pattern of Te. The diffraction pattern has three-fold symmetry
indicating that the face used for the diffraction experiment is perpendicular to the c-axis.
be achieved within a few minutes. We found that the surfaces resulting from this polishing
method are reasonably clean but still have remaining visbible features resembling “pocks”.
In fact, returning to standard, purely mechanical polishing methods produced slightly better results. This might be due to the much improved polishing technology since 1970 (the
year when Ref. [151] on Te sample preparation was published).2
We confirmed the orientation of the sample by performing a Laue diffraction on
the first prepared surface (which we estimated to be perpendicular to the c-axis). Figure
8.6 shows the Laue diffraction pattern of the Te sample. The pattern shows three-fold
symmetry, indicative of the only three-fold symmetry in the Te lattice — the three-fold
2
The polishing cloths, turntables and glass surfaces in our apparatus are probably much better than the
polishing machines in the 70’s.
166
screw axis along the c-axis.3 Upon closer inspection, the diffraction pattern is not completely
symmetric. The reason for this skewed pattern lies in a combination of misalignment of the
surface to the x-ray source and the fact that the surface may not be exactly perpendicular
to the c-axis.
As discussed in Section 4.3.2, it is necessary to measure the reflectivity of a surface
containing the c-axis in order to efficiently extract both the real and imaginary part of the
dielectric function of a uniaxial material. We therefore polished a surface perpendicular to
the surface previously polished (which we now know is perpendicular to the c-axis) using
standard mechanical polishing techniques. As mentioned above, the surface quality we
obtain is almost optical. There are a few remaining visible scratches but the flatness is
sufficient for our experiment, i.e. there is a clean specular reflection from the surface.
8.2
The reflectivity changes due to the coherent phonon modes in Te are on the order of 0.1 –
10%. It is therefore necessary to use the multi-shot setup described in Section 4.3.4 to resolve
these changes. Furthermore, the maximal Brewster angle for Te within the wavelength range
of the broadband probe is about 81o . It is therefore necessary to choose one angle of the
two angles above that value. For this experiment we chose the lower angle as 49.5o and
the higher angle as 83.5o. Using the methods described in Section 4.3.4 we determined the
dielectric constant of the native oxide layer on Te (TeO2 ) to be ε = 5.0.4 In matching the
correction factors (as described in Section 4.3.4), we found different values for the oxide
layer thickness for different spots on the sample. The values range from 5.5 – 8.5 nm. It has
been reported in previous work that surface roughness effectively influences the reflectivity
of an interface in the same way as a dielectric thin layer (like the oxide layer) where different
degrees of roughness can be taken account of with different layer thicknesses [153]. Given
the imperfections on the sample surface, it is not surprising that we find a range of oxide
3
4
Three-fold screw axes do in fact cause three-fold symmetries in Laue x-ray diffraction patterns [152].
To our knowledge there is no reported value for ε of TeO2 in the literature.
167
layer thicknesses at different spots on the sample. We measured ε(ω) for pump fluences
ranging from 2.0 to 5.6 mJ/cm2 .5 The highest fluence of 5.6 mJ/cm2 corresponds to an
excitation of about 6% of all valence electrons assuming linear absorption of the 800 nm
pump pulse.
8.2.1
The Ordinary ε(ω)
Figure 8.7 shows the evolution of the ordinary part of ε(ω) of Te with time following a
strong pump excitation with a pump fluence of F = 5.6 mJ/cm2 . The solid circles indicate
the measured values for Re[ε(ω)] and the hollow circles indicate the measured values for
Im[ε(ω)]. The solid and dotted curves indicate the literature values for the ordinary part of
ε(ω) of Te [21]. Figure 8.7(a) shows ε(ω) at the negative time delay of −500 fs. The graph
shows excellent agreement between our data and the literature data. We omit error bars
here for sake of clarity. The magnitude of the error bars in this experiment are on the order
of the marker size and hence do not add significant information. Figure 8.7(b) shows ε(ω)
at a time delay of 220 fs. Both real and imaginary part of ε(ω) redshift significantly on this
time scale. As a consequence, the values of ε(ω) around 1.7 eV increase sharply from about
20 to 60 for Im[ε(ω)] and 35 to 40 for Re[ε(ω)]. For a quantitative evaluation of the redshift
we choose the zero-crossing (ZC) of Re[ε(ω)] which represents the bonding-antibonding split
(BAS) of Te [24]. The BAS redshifts by about 0.4 eV from 2.5 to 2.1 eV within 220fs after
pump excitation. Figure 8.7(c) shows ε(ω) 370 fs after pump excitation. The gray-shaded
symbols indicate the ε(ω) values for the previously shown time delay of 220 fs (Fig. 8.7(b)).
The ε(ω) partly recovers: the BAS blueshifts back by about 0.15 eV to 2.25 eV and the low
energy end of Im[ε(ω)] sinks back to about 40. Figure 8.7(d) shows ε(ω) at a time delay of
520 fs. Again, the gray-shaded symbols indicate the ε(ω) values values for the previously
shown time delay which in this case is 370 fs. The ε(ω) redshifts again by about 0.1 eV to
2.15 eV and the low energy end of Im[ε(ω)] increases again to values of 40. It becomes clear
at this point that ε(ω) is undergoing an oscillatory motion with a half period of about 150
5
To be consistent with the literature to date, in this chapter we choose to quote fluences not as the peak
of the Gaussian profile, but as the total energy divided by the 1/e2 area of the pump spot.
168
60
Te (ord)
−500 fs
5.6 mJ/cm2
40
dielectric function
dielectric function
60
20
0
20
0
(a)
−20
1.5
Te (ord)
220 fs
5.6 mJ/cm2
40
(b)
2.0
2.5
3.0
−20
1.5
3.5
2.0
energy (eV)
40
20
0
3.5
Te (ord)
520 fs
5.6 mJ/cm2
40
20
0
(c)
−20
1.5
(d)
2.0
2.5
3.0
−20
1.5
3.5
2.0
2.5
3.0
3.5
energy (eV)
energy (eV)
60
60
Te (ord)
670 fs
5.6 mJ/cm2
40
dielectric function
dielectric function
3.0
60
Te (ord)
370 fs
5.6 mJ/cm2
dielectric function
dielectric function
60
20
0
Te (ord)
820 fs
5.6 mJ/cm2
40
20
0
(e)
−20
1.5
2.5
energy (eV)
(f)
2.0
2.5
energy (eV)
3.0
3.5
−20
1.5
2.0
2.5
3.0
3.5
energy (eV)
Figure 8.7: Ordinary part of the dielectric function of Te for an excitation with a fluence
of 5.6 mJ/cm2 — • = Re[ε(ω)], ◦ = Im[ε(ω)]. The solid and dashes curves show real and
imaginary part of the ordinary part of ε(ω) from previous cw measurements [21]. The
gray-shaded symbols indicate the ε(ω) for the previous time delay respectively.
fs. This is further validated by Fig. 8.7(e) where the BAS redshifts again by about 0.1 eV
and the low energy end of Im[ε(ω)] falls to 40, and Fig. 8.7(f ) where the BAS blueshifts
again by about 0.05 eV and the low energy end of Im[ε(ω)] rises again to about 50.
We also measured ε(ω) for two pump fluences lower than 5.6 mJ/cm2 . The ε(ω)
169
zero crossing of Re[ε] (eV)
2.6
2.4
2.2
2.0
−0.5
0.0
0.5
1.0
1.5
2.0
time delay (ps)
Figure 8.8: Zero-crossing of real part of ordinary ε(ω) vs. time for different excitation
fluences: F = 5.6 mJ/cm2 (solid curve); F = 2.5 mJ/cm2 (dotted curve); F = 2.0 mJ/cm2
(dashed curve).
dynamics are very similar to the high fluence case but less pronounced. Rather than showing
the entire set of data for all fluences and all time delays it is more instructive to plot a specific
feature of ε(ω) vs. time to get a better feeling for the temporal evolution of ε(ω). Fig. 8.8
shows the ZC of Re[ε(ω)] vs. time for three different excitation fluences. The ZCs at negative
time delays are not exactly equal for all fluences. This discrepancy is due to the fact that
the 1 kHz pump pulse train is heating the sample more for higher fluences than for lower
fluences. Higher temperatures lead to a lower BAS partly because the occupation of the
antibonding states is higher (see Section 5.2.2). The ZC rapidly drops immediately after
the excitation and oscillates after that. For the highest fluence of 5.6 mJ/cm2 , the ZC drops
from its initial value of 2.5 eV to 2.1 eV at 220 fs and oscillates around a new equilibrium
position. The inital peak-to-peak amplitude is about 0.2 eV. The equilibrium position
partly recovers from about 2.25 eV right after excitation to about 2.32 eV at 2 ps where
it stabilizes. The dynamics are less pronounced but of the same shape for lower fluences.
170
For F = 2.5 mJ/cm2 , the ZC drops by about 0.15 eV and the subsequent oscillations have
an amplitude of about 0.05 eV. As the pump fluence is lowered further the amplitude of
the oscillations continues to decrease as indicated by the dotted curve for F = 2.0 mJ/cm2
— the peak-to-peak amplitude is now about 0.02 eV. From these data it is evident that
we chose the time delays corresponding to the extrema in the ZC oscillations for the ε(ω)
graphs in Fig. 8.7.
8.2.2
The Extra-Ordinary ε(ω)
As described in Section 4.3.2, our technique allows the measurement of both, the ordinary
and extra-ordinary part of ε(ω). Figure 8.9 shows the extra-ordinary part of ε(ω) at various
time delays for a pump fluence of 5.6 mJ/cm2 . The solid circles indicate the measured values
for Re[ε(ω)] and the hollow circles indicate the measured values for Im[ε(ω)]. The solid and
dotted curves indicate the literature values for the extra-ordinary part of ε(ω) of Te [21].
Figure 8.9(a) shows ε(ω) at a negative time delay of −500 fs. The graph shows excellent
agreement between our data and the literature data. The extra-ordinary ε(ω) of Te evolves
quite similarly to the ordinary ε(ω) as shown in Fig. 8.7. At 280 fs after excitation the ZC
of Re[ε(ω)] redshifts from an intial value of 2.17 eV to 1.9 eV. This is 12% change is slightly
less than the 16% shift in the ordinary case. Although the initial redshifts are comparable,
the following oscillatory modulation of ε(ω) is much less pronounced in the extra-ordainry
case, as shown in Figs. 8.9(b) − (f ). The gray-shaded symbols in each plot represent the
time delay data from the previous one. The imaginary part undergoes only slight changes.
The real part blueshifts and redshifts in an oscillatory manner with an amplitude of about
0.05 eV.
Again, it is useful to plot the ZC of the extra-ordinary Re[ε(ω)] vs. time delay
as shown in Fig. 8.10 for three different fluences: F = 5.6 mJ/cm2 (solid curve), F =
4.1 mJ/cm2 (dotted curve), and F = 2.5 mJ/cm2 (dashed curve). The ZCs at negative
time delays are exactly equal in this case. Evidently, heating does not influence the extraordinary ε(ω) to the same degree as in the ordinary case as shown in Fig. 8.8. Other than
171
70
70
Te (ext)
−500 fs
5.6 mJ/cm2
Te (ext)
280 fs
5.6 mJ/cm2
50
dielectric function
dielectric function
50
30
10
−10
30
10
−10
(a)
−30
1.5
(b)
2.0
2.5
3.0
−30
1.5
3.5
2.0
energy (eV)
70
dielectric function
dielectric function
3.5
Te (ext)
580 fs
5.6 mJ/cm2
50
30
10
−10
30
10
−10
(d)
(c)
−30
1.5
2.0
2.5
3.0
−30
1.5
3.5
2.0
energy (eV)
2.5
3.0
3.5
energy (eV)
70
70
Te (ext)
730 fs
5.6 mJ/cm2
Te (ext)
880 fs
5.6 mJ/cm2
50
dielectric function
50
dielectric function
3.0
70
Te (ext)
430 fs
5.6 mJ/cm2
50
30
10
−10
30
10
−10
(f)
(e)
−30
1.5
2.5
energy (eV)
2.0
2.5
energy (eV)
3.0
3.5
−30
1.5
2.0
2.5
3.0
3.5
energy (eV)
Figure 8.9: Extra-ordinary part of the dielectric function of Te for an excitation with a
fluence of 5.6 mJ/cm2 — • = Re[ε(ω)], ◦ = Im[ε(ω)]. The solid and dashes curves show
real and imaginary part of the extra-ordinary part of ε(ω) from previous measurements [21].
The gray-shaded symbols indicate the ε(ω) for the previous time delay respectively.
that, the dynamics of the extra-ordinary ε(ω) are very similar to the ordinary case. The
ZC rapidly drops immediately after the excitation and oscillates after that. As indicated in
Fig. 8.9 the initial drop is of the same relative magnitude as in the ordinary case. For the
highest fluence of 5.6 mJ/cm2 , the ZC drops from its initial value of 2.17 eV to 1.9 eV at
172
zero crossing of Re[ε] (eV)
2.2
2.1
2.0
1.9
1.8
−0.5
0.0
0.5
1.0
1.5
2.0
time delay (ps)
Figure 8.10: Zero-crossing of real part of extra-ordinary ε(ω) vs. time for different excitation
fluences: F = 5.6 mJ/cm2 (solid curve); F = 4.1 mJ/cm2 (dotted curve); F = 2.5 mJ/cm2
(dashed curve).
220 fs and oscillates around the new equilibrium position. The amplitude of the subsequent
oscillations is significantly smaller. The inital peak-to-peak amplitude is about 0.05 eV,
which corresponds to a modulation of about 2% (compared to 8% for the ordinary case).
The dynamics are less pronounced but of the same shape for lower fluences. For F = 4.1
mJ/cm2 the ZC drops by about 0.18 eV and the following oscillations have an amplitude of
about 0.04 eV. For F = 2.5 mJ/cm2 the initial drop is about 0.7 eV and the peak-to-peak
amplitude is about 0.02 eV.
8.3
Discussion
In this section we present a detailed analysis of the data presented in the previous section.
First, we discuss the detailed phonon dynamics induced by the femtosecond laser excitation.
Second, we discuss the influence of the lattice vibrations on the electronic configuration of
173
Te.
8.3.1
Detailed Phonon Dynamics
The most prominent feature in the ε(ω) behavior is the oscillatory motion following the
pump excitation. To quantitatively track the exact value of the BAS oscillation frequency
we show the Fourier transforms of the BAS time traces in Figs. 8.8 and 8.10 in Fig. 8.11(a)
and (b). Clearly, the peak of the Fourier transforms redshifts with increasing fluence for
both the ordinary and the extra-ordinary part of ε(ω). Fig. 8.11(c) shows the position of the
peaks for ordinary and extra-ordinary ε(ω) vs. fluence. The lines are linear regression fits
through the data. Both fits give values of about 3.5 THz when extrapolating to zero pump
fluence, which is very close to the Raman value of the A1 -mode of 3.6 THz and within the
accuracy of our measurement [145]. We obtain a slope of 0.03
of ε(ω) and a slope of 0.08
THz
mJ/cm2
THz
mJ/cm2
for the ordinary part
for the extra-ordinary part of ε(ω). This softening of the
A1 -mode in Te has been observed before by Hunsche et al. [78] and we describe the details
of this process in Section 5.2.2. The data for the ordinary part of ε(ω) is in good agreement
with the results of Hunsche et al., where a softening of about 0.04
THz
mJ/cm2
is reported[78].
Hence, we clearly find that the ordinary part of ε(ω) is modulated dominantly by the A1
phonon mode which in turn is effectively driven by DECP (see Section 5.2.2).
The data for the extra-ordinary ε(ω) on the other hand is in slight disagreement
with the results reported in Ref. [78]. The slope we obtain for the fluence dependent
frequency shift is significantly more negative in this case. We attribute this discrepancy
to other impulsively excited modes in Te. We know from Section 8.1.1 that there are 4
Raman-active optical phonon modes in the Te lattice besides the A1 -mode: ETO/LO and
ETO/LO . All Raman-active modes can be excited by ISRS as we described in Section 5.2.2.
In fact both, ETO and ELO have been observed in previous experiments [154]. The higher
frequency E modes were not observed, however, due to lack of temporal resolution [154].
The resonance frequencies for these modes are above 4 THz which makes them very hard to
detect for our experiment as well. As opposed to all previous experiments, our experiment
174
(a)
(b)
1.0
1.0
ext
0.8
FT intensity (norm.)
FT intensity (norm.)
ord
2
5.6 mJ/cm
2.5 mJ/cm2
2.0 mJ/cm2
0.6
0.4
0.2
0.0
2.0
2.5
3.0
3.5
4.0
4.5
0.8
5.6 mJ/cm2
4.1 mJ/cm2
2.5 mJ/cm2
0.6
0.4
0.2
0.0
2.0
5.0
2.5
3.0
3.5
4.0
4.5
5.0
frequency (Thz)
frequency (THz)
(c)
3.5
frequency (THz)
3.4
3.3
3.2
3.1
3.0
1
2
3
4
5
6
fluence (mJ/cm2)
Figure 8.11: Frequency dynamics of the coherent phonon modes in Te. (a) shows Fourier
transforms of time resolved zero-crossing traces for the ordinary part of ε(ω) at different
excitation fluences: F = 5.6 mJ/cm2 (solid curve); F = 2.5 mJ/cm2 (dotted curve); F = 2.0
mJ/cm2 (dashed curve). (b) shows Fourier transforms of time resolved zero-crossing traces
for the extra-ordinary part of ε(ω) at different excitation fluences: F = 5.6 mJ/cm2 (solid
curve); F = 2.5 mJ/cm2 (dotted curve); F = 2.0 mJ/cm2 (dashed curve). (c) shows the
frequencies of the ZC oscillations vs. fluence for the ordinary ε(ω) (•) and the extra-ordinary
case (◦). The lines are linear regression fits through the data points.
provides information on the full dielectric tensor of Te for the first time. It is hence possible
to distinguish the effects of the different phonon modes on each element of the dielectric
tensor. It is reasonable to expect that the ordinary part of ε(ω) is more affected by phonon
modes where the lattice distortion is directed perpendicular to the c-axis (and vice versa for
the extra-ordinary ε(ω)). As shown in Fig. 8.3 the following modes all vibrate perpendicular
to the c-axis: A1 , A2 , ELO , and ELO . The only modes vibrating along the c-axis are: ETO
175
and ETO . The A2 -mode is only infra-red active and hence not excited in our experiment.6
The higher frequency ETO/LO -modes are not detectable in our experiment due to temporal
resolution, as described above. The remaining modes are therefore: A1 , ETO , and ELO .
Let us first consider the ordinary part of ε(ω) in more detail. DECP is a much more
efficient excitation process than ISRS, because ISRS is a higher order nonlinear optical
process which requires phasematching (see Sections 5.2.2 and 2.3.1). The ordinary ε(ω) is
therefore completely dominated by the A1 -mode. This is in agreement with the findings of
Dekorsy et al. in 1995 and Hunsche et al. in 1996. In the extra-ordinary case on the other
hand, the effect of the A1 -mode on ε(ω) is suppressed due to the fact that the vibration
is purely perpendicular to the c-axis. The lattice stays mostly fixed along the c-axis. The
only phonon mode vibrating along the c-axis is the ETO -mode (remember that we cannot
detect the ETO -mode). Now, even though ISRS is much less efficient than DECP, the ETO mode influences the extra-ordinary ε(ω) much more efficiently and hence competes with the
A1 -mode.
To illuminate this point we plot the maximum value of the Fourier coefficients (as
obtained from the Fourier transforms in Fig. 8.11) vs. fluence in Fig. 8.12. The solid black
circles indicate the Fourier coefficient maxima for the ordinary case and the whites circles
indicate the data for the extra-ordinary case. Clearly, the oscillations in the ordinary ε(ω)
become more pronounced with increasing fluence causing the Fourier coefficient to increase
by an order of magnitude as the fluence is raised from 2.0−5.6 mJ/cm2 . The amplitude of the
Fourier coefficient of the A1 -mode increases roughly linearly with fluence in agreement with
previous results [78]. In the extra-ordinary case on the other hand, the Fourier coefficients
increases only weakly with fluence. This finding helps us in understanding the strongly
negative slope for the extra-ordinary ε(ω) case in Fig. 8.11(c). The fluence dependence of
6
The A2 -mode can be excited via the Dember field effect, but only on surfaces perpendicular to the c-axis
[154]. The Dember field is built up rapidly due to the strong carrier density gradient at the surface. The
electron and hole diffusion coefficients are sufficiently different for a strong field to rapidly built up due to
ambipolar diffusion. Since the internal polarization caused by the A2 -mode is perpendicular to the c-axis
(see Fig. 8.3), the driving field has to be directed perpendicular to the c-axis as well. Hence, the Dember
field effect can only excite A2 -modes if the surface is perpendicular to the c-axis.
176
peak Fourier coeff. (norm.)
1.2
0.8
0.4
0.0
1
2
3
4
5
6
fluence (mJ/cm2)
Figure 8.12: Maximum value of Fourier transform coefficients for transforms of time resolved
zero-crossing traces of ε(ω) for both ordinary (•) and extra-ordinary (◦) case. The lines are
guides to the eye.
ISRS is superlinear because ISRS involves a nonlinear wavemixing process. Therefore, it
can be expected in the extra-ordinary case that the influence of the ETO -mode becomes
comparable to, or even dominates, the suppressed influence of the A1 -mode for sufficiently
high fluences. The resonance frequency of the ETO -mode is 2.94 THz. We conclude that
the strong fluence dependence of the oscillations of the extra-ordinary ε(ω) are a mix of
softening of the A1 -mode and an increased contribution of the lower frequency ETO -mode.
The accuracy of our experiment does not allow a distinction of the individual phonon modes,
but rather returns a more redshifted frequency.
8.3.2
The Effects on the Bandstructure
Let us turn to the effects of the lattice dynamics on the electronic configuration of Te.
As described in Section 8.1.1 the bandstructure is affected dramatically by the atomic
movement along the direction of the A1 -mode. At x = 0.3 d the indirect gap closes, giving
177
rise to a semiconductor-to-metal transition. From Fig. 8.8 we find that the BAS of Te drops
by 0.4 eV due to the pump excitation. The BAS can be interpreted as the average separation
of electronic states in the conduction and valence band weighted by the respective oscillator
strength. Hence, it is not unreasonable to assume that the bandgap redshifts by an amount
on the same order of magnitude as the BAS. We know from Fig. 8.4, on the other hand, that
the bandgap in Te is only 0.3 eV wide which is significantly less than the 0.4 eV redshift
of the BAS. Therefore we have evidence for a closed indirect gap, or in other words for an
ultrafast semiconductor-to-metal transition driven by a coherent phonon mode.
For a true metal one would expect ε(ω) to take on a Drude-like shape. The ε(ω)
in Fig. 8.7(b) (corresponding to the minimum of the BAS), on the other hand, clearly shows
a ZC of Re[ε(ω)] indicating a strong remaining transition in the system. As described in
Section 2.1.1, the fundamental requirement for the Drude-model to be applicable is an electron dispersion which is close to parabolic at the Fermi energy (or the energy of the most
energetic electrons). This is not necessarily the case for Te at x = 0.3 immediately after
excitation. As shown in Fig. 8.5, the electrons at the top of the valence band have actually
negative parabolic dispersion. In equilibrium (i.e. at long time delays), the electrons would
scatter from the valence band maximum at the H-point in the Brillouin zone to the conduction band minimum at the A-point because these states are energetically favorable. After
this equilibration, the most energetic electrons occupy the conduction band minimum at
the A-point and it is possible to approximate the electron dispersion there with a parabola.
Electron transfer from the H-point to the A-point in the Brillouin zone requires
electron-phonon scattering events to provide the necessary electron momentum. In III-V
semiconductors, typical electron-phonon scattering times are on the order of hundreds of
fs [91]. We can take this time scale as an order of magnitude estimate for electron-phonon
scattering in Te.7 However, the time for which the BAS drops by more than 0.3 eV — the
time for which the band are potentially crossed — is extremely short. It less than 100 fs for
the first cycle of the oscillation and it is barely existent for the second cycle. Thereafter,
7
To our knowledge, there is no measurement of the electron-phonon scattering rate in Te reported to
date.
178
the phonon amplitude is too small to drive the BAS low enough. Therefore, the time for
which the indirect gap is closed is too short to have an efficient transfer of electrons into the
conduction band. Hence, we have indicative evidence for an intruiging state of matter where
the bands of the material are crossed for too short a time to allow carrier equilibration and
thus Drude-like dielectric behavior. The ε(ω) is therefore semiconductor-like throughout.
An appropriate name for this phase might be frustrated metal.
Quantitative vs. Qualitative Statements
The dielectric function is an excellent optical signature to probe bandstructure changes as
described in Section 2.1.2. Theoretical calculations of the kind described in that section can
predict the ε(ω) resulting from a given bandstructure. It is hence possible to make fairly
accurate statements about the bandstructure from ε(ω) data. Matching dielectric function
predicitions to the experimental data could even allow to quantitatively extract the spatial
amplitude of the A1 -phonon mode. At the time of writing of this thesis, collaborative
efforts with several theoretical groups were under way to make more quantitative statements
about the bandstructure changes during coherent phonon motion. Here, we have to restrict
ourselves to quite qualitative statements due to the lack of calculations. It is clear however,
that this data is unique in the sense that it provides a first look into detailed electronic
configuration changes due to coherent phonons.
8.4
Conclusions
We have measured the full dielectric function response of Te to strong photo-excitation.
The ordinary part of ε(ω) shows strong oscillatory features due to the coherently excited
A1 -mode of the Te lattice. We directly track the BAS of the material and hence provide
the first direct observation of DECP. The magnitude of the decrease in the BAS hints
towards an intruiging extreme non-equilibrium state of matter — a solid with crossed bands
which is non-metallic. Furthermore, measuring individual dielectric tensor elements allows
179
a distinction of the effect of various phonon modes on each element in the dielectric tensor.
For instance, the extra-ordinary part of ε(ω) shows the influence of the ETO -mode on the
dielectric tensor of Te. This body of data invites new, more detailed theoretical work on
coherent phonons in solids.
Chapter 9
Summary and Outlook
The experimental results presented in this thesis clearly demonstrate the power of the
femtosecond time resolved ellipsometry technique developed in our laboratory. Directly
measuring the real and imaginary part of the dielectric function over a broad energy range
provides a great wealth of information on ultrafast carrier and lattice dynamics in photoexcited solids, far exceeding the perspective gained from typical reflectivity or transmissivity
experiments to date.
Ultrafast Phase-Transformations in a-GaAs
Using FTRE, we were able to illuminate on the nature of the structural phase changes in aGaAs following femtosecond laser excitation. Having performed the first ε(ω) measurement
of heated a-GaAs we were able to confirm that c-GaAs undergoes a nonthermal structural
transition to a disordered state with identical optical properties as heated a-GaAs. Furthermore we estimated the crystallization energy of GaAs to be about 3.1 kcal per mole of
GaAs molecules. For fluences well above the damage fluence Fth-a , we confirm that a-GaAs
undergoes a semiconductor-to-metal transition to a metallic phase similar to that reached
by the crystalline material. From the ε(ω) data we also deduce that the transition is mainly
structurally driven because it takes longer than the time resolution of the setup even for the
highest fluences used in our experiments — it takes 170 fs at 14 Fth-a . Future experiments
180
Chapter 9: Summary and Outlook
181
at even higher fluences should investigate the possibility of a purely electronically driven
band gap collapse — a collapse within the duration of the pump pulse.
Ultrafast Phase Dynamics in GeSb Thin Films
The time resolved measurement of ε(ω) enabled us to successfully disprove a recent claim of
an ultrafast disorder-to-order transition from Ref. [82]. The material is not in the crystalline
state 200 fs after excitation, but the ε(ω) data reveal a transition to a fluence-independent
state which corresponds to a new non-thermal phase. For fluences below the crystallization
fluence Fcr we successfully modelled the behavior of ε(ω) assuming a propagating liquid-solid
interface. We were able to extract heat diffusion speeds and resolidification rates for the
material after excitation with fs laser pulses. These results open the door to more detailed
ε(ω) studies of Sb-rich thin films to optimize these exciting materials for their applications
in re-writable optical data storage devices.
Coherent Phonons in Te
We measured the spectral dielectric function response of Te to strong photo-excitation.
The ε(ω) shows strong oscillatory features due to the coherently excited A1 -mode of the Te
lattice. We report the first direct observation of DECP by tracking the bonding-antibonding
split of the material with fs time resolution. The data reveals indicative evidence of an
intruiging, extreme non-equilibrium state of a solid, where the bands are crossed but no
metallic behavior can be detected. This frustrated metallic phase can only exist due to the
fact that the bands cross for such a short time that there is not enough time for efficient
electron scattering. Furthermore, we performed the first fs time resolved measurement of all
dielectric tensor elements in a uniaxial crystal. The detailed ε(ω) data allow us to distinct
the effects of different phonon modes on specific elements in the dielectric tensor. This
body of data invites new, more detailed theoretical work on coherent phonons in solids.
Furthermore, there are several other materials such as Ti2 O3 which allow the excitation
of A1 -phonon-modes as well as exhibiting metal-insulator transitions in thermodynamic
Chapter 9: Summary and Outlook
182
equilibrium. Future experiments should study a variety of these materials to gain a deeper
understanding of coherently driven phase transitions.
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