Download Ultrafast Dynamics of Coherent Optical Phonons in Bismuth

Thèse de Doctorat de l’ÉCOLE POLYTECHNIQUE
Spécialité: Physique
présentée par
Thomas Garl
pour obtenir le grade de
Docteur de l’École Polytechnique
Ultrafast Dynamics
of Coherent Optical Phonons
in Bismuth
soutenue publiquement le 4 juillet 2008 devant le jury composé de
M.
M.
M.
M.
M.
M.
Antoine ROUSSE
Eric COLLET
Marino MARSI
Guillaume PETITE
Olivier UTEZA
Klaus SOKOLOWSKI–TINTEN
Ecole Polytechnique, France
Université de Rennes I, France
Université Paris Sud, France
Ecole Polytechnique, France
Université de Marseille, France
Universität Duisburg–Essen,
Allemagne
Directeur de thèse
Rapporteur
Rapporteur
Président du jury
Examinateur
Examinateur
Thèse préparée au Laboratoire d’Optique Appliquée UMR 7639
ENSTA - Ecole Polytechnique - CNRS
ii
Abstract
The work presented in this thesis is devoted to the study of femtosecond laser induced coherent atomic motion (phonons) . The investigation of phonons is a field of
fundamental importance, as this atomic motion can play a key role in the dynamics
of phase transitions.
In the framework of this thesis, time-resolved measurements of the reflectivity
in the semi-metal bismuth close to the damage threshold have been carried out
which allow for an investigation of the dynamics of the subtle atomic displacements.
Due to the high temporal resolution of 35 fs and the high sensitivity in detecting
reflectivity changes of ∆R/R0 = 10−5 , these experiments showed two novel aspects
of the reflectivity dynamics. The results also contain the signature of a transient
state reached on a ps-time scale that has never been observed before.
The detailed study of the reflectivity dynamics showed for the first time that during the very first moments of the excitation there is a negative change in reflectivity.
Furthermore, a negative change in reflectivity occurring some picoseconds after excitation has been measured. Theoretical considerations presented in this work allow
to associate these two effects with a subtle coherent displacement of atoms during
the application of the pump pulse, and with the interaction of the heated electron system with the cold lattice. The changes in phonon frequency and damping
constant with temperature and excitation fluence were studied experimentally and
theoretically. The results of a measurement carried out with two subsequent pump
pulses showed that the bismuth crystal remains in the solid state even though the
energy deposit by the pump pulse is high enough to induce a transition to the liquid
state.
In order to gain a better understanding of the physical origins of the reflectivity
changes and to verify the theoretical model, we carried out a double-probe experiment which allowed for the recovery of the real and imaginary parts of the dielectric
function. The results indicate a periodic change of the band structure due to atomic
motion. The changes of the real and imaginary parts of the dielectric function show
for the first time, that the bismuth sample attains a transient state after some picoseconds. This state neither corresponds to a heated solid sample nor to bismuth
in the liquid state, and its physical origin remains unknown.
iii
Abrégé
Ce travail de thèse a porté sur l’étude des mouvements atomiques cohérents (phonons)
induits par une impulsion laser femtoseconde. L’étude des phonons est un domaine
de recherche fondamental, car ces mouvements atomiques peuvent jouer un rôle clé
dans la compréhension des transitions de phase : effets précurseurs, dynamique,
existence de phases transitoires.
Dans ce cadre, des mesures de réflectivité du semi-métal bismuth après une
excitation laser proche du seuil de dommage du matériau ont été réalisées afin
d’étudier la dynamique des déplacements fins des atomes. Ces expériences ont mis en
évidence deux nouveaux effets de la dynamique de la réflectivité grâce à une haute
résolution temporelle de 35 fs et une grande sensibilité en mesurant des changements de réflectivité de ∆R/R0 = 10−5 . Un état transitoire du bismuth atteint
quelques picosecondes après l’excitation laser, et jamais observé jusqu’à présent, a
été découvert.
L’étude détaillée de la dynamique de la réflectivité a mis en évidence pour la
première fois une chute ultrabrève de la réflectivité qui se déroule pendant les premiers instants de l’excitation. De même, un changement négatif de la réflectivité
apparaissant quelques picosecondes après l’excitation a été mis en évidence. Les
études théoriques réalisées pendant la thèse permettent d’associer ces deux effets,
qui restaient inexplicables avec les théories existantes, à un déplacement cohérent
fin pendant l’application de l’impulsion de pompe et l’interaction du système des
électrons chauffés avec le réseau froid. Les changements de la fréquence et de
l’amortissement du phonon avec la température et le flux d’excitation ont été étudiés
expérimentalement et théoriquement de manière systématique. Grâce à une expérience réalisée avec deux impulsions de pompe, nous avons pu montrer que l’échantillon
reste dans l’état solide alors que l’énergie transférée par l’impulsion laser est suffisamment élevée pour induire une transition vers l’état liquide.
Afin de mieux comprendre les origines physiques du changement de réflectivité et
pour vérifier la validité de notre théorie, nous avons effectué une expérience à deux
impulsions de sonde et déterminé la fonction diélectrique à partir des deux mesures
simultanées de réflectivité. Les résultats contiennent la signature d’un changement
périodique de la structure de bande due aux mouvements atomiques. Les changements de la partie réelle et imaginaire de la fonction diélectrique montrent, pour la
première fois, que le bismuth atteint un état transitoire après quelques picosecondes,
qui ne correspond pas à un état solide chauffé par l’énergie laser, ni à un état liquide.
La nature de cette phase transitoire reste inconnue actuellement.
iv
Contents
Abstract
iii
Abrégé
iv
Acknowledgements
ix
I
Introduction
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1 Research framework
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II Theoretical considerations
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2 Interaction of light with solids
2.1 Optical properties of solids . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The electromagnetic origin of optical properties . . . . . . .
2.1.2 Reflection and refraction of light at an interface . . . . . . .
2.1.3 Optical properties of anisotropic crystals . . . . . . . . . . .
2.2 Optical properties and electronic structure . . . . . . . . . . . . . .
2.2.1 Intraband contributions to the dielectric function: The Drude
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Interband contributions to the dielectric function: The classical oscillator model . . . . . . . . . . . . . . . . . . . . . . .
3 Coherent optical phonons in bismuth crystals
3.1 Phonons - vibrations of the crystal lattice . . . . . . . . .
3.2 Excitation of coherent optical phonons . . . . . . . . . . .
3.2.1 Impulsive Excitation of coherent phonons . . . . . .
3.2.2 Displacive excitation of coherent phonons . . . . . .
3.3 Detection of coherent phonons . . . . . . . . . . . . . . . .
3.4 Properties of coherent optical phonons in bismuth crystals
3.4.1 Crystal structure and phonon modes . . . . . . . .
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Contents
3.4.2
3.4.3
Symmetry properties of optical phonon modes in bismuth . . . 34
Previous work on coherent optical phonons in bismuth . . . . 36
4 A complete model for transient reflectivity in laser-excited bismuth
4.1 Transient properties of a laser-excited solid . . . . . . . . . . . . . .
4.1.1 Two-temperature model . . . . . . . . . . . . . . . . . . . .
4.1.2 Relaxation times: Quasi-equilibrium electron and lattice temperatures and the validity of the TTM . . . . . . . . . . . .
4.1.3 Excitation of electrons . . . . . . . . . . . . . . . . . . . . .
4.1.4 Absorption of energy and electron temperature . . . . . . .
4.2 Laser-induced forces and atomic motion . . . . . . . . . . . . . . .
4.2.1 The stress tensor and related forces . . . . . . . . . . . . . .
4.2.2 Laser-induced atomic motion . . . . . . . . . . . . . . . . .
4.3 Non-linear phenomena related to electron-lattice-equilibration . . .
4.3.1 Anharmonicity of vibrations and shift in equilibrium position
due to lattice heating . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Phonon decay . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Red-shift of the phonon frequency . . . . . . . . . . . . . . .
4.4 Transient optical properties . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Dielectric function . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Transient reflectivity . . . . . . . . . . . . . . . . . . . . . .
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III Experiments and experimental results
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5 Experimental techniques and setups
5.1 Ultrafast measurements with the pump-probe technique
5.2 The femtosecond laser system . . . . . . . . . . . . . .
5.3 Measuring transient reflectivity . . . . . . . . . . . . .
5.3.1 Single- and double-probe setup . . . . . . . . .
5.3.2 High-sensitivity detection system . . . . . . . .
5.3.3 Double-pump setup . . . . . . . . . . . . . . . .
5.4 Recovery of the dielectric function . . . . . . . . . . . .
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6 Reflectivity measurements of coherent optical phonons in bismuth
6.1 Single probe optical measurements . . . . . . . . . . . . . . . . .
6.1.1 Experimental results . . . . . . . . . . . . . . . . . . . . .
6.1.2 Analysis and discussion . . . . . . . . . . . . . . . . . . . .
6.2 Fluence dependence of reflectivity dynamics . . . . . . . . . . . .
6.2.1 Experimental results . . . . . . . . . . . . . . . . . . . . .
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6.2.2
6.2.3
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Analysis and discussion . . . . . . . . . . . . . . . . . . . . .
Accuracy of fluence measurements and the damage threshold
of bismuth . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Temperature dependence of reflectivity dynamics . . . . . . . . . .
6.3.1 Experimental results . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Analysis and discussion . . . . . . . . . . . . . . . . . . . . .
Reflectivity dynamics under double-pump excitation . . . . . . . . .
6.4.1 Experimental results . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Analysis and discussion . . . . . . . . . . . . . . . . . . . . .
Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . .
7 Ultrafast dynamics of the dielectric function in bismuth
7.1 Measurement of the unperturbed dielectric function .
7.2 Time-resolved measurement of the dielectric function
7.3 Error analysis . . . . . . . . . . . . . . . . . . . . . .
7.4 Analysis and discussion . . . . . . . . . . . . . . . . .
7.5 Summary and conclusion . . . . . . . . . . . . . . . .
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IV Conclusions and perspectives
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8 Summary and outlook
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V Résumé en Français
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9 Contexte de travail de recherche
10 Considérations théoriques
10.1 Interaction laser–matière . . . . . . . . . . . . . . . . . . . .
10.1.1 L’origine électromagnétique des propriétés optiques .
10.1.2 Réflexion et réfraction de la lumière à une interface .
10.1.3 Propriétés optiques des cristaux anisotropes . . . . .
10.1.4 Propriétés optiques et structure électronique . . . . .
10.2 Phonons optiques cohérents dans le bismuth . . . . . . . . .
10.2.1 Excitation des phonons optiques cohérents . . . . . .
10.2.2 Détection des phonons optiques cohérents . . . . . .
10.2.3 Structure cristalline et modes normaux du bismuth .
10.3 Un modèle complet de la réflectivité du bismuth photoexcité
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viii
Contents
10.3.1 Propriétés transitoires d’un solide excité par une impulsion
laser ultra–brève . . . . . . . . . . . . . . . . . . . . . . . . . 152
10.3.2 Forces induites par le laser et mouvement atomique . . . . . . 153
10.3.3 Relaxation du phonon et décalage vers le rouge de la fréquence 154
10.3.4 Changements des propriétés optiques . . . . . . . . . . . . . . 155
11 Résultats expérimentaux
11.1 Dispositifs expérimentaux . . . . . . . . . . . . . . . . . . . . . . .
11.2 Mesures de réflectivité après excitation optique simple . . . . . . . .
11.2.1 Mesures pompe–sonde simple . . . . . . . . . . . . . . . . .
11.2.2 Mesures en fonction de la fluence . . . . . . . . . . . . . . .
11.2.3 Mesures en fonction de la température . . . . . . . . . . . .
11.3 Mesures de réflectivité après deux impulsions de pompe . . . . . . .
11.4 Dynamique ultra-rapide de la fonction diélectrique du bismuth . . .
11.4.1 Mesure résolue en temps de la fonction diélectrique à 800 nm
11.4.2 Analyse et discussion . . . . . . . . . . . . . . . . . . . . . .
12 Conclusions et perspectives
VI Appendix
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A Derivatives of the Drude dielectric function and reflectivity
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B Material properties of of bismuth
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References
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Acknowledgements
The fact that the cover page of this manuscript only shows one author’s name may
suggest that this work has been done by a single person. In fact, I am sure that
every scientist knows that such a project always requires a lot of support by others
and good team working. During the three years of my PhD-work at the Laboratoire
d’Optique Appliquée, I was lucky to have a lot of nice people around that would help
me with dozens of different things and without whom achieving a doctor’s degree
would not have been possible.
First of all, I express my gratitude towards Gérard Mourou, the former director of
the LOA, for giving me the opportunity to work at this great place at the forefront
of science.
I want to thank the members of the jury Eric Collet, Marino Marsi, Guillaume
Petite, Olivier Uteza, and Klaus Sokolowski-Tinten, who honoured me with their
participation in the defence of this thesis and their interest in the work that I
presented. My special thanks go to Marino Marsi and Eric Collet who have accepted
to examine and evaluate the manuscript as rapporteurs, and to Guillaume Petite for
being président du jury.
I express my sincere gratitude to Antoine Rousse, the director of my thesis and
the head of groupe PXF. It was a pleasure to work in his group and I thank him
for having confidence in me and for being an excellent and experienced advisor. His
support, his comments, his suggestions and the discussions we had were vital for
this work and always allowed me to make progress.
I consider myself extremely lucky to have worked together closely with Davide
Boschetto. I learned a lot working on experiments with him. He trained me on
basically everything, answered (almost) all of my zillion questions and made uncountable helpful suggestions. During the numerous experiments we did together,
both his positive, happy attitude and his will to advance were a great help and
inspiration. I want to thank him for his sincere and open-minded character, and of
course for the stimulating and pleasant working atmosphere, that he even managed
to keep friendly when it got late and the student got hungry (probably because he
learned quickly that I can be motivated to work late as long as there is dinner at
7pm).
A great contribution to this work originates from the collaboration with our colleagues from overseas, Andrei Rode and Eugene Gamaly, and I am very grateful for
the possibility to work with such bright and experienced scientists. I want to thank
Andrei for the participation in our experiments in Palaiseau and for countless fruitful discussions about our topic. Furthermore, I thank him for being the ’organiser
in chief’ of my stay in Canberra, which was certainly one of the highlights of the
ix
past years. I thank Eugene for his indispensable contributions to the theoretical
part of this work, as well as for the support he provided during the writing of the
manuscript. His detailed answers to all my questions that he provided with great
patience and passion were a very important help.
I am very grateful to Jean Etchepare and Olivier Albert for their contributions to
the experiments and discussions and for being friendly colleagues with an open ear
and a helping hand. I thank Guy Harmoniaux and Armindo Dos Santos for letting
me benefit from their skills and experience and for their support.
I would also like to thank all the other members of the PXF group that I did not
mention yet: David Glijer for his help with the optical experiments and for sharing
the office with me, Kim Ta Phuoc and Romuald Fitour for giving me the pleasure
to work with them on experiments with the betatron source and Nikolai Artemiev,
with whom I worked in salle bordeaux on the plasma source experiment, and who
did a great job in introducing me to everything he set up. I also would like to thank
Felicie Albert, Guillaume Debourg and Barbara Mansart.
I am indebted to many other people at LOA to whom I would like to express my
gratitude. One of the most important ingredients for our experiments was provided
by Laura Antonucci and Gilles Rey: infrared laser light. I thank them for making
sure that the photons coming out of salle rouge always travelled at the speed of
light and for taking care of everything else that had to do with the laser system.
For performing dozens of spider measurements in our lab, I want to thank Brigitte
Mercier. I am very thankful to Jean-Lou Charles et Mikael Martinez from the
workshop for providing us with special mechanical parts on almost fs-time scales
and I appreciate their good sense of humour that made going to the workshop a
pleasure. I am very grateful to the people of la cellule that provided us with all
kinds of electronic and mechanical devices: Thierry Lefrou, Denis Douillet, Gregory
Iaquaniello and Pascal Rousseau. I also want to thank the secretaries at LOA that
helped me a lot with all the paperwork there is to do in a french research lab:
Patricia Toullier, Cathy Sarrazin, Sandrine Bosquet, Valérie Ferragne, and Octavie
Verdun. I thank Fatima Alahyane, Pierre Zaparucha and Arnaud Chiron for taking
care of any computer problem that would occur. Special thanks go to Alain Paris,
not only for being a great help with computer issues, too, but also for watching out
that I don’t drown in the swimming pool during lunch break and for giving me a
calm office which was a great help writing this manuscript.
l would like to thank the people of the AG Röntgenoptik in Jena for the possibilty
to work on a joint experiment in their lab: Eckhard Förster, Ingo Uschmann, Tino
Kämpfer, Sebastian Höfer, Robert Lötzsch, and Ulf Zastrau.
I thank Jennifer K. Barry for her helpful suggestions and corrections of style and
language.
x
Finally, I would probably never have achieved this without the support of my
family and friends. I want express my deep gratitude to my parents and my sisters
for their unconditional support and for being there. I am lucky to have met a lot of
nice and interesting people during my time in Paris, thanks for sharing all the nice
moments with me (you know who you are!). And, last but not least, I deeply thank
meinem Goldfisch Bernadette for going through all this with me . . .
. . . merci à tous, herzlichen Dank, thanks everybody!
I gratefully acknowledge financial support by the 6th framework program of the
European Union through the Marie Curie Training Network “FLASH”.
xi
xii
Contents
Part I
Introduction
1
1 Research framework
The invention of the laser almost 50 years ago [1] marked the beginning of revolutionary developments in many areas of science and engineering. Since its advent,
laser technology has found an enormous variety of applications in physics, chemistry,
biology, material processing, medicine and meteorology. Amongst the numerous possibilities to benefit from the distinct qualities of laser light, the use of pulsed lasers
represents an important field. Continuous development of laser sources has rapidly
lead to the decrease of the temporal width of laser pulses, which went below the
duration of a picosecond for the first time in 1976 [2]. Nowadays the pulse duration
has almost reached the limit of one cycle of an electromagnetic wave in the visible
range which is a few femtoseconds [3].
As a consequence of the short pulse durations, the intensities of the electromagnetic fields compared to continuous–wave light sources can be extreme. These two
qualities of fs–laser pulses have created a new scientific field, called ultrafast phenomena, which spans traditional scientific disciplines of physics, chemistry and biology.
With the availability of fs–light sources, researchers possess tools that have a temporal resolution high enough to investigate atomic motion, phase transitions or the
formation and breaking of chemical bonds in the time domain. By stroboscopically
probing with fs–laser pulses, theses processes, which occur on time–scales ranging
from a few femtoseconds to tens of picoseconds, can be resolved and analysed. Furthermore, the high intensities of the laser sources not only open the way to examine
previously unknown phenomena, but also allow for the development of secondary
sources of radiation and particles with fs–resolution, such as higher harmonics [4],
x–rays [5] and electrons [6].
A major axis in the research field of ultrafast phenomena is the excitation and
detection of coherent lattice vibrations. In the past two decades, generation of large
THz lattice vibrations with a high degree of spatial and temporal coherence has
been observed in a vast variety of transparent and opaque materials among which
are semi–metals [7], transition metals [8], cuprates [9], insulators [10], and semi–
conductors [11]. While phonons have always been a subject of great interest in solid
state physics due to their relation to transport phenomena, the investigation of coherent phonons is of particular interest: the ability to drive and control coherent
lattice vibration via an external photon flux opens a number of interesting appli-
3
4
1. Research framework
cations such as the possibility to induce particular phase transitions (non–thermal
melting [12], paraelectric–to–ferroelectric [13], or insulator–to metal transitions [14]),
the selective opening of the “caps” of nano–tubes in non–equilibrium conditions [15],
or providing a basis for SASER (sound amplification by stimulated emission of radiation) [16]. The great interest in the challenge to investigate, understand and control
ultrafast structural changes is underlined by the enormous efforts that are made
in the development of sub–ps x–ray sources with high brilliance. The realisation of
free–electron lasers as XFEL at DESY in Hamburg or LCLS in Stanford, demanding
budgets of almost a billion Euros, illustrates the importance of this research for the
scientific community.
This work is focused on experiments that investigate coherent optical phonons in
the semi–metal bismuth, which have previously been the subject of several theoretical and experimental investigations. The interest in bismuth is manifold: it has a
relatively simple crystal structure with two atoms in a trigonal unit cell that can be
derived from a cubic structure by applying two slight distortions. Theoretical studies indicate that slightly changing the crystal structure by either increasing the shear
angle or internally displacing the atoms changes the electronic configuration from
semi–metallic to metallic or from semi–metallic to semi–conductor, respectively [17].
It has also been shown that quantum confinement converts bismuth from a semi–
metal to a semi–conductor [18, 19]. Furthermore, a transition to a metallic state can
be induced at high pressures [20]. Coherent atomic motion in bismuth can be studied with different experimental techniques: time resolved optical spectroscopy [21],
which is sensitive to the valence electrons, and time–resolved x–ray diffraction [22],
which is sensitive to the inner electrons and therefore allows for a direct access to
atomic displacement. Previous studies of optical phonons in bismuth yielded a variety of interesting results. Nevertheless, crucial questions concerning the mechanism
of excitation remain unanswered and the properties of the laser–excited state of
bismuth are not understood.
This manuscript presents a detailed study of the dynamics of laser–excited bismuth crystals. The electron and the lattice dynamics following excitation of bismuth
crystals have been studied by measuring the transient changes in reflectivity and dielectric function with a temporal resolution as good as 35 fs. Besides the analysis
of the properties of coherent phonons in bismuth that have been studied in a wide
range of temperatures and excitation fluences, the experiments allowed us to uncover
a novel effect associated with coherent lattice dynamics that cannot be understood
in the light of the existing theories: a sharp drop in reflectivity before the onset of
reflectivity oscillations. This drop is related to a coherent displacement of atoms
during the pump pulse that changes the polarisation–related part of the dielectric
function. Furthermore, new results revealing the optical properties of the transient
5
state that is established after ∼ 20 ps are presented. In particular, measurements of
the transient reflectivity of bismuth, excited with a single pump pulse or two subsequent pump pulses showed that the sample does not undergo a solid–to–liquid phase
transition. The recovery of the dielectric function from a double–probe experiment
suggested that the transient state of the sample after electron–lattice equilibration
is neither described by the optical properties of solid bismuth nor liquid bismuth.
The determination of the changes in dielectric function as a function of temperature showed that the changes in optical properties cannot be attributed to lattice
heating. These results raise new questions concerning the photo–induced state in
bismuth, based on the observation that the pump pulse not only transfers heat to
the system, but also creates a new electronic state that differs from the states in
equilibrium conditions.
Organisation of the dissertation
The manuscript is structured as follows:
In chapter two, the basics of laser–matter interaction are reviseted. The link
between the optical properties of a material and its structure and its electronic configuration is explained. The dielectric function, which is the crucial optical property
for this work, is introduced along with two simple models that allow conclusions
to be made about the link between the electronic and optical properties of a material. Furthermore, the special characteristics of the optical properties of anisotropic
media such as bismuth are briefly considered.
Chapter three reviews two theories of generation and detection of coherent phonons,
that have previously been applied to experimental results on coherent phonons in
bismuth. The main aspects of the theories as well as their limitation are discussed.
In addition, salient results of previous work on optical phonons in bismuth will be
briefly summarised.
In chapter four, theoretical considerations are presented that are intended to explain experimental results of this dissertation. Starting from first principles, expressions that relate the properties of coherent phonons in bismuth to transient
reflectivity are derived, and explanations to novel experimental observations are
offered.
The experimental techniques that were used for the investigation of coherent optical phonons in bismuth are presented in chapter five. The pump–probe technique,
a general concept to investigate ultrafast phenomena, is introduced and the different
experimental setups are described. The chapter also contains the description of a
6
1. Research framework
way to recover the dielectric function from two simultaneous reflectivity measurements and considerations for the ideal experimental conditions based on a numerical
simulation of the errors that occur in recovery.
In chapter six, the measurements of the transient reflectivity of optically excited
bismuth are presented. The results of different series of measurements are carefully
analysed and discussed in the light of our theoretical considerations. The properties
of coherent optical phonons that can be derived from the optical measurements are
presented as functions of excitation fluence and temperature and for different crystal
structures and orientations. Finally, a measurement of the reflectivity dynamics of
bismuth after excitation with two subsequent laser pulses is presented, which allows
one to draw conclusions about the transient properties of bismuth after electron–
lattice equilibration.
Chapter seven presents the dynamics of the dielectric function of laser–excited
bismuth. The change of the real and imaginary parts as recovered from reflectivity
measurements is investigated and the error of recovery is carefully analysed. The
results are discussed in relation to an ellipsometry measurement that investigates the
temperature dependence of the real and imaginary part of the dielectric function.
Chapter eight summarises the results of this work and highlights the most important conclusions. In addition, a number of future experiments that are aimed
at a further understanding of the ultrafast dynamics in fs–laser–excited bismuth is
proposed.
Due to higher degree of manageability and beauty of form, cgs–units are used
throughout the theoretical parts of the manuscript. However, in some cases SI–units
are used in order to be able to compare the results and considerations to related
work more easily. These cases predominantly occur in the chapters that report
experimental results, e.g. laser fluences are always given in mJ/cm2 , not erg/cm2 .
Part II
Theoretical considerations
7
2 Interaction of light with solids
2.1 Optical properties of solids
It is the aim of any optical experiment to gain information about a sample by measuring its optical properties. In this work, light–induced atomic motion is examined
by determining changes in reflectivity and dielectric function. It is thus necessary to
understand both how matter affects light in order to know what one can learn about
a material from its optical properties, and to comprehend how light affects matter
to be able to reveal the processes which are responsible for the measured changes.
This chapter introduces the basic optical properties of solids such as the refractive
index and the dielectric function and summarises the principals of light–matter interaction. The Fresnel formulae, which connect these properties to reflection are
presented, and the dependence on crystal symmetry is pointed out. Finally, the
relationship between dielectric function and electronic structure is presented on the
basis of two common models, which can be used to describe different contributions
to absorption of light by crystal electrons.
2.1.1 The electromagnetic origin of optical properties
The optical properties of a material arise from the nature of light as an electromagnetic wave as well as from the interaction of electric and magnetic fields with the
material. In a vacuum, an electromagnetic wave can be described by the temporal
and spatial evolution of two vectors, namely the electric field E and the magnetic
flux density B. In order to describe the influence of the fields on a material, another
set of vectors is needed: The electric displacement D and the magnetic field H. The
evolution of these fields in an arbitrary medium is governed by Maxwell’s equations,
which relate the space and time derivatives of the four vectors to the free charge
9
10
2. Interaction of light with solids
density ρ and the free current density j:
∇·D =
4πρ ,
∇·B =
0,
1 ∂B
∇×E = − ·
,
c ∂t
1 ∂D 4π
∇×H =
·
+
j.
c ∂t
c
(2.1)
(2.2)
(2.3)
(2.4)
The Maxwell–equations are presented in Gaussian units. The constant c denotes
the velocity of light in vacuum and is approximately 3 · 1010 cm/s. The influence of
the medium on the light field is dictated by the material equations
j = σ·E,
(2.5)
D = ·E,
(2.6)
B = µ·H.
(2.7)
Here σ is called the specific conductivity, stands for the dielectric constant and
µ for the magnetic permeability. Substances for which σ is negligibly small or zero
are called insulators or dielectrics, whereas conductors have a specific conductivity
which is higher. Other classes of materials called semi–conductors are insulators at
absolute zero and exhibit a conductivity which increases with temperature over a
wide range. The latter behaviour can also be observed in semi–metals like bismuth,
however, the conductivity of semi–metals is non–zero at T = 0.
The material equations allow for a unique determination of the field vectors from a
given distribution of charges and currents. The functional forms of D and H can be
very complex. However, it is possible to describe the interaction of field and matter
with a simple model which is adequate for most practical cases. For this purpose,
each of the vectors D and H is expressed as the sum of two terms, of which one
takes into account the contribution of the vacuum field and the other the influence
of matter:
D = E + 4πP ,
(2.8)
H = B − 4πM ,
(2.9)
where the electric polarisation P is the macroscopically averaged electric dipole
moment arising from separation of bound charges in a medium. The magnetic dipole
polarisation (or magnetisation) M is the macroscopically averaged magnetic dipole
moment due to induced bound currents. Higher order moments such as quadrupoles
are usually very small and can be neglected in most cases. For sufficiently weak field
2.1. Optical properties of solids
11
strengths one can assume P and M to be linear in E and H respectively:
P = χe E ,
(2.10)
M = χm H .
(2.11)
The factors χe and χm are called the linear electric and magnetic susceptibilities,
they are related to the dielectric constant and the magnetic permeability by the
formulae
= 1 + 4πχe ,
(2.12)
µ = 1 + 4πχm .
(2.13)
For isotropic materials, the susceptibilities are scalars; otherwise, they are second–
rank tensors that relate each component of P (or M) to the components of E (or
B).
In order to describe the linear optical response of a medium, a set of differential
equations can be derived from Maxwell’s equations using 2.6 and 2.7 and then
eliminating either E or H:
∇2 E −
µ ∂ 2 E
= 0,
c2 ∂t2
∇2 H −
µ ∂ 2 H
= 0.
c2 ∂t2
(2.14)
The case considered here is characterised by the absence of charges and currents,
i.e. j = 0 and ρ = 0. One possible solution to these standard equations of wave
motion is a set of coupled transverse electric and magnetic waves whose frequency ω
is proportional to the magnitude of the wave vector k:
c
ω = √ k.
(2.15)
µ
The phase velocity of the wave is
v=
c
c
ω
=√ = ,
k
µ
n
(2.16)
√
where n = µ denotes the refractive index of the medium. The linear response of
a material to light is fundamentally governed by the refractive index n, which is a
function of the frequency ω, since both the dielectric constant and the the magnetic
permeability depend on ω:
p
n(ω) = (ω)µ(ω) .
(2.17)
For the optical part of the electromagnetic spectrum, the magnetic permeability
can be taken to be µ = 1 [23]. As a result, the optical response of a medium is
solely determined by its response to an oscillating electric field, whereas the oscillating magnetic field has no effect on the material. Thus the optical refractive index
12
2. Interaction of light with solids
depends only on the dielectric function (ω), which is the frequency–dependent dielectric constant:
p
n(ω) = (ω) .
(2.18)
If propagation of light through a semi–conductor or a metal is considered, absorption
of electromagnetic radiation has to be taken into account. In this case, the specific
conductivity is not zero, so an additional term appears in 2.14. In this case, the
wave equations are:
∇2 E −
µ ∂ 2 E 4πµσ ∂E
= 0,
− 2
c2 ∂t2
c
∂t
µ ∂ 2 H 4πµσ ∂H
= 0.
− 2
c2 ∂t2
c
∂t
∇2 H −
(2.19)
The solution of these equations is formally identical with the corresponding ones
in the case of non–conducting media if the dielectric constant is replaced by the
complex quantity
4πσ
ˆ = + i
.
(2.20)
ω
For ease of notation the “hat” will be left out from now on and this complex number
will be referred to as the as the dielectric function consisting of a real and imaginary
part = re + iim . In analogy to the non–absorbing case, a complex index of
√
refraction is defined as n̂ = = η + iκ (which will be displayed without the hat
from now on as well), which consists of two real quantities η and κ of which the
latter is usually called the attenuation index.
Considering the simplest solution to the wave equation for a conducting medium,
a monochromatic plane wave of the form
ω
ω
E = E0 · e−κ c x ei(η c x−ωt) ,
(2.21)
it can be seen that the intensity, which is proportional to the time average of E2 ,
varies according to the equation
ω
I(x) = I(0)e−2κ c x .
(2.22)
The intensity of the electric field decreases exponentially with distance of propagation through the medium, and the rate of attenuation depends on the imaginary
part of the refractive index. Now the absorption coefficient α and its inverse, the
absorption length dabs , which is the distance over which the intensity drops to 1/e
of its initial value, can be defined:
α=
2ω
4π
κ=
κ,
c
λ0
dabs =
1
λ0
=
.
α
4πκ
(2.23)
The skin depth ls is defined as the length over which the current density decreases
by a factor 1/e, therefore it is twice as long as the absorption depth defined in
equation 2.23.
2.1. Optical properties of solids
13
In terms of the dielectric function, the real and imaginary parts of the complex
refractive index are given by
re = η 2 − κ2 ,
im = 2ηκ .
(2.24)
Expressions for η and κ as functions of the real and imaginary part of the dielectric
function can be derived from the above formulae:
s q
1
2
2
η =
re + re + im ,
(2.25)
2
im
κ = r (2.26)
.
p
2
2
2 re + re + im
In the above section, the fundamentals of light-matter interaction have been summarised, and the link between polarisation, magnetisation, and the dielectric function (or refractive index) of a medium has been presented. Now it is instructive to
examine how light waves behave at an interface between to different media and to
relate reflection and transmission of light to the fundamental optical property (ω)
as well as to geometrical conditions.
2.1.2 Reflection and refraction of light at an interface
Once the dielectric function of a material is known, it is possible to deduce all of
the linear optical properties from it. For the experiments presented in this work,
the relation between the reflectivity of a material and its dielectric function (ω) is
of special interest. These two quantities are related by the Fresnel formulae which
will be considered in this section.
When an electromagnetic wave passes through an interface between two media
of different optical properties, it is split into a transmitted wave propagating into
the second medium and a reflected wave propagating back into the first medium, as
depicted in figure 2.1. The relation between the angles of incidence, reflection, and
transmission depends on the indices of refraction or the dielectric functions of the
two media. The amplitudes of the corresponding electric and magnetic fields depend
on n or as well, and they are functions of the polarisation and the angle of incidence
of the light. These relations can be deduced from the boundary conditions of the
fields at an interface, and their derivation can be found in many standard textbooks
on electromagnetism or optics (e.g. [24] and [25]).
The first consequence of the boundary conditions, which can be derived using
the integral versions of Maxwell’s equations, is that all three beams lie in the same
plane perpendicular to the interface, which is referred to as the plane of incidence.
14
2. Interaction of light with solids
kr
ki
qr qi
medium 1
e1
medium 2
e2
qt
kt
Figure 2.1: Wave vectors of incident (ki ), reflected (kr ) and transmitted (kt ) beams at
an interface between two media with different refractive indices
The relation between the angles of the beams with respect to the surface normal are
expressed by the law of reflection and the law of refraction:
θi = θr ,
n1 · sin θi = n2 · sin θt ,
(2.27)
of which the latter is also called Snell’s law. The relation of incident, reflected and
transmitted field amplitudes Ei , Er and Et are governed by the Fresnel formulae [23]
p
√
1 cos θi − 2 − 1 sin2 θi
p
Er,s = rs · Ei,s = √
· Ei,s ,
(2.28)
1 cos θi + 2 − 1 sin2 θi
√
2 1 cos θi
p
Et,s = ts · Ei,s = √
· Ei,s ,
(2.29)
1 cos θi + 2 − 1 sin2 θi
p
2 cos θi − 1 (2 − 1 sin2 θi )
p
· Ei,p ,
(2.30)
Er,p = rp · Ei,p =
2 cos θi + 1 (2 − 1 sin2 θi )
22 cos θi
p
Et,p = tp · Ei,p =
· Ei,p .
(2.31)
2 cos θi + 1 (2 − 1 sin2 θi )
Here, the indices s and p denote components of the field which are s– and p–polarised,
meaning perpendicular and parallel to the plane of incidence, respectively. The
power reflectivity and transmittivity is then given by the absolute square of the
2.1. Optical properties of solids
15
1 ,0
0 ,9
R e fle c tiv ity
0 ,8
0 ,7
0 ,6
0 ,5
0 ,4
0 ,3
0
1 0
2 0
3 0
4 0
5 0
6 0
Angle of incidence / °
7 0
8 0
9 0
Figure 2.2: Reflectivity of an absorbing isotropic crystal for s–polarised (solid curve) and
p–polarised light (dashed curve) as a function of the incident angle.
Fresnel coefficients rs , rp , ts , and tp , which are complex numbers for a typical material. In the case of an interface between a medium with n = 1 and another with
√
√
n = = re + iim , the reflectivity for s– and p–polarised light is expressed as:
cos θ − p + i − sin2 θ 2
i
re
im
i
p
(2.32)
Rs = ,
cos θi + re + iim − sin2 θi ( + i ) cos θ − p + i − sin2 θ 2
re
im
i
im
i
p re
Rp = (2.33)
.
(re + iim ) cos θi + re + iim − sin2 θi Figure 2.2 shows an example of the reflectivity of s– and p–polarised light of an
isotropic absorbing crystal. For s–polarised light, the reflectivity monotonically increases with the incident angle. In the case of p–polarisation, there is a minimum at
the Brewster angle αB . For non–absorbing media, the reflectivity at αB is zero, in
the depicted case, it is slightly higher than 30%. At normal incidence, the electromagnetic fields are parallel to the surface, and the boundary conditions of the fields
at the sample surface do not depend on polarisation. Therefore, the reflectivities for
s– and p–polarised light are equal.
2.1.3 Optical properties of anisotropic crystals
In the last section, formulae for the reflectivity have been derived under the assumption that the crystals are isotropic and thus have a dielectric function that can be
16
2. Interaction of light with solids
treated as a scalar. In particular, this means that the direction of the vectors D
and E is the same, as can be seen from equation 2.6. In order to take account of
anisotropy, this assumption has to be dropped and replaced by a relation in which
each component of D is related to the components of E

Dx = xx Ex + xy Ey + xz Ez

D = ·E,
(2.34)
Dy = yx Ex + yy Ey + yz Ez

Dz = zx Ex + zy Ey + zz Ez
in which the dielectric function is a second–rank tensor with nine elements. As a
result, the refractive index is a tensor as well, and equation 2.18 stays valid for the
components of n and . It can be shown that even for an anisotropic crystal the
dielectric tensor must be symmetric, so that the number of individual components is
reduced to six [23]. Furthermore, a transformation of the coordinate system allows
to express the material equation 2.34 in a system of principle dielectric axes such
that it takes the simple form [24]:
Dx = x Ex ,
Dy = y Ey ,
Dz = z Ez .
(2.35)
Here, x , y , and z are called the principal dielectric constants, and it can be seen
that E and D have different directions unless the orientation of E coincides with
one of the principal axes.
If, in addition to anisotropy, the crystal is absorbing, the conductivity has to
be taken into account via the conductivity tensor σ, which is also symmetric. In
general, the principal axes of the dielectric tensor and the conductivity tensor are
not the same, making the theory of propagation of light very complex. However,
they coincide for crystals having a symmetry which is as high or higher than the
symmetry of orthorhombic crystals. This is also the case for bismuth crystals which
are rhomboedric.
Another feature of the crystalline state of bismuth is that it is uniaxial. Such
materials have crystal symmetries that allow for a distinction between a c–axis and
an ab–plane. Hence, the number of individual elements of the dielectric function is
reduced to two. For an ordinary wave, which is an electric field perpendicular to the
optical axis, optical properties are described by o . In the case of an extraordinary
wave consisting of an electric field parallel to the optical axis, the optical properties
are described by e . Reflectance from planes parallel to the optical axis (basal plane)
of a uniaxial crystal is depicted in figure 2.3. If the incident light is s–polarised, it
is an ordinary wave, and the amplitude of the reflected wave is expressed as [26]:
p
cos θ − n2o − sin2 θ
r
p
Ey =
· Eyi .
(2.36)
2
2
cos θ + no − sin θ
2.2. Optical properties and electronic structure
17
Figure 2.3: Reflection of an ordinary (Ey ) and an extraordinary wave (z–component of
Exz ) off the basal plane of a uniaxial crystal, whose optical axis is indicated by the dashed
line
√
Using no = o shows the analogy the reflectivity of an isotropic crystal described
by equation 2.32. In the case of p–polarised light the expression becomes more
complicated due to the fact that the wave sees an “effective refractive index” because
the electric field has components that are parallel and normal to the optical axis:
p
no ne cos θ − n2e − sin2 θ
r
i
p
Exz =
· Exz
.
(2.37)
2
2
no ne cos θ + ne − sin θ
2.2 Optical properties and electronic structure
In general, electromagnetic waves interact with all the charged constituents in a
material. If we think of a medium as a system of valence electrons and ions, the
interaction of light with it should consist of contributions from both subsystems.
Due to the fact that the ion mass is much higher than the electron mass, the fields
have a negligible effect on the ions. Therefore the optical properties are solely
determined by the processes of interaction between the crystal electrons and the
electromagnetic waves.
Depending on the class of material, there are different contributions to the dielectric function which originate from the various ways crystal electrons can interact
with an electromagnetic field. In a metal, the dominating contribution arises from
free–carrier absorption. The electrons undergo intraband transitions, processes in
18
2. Interaction of light with solids
which they are excited from an occupied to an unoccupied state in the same band.
To absorb photon energy, the electron has to gain momentum and for the sake
of momentum conservation a third particle, e.g. a phonon or an impurity has to
be involved to carry away the additional momentum. In semi–conductors, characterised by a relatively small band–gap between the highest occupied energy band
(valence band) and the lowest non–occupied band (conduction band), the dominant
processes are interband transitions, in which an electron is excited from the valence
to the conduction band. Because of the different nature of these contributions, it is
useful to write the dielectric constant as
(ω) = 1 + 4π(χinterband + χintraband ) .
(2.38)
In this section, the two different contributions will be examined closer. A microscopic theory would involve a calculation of the power loss due to absorption, which
is given by the product of the photon energy and the transition probability per
unit time. Expressions for re and im can be derived from the connection of the
absorption coefficient to the imaginary part of the dielectric function with the help
of the Kramers–Kronig–relations [24]. However, the dielectric function can also be
described by simple models which are able to reproduce the response of the electron system to light at optical frequencies reasonably well. Since for the analysis
of the results presented in this work these models are sufficient, a description of
the quantum–mechanical or semi–classical derivations of microscopic theory is out
of the scope of this work. The reader is referred to textbooks on solid state or
semi–conductor physics that treat this subject, e.g. [27] or [28].
2.2.1 Intraband contributions to the dielectric function: The
Drude model
The interband contribution to the dielectric function will be described in the scope
of the Drude model, developed by Paul Drude in 1900 in order to explain the conductivity of metals. The basic assumption of this model is that the valence electrons
in a metal are detached from the atoms and can propagate almost freely through
the metal while the ions are immobile. The electrons are treated as a dilute gas
and interact with ions via collisions. Electromagnetic interaction of electrons with
other electrons or ions is neglected. The probability of collision per unit time is 1/τ ,
and the time τ is known as the relaxation time of the electrons. In the presence of
an applied electrical field E, the equation of motion of an electron with charge e is
taken to be [29]
dp p
+ + eE = 0 .
(2.39)
dt
τ
2.2. Optical properties and electronic structure
19
If the electric field is oscillating with a frequency ω, equation 2.39 leads to the
following expression for p:
eE(ω)
p(ω) = 1
.
(2.40)
− iω
τ
Since the current density j = −ne ep/me = σE, with me denoting the electron mass
and ne the number of electrons per unit volume, one can derive
σ(ω) =
ne e 2 τ
.
me (1 − iωτ )
(2.41)
Using this result together with equation 2.20 leads to the Drude form of the dielectric
function:
ne e2 τ
(ω) = 1 + 4πi ∗
,
(2.42)
me (1 − iωτ )
which, after the separation of real and imaginary parts can be displayed in the
following way:
= re + iim
ωp2
ωp2
νe−ph
=1− 2
+ i 2
.
2
2
ω + νe−ph
ω + νe−ph ω
(2.43)
Here, ωp = (4πe2 ne /m∗e )1/2 is the plasma frequency, and νe−ph = 1/τ is the electron–
phonon momentum exchange rate. In the last two equations, the so–called electron
effective mass m∗e is used instead of the free electron mass me , because the former
takes into account effects of the periodic lattice potential which alter the response
of a crystal electron from that of a free one.
With the Drude model, important features of the dielectric functions of good
metals such as copper or aluminium can be described. However, in a great number
of materials the assumptions of the Drude model are not valid and contributions of
bound electrons have to be taken into account. This can be done with the classical
oscillator model presented in the next section. Due to the large amount of carriers
excited by a high–intensity laser pulse, the Drude form of the dielectric function
is well–suited to explain properties of femtosecond–laser–excited bismuth at the
wavelength used in the experiments of this work.
2.2.2 Interband contributions to the dielectric function: The
classical oscillator model
When examining electrons in a material that cannot move as freely as carriers in
a metal, one can think of them as charged particles of mass me and charge e attached to an ion of infinite mass by a spring. Therefore, each electron corresponds
20
2. Interaction of light with solids
to an harmonic oscillator characterised by a resonance frequency ωi and a damping coefficient Γi . The dielectric function can now be calculated by evaluating the
polarisation induced by an electric field oscillating with a frequency ω. Then, the
displacement x(ω) of the bound electron in this model, which is also referred to as
the Lorentz–model, is given by
d2 x
e
dx
+ ω02 x +
E(ω) = 0 .
+Γ
2
dt
dt
me
(2.44)
In the above equation, Γ denotes the damping coefficient and ω0 the resonance
frequency of the oscillator. Solving the equation yields the resulting dipole moment
p(ω) = ex(ω) =
e2 E(w)
.
me (ω02 − ω 2 + iωΓ)
(2.45)
Taking into account that there are ne electrons per unit volume, and using equation 2.10, an expression for χ is retrieved, and the dielectric function can be expressed
as:
4πne e2
(ω) = 1 +
.
(2.46)
me (ω02 − ω 2 + iωΓ)
The separation of real and imaginary parts results in
(ω) = 1 −
4πne e2
ω 2 − ω02
4πne e2
ωΓ
+
i
·
. (2.47)
2
2
me (ω 2 − ω0 )2 + (ωΓ)2
me (ω 2 − ω0 )2 + (ωΓ)2
The above equation is valid for a crystal in which every electron shows the same
behaviour. To take into account different groups of electrons, the model can be
extended to a multiple oscillator model. One can assume that a group of electrons
j, which is a fraction fj of the total number of electrons, corresponds to a set
of oscillators with resonant frequency ωj and damping coefficient Γj . Then, the
dielectric function can be expressed as:
(ω) = 1 +
4πne e2 X
fj
.
2
2 + iωΓ
me
ω
−
ω
j
j
j
(2.48)
Equation 2.46 is a very general concept for the dispersion of the dielectric constant,
and it can be used to model a variety of processes which differ substantially in their
physical origin. When comparing the equation of motion of the Drude model in 2.39
and the resulting expression for the dielectric function in 2.42 to the corresponding
equations of the Lorentz–model, it can be seen that by setting ωj = 0 and Γ = 1/τ ,
the Drude expressions can be derived from the equations of the Lorentz–model. This
is due to the fact that in a metal the electrons do not experience a repulsive force
and that energy dissipation is taken into account by the collision time, so that the
electrons behave like oscillators with eigenfrequency zero.
3 Coherent optical phonons in
bismuth crystals
Since laser pulses with durations sufficiently shorter than the oscillation period of
fundamental lattice vibrations are available, there has been a large number of reports
on coherent excitation of these modes in a variety of different materials. Among these
materials is the semi-metal bismuth, which has optical phonon modes that have been
subject of numerous publications. One of the distinctive features of this material is
that despite the fact that it has been intensively studied for its unique electrical and
thermal properties and their applications, the investigation of its coherent phonon
modes has produced considerable scientific debates. A striking example is the lack
of clarity concerning the mechanism of excitation of coherent lattice vibrations,
which was seemingly resolved by the introduction of a new theoretical approach
but then again challenged by later experimental findings. Likewise, the origin of
an observed shift of the phonon frequency was highly controversial until recently.
Still, numerous questions concerning experimental observations of coherent optical
phonons in bismuth remain unanswered.
It is thus the goal of this chapter to familiarise the reader with the concepts of
excitation and detection of optical phonons and to summarise previous experimental
results. Following a general introduction covering the very basics of lattice vibrations, an overview of two theoretical approaches for excitation of coherent phonons
in solids will be delivered. Thereafter, it is presented how coherent lattice vibrations
alter the optical properties of solids according to the two theories, and thus allow
for detection by measuring the transient changes in optical reflectivity. Finally, the
properties of optical phonons in bismuth will be described, and an overview of salient
results of previous experiments will be given.
3.1 Phonons - vibrations of the crystal lattice
The vibrations of atoms in a crystal lattice of a solid around their equilibrium
positions can be decomposed into a linear combination of normal modes, which
have their origin in crystal symmetry. Every normal mode is characterised by a
21
22
3. Coherent optical phonons in bismuth crystals
ω( k )
o p tic a l b r a n c h
a c o u s tic m o d e
a c o u s tic b r a n c h
o p tic a l m o d e
- π/ a
π/ a
k
Figure 3.1: Diatomic linear chain: Acoustic and optical mode (left) and dispersion relation
of phonons (right), a is the inter-atomic distance in the chain.
frequency ν and a wave vector k, and a boson with energy E = h · ν and momentum
p = h · k can be associated to each of them. These bosons are called phonons and
represent the quanta of energy of lattice vibrations.
As a first approach, information about the classical normal modes can be obtained
by solving the equation of motion for atoms in a one-dimensional crystal, i.e. a linear
chain of atoms, by using an harmonic approximation for the potential energy of the
atoms. The two simplest cases are a crystal with a single atom per elementary cell
(monoatomic chain), and a crystal with two atoms per elementary cell (diatomic
chain). Solutions for both cases describe waves with a wave vector k and frequency
ω propagating along the chain with a phase velocity c = ω/k and a group velocity
v = ∂ω/∂k. They differ in their relation between frequency ω and wave vector k:
for the mono-atomic chain the dispersion relation has a single branch whereas in
the case of the diatomic chain there are two branches. The dispersion relation is
depicted in figure 3.1. The lower branch is referred to as the acoustic branch because
its dispersion relation is of the form ω = ck for small k which is characteristic of
sound waves. The upper branch is named optical branch because the optical modes
can interact with electromagnetic radiation. In contrast to the acoustic phonon, the
optical phonon’s frequency is non-zero at k = 0.
Solution of the equation of motion for an atom in a three-dimensional crystal
with a basis of p atoms leads to a dispersion relation with 3p different branches, 3 of
them corresponding to acoustic and 3p − 3 to optical normal modes. The dispersion
relation of phonons in bismuth, which is a crystal with two atoms per elementary
cell, is depicted in figure 3.2. For each type of phonon, optical and acoustical,
there is one longitudinal and one transverse mode, meaning that the direction of
3.2. Excitation of coherent optical phonons
23
atomic displacement is parallel or perpendicular to the direction of propagation,
respectively. Due to the uniaxial crystal symmetry both transverse modes are double
degenerate such that there is a total number of 2 + 4 = 6 branches.
B is m u th a t 7 5 K
tr ig o n a l d ir e c tio n
tr ig o n a l d ir e c tio n
F re q u e n c y / T H z
3 ,0
2 ,0
1 ,0
0 ,0
0 ,0
0 ,2
0 ,4
0 ,6
0 ,8
1 ,0
R e d u c e d w a v e v e c to r
Figure 3.2: Dispersion relation of phonons in bismuth at 75 K determined with neutron
scattering (from [30]). LO and TO stand for longitudinal and transverse optical phonons,
LA and TA for longitudinal and transverse acoustic phonons, respectively.
3.2 Excitation of coherent optical phonons
Two basic features can be associated with optical phonons that are excited by an
ultrashort laser pulse. First of all, the atoms are excited coherently, meaning that
all the atoms oscillate with a constant phase relation. This makes it possible to
observe their transient movement in real-time and not just a time-averaged mean
displacement of atoms. Second, the excitation takes place at a well-defined time
t = 0 corresponding to the time where the pump pulse reaches the sample surface,
which gives the opportunity to investigate the dynamics of relaxation of the system.
In the following, two theoretical concepts for excitation of coherent optical phonons
shall be described; impulsive stimulated Raman scattering (ISRS) and displacive excitation of coherent phonons (DECP). To a certain extent this is a “historical” ap-
24
3. Coherent optical phonons in bismuth crystals
proach: while the former has been successfully applied to explain coherent phonon
generation in liquids and transparent crystals since the 1970s [31, 32], the latter was
proposed in 1990 to explain observations made in certain opaque materials that,
at a first glance, did not coincide with ISRS [33]. Later, additional experimental
evidence lead to a theoretical study that rendered DECP a special case of ISRS,
emphasising the close connection between these mechanisms. Despite the fact that
DECP cannot explain important experimental results of this work, it will be treated
here for its comprehensive straight-forward approach and certain parallels to the
model describing the transient reflectivity perturbed by a coherent phonon field,
which will be developed below.
Both theories are based on the same general equation of motion of the phonon,
describing the time dependance of the normal coordinate q of the atom with mass
m by the means of a driven harmonic oscillator:
d2 q
dq
F (t)
+ 2γ + ω02 q =
,
2
dt
dt
m
(3.1)
where ω0 is the frequency of the phonon, γ is the damping constant which is phenomenologically introduced to take into account the multiple decay mechanisms of
the phonon, and F (t) denotes the external force due to the electric field of the
pump laser pulse. The damping constant γ is related to the damping time τ via
γ = 1/τ . There are phase-destroying mechanisms as well as population-decreasing
mechanisms which contribute to the damping time [34].
As we will see in the next sections, the main difference between the two theoretical
approaches is the underlying mechanism of the driving force. It leads either to an
impulsive force, which can be described by a δ-function if the laser pulse is sufficiently
short compared to the phonon period, or a displacive force of step-like form like a
Heaviside-function. This difference can be illustrated by the simple analogy of a
pendulum: while the force associated with the Raman-based excitation scheme can
be thought of as a kick-off changing the kinetic energy, the displacive mechanism
modifies the potential energy of the pendulum.
3.2.1 Impulsive Excitation of coherent phonons
In the framework of ISRS, optical phonons are excited by a Raman process, which
is a two-photon scattering process. The two different kinds of Raman scattering are
illustrated in figure 3.3 a): a material in the state |gi undergoes a transition to the
final state |ni via a virtual level associated with an excited state |n0 i by absorbing a
photon with frequency ω and emitting a photon with frequency ωs , while the energy
difference h̄ω0 = h̄(ω − ωs ) is carried away by a phonon with frequency ω0 . This is
3.2. Excitation of coherent optical phonons
E
25
a)
b)
n'
n'
I(w)
virtual levels
hw
hw
n
n
g
w2- w1= w0
hwa
hws
vibrational levels
g
w1
wl
w2
w
Figure 3.3: a) Stokes and anti-Stokes scattering process. b) Spectral profile of a laser
pulse with central frequency ωl and a couple of frequencies ω1 and ω2 contained in the
spectral width
referred to as Stokes scattering, and the scattered light is called Stokes line. The
much weaker anti-Stokes line is created during the anti-Stokes scattering process,
where the transition goes from |ni to |gi via a virtual level. In this process one
photon of frequency ω is absorbed, one of frequency ωa is emitted, and the energy
difference is compensated by annihilation of a phonon of frequency ω0 = ωa − ω.
Stimulated Raman scattering may be achieved in two different ways. One method
is to focus an intense laser of frequency ωl into a medium. If it has a Raman active
vibrational mode, which means that a change of polarisability is induced by the
vibration, coherent emission at a frequencies ωl − ω0 and ωl + ω0 can be observed,
where ω0 is the frequency of vibration. Another method is to spatially overlap two
laser beams of frequencies and wave vectors (ω1 , k1 ) and (ω2 , k2 ), which fulfil the
condition
ω2 − ω1 = ω0
(3.2)
in the material. By stimulated scattering, light with frequency ω1 is amplified while
the output at ω2 gets weaker, and a coherent vibrational wave characterised by
frequency ω0 and wave vector k0 = k2 − k1 is generated inside the material [35].
If a pulsed laser is used, the two frequencies can also be contained in the spectral
width ∆ωl of the laser and may therefore be taken from one single pulse, a situation
illustrated in figure 3.3 b). If ∆ωl ω, there is a large number of couples of
frequency ω1 and ω2 which fulfil equation 3.2.
The problem of coupled electromagnetic and lattice waves can be solved with a
classical model which leads to expressions for the normal coordinate q. The key
assumption is that the polarisability α of the material is not constant, but depends
26
3. Coherent optical phonons in bismuth crystals
on the inter-atomic distance. In the Placzek model [36], α is expressed as a linear
function of q according to the equation:
∂α
α(t) = α0 +
q(t) .
(3.3)
∂q 0
Here, α0 is the polarisability of the material for an inter-atomic distance fixed at its
equilibrium value. If an external optical field E(z, t) is applied (for simplicity, we
consider a linear polarised wave propagating in z-direction), a dipole moment will
be induced. It is expressed as:
P(z, t) = α · E(z, t) .
(3.4)
The energy needed to establish the oscillating dipole moment P is:
1
1
W = P(z, t) · E(z, t) = (α · E(z, t)) · E(z, t) .
(3.5)
2
2
Thus, the external field exerts a force on the vibrational degree of freedom which is
given by:
dW
1 ∂α
F(t) =
=
: E(z, t)E(z, t) .
(3.6)
dq
2 ∂q 0
Inserting the expression for F(t) into the equation of motion 3.1 we obtain:
dq 2
dq
∂α
1
2
: E2 (z, t) ,
+ 2γ + ω0 q = N
2
dt
dt
2
∂q 0
(3.7)
where N is the number of oscillators per unit volume and α is the differential polarisability tensor which can be expressed as:


αxx αxy αxz
α = αyx αyy αyz  .
(3.8)
αzx αzy αzz
The elements of this tensor are derived from crystal symmetry and are zero or nonzero depending on the selection rules which limit the number of allowed modes that
can be excited via a Raman process. This allows for the excitation of the desired
modes or combination of modes by choosing the right polarisation of the pump light.
Once several modes are excited, the desired mode can be probed depending on the
choice of polarisation of the probe light.
We now assume that incident and scattered light, as well as the material vibration,
are linearly polarised in y-direction, so that the only element that has to be considered is αyy . In addition, we assume an optically isotropic, dispersion-free medium.
Then equation 3.7 can be rewritten as:
∂ 2q
∂q
1
+ 2γ
+ ω02 q = N α0 E 2
2
∂t
t
2
where α0 = (∂αyy /∂q)0 .
(3.9)
3.2. Excitation of coherent optical phonons
27
For convenience and in good agreement with experimental reality, we consider the
incident pump laser pulse to be described by a Gaussian
E = A · e−(t−zn/c)
2 /(2τ 2 )
l
cos [ωl (t − zn/c)]
(3.10)
with electric field amplitude A, pulse duration τl and central frequency ωl . Now the
equation of motion is expressed as:
∂ 2q
1
∂q
2
2
+ ω02 q = N α0 A2 e−(t−zn/c) /(τl ) ,
+ 2γ
2
∂t
t
4
(3.11)
where a high-frequency term on the right side has been neglected as it does not
contribute effectively to the driving of the vibrational mode. Equation 10.37 shows
that a spatially uniform, temporally impulsive force is exerted on a Raman active
vibrational mode. If we define t = 0 as the time at which the center of the pump
pulse arrives at the sample surface located at z = 0, a solution with a Green’s
function method for small γ yields [35]:
q(z > 0, t > 0) = q0 e−γ(t−zn/c) sin [ω0 (t − zn/c)] .
(3.12)
The expression for the vibrational amplitude is:
√
q0 =
2π
π
2 2
2 2
N α0 A2 τl e−ω0 τl /4 =
FN α0 e−ω0 τl /4 ,
4ω0
ω0 nc
(3.13)
√
where F = ncA2 τl /(8 π) is the integrated intensity of the pulse, which is often
referred to as laser fluence.
The result of this derivation merits some discussion: it can be seen from equation 3.12, that through ISRS an ultrashort laser pulse produces a vibrational wave,
that has a well-defined, spatially uniform phase if the pump pulse is spatially uniform in the transverse direction. This means that, in fact, the phonon is coherent,
which is a fundamental feature since it enables us to resolve the oscillations in time,
as we will see below. The amplitude of the phonon is proportional to the fluence
2 2
and depends on the product of frequency ω0 and pulse length τl in the term e−ω0 τl /4 ,
so the shorter the pulse duration is, the more the atoms are displaced from their
equilibrium positions. Furthermore, the oscillations are described by a sinusoidal
function, which is characteristic for excitation with an impulsive force. As we will
see, this is not the case in the DECP model, which is an important difference because the initial phase has been a widespread argument in literature of the past two
decades to support either an impulsive or a displacive type of excitation.
28
3. Coherent optical phonons in bismuth crystals
E
s
ed
cit
x
e
ground
u0
e
tat
state
u1
x
Figure 3.4: Schematic illustration of DECP: Excitation of carriers changes the atom’s
equilibrium coordinate from its initial position u0 to u1 on an excited potential energy
surface kick-starting the atomic vibration.
3.2.2 Displacive excitation of coherent phonons
The DECP-model was derived by Zeiger and co-workers in the early 90s and aimed
at explaining the characteristics of time-resolved optical measurements of coherent
optical phonons in opaque crystals like Sb, Bi, Te and Ti2 O3 . In time-resolved reflectivity measurements, oscillations with the frequency of the A1g -mode had been
observed while the Eg -mode had not been detected, although both modes are Raman active. This discrepancy with ISRS, together with the fact that the oscillations
showed a cosine-like behaviour, led the authors to the alternative excitation mechanism [21], which shall be presented here.
In the DECP process, the equilibrium positions of the atoms are substantially
altered due to high-density photo-excitation of carriers induced by the pump light,
as it is shown in fig 3.4. The electronically excited system then goes into a quasiequilibrium state in a time which is short compared to the equilibration time of
the lattice. This leads to an excitation of A1g atomic vibrations with k ∼ 0. Since
this approach applies to absorbing materials, laser excitation does not only create
a great number of free carriers but also changes the electron temperature Te , which
can also be the reason for phonon excitation. The theory does not distinguish if
the rise in electron temperature or excited carrier density is the dominant source of
excitation. The result for ∆R(t) is similar if the nuclear coordinate q0 is assumed
to be proportional to either n(t) or Te . In the following, the phenomenological
derivation of DECP as it has been performed in the original work by Zeiger et al. will
be presented, where the process of phonon generation is associated with the laser-
3.2. Excitation of coherent optical phonons
29
induced change in carrier density n(t). A microscopic treatment can be found in [37].
The source of phonon excitation is a change in the quasi-equilibrium coordinate
q0 (t), where q0 (0) defines the position before the arrival of the pump pulse. It is
assumed that q0 (0) is linearly related to the density of excited carriers n(t) according
to [21]
q0 (t) = ζ · n(t) ,
(3.14)
where ζ is a constant of proportionality. The rate of change of n(t) is taken to be
dn(t)
= ρP (t) − βn(t) ,
dt
(3.15)
which is the sum of the rate of generation of carriers in excited bands and the electron
relaxation rate. The augend is assumed to be proportional to the power density P (t),
while the addend is governed by the time constant of electron relaxation, which is
the inverse of β. The intensity profile of the ultrashort laser pulse can be expressed
as
P (t) = Epump · g(t) ,
(3.16)
where Epump stands for maximum intensity and g(t) denotes the normalised envelope
function
Z∞
g(t)dt = 1 .
(3.17)
−∞
Hence, the density of free carriers is
Z∞
j(t) = ρEpump
g(t − τ )e−βτ dτ .
(3.18)
0
Supposing that interband transitions are the major contribution to the absorption
of the exiting laser pulse, ρ can be written as
ρ=f
2κ
(1 − R) ,
h̄c
(3.19)
where R denotes the reflectivity, κ the imaginary part of the refractive index, and f
the fraction of electrons transferred in intraband transition that are left in excited
bands after the electron system has returned to a quasi-equilibrium state.
The constant of proportionality ζ, appearing in equation 3.14, can be expressed
as a function of material parameters considering a thermodynamic description of
quasi-equilibrium [21]:
2∆
.
(3.20)
ζ=−
qeq (0)µω02 n0
30
3. Coherent optical phonons in bismuth crystals
Here, µ is the (nuclear) effective mass per unit cell, n0 is the number of unit cells per
unit volume, qeq is the equilibrium displacement from a higher symmetry structure
(see section 3.4.1) and ∆ is the characteristic energy splitting in the band structure
due to a symmetry-lowering displacement in the crystal.
The temporal evolution of the normal coordinate is described by the following
equation:
d2 q
dq
(3.21)
+ 2γ + ω02 q = ω02 q0 (t) = ω02 ζ n(t) .
2
dt
dt
Under the assumption that the frequency of the phonon mode ω0 does not depend
on the level of excitation, the solution can be written as:
ω 2 ζρEpump
q(t) = 2 0 2
ω0 + β − 2γβ
Z∞
β0
−βτ
−γτ
g(t − τ ) e
−e
cos(Ωt) − sin(Ωt)
,
Ω
(3.22)
0
where
q
Ω = ω02 − γ 2 ,
and
β0 = β − γ .
(3.23)
Equation 3.22 shows that the nuclear displacement associated with the normal mode
consists of two contributions, an exponentially decaying and an oscillatory part. The
decaying part can be attributed to the change of equilibrium position due to the large
number of excited carriers. It decreases when the electrons relax and finally returns
to zero. The oscillatory part is associated with a damped oscillation characterised
by a frequency Ω and a damping constant γ.
The maximum displacement depends on the form of the pump pulse. It attains its
maximum value when the envelope function can be considered as a delta-function
g(t) = δ(t), which is valid if the pulse duration is largely inferior to the oscillation
period. If on the other hand it is longer than the oscillation period, the oscillatory
part is suppressed. If the electrons relax quickly to the fundamental state, meaning
that β is large, the fraction ω02 /(ω0 − β)2 → 0 and therefore q0 (t) → 0. A small value
of β produces oscillations with large amplitudes, and due to the fact that β 0 /Ω is
negligible, the oscillatory part has a cosine-like form in contrast to a the sinusoidal
behaviour of the ISRS-mechanism. In this case, ω02 /(ω0 − β)2 → 1 and the phonon
amplitude can be expressed as:
q0 = ζρEpump = ζnmax = −
2∆
nmax .
qeq (0)µω02 n0
(3.24)
It can be seen that it is proportional to the maximum excited carrier density nmax
at t = 0.
3.3. Detection of coherent phonons
31
3.3 Detection of coherent phonons
There are several possibilities to detect coherent phonons in crystals in the time
domain. One of them is to observe changes in the optical properties that are induced
by the coherent lattice displacement in the investigated material. In this work, timeresolved reflection measurements are presented, and the attention shall be focused
on this class of detection.
The basic idea behind all theoretical modelling of phonon-induced reflectivity
changes is to assume that the changes with respect to unperturbed reflectivity ∆R
are small. In this case, they can be expressed in the following way:
∂R
∂R
∆R(t) =
· ∆ξre (t) +
· ∆ξim (t) .
(3.25)
∂ξre 0
∂ξim 0
Here, ξre and ξim denote the real and imaginary parts of the respective optical
property considered, for example the dielectric function or the refractive index. The
index zero indicates that the derivatives are determined from the unperturbed values
of this property. The temporal variation of the real and imaginary parts ∆ξre (t) and
∆ξim (t) is then related to underlying material properties that exhibit changes due
to laser excitation and phonon generation.
In the TSRS approach, the transient reflectivity is expressed via phonon-induced
changes of the complex linear susceptibility χ = χre + iχim :
∂R
∂R
∆R(t) = |∂(χre + iχim )/∂q|
cos η +
sin η · q(t) ,
(3.26)
∂χre
∂χim
where η is the phase of ∂χ/∂q. This formula only accounts for reflectivity changes
caused by coherent lattice displacement to which the change in reflectivity is proportional, and effects related to the excitation of electrons are neglected. For this reason,
the formula cannot be used to model the transient reflectivity in photo-excited bismuth. In addition, the values of the elements of ∂χ/∂q are not known, so that this
expression cannot be used to quantitatively predict the oscillatory contribution to
the reflectivity changes.
In the framework of DECP, transient reflectivity changes are related to three
effects: the excitation of electrons, the change of electron temperature and the
coherent atomic motion. The variation of reflectivity is expressed as:
1
∂R
∂R
∂R
∆R(t)
=
n(t) +
Te (t) +
q(t) ,
(3.27)
R
R
∂n
∂Te
∂q
where R is the unperturbed reflectivity at t < 0 before the arrival of the pump
pulse. As it has been discussed in section 3.2.2, there is no distinction between
32
3. Coherent optical phonons in bismuth crystals
effects related to n(t) and those related to Te (t), and n(t) is assumed to be the
dominant source driving q(t), so that the term proportional to Te in equation 3.27
can be neglected. Considering the fact that the measured reflectivity changes are
convoluted with the pulse duration of the laser one obtains from equations 3.18, 3.22
and 3.27:
∆R
=A
R
Z∞
G(t − τ )e−βτ dτ
0
+B
ω02
+
ω02
β2 −
Z∞
2γβ
(3.28)
G(t − τ ) e−βτ − e−γτ
0
cos(Ωτ ) −
β
sin(Ωτ )
Ω
dτ ,
0
where
Z∞
g(t − τ )g(τ )dτ
G(t) =
(3.29)
−∞
is the auto-correlation function of the probe pulse and the coefficients
∂R
∂re
∂R
∂im
1
+
ρEpump ,
A=
R
∂re
∂n
∂im
∂n
and
1
B=
R
∂R
∂re
∂re
∂q
+
∂R
∂im
∂im
∂q
(3.30)
ζρEpump
(3.31)
are material-specific. The notation ∆R expresses that the reflectivity is averaged
over the pulse duration. The discussion of equation 3.22 can be applied to the
proportions of the decay constants β and γ as well as the magnitudes of pulse
duration and oscillation period of the phonon. If the pulse duration is largely inferior
to the oscillation period, G can be taken as a δ-function and we obtain
∆R
ω02
β0
−βτ
−γτ
−βτ
= Ae
+B 2
e
−e
cos(Ωτ ) − sin(Ωτ )
. (3.32)
R
ω0 + β 2 − 2γβ
Ω
According to this equation, the temporal behaviour of the reflectivity change is
composed of two contributions. The first term, which is proportional to the pump
energy via the coefficient A, consists of an exponential decay characterised by the
transfer rate of electrons from excited states to the ground state. It describes the
non-oscillatory component of transient reflectivity which has been observed in all the
materials in which DECP is believed to be the excitation mechanism. The second
term, proportional to the pump energy and to the constant of proportionality ζ,
introduced in equation 3.14, is a sum of two components. The exponential decay
can either counterbalance or intensify the feature of transient reflectivity associated
3.4. Properties of coherent optical phonons in bismuth crystals
33
with the decay of excited carriers depending on the signs of the two coefficients A
and B. The trigonometric terms describe an oscillation at the phonon frequency
whose amplitude decays exponentially, and the associated time constant represents
the damping of the phonon mode.
As we will see in the following, it is not possible to describe the complicated transient behaviour of reflectivity changes in photo-excited bismuth in a satisfactory
manner with the formula derived above. In particular, behaviour of the reflectivity during the excitation as well as after several ps cannot be understood in the
framework of these approaches. A treatment that includes a variety of additional
important processes that affect material properties will be presented in chapter 4.
3.4 Properties of coherent optical phonons in
bismuth crystals
3.4.1 Crystal structure and phonon modes
The crystal symmetry of bismuth is the so-called α-arsenic or A7-structure with
the point group 3m. This structure is typical for group-V-elements. It is also the
common phase of arsenic and antimony, and can be found in phosphorus under
pressure. It can be derived from a simple cubic lattice with one atom per lattice
site by applying to independent distortions. First, a shear along the body diagonal
transforms the unit cube into a rhomboedron, then an internal displacement of
the two atoms along the body diagonal in opposite directions results in a lattice
with trigonal symmetry and two atoms per unit cell, as it is depicted in figure 3.5.
The distortion to a less symmetric structure is stabilised by the so-called PeierlsJones-mechanism [38], introducing a small band-gap over a certain range in the
Brillouin-zone. The band-gap is responsible for several characteristics of bismuth,
e.g. its semi-metallic behaviour or the low, highly anisotropic electron effective mass.
The parameters of the unit cell are [39]:
a = b = c = 4.746 Å ,
α = β = γ = 57.23◦ .
If the elementary cell is chosen such that the two atoms have the same distance
from the center of the cell , their position on the body diagonal cdia can be expressed
relative to the origin O with the parameter u0
cdia = 11.862 Å
u0 = ±0.234 ,
so that there is an atom at u0 and another at (1 − u0 ). In this structure, there are
two optical phonon modes:
34
3. Coherent optical phonons in bismuth crystals
Figure 3.5: Trigonal unit cell of a bismuth crystal (left), direction of atomic displacement
of A1g -phonon mode (middle) and one of the two elementary directions of the Eg -phonon
mode (right)
• the fully-symmetric A1g -mode which is also referred to as the breathing mode,
is a longitudinal optical mode consisting of a displacement of the two atoms
along the body diagonal in opposite directions with a frequency of 2.92 THz
corresponding to a vibrational period of 342 fs,
• the Eg -mode, a transverse optical mode which is doubly degenerate in the
x-y-plane, consists of a displacement of the two atoms in opposite directions
either parallel to the x- or y-axis. This mode has a frequency of 2.22 THz
corresponding to a vibrational period of 450 fs.
The direction of atomic motion associated with these two modes with respect to the
symmetry of the unit cell is shown in figure 3.5.
3.4.2 Symmetry properties of optical phonon modes in bismuth
Both optical phonon modes in bismuth are Raman active, the corresponding selection rules and symmetry properties can be derived from crystal symmetry. The
3.4. Properties of coherent optical phonons in bismuth crystals
35
Raman tensors for the symmetry considered here are [40]:

a 0 0
A1g : χR =  0 a 0  ,
0 0 b




0 e 0
e
0 −d
Eg(2) : χR =  e 0 d  .
: χR =  0 −e 0  ,
0 d 0
−d 0
0

Eg(1)
(3.33)
These tensors are based on a hexagonal lattice, where the trigonal axis is along [001].
The constitution of the Raman tensors has important consequences for the excitation and the detection of the corresponding vibrational mode, and different phonon
modes can be separated on the basis of the symmetry of the tensors. The case relevant for the experiments presented in this work shall be considered in the following:
we assume a bismuth single crystal with a surface perpendicular to the trigonal axis
that is excited and probed in standard reflection geometry as shown in figure 3.6.
The pump laser is polarised with an angle ϕ with respect to the optical plane, and
the probe laser is s-polarised. In stimulated Raman scattering, the normal coordinate is proportional to the Raman tensor “surrounded“ by two components of the
electrical field [41]:
∂χuv
q ∝ E1
E2 ,
(3.34)
∂q
where E1 and E2 are contained in the pump pulse and their frequencies fulfil the
condition 3.2. Considering an electrical field polarised in the x-y-plane of the form
Eϕ = E0 · (cos ϕ, sin ϕ, 0), where ϕ denotes the angle between the field vector and
the x-axis as shown in figure 3.6, and using the tensors 3.33 in 3.34, one can easily
derive that the excitation of the A1g -modes is independent of polarisation. In contrast, the excitation of the Eg -mode does depend on the polarisation of the pump
pulse:
q(Eg(1) ) ∝ e cos(2ϕ) ,
q(Eg(2) ) ∝ e sin(2ϕ) ,
(3.35)
(1)
(2)
meaning that the symmetry of superposition of Eg and Eg follows the pump
polarisation, but the total displacement is independent of it, due to the fact that
cos2 (2ϕ) + sin2 (2ϕ) = 1.
For the detection of the coherent phonon, the polarisation plays an important role
too. The reflectivity change depends on the Raman tensor and the applied electric
field, and its proportionality can be expressed as [42]:
∂χuv
r
∆R ∝ E
Ei q,
(3.36)
∂q
36
3. Coherent optical phonons in bismuth crystals
Figure 3.6: Left: Experimental geometry in which the probe pulse is s-polarised and the
pump polarisation angle ϕ is measured with respect to the optical plane. The trigonal
axis of the crystal is parallel to the z-axis. Right: Components of the electric pump field
polarised in the x-y-plane
with an incident and reflected electric field Ei and Er . It can be seen that reflectivity changes induced by the symmetric A1g -mode are the same for all polarisations;
therefore, they can be detected with any probe polarisation. This kind of measurement is referred to as isotropic reflectivity measurement. The Eg -mode can
be observed in isotropic configuration as well, however, in some experiments it has
been detected with an anisotropic reflectivity measurement. In this configuration,
the probe beam is analysed as two orthogonal components after reflection from the
sample surface and detected with matched photodiodes, a technique often referred
to as electro-optic (EO) sampling. For the anisotropic reflectivity change we can
write:
∂χuv
∂χuv
r
i
r
∆Ru − ∆Rv ∝ Eu
E q − Ev
Ei q.
(3.37)
∂q
∂q
In this geometry, the isotropic response of the A1g -phonon is eliminated and the
Eg -signal, which is usually weaker, can be recorded separately.
3.4.3 Previous work on coherent optical phonons in bismuth
Up to the present day, there are large number of publications on coherent lattice
vibrations in bismuth that report a variety of interesting results. In this section, the
most important and most relevant to this work will be presented, and, if present,
similarities or differences of bismuth to other materials of identical or similar crystal
symmetry (e.g. antimony or tellurium) will be pointed out.
3.4. Properties of coherent optical phonons in bismuth crystals
37
A1g -mode
The first observation of coherent A1g -phonons in Bi and Sb was reported by Cheng et.
al [7]. In the case of Bi, optical excitation of single crystals, cleaved along the trigonal
face, and thin films of various thickness resulted in reflectivity oscillations with a
frequency of 2.9 THz. By measuring the initial phase, it was seen that the oscillations
had a cosinusoidal form [21]. Later, it was shown that the phonon amplitude could
be controlled optically by applying two pump pulses successively [43]. The amplitude
could be enhanced or cancelled by varying the delay between the two pump pulses,
thus making it possible to to observe the non-oscillatory component of transient
reflectivity alone. The presumably exclusive excitation of fully-symmetric modes
led to the development of DECP, which also provided an explanation for the initial
phase of the oscillation that the authors found to be contradictory to ISRS.
Eg -mode
The first experiments on coherent phonon oscillation in Bi, Sb, Te and Ti2 O3 failed
to observe the Eg -mode. A coherent excitation of this phonon in Bi was reported
for the first time for experiments with polycrystalline thin films, where it was found
to be superimposed on the A1g -oscillations in an anisotropic reflectivity measurement [43]. Coherent oscillation in modes other than the fully-symmetric ones were
also found in Sb and Te [44, 45]. A later, more detailed study performed with
bismuth crystals cut perpendicular to the c-axis at temperatures between 8 K and
300 K showed that at low temperature both optical modes were excited as coherent
phonons [46]. In the experiment, polarisation dependence of the phonon amplitudes
in isotropic and anisotropic geometries that are described by equations 3.35 was
found at a temperature of 8 K. According to the authors, this gives a strong support
for Raman-scattering based excitation mechanism. The absence of the Eg -phonon
at room temperature was attributed to non-linear enhancement of its damping with
rising temperature and the possibility of an over-damping of this mode was considered.
Mechanism of excitation of coherent phonons
To overcome the disagreement between the experimental observations described
above, Garret and co-workers first proposed that DECP is not a distinct mechanism, but a particular case of stimulated Raman scattering [44]. In this publication,
as in the review by Merlin [47], it is shown that DECP can be rendered as a special case of resonant stimulated Raman scattering which is referred to as transient
stimulated Raman scattering (TSRS). Here, the driving force can be of impulsive or
38
3. Coherent optical phonons in bismuth crystals
displacive character, depending on the coupling of phonons to virtual or real charge
density fluctuations. Still, the difference in initial phase of nearly π/2 suggests different couplings to the electronic states. A possible explanation for this discrepancy
is an anisotropic shift of the vibrational potential which is different for the trigonal
axis for the A1g -mode and the binary and bisectrix axes for the Eg -mode.
Oscillation frequency of phonon modes
In experiments where bismuth crystals were optically excited with fluences of several
mJ/cm2 , it was observed that the frequency of the A1g -mode exhibits a red-shift
that is approximately proportional to the pump fluence [48]. Similar observations
were made examining phonons in Tellurium [49]. Another study investigating the
behaviour of the frequency at high excitation yielded a transient red-shift, the frequency being reduced by almost 10% in the beginning of the oscillation and increasing with time [50]. The origin of this red-shift has been particularly controversial
as it can be attributed to an anharmonicity of the the lattice potential [51] leading
to an explicit dependence of vibration frequency on vibration amplitude, as well as
to electronic softening, i.e. a reduction of the inter-atomic bonding strength due to
excitation of electrons from valence to conduction bands [52]. Murray et al. performed double-pump experiments to excite coherent phonons that varied in amplitude but oscillated on the same excited potential energy surface [53]. By separately
controlling vibration amplitude and excited carrier density, the dependence of the
A1g -frequency on the vibration amplitude could be elucidated. While the effect of
electronic softening is clearly visible through a frequency which is decreasing with
increasing pump fluence, the dependance on oscillation amplitude is very weak and
the authors estimated the contribution of anharmonicity to be on the order of 1%,
a result which could also be confirmed theoretically [54].
Time-resolved x-ray diffraction measurements
With the advent of x-ray sources that can deliver pulses with a duration below a picosecond, it has become feasible to resolve atomic motion with time-resolved Braggdiffraction. Since the intensity of a diffracted signal directly depends on the positions
of the atoms [55], it is possible to measure atomic displacements as a function of
time. In the first experiments perfomed with x-ray pulses generated by fs-laserplasma-interaction the amplitude of oscillation could be measured and was found
to be between 0.15 Å and 0.25 Å [55, 22] for excitation fluences of several mJ/cm2 .
A later study using an accelerator-based femtosecond x-ray source [56] could confirm the excitation fluence dependent behaviour of the A1g -frequency that had been
found with optical measurements before. It was shown that an increase of excited
3.4. Properties of coherent optical phonons in bismuth crystals
39
electrons leads to a reduction of the Peierls-distortion. In addition, by calculating
the curvature of the inter-atomic potential and comparing it to results obtained
with density-functional-theory, it was pointed out that the atomic motion is welldescribed by a purely harmonic potential. This result strongly supports electronic
softening as the major mechanism responsible for the red-shift of the A1g -frequency.
40
3. Coherent optical phonons in bismuth crystals
4 A complete model for transient
reflectivity in laser-excited bismuth
In the last chapter, two models for excitation of coherent phonons, as well as the associated approach to model phonon induced reflectivity variations, were introduced.
Both models, ISRS and DECP, are frequently used to explain the excitation of coherent phonons in literature, even though in certain cases the excitation mechanism
is controversial. In the framework of ISRS, reflectivity changes are solely related to
atomic movement, while other laser induced processes that can change the optical
properties of a sample are not taken into account. Thus, it is not possible to describe
the entire dynamics of reflectivity changes after laser excitation with this theoretical
approach only. In the framework of DECP, the excitation of electrons and the rise in
electron temperature are considered, and an approach to model transient reflectivity
as a superposition of electron-related changes and changes caused by the coherent
atomic vibration is offered.
However, the new experimental results presented in this work go beyond the existing theories and show features that cannot be explained with either of the two
models, namely an initial negative peak in the transient reflectivity signal, and a
negative change in reflectivity that appears after several picoseconds. In the following sections, the interconnected processes of electron heating by laser absorption,
interaction within and between the electronic and lattice subsystems, the forces driving the atomic motion, and the changes of optical properties induced by these effects
will be considered. It will be shown that the initial drop in reflectivity can be related
to a coherent displacement of atoms caused by the polarisation force during the excitation pulse, and that the oscillating part of the reflectivity is related to optical
phonons, which are excited by the electron temperature gradient, through electronphonon coupling. The following considerations can be applied to a great variety of
materials. Nevertheless, numerical estimates and order-of-magnitude considerations
will be restricted to the case of photo-excited bismuth.
41
42
4. A complete model for transient reflectivity in laser-excited bismuth
4.1 Transient properties of a laser-excited solid
Due to the very high power densities that can be achieved by focusing an ultrashort
laser pulse onto the surface of an absorbing material, the energy deposited in the
absorption layer can be huge. This leads to a high number of excited carriers that
are present right after the absorbed laser pulse. In general, it can be assumed
that an absorbed energy density of several hundred nJ/cm2 during the typical pulse
duration of some tens of femtoseconds excites a number of electrons on the order of
one per atom. In this section, the highly non-equilibrium properties and dynamics
of a solid will be considered.
The temporal evolution of the processes that arise when a sub-picosecond laser
pulse is absorbed can be divided into three intervals depicted in figure 4.1. Initially,
the laser energy is absorbed by the valence electrons in a volume that is defined
by the illuminated surface of the crystal and the penetration depth of the light.
A highly non-equilibrium distribution of electron energy is created that cannot be
described by Fermi-Dirac statistics in contrast to the energy distribution of the nonexcited state. The qualitative form of the distribution of excitations is sketched in
figure 4.1 a). Its rectangular shape is determined by the photon energy h̄ω of the laser
pulse and the absorbed energy density. The photons are absorbed by a fraction of
the valence electrons. This leads to the occupation of states above the Fermi energy
F while empty states below F are created. At this point, it is not possible to define
an electron temperature Te due to the high degree of non-equilibrium.
Subsequent to the excitation, two different processes take place simultaneously.
One is the thermalisation of electrons through collisions with each other. After a
time τe−e , the excess energy carried by the optically excited electrons is redistributed
within the entire electronic system and a thermal equilibrium among the electrons
is created. Once thermal equilibrium in the electronic system is reached, the energy
distribution can again be described by a Fermi-Dirac distribution, as shown on the
left side of figure 4.1 b). The hot electron bath can be characterised by an electronic
temperature Te , which is much higher than the temperature of the lattice Tl . The
electrons’ energy has increased and the spread of the distribution around the Fermi
energy is wider than at room temperature. Due to the small electron heat capacity,
maximum temperatures of several thousands of Kelvin can be reached in the electron
system. The second process occurring directly after excitation is ballistic motion of
electrons into deeper parts of the crystal. Due to the absorption profile of the laser
pulse, a spatial energy gradient is established, which is the source of transport effects
that carry away the energy from the excited region. Electrons penetrate into the
crystal at velocities comparable to the Fermi-velocity (∼ 108 cm/s) and distribute
energy over distances that can be several times larger then the laser penetration
4.1. Transient properties of a laser-excited solid
43
f
a)
1
Te Tl = Tt<0
hw
0
f
eF
E
1
Te > Tl
b)
0
f
eF
E
1
Te = Tl
c)
0
eF
E
Figure 4.1: Laser excitation and following electron dynamics: a) Arrival of the laser pulse
at t = 0 and formation of a non-equilibrium electron distribution, ballistic motion of
electrons while the lattice is still cold; b) electron distribution after thermalisation of the
electronic system and onset of diffusive motion; c) Electron-lattice equilibration through
electron-phonon coupling, start of thermal heat conduction.
depth, i.e. between several tens of nm and more than 100 nm depending on the
material [57].
After thermalisation of the electronic system, the electron bath is cooled by
electron-phonon interactions. The excited electrons lose their energy to the cold
lattice due to electron-phonon coupling, and the time in which the energy transfer
occurs is governed by the coupling strength. At the same time, electrons lose their
energy by another mechanism. Diffusive motion moves the electrons deeper into
the material due to the electron temperature gradient between the photo-excited
region and the rest of the sample. The speed of this diffusion is much slower than
in the case of ballistic motion. After a certain time, which is usually on the order
of a few ps, the electrons and the lattice have a common temperature, a situation
depicted in figure 4.1 c). The transport of energy now takes place in form of thermal
diffusion driven by the thermal gradient between the now equilibrated excited region
and the deeper parts of the material.
44
4. A complete model for transient reflectivity in laser-excited bismuth
4.1.1 Two-temperature model
Once thermal equilibrium among the excited electrons is established, the excited
solid can be considered as a system composed of a hot electron bath in a cold lattice.
It can be characterised by two separate temperatures: the electron temperature Te
and the lattice temperature Tl , as described above. If both are high with respect
to the Debye temperature, Te TD and Tl TD , electron-phonon energy transfer
is assumed to be directly proportional to the difference of electron temperature and
lattice temperature. Hence, it can be expressed in the form g · (Te − Tl ), where g
is the electron-phonon coupling constant. In the two-temperature model (TTM),
originally proposed by Anisimov et al. [58], this term is connecting a set of two
coupled diffusion equations, which model the evolution of Te and Tl in time and
sample depth z:
∂Te
∂
∂Te
Ce
=
κe
− g · (Te − Tl ) + P (z, t) ,
∂t
∂z
∂z
∂Tl
∂
∂Tl
=
κl
+ g · (Te − Tl ) .
Cl
∂t
∂z
∂z
(4.1)
(4.2)
Here, Ce and Cl are the respective heat capacities of the electrons and the lattice, while κe and κl stand for the thermal conductivities of the electrons and the
lattice. The source term P (z, t) describes the absorbed energy density, which is
determined by the optical penetration depth and the absorbed fraction of the incident intensity, assuming an exponential variation of the absorption with depth z.
The basic conditions for the validity of the two-temperature model are that electronelectron scattering is a very fast process compared to electron-phonon scattering, and
that the coupling between electrons and lattice can be captured by the linear term
g · (Te − Tl ). Thermal heat diffusion is not taken into account, due to the fact that
it takes place on time scales much slower than the processes the two-temperature
model is supposed to describe.
The TTM-equations can be solved numerically to predict the time dependence of
Te and Tl . The thermalisation time, which is on the order of several picoseconds,
depends on the electron-phonon coupling constant and the absorbed energy. An
example for the time evolution of electron and lattice temperature in bismuth following an excitation typical for the measurements presented in this work is shown in
figure 4.2. Here, the maximum electron temperature Te,max at the end of the pump
laser pulse has been estimated with equation 4.16 (see below), and the evolution of
Te and Tl has been calculated under the assumption that at t = 0, Tl corresponds
to room temperature and Te = Te,max .
4.1. Transient properties of a laser-excited solid
45
3 0 0 0
T
e
te m p e ra tu re / K
2 5 0 0
2 0 0 0
1 5 0 0
1 0 0 0
T
l
5 0 0
0
1 0
2 0
3 0
4 0
5 0
tim e / p s
Figure 4.2: TTM-calculation of time evolution of lattice and electron temperature in
bismuth after excitation with a 40 fs laser pulse of 3 mJ/cm2
4.1.2 Relaxation times: Quasi-equilibrium electron and lattice
temperatures and the validity of the TTM
As described in the preceding section, one of the conditions that has to be fulfilled
if one wants to apply the two-temperature model is that thermal equilibrium is
established in the electron subsystem. Therefore, it is necessary to evaluate the time
in which electron-electron collisions lead to an energy distribution which locally is in
equilibrium. This can be done by determining the rate of collisions between excited
electrons. The collision rate for a strongly correlated degenerate electron gas can be
expressed in the form [59, 60]
νe−e = c · ωp
e
,
F
(4.3)
where F is the Fermi-energy and e is the electron energy in excess of the Fermilevel. The exact value of the proportionality factor c depends on the material, but as
an approximation, it can be taken to be on the order of unity [61]. The Fermi-energy
F = 5.2 eV and the plasma frequency ωp = 1.31 · 1016 s−1 can be extracted from
optical data (e.g. ellipsometry measurement of the dielectric function presented in
chapter 7 or [39, 62]). Thus for the experimental conditions of this work in which
46
4. A complete model for transient reflectivity in laser-excited bismuth
e ≈ 0.02F , the electron-electron equilibration time te−e = (νe−e )−1 is on the order
of a few fs.
In a similar manner, phonon-phonon interactions lead to the establishment of
an equilibrium in the phonon subsystem. Once the equilibrium is achieved, the
lattice temperature Tl can be defined. The effective phonon-phonon collision rate
can be approximated by the first-order term that is linear in temperature in the
quantum-mechanical expression containing the interaction matrix elements [63, 64]:
νph−ph ≈ ωD
Tl
,
TD
(4.4)
where ωD and TD are Debye frequency and temperature. In the case of bismuth,
ωD = 1.56 · 1013 s−1 and TD = 119 K, so that the phonon-phonon relaxation time
tph−ph = (νph−ph )−1 is in the range from 15 fs to 25 fs for temperatures between room
temperature and slightly below the melting temperature Tmelt = 544.5 K [39].
The different temperatures of the lattice and the electron subsystems lead to the
transfer of energy from the electrons to the lattice. The heating time is determined
by the electron-phonon energy transfer rate [65, 66], which can be approximated as
energy
νe−ph
≈
2
h̄ωD
Tl 2
.
F TD 2
(4.5)
energy −1
In the conditions of our experiments, the equilibration time tenergy
e−ph = (νe−ph )
ranges from several ps to tens of ps.
As described in the previous section, heat diffusion is not considered in the framework of the two-temperature-model. The time for cooling the skin depth of roughly
30 nm can be estimated as:
tcool ≈ ls2 /D ,
(4.6)
where D is the diffusion coefficient. For bismuth, D = 0.067 cm2 /s [62], which leads
to tcool ≈ 119 ps. Therefore, it can be regarded as a good practice to neglect heat
diffusion during an observation period of at most 35 ps.
The above estimations show that the main non-equilibrium process is electronphonon temperature equilibration, while electron-electron and phonon-phonon equilibration can be considered to follow the slower evolution of lattice heating adiabatically. Therefore, the assumptions of the two-temperature-model are valid in our
case, and the model can be used to describe the evolution of temperatures after
laser-excitation.
4.1.3 Excitation of electrons
In order to estimate the number density of excited carriers at the end of the pump
laser pulse and its dependence on time, an avalanche-like model is derived in the
4.1. Transient properties of a laser-excited solid
47
following way. It is assumed that all absorbed laser energy excites electrons from
the valence to the conduction band, and thus changes the number density of excited
carriers and the electron temperature.
If one neglects all losses, the absorbed electron energy density rate is expressed
as [65]:
∂e
I(z, t)
= 2A ·
,
(4.7)
∂t
ls
where A = 1 − R is the absorption coefficient and I(z, t) = I0 (z = 0, t) · e−2z/ls is
the transient laser intensity. It is now assumed that an electron can be transferred
from the valence to the conduction band if it gains an energy ∆ equal to the overlap
of the two bands, which is on the order of ∼ 0.2 eV in the case of bismuth [39].
If an electron gains an energy which exceeds the overlap of bands, it is capable of
transferring the energy excess by electron-electron collisions to other electrons and
thus creating an avalanche of excited electrons. The electron excitation rate can be
estimated as:
1 ∂e
2A
we ≈ ·
=
· I(t)
(4.8)
∆ ∂t
∆ · ne · ls
From this, the growth rate of excited electrons can be calculated as follows:
2A
dne
= w e · ne =
I(t),
dt
∆ · ls
(4.9)
so that if n0 denotes the number density of free carriers in the unperturbed medium,
the number density of electrons at the pulse end reads:
2A
ne (tp ) = n0 +
·
∆ · ls
Ztp
I(t) · dt = n0 +
2AF (tp )
.
∆ · ls
(4.10)
0
It can be seen that the number of excited electrons after the end of the laser pulse is
proportional to the electron temperature Te . From that point, he number of excited
electrons can be expressed by the following rate equation:
dne
= −γrec · ne .
(4.11)
dt t>tp
Due to the fact that for bismuth the recombination rate γrec is unknown to the best
of our knowledge, it is assumed from now on that after the end of the pump pulse its
proportionality to Te stays valid, so that the change in number density of electrons
can be written as:
∆ne (t)
∆Te (t)
∝
,
(4.12)
ne
Te, max
where Te, max is the maximum electron temperature at the end of the pump pulse.
48
4. A complete model for transient reflectivity in laser-excited bismuth
4.1.4 Absorption of energy and electron temperature
The absorbed energy density, which has its maximum at the end of the pump laser
pulse, can be estimated with sufficient accuracy if one assumes that the laser excitation does not significantly change the optical properties. Using the optical properties
of the unexcited material, the absorbed energy density can be expressed as:
abs =
AF (tp )
2AF (tp )
=
,
dabs
ls
(4.13)
where A is the absorption coefficient, which can be retrieved from the dielectric
function of the unperturbed material, F (tp ) is the fluence delivered to the sample
at the end of the laser pulse, and ls is the skin depth. The electron temperature Te
can be estimated using the fact that, after quick thermalisation, the absorbed laser
energy is stored in a thermal bath of electrons with an energy density
Q = Ce ne Te ,
(4.14)
where Ce and ne are electron heat capacity and number density, respectively. In
the Sommerfeld theory of metals, the heat capacity of the free-electron gas Ce is
expressed as [29]:
2
π 2 kB
Te
Ce =
.
(4.15)
2 F
Here, kB denotes the Boltzmann-constant and F the Fermi-energy. By equating
4.13 - 4.15, the following expression for the electron temperature can be derived:
Te2 =
4F AF (tp )
.
2
π 2 ne ls kB
(4.16)
For our experimental conditions, an estimation of the maximum electron temperature with equation 4.16 results in an electron temperature of several thousand
Kelvin, which is a magnitude that has been reported for other materials at comparable excitation fluences [57]. The excited carrier density calculated with equation 4.10 is on the order of 1023 cm−3 . Therefore, it is reasonable to assume that
bismuth is rather metal-like after strong optical excitation.
4.2 Laser-induced forces and atomic motion
In this section, the forces acting on the photo-excited sample during and after the
excitation pulse will be examined. For this purpose, the stress tensor will be used,
which allows for the derivation of expressions for the different contributions to the
4.2. Laser-induced forces and atomic motion
49
laser-induced forces on the atoms. The forces, whose magnitudes are different depending on the time-scale with respect to the inverse of the phonon period, are
responsible for the atomic motion. In order to estimate the change in optical properties due to the atomic motion later in this chapter, the effect of the forces on the
atoms will be considered in the following section.
4.2.1 The stress tensor and related forces
In order to estimate the forces that act on a material, the field–and–matter stress
tensor can be used. The force acting on a volume dV of a sample is the change
of momentum per unit time. As a consequence of conservation of momentum, this
change is equal to the amount of momentum that enters and leaves through the
surface of dV . If the momentum flux tensor is denoted −σik , this can be expressed
as
Z
I
fi dV = σik dfk ,
(4.17)
where fi denotes a component of the force and the integration on the right-hand
side has to be performed over the surface of dV . The tensor σik is called the stress
tensor. It takes into account contributions of material pressure and radiation (field)
pressure, and its form depends on the characteristics of the material. From the
stress tensor, the force acting on the material can then be calculated as
fi =
∂σik
.
∂xk
(4.18)
For simplification, the case of an isotropic crystal shall be considered here. In this
case, the expression for σik is [23]
∂jk
E 2 ik δik E 2
Ei Dk
+
,
(4.19)
σik = −P · δik −
na
δik +
8π
8π
∂na T
4π
where the electric field displacement vector has the form Dk = kj Ej . In the above
equation, na denotes the atomic density and δik the Kronecker–delta. We assume
that the dielectric tensor that is modified by the laser pulse consists of two terms
jk = D · δjk + pjk ,
(4.20)
a Drude-like term D and a polarisation term p . As a further difference to the
equilibrium case, we consider the pressure to be composed of a contribution from
the laser-affected electron and the lattice subsystems P = Pe + Pl .
In accordance with 4.18, the expression for the force derived from 4.19 is
∂P
E 2 ∂ik
∂ E2
∂ E2
∂jk
∂ Ei Dk
fi = −
−
− ik
+
na
+
, (4.21)
∂xk 8π ∂xk
∂xk 8π ∂xk 8π
∂na T
∂xk 4π
50
4. A complete model for transient reflectivity in laser-excited bismuth
which, after simplification and rearranging, can be written as
∂P
E 2 ∂ik
∂ E2
∂jk
fi = −
−
+
na
.
∂xk
8π ∂xk
∂xk 8π
∂na T
(4.22)
Taking into account that for the Drude-like function the relation
∂jk
na
= D − 1
∂na
holds, using 4.20 results in
∂pik E 2 D − 1 ∂E 2
∂P
−
+
= fitherm + fipol + fipond .
fi = −
∂xi
∂xk 8π
8π ∂xi
(4.23)
The resulting force fi is the sum of three contributions. The first term describes
the thermal force. The gradient of electronic pressure is a manifestation of the
electrostatic interaction between electrons and ions during the pulse time when
only electrons are excited while the lattice remains cold. The second term is the
polarisation force, which is associated with the field-induced dipole moments, and
the third term is the ponderomotive force. The polarisation and ponderomotive
forces are effective during the laser pulse only, while the thermal force is effective
until the temperature gradients have smoothened spatially.
At this point it is instructive to compare the magnitude of the different contributions to each other. The thermal force on a single atom can be approximated
as
1 ne Te
F therm ≈
·
,
(4.24)
na ls
where the pressure gradient has been expressed as a ratio of electron temperature
and skin depth, and the division by the atomic density na is done to transform the
volume force (or “force density”) in equation 4.23 into the force on a single atom.
In the conditions of the experiments presented in this work (laser fluence 7 mJ/cm2 ,
tp = 40 fs, I = 1.75 · 1011 W/cm2 , I/c = 5.83 · 10−7 erg/cm3 , Te,max = 0.43 eV;
for material properties of bismuth see appendix B), the value of the force at the
end of the laser pulse, where the electron temperature has its maximum Te,max , is
F therm = 4.66 · 10−7 dyne.
To calculate the polarisation force, the polarisability tensor in the Placzek form
is used
∂αik
αik = α0,ik +
· xl ,
(4.25)
∂xl 0
where xl denotes the displacement in l-direction. Then the polarisation term in
equation 4.20 is
∂αik
p
ik = 1 + 4πα0,ik + 4π
· xl .
(4.26)
∂xl 0
4.2. Laser-induced forces and atomic motion
The polarisation force is reduced to the following:
∂αik
E2
pol
fi =
·
.
∂xk 0 2
51
(4.27)
It shall be assumed that (∂αik /∂xk )0 ≈ α0 /d, where d is the inter-atomic distance. The unperturbed polarisability can be estimated with the Lorentz-Lorentz
formula [29], α0 = 3( − 1)/4π( + 2). Now the polarisation force acting on a single
atom can be approximated as:
F pol ≈
4π α0 I
·
· .
na d c
(4.28)
Here, the average intensity during the laser pulse I = cE 2 /8π has been introduced.
The magnitude of the force is F pol = 1.64 · 10−7 dyne.
Finally, the ponderomotive force can be approximated by replacing the derivative
of the squared electric field by the ratio of average intensity and skin depth, so that
for the force on a single atom we get
F pond ≈
1 D − 1 I
·
· .
na
ls
c
(4.29)
In the experimental conditions used above, this force is F pond = 1.50 · 10−8 dyne.
Thus, at the end of the pump pulse, the thermal force is the largest contribution to
the laser-induced forces. The ponderomotive force is an order of magnitude lower
than the other two forces and can therefore be neglected as a first approximation.
However, one has to take into account the different temporal evolution of the forces.
If a flat-top-hat intensity distribution in time is assumed for the intensity profile of
the laser, the polarisation force dominates during the first few per cents of the pulse
time. Then, the thermal force increases linearly with time, before it finally becomes
the major contribution.
4.2.2 Laser-induced atomic motion
As already mentioned in chapter 3, the atomic motion can be described by a driven
harmonic oscillator
d2 qk
dqk
Fklaser
2
+
2γ
+
ω
q
=
,
(4.30)
k
0
dt2
dt
m
where Fklaser = Fktherm + Fkpol + Fkpond is the sum of the laser-induced forces.
The relative strength of the elastic force and the imposed forces is different on
a short time scale t ω0−1 and a long time scale t ω0−1 . This has important
consequences concerning their action on the atoms and the associated atomic displacement, that will be presented in the following paragraphs.
52
4. A complete model for transient reflectivity in laser-excited bismuth
Fast atomic displacement
The elastic force, which is the restoring force occurring due to atomic displacement,
is not effective on a time scale which is short compared to the phonon period.
Correspondingly, the first term on the left-hand side of equation 4.30 dominates,
and the equation is reduced to
d2 qk
Fklaser
.
≈
dt2
m
(4.31)
The solution to this equation is
1
qk (t) =
m
Ztp
0
dt0
Zt0
Fklaser (t00 )dt00 ≈
Fklaser t2p
.
2m
(4.32)
0
Thus on a short time scale, the laser-imposed forces produce a coherent displacement
of atoms. The above qk is what we refer to as rapid initial displacement which
manifests itself in a sharp drop in reflectivity during the pump pulse as we will see
below. Equation 4.32 allows one to estimate the magnitude of the fast coherent
displacement. Using the above parameters, valid for our experiments with bismuth,
one obtains that, during the pump pulse, the bismuth atoms are displaced less than
10−10 cm which is approximately one tenth of the cold phonon’s amplitude.
Quasi-harmonic vibrations
In order to relate the laser induced forces to the forces responsible for the “cold”
phonon in unexcited bismuth, it is necessary to know the amplitude of oscillation of
the cold phonons. It can be estimated on the basis of the adiabaticity principle (for
T < TD ) as the following [64]:
21
h̄
(4.33)
q0 ≈
m · ω0
For bismuth, one obtains an amplitude of q0 = 4.01 · 10−10 cm. From this, the
magnitude of the elastic force appearing in equation 4.30 can be calculated, F elast =
mω02 q ∼ 4.93 · 10−5 dyne. As one can see, the laser-induced forces are much smaller
than the elastic force, ω02 q0 Fklaser /m. Therefore, the laser-induced forces act as a
small perturbation that slightly affect the “cold” atomic vibrations on a long time
scale t ω0−1 . Hence, it is reasonable to seek a solution similar to that for the
damped harmonic oscillator, whose amplitude varies slowly in comparison with the
phonon period:
1
2
2 2
q(t) ≈ Q(t) · exp i ω0 − γ
t − γt + iϕ .
(4.34)
4.3. Non-linear phenomena related to electron-lattice-equilibration
53
It shall be assumed, that the displacement due to perturbation oscillates with the
same frequency as the unperturbed system and thus only affects the oscillation
amplitude. The slowly varying amplitude can be determined using the applied
force:
1
F therm
∆qk = k
· exp i ω02 − γ 2 2 t − γt + iϕ .
(4.35)
2mω0 γ
The atomic vibrations begin when the electrostatic field transfers energy to the
atoms. Therefore, the phase ϕ is needed to adjust the above solution to the initial
atomic displacement.
4.3 Non-linear phenomena related to
electron-lattice-equilibration
It has been shown in the preceding sections, that a strong, ultrashort excitation of a
material gives rise to a fascinating interplay of the electronic and lattice subsystems
along with laser-induced forces that are responsible for subtle atomic displacements
and the excitation of atomic vibrations. In the following, the evolution of coherent
phonons with regard to phenomena that arise in the process of electron-to-lattice
energy transfer and lattice heating will be treated. These phenomena include a
change of inter-atomic potential due to lattice heating, the decay of phonons, and a
red-shift in phonon frequency.
4.3.1 Anharmonicity of vibrations and shift in equilibrium
position due to lattice heating
Up to this point, damped harmonic oscillations of atoms were considered taking
into account the dependence of the thermal force on excitation of electrons and
lattice temperature. However, after the end of the pump pulse, the initially excited
harmonic phonons are gradually transformed due to a rise in lattice temperature
caused by energy transfer from the electrons to the lattice.
After the establishment of the lattice temperature, the energy distribution of
phonon modes is in accordance with a Boltzmann distribution φph = exp(−∆Uel /kB T ).
In the harmonic approximation, there is neither an interaction of phonons nor damping of the phonon mode, and the oscillation amplitude does not change with time.
The equilibrium position of vibrating atoms does not depend on the temperature
and the average displacement from the equilibrium position is zero [67]. However,
54
4. A complete model for transient reflectivity in laser-excited bismuth
the mean square displacement is proportional to temperature:
R∞
hq 2 i =
q 2 · exp (−∆Uel /kB T ) dq
−∞
R∞
=
exp (−∆Uel /kB T ) dq
kB T
,
2
M ωph
(4.36)
−∞
a result which qualitatively complies with calculations of the average amplitude of
atomic vibrations at temperatures that exceed the Debye-temperature, M ω 2 hq 2 i ≈
kB T , where ωD = kB TD /h̄ is the Debye-frequency [59, 68].
When the lattice temperature increases, phonon-phonon interactions become more
important. The atomic vibrations cannot be described with a harmonic approach
and higher-order terms have to be taken into account. The expression for the perturbation in the inter-atomic potential with a third-order term is [67]
∆Upert = C2 q 2 − C3 q 3 , with
1 ∂ 2U
1 ∂ 3U
b
b
C2 =
≈ 2 , and C3 =
≈ 3.
2
3
2 ∂q
2d
6 ∂q
6d
(4.37)
Here, d denotes the inter-atomic distance and b the binding energy. The distribution
function, which includes the anharmonic term, reads
C2 q 2 − C3 q 3
φnl = exp −
,
(4.38)
kB T
and the expression for the average displacement from the cold equilibrium position
that increases with increasing lattice temperature, is:
hqnl i =
3C3 kB T
kB T
=
·d.
2
4C2
2b
(4.39)
At high temperatures the change in the average atomic position becomes a considerable part of the oscillation amplitude. Then the atomic motion loses its coherent
character and is randomised. If equation 4.39 is used to estimate the displacement
at the melting temperature (Tmelt = 544.5 K [39]), it coincides with the average
displacement from the Lindemann criterion of melting [69].
4.3.2 Phonon decay
In general, the relaxation of a coherent phonon mode is related to scattering with
the elements of the incoherent population of the system, including thermal phonons,
lattice defects, electrons etc. The nature of the scattering process depends on the
4.3. Non-linear phenomena related to electron-lattice-equilibration
55
ω( k )
ωL
O
ωL O / 2
- π/ a
π/ a
0
k
Figure 4.3: Decay of a zone-center optical phonon of energy h̄ω to two acoustic phonons
of energy h̄ω/2 under conservation of energy and momentum.
involved scattering partners, there can be elastic and inelastic scattering that contributes to the relaxation of the coherent mode. In an elastic scattering process, the
phonon momentum is randomised without energy transfer, which leads to a loss of
phase coherence. This is the case for scattering by lattice defects [70]. In an inelastic
scattering process, the momenta and the energies of the scattering partners change,
and this process is referred to as phonon decay or energy relaxation.
In semi-metals and semi-conductors, the relaxation of coherent optical phonons
is dominated by inelastic scattering with thermal phonons [71, 72]. This phononphonon scattering can be thought to consist only of the decay of a coherent optical
phonon of energy h̄ω into two acoustic phonons of energy h̄ω/2 with opposite wave
vectors, a process depicted in figure 4.3. Processes involving decay into more than
two phonons are possible as well, and indeed they occur with a much lower probability and can therefore be neglected as a first approximation. The phonon dynamics
can be described within an oscillator model where the lattice potential has a certain
anharmonicity that determines the phonon-phonon decay. The third-order anharmonic term induces the three-phonon decay process, the fourth-order term gives rise
to a four-phonon decay and so on.
The probability of the decay of an optical phonon into two acoustic phonons per
unit time can be calculated with the help of the quantum perturbation theory [64].
In a simplified manner, one can estimate the perturbation Hamiltonian H 0 as the
third term in the series of potential expansion in powers of atomic displacement
from equation 4.36. The scaling for the probability of phonon decay is
w ∝ |H 0 |2 ·
1
.
h̄ · T
(4.40)
56
4. A complete model for transient reflectivity in laser-excited bismuth
2
The average phonon energy is taken as M ωD
hq 2 i ≈ h̄ω ≈ kB T , where ωD is the
Debye-frequency and kB the Boltzmann–constant. The third-order term in the perturbation potential can be expressed as
1
H ≈
6
0
∂ 3U
∂q 3
b
· hq i ≈ 3
6d
2
3
2
T
2
M ωD
32
.
(4.41)
Now it is possible to express the probability of phonon decay (or the optical phonon
decay rate) as a function of the basic properties of a solid. For temperatures in
excess of the Debye-temperature, kB T h̄ωD , the single phonon decay rate is
γdecay
h̄
≈C·
M d2
kB T
h̄ωD
2
,
(4.42)
where C is a dimensionless constant of proportionality. For bismuth at TD = 119 K,
the decay rate is of the order of ps−1 (assuming C to be close to unity), which is in
agreement with experimental observations.
4.3.3 Red-shift of the phonon frequency
As mentioned in section 3.4.3, the oscillation frequency of the A1g -mode in bismuth
shows a red-shift that depends on the level of excitation. In addition, the frequency
is chirped, meaning that it changes with time. Directly after excitation, the redshift has a maximum. Then, with increasing delay between pump and probe beam,
it emerges towards the unperturbed A1g -frequency. It has been shown that it is
reasonable to attribute this fluence and time dependent red-shift to electronic softening, which is a reduction of inter-atomic bond strength due to carrier excitation.
In order to relate the shift in phonon frequency to the level of excitation, we make
use of a simplified form of the empirical chemical pseudo-potential [73, 74]:
V (r) = vr · e−θ · r − va · e−λ · r .
(4.43)
Here, vr , θ, and va , λ characterise the repulsive and attractive contributions to the
potential, respectively. If the solid is excited, the attractive part of the inter-atomic
potential decreases with respect to the repulsive part, thus effectively the repulsion
increases. It follows from equation 4.43, that in the heated solid the mean interatomic distance at which the potential has its minimum can be expressed through
the binding energy as the following
1
(s − 1) · va
T
≈ d0 +
,
(4.44)
d = · ln
λ
s · b
λ · b,0
4.4. Transient optical properties
57
where b is the binding energy (the subscript 0 denotes the unexcited case) and
s = θ/λ. So in the excited state at b − Te , the inter-atomic distance increases. The
attractive part of the potential is mostly affected, so the magnitude of the increase
depends on the gradient of this part:
dq ≈
1 Te
·
.
λ b,0
(4.45)
The excited phonon frequency can be recovered using the second derivative of the
potential:
2 ∂ V
λ2 · s · |b |
1
2
·
=
.
(4.46)
ω =
m
∂r2 r=d
m
The phonon frequency of the excited solid can then be expressed as:
Te
2
2
ω ≈ ω0 1 −
.
|b,0 |
(4.47)
It can be seen that the phonon frequency directly depends on the level of electron
excitation via the electron temperature Te . As the electron temperature has its
maximum at the end of the pump pulse and then decreases due to electron-phonon
interaction, the increase of phonon frequency with time can be understood in the
framework of this model. In the same manner, the dependence of phonon frequency
on excitation fluence is understandable, since with increasing fluence the maximum
lattice temperature increases as well.
4.4 Transient optical properties
In this section, the changes in the optical properties that are induced by the pump
laser pulse will be examined. The main goal of this section is to derive expressions
that allow for the formation of a model of the transient reflectivity changes that will
be presented in chapter 6. As the reflectivity depends on the dielectric function , we
will at first consider how the dynamics of the dielectric function change, taking into
account the laser-induced processes described above. Once the transient changes
of the real and imaginary part of are known, the transient reflectivity can be
expressed through these changes.
4.4.1 Dielectric function
As mentioned above, we assume that the dielectric tensor, modified by the laser,
consists of two terms, a Drude-like term and a polarisation term, as it is noted in
58
4. A complete model for transient reflectivity in laser-excited bismuth
equation 4.20. The real and imaginary parts of the dielectric tensor can then be
expressed as:
jk = D · δjk + pjk ≡ re + i · im ,
re = pjk + D
im = D
re ;
im .
(4.48)
The Drude term, in its conventional form (see section 2.2.1), is written as
D = re + iim
ωp2
ωp2
νe−ph
=1− 2
+ i 2
.
2
2
ω + νe−ph
ω + νe−ph ω
(4.49)
This contribution depends on the number density of conductivity electrons ne and
on the electron effective mass m∗e through the plasma frequency
ωp =
4πe2 N
m∗e
12
.
(4.50)
Even though the electronic properties of unexcited bismuth are well-studied and
available in literature [75], it is unknown to the best of our knowledge how the
electron effective mass changes during the excitation and the heating of a solid and
the subsequent processes. For unexcited solid bismuth, m∗e is characterised by a
considerable anisotropy. The magnitude of the elements of the electron effective
mass tensor lie in the range from less than a percent of the free electron mass me to
1.36 · me [76, 77], depending on the electron wave vector. In contrast, it is established
that in liquid bismuth the effective mass is equal to that of the free electron [78].
In the following, any changes of the electron effective mass m∗e are ignored and it is
assumed that in photo-excited bismuth the effective mass in equation 4.50 can be
approximated by the mass of the free electron.
Without any prior knowledge of the behaviour of the dielectric function under intense photo-excitation, it is reasonable to assume small changes in dielectric function
due to the fact that the laser-induced reflectivity changes are small. The changes of
the real and imaginary parts of the dielectric function can then be expressed as
∂D
∆ne
∂D
∆νe−ph
p
p
re
re
D
∆re = ∆jk + ∆re = ∆jk +
·
+
·
,
∂ ln ne 0 ne,0
∂ ln νe−ph 0 νe−ph,0
(4.51)
∂D
∆ne
∂D
∆νe−ph
im
im
D
∆im =
∆im
=
·
+
·
.
∂ ln ne 0 ne,0
∂ ln νe−ph 0 νe−ph,0
Here, the subscript “0” denotes that the derivatives are calculated from the unperturbed function and that changes in ne and νe−ph are considered with respect to their
unperturbed values. The derivatives of re and im with respect to electron number
and electron-phonon momentum exchange rate are presented in appendix A.
4.4. Transient optical properties
59
In order to estimate the changes with the help of equation 10.45, one has to know
the magnitudes of ∆ne and ∆νe−ph . The latter is responsible for the dependence of
the optical properties on phonon amplitude and frequency. Electron-phonon coupling represents the Coulomb interaction between the electron charge e and the
average dipole electric field that is created by atomic displacement, Eph ≈ eq/d3 .
The energy of electron-phonon interaction is ph ≈ qeEph ≈ e2 q 2 /d3 . The electronphonon momentum exchange rate can be expressed through the energy of interaction [68]
νe−ph ≈
ph
e2 q 2
≈
≈ nph q 2 ve .
h̄
h̄d3
(4.52)
Here, ve denotes the electron velocity and nph denotes the “cold” phonons’ number
density, which can be approximated in a conventional scaling as nph ≈ na ≈ d−3 .
The number density of phonons at Tl > TD can be approximated as nph ≈ na T( t)/TD .
Supposing an electron velocity ve = 108 cm/s and a phonon amplitude q = 10−9 cm,
one can see that the electron-phonon momentum exchange rate is comparable to the
phonon frequency. The above scaling gives reasonable estimates and qualitatively
agrees with derivations based on a quantum-mechanical [64] or kinetic approach [68].
The perturbation of the electron-phonon coupling rate due to laser excitation can
then be expressed through the variations in lattice temperature and phonon amplitude:
∆Tl (t) 2∆q(t)
∆νe−ph
=
+
,
νe−ph,0
T0
q0
(4.53)
where the first term relates to the contribution from incoherent phonons (the phonon
number nph is proportional to the lattice temperature) and the second to the contribution from coherent phonons. The polarisability-related part in equation 10.45
can be expressed with the help of 4.26 as
∆pik
= 4π
∂αik
∂xl
· ∆ql (t) .
(4.54)
0
With the above considerations and those from section 4.1.3, it is possible to express
the dielectric function of a swiftly excited metal-like solid as a function of the changes
in the number density of conductivity electrons and the variation of the electronphonon momentum exchange rate. The transient changes following excitation are,
in turn, expressed as explicit functions of electron temperature, lattice temperature
and atomic vibrations.
60
4. A complete model for transient reflectivity in laser-excited bismuth
4.4.2 Transient reflectivity
In this section, the directly measurable time-dependent reflectivity changes of a
laser-excited solid will be considered. For this purpose, the reflection coefficient
√
− 1 2
R = √
(4.55)
+ 1
is expressed through the real and imaginary parts of the dielectric function =
re + i · im as
p
|| + 1 − 2(|| + re )
p
,
(4.56)
R=
|| + 1 + 2(|| + re )
p
where || = 2re + 2im . A small variation of reflectivity then reads:
∂R
∂R
p
D
∆R =
· ∆re + ∆jk +
· ∆D
(4.57)
im .
∂re 0
∂im 0
By inserting the variations of the dielectric function from 10.45, the first order
reflectivity variation can be expressed through the changes in polarisation, in number
of conduction electrons and in electron-phonon momentum exchange rate
∆R = Cpol · ∆pjk + Cne ·
∆ne
∆νe−ph
+ Cνe−ph ·
.
ne,0
νe−ph,0
The first coefficient Cpol is the derivative of unperturbed reflectivity
∂R
Cpol =
,
∂re 0
(4.58)
(4.59)
and the coefficients Cne and Cνe−ph are combined from the derivatives of the unperturbed reflectivity and dielectric function:
∂D
∂R
∂D
∂R
re
im
Cne =
·
+
·
,
∂re 0
∂ ln ne 0
∂im 0
∂ ln ne 0
(4.60)
∂R
∂D
∂R
∂D
re
im
Cνe−ph =
·
+
·
.
∂re 0
∂ ln νe−ph 0
∂im 0
∂ ln νe−ph 0
In bismuth, the reflectivity changes are on the order of ∆R/R0 ∼ 10−3 for excitations with several mJ/cm2 and can therefore be considered small changes that
are described reasonably well with expansion 4.58. The coefficient Cpol is negative
(the values of the coefficients are presented in appendix A). Thus, a small atomic
displacement occurring during the excitation pulse creates a negative contribution
to the reflectivity change. The coefficients Cne and Cνe−ph are positive and negative,
4.4. Transient optical properties
61
respectively. Accordingly, the excitation of electrons leads to a positive change in
reflectivity that gradually decreases due to the recombination of excited carriers, but
remains positive all the time. The third contribution is negative and is the result
of the effect of atomic vibrations on the optical properties and a change in lattice
temperature. After the energy transfer from the electrons to the lattice, the positive
contribution due to electron excitation is reduced due to the lower electron temperature. Thus the only contribution remaining is the negative third term, which gives
rise to a negative change in reflectivity.
Finally, the reflectivity change can be expressed as an explicit function of the
change in the polarisation-related part of the dielectric function, lattice and electron
temperatures and the coherent atomic displacement due to oscillations on the basis
of equation 4.58
∆R/R0 = A · ∆pol + B · ∆Te + C · ∆Tl + D · ∆q ,
(4.61)
where the coefficients four coefficients A, B, C and D can be calculated by combining
the coefficients in 4.59 and 4.60 with the unperturbed values of electron density and
electron-phonon momentum exchange rate.
62
4. A complete model for transient reflectivity in laser-excited bismuth
Part III
Experiments and experimental results
63
5 Experimental techniques and
setups
The investigation of ultrafast processes demands sophisticated experimental schemes.
The experiments described in this work examine sub-nanometer atomic displacement
in a layer of several nanometers below the crystal surface. The physical processes
that lead to coherent atomic motion and the effects that are induced by it take
place on a time-scale ranging from a few femtoseconds to several tens of picoseconds. It is an experimental challenge to uncover the tiny reflectivity changes that
are induced by such subtle atomic motion, and it requires powerful tools that have
to be adeptly combined. In this chapter, the tools and techniques that were used
to perform time resolved measurements of optical properties shall be described. In
particular, the pump-probe technique has been employed to perform time-resolved
reflectivity measurements of laser-excited bismuth crystals with femtosecond time
resolution. The measurements were carried out with high temporal resolution and
low noise due to the use of a high-power laser that delivers a stable train of ultrashort laser pulses and a high-sensitivity detection system that will both be subject
of the following sections. In addition, the different setups that were used to perform
single-probe, double-probe and double-pump measurements will be described and
a method to extract the dielectric function of a material from two simultaneous
reflectivity measurements will be presented and discussed.
5.1 Ultrafast measurements with the pump-probe
technique
The main difficulty that the investigator of ultrafast processes has to circumvent, is
the limited time resolution of detectors. While femtosecond laser pulses are available
and thus processes on these time scales can be excited, there is no possibility to
resolve them directly. A common method to investigate ultrafast dynamics is to
use streak cameras. These instruments are able to measure the transient variations
of the light’s intensity by transforming the temporal profile of a light pulse into a
spatial profile on a detector. The temporal resolution of the best streak cameras
65
66
5. Experimental techniques and setups
Figure 5.1: Schematic representation of the pump-probe technique
is limited to several hundreds of femtoseconds, a time span which corresponds to
the period of the optical phonons investigated in this work. In order to resolve
the dynamics of the phonons properly, it is necessary to record several data points
per period. Due to their limited temporal resolution, this cannot be achieved with
streak cameras.
If the intensity changes of reflected laser beams are measured with photodiodes,
the limiting factor for the time resolution is the rise time, which is defined as the
time the output current of the diode needs to develop after an instantaneous rise in
input light power. Even for the fastest photodiodes, the rise times are not less than
several picoseconds, standard photodiodes usually have characteristic rise times on
the order of nanoseconds.
A way to deal with this problem is to measure the reflectivity with the pumpprobe technique which is illustrated in figure 5.1. One laser pulse, referred to as the
pump pulse, is inducing the dynamic processes that change the optical properties
of the sample. After a known time delay ∆t, a second laser pulse, the probe pulse,
arrives at the excited area of the sample. A detector then measures the reflectivity
of the sample at this well-defined moment. If the delay between the pump and
probe is varied by simply changing the optical path length of either one of the
pulses, it is possible to record a set of measurements corresponding to different
times after the excitation and thus reveal how the reflectivity varies with time. In
this configuration, the fundamental limit of time resolution is the duration of the
probe pulse. The magnitude of the changes in reflectivity depends on the material
and the level of excitation. In many cases, these changes are small, and therefore,
the signal is averaged over a large number of pulses for a given time delay. This
5.2. The femtosecond laser system
67
Figure 5.2: Schematic representation of the kHz-laser setup
method can also be used in order to compensate for intensity fluctuations in the
probe pulse.
The pump-probe technique is not restricted to reflectivity measurements, but
rather a general concept for the detection of changes that cannot be resolved even
with the fastest detectors. Ultrafast processes in transparent solids can also be
investigated by measuring temporal changes in transmittivity or by detecting farinfrared light in the terahertz frequency range that is generated by IR-active coherent
phonons [42]. Furthermore, other sources of radiation can be used to probe ultrafast
dynamics in material media. In the past ten years, great progress has been made in
producing x-ray pulses in the femtosecond regime. Accelerator-based sources as well
as x-rays produced by laser-plasma-interaction are used in pump-probe experiments
delivering information about the structure of matter and the time scales on which
possible changes occur. In addition, pulsed electron sources can deliver ultrashort
electron bunches that can be used for time-resolved diffraction experiments.
5.2 The femtosecond laser system
The laser system that provides the ultrashort pulses that are needed for our experiments is depicted in figure 5.2. It is based on titanium-doped sapphire crystals
(Ti:Sa) and uses the chirped pulse amplification technique (CPA) [79]. It delivers
10 mJ-pulses of 35 fs at a central wavelength of 800 nm, and at a repetition rate
of 1 kHz. The laser light can be directed to different experiments and is used for
68
5. Experimental techniques and setups
Duration
Spectral width
Energy
Peak power
Repetition rate
Average power
Pump power
Oscillator Stretcher Pre-amp 4-pass amp
15 fs
300 ps
167 ps
167 ps
60 nm
60 nm
32 nm
32 nm
3 nJ
0.8 nJ
1.4 mJ
18 mJ
200 kW
2.6 W
8.4 MW
108 MW
100 MHz 100 MHz
1 kHz
1 kHz
300 mW
80 mW
1.4 W
18 W
3.4 W
8.5 W
80 W
Compressor
35 fs
32 nm
10 mJ
0.25 TW
1 kHz
8W
Table 5.1: General performances of the elements of the laser system
different applications like the generation of x-rays by laser-plasma interaction or the
generation of higher harmonics. For these kinds of applications, high energy pulses
are necessary, for our purposes, a small portion of this energy is enough.
In the following, the different elements of the laser system will be briefly described.
The first element of the chain is a diode-pumped Ti:Sa oscillator (Spectra Physics
Milenia) which delivers laser pulses at an energy of a few nJ, with a duration of 15 fs,
and at a repetition rate of 100 kHz. Before amplification, the pulse is stretched to a
duration of 300 ps with an Offner-type stretcher. The pre-amplifier of the laser chain
is a regenerative amplifier, which is pumped with 8.5 W by a frequency-doubled
Nd:YLF laser (BMI YLF) at a repetition rate of 1 kHz. The laser cavity of this
amplifier consists of a Pockels-cell, a dielectric polariser and a Ti:Sa crystal as the
amplifying medium. The next element of the laser chain is a four-pass cryogenic
cooled Ti:Sa amplifier, which is pumped by four frequency-doubled Nd:YLF lasers
(Thales Jade) delivering 20 W of laser power at 532 nm each. The pulses, which
have an energy of 18 mJ and a duration of 167 ps, are then compressed in a gratingbased compressor, resulting in 10 mJ-pulses of 35 fs duration. The performances of
the elements of the laser chain are summarised in table 5.1.
5.3 Measuring transient reflectivity
In this section, the experimental setups that were used to conduct time-resolved
measurements of reflectivity in photo-excited bismuth are described. Three different
configurations were utilised: a single-probe, a double-probe and a double-pump
setup. With the single-probe setup, transient reflectivity changes of a single probe
beam can be recorded. The double-probe setup enables one to measure reflectivity
changes at two different incident angles at the same time, which allows for recovery
of changes in the dielectric function. In both setups, photo-excitation of the sample
5.3. Measuring transient reflectivity
69
Figure 5.3: Schematic representation of the double-probe setup
is achieved with a single pump pulse arriving at a time delay ∆t with respect to the
probe pulse(s). With the double-pump setup it is possible to excite the sample with
two successive pump pulses with a variable time delay and record the photo-induced
reflectivity changes with a single probe.
5.3.1 Single- and double-probe setup
The scheme of the experimental setup, which is used to investigate ultrafast dynamics of reflectivity in bismuth, is presented in figure 5.3. The setups for single-probe
and double-probe measurements are essentially the same, the main difference is
that for double-probe measurements the single-probe setup is extended by adding
a second probe path with identical properties. The laser beam is coming from the
system described in the previous section. The energy delivered to our reflection
experiments is just a small portion of the total energy of the laser, the main part is
directed to another experiment that can be performed at the same time. The beam
has a diameter of 4 cm and the energy per pulse is 60 µJ, which is still several times
more than actually needed for the experiment. Before entering the setup, the beam
size is reduced with a 4 mm pin-hole. A tiny portion (less than 1%) of the light
is deflected by a 1 mm-thick glass plate (GP) and focused onto a photodiode that
70
5. Experimental techniques and setups
provides a reference signal that is used to take into account intensity fluctuations of
the laser. The main beam passes a beam splitter (BS1) and is divided into a pump
and a probe path containing 80% and 20% of the laser energy, respectively. The
intensities of both can be adjusted with variable neutral density filters, which are
not indicated in the scheme.
The pump beam then passes an optical chopper (Scitec Instruments 300CD) operating at 500 Hz which allows for lock-in detection of the signals. It is focused onto
the sample by an anti-reflection coated BK7 lens (L1) with a focal length of 500 mm
resulting in a spot size of 125 µm FWHM. The delay between the pump and the
probes can be varied with a delay line, created with two perpendicular mirrors that
can be translated with a stepper motor and thus change the optical path length of
the pump beam. The step size of the stepper motor is 1 µm, thus each step changes
the optical path length by 2 µm corresponding to a temporal delay of 6.7 fs.
The probe beam passes a half-wave plate (λ/2) after beamsplitter BS1, which
changes the polarisation of the laser light to s-polarisation. In order to minimise
residual p-polarised light, a high-contrast polariser (P) is used behind the halfwave plate. Afterwards the beam is divided into two equal portions with a second
beamsplitter (BS2). The beam named “probe 2” in figure 5.3 can be delayed with
a delay stage in order to perfectly synchronise the arrival of the two probes on the
sample, which is very important for an exact determination of the dielectric function,
as we will see in chapter 7. To enable an accurate synchronisation, a stepper motor
with a step size of 0.1 µm is used, which corresponds to 0.67 fs of temporal delay.
The two probe beams are focused onto the sample with identical anti-reflection
coated BK7 lenses (L2) of 100 mm focal length, the focal spot size of the probe is
40 µm FWHM.
The superposition of the three laser spots on the sample is carefully monitored
using a CCD-camera (Ganz 25C) with a 4.5×-zoom objective. The camera is also
used to determine the spot sizes of the pump and the probe beams. The pump spot
diameter is chosen to be more than three times larger than the probe spot in order
to compensate for the Gaussian spatial intensity profile and therefore assure that
the smaller probe beams probe a preferably equally excited region of the sample.
The energies of the pump pulse can be varied from a few nJ up to 3 µJ which
corresponds to a maximum excitation fluence of 45 mJ/cm2 largely exceeding the
damage threshold for bismuth. It is important that the fluence of the probe pulses
is much lower than that of the pump beam in order to avoid excitation of the sample
by the probe. The usual probe fluence in this experiment is 100 nJ/cm2 , therefore,
it is more than one order of magnitude lower than the lowest pump fluences applied.
The reflected probe beams and the reference signal are detected with identical
Si-PIN-photodiodes (Hamamatsu S1232) and the photocurrents are measured with
5.3. Measuring transient reflectivity
71
two digital lock-in amplifiers (Stanford SR830). Additional high-contrast polarisers
are used before the signal photodiodes in order to reduce the detection of scattered
light from the pump. A further improvement of the signal is achieved by placing pinholes and lenses in front of the signal photodiodes such that the divergent reflected
beams are parallel behind the lenses, and effects of the residual pump light are
minimised. The elements from the pin-hole to the photodiodes are covered with
black aluminium shielding material, which assures that only the light which enters
the detection path through the pin-holes is detected.
For single probe measurements, the incident angle of the probe beam is as close
to normal incidence as possible, and the choice of incident angles for the doubleprobe configuration is considered in section 5.4. Data acquisition and control of
experimental parameters like position of the stepper motors or integration time of
the lock-in amplifiers is done with a LabVIEW-program.
5.3.2 High-sensitivity detection system
In bismuth, the laser-induced variations in reflectivity ∆R/R0 (where R0 denotes
unperturbed reflectivity) are of the order of 10−4 to 10−3 depending on the level of
excitation. When measuring such small variations, a considerable error can be introduced by intensity fluctuations of the laser. To minimise the influence of fluctuations,
it is better to measure the reflectivity variations instead of absolute reflectivity. This
can be achieved by measuring the difference of the photocurrent from the signal photodiode with respect to the reference photodiode, ∆I = Isig − Iref . The photocurrent
of the signal photodiode can be expressed as Isig = fsig · c · Rt>0 · I, where fsig is the
fraction of laser light in the signal path, c is a proportionality constant taking into
account characteristics of the photodiode, Rt>0 is the reflectivity at some time after
excitation, and I is the intensity of the laser light. Before starting an acquisition, the
signal of the reference diode is adjusted such that is has the same magnitude as the
ones created by reflection of the probe beams from the unperturbed sample in the
two signal diodes. This is done by adjusting the intensity of the reference beam with
a circular variable neutral density while monitoring the difference signal with the
lock-in amplifier at 1 kHz until it is minimised. This assures that the photocurrent
of the reference diode Iref corresponds to the unperturbed reflectivity R0 = Rt<0 ,
which can be expressed as Iref = fref · c · R0 · I. Here, fref denotes the fraction of the
laser light in the reference path, which, after adjusting the signals, has the same
magnitude as fsig . The pump beam is chopped at 500 Hz, so the changes in ∆I
occur at this frequency. This allows for the detection of the changes with the lock-in
amplifier by using a 500 Hz-reference signal, which is phase-locked to the repetition
frequency of the laser. The difference ∆I is acquired and afterwards divided by the
72
5. Experimental techniques and setups
value of the unperturbed reflectivity signal obtained at 1 kHz
∆I
Isig − Iref
c · (fsig Rt>0 − fref R0 ) · I
∆R
=
=
=
,
Iref
Iref
fref · c · R0 · I
R0
(5.1)
resulting in reflectivity change with respect to unperturbed reflectivity. It can be
seen from 5.1 that theoretically, the signal does not depend on the intensity of
the laser. Under realistic experimental conditions, spatial variations of the laser
lead to different pulse-to-pulse intensity distributions in the reference and the signal
paths, therefore the fractions fsig and fref in the nominator and the denominator are
not equal and cannot be cancelled. The same applies to the constants c. Due to
the fact that even though the photodiodes are identical in construction, there is a
certain tolerance on their response as a function of intensity, so that each variation
of intensity (even if they are of exactly equal magnitude in the signal and reference
path) introduces an uncertainty in ∆I. In addition, the magnitude of the reference
signal is determined at a repetition rate of 1 kHz, whereas the acquisition of the
signal is done at half this frequency under the assumption that this does not affect
the measurement.
A crucial point for a good detection is the synchronisation of the chopper and
the laser, because it has to be assured that every second pulse of the pump beam
is blocked completely. Any residual pump signals at 1 kHz, even small portions of
unwanted laser pulses which pass the chopper, degrade the signal-to-noise ratio. To
ensure perfect synchronisation of the chopper and the laser, the chopper control unit
is triggered with the same 500 Hz-signal that is used as a reference for the lock-in.
The rotating speed of the chopper and its phase with respect to the trigger are controlled in real time and are automatically adjusted. Before starting an acquisition,
a photodiode is used to control the synchronisation, and the phase of the chopper
with respect to trigger frequency is changed if necessary.
The choice of the photodiode plays an important role for the detection system
as well. This is illustrated in figure 5.4: a) shows the intensity of the probe path
of the experiment as a function of time, b) shows the intensity of the pump path
and the 500 Hz-trigger signal that is used for the chopper and the lock-in amplifier.
In c) variations of the probe intensity are shown for an increase or a decrease in
reflectivity. In panel d) the probe signal that can be measured at a high-impedance
input of an oscilloscope or a lock-in in the absence of the pump beam is depicted.
The decay time of the signal depends on the capacity of the photodiode and the
input impedance. Panel e) shows how a probe signal created by probe pulses as
in c) would look like, and f) shows the difference between the signal of the probe
photodiode and the reference for the probe signal of panel e). It can be seen that
by adjusting the decay of the signal such that it corresponds to the repetition rate
5.3. Measuring transient reflectivity
a)
I
b)
73
d)
1ms
U
t
e)
1ms
t
U
I
c)
t
f)
t
DU
I
t
t
Figure 5.4: a) Probe pulses at a repetition rate of 1 kHz; b) pump pulses at a repetition
rate of 500 Hz (black line) and trigger signal for the chopper and the lock-in (green dotted
line); c) Intensities of probe pulses if pumped at 500 Hz in the case of an increase (first
pulse) and a decrease (third pulse) of reflectivity (red line) and intensities of probe pulses
without pump (black dotted line); d) signal of probe pulses without pump; e) signal of
probe pulses with pump (red line) and without pump (black dotted line); f) difference of
probe and reference signal
of the laser, a high resolution in the detection of the spectral component at 500 Hz
is achieved. By adjusting the decay of the signal, the detectable component in the
frequency domain is maximised due to the fact that the amplifier integrates the
signal during the chosen time constant of its low-pass filter.
The accuracy of detection can be estimated by measuring the noise of the system
at negative delays, i.e. before the arrival of the pump pulse. At t < 0, the fluctuations
of ∆R/R0 of the system measured with a lock-in time constant of τLI = 1 s are
δ = 10−5 . The measured reflectivity changes are on the order of 10−3 . In this
configuration, a complete measurement with good signal quality covering a time
span of 25 ps performed at intervals of 40 fs can be completed in 20 to 25 minutes.
74
5. Experimental techniques and setups
Figure 5.5: Schematic representation of the double-pump setup
5.3.3 Double-pump setup
An interesting question that arises when investigating ultrafast dynamics of photoexcited material is how the medium that has already been excited behaves when
another optical excitation is applied. For example, the investigation of the dynamics
of optical phonons in a material several picoseconds after a first excitation can yield
information about a possible transient state or a phase transition that could have
been induced by the first excitation. For this purpose, the double-probe setup
can be slightly modified in order to be able to excite a sample with two intense
pump pulses that can be delivered with a variable time delay while monitoring the
reflectivity changes with a probe beam. The resulting double-pump setup is depicted
in figure 5.5. The main difference between the double-pump and the double-probe
setup is that the probe beams of the double-probe setup are used as pump beams
in the double pump setup and that the pump beam of the double-probe setup is
used as the probe beam. This is achieved by modifying the energy of the pump and
probe beams and exchanging the positions of the chopper and the optics that change
the polarisation of the probe (λ/2, P). The pump beams are focused with identical
500 mm lenses, and the probe beam is focused with a lens of 100 mm focal length.
In this configuration, pump spots with a diameter of 125 µm and a probe spot with
5.4. Recovery of the dielectric function
75
a diameter of 40 µm, as in the double-probe setup, are achieved. The detection of
reflectivity changes, as well as the control of the experiment and its parameters is
done in the same way as for the setups described in section 5.3.1. The investigated
material can be excited by two successive laser pulses with more than 20 mJ/cm2
each, the time between the probe pulses can be varied between zero and several tens
of picoseconds.
5.4 Recovery of the dielectric function
In contrast to other optical properties of a material like reflectivity or transmittivity,
the dielectric function cannot be measured directly. As the real and imaginary parts
of the dielectric function are linked to reflectivity and transmittivity via the Fresnelformulae 2.28 - 2.31, they can be determined from two independent measurements.
For example, if the reflectivities for s- and p-polarisation at a given angle of incidence
are known, one has enough information to derive the real and imaginary part of the
dielectric function from it. A widespread technique used to measure static optical
properties is ellipsometry [80]. In a typical ellipsometry measurement, changes of
polarisation of light are measured at a fixed angle of incidence. The dielectric
function can then be numerically determined using the Fresnel equations. Usually, a
large number of data points are acquired by rotating the polarisation and repeating
the measurement for several angles of polarisation of the incident beam, which allows
for the determination of the dielectric function with a great accuracy.
Another possibility is to measure reflectivities at a given polarisation for at least
two different angles. This time-resolved dual-angle reflectometry method has been
successfully applied previously in single-wavelength configuration [81] as well as with
white-light femtosecond laser pulses [82]. If the incident angles are denoted α1 and
α2 , the relation between the real and imaginary part of the dielectric function and
the two measured reflectivities R1 and R2 can be expressed as the following:
R1 = f (α1 , re , im )
R2 = f (α2 , re , im ) ,
(5.2)
where the function f is defined by 2.32 and 2.33 for s- and p-polarised light, respectively. This is a system of two equations, where the two unknown variables are
re and im while R1 , R2 and the incident angles are measured. As f is a non-linear
function of re and im , the equations can not be solved directly. There are several
ways to find numerical solutions to this problem. A very powerful one is the NewtonRaphson-method (NRM) [83], which can be used to solve a system of N functional
relations involving variables xi , i = 1, 2, . . . , N :
gi (x) = 0,
i = 1, 2, . . . , N.
(5.3)
76
5. Experimental techniques and setups
Figure 5.6: Determination of the error in recovery. The reflectivities (R1 , R2 ) are calculated from the unperturbed dielectric function (re , im ) (black dotted arrows). Then,
the error in recovery is estimated by taking into account the error in reflectivity measurement δ. Here, two of the four pairs of reflectivities are shown: (R1+ , R2+ ) lead to
(re,++ , im,++ ) (orange arrows) and (R1− , R2+ ) to (re,−+ , im,−+ ) (green arrows). The
error is the maximum difference to the unperturbed values.
In our case, the function vector g has two components gi = f (αi , x) − Ri where the
index i is 1 or 2 and the vector x = (re , im ) denotes the two-component dielectric
function. In the neighbourhood of x, each of the functions gi can be approximated
in a Taylor series
g(x + δx) = g(x) + J · δx + O(δx2 ),
(5.4)
where J is the Jacobian matrix whose N 2 elements are:
Jij =
∂gi
.
∂xj
(5.5)
If terms of order δx2 and higher are neglected and g(x + δx) is set to zero, one
obtains a set of linear equations for the corrections δx that can be solved easily:
J · δx = −g .
(5.6)
Starting with an initial guess for x, the first calculation of δx moves each function
closer to zero simultaneously. The calculated corrections are added to the solution
vector xnew = xold + δx and the process is iterated until the convergence criteria are
fulfilled. In the implementation of the NRM used in this work, convergence of the
iteration is fulfilled when the corrections for a variable xi fall below a 10−5 · xi , a
condition which is usually fulfilled after less then ten iterations.
5.4. Recovery of the dielectric function
77
When extracting the dielectric function from the minimal number of two reflectivity measurements, the experimental parameters have to be chosen carefully in order
to minimise the error in re and im that is originating from the uncertainty of reflectivity measurement and translated through the inversion of the Fresnel formulae.
For example, it is intuitively understandable that the choice of a pair of incident
angles of the two probe beams α1 and α2 that are very close to each other leads to
a big error in dielectric function because the absolute reflectivities are almost equal.
In order to find the optimal experimental configuration, a simulation of the
uncertainty in dielectric function, which occurs when it is recovered from reflectivity measurements with a given error, has been carried out. The procedure is
sketched in figure 5.6. The computer program, which uses the NRM to solve the
problem of two equations with two unknown variables, first calculates the reflectivities R1 and R2 from the dielectric function for a given pair of angles α1 and
α2 . Here, the unperturbed values for re and im that were determined with ellipsometry (re = −16.25 ; im = 15.4 ; see chapter 7) are used. Then the inversion of the Fresnel equations is performed with four different pairs of reflectivities
(R1+ , R2+ ), (R1+ , R2− ), (R1− , R2+ ), (R1− , R2− ), where
Ri+ = (1 + δ) · R(αi )
and
Ri− = (1 − δ) · R(αi ) ,
i = 1, 2
are absolute reflectivities with a maximum deviation from their actual values given
by the uncertainty δ in measuring ∆R/R. Using the results of these inversions,
(re,++ , im,++ ) obtained from (R1+ , R2+ ), (re,+− , im,+− ) obtained from (R1+ , R2− )
etc., the differences from the actual values of the dielectric function δre,++ =
|re,++ − re | and δim,++ = |im,++ − im | etc. are calculated for all four combinations. The biggest deviation is taken as the error in recovery of the dielectric
function δre and δim . This calculation is repeated for all possible combinations of
incident angles 0◦ ≤ αi < 90◦ , α1 6= α2 , in steps of 0.25◦ .
The results of this simulation are presented in figure 5.7. The sum of the squared
relative errors of real and imaginary part of the dielectric function δsum = (δre /re )2 +
(δim /im )2 is presented as a function of the two incident angles. δsum is plotted on
a logarithmic false colour scale. The calculation has been performed for an error in
measurement of reflectivity of δ(∆R/R0 ) = 10−5 and δ(∆R/R0 ) = 10−4 , depicted
on the left and the right side of the figure, respectively. For our experimental conditions, only the former is relevant; nevertheless, it is important to see how the
recovery of the dielectric function would suffer from a less accurate measurement of
reflectivity. The black line from bottom left to top right is for α1 = α2 , where δsum
is set to zero.
The calculations show that the choice of incident angles has an enormous influence
on the quality of recovery of the dielectric function. The sum of squared errors
78
5. Experimental techniques and setups
Figure 5.7: Calculation of error in recovery of the dielectric function that would occur due
to an uncertainty in reflectivity measurement of 10−5 (left) and 10−4 (right). The sum of
the squares of relative errors of real and imaginary part δsum (see text) is plotted as a
function of the two incident angles.
δsum varies over a range of three orders of magnitude. The red zones indicate that
the program failed to calculate the dielectric function from the pairs of reflectivities
described above. For these combinations of angles, the Newton-Rahphson algorithm
does not converge and the program sets δsum = 1. The graphs also show that there
is a relatively wide range of angles where the error is low, which is the case around
the minimum (26◦ < α1 < 41◦ , 0◦ < α2 < 5◦ ) for δ(∆R/R0 ) = 10−5 . As expected,
a choice of two angles which are too close leads to a large error in recovery of the
dielectric function. If α1 ≈ α2 the NRM diverges. In the green and light blue zones
close to the diagonal corresponding to a difference between α1 and α2 of less than 1◦ ,
the error is almost two orders of magnitude higher than the lowest possible value.
The simulation done with an error in reflectivity of δ(∆R/R0 ) = 10−4 yields
similar results. The main differences are that δsum is approximately one order of
magnitude higher and that the red zones are larger. Both are comprehensible as
the uncertainty in reflectivity measurement in this case was taken to be one order
of magnitude higher.
Another important point that has to be considered is the loss of time resolution
due to the projection of the wave fronts of the different beams onto the sample surface. The bigger the difference in incident angles, the bigger the temporal mismatch
between the beams. In order to have a good compromise between time resolution
and accuracy in recovery of the dielectric function, we chose 19.5◦ and 34.5◦ for the
two probe beams and an angle of 22◦ for the pump beam. With this setup, the temporal mismatch between the two probes is ∼ 40 − 45 fs and therefore comparable to
5.4. Recovery of the dielectric function
79
the pulse duration of the laser. The error in recovery predicted by the simulation
for this configuration is (δre /re ) = 4.0 · 10−3 , (δim /im ) = 2.5 · 10−2 and therefore
δsum = 2.5 · 10−3 . In order to check if the calculation of the dielectric function from
measured experimental data is correct, some measurements have been repeated with
a second pair of angles (α1 = 14.5◦ , α2 = 29.5◦ ). As the values for re and im should
not depend on the angles of incidence at which the reflectivities are measured, a
comparison of the results at two different pairs of angles can be used to test the
validity of the experimental method.
The significance of the estimations presented above is limited, inasmuch as only
the recovery of the unperturbed dielectric function is considered. However, without
having acquired any experimental results, it is not clear what the magnitude of
changes in dielectric function is. Usually, one would expect changes in the dielectric
function that are of the same order of magnitude than the reflectivity changes if an
optical excitation is applied. Under these circumstances, due to the small changes
of reflectivity, the simulation can be considered a good indication of the real errors
of re and im even in the photo-excited case.
80
5. Experimental techniques and setups
6 Reflectivity measurements of
coherent optical phonons in
bismuth
In this chapter, the femtosecond time-resolved measurements of the reflectivity dynamics after laser-excitation below the damage threshold in bismuth are presented.
The measurements have been performed with the single-probe setup presented in
chapter 5. The results are discussed in relation to previous investigations of optical
phonons in bismuth and in the light of the theoretical considerations of chapter 4.
6.1 Single probe optical measurements
6.1.1 Experimental results
The time-dependent reflectivity dynamics induced by excitation with two different
pump fluences are depicted in figure 6.1. These measurements, which show the relative reflectivity change ∆R(t)/R0 , where R0 denotes the unperturbed reflectivity at
negative time delays, are taken from a series of measurements that were done with
a laser pulse duration of 35 fs and with a probe beam that was close to normal incidence (10◦ ). The sample was a single crystal of bismuth whose (111)-axis (the c-axis)
was perpendicular to the surface, and the experiment was carried out at room temperature. The traces contain characteristic features that were observed at all pump
fluences in the range from 1.5 mJ/cm2 to 15.0 mJ/cm2 . At the very beginning, during excitation, the reflectivity drops below the unperturbed value for a short instant,
then it rapidly increases above the initial level and reaches a maximum at about
200 fs after the excitation pulse. The width of the drop roughly corresponds to the
pulse duration, it is 45 fs FWHM for an excitation with 6.7 mJ/cm2 . The reflectivity
then shows a behaviour that can be described by a superposition of an oscillatory
and a non-oscillatory component. The former is characterised by a damped sinusoidal oscillation with a fluence-dependent frequency, and the latter by a rapid rise
on a fs-scale followed by a slow decrease on a ps-scale. The frequency of oscillation in
the case of excitation with 2.7 mJ/cm2 is 2.90 THz, which coincides with the value
81
82
a )
6. Reflectivity measurements of coherent optical phonons in bismuth
1 ,5
1 ,5
1 ,0
/ R
0
1 ,0
∆R
0 ,5
/ R
1 0
3
0

0 ,0
0 ,5
∆R
-0 ,5
0 ,0

0 ,5
1 ,0
1 ,5
2 ,0
1 0
3
T im e / p s
0 ,0
-0 ,5
0
2
4
6
8
1 0
1 2
1 4
1 6
1 8
2 0
2 2
2 4
2 6
2 8
3 0
T im e / p s
b )
1 ,5
1 ,5
0
1 ,0
/ R
1 ,0
∆R
0 ,5
/ R
1 0
3
0

0 ,0
0 ,5
∆R
-0 ,5
0 ,0

0 ,5
3
1 0
1 ,0
1 ,5
2 ,0
T im e / p s
0 ,0
-0 ,5
0
2
4
6
8
1 0
1 2
1 4
1 6
1 8
2 0
2 2
2 4
2 6
2 8
3 0
T im e / p s
Figure 6.1: Transient reflectivity change in bismuth following an excitation with
6.7 mJ/cm2 (upper panel) and 2.7 mJ/cm2 (lower panel). The graphs show the experimental curves (black lines) and curves resulting from fits according to equation 6.1 (red
dotted lines). The insets show a “zoom” on the first 2.5 ps of the reflectivity dynamics.
for the A1g optical mode of bismuth, 2.92 THz [84]. In the case of 6.7 mJ/cm2 , the
frequency has an initial value of 2.86 THz. The decrease gets bigger with increasing
excitation fluence, which is in agreement with previous experiments [48]. In addition, the frequency shows a time-dependent behaviour: with increasing time delay,
the frequency increases towards the unperturbed A1g mode frequency of 2.92 THz.
At about 10 ps after excitation, the signal drops below the unperturbed level and,
after having reached its minimum value, it remains almost constant. Then it slowly
returns to the unperturbed value in roughly 4 ns, which indicates that the sample
completely recovers and returns back to the initial, unperturbed state before the
next excitation pulse arrives.
6.1. Single probe optical measurements
83
6.1.2 Analysis and discussion
The initial sharp drop in reflectivity and the negative reflectivity change after 10 ps
are novel observations that cannot be understood in the light of the previous theoretical models presented in chapter 3. In what follows, the complex features of
transient reflectivity will be related to the theoretical considerations of chapter 4.
The reflectivity changes of the entire observation period can be theoretically reproduced and understood in the framework of the underlying considerations of our
theory considered in chapter 4. Firstly, the initial drop in reflectivity is a result
of the interplay of the different contributions to ∆R/R0 that compete during the
excitation pulse. The polarisation components are present from the very beginning
of the laser pulse, creating a negative contribution, while the number of free carriers, which is proportional to the electron temperature, increases during the pulse.
Therefore, the first negative drop in reflectivity should always be present at the
beginning of an excitation pulse until the positive contribution from the ne -related
term becomes dominant. The amplitude of the initial drop was found to be approximately proportional to the laser intensity, which is in agreement with the theory:
∆p is proportional to the fast atomic displacement (section 4.2.1), which is proportional to the laser-induced force, of which the polarisation-dependent contribution
is in turn proportional to the laser intensity, F pol ≈ 4πα0 I/(na dc) (equation 4.28).
In addition, the initial drop could only be resolved with the maximum time resolution of 35 fs. With longer laser pulses or larger incident angles this feature is
less pronounced, and with a time resolution of 50 fs the drop could not be resolved
anymore, which qualitatively agrees with the theory. The magnitude of the negative
reflectivity change occurring after ∼ 20 ps increases with increasing laser fluence.
This is also in agreement with the theory, due to the fact that it attributes the
change to electron-lattice temperature equilibration.
The red dotted lines in figure 6.1 are fits that have been performed on the basis
of the following equation
∆R/R0 = A · ∆pol + B · ∆Te + C · ∆Tl + D · ∆q ,
(6.1)
which has been derived in chapter 4. Here, the four coefficients A, B, C and D
are fitting parameters that have the same signs as the corresponding coefficients
in equation 4.58: A, C, and D are negative and B is positive. The electron
temperature at the end of the pulse is calculated with the help of equation 4.16,
2
Te2 = 4F AF (tp )(π 2 ne ls kB
), for 2.7 mJ/cm2 , Temax = 2835 K and for 6.7 mJ/cm2 ,
Temax = 4450 K. The temporal evolution of Te and Tl is estimated with the twotemperature model. The maximum lattice temperature is reached after equilibration of electron and lattice temperatures after ∼ 20 ps and is respectively 702 K and
84
6. Reflectivity measurements of coherent optical phonons in bismuth
0
-5
∆R / R
R T
-1 0
-1 5
1 0
3

-2 0
-2 5
-3 0
3 0 0
3 5 0
4 0 0
T e m p e ra tu re / K
4 5 0
5 0 0
Figure 6.2: Relative change in reflectivity ∆R/RRT as a function of sample temperature,
where RRT denotes the reflectivity at room temperature (dots); linear fit to the data
(dotted line).
1273 K if electron diffusion is neglected. The amplitude of oscillation is taken to
be proportional to the maximum electron temperature, and the temporal evolution
of the frequency is modelled according to equation 4.47, ω 2 ≈ ω02 (1 − (Te /|b,0 |)).
The agreement between the theoretical and the experimental curves is excellent.
However, at time delays larger than 30 ps, the experimental curve starts to evolve
towards the unperturbed value while the theoretical one remains at its minimum
reflectivity value. This slight discrepancy probably relates to heat diffusion from the
skin layer, which was not included in the calculations.
A remarkable finding is the elevated lattice temperatures after electron-lattice
equilibration, which are higher than the melting temperature of 544.5 K [39]. In
addition, the deposited energy densities, 0.48 kJ/cm3 in the case of 2.7 mJ/cm2
pump fluence and 1.19 kJ/cm3 in the case of 6.7 mJ/cm2 , are comparable to the
enthalpy of melting in equilibrium which is 0.5 kJ/cm3 [62]. This could lead to
the conclusion, that the evolution of the lattice temperature over the melting point
should result in the melting of the crystal. However, it has been demonstrated that
this is not mandatory in the case of ultrafast heating. In previous experiments
that were performed with ice [85], gallium [81], and aluminium [86, 87], samples
were overheated to Tlmax /Tmelt = 1.07, 2.67, 9.40, respectively, where Tlmax denotes
the maximum lattice temperature and Tmelt denotes the melting temperature. The
6.1. Single probe optical measurements
85
investigations yielded that these materials do not melt during times of 200 ps, 20 ps
and 3.5 ps, respectively. These times are longer than electron-lattice equilibration
time when melting should occur. Therefore, it is possible that in our case, where the
ratio of maximum temperature and lattice temperature is 1273 K/544.5 K = 2.34,
the sample is not molten after the electrons have transferred their energy to the
lattice. Nevertheless, the reflectivity of liquid bismuth is lower than the reflectivity
of solid bismuth, so the possibility that the decrease in reflectivity is due to melting
cannot be excluded solely on the basis of a reflectivity measurement at a single
wavelength.
In order to compare the reflectivity change that occurs after electron-lattice equilibration to a change in reflectivity caused by simply heating an unperturbed bismuth
crystal, a standard ellipsometry measurement has been carried out, where the reflectivity has been measured as a function of the sample temperature. The relative
change in normal-incidence reflectivity ∆R/RRT = (R(T ) − RRT )/RRT for temperatures between room temperature and 475 K is depicted in figure 6.2.
Here, RRT denotes the reflectivity at room temperature. The measurement has
been repeated several times, and the data points correspond to the average value
and the error bars to the standard deviation. The measurement that has been
performed with the same sample as the time-resolved measurements shows that the
reflectivity linearly decreases with increasing temperature in the probed range of
temperatures. However, the temperature decrease is small, and the slope that can
be obtained by fitting with a linear function of temperature is ∼ −0.15 · 10−3 K−1 .
That means, that a decrease in reflectivity of ∆R/R0 = 0.3 · 10−3 , as it has been
measured in the time-resolved measurements after ∼ 25 ps, would correspond to
a rise in temperature of ∼ 2 K, if it was caused by simply heating the sample in
equilibrium. At a first glance, this result seems to contradict several features of
the applied model. In fact, it can be associated with possible shortcomings of the
theory:
• the motion of electrons into the bulk is not taken into account. Thus, the
lattice temperature after electron-lattice equilibration as calculated with the
two-temperature model is probably too elevated. Indeed, recent experiments
performed with a time-resolved x-ray diffraction technique showed that coherent oscillations are present much deeper in the crystal than the skin layer,
exceeding the magnitude of the skin layer several times [88]. This is an indication for the presence of excited electrons up to a much bigger depth than
the skin layer. In the framework of our theory, it could be interpreted as an
electron gradient over a bigger part of the bulk as assumed above which excites
coherent phonons over its whole length. The presence of atomic vibrations in
86
6. Reflectivity measurements of coherent optical phonons in bismuth
a much bigger volume than that defined by the pump spot size and the skin
layer leads to a distribution of the laser energy to a bigger volume therefore
effectively lowering the lattice temperature in the excited part of the crystal.
• The estimations make use of certain physical properties of bismuth that may
drastically change upon strong electronic excitation. For instance, the electron diffusion and energy transport properties could be significantly altered
due to the highly non-equilibrium state of the sample after the strong optical
excitations applied in the experiments. Furthermore, it is thinkable that the
assumption that the electron effective mass equals the free electron mass does
not describe the physical reality of a strongly excited solid.
In conclusion, it can be stated that despite its partially crude simplifications, our
theory describes the basic processes that lead to the complex transient behaviour of
reflectivity in bismuth reasonably well. However, the disagreement of theoretically
calculated lattice temperature after electron-lattice equilibration, and the estimation
of this particular temperature from the reflectivity change indicates limitations of
this approach. This disagreement is an indication for the presence of a mechanism
carrying away the energy from the skin layer as indicated above or another physical
process lowering the reflectivity that has to be identified.
6.2 Fluence dependence of reflectivity dynamics
In the last section, two measurements of the reflectivity dynamics in laser excited
bismuth have been presented. The pump pulses applied to the sample in this experiment lead to the excitation to a highly non-equilibrium state which, according
to the calculations with the two-temperature model, results in a temperature after
electron-lattice equilibration that is higher then the equilibrium melting temperature. The fact that no damage of the sample was observed raises the question of
how far from the damage threshold these measurements have been carried out and
how the dynamics change if one approaches the level of excitation which leads to
disordering of the crystal. In the following section, measurements of the reflectivity
Figure 6.3 (facing page): Transient reflectivity changes in bismuth at room temperature
for excitation fluences ranging from 3.7 mJ/cm2 to 21.0 mJ/cm2 . In the upper graph,
the four different measurements are displayed separately for delay times up to 30 ps. For
clarity, the curves are set off vertically and the dotted lines indicate ∆R/R0 = 0 for each
curve. The lower graph shows the first few oscillations of the same four measurements
without the offset.
6.2. Fluence dependence of reflectivity dynamics
87
a )
2
2
3 .7 m J /c m
0
2
2
6 .9 m J /c m
0
/ R
0
2
3
0
1 0
1 5 .0 m J /c m
2
∆R

2
0
2 1 .0 m J /c m
0
2
4
6
8
1 0
1 2
1 4
1 6
1 8
2 0
2 2
2 4
2
2 6
2 8
3 0
T im e / p s
b )
3 .7
6 .9
1 5 .0
2 1 .0
0
8
2
2
2
2
1 0
3
/ R
4
m J /c m
m J /c m
m J /c m
m J /c m

∆R
0
-4
-0 ,4
-0 ,2
0 ,0
0 ,2
0 ,4
0 ,6
0 ,8
T im e / p s
1 ,0
1 ,2
1 ,4
1 ,6
1 ,8
2 ,0
88
6. Reflectivity measurements of coherent optical phonons in bismuth
dynamics that cover a larger span of excitation fluences up to the excitation close
to the damage threshold and slightly above will be presented.
6.2.1 Experimental results
A series of measurements of ∆R/R0 for different excitation fluences ranging from
3.7 mJ/cm2 to 21.0 mJ/cm2 is shown in figure 6.3. The upper graph shows the
transient reflectivity changes for each fluence for a time span after optical excitation
of 30 ps, where the traces are set off vertically. The lower graph shows the first
few oscillations of the same four traces in a single plot. The sample was the same
bismuth crystal as in the measurements of the previous section, and the experiment
has been performed at room temperature. The applied laser energy was below the
melting threshold (see section 6.2.3) for all four traces. The reflectivity dynamics can
be described in a similar way to the measurements presented in figure 6.1. At first,
there is a narrow drop in ∆R/R0 . This feature is only visible in the measurement
corresponding to the highest fluence, but it is much less pronounced compared to the
measurement presented in figure 6.1. The reason for this difference is that this set of
measurements was recorded with a lower temporal resolution of ∼ 50 fs which is due
to a longer pulse duration. Then, the reflectivity rises and shows damped oscillations
with a frequency close to the A1g -mode frequency. The reflectivity decreases, drops
below the the unperturbed value of reflectivity, ∆R/R0 = 0, and finally reaches its
minimum value after ∼ 20 ps before slowly increasing towards the unperturbed level
on a nanosecond time-scale.
As it can be seen from the presented results, all of the features of the transient
reflectivity changes depend on excitation fluence: the amplitude and damping of the
oscillations, the red-shift of the A1g frequency, the time when the relative reflectivity
change crosses the zero-level and the value of the minimum. In the next section,
these dependences on the level of excitation will be examined more closely.
6.2.2 Analysis and discussion
As it can be seen from the lower panel in figure 6.3, the amplitude of the reflectivity oscillations increases with increasing pump fluence. In order to estimate the
amplitudes for different fluences, least-square fits to the reflectivity signals with a
damped harmonic oscillator superposed to an exponential decay,
∆R/R0 = Ae · e−βt + Aq · e−γt · sin(2πνt + φ) + B ,
(6.2)
have been performed for each fluence. Here, Ae , Aq , β, γ, ν, φ and B are fitting
parameters. The above function is not suited to reproduce the transient reflectivity
6.2. Fluence dependence of reflectivity dynamics
89
8
) 1 0
3
7

A m l l i t u d e o f o s c i l l a t i o n ( ∆R / R
0
6
5
4
3
2
1
0
0
5
1 0
F lu e n c e / m J c m
1 5
2 0
-2
Figure 6.4: Amplitude of the reflectivity oscillations as a function of the pump fluence
(dots) and a linear fit to the data points (dotted line).
changes of the whole observation period of several tens of ps. This is due to the fact
that the electronic part (the first term) is taken to be an exponential decay while
it should be described by a term governed by the electron temperature in the twotemperature model and that the frequency is not chirped. However, it is possible
to fit this function with good accuracy to the first few oscillations, which allows for
an exact determination of the oscillation amplitude Aq , the frequency ν, and the
damping constant γ.
The amplitude of oscillations as a function of pump fluence is shown in figure 6.4.
The error bars that result from the fit get bigger for higher fluences. This can be understood by considering that with higher fluences, the physical reality deviates more
and more from equation 6.2. For instance, the frequency chirp gets bigger and thus
so does the error that is induced by fitting with a fixed frequency. The oscillation
amplitude increases linearly with the excitation fluence until ∼ 12 mJ/cm2 , but for
higher levels of excitation there is a slight deviation from the linear behaviour. The
linear dependence was also found in previous measurements performed by DeCamp
et al. [48]; however in these experiments a saturation of the amplitude was reported
for fluences higher than 8 mJ/cm2 .
The A1g -frequency as a function of the fluence is shown in figure 6.5. Here, the
data has been obtained by fitting the function 6.2 to the first two oscillations in order
to consider the frequency shortly after optical excitation. The error bars result from
90
6. Reflectivity measurements of coherent optical phonons in bismuth
3 ,0
2 ,9
2 ,8
2 ,6
2 ,5
2 ,4
2 ,3
A
1 g
fre q u e n c y / T H z
2 ,7
2 ,2
2 ,1
2 ,0
0
5
1 0
F lu e n c e / m J c m
1 5
2 0
-2
Figure 6.5: A1g -mode frequency directly after the pump pulse as a function of the pump
fluence (black dots) and a theoretical curve calculated with equation 4.47 (dotted line).
the fits, and the error increases with fluence due to the same reasons as described
for the error bars in figure 6.4. The red-shift of the frequency gets bigger with
increasing fluence, a result which is in agreement with previous findings [48, 50].
For the highest fluence in this series of measurements, 21 mJ/cm2 , the decrease in
frequency is ∼ 25%, indicating the enormous altering of the lattice potential due to
electronic excitation.
The dotted line in figure 6.5 has been calculated with equation 4.47, ω 2 ≈
2
ω0 (1 − Te /|b,0 |), without any additional numerical coefficient. For fluences up to
7 mJ/cm2 , the red-shift is reproduced in a satisfactory manner, but for higher fluences the data points significantly deviate from the prediction of section 4.3.3. It
should be noted here that the power densities applied to the bismuth samples at
fluences of several mJ/cm2 lead to a highly non-equilibrium transient state and
overheat the sample to a factor of more than Tlmax /Tmelt = 3 (for 10 mJ/cm2 , for
instance). Taking into account that the theoretical curve was calculated using a
simplistic form of an inter-atomic potential and that the squared frequency was approximated to decrease linearly with electron temperature, the coincidence of the
red-shift theoretically calculated without any fitting parameters and measured redshift for fluences < 10 mJ/cm2 is amazing. When even higher fluences are applied,
the energetic conditions get closer to the onset of disordering of the sample and
non-linear effects affect the atomic oscillations.
6.2. Fluence dependence of reflectivity dynamics
2 ,9 0
3 ,0
2 ,8 8
2 ,8
2 ,8 6
2 ,4
4 .0
6 .7
1 3 .3
2 0 .0
2 ,2
2 ,0
m J /
m J /
m J /
m J /
c m
2
c m
c m
c m
2
2
2
F re q u e n c y / T H z
2 ,6
F re q u e n c y / T H z
91
2 ,8 4
2 ,8 2
2 ,8 0
2 ,7 8
1 ,8
2 ,7 6
0 ,0
0 ,6
1 ,2
1 ,8
T im e / p s
2 ,4
3 ,0
0 ,0
0 ,6
1 ,2
1 ,8
2 ,4
3 ,0
T im e / p s
Figure 6.6: Left: temporal dependence of the A1g -mode frequency for excitation fluences
of 4.0 mJ/cm2 (black crosses), 6.7 mJ/cm2 (red triangles), 13.3 mJ/cm2 (green squares)
and 20.0 mJ/cm2 (blue dots). Right: temporal dependence of the A1g -mode frequency for
6.7 mJ/cm2 (red triangles) and a fit according to equation 4.47 (black line).
In order to examine the chirp of the phonon frequency and its dependence on
the fluence, one has to estimate the frequency as a function of time. This has
been done by fitting 6.2 to the reflectivity signals with different “time windows”.
The function is fitted to two oscillations, for example, between the first maximum
at t1 and the third maximum at t3 . The resulting frequency is associated with
the time t1 + (t3 − t1 )/2 corresponding to the middle of the interval defined by
t1 and t3 . Then the time window is moved and the fit is performed between the
second and fourth maximum and so on. The results for four different fluences are
presented on the left side of figure 6.6. The data is derived from an earlier series of
measurements; therefore the frequencies directly after excitation slightly differ from
the data of figure 6.5 due to uncertainties in the measurement of the pump fluence.
The temporal behaviour of the frequency has a strong dependence on the excitation
fluence. The frequency is lowest directly after excitation and then evolves towards
the unperturbed A1g -frequency of 2.92 THz, while the temporal evolution changes
with fluence. The right side of figure 6.6 only shows the data for 6.7 mJ/cm2 . The
black line is a theoretical curve for which the temporal behaviour of the frequency
has been estimated on the basis of equation 4.47. In the equation, the electron
temperature has been taken to decay like p1 · exp(−t/τ ) + p2 where p1 , p2 and τ
were fitting parameters. As mentioned before, the electron temperature does not
decrease exponentially, however, for the first 3 ps an exponential decrease is a good
approximation. The temporal behaviour is excellently reproduced with the fit which
92
6. Reflectivity measurements of coherent optical phonons in bismuth
4 4 0 fs
5 2 0 fs
1 2
1 0
∆R / R
8
6
1 0
3

4
2
0
0 ,0
0 ,2
0 ,4
0 ,6
0 ,8
1 ,0
1 ,2
1 ,4
1 ,6
T im e / p s
Figure 6.7: Transient reflectivity changes in bismuth after optical excitation with a fluence
of 28.5 mJ/cm2 .
results in p1 = 0.16 eV and p2 = 0.14 eV. The two parameters p1 and p2 represent
the maximum electron temperature, Temax = (0.16 + 0.14) eV/kB = 3500 K. The
offset p2 can be regarded as the common temperature of electrons and lattice after
equilibration Tlmax = 0.14 eV/kB = 1600 K. These two values are in reasonable
agreement with the results of the two temperature model (Temax = 4450 K and
Tlmax = 1273 K).
As in the case of the theoretical curve in figure 6.5, the temporal behaviour of the
frequency in figure 6.6 cannot be modelled with equation 4.47 for fluences higher
than 6.7 mJ/cm2 , which is due to the fact that for both dependencies the same
theoretical considerations have been used.
The red-shift of the frequency and the temporal chirp become even more intense
when a fluence above the melting threshold is applied. Figure 6.7 shows a measurement of the reflectivity dynamics in bismuth after excitation with 28.5 mJ/cm2 .
When applying this power density at the repetition rate of 500 Hz to the sample,
cumulative damages could be observed after ∼ 10 s. The small accessible size of
the sample surface did not allow for the exposure of a fresh area on the surface
with each pump pulse or a small number of pulses. As a compromise, the sample
has been exposed for 5 s corresponding to 2500 pump pulses and then moved in
order to have an undamaged spot on the surface for the next acquisition. Like this,
the first 1.6 ps of the reflectivity dynamics could be put together from a reasonable
6.2. Fluence dependence of reflectivity dynamics
93
1 ,4
D a m p in g c o n s ta n t / p s
-1
1 ,2
1 ,0
0 ,8
0 ,6
0 ,4
0 ,2
0 ,0
0
5
1 0
F lu e n c e / m J c m
1 5
2 0
-2
Figure 6.8: Damping constant as a function of the excitation fluence (dots) and linear fit
to the data (dotted line).
number of acquisitions. The time period from the first to the second maximum is
520 fs corresponding to a frequency of 1.92 THz, and from the second to the third
maximum it is 440 fs corresponding to 2.27 THz. To the best of our knowledge, this
is the largest red-shift of the A1g -mode in bismuth ever measured, and it confirms
results previously observed by Sokolowski-Tinten et al. in a time-resolved x-ray
diffraction measurement [22]. Here, the sample was a 50 nm thick bismuth film and
the excitation fluence was more than a factor of two lower. The fact that a similar
reduction of frequency to 2.12 THz was observed after the excitation with a lower
energy is probably due to the different spatial distributions of energy in bulk crystals
and thin films where the diffusion of electrons is limited to a small depth. The high
damping of the oscillations in figure 6.7 illustrates that the heating of the lattice due
to transfer of energy from the excited electrons and thermal expansion transforms
the initially harmonic vibrations of atoms into a strongly non-linear chaotic motion
manifesting the onset of destruction of the solid phase.
The damping constant of the reflectivity oscillations shown in figure 6.3 has been
determined by fitting the function 6.2 to the transient reflectivity signals and taking
into account the chirp of the frequency. Again, the fitting function is not suited to
describe the temporal behaviour of reflectivity for the entire presented observation
period of 30 ps, but it is possible to produce fits to the first few oscillations that allow
for the estimation of the damping with sufficient accuracy. The result of the fits, the
94
6. Reflectivity measurements of coherent optical phonons in bismuth
) 1 0
3
0 ,0
-0 ,5
o f r e l f e c t i v i t y c h a n g e ( ∆R / R
-1 ,0
M in im u m
0

-4 ,5
-1 ,5
-2 ,0
-2 ,5
-3 ,0
-3 ,5
-4 ,0
-5 ,0
-5 ,5
0
5
1 0
F lu e n c e / m J c m
1 5
2 0
-2
Figure 6.9: Minimum reflectivity change ∆R/R0 after ∼ 20 ps as a function of the excitation fluence (dots) and linear fit to the data (dotted line).
damping constant γ as a function of the pump fluence, are depicted in figure 6.8,
and the error bars result from the fitting process. The magnitudes of the damping
constants are ranging from 0.28 ps−1 for 1.5 mJ/cm2 to 1.21 ps−1 for 21.0 mJ/cm2 ,
which is in agreement with previously reported results [48]. The dotted line in the
plot is a linear fit to the data which shows that the damping constant increases
almost linearly with the fluence.
Finally, the negative change in reflectivity occurring several picoseconds after excitation shall be analysed. It can be seen from figure 6.3 that the time when the
reflectivity change crosses the zero as well as the magnitude of negative reflectivity change that is reached after ∼ 20 ps depend on the pump fluence. The time
when the reflectivity drops below its unperturbed value is reduced from ∼ 8 ps for
3.7 mJ/cm2 to ∼ 1.5 ps for 21.0 mJ/cm2 . This can be understood in the framework of the two-temperature model where the transfer of energy from the excited
electrons to the lattice is governed by the linear term g · (Te − Tl ). Due to the fact
that the maximum electron temperature is proportional to the fluence, and that for
fluences of several mJ/cm2 the initial lattice temperature is negligible compared to
the electron temperature, the electron-lattice coupling linearly increases with excitation fluence. Therefore, the higher the fluence, the quicker the energy from the
initially excited electrons is transferred to the lattice, and in the framework of the
6.2. Fluence dependence of reflectivity dynamics
95
theoretical considerations of chapter 4 this increase in lattice temperature leads to
the decrease in reflectivity.
The fluence dependence of the minimum reflectivity that is attained after ∼ 20 ps
is depicted in figure 6.9. The negative change in reflectivity increases with the pump
fluence, and the dotted line shows that the dependence can be described by a linear function reasonably well. To these observations, the discussion of section 6.1.2
applies: if the negative change in reflectivity is attributed to an increase in lattice
temperature, it can be understood in the framework of the theoretical considerations
of chapter 4. The fluence dependence of the maximum reflectivity change is then
related to the dependence of lattice temperature Tlmax after electron-lattice equilibration. However, when calculated with the two-temperature model, the Tlmax is several
times higher than the melting temperature. In contrast, the reflectivity change of
∆R/R0 = −5 · 10−3 that is observed after 20 ps for a fluence of 21.0 mJ/cm2 would
correspond to a temperature increase of 30 K if it was compared to the change in
reflectivity that occurs during heating in equilibrium (see figure 6.2). This discrepancy underlines the need for additional experiments that will be discussed at the
end of this chapter.
6.2.3 Accuracy of fluence measurements and the damage
threshold of bismuth
When comparing results from different series of measurements, it can be seen that
several features of reflectivity oscillations differ in their magnitude even though the
excitation fluence is the same. For example, the amplitude of the upper curve
in figure 6.1 and the second curve from above in figure 6.3 do not coincide, even
though the excitation fluence is similar. These variations are due to the uncertainty
in measurement of the excitation fluence that will be evaluated here.
The excitation fluence is usually measured by evaluating the spot size of the
pump pulse and the energy of a laser pulse and then dividing the pulse energy by
the spot area. If the probing is done with a smaller spot size than the pump spot,
a systematic error is introduced by neglecting the Gaussian intensity profile of the
laser beam. In the experiments presented here, the pump and probe spots had a
diameters of 125 µm and 40 µm, respectively. If the probe spot is aligned such that
it is exactly in the middle of the pump spot, the power density on the probed area
is higher than the average intensity over the pumped area. Thus, for the evaluation
of the pump fluence, a factor has to be considered which takes into account the
overlap of the Gaussian profiles of the pump and the probes. Still, the uncertainties
in power measurement and determination of the spot size introduce a considerable
error. In our case, the energy per pulse could be measured with an accuracy of
96
6. Reflectivity measurements of coherent optical phonons in bismuth
5%. Concerning the error on the pump spot diameter, it is reasonable to assume an
error of 10%. The calculation of the error on fluence with the laws of propagation
of uncertainty results in a relative error in the fluence F of 20%. Considering that
by translating or realigning the sample it can be moved out of focus, introducing an
error in the spot size, and that the superposition of pump and probe is not perfect,
it can be understood that the uncertainty in F can be even higher.
When performing sets of measurements with increasing fluence, irreversible damage to the samples, which can be seen as dark grey spots on the silverish bismuth
surface, occurred for fluences of Fth = 24 mJ/cm2 and higher. Due to the fact that
the pumping was done at a rate of 500 Hz, the damage threshold for single shot excitation was not determined in this experiment. The estimated damage threshold is
in good agreement with the value reported by DeCamp et al., which is ∼ 25 mJ/cm2
(for pulses with a central wavelength of 790 nm at a kHz repetition rate) [48], and
the one reported by Misochko et al., which is 22 mJ/cm2 (pulses with a central
wavelength of 800 nm at a 100 kHz repetition rate) [89].
6.3 Temperature dependence of reflectivity dynamics
In section 6.2, measurements of the transient reflectivity in bismuth have been presented for different excitation fluences below the damage threshold. The changes in
the properties of coherent phonons due to the variation of the level of excitation, and
therefore of the maximum electron temperature, at the end of the pump pulse and
the interplay between the electron and lattice subsystems have been analysed and
discussed. Now it is instructive to investigate how the dynamics change if the initial
lattice temperature is varied either by cooling the sample below room temperature
or by heating it towards the melting temperature. In what follows, measurements of
the reflectivity dynamics that were performed at initial sample temperatures ranging
from 50 K to 510 K are presented.
Figure 6.10 (facing page): Transient reflectivity changes in bismuth for initial crystal
temperatures ranging from 50 K to 510 K. In the upper graph, the five different measurements are displayed separately for delay times up to 30 ps. For clarity, the curves are
offset and the dotted lines indicate ∆R/R0 = 0 for each curve. The lower graph shows
the first few oscillations of the same five measurements without the offset.
6.3. Temperature dependence of reflectivity dynamics
97
a )
4
5 1 0 K
0
5
3 9 0 K
0
5
2 9 0 K
/ R
0
0
1 0
3
5

1 7 0 K
∆R
0
5
5 0 K
0
0
2
4
6
8
1 0
1 2
1 4
1 6
1 8
2 0
2 2
2 4
2 6
2 8
3 0
T im e / p s
b )
5 0
1 7 0
2 9 0
3 9 0
5 1 0
2 0
1 5
1 0
3
/ R
0
2 5
1 0

K
K
K
K
K
∆R
5
0
-5
-0 ,4
-0 ,2
0 ,0
0 ,2
0 ,4
0 ,6
0 ,8
T im e / p s
1 ,0
1 ,2
1 ,4
1 ,6
1 ,8
2 ,0
98
6. Reflectivity measurements of coherent optical phonons in bismuth
6.3.1 Experimental results
Figure 6.10 shows measurements of the reflectivity dynamics in bismuth after photoexcitation with a fluence of 6.7 mJ/cm2 for different initial sample temperatures.
The experiments have been carried out with the single-probe setup described in
section 5.3.1. The graphs show the results for two different series of measurements:
a first one in which the sample was heated from room temperature close to melting
temperature, and a second one in which the sample was cooled from room temperature down to 50 K. The heating of the sample was achieved with a resistance
cartridge heater, and the cooling was done in a closed-cycle cryostat. In both cases,
the sample temperature was measured with a thermocouple. The traces of both
series acquired at room temperature show a good agreement. The sample that was
used for the measurements is the same single crystal of bismuth cut perpendicular
to the (111)-axis that has been used for the experiments described above. In the
upper part of the figure, the measurements at five different initial sample temperatures are shown for an observation period of 30 ps with the traces set off vertically
for clarity. The lower panel shows the first two picoseconds of the same five traces
in one graph allowing one to compare the amplitudes and to see the difference in
frequency directly.
In general, the reflectivity dynamics consist of the same elements as described
above, a sharp drop, a rise in reflectivity, damped oscillations with the A1g -mode
frequency and a negative change in reflectivity that occurs after several picoseconds.
However, there are striking differences between the measurements. For example,
the damping of the oscillations is much lower at low temperatures, which leads
to a reflectivity signal with oscillations still visible at ∼ 35 ps (not shown in the
figure). Furthermore, the oscillation frequency is temperature-dependent and the
non-oscillatory component in the reflectivity signals changes with temperature, too.
In addition, it can be seen that the reflectivity oscillations measured at 50 K are
modulated with a frequency different from the A1g -frequency.
The initial sharp drop is less pronounced compared to what is presented in figure 6.1. As in the case of the fluence-dependent set of measurements described
above, this is caused by a lower temporal resolution. The features of the reflectivity
dynamics and their dependence on the initial sample temperature will be considered
in the following section.
6.3.2 Analysis and discussion
As a first part of the analysis, the frequency of the reflectivity oscillations will be
considered. The lower part of figure 6.10 illustrates that the frequency changes with
6.3. Temperature dependence of reflectivity dynamics
5 0
1 7 0
2 9 0
3 9 0
5 1 0
0 ,1 4
F T In te n s ity / a .u .
0 ,1 2
0 ,1 0
99
K
K
K
K
K
0 ,0 8
0 ,0 6
2 ,1 2 T H z
0 ,0 4
0 ,0 2
0 ,0 0
1 ,5
2 ,0
2 ,5
3 ,0
3 ,5
4 ,0
F re q u e n c y / T H z
Figure 6.11: Fourier transformed spectra obtained form the transient reflectivity signals
shown in figure 6.10.
temperature. The upper part of the figure shows that at the lowest temperature,
which was 50 K, the oscillations are a superposition of two oscillations with different frequencies. Figure 6.11 shows the Fourier transformed (FT) spectra that
were obtained from the time-domain data at different temperatures. The spectrum
obtained from the reflectivity oscillations at 50 K has a main peak at 3.02 THz,
a frequency which coincides with the frequency of the LO-phonon (A1g -mode) at
~k = 0 measured with neutron scattering (see figure 3.2 or [30]) at 75 K. In addition,
there is a small peak with a central frequency of 2.12 THz. This frequency is close
to that of the TO-phonon (Eg -mode) at ~k = 0 measured with neutron scattering,
which is 2.23 THz. This peak is only present in the spectrum that corresponds to
a sample temperature of 50 K. The fact that the Eg -mode could be observed with
an isotropic reflectivity measurement at a low temperature as well as its absence
for temperatures of 170 K and higher is in qualitative agreement with a previous
experiment reported by Ishioka et al. [46].
The central frequency of the A1g -peak decreases with increasing temperature.
The peaks have an asymmetric shape: they are much broader to the red side of the
spectrum. This can be attributed to the chirp of the frequency, which is present at
all temperatures.
In order to analyse the dependence of the A1g -frequency at the beginning of the
oscillations on the lattice temperature more thoroughly, the data has been fitted
100
6. Reflectivity measurements of coherent optical phonons in bismuth
3 ,0
2 ,8
2 ,7
A
1 g
fre q u e n c y / T H z
2 ,9
2 ,6
2 ,5
0
5 0
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
3 5 0
4 0 0
4 5 0
5 0 0
5 5 0
T e m p e ra tu re / K
Figure 6.12: A1g -mode frequency directly after the pump pulse as a function of the initial
sample temperature (black dots) and a theoretical curve fitted with equation 6.4 (dotted
line).
with the same function (equation 6.2) that has been used in section 6.2.2. As above,
this function does not reproduce the transient reflectivity changes for the whole
observation period, but it can be used to produce reasonably good fits to the first
2-6 oscillations allowing for the estimation of amplitude, frequency and damping of
the oscillations. The frequency of the reflectivity oscillations obtained from the fits
to the first two oscillations as a function of the initial sample temperature is depicted
in figure 6.12, and the error bars result from the fit. The frequency decreases with
temperature, starting from 2.95 THz at 50 K and reaching 2.58 THz at 510 K. The
reason that these frequencies do not coincide with the frequencies corresponding
to the peaks in the Fourier-transformed spectra is that they are estimated for the
period directly after excitation, while the spectra cover the whole observation period.
The decrease can be understood considering the binding energy of a solid. If it is
heated in equilibrium conditions, meaning that lattice and electron temperature
are the same, the binding energy decreases along with an increase of the interatomic distance. The phonon frequency scales with binding energy and inter-atomic
distance as ω 2 = b /(M d2 ). Therefore both the changes in inter-atomic distance,
d ≈ d0 + T /(λb,0 ), and in binding energy, b ≈ b,0 − kB · T , contribute to the
frequency change as the following:
T
T
2
2
· 1+
.
(6.3)
ω = ω0 1 −
b,0
λ · d0 · b,0
6.3. Temperature dependence of reflectivity dynamics
101
0 ,9
D a m p in g c o n s ta n t / p s
-1
0 ,8
0 ,7
0 ,6
0 ,5
0 ,4
0 ,3
0 ,2
0
5 0
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
3 5 0
4 0 0
4 5 0
5 0 0
5 5 0
T e m p e ra tu re / K
Figure 6.13: Damping constant of the reflectivity oscillations as a function of initial lattice
temperature (dots), fit assuming a squared-temperature law (dotted line) and a fit with
the exponent as fitting parameter (solid line, see text).
Keeping only the first order terms in the expansion of the above squared frequency
in a series of a small parameter, T /b,0 , the red-shift of the phonon frequency as a
function of temperature can be approximated as
T
2
2
2
ω = ω0 1 − 1 +
.
(6.4)
λ · d0 b,0
The dotted line in figure 6.12 is a fit to the frequency data with the equation 6.4.
The result for the fitting parameter λ is 2.7 · 108 cm−1 , which is approximately one
tenth of the inverse of the inter-atomic distance. The fitted curve reproduces the dependence of the A1g -frequency as a function of temperature in reasonable agreement,
with most of the data points lying on the curve within their error bars.
With the same procedure as in section 6.2.2, the damping constants have been
estimated from the reflectivity measurements at each temperature. The results are
shown in figure 6.13. The damping constant increases with temperature, ranging
from 0.25 ps−1 at 50 K to 0.83 ps−1 at 510 K. Concerning the three data points
at room temperature and lower, this is in good agreement with a previous experiment on coherent phonons at low temperatures reported by Hase et al. [71]. In
section 4.3.2, the probability of the decay of an optical phonon into two acoustic
phonons has been considered. For temperatures T TD , the two-phonon decay rate can be expressed as proportional to the squared temperature resulting in
102
6. Reflectivity measurements of coherent optical phonons in bismuth
) 1 0
3
1 6
1 4
A m l l i t u d e o f o s c i l l a t i o n ( ∆R / R
0

1 2
1 0
8
6
4
2
0
5 0
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
3 5 0
4 0 0
4 5 0
5 0 0
5 5 0
T e m p e ra tu re / K
Figure 6.14: Amplitude of reflectivity oscillations as a function of initial lattice temperature.
γdecay ∝ (kB T /(h̄ωD ))2 (equation 4.42). The equation has been used to perform a
fit to the data of figure 6.13, which is plotted as the dotted line. The data points
for 50 K and 170 K have not been taken into account because they are below or to
close to the Debye-temperature. The fit with a squared-temperature function does
not reproduce the temperature dependence. Fitting the data with the exponent as
a fitting parameter results in a power law of 1.35, which is plotted as the solid line
in the figure. This suggests that the two-phonon decay process is mixed with other
dissipation processes such as multi-phonon interaction.
As a final point, the dependence of the amplitude on temperature shall be considered. As in section 6.2.2, the amplitude of the reflectivity oscillations was obtained
by fitting function 6.2 to the reflectivity signals. The results are shown in figure 6.14
and show an interesting dependence on temperature: for temperatures lower than
room temperature, the amplitude linearly decreases with temperature, but for temperatures higher than room temperature, there is no change in amplitude within the
error bars. A previous study on coherent optical phonons in bismuth at temperatures below room temperature yielded a temperature dependence of the amplitude
which can be described as a linear increase from 8 K to 150 K followed by an abrupt
drop at temperatures higher than 150 K [46].
Furthermore, this report found a similar temperature dependence for the ampli-
6.4. Reflectivity dynamics under double-pump excitation
103
tude of oscillation (parameter Aq in equation 6.2) and the amplitude of the electronic
part of the reflectivity dynamics (parameter Ae ). A similar dependence is not likely
to have been observed in our case, and it has to be considered that this previous
experiment was carried out at excitation fluences three orders of magnitude lower
than in the experiments reported here. In the low-temperature study by Hase et
al. the amplitude increased with decreasing temperature, though not up to nearly
a factor 4 at 50 K compared to the amplitude at room temperature [71].
The fact that the amplitude increases with decreasing temperature underlines
the interpretation of coherent lattice motion as a process related to the excitation
of electrons. The cooling of the sample leads to an electronic configuration with
more electrons in the valence and less in the conduction band. Therefore, optical
excitation at lower temperatures leads to a greater number of excited electrons and
thus to a stronger electron pressure gradient which, in turn, leads to larger atomic
displacements.
The above results have been obtained in bismuth crystals at temperatures between
50 K and 510 K. It should be noted, that even at the highest temperature, which
is only ∼ 35 K below the melting temperature, no damage of the sample could be
observed. However, the drop in reflectivity below the unperturbed value after several
ps that reaches its minimum after ∼ 20 ps could indicate a transition to the liquid
state that has a lower reflectivity. An experiment which helps to identify the phase
of the crystal after electron-lattice equilibration is presented in the next section.
6.4 Reflectivity dynamics under double-pump
excitation
The experiments presented in the previous sections allowed for the analysis of the
complex dynamics of reflectivity in bismuth after excitation with an intense fslaser pulse. It was shown how the subtle atomic displacements during and after
the laser pulse as well as the interplay of the lattice and electron subsystem are
imprinted in the optical properties of the sample. However, the discrepancy in lattice
temperatures determined from calculations with the two-temperature model and
experimental estimations demand for additional information on the state of bismuth
that is present after some tens of picoseconds. For this reason, an experiment in
which two pump pulses with a delay of 25 ps will be presented in this section.
The excitation of the sample, which already is in an excited state, allows one to
draw further conclusions about the nature of a transient state after electron-lattice
equilibration and to answer the question if the sample is molten.
104
a )
6. Reflectivity measurements of coherent optical phonons in bismuth
4
3
/ R
0
2
∆R
1
1 0
3

x
0
x
-1
-2
0
5
1 0
1 5
2 0
2 5
3 0
3 5
4 0
4 5
5 0
T im e / p s
b )
4
c )
4
S ig n a l a f te r 1 s t p u m p p u ls e
S ig n a l a f te r 2 n d p u m p p u ls e
S ig n a l a f te r 1 s t p u m p p u ls e
S ig n a l a f te r 2 n d p u m p p u ls e
3
0
3
2
∆R
∆R
/ R
/ R
0
2

1
1
1 0
1 0
3
3

0
0
-1
-1
0
5
1 0
1 5
2 0
T im e a fte r a r r iv a l o f fir s t/s e c o n d p u m p p u ls e / p s
2 5
0
1
2
3
4
5
6
T im e a fte r a r r iv a l o f fir s t/s e c o n d p u m p p u ls e / p s
Figure 6.15: Reflectivity dynamics in bismuth in a double-pump experiment. Upper
graph: transient reflectivity of the whole observation period of 50 ps, lower graphs: signal
after first excitation from the upper graph (data points between black cross and red cross)
superimposed onto the signal after second excitation from the upper graph (data points
after the red cross).
6.4.1 Experimental results
The experiment has been carried out with the double-pump setup presented in
chapter 5. The incident angles of the two pump beams were 15◦ and 30◦ , respectively,
and the probe beam was orientated with an incident angle of 22◦ . The pump power
was 6.9 mJ/cm2 in the case of the beam arriving at a lower angle. In order to
compensate for the fact that the absorption decreases with increasing incident angle,
the fluence of the pump beam arriving at 30◦ was 8.0 mJ/cm2 , so that the absorbed
energy density was equal for both pump beams. This was verified by performing
single pump measurements with either the pump beam at the smaller angle or the
pump beam at the bigger angle and adjusting the intensities of the pump beams
such that the amplitude of oscillation was equal in both cases. The sample was the
same as in the measurements presented above and the measurement was carried out
6.4. Reflectivity dynamics under double-pump excitation
105
at room temperature.
The transient changes in reflectivity that were recorded applying the first pump
pulse and the second pump pulse 25 ps later are shown in figure 6.15. The upper
panel shows the reflectivity signal for the whole observation period of 50 ps. After
the first pump pulse, the reflectivity changes in the same manner as described above:
after a fast increase in reflectivity reaching its maximum at approximately 200 fs,
there is a decrease in reflectivity superimposed on damped harmonic oscillations
with a frequency close to the A1g -mode frequency. At ∼ 10 ps, the reflectivity drops
below its unperturbed value and reaches a minimum at ∼ 20 ps. The magnitude
of this minimum in ∆R/R0 equals 0.8 · 10−3 . Then the second pump pulse induces
reflectivity dynamics that are very similar to what follows the first pulse, with the
difference that the base line of these changes in reflectivity seems to be the minimum
of ∆R/R0 = 0.8 · 10−3 that is reached 20 ps after the first pump pulse. The narrow
initial drop in reflectivity could not be observed, and as above, this is due to a higher
pulse duration in combination with a reduction of temporal resolution due to the
difference in incident angles. The slight differences in the two signals can only be
seen when both are plotted in the same graph which is shown in the lower plots
in figure 6.15. Here, the reflectivity changes after the second pump pulse, meaning
from t = 25 ps, have been moved and superimposed onto the reflectivity changes
after the first pulse. It can be seen that the frequency of the oscillations following
the second pulse is lower than the frequency of the oscillations following the first
pulse, and that the relative minima that are reached ∼ 20 ps after the respective
pump pulse have different magnitudes.
6.4.2 Analysis and discussion
The reflectivity traces have been analysed using the same fitting function 6.2 consisting of a single exponential decay superimposed onto damped harmonic oscillations
that has been used in the previous sections. The oscillations that occur after the
first pump pulse are characterised by an initial frequency of 2.82 THz and a damping constant of 0.43 ps−1 , which is in agreement with the frequency observed after
excitation with 6.9 mJ/cm2 shown above. The oscillations that are present after
the second pump pulse have a frequency of 2.75 THz, and the damping constant is
0.48 ps−1 . At t = 25 ps, after the first excitation, it is possible to excite coherent
oscillations with a second pump pulse that lead to very similar reflectivity dynamics. This shows that after electron-lattice equilibration which leads to heating of
the lattice the sample is not molten. In fact, if the reflectivity dynamics after the
second pump pulse are compared to what has been measured for different initial
sample temperatures, one could come to the conclusion that it is a simple rise in
106
6. Reflectivity measurements of coherent optical phonons in bismuth
lattice temperature that is responsible for the red-shift, the increased damping, and
the lower value of the minimum that is attained after ∼ 20 ps. If the sample was
just heated in equilibrium, the frequency and damping constant measured after the
second pump pulse of 2.75 THz and 0.48 ps−1 , respectively would correspond to a
sample temperature of ∼ 330 K (compare to figures 6.12 and 6.13). This estimated
increase in temperature after excitation with 6.9 mJ/cm2 does not agree with the
maximum lattice temperature of 1273 K calculated with the two-temperature model.
It does not agree with the rise in temperature that one could estimate comparing the
magnitude of the minimum after the first pulse of ∆R/R0 ≈ −10−3 to the data on
the decrease in reflectivity with temperature of figure 6.2, which would correspond
to an increase in temperature of a few Kelvin.
In conclusion, the above results show that the process of electron-lattice energy
transfer after strong optical excitation does not result in the melting of the sample
even though a calculation with the two-temperature model suggests that the maximum lattice temperature, which is reached tens of picoseconds after excitation, is
significantly higher than the melting temperature. However, the inconsistent results
concerning the sample temperature after electron-lattice equilibration reveal that
the transient state of bismuth cannot be characterised on the basis of single-probe
reflectivity measurements alone.
6.5 Summary and conclusion
In this chapter, a detailed analysis of the reflectivity dynamics in bismuth after excitation with femtosecond pump pulses was presented. The energy density was varied
between a few mJ/cm2 up to the melting threshold of 24 mJ/cm2 . In a second series
of measurements the dynamics for different sample temperatures ranging from 50 K
to 510 K were investigated. The reflectivity oscillations recorded with a high accuracy allow one to uncover the subtle atomic motion, which is induced by the action
of the laser pulse, whose duration is shorter than all relaxation times. First, a coherent atomic displacement is induced by the polarisation force and electron pressure
force that can be resolved as a sharp negative drop in reflectivity. The excitation
of electrons leads to an increase in reflectivity, and reflectivity oscillations with a
gradually decreasing amplitude can be observed while the electrons transfer their
energy to the lattice. The initially harmonic vibrations of atoms are transformed
into a chaotic motion, which represents the onset of disordering of the solid phase.
A temperature dependence of the damping rate, which is close to the dependence
of the rate of decay of an optical phonon into two acoustic phonons, was observed,
confirming the interpretation of the inverse damping rate as the phonon lifetime.
6.5. Summary and conclusion
107
The observed dependence of the red-shift on sample temperature and excitation fluence was observed and modelled using a simple approximation of the inter-atomic
potential.
After electron-lattice equilibration, the reflectivity of the sample is between those
of solid and liquid, and calculations show that the lattice temperature is significantly
higher than the melting temperature. However, double pump measurements indicate
that melting does not occur at this stage, which is in accordance with results reported
for other materials in the case of ultrafast heating.
It is clear from these results that a deeper insight into internal properties of a solid
in this transient state is required. One of the possibilities for shedding more light on
the properties of a strongly excited solid during electron-lattice equilibration and its
state afterwards is the determination of the dielectric function, which is the subject
of the following chapter.
108
6. Reflectivity measurements of coherent optical phonons in bismuth
7 Ultrafast dynamics of the dielectric
function in bismuth
In this chapter, the results of the dual-angle reflectivity measurements are presented.
On the basis of the determination of the unperturbed dielectric function with ellipsometry that is presented in the first section, it is possible to recover the transient
dielectric function at 800 nm from the reflectivity data. The validity of the results
is investigated and the sources and magnitudes of the errors are carefully analysed.
In order to compare the dielectric function of the bismuth sample in the state that
is reached after ∼ 20 ps to that of bismuth heated in equilibrium, a measurement of
the unperturbed dielectric function as a function of crystal temperature is presented.
7.1 Measurement of the unperturbed dielectric
function
As described in section 5.4, the dielectric function can be recovered from two individual measurements of reflectivity at different angles. The reflectivities R1 and R2
measured at the incident angles α1 and α2 , and the real and imaginary parts of the
dielectric function are linked by a system of two equations
R1 = f (α1 , re , im )
R2 = f (α2 , re , im ) .
(7.1)
Here, R1 and R2 denote absolute reflectivities, and the function f denotes the
Fresnel-formula for the chosen polarisation. Due to the fact that, in our experiment transient, relative reflectivity changes ∆R/R0 are measured, the unperturbed
reflectivity R0 is needed to calculate the absolute transient reflectivities from the
measured data for the inversion of the Fresnel-formulae.
A way to determine the unperturbed reflectivity is to calculate it with the Fresnel
formula from the unperturbed dielectric function which, in turn, can be measured
with ellipsometry. For anisotropic samples, the so-called generalised ellipsometry
technique has to be used to fully determine the optical properties. For an arbitrary
orientation of the crystal with respect to the sample surface, the sample can mix
s- and p-polarised light (cross-polarisation), which can be measured to determine
109
110
7. Ultrafast dynamics of the dielectric function in bismuth
the optical properties of the material. In case of a uniaxial crystal whose optical
axis is perpendicular to the sample surface, the cross-polarisation reflection coefficients are zero. In this case, an ellipsometry measurement can be used to measure
the pseudo-dielectric function of the material in this particular configuration. The
pseudo-dielectric function is an approximation of the projection of the dielectric
function tensor along the direction that is defined by the intersection of the sample
surface and the plane of incidence [90]. This means that in our case, the pseudodielectric function measured with ellipsometry is an approximation of the ordinary
dielectric function, which is the component of the dielectric function tensor describing interaction with electromagnetic fields perpendicular to the optical axis. The
fact that the component of the dielectric function tensor along the intersection of the
plane of incidence and the sample surface is the most important can be understood if
one considers that, due to Snell’s law, the penetrating light beam is nearly normal to
the sample surface, and therefore, the component of the electric field perpendicular
to the surface is much smaller than the component parallel to the surface.
The pseudo-dielectric function hi for a uniaxial crystal whose optical axis is perpendicular to the sample surface can be expressed in an expansion as [91]
hi ≈ o −
∆
,
o − 1
(7.2)
where ∆ = e − o denotes the difference of extraordinary and ordinary dielectric
function. The expansion 7.2 has been truncated to first order in ∆. The pseudodielectric function hi depends on both e and o . It also depends on the angle of
incidence, but this dependence is a minor contribution in the second-order terms
that are neglected in the above expression 7.2. By separating the real and imaginary part of hi, and using literature values for o and e of bismuth [92, 93], one
can see that the difference between the pseudo-dielectric function and the ordinary
dielectric function in bismuth at 800 nm for our geometrical conditions is less then
one per cent. It is therefore reasonable to assume that the dielectric function measured with ellipsometry, which is the pseudo-dielectric function depending on both
ordinary and extraordinary dielectric functions, corresponds to the ordinary dielectric function in this case. This is important because the dielectric function measured
with ellipsometry is used to determine the absolute reflectivities of the sample in
the time-resolved measurements presented in the next section.
The result of the ellipsometry measurement is depicted in figure 7.1. The graph
shows the real and imaginary parts of the dielectric function of our bismuth sample
for photon energies ranging from 0.56 eV to 3.5 eV, corresponding to a range of
wavelengths between 2.20 µm and 350 nm. The measurement has been carried out
at two different incident angles, 50◦ and 70◦ . The results for both angles coincide,
7.1. Measurement of the unperturbed dielectric function
1 0 0
εr e
9 0
εi m
8 0
D ie le c tr ic fu n c tio n
m e a s u r e d w ith
e llip s o m e tr y
}
εr e
7 0
}
o r d in a r y
}
e x tr a o r d in a r y
εi m
6 0
εr e
5 0
εi m
4 0
111
[9 2 ]
[9 2 ]
3 0
2 0
1 0
0
-1 0
-2 0
-3 0
0 ,5
1 ,0
1 ,5
2 ,0
2 ,5
3 ,0
3 ,5
P h o to n e n e rg y / e V
Figure 7.1: Dielectric function of bismuth as a function of photon energy. The lines are
data from the ellipsometry measurement (see text), the dots and crosses show the ordinary
and extraordinary dielectric function as published in [92].
confirming that the dependence on the incident angles is negligible, as discussed
above. For the experiments presented here, only the wavelength of the laser, which
is 800 nm corresponding to a photon energy of 1.55 eV, is important, however, the
acquisition of a large spectrum allows for comparison with measurements of the dielectric constant in literature. The black and red lines are the dielectric function
measured with ellipsometry. The dots in the graph show the ordinary dielectric
function as published by Lenham et al. [92], and the crosses correspond to the data
of the extraordinary dielectric function in the same publication. The measurements
of the pseudo-dielectric function and the ordinary dielectric function by Lenham
agree qualitatively. However, for certain ranges of photon energies, there is a systematic deviation from the data of [92]. For instance, the imaginary part measured
with ellipsometry is lower throughout almost the whole depicted range of photon
energies. The real part shows good agreement for energies greater than 1.55 eV,
but is systematically lower in the infrared. These deviations are likely to be due
to different surface qualities. On the one hand, the surface roughness and thus the
method of surface preparation has an influence on the determination of optical properties [94]. On the other hand, bismuth oxidises when exposed to air, and therefore,
differences in the optical properties of a bismuth sample can also be caused by oxide
112
7. Ultrafast dynamics of the dielectric function in bismuth
layers. The sample that was used in the experiments presented in this work did
not show any sign of oxidation, which would normally appear as a yellow layer of
bismuth trioxide. However, it is sure that an oxide layer is present at the sample
surface, which has been identified through its characteristic diffraction pattern in an
electron diffraction experiment performed with the same sample. Nevertheless, we
assume that its thickness is negligible compared to the skin layer and therefore does
not affect the measurements. The dielectric function at 800 nm (indicated by the
vertical dotted line) of our sample was found to be re = −16.25 and im = 15.40. In
the next section, we will see how the dielectric function at this wavelength changes
after optical excitation.
7.2 Time-resolved measurement of the dielectric
function
For the investigation of the dynamics of the dielectric function at 800 nm in bismuth, dual-angle reflectivity measurements have been carried out at room temperature with the double-probe setup described in section 5.3.1. The sample was the
same bismuth single crystal with the (111)-axis perpendicular to the surface as in
the experiments presented in chapter 6. The transient reflectivity signals obtained
after excitation with 6.9 mJ/cm2 are shown in the upper panel of figure 7.2. The
curves measured at incident angles of α1 = 19.5◦ and α2 = 34.5◦ are similar to the
measurements described in the previous chapter. However, there are features in the
transient reflectivity signals that depend on the incident angle. The signal measured
at 19.5◦ has a bigger amplitude of oscillation, and the negative reflectivity change
that reaches a minimum after ∼ 20 ps is more pronounced. In contrast, frequency
and damping of the oscillations are the same at both angles of incidence, and the
reflectivity at both angles crosses the unperturbed value ∼ 7 ps after excitation.
Due to the fact that the reflectivity changes were measured with s-polarised
laser beams, only the ordinary part of the dielectric function was probed (see section 2.1.3). Thus the inversion of the Fresnel formulae using the dual-angle reflectivity data of this experiment results in the transient ordinary dielectric function,
which will, for simplicity, be referred to as dielectric function from now on.
The transient dielectric function was recovered from the two reflectivity measurements by using the Newton-Raphson algorithm presented in section 5.4. Due to
the fact that the experiment measures variations in reflectivity with respect to unperturbed reflectivity, but the Fresnel equations relate absolute reflectivity to the
dielectric function, the acquired reflectivity data has to be converted into absolute
reflectivities. The absolute transient reflectivities are R1 (t) = (1 + ∆R1 /R1,0 ) · R1,0
7.2. Time-resolved measurement of the dielectric function
a)
113
φ=19,5°
φ=34,5°
5
4
/ R
0
3
∆R
2
1 0
3

1
0
-1
b)
0
2
4
6
8
1 0
1 2
1 4
1 6
Time / ps
1 8
2 0
2 2
2 4
2 6
2 8
3 0
4 0
3 5
Im(ε) of liquid Bi
D ie le c tr ic fu n c tio n
3 0
2 5
εim
2 0
Im(ε) of solid Bi
1 5
Re(ε) of liquid Bi
-1 5
Re(ε) of solid Bi
εre
-2 0
0
2
4
6
8
1 0
1 2
1 4
1 6
Time / ps
1 8
2 0
2 2
2 4
2 6
2 8
3 0
Figure 7.2: Upper panel: dual-angle measurement of the transient reflectivity changes,
pump fluence 6.9 mJ/cm2 . Lower panel: recovered transient dielectric function (black
curves), re and im of solid bismuth (black lines), re and im of liquid bismuth (blue
dotted lines).
and R2 (t) = (1 + ∆R2 /R2,0 ) · R2,0 . Here, the first subscript 1 or 2 denotes the incident angles α1 and α2 and the subscript 0 denotes the unperturbed reflectivities
R1,0 = R0 (α1 ) and R2,0 = R0 (α2 ) at the respective angle, .
The result of the recovery is shown in the lower plot of figure 7.2. For t < 0,
the values of the dielectric function correspond to those measured with ellipsometry.
After the arrival of the pump pulse, both real and imaginary parts show oscillations
with the same frequency as the reflectivity oscillations. The oscillations of re and
114
7. Ultrafast dynamics of the dielectric function in bismuth
im are superimposed on a non-oscillatory component. The real part decreases and
reaches a minimum at the same time when the reflectivities reach a maximum,
which is at ∼ 200 fs. At about 4 ps, the base line of the oscillations crosses the
unperturbed value and at ∼ 20 ps the real part reaches a maximum of trans
=
re
−13.80. The imaginary part im increases after the arrival of the pump pulse and
reaches a maximum ∼ 200 fs, the time when the reflectivities reach a maximum. The
non-oscillatory component decreases, crosses the unperturbed value at ∼ 4 ps and
reaches a minimum value of trans
= 11.30 after ∼ 20 ps. The changes of the real and
im
the imaginary parts are of different signs throughout the whole observation period.
A maximum in transient reflectivity corresponds to a maximum in imaginary part
but to a minimum in real part. The values that are reached after ∼ 20 ps stay
approximately constant for the remaining 10 ps of the observation period.
Similar results have been obtained for fluences between 1.5 mJ/cm2 and 14 mJ/cm2 .
The characteristics of the dynamics of the dielectric function change in a similar way
to the transient reflectivity signals. With increasing fluence, the maximum change
directly after excitation increases, as well as the amplitude of the following oscillations. The frequency and the damping of the oscillations of re and im correspond to
the frequency and the damping of the reflectivity oscillations, therefore the red-shift
of the oscillation frequency of re and im and the damping increases with fluence.
The values of the transient state reached after ∼ 20 ps also change with fluence:
with increasing fluence, re increases and im decreases.
7.3 Error analysis
Before discussing the measurements of the transient dielectric function in bismuth,
it is necessary to investigate the validity of the results and estimate the order of
magnitude of error. In this section, a detailed analysis of the error in dielectric
function and its sources will be presented.
As a first test to validate the calculation of the dielectric function from the two reflectivity measurements, the inverse of recovery was performed. From the recovered
real and imaginary part, the reflectivity was calculated with the Fresnel formula and
compared to the originally measured reflectivity. The measured reflectivities can be
reproduced with an accuracy of 10−12 , the marginal deviation is probably due to
round-off errors in the process of calculation.
Due to the fact that the dielectric function is not measured directly, but numerically calculated from two reflectivity measurements, the error in re and im cannot be
determined directly. However, it is possible to estimate the error in dielectric function by identifying the sources of error that affect the process of recovery. There
7.3. Error analysis
115
are three main sources of error in dielectric function: an uncertainty in reflectivity, an uncertainty in measuring the incident angles and an uncertainty in temporal
superposition of the two probe beams.
In addition, the imperfect spatial superposition of the probe beams causes an error due to the fact that the two beams do not probe equally excited regions on the
sample. An example for a measurement performed with poor spatial superposition
of the probe beams is shown in figure 7.3 a). While the first maxima of both reflectivity signals occur at the same time, the two signals significantly dephase during a
few ps after excitation. It can be seen that the signal measured at 29.5◦ oscillates
with a lower frequency than the one measured at the lower incident angle. Therefore
it can be concluded that the probe spots were not perfectly superposed, and that
the beam at the larger incident angle probed a region closer to the maximum of
the Gaussian spatial profile of the pump beam, leading to a larger red-shift of the
phonon frequency. When trying to recover the dielectric function from such signals,
the algorithm used to invert the Fresnel equations does not converge. This is understandable as the two probe beams do not probe the same dielectric function due to
different levels of excitation, so that a physical solution does not exist. In order to
achieve a good spatial superposition of the two probe beams, one has to make sure
that the maxima and minima of the oscillations of both reflectivity signals coincide
from t = 0 until the time when the oscillations are damped out. We assume that,
in this case, the error due to a lack of spatial superposition is negligible.
In what follows, the error induced by the uncertainty in reflectivity, the uncertainty in incident angles and uncertainty in temporal superposition will be estimated. For the estimation, the measurement presented in the preceding section will be used. The error induced by the uncertainty in reflectivity was calculated in the same way as in the simulation presented in section 5.4: for each
time t, the dielectric function has been calculated with the measured reflectivities
R1 (t) = (1 + ∆R1 (t)/R1,0 ) · R1,0 and R2 (t) = (1 + ∆R2 (t)/R2,0 ) · R2,0 , and then with
all four combinations of (1 + ∆R1 (t)/R1,0 ± δ) · R1,0 and (1 + ∆R2 (t)/R2,0 ± δ) · R2,0 ,
where δ is the relative error in measuring ∆R/R0 . The maximum difference between
the solutions re (t) and im (t) obtained with R1 (t) and R2 (t) and the four solutions
subjected to errors is taken as the error of the dielectric function. For the estimation, the uncertainty in measurement of the relative reflectivity changes, δ = 10−5 ,
has been used.
In a similar way, the uncertainty in incident angles of the probe beams was taken
into account. The dielectric function was calculated from (1 + ∆R1 (t)/R1,0 ) · R(α1 )
and (1+∆R2 (t)/R2,0 ) · R(α2 ), and then for all four combinations of (1+∆R1 (t)/R1,0 )
· R(α1 ± δα) and (1 + ∆R2 (t)/R2,0 ) · R(α2 ± δα), where δα denotes the absolute error
in incident angles. As above, the maximum difference between the solutions obtained
116
7. Ultrafast dynamics of the dielectric function in bismuth
with R1 (t) and R2 (t) and those obtained at the “wrong angles” has been taken as
the error in dielectric function. The uncertainty in the measurement of the incident
angles was estimated to be δα = 0.5◦ .
The uncertainty in temporal superposition was taken into account in the following
way: one of the two measurements was shifted a time δt to the positive or negative,
as illustrated in figure 7.3 b). For example, to shift a measurement performed with
time steps of 40 fs a time δt = 5 fs to the right (positive),
∆R+δt (t)
∆R(t)
5
∆R(t + 40 fs) ∆R(t)
=
+
·
−
R0
R0
40
R0
R0
was calculated for each data point. Then, the dielectric function was obtained on
the basis of the real measured data (the black and the red curves in figure 7.3), and
then with the the data where one signal has been moved to the positive (black and
blue curves) and negative (black and orange curves). The maximum difference between the dielectric function values obtained from the unshifted data and the shifted
data has been taken as the error. The temporal superposition in the experiment
was assured by adjusting the optical paths of the two probe beams such that the
reflectivity maxima and minima exactly coincide. We assume that the uncertainty
in temporal superposition is δ = 5 fs.
The plots c) - h) in figure 7.3 show the errors originating from each source of
errors affecting the accuracy of recovery of the dielectric function. Plots c) and d)
show the first few oscillations of the real and imaginary part, and the error bars
correspond to the error induced by the uncertainty in reflectivity. The error in the
real part is (δre )δR ∼ 0.2 throughout the whole observation period. The error in
the imaginary part is (δim )δR ∼ 0.3 at almost every data point, except for the first
few minima where it is between 0.9 and 2.0.
The uncertainty in incident angles leads to errors in dielectric function that are
higher at the first few minima of re and maxima of im and have a value that
is approximately constant at the other data points. Plots e) and f) show their
magnitude during the first ps after excitation. At the first minimum of re , (δre )δα ∼
1.0; and at the first maximum of im , (δim )δα ∼ 9.3. The errors of the data points
Figure 7.3 (facing page): Above: a) dual-angle reflectivity signal acquired with poorly
aligned probe beams; b) dual-angle reflectivity measurement (black and red curve) and
reflectivity curves for the estimation of uncertainty in temporal superposition by shifting
one probe ±5 fs in time (blue and orange curves). Plots c) - h): Comparison of the
error in re induced by c), d) uncertainty in measurement of reflectivity; e), f) error in
measurement of the incident angles; and g), h) uncertainty in temporal superposition of
the probe beams.
7.3. Error analysis
a)
117
b)
α1 = 14,5°
α2 = 29,5°
8
α1 = 19,5°
α2 = 34,5°
α2 = 34,5° ( ' +5fs ' )
4
α2 = 34,5° ( ' -5fs ' )
6
3
/ R
/ R
0
0
4
2
∆R
∆R
2

1 0
1 0
3
3

0
1
-2
0
-4
0 ,0
c) - 1 4
0 ,5
1 ,0
1 ,5
2 ,0
Time / ps
2 ,5
3 ,0
3 ,5
4 ,0
d) 5 5
0 ,0
0 ,1
0 ,2
Time / ps
0 ,3
0 ,4
0 ,5
5 0
-1 6
4 5
4 0
I m ( ε)
R e ( ε)
-1 8
-2 0
3 5
3 0
2 5
-2 2
2 0
1 5
-2 4
1 0
-0 ,5
0 ,0
e) - 1 4
0 ,5
Time / ps
1 ,0
1 ,5
f)
-0 ,5
0 ,0
Time / ps
0 ,5
1 ,0
1 ,5
-0 ,5
0 ,0
Time / ps
0 ,5
1 ,0
1 ,5
5 5
5 0
-1 6
4 5
4 0
I m ( ε)
R e ( ε)
-1 8
-2 0
3 5
3 0
2 5
-2 2
2 0
1 5
-2 4
1 0
-0 ,5
0 ,0
g) - 1 4
0 ,5
Time / ps
1 ,0
1 ,5
h) 5 5
5 0
-1 6
4 5
4 0
I m ( ε)
R e ( ε)
-1 8
-2 0
3 5
3 0
2 5
-2 2
2 0
1 5
-2 4
1 0
-0 ,5
0 ,0
0 ,5
Time / ps
1 ,0
1 ,5
-0 ,5
0 ,0
0 ,5
Time / ps
1 ,0
1 ,5
118
7. Ultrafast dynamics of the dielectric function in bismuth
around the following minima have values that are comparable, but get smaller with
increasing time. All other data points have approximately the same error due to
uncertainty in incident angles, which is ∼ 0.2 in the case of the real part and ∼ 0.3
in the case of the imaginary part.
The biggest contribution to the error is due to the uncertainty in temporal superposition of the probe beams, which is shown in figures 7.3 g) and h). The maximum
errors can be found at the first minimum of re and the first maximum of im . Their
magnitudes are (δre )δt ∼ 2.4 and (δim )δα ∼ 15.0, which corresponds to relative
errors of ∼ 10% for the real and ∼ 35% for the imaginary part, respectively. With
increasing time, both the errors in real and imaginary parts decrease and reach minimum values of less than one per cent after ∼ 8 ps. The fact that the error induced
by the uncertainty in temporal superposition decreases with time and stays at a
low level after several ps is related to the amplitude of reflectivity oscillations. At
delays of ∼ 10 ps the oscillations are almost damped out, and the reflectivity does
not change rapidly with time. Thus, a shift in time of a few fs at this stage does
not affect the recovery of the dielectric function much, whereas during the first few
ps , the reflectivity changes rapidly and thus even a small shift in time of one curve
causes great deviations.
The validity of the results of re and im can be confirmed in a reliable way by
performing the same measurement at two different pairs of incident angles. As
the dielectric function (in contrast to reflectivity) does not depend on the incident
angles of the probe beams, the recovery of re and im should yield the same result for
both pairs of angles. In order to verify the result of recovery shown in figure 7.2, a
second measurement at the same pump fluence of 6.9 mJ/cm2 has been carried out.
Here, the incident angles were α1 = 14.5◦ and α2 = 29.5◦ . The transient dual-angle
reflectivity measurements carried out at two different pairs of incident angles are
shown in figure 7.4 where a) shows the measurement performed at (α1 = 19.5◦ , α2 =
34.5◦ ) (which is the same as the one in figure 7.2) and b) shows the measurement
performed at (α1 = 14.5◦ , α2 = 29.5◦ ). The dielectric function recovered from both
measurements are shown in plots c) - h), where the black curves are recovered from
the measurement shown in panel a) and the blue curves are recovered from the
measurement shown in panel b). The error bars of the data of the black curves
present the total error that is calculated from the errors originating from three
Figure 7.4 (facing page): Above: dual-angle reflectivity measurement at a) (19.5◦ , 34.5◦ ),
and at b) (14.5◦ , 29.5◦ ). Plots c) - h): dielectric function recovered from measurement
in a) (black curve) and b) (blue curve), plots c) and e) show re and im directly after
excitation, d) and f) after ∼ 20ps, plots g) and h) show the whole observation period.
7.3. Error analysis
a)
119
5
b) 5
α1 = 19,5°
α1 = 14,5°
α2 = 34,5°
4
α2 = 29,5°
4
3
0
0
3
2
∆R
∆R
/ R
/ R
2


1
1 0
1 0
3
3
1
0
0
-1
-1
0
c)
5
1 0
Time / ps
1 5
2 0
2 5
3 0
-1 4
-1 6
0
d)
5
1 0
Time / ps
1 5
2 0
2 5
3 0
-1 3 ,5
R e ( ε)
R e ( ε)
-1 8
-2 0
-1 4 ,0
-2 2
-2 4
-1 4 ,5
0 ,0
0 ,5
1 ,0
Time / ps
e) 7 0
1 ,5
f)
2 1 ,0
2 1 ,5
2 2 ,0
2 2 ,5
2 1 ,0
2 1 ,5
2 2 ,0
2 2 ,5
Time / ps
1 2 ,0
6 0
5 0
I m ( ε)
I m ( ε)
1 1 ,5
4 0
3 0
1 1 ,0
2 0
1 0
1 0 ,5
g)- 1 4
0 ,0
0 ,5
1 ,0
Time / ps
1 ,5
h)7 0
Time / ps
6 0
-1 6
R e ( ε)
5 0
I m ( ε)
-1 8
4 0
3 0
-2 0
2 0
-2 2
1 0
0
5
1 0
1 5
Time / ps
2 0
2 5
3 0
0
5
1 0
1 5
Time / ps
2 0
2 5
3 0
120
7. Ultrafast dynamics of the dielectric function in bismuth
different sources that were estimated above. The square root of the sum of squared
absolute errors is taken to be the total error, so that, for example, the total error
1/2
, where the subscripts δR,
of the real part is δre = (δre )2δR + (δre )2δα + (δre )2δt
δα and δt denote the errors induced by uncertainty in reflectivity measurement,
uncertainty in the measurement of the incident angles, and uncertainty in temporal
superposition of the probe beams, respectively.
Concerning the real part, both measurements are in good agreement. In figure 7.4,
plot c) shows the results for a short period of a little less than 2 ps directly after excitation, and plot d) shows the results after ∼ 20 ps, where the real part has reached
the maximum. With the exception of a few data points, all data points recovered
from the measurement in 7.4 b) lie within the error bars of the measurement recovered from 7.4 a), which is the case for the whole observation period including the
data range not shown in the two graphs c) and d). The values of the real part recovered from a) are slightly higher than the values of the real part recovered from b)
throughout the whole observed time span of 30 ps, shown in plot g), but both curves
are close to each other and agree well. The results for the imaginary part that are
compared in plots e) and f) are in good agreement as well, with the exception that
the data points around the first few maxima show a considerable deviation. Nevertheless, most of the data points of the whole observation period coincide within the
errors. As in the case of the real part, the values of im recovered from a) are slightly
shifted with respect to those recovered from b) throughout the whole observation
period.
The above considerations show that the dielectric function has been determined
from two reflectivity measurements with a good accuracy, that the results are reproducible and that they do not depend on the geometry of the experimental setup.
The real part of dielectric function can be recovered with a relative error of less than
two per cent, and the imaginary part with a relative error of less than five per cent.
These numbers are valid for the great majority of data points, except for the first
few local extrema, where the errors in real and imaginary parts can be up to ∼ 10%
and ∼ 40%, respectively.
7.4 Analysis and discussion
The laser-induced changes in the dielectric function are a manifestation of the significant coherent lattice displacement that the bismuth crystal experiences after strong
optical excitation. As already observed in experiments with tellurium [95, 96], the
dielectric function of bismuth shows considerable changes oscillating with the phonon
frequency. As the lattice configuration determines the band structure on which the
7.4. Analysis and discussion
121
electrons are distributed, the oscillations of the dielectric function can be interpreted
as evidence of the periodic changes of the whole band structure due to the motion
of the atoms.
A remarkable feature of the oscillations of the dielectric function is that the real
and imaginary part change in opposite directions with respect to the liquid phase
throughout the whole observation period. In the measurement performed with a
fluence of 6.9 mJ/cm2 , shown in figure 7.2, the real part decreases from its unperturbed value of −16.25 to a smaller, more metallic value and reaches a minimum of
−22.0 ± 0.2 at about 200 fs. Thus, the real part evolves away from the value of re
of liquid bismuth, which is 11.0. At the same time, the imaginary part increases,
indicating a more absorbing state, and reaches a maximum of 42 ± 18 at 200 fs.
Therefore, the imaginary part evolves towards the value of im of liquid bismuth,
which is 28.9. Due to the large error of im at this time, it is not clear if it crosses
liquid value at the maximum or reaches a value that is slightly below. Then, the
real and imaginary parts show damped oscillations superposed to a non-oscillatory
component. In the case of the real part, this component, which has its minimum
shortly after excitation, increases, crosses the unperturbed value a few ps after excitation, and evolves towards the value of re of liquid bismuth without reaching it.
The non-oscillatory component of the imaginary part, which has a maximum shortly
after excitation, decreases, crosses the unperturbed value a few ps after excitation,
and then reaches a minimum which is lower than im of unperturbed solid bismuth.
Thus, this behaviour represents an evolution away from the value of im of liquid
bismuth.
The observed changes in dielectric function can be partially understood in the
framework of the theory presented in chapter 4. The changes in real and imaginary
part can be expressed through the changes in number density of excited electrons
and the changes in electron-phonon momentum exchange rate, as expressed in equation 10.45. The changes in the real part can be expressed as
∂D
∆ne
∂D
∆νe−ph
re
re
∆re =
·
+
·
,
∂ ln ne 0 ne,0
∂ ln νe−ph 0 νe−ph,0
where the contribution from the initial coherent displacement (polarisation-related
part) has been neglected because here it could not be resolved due to an insufficient
temporal resolution. An increase in number density of excited electrons leads to a
negative change in the real part due to the fact that (∂D
re )/(∂ ln ne )0 = −17.4 (see
appendix A), and the oscillations can be related to the perturbation of νe−ph that is
caused by the coherent atomic vibrations. An increase of the lattice temperature,
which according to equation 4.53 leads to a positive change in νe−ph , can be the
reason for a positive change in re , however this contribution should be small due
122
7. Ultrafast dynamics of the dielectric function in bismuth
to the fact that (∂D
re )/(∂ ln νe−ph )0 = 0.032, which is a small positive number. The
changes in the imaginary part can be expressed as
∂D
∆ne
∂D
∆νe−ph
im
im
∆im =
·
+
·
.
∂ ln ne 0 ne,0
∂ ln νe−ph 0 νe−ph,0
Hence, an increase in number density of excited electrons creates a positive change
as (∂D
im )/(∂ ln ne )0 = 15.4, and as above, the oscillations can be related to the
phonon-induced changes in νe−ph . However, the negative change in im that is most
pronounced at ∼ 20 ps cannot be explained with the above equation. An increase
in lattice temperature creates a positive change in νe−ph which leads to an increase
in im due to the fact that (∂D
im )/(∂ ln νe−ph )0 = 0.87.
There is not a single moment in time where both the real and imaginary parts of
the dielectric function have values that lie between those for solid and liquid. We can
therefore exclude that melting occurs during the observed period of 30 ps, because
melting or incomplete melting should lead to an evolution of both real and imaginary
parts towards the values of liquid bismuth, as it has been observed in gallium [81]
or aluminium [98], for instance. This result strongly supports the results of the
double-pump experiment presented in section 6.4, which also lead to the conclusion
that the sample does not melt after irradiation with an energy density of several
mJ/cm2 . The values of the dielectric function of solid and liquid bismuth, as well
as of the state reached ∼ 20 ps after excitation with 6.9 mJ/cm2 , are summarised
in table 7.1.
In a similar way to the negative change in reflectivity reached after ∼ 20 ps, the
values of the real and imaginary parts at that time depend on the fluence. With
increasing fluence, the real part reached after ∼ 20 ps increases, while the imaginary
part decreases. Figure 7.5 illustrates this dependence of the level of excitation. In
the graph, re /re,0 and im /im,0 are shown, where re,0 and im,0 denote the real and
imaginary part of the unperturbed solid bismuth sample. The evolution of the real
part towards the value of the liquid state for which re,liquid /re,0 = 0.68 is clearly
visible. Excitation with 13.8 mJ/cm2 results in re = −8.93, which is higher than
the value of the liquid state. At the same time, the imaginary part evolves away
re
im
||
Solid bismuth
-16.25 15.40 22.39
Liquid bismuth -11.0 28.9 30.92
Transient state -13.80 11.30 17.84
Table 7.1: Dielectric function of bismuth in solid, liquid and transient state (see text) at
800 nm. The values of the liquid state are taken from [97].
7.4. Analysis and discussion
123
εr e / εr e
1 ,0
εi m / εi m
,0
,0
0 ,9
ε / ε0
0 ,8
0 ,7
0 ,6
0 ,5
0
2
4
6
8
F lu e n c e / m J c m
1 0
1 2
1 4
-2
Figure 7.5: Real part (solid circles) and imaginary part (open circles) of the dielectric function of bismuth ∼ 20 ps after excitation as a function of the excitation fluence, normalised
with respect to the unperturbed values of the solid state re = −16.25 and im = 15.4.
from the liquid state, for which im,liquid /im,0 = 1.88
The fact that the values of re and im are different from the dielectric functions
of both the solid and the liquid phase in equilibrium conditions does not exclude
a simple heating of the lattice. Indeed, a solid-to-liquid phase transition is of the
first order type, therefore an abrupt change in the response function of the crystal
is expected to occur during this phase transition. In order to investigate the evolution of the unperturbed dielectric function of our solid bismuth sample when the
temperature evolves towards the melting temperature, an additional ellipsometry
measurement was carried out where the real and imaginary parts of at 800 nm as
a function of the crystal temperature were determined. Figure 7.6 shows the result.
The real and imaginary parts of the dielectric function, normalised with respect to
their values at room temperature, are shown for a temperature range between room
temperature and 475 K, which is 70 K below melting temperature. The experiment
has been repeated three times, the depicted data points are the average values and
the error bars are the standard deviations. The real part does not show any significant changes with temperature within the error bars, whereas the imaginary part
increases with temperature. The changes in the real and imaginary part of the dielectric function with temperature can be qualitatively understood in the framework
of our theory. As considered above, a rise in lattice temperature causes a positive
124
7. Ultrafast dynamics of the dielectric function in bismuth
1 ,2 0
εr e
εi m
1 ,1 5
ε / εR
T
1 ,1 0
1 ,0 5
1 ,0 0
0 ,9 5
3 0 0
3 5 0
4 0 0
T e m p e ra tu re / K
4 5 0
5 0 0
Figure 7.6: Real part (solid circles) and imaginary part (open circles) of the dielectric
function of solid bismuth as a function of crystal temperature, normalised with respect to
their values at room temperature.
change in electron-phonon momentum exchange rate. Therefore, one would expect a
positive change in im which was clearly observed. The change in re with increasing
temperature is also expected to be positive, but much smaller than the change in im .
Experimentally, no change in re could be observed, however, it is possible that the
magnitude of positive change is smaller than the error bars and therefore not visible.
A comparison of figure 7.5 and figure 7.6 shows that the transient state, which is
attained after ∼ 20 ps, is different from a solid heated in equilibrium conditions.
It can be concluded from these results that the bismuth crystal is in a transient
state after photo-excitation and electron-phonon equilibration, that is neither described by a warm solid bismuth crystal nor liquid bismuth. It is instructive to
compare this result to available findings obtained with time-resolved x-ray diffraction. In an experiment reported in [99], the (222)-diffraction peak of a 50 nm thick
bismuth film was investigated after an excitation with an 800 nm laser pulse delivering 6 mJ/cm2 fluence. In addition to the oscillating changes in diffraction intensity
attributed to the A1g -oscillation, a shift in the Bragg-angle was observed, starting
a few ps after excitation and continuously growing until 20 ps. This shift was attributed to a change in the distance of atomic layers due to thermal expansion, and
an increase in temperature of ∼ 170 K was estimated from the data. The same
change in temperature was obtained by analysing the reduction of diffracted inten-
7.4. Analysis and discussion
125
sity due to the Debye-Waller effect. In our case, it is likely that the increase in
temperature is much smaller due to the fact that in a bulk bismuth crystal the energy can be distributed over a larger layer. However, the dielectric function ∼ 20 ps
after excitation suggests that there is an additional mechanism that is responsible
for the observed changes of the optical properties, while the time-resolved x-ray measurements suggest that a structural transformation is not likely to have occurred.
Due to the fact the the present data is limited to a single wavelength, the possibilities to analyse the electronic configuration of the system after photo-excitation
are limited. The knowledge of the transient dielectric function at several different
wavelengths, or over a wide spectral range like the visible range, would allow for
deeper insights into the changes of the band structure and the connection to coherent atomic displacement. In the light of the available data, the transient state
attained after ∼ 20 ps seems to be characterised by a less metallic state with a lower
concentration of carriers than the unexcited state. The availability of the dielectric
function over a wide spectral range would also allow for modelling of the dielectric
function as the sum of a Drude and a Lorenz contribution, which could allow for
conclusions about to which degree a strongly excited bismuth crystal can really be
described by a Drude dielectric function.
However, the performances of experimental setups that can measure the dielectric
function over a wider spectral range have to be improved. In previous dual-angle
reflectometry experiments using white light fs-laser pulses created by non-linear laser
solid interaction [82], an error in measuring absolute reflectivities of 5% was reported.
While for other materials like GaAs the change in reflectivity after optical excitation
is an order of magnitude higher than this error, the reflectivity changes in bismuth
are much smaller and therefore demand a much higher accuracy of the experimental
apparatus. Another possibility to measure the dielectric function of bismuth over a
greater range of wavelengths would be the use of an optical parametrical amplifier
(OPA) as a source of probe laser light. Systems that can deliver laser pulses with a
duration of several tens of femtoseconds, tuneable from ∼ 200 nm to several µm are
commercially available and could be used to measure the dielectric function over a
spectral range by repeating the measurement at a certain number of wavelengths.
If an accuracy comparable to the accuracy achieved at 800 nm in our experiments
could be achieved for a certain range of wavelengths, the acquisition of the dielectric
function over this spectral range would just be a matter of experimental control and
acquisition time.
126
7. Ultrafast dynamics of the dielectric function in bismuth
7.5 Summary and conclusion
In this chapter, a measurement of the transient dielectric function of bismuth after
optical excitation with a fluence of several mJ/cm2 has been presented. The real
and the imaginary parts of the dielectric function were recovered from a dual-angle
reflectivity measurement. The real part of the dielectric function shows changes
that can be described by damped oscillations superimposed to an non-oscillatory
component. It drops below its unperturbed value directly after excitation, then
oscillates with the same frequency and damping as the reflectivity. After 4 ps, it
crosses the unperturbed value and at ∼ 20 ps it reaches a maximum. The maximum
value of the real part stays approximately constant for the following 10 ps. The
imaginary part can also be characterised by damped oscillations superimposed to
a non-oscillatory component, but here the changes are in the opposite direction.
Directly after excitation, the imaginary part increases, then it oscillates with the
same frequency as the reflectivity oscillations. It decreases, crosses the unperturbed
value at 4 ps and then reaches a minimum. For the following 10 ps, the imaginary
part roughly stays constant.
There is no evidence for a transition to the liquid phase, because the values of the
real and imaginary parts do not coincide with the values of liquid bismuth and do
not exhibit an evolution towards the liquid values at the same time. The transient
state attained at 20 ps is characterised by a dielectric function, whose real part is in
between those of solid and liquid bismuth. In contrast, the imaginary part is smaller
than that of the solid state, which in turn is smaller than that of the liquid state,
meaning that the imaginary part has evolved away from the liquid state.
A comparison with a measurement of the unperturbed dielectric function at
800 nm as a function of the sample temperature showed that heating of the lattice
cannot be the reason for the observed changes in the transient dielectric function
after 20 ps. Therefore, the transient state can be neither described by a liquid nor
a heated solid.
The results demand for a profound investigation of the laser-induced changes of
the optical properties of bismuth and their connection to atomic displacements. For
a deeper understanding, the measurement of the dielectric function over a wider
spectral range would be desirable. Accordingly, it would be interesting to carry
out measurements of the transient dielectric function of bismuth as a function of
temperature, as well as to examine the dielectric function of liquid bismuth after
photo-excitation to the results presented here.
In conclusion, it is shown that the complex ultrafast dynamics of the dielectric
function at 800 nm in bismuth consists of a fast evolution to a more metallic state
during the first ∼ 200 fs after excitation followed by an evolution to a transient state
7.5. Summary and conclusion
127
that is reached after ∼ 20 ps, which seems to be characterised by lower electron density and lower absorption. The fact, that in a simple system like bismuth, the pump
laser does not only transfer heat to the system, but creates new electronic states,
opens general questions concerning the photo-induced transient state in matter.
128
7. Ultrafast dynamics of the dielectric function in bismuth
Part IV
Conclusions and perspectives
129
8 Summary and outlook
The study of coherent optical phonons in bismuth presented in this work could be
carried out on the basis of an experimental setup that allows for the measurement of
transient reflectivity changes of a photo-excited sample with a temporal resolution
as good as 35 fs and an accuracy of ∆R/R0 = 10−5 . This enables not only a detailed
analysis of the reflectivity dynamics in bismuth after excitation with a fs-laser pulse,
but also the recovery of the single-wavelength dielectric function with good accuracy.
In a first series of measurements, the reflectivity dynamics in bismuth after optical excitation with a fs-laser pulse centred at 800 nm has been investigated. The
excitation fluences ranged from 1.5 mJ/cm2 to 21 mJ/cm2 . Due to the high temporal resolution, a novel feature could be observed: a narrow drop in reflectivity
occurring directly before the reflectivity oscillations. It was shown that this feature,
which can only be resolved if the time resolution is higher than 40 fs, relates to a subtle coherent atomic displacement caused by the polarisation force during the pump
pulse. In contrast to previous work which focused mainly on the first ps after the
arrival of the pump pulse, we investigated the complex transient reflectivity changes
in bismuth for several nanoseconds, and observed that ∼ 10 ps after excitation, the
reflectivity drops below the unperturbed level and reaches a minimum after ∼ 20 ps.
Then, it increases up to the unperturbed value on a ns-time-scale, and reaches the
unperturbed value after ∼ 4 ns.
The transient reflectivity changes were interpreted in the framework of theoretical considerations that take into account the excitation of electrons through the
absorption of laser energy, the laser-induced forces acting on the atoms during and
after the pump pulse, and the interaction of the electron and lattice subsystems
by electron-phonon coupling. The coherent displacement of atoms during the laser
pulse creates a negative contribution to the reflectivity. This contribution is compensated on a fs-time-scale by the positive contribution to reflectivity due to the
excitation of electrons that increases during the pump pulse and has a maximum at
the end of the pump pulse while the lattice remains cold. Coherent harmonic oscillations of the practically cold lattice are generated by the thermal force, which was
expressed through the electron temperature gradient present after strong electronic
excitation in the skin layer. The oscillations with a frequency corresponding to the
A1g -mode frequency are imprinted into the reflectivity changes via a perturbation
131
132
8. Summary and outlook
of the electron-phonon coupling rate caused by the coherent displacement. The amplitude of oscillation gradually decreases while the electrons transfer their energy to
the lattice.
The characteristics of reflectivity oscillations and their dependance on the excitation fluence was analysed. The red-shift and the chirp of the frequency of oscillation
were related to a softening of the inter-atomic potential due to electronic excitation
in accordance with previous work, and could be theoretically reproduced on the
basis of a simple model. The temporal evolution of the electron and lattice temperatures was determined with the two-temperature model. The strong electronic
excitation resulted in maximum electron temperature of several thousand Kelvin,
and a temperature of the electrons and the lattice after energy equilibration that is
higher than the equilibrium melting temperature of solid bismuth. Despite the high
temperature, no evidence for a transition to the liquid phase could be found.
A second series of measurements was carried out to investigate the temperature
dependance of the reflectivity dynamics in bismuth. Transient reflectivity signals
were recorded in bismuth at temperatures ranging from 50 K to 510 K, which is
close to the melting temperature. The phonon frequency exhibited a red-shift that
increased with temperature. This behaviour could be theoretically reproduced by
considering a reduction of the binding energy and an increase in inter-atomic distance
due to the heating of the lattice. The observed increase of the damping rate with
temperature was close to the dependence of the rate of optical phonon decay into
two acoustic phonons, confirming the interpretation of the inverse damping rate as
the lifetime of phonons. Even at the highest temperature, no melting was observed
after excitation with several mJ/cm2 .
The ultrafast dynamics of laser-excited bismuth were further examined by carrying
out a double-pump experiment. Here, the sample was excited with a first pump
pulse, and after 25 ps, a second pump pulse was applied. The reflectivity dynamics
after the second pump pulse were found to be very similar to those after the first
pump pulse, suggesting that the sample is still in the solid state after electron-lattice
equilibration, even though the two-temperature model suggests that the crystal is
overheated to more than two times the melting temperature.
Despite the general good agreement between the theory and the results, it is not
possible to determine to which degree the heating of the lattice through energy
transfer from the excited electrons is the origin of the negative change in reflectivity
after ∼ 10 ps. A measurement of the decrease in reflectivity of the unperturbed
bismuth sample heated in equilibrium conditions shows that the negative change in
reflectivity that reaches a maximum ∼ 20 ps after photo-excitation would correspond
to a rise in temperature of a few degrees if it was caused by equilibrium lattice
heating.
133
In order to investigate the validity of our theory and elucidate the origin of the
negative change in reflectivity after electron-lattice equilibration, we recovered the
transient dielectric function at 800 nm from two simultaneous reflectivity measurements. The results showed that the changes in the real and imaginary parts have
opposite signs throughout the whole observation period of 30 ps, and that at no time
the transient dielectric function coincided with the dielectric function of liquid bismuth. Shortly after excitation, the real part decreases, which is an evolution away
from the liquid state, and the imaginary part increases towards the value of liquid
bismuth. Both parts show damped oscillations at the phonon frequency. They cross
the unperturbed values and reach transient state values after ∼ 20 ps which stay
approximately constant for the next 10 ps. The real part of the dielectric function
of this transient state is between those for liquid and solid bismuth. In contrast, the
imaginary part is lower than the value for solid bismuth, which presents an evolution away from the liquid value. In order to compare the transient state ∼ 20 ps
after photo-excitation to a solid heated in equilibrium conditions, an ellipsometry
measurement determining the real and imaginary parts of the unperturbed dielectric function at 800 nm as a function of the crystal temperature was carried out.
It clearly shows that the transient state reached after ∼ 20 ps characterised by an
increase of the real part and a decrease of the imaginary part cannot be described by
a heated solid, because with increasing temperature the real part of the unperturbed
dielectric function does not show significant changes while the imaginary part increases. A possible interpretation of this result would be that it is the signature of
a less metallic state with a lower density of carriers.
The results presented raise the question of how the properties of the transient
state reached on a time-scale of tens of picoseconds after excitation can be characterised. While the straightforward picture of a solid experiencing strong electronic
excitation and energy transfer from the electrons to the lattice resulting in a heated
solid can be doubted, neither the transformation from semimetal to an even less
metallic state nor the opening of a band-gap can be excluded. The measurement
of the transient dielectric function over a wide spectral range would allow for an
unambiguous characterisation of the transient state. Due to the fact that for the
recovery of the dielectric function the transient reflectivity changes in bismuth have
to be determined with a high accuracy, this measurement represents a considerable
experimental challenge.
Moreover, it would be instructive to perform such a measurement as a function
of the sample temperature. In particular, the intriguing fact that even close to the
melting temperature a deposition of several mJ/cm2 does not lead to melting merits
further investigation. An important question linked to this phenomenon concerns
the diffusion of electrons into the bulk after the absorption of the laser pulse. Up to
134
8. Summary and outlook
this point, it is not clear which portion of energy is carried away by electrons into
deeper parts of the bulk and how the diffusivity changes upon electronic excitation.
The investigation of ultrafast dynamics in thin bismuth films of variable thickness
would allow one to estimate which effect a confinement of the electrons in a thin layer
has on the dynamics that follow strong excitation of electrons due to absorption of
laser energy.
Even though the high temporal resolution in the experiments presented here allowed for the uncovering of a novel effect, a fast coherent displacement of the atoms
during the pump pulse, the availability of shorter laser pulses would be desirable.
An ideal configuration for the examination of this subtle atomic displacement would
be the availability of pump pulses with a duration of several tens of femtoseconds
in combination with probe pulses with duration of a few fs. This would enable an
investigation of the first response of the atoms to the pump laser light through transient reflectivity changes that are not convoluted with a temporal beam profile that
has the same width as the pump pulses.
Promising possibilities to conduct further measurements on coherent lattice dynamics will be provided through advanced fs-X-ray sources. Recent proposals suggest the feasibility of generating bright X-ray pulses with a duration of the order
of 1 fs in a free–electron laser [100]. In combination with pump laser pulses with a
duration of a few femtoseconds, researchers would be equipped with an experimental
tool that allows for the exact tracking of atomic motion with a temporal resolution
highly superior to the typical period of atomic oscillations. Hence, the coherent displacement observed for the first time in our experiments could be spatially resolved
by analysing the transient diffracted intensity of the corresponding Bragg-peaks.
Another challenging experiment would be the investigation of the Eg -phonon
mode with time-resolved X-ray diffraction. Up to this point, it remains obscure
why this mode is only visible at very low temperatures whereas the A1g -mode can
be excited at temperatures ranging from ∼ 5 K up to close to the melting temperature. Time-resolved X-ray diffraction at low temperature could also elucidate
wether a coupling between the two modes that was found in DFT-calculations can
be confirmed experimentally.
Furthermore, future studies could include a lot of different materials. An interesting question that arises due to the discovery of the initial coherent displacement
preceding the coherent A1g -mode oscillations is if this effect can be observed in
other materials than bismuth. The theory presented in chapter 4 is not restricted
to bismuth, but should be applicable to a variety of different solids. As a first step,
one could investigate if the novel effects observed in bismuth can also be found in
materials with a similar crystal structure, for example antimony.
We can conclude that despite the recent advances in the understanding of ultrafast
135
dynamics in solids, of which a tiny part is presented in the present work, alone in
the case of bismuth there are still a lot of answered questions and a great number
of unravelled mysteries. This is good news for the community of ultrafast research
— we won’t get bored.
136
8. Summary and outlook
Part V
Résumé en Français
137
9 Contexte de travail de recherche
aaa L’invention du laser il y a presque 50 ans [1] a rendu possible des développements
révolutionnaires dans beaucoup de domaines de science et d’ingénierie. Cette technologie a ouvert des larges domaines d’applications en physique, chimie, biologie,
traitement des matériaux, médecine et météorologie. Parmi les nombreuses propriétés particulières de la lumière laser, la production d’impulsions de durées de
plus en plus brèves est un domaine d’activité en plein développement. Des durées
inférieur à une picoseconde ont été obtenues pour la première fois en 1976 [2].
Aujourd’hui la durée d’impulsion a presque atteint la limite d’un cycle d’un onde
électromagnétique qui est quelques femtosecondes dans le visible [3].
Grâce aux sources de lumière femtoseconde, les chercheurs disposent d’outils avec
une résolution temporelle qui est assez élevée pour étudier le mouvement atomique, les transitions de phase ou la formation et le bris des liaisons chimiques dans
le domaine temporel. En sondant stroboscopiquement avec des impulsions femtoseconde, ces processus qui apparaissent sur une échelle temporelles de quelques
femtosecondes à plusieurs dizaines de picosecondes peuvent être résolus et analysés.
Une autre conséquence des durées d’impulsions courtes est l’augmentation significative de l’intensité du champ électromagnétique qui peut être focalisé sur une cible.
Ces intensités permettent le développement des sources secondaires de radiation
et de particules comme les harmoniques d’ordre élevé [4], les rayons X [5] et les
électrons [6].
Un axe principal de recherche dans le domaine des phénomènes ultra–rapide est
l’excitation et la détection des vibrations cohérentes du réseau. Pendant les deux
dernières décennies, la génération des vibrations THz, caractérisées par un degré
élevé de cohérence spatiale et temporelle, a été démontré pour une grande variété de
matériaux transparents et opaques. Parmi ces matériaux sont des semi–métaux [7],
des métaux de transition [8], des cuprates [9], des diélectriques [10] et des semi–
conducteurs [11]. Alors que les phonons ont toujours été un sujet important dans
la physique de l’état solide dû à leur relation avec les phénomènes de transport,
l’étude des phonons cohérents revêt un intérêt particulier: Il est possible d’induire
et de contrôler les vibrations atomiques pour déclencher des transitions de phase
(fusion non–thermique [12], paraeléctrique–ferroeléctrique [13], isolant–métal [14]),
pour ouvrir d’une façon sélective des caps des nano–tubes dans des conditions
139
140
9. Contexte de travail de recherche
non–équilibre [15], ou pour créer la base pour le SASER (amplification du son par
émission stimulée de radiation) [16]. Un grand intérêt est de comprendre et de
contrôler des changements structurels ultra–rapides, qui représente l’axe majeur de
devéloppement des sources X femtoseconde intense dans le monde. La réalisation
des lasers à électrons libre comme le XFEL à Hambourg ou LCLS à Stanford, avec
des budgets proches d’un milliard d’euros, souligne l’importance de ce genre de
recherche pour la communauté scientifique.
Ce travail se concentre sur des expériences de phonons optiques cohérents dans le
semi–métal bismuth, qui ont été le sujet de plusieurs études théoriques et expérimentales préliminaires. Le choix de ce matériel est motivé par plusieurs qualités particulières: Le bismuth a une structure simple avec deux atomes dans la maille
élémentaire qui a une symétrie trigonale. Cette structure peut être dérivée d’une
structure cubique en modifiant deux paramètres de la structure, la distance entre les deux atomes dans la maille et l’angle de cisaillement de la maille. Des
études théoriques indiquent que la variation de la distance des atomes change la
configuration électronique de semi–métallique à métallique, et que la variation de
l’angle de cisaillement induit une transition de l’état semi–métallique à l’état semi–
conducteur [17]. Il a été démontré également que sous des conditions de confinement quantique le bismuth se transforme en semi–conducteur [18, 19]. En outre,
une transition à un état métallique peut être induit à haute pression [20]. Le mouvement atomique dans le bismuth peut être étudié en utilisant deux techniques
complémentaires: La spectroscopie optique résolue en temps, qui permet d’obtenir
des informations sur la dynamique des électrons, et la diffraction x résolue en temps,
qui permet d’étudier la dynamique de la structure. Des études préliminaires des
phonons optiques cohérents dans le bismuth ont produit une grande variété de
résultats intéressants. Néanmoins, des questions cruciales, par exemple en ce qui
concerne le mécanisme d’excitation des phonons cohérents, sont sans réponse et les
propriétés le l’état excité du bismuth reste inconnu.
Ce manuscrit présente une étude détaillé de la dynamique des cristaux de bismuth
excité par un laser. La dynamique des électrons et du réseau qui suit l’excitation
du cristal a été examinée en mesurant les changements de la réflectivité et de la
fonction diélectrique avec une résolution temporelle de 35 fs. Ces expériences nous
ont permis de mener une analyse des propriétés des phonons cohérents dans le bismuth en fonction de la température et la fluence d’excitation. Nous avons mis en
évidence un effet nouveau associé à la dynamique cohérente du réseau qui ne peut
pas être expliqué avec les théories préliminaires: Une diminution très brève de la
réflectivité dans les tous premiers instants de l’excitation laser. Cette diminution
est due à un déplacement cohérent des atomes pendant l’application de l’impulsion
de pompe qui change la partie de la fonction diélectrique liée à la polarisation. De
141
plus, des nouveaux résultats sur les propriétés optiques d’un état transitoire qui
est établi après ∼ 20 ps sont présentés. Des mesures de la réflectivité du bismuth,
excité avec une seule impulsion de pompe ou deux impulsions de pompe successives
montrent que l’échantillon ne subit pas de transition de l’état solide à l’état liquide. La récupération de la partie réelle et imaginaire de la fonction diélectrique à
partir d’une expérience double sonde suggère que l’état transitoire qui est atteint
après l’équilibration électrons–réseau n’est ni caractérisé par un solide ni par un
liquide. La détermination des changements de la fonction diélectrique a montré que
les changements des propriétés optiques ne peuvent pas être dus à un échauffement
du réseau. Ces résultats suscitent des nouvelles questions concernant un état photoinduit dans le bismuth. Il a été observé que la pompe optique ne transfère pas
seulement de l’énergie au système, mais crée un état électronique nouveau qui est
différent de l’état à l’équilibre.
L’organisation du manuscrit est décrite à la fin du chapitre 1.
142
9. Contexte de travail de recherche
10 Considérations théoriques
Dans ce chapitre, les considérations théoriques des chapitres 2 à 4 seront résumées.
La première section traite les bases de l’interaction laser–matière. La deuxième
section regroupe les propriétés des phonons optiques dans le bismuth et décrit les
théories préliminaires d’excitation et détection des phonons dans les solides. Dans la
troisième section, les principes de la nouvelle théorie qui a été développé dans le cadre
de cette thèse afin d’expliquer les nouveaux résultats obtenus lors des expériences,
sont récapitulés.
10.1 Interaction laser–matière
Dans le cadre de ce travail, le mouvement atomique photoexcité est examiné en
déterminant les changements de la réflectivité et la fonction diélectrique. Il est donc
nécessaire de comprendre l’interaction de la lumière avec le matériel. Dans la suite,
les propriétés optiques de base comme l’index de réfraction et la fonction diélectrique
seront introduits, et les principes de l’interaction lumière–matière seront récapitulés.
10.1.1 L’origine électromagnétique des propriétés optiques
A vide, une onde électromagnétique peut être décrite par l’évolution spatio–temporelle
de deux vecteurs, le champ électrique E et l’induction magnétique B. Afin de
décrire l’influence de ces champs sur un matériel, une deuxième paire de vecteurs
est nécessaire, l’induction électrique D et le champ magnétique H. L’évolution de
ces champs est définie par les équations de Maxwell, qui attribuent les dérivées des
quatre vecteurs par rapport à l’espace et au temps à la densité de charge électrique
ρ et la densité de courant j:
∇·D =
∇·B =
4πρ ,
0,
1 ∂B
∇×E = − ·
,
c ∂t
1 ∂D 4π
·
+
j.
∇×H =
c ∂t
c
(10.1)
(10.2)
(10.3)
(10.4)
143
144
10. Considérations théoriques
Ici, la constante c est la vitesse de la lumière. L’influence du milieu sur le champ de
lumière est dictée par les équations suivantes:
j = σ·E
(10.5)
D = ·E,
(10.6)
B = µ·H.
(10.7)
dans lesquelles ρ est la conductivité spécifique, est la constante diélectrique et µ
est la perméabilité magnétique.
Afin de décrire la réponse optique linéaire d’un matériau, il est possible de déduire
deux équations différentielles à partir des équations de Maxwell en éliminant ou E
ou H:
∇2 E −
µ ∂ 2 E 4πµσ ∂E
− 2
= 0,
c2 ∂t2
c
∂t
∇2 H −
µ ∂ 2 H 4πµσ ∂H
− 2
=0
c2 ∂t2
c
∂t
(10.8)
Une solution possible de ces équations est une paire d’ondes électrique et magnétique
dont la fréquence est proportionnelle à l’amplitude du vecteur d’onde k:
c
ω = √ k.
µ
(10.9)
La vitesse de phase de cette onde est
v=
ω
c
c
=√ = ,
k
µ
n
(10.10)
√
ou n = µ décrit l’index de réfraction du milieu, qui est une fonction de la fréquence
ω, due au fait que la constante diélectrique et la perméabilité magnétique dépendent
de ω:
p
n(ω) = (ω)µ(ω) .
(10.11)
Grâce au fait que pour les fréquences optiques du spectre électromagnétique la
perméabilité magnétique µ = 1 [23], la réponse optique d’un milieu est uniquement déterminée par la réponse à un champ électrique oscillant. Pour cette raison,
l’index de réfraction optique ne dépend que de la fonction diélectrique:
n(ω) =
p
(ω) .
(10.12)
Il faut noter que l’index de réfraction ainsi que la fonction diélectrique sont des
nombres complexes, décrits par les expressions suivantes:
n̂ =
√
= η + iκ ,
ˆ = + i
4πσ
= re + iim .
ω
(10.13)
10.1. Interaction laser–matière
145
L’index de réfraction complexe est composé de deux quantités η et κ. De la même
façon, la fonction diélectrique consiste de deux quantités, la partie réelle re et la
partie imaginaire im . Dans la suite, les accents circonflexe seront omis afin d’assurer
une notation simplifiée.
Si on considère la solution la plus simple des équations 10.8, qui est une onde
plane monochromatique,
ω
ω
E = E0 · e−κ c x ei(η c x−ωt) ,
(10.14)
on peut voir que l’intensité, qui est proportionnelle à la moyenne temporelle de E2 ,
varie en accord avec l’équation
ω
I(x) = I(0)e−2κ c x .
(10.15)
L’intensité I diminue d’une façon exponentielle avec la distance de propagation x
dans le milieu, et le taux d’atténuation dépend de la partie imaginaire de l’index
de réfraction. Maintenant, le coefficient d’absorption α, et l’inverse, la longueur
d’absorption dabs , peuvent être définis:
α=
4π
2ω
κ,
κ=
c
λ0
dabs =
1
λ0
=
.
α
4πκ
(10.16)
L’épaisseur de peau ls est définie comme la longueur sur laquelle la densité de courant
diminue d’un facteur 1/e, elle est donc égale deux fois la profondeur d’absorption.
10.1.2 Réflexion et réfraction de la lumière à une interface
Une fois la fonction diélectrique d’un matériel connue, il est possible d’en dériver
toutes les propriétés optiques linéaires. Pour les expériences présentées dans ce
travail, la relation entre la réflectivité d’un matériau et sa fonction diélectrique est
particulièrement importante. Ces deux quantités sont liées par les équations de
Fresnel, avec lesquelles il est possible de déduire des expressions pour la réflectivité
en fonction de la fonction diélectrique qui seront présentées dans ce paragraphe.
Quand une onde électromagnétique franchit l’interface entre deux milieux, elle est
séparée en une onde réfractée et une onde réfléchie (cf. figure 2.1). Les relations
entre les angles des rayons incidents (θi ), réfractés (θt ) et réfléchis (θr ) par rapport à
la normale de la surface sont exprimées par la loi de réflexion et la loi de réfraction
θi = θr ,
n1 · sin θi = n2 · sin θt ,
(10.17)
qui sont également appelées les lois de Snell-Descartes. Les relations entre les
amplitudes des champs incidents, réfractés et réfléchis dépendent de la fonction
146
10. Considérations théoriques
diélectrique ainsi que de l’angle d’incidence et de la polarisation de la lumière et
sont exprimées par les équations des Fresnel, 2.28 à 2.31. Dans le cas d’une inter√
√
face entre un milieu n = 1 et un autre n = = re + iim , les réflectivités pour
les polarisations s et p sont exprimées par:
Rs
Rp
cos θ − p + i − sin2 θ 2
i
im
i
p re
= ,
cos θi + re + iim − sin2 θi ( + i ) cos θ − p + i − sin2 θ 2
re
im
i
re
im
i
p
= .
(re + iim ) cos θi + re + iim − sin2 θi (10.18)
(10.19)
10.1.3 Propriétés optiques des cristaux anisotropes
Contrairement aux cristaux isotropes, les cristaux anisotropes ne peuvent pas être
décrits par une fonction diélectrique scalaire. Il est possible de montrer que le
tenseur de la fonction diélectrique qui est composé de neuf éléments est symétrique.
En conséquence, le nombre d’éléments indépendants est réduit à six [23]. De plus,
il est possible de trouver un système de coordonnées tel que le tenseur peut être
exprimé par trois éléments indépendents, x , y et z , qui décrivent les propriétés
optiques le long les trois axes diélectriques principaux [24].
Dans le cas des cristaux uniaxiaux dont le bismuth fait partie, le tenseur de la
fonction diélectrique est encore réduit et composé de deux éléments indépendants.
En cas d’une onde ordinaire, caractérisée par un champ électrique perpendiculaire
à l’axe optique, les propriétés optique sont décrites par la fonction diélectrique ordinaire o . En cas d’une onde extraordinaire, dont le champ électrique est parallèle
à l’axe optique, les propriétés optiques sont décrites par la fonction diélectrique extraordinaire e . La réflexion des ondes ordinaires et extraordinaires est tracée dans
la figure 2.3, et les formules des amplitudes des champs réfléchis en fonction de la
polarisation de la lumière sont présentées dans le paragraphe 2.1.3.
10.1.4 Propriétés optiques et structure électronique
En fonction du matériau, les contributions à la fonction diélectrique peuvent être
d’origines différentes. Dans un métal, la contribution dominante est liée à l’absorption
des porteurs libres qui subissent des transitions intrabandes. Dans les semi–conducteurs, les processus dominants sont les transitions interbandes. Dans la suite, nous
admettons que la fonction diélectrique du bismuth photo-excité puisse être décrite
par le modèle de Drude. Ce modèle a été développé afin d’expliquer la conductivité
10.2. Phonons optiques cohérents dans le bismuth
147
des métaux. La forme de Drude de la fonction diélectrique est la suivante:
(ω) = 1 + 4πi
ne e2 τ
.
m∗e (1 − iωτ )
(10.20)
Ici, ne est la densité d’électrons, e la charge d’un électron, τ l’inverse de la probabilité
de collision électron–électron, m∗e la masse efficace de l’électron et ω la fréquence de
la lumière. Après une séparation de la partie réelle et la partie imaginaire, peut
être écrit de la façon suivante:
= re + iim = 1 −
ωp2
ωp2
νe−ph
+
i
.
2
2
ω 2 + νe−ph
ω 2 + νe−ph ω
(10.21)
Dans cette équation, ωp = (4πe2 ne /m∗e )1/2 est la fréquence de plasma, et νe−ph = 1/τ
est la fréquence de collision électron–phonon.
10.2 Phonons optiques cohérents dans le bismuth
Dans cette section, les principes de l’excitation et la détection de phonons optiques
cohérents seront résumés. Deux théories différentes, displacive excitation of coherent
phonons (DECP) et impulsive stimulated Raman scattering seront introduites et les
possibilités de modéliser la dynamique de la réflectivité offertes par ces théories
seront discutées. Puis, les propriétés des phonons optiques dans le bismuth seront
présentées.
10.2.1 Excitation des phonons optiques cohérents
Dans ce paragraphe, deux théories préliminaires de phonons cohérents, le DECP
et l’ISRS sont introduits. Les deux théories sont basées sur l’hypothèse que la
dépendance temporelle de la coordonnée normale atomique q peut être décrite par
l’équation de l’oscillateur harmonique forcé:
d2 q
dq
F (t)
+ 2γ + ω02 q =
.
2
dt
dt
m
(10.22)
dans laquelle ω0 est la fréquence du phonon, γ la constante d’amortissement, et F (t)
la force externe créée par l’impulsion de pompe. La différence principale entre ces
deux théories est le mécanisme physique responsable de la force externe, qui même
ou à une force impulsive ayant une forme qui peut être décrite par une fonction
δ, ou une forme displacive qui peut être décrite par une fonction Heaviside. Cette
différence peut être expliquée en utilisant l’analogie du pendule: la force associée à
l’excitation impulsive change l’énergie cinétique alors que le mécanisme displacive
modifie l’énergie potentielle du pendule.
148
10. Considérations théoriques
Excitation impulsive des phonons cohérents
Dans le cadre d’ISRS, le phonon est excité par un processus Raman comme il est
décrit dans le paragraphe 3.2.1. En utilisant un modèle classique basé sur l’hypothèse
que la polarisibilité d’un matériel dépend de la distance inter-atomique (modèle de
Platzeck [36]), l’équation de mouvement 10.22 devient
∂ 2q
1
∂q
2
2
+ ω02 q = N α0 A2 e−(t−zn/c) /(τl ) ,
+ 2γ
2
∂t
t
4
(10.23)
dans laquelle N est la densité atomique et A l’amplitude de l’impulsion de pompe
Gaussienne, dont l’expression du champ électrique est:
2 /(2τ 2 )
l
E = A · e−(t−zn/c)
cos [ωl (t − zn/c)] .
(10.24)
Dans cette dernière équation, τl est la durée d’impulsion est ωl la fréquence centrale. Equation 10.23 montre que la force qui est active sur un mode Raman est
spatialement uniforme et temporellement impulsive. Si nous définissons t = 0 comme
l’instant auquel le centre de l’impulsion de pompe arrive à la surface de l’échantillon
localisée à z = 0, il est possible de trouver la solution suivante [35]:
q(z > 0, t > 0) = q0 e−γ(t−zn/c) sin [ω0 (t − zn/c)] .
(10.25)
L’expression de l’amplitude de vibration est:
√
2π
π
2 2
2 2
(10.26)
q0 =
N α0 A2 τl e−ω0 τl /4 =
FN α0 e−ω0 τl /4 .
4ω0
ω0 nc
√
Ici, F = ncA2 τl /(8 π) est l’intensité intégrée de l’impulsion, qui est souvent appelée
la fluence.
Les équations précédentes montrent que par ISRS une impulsion ultra–courte
produit une onde de vibration avec une phase bien définie qui est uniforme dans
la direction transversale. Le phonon est donc cohérent, ce qui est une qualité fondamentale qui nous permet de résoudre les oscillations dans le domaine temporel.
L’amplitude du phonon est proportionnelle à la fluence et dépend du produit de
la fréquence ω0 et de la durée d’impulsion. En conséquence, la plus courte est
l’impulsion, le plus les atomes sont déplacés de leur position d’équilibre. De même,
les oscillations sont décrites par une fonction sinusoı̈de, ce qui est un caractéristique
du modèle ISRS. Nous verrons dans la suite que ce n’est pas le cas pour le modèle
DECP. Il s’agit d’une différence importante car dans la littérature la phase initiale de l’oscillation est souvent utilisée comme argument pour soutenir un des deux
mécanismes d’excitation.
10.2. Phonons optiques cohérents dans le bismuth
149
Excitation displacive des phonons cohérents
Le théorie de DECP admet que le phonon soit excité par une modification considérable de la position d’équilibre des atomes due à l’excitation d’une grande densité de porteurs par la lumière de pompe. Si la durée de l’impulsion d’excitation
est bien inférieure à la période de vibration des noyaux, les vibrations atomiques
peuvent être cohérentes.
En utilisant une des hypothèses générales de cette théorie, que le déplacement
atomique q0 (t) est lié d’une façon linéaire à la densité des porteurs n(t),
q0 (t) = ζ · n(t) ,
(10.27)
l’équation de mouvement des atomes 10.22 devient:
d2 q
dq
+
2γ
+ ω02 q = ω02 q0 (t) = ω02 ζ n(t) .
2
dt
dt
(10.28)
Ici, ζ est une constante de proportionnalité. En admettant que la fréquence du
phonon ne dépende pas du niveau d’excitation, la solution de cette équation peut
être écrite
ω 2 ζρEpump
q(t) = 2 0 2
ω0 + β − 2γβ
avec
Z∞
β0
−βτ
−γτ
, (10.29)
g(t − τ ) e
−e
cos(Ωt) − sin(Ωt)
Ω
0
q
Ω = ω02 − γ 2 ,
and
β0 = β − γ .
(10.30)
Dans ces équations, Epump est l’intensité maximale de la pompe et g(t) ainsi que β
sont définis dans le paragraphe 3.2.2.
L’équation 10.29 montre que le déplacement nucléaire associé avec le mode normale est composé de deux contributions, une partie oscillatoire et une partie exponentielle. Le premier peut être associé aux vibrations atomiques excitées par le
déplacement rapide dû à l’excitation des électrons et le dernier à un changement de
la position d’équilibre qui décroı̂t avec la relaxation des électrons. Dans le cas du bismuth, le facteur β 0 /Ω est négligeable, les oscillations ont donc une forme cosinusoı̈de
contrairement au cas de la théorie ISRS.
10.2.2 Détection des phonons optiques cohérents
Parmi les nombreuses possibilités de sonder les phonons cohérents, celle qui est
utilisée dans ce travail est d’observer les changements de la réflectivité induits par
les déplacements cohérents des atomes. L’idée de base de la modélisation théorique
des changements de la réflectivité est d’admettre que ceux-ci soient petits par rapport
150
10. Considérations théoriques
à la réflectivité non-perturbée. Dans ce cas ils peuvent être exprimés de la façon
suivante:
∂R
∂R
∆R(t) =
· ∆ξre (t) +
· ∆ξim (t) .
(10.31)
∂ξre 0
∂ξim 0
Ici, ξre et ξim sont les parties réelle et imaginaire de la propriété optique considérée,
par exemple la fonction diélectrique ou l’index de réfraction. L’index zéro indique
que les dérivées sont déterminées avec les valeurs non-perturbées de cette propriété.
Les variations temporelles de la partie réelle ∆ξre (t) et de la partie imaginaire ∆ξim (t)
sont liées aux propriétés du matériel qui changent due à l’excitation laser et la
génération des phonons.
Dans l’approche ISRS, les changements de la réflectivité peuvent être exprimés
par les changements de la susceptibilité linéaire complexe χ = χre + iχim induits par
le phonon:
∂R
∂R
∆R(t) = |∂(χre + iχim )/∂q|
cos η +
sin η · q(t) ,
(10.32)
∂χre
∂χim
où η est la phase de ∂χ/∂q. Cette formule ne peut exprimer que des changements
de la réflectivité dus aux déplacements cohérents du réseau, tous les effets liés à
l’excitation sont négligées. De même, les valeurs des éléments de ∂χ/∂q ne sont pas
connues, cette expression ne peut donc pas être utilisée pour prédire quantitativement les changements de la réflectivité.
Dans le cadre de DECP, les changements de la réflectivité sont admis d’être induits par trois effets: l’excitation des électrons, le changement de la température
électronique et le mouvement atomique. La théorie ne fait pas la distinction entre les changements produits par l’excitation des électrons et le changement de la
température électronique Te et en négligeant le terme lié à Te , les variations de la
réflectivité sont exprimées comme:
1
∂R
∂R
∂R
∆R(t)
=
n(t) +
Te (t) +
q(t) ,
(10.33)
R
R
∂n
∂Te
∂q
où R est la réflectivité non–perturbée à t < 0 avant l’arrivée de l’impulsion de
pompe. Comme il est montré dans le paragraphe 3.3, il est possible d’exprimer les
changements de la réflectivité en considérant que le signal mesuré est convolué avec
la durée d’impulsion de sonde de la manière suivante:
∆R
ω02
β0
−βτ
−βτ
−γτ
= Ae +B 2
e
−e
cos(Ωτ ) − sin(Ωτ )
. (10.34)
R
ω0 + β 2 − 2γβ
Ω
Cette équation, dans laquelle A et B dont les expressions sont données dans 3.3
dépendent de l’échantillon, est dérivée sur l’hypothèse que la durée d’impulsion de
10.2. Phonons optiques cohérents dans le bismuth
151
sonde est bien inférieure à la période du phonon. Le premier terme, qui est proportionnel à l’énergie de pompe via le coefficient A est une exponentielle caractérisée
par le taux de relaxation des électrons β et représente la partie non–oscillatoire. Le
deuxième terme, proportionnel à l’énergie de pompe via le coefficient B, est la somme
de deux composants. L’exponentielle peut équilibrer ou amplifier le comportement
associé à l’excitation et la relaxation des porteurs, ce qui dépend des signes des
deux coefficients A et B. Le terme trigonométrique décrit une oscillation amortie et
la constante de temps associée présente l’amortissement du phonon. Comme nous
verrons dans la suite, il n’est pas possible de décrire le comportement compliqué
des changements de la réflectivité dans le bismuth photo–excitée avec les formules
présentées ci-dessus. En particulier, le comportement de la réflectivité, pendant
et quelques ps après l’excitation ne peut pas être compris dans le cadre des deux
théories présentées dans ce paragraphe. Il est donc nécessaire de développer une
nouvelle théorie qui est capable de reproduire tous les changements de réflectivité
que nous avons observés lors de nos expériences. Cette approche théorique sera
présentée dans le paragraphe 10.3.
10.2.3 Structure cristalline et modes normaux du bismuth
La symétrie cristalline du bismuth est la structure A7 avec le “point group” 3m.
Cette structure peut être dérivée d’une structure cubique en appliquant deux distorsions indépendantes. D’abord, une élongation le long la diagonale de la maille
élémentaire transforme la maille cubique en rhomboédrique, puis, un déplacement
interne des deux atomes sur la diagonale de la maille dans des directions opposées
résulte en un réseau trigonale avec deux atomes par maille élémentaire, présenté
dans la figure 3.5. Les paramètres de la maille élémentaire sont les suivants:
a = b = c = 4.746 Å ,
α = β = γ = 57.23◦ .
Si la maille élémentaire est choisie telle que les deux atomes ont la même distance
du centre de la maille, leurs positions sur la diagonale cdia peuvent être exprimées
par rapport à l’origine O avec le paramètre u0
cdia = 11.862 Å
u0 = ±0.234 ,
un atome étant à u0 et l’autre à (1 − u0 ).
Dans cette structure, il y a deux modes de phonons optiques:
• le mode longitudinal symétrique A1g qui est aussi appelé mode de respiration
(breathing mode), qui présente un déplacement des deux atomes le long la
diagonale avec une fréquence propre de 2.92 THz,
152
10. Considérations théoriques
• le mode transversal Eg , doublement dégénéré dans le plan x-y, qui présente un
déplacement des deux atomes dans des directions opposées. Ce mode a une
fréquence de 2.22 THz.
Les mouvements atomiques associés à ces deux modes sont présentés dans la figure 3.5. Les propriétés de symétrie des deux modes optiques du bismuth, ainsi que
les résultats préliminaires qui se trouvent dans la littérature sont présentés dans les
paragraphes 3.4.2 et 3.4.3.
10.3 Un modèle complet de la réflectivité du bismuth
photoexcité
Dans les paragraphes 10.2.2 et 10.2.1 deux théories d’excitation et de détection des
phonons cohérents ont été introduites. Aucune des deux théories n’est susceptible
d’expliquer certains nouveaux résultats obtenus dans le cadre de ce travail de thèse.
Dans la suite, nous développons donc un nouvelle approche théorique en tenant en
compte tous les processus pertinents pendant et après l’impulsion de pompe. Nous
considérons l’échauffement du système électronique pendant l’excitation optique,
l’interaction entre les électrons et le réseau et les forces qui déclenchent les mouvements atomiques. Puis, nous estimons les changements des propriétés optiques dus
à ces processus.
10.3.1 Propriétés transitoires d’un solide excité par une
impulsion laser ultra–brève
L’évolution temporelle des processus qui sont déclenchés par une excitation optique
ultra–brève est tracée dans la figure 4.1. Au début, l’énergie du laser est absorbée
par les électrons dans la bande de valence dans un volume défini par la tache focale
est la profondeur de pénétration. Après la thermalisation, le système électronique
peut être caractérisé par la température électronique Te , qui est bien supérieure à
la température du réseau Tl . L’ensemble des électrons chauffés est refroidi par interaction électron–phonon qui mène à un échauffement du réseau et par des autres
processus comme la diffusion des électrons. Après un certain temps qui est normalement de l’ordre de quelques picosecondes, les électrons et le réseau ont atteint une
température commune.
Les changements des deux températures Te et Tl peuvent être décrits avec le
modèle à deux températures (two-temperature model, TTM), qui est présenté
dans le paragraphe 4.1.1. Ce modèle permet de calculer l’évolution spatio–temporelle
10.3. Un modèle complet de la réflectivité du bismuth photoexcité
153
des deux températures avec un système de deux équations différentielles couplées.
Le modèle assume que la diffusion dans le système électronique est un processus
beaucoup plus rapide que l’interaction électron–phonon. Il est montré dans le paragraphe 4.1.2 que cette hypothèse est valable dans nos conditions et que la diffusion
de chaleur peut être négligée.
Afin de calculer la température électronique directement après l’application de
l’impulsion de pompe, nous avons dérivé un modèle d’avalanche qui assume que
toute l’énergie de laser absorbée excite des électrons de la bande de valence à la bande
de conduction. De plus, il est assumé que les électrons munis d’énergies supérieures
au gap sont capables de transférer l’énergie excédentaire à d’autres électrons et créent
une avalanche d’électrons excités. Cela nous permet de calculer la température
électronique ainsi que la densité des électrons avec les formules simples suivantes
(cf. paragraphes 4.1.3 et 4.1.4):
Te2 =
4F AF (tp )
,
2
π 2 ne,0 ls kB
ne (tp ) = ne,0 +
2AF (tp )
.
∆ · ls
(10.35)
Ici, f est l’énergie de Fermi, A est l’absorption du bismuth à la longueur d’onde
concernée, F est la fluence de l’impulsion de pompe, tp la durée d’impulsion, ne,0
est la densité électronique avant l’excitation, ls est l’épaisseur de peau, kB est la
constante de Boltzmann et ∆ est l’énergie du gap.
En combinant le modèle à deux températures avec ces deux formules et en assumant que les changements de ne sont proportionnels aux changements de Te après
l’impulsion de pompe, il est possible de calculer théoriquement toute l’évolution
temporelle de Te , Tl et ne .
10.3.2 Forces induites par le laser et mouvement atomique
Afin d’estimer les forces qui sont actives pendant et après l’impulsion de pompe,
nous avons utilisé le “field-and-matter stress tensor” [23], qui tient en compte des
contributions de pression dans le matériel et de pression due à l’irradiation. En
assumant que le tenseur diélectrique jk consiste d’une contribution de Drude D · δjk
et une contribution due à la polarisation pjk , le calcul mène à cette expression de la
force sur les atomes (cf. paragraphe 4.2.1):
fi = −
∂P
∂p E 2 D − 1 ∂E 2
− ik
+
= fitherm + fipol + fipond .
∂xi
∂xk 8π
8π ∂xi
(10.36)
Ici, fi est une composante du vecteur de la force et P est la pression. La force est
composée de trois contributions. Le premier terme est la force thermique, qui est
proportionnel au gradient de la température électronique. Le deuxième terme est la
154
10. Considérations théoriques
force associée aux moments dipolaire induits par le champ électrique de l’impulsion
de pompe, et le troisième terme est la force ponderomotrice. Les deux derniers
termes ne sont différents de zéro que pendant l’impulsion de pompe alors que la
force thermique reste active jusqu’à ce que le gradient de la température électronique
devienne négligeable. Si on estime l’amplitude de ces trois forces, on voit que la
force ponderomotrice est négligeable par rapport aux autres, car son amplitude est
un ordre de grandeur plus faible.
Comme il est mentionné dans le paragraphe 3, le mouvement atomique peut-être
décrit par un oscillateur harmonique forcé
d2 q k
Fklaser
dqk
2
q
=
+
ω
,
+
2γ
0 k
dt2
dt
m
(10.37)
où Fklaser = Fktherm + Fkpol + Fkpond est la somme des forces induites par le laser.
L’amplitude relative de ces forces est différente sur une échelle de temps courte, t ω0−1 , et une échelle de temps plus longue, t ω0−1 . Cela entraı̂ne des conséquences
importantes en ce qui concerne leur effet sur les atomes et le déplacement atomique
associé.
Pour l’échelle de temps caractérisée par t ω0−1 , la force élastique n’est pas
effective, l’équation 10.37 est donc réduite à
d2 qk
Fklaser
≈
.
dt2
m
(10.38)
La solution de cette équation est
1
qk (t) =
m
Ztp
0
dt0
Zt0
Fklaser (t00 )dt00 ≈
Fklaser t2p
.
2m
(10.39)
0
Il y a donc un déplacement cohérent au début de la dynamique qui est produit par
les forces insuites par le laser, que nous appelons le déplacement atomique initial.
Pour l’échelle de temps caractérisée par t ω0−1 , on peut trouver une solution
en assumant que les forces induites par le laser agissent comme une perturbation
légère, ce qu’il mène à l’expression
1
F therm
∆qk = k
· exp i ω02 − γ 2 2 t − γt + iϕ .
(10.40)
2mω0 γ
pour l’amplitude des oscillation harmoniques.
10.3.3 Relaxation du phonon et décalage vers le rouge de la
fréquence
Nous assumons que l’interaction phonon-phonon est le mécanisme dominant responsable pour la relaxation des phonons optiques (cf. paragraphe 4.3.2 et figure 4.3).
10.3. Un modèle complet de la réflectivité du bismuth photoexcité
155
Nous pouvons donc trouver une expression pour la constante d’amortissement en
calculant la probabilité de relaxation d’un phonon de l’énergie h̄ω en deux phonons
optiques de l’énergie h̄ω/2, ce qu’il mène à
γdecay
h̄
≈C·
M d2
kB T
h̄ωD
2
,
(10.41)
où M est la masse atomique, d est la distance inter-atomique et ωD la fréquence de
Debye.
Les changements de la fréquence avec la température et la fluence d’excitation
ont été considérés en utilisant un modèle simple du potentiel inter-atomique décrit
dans le paragraphe 4.3.3. Les expressions de la fréquence sont
Te
2
T
2
2
2
2
ω ≈ ω0 1 −
et ω = ω0 1 − 1 +
,
(10.42)
|b,0 |
λ · d0 b,0
où b est l’énergie de liaison de l’atome non-perturbé et λ un paramètre du modèle
du potentiel inter-atomique.
10.3.4 Changements des propriétés optiques
Fonction diélectrique
Comme mentionné ci-dessus, nous assumons que le tenseur diélectrique modifié par
le laser consiste de deux termes, un terme de Drude et un terme lié à la polarisation.
Les parties réelles et imaginaires de la fonction diélectrique peuvent donc être écrites
comme
jk = D · δjk + pjk ≡ re + i · im ,
(10.43)
re = pjk + D
im = D
re ;
im .
L’expression du terme de Drude est:
D = re + iim
ωp2
ωp2
νe−ph
=1− 2
+ i 2
.
2
2
ω + νe−ph
ω + νe−ph ω
(10.44)
Donc:
∆re =
∆im =
∆pjk
+
∆D
re
∆D
im
=
=
∆pjk
∂D
re
+
∂ ln ne 0
∂D
im
∂ ln ne 0
∆ne
∂D
re
·
+
ne,0
∂ ln νe−ph 0
∆ne
∂D
im
·
+
ne,0
∂ ln νe−ph 0
∆νe−ph
,
νe−ph,0
∆νe−ph
·
.
νe−ph,0
(10.45)
·
156
10. Considérations théoriques
où l’index 0 indique que les dérivées sont calculées avec les valeurs du cas non-excité
et que les changements de ne et νe−ph sont considérés par rapport à leurs valeurs
non-perturbés. Les dérivées de re et im se trouvent dans l’appendice A.
L’expression pour ∆νe−ph est:
∆νe−ph
∆Tl (t) 2∆q(t)
=
+
.
νe−ph,0
T0
q0
(10.46)
Avec ce qu’il est considéré ci-dessus, il est possible d’exprimer la fonction diélectrique
d’un matériau qui est excité avec une impulsion laser ultra-brève comme la fonction
des changements de la densité électronique et de la variation de la fréquence de
collision électron-phonon.
Reflectivité
Les changements de la réflectivité qui peuvent être mesurés directement sont calculés
à partir de
√
− 1 2
(10.47)
R = √
+ 1
où = re + i · im . On trouve l’expression suivante pour une variation légère de la
réflectivité:
∆νe−ph
∆ne
∆R = Cpol · ∆pjk + Cne ·
+ Cνe−ph ·
,
(10.48)
ne,0
νe−ph,0
où les coefficients Cpol , Cne et Cνe−ph , qui sont définis par les équations 4.59 et 4.60,
dépendent du matériel. Les valeurs des coefficients se trouvent dans l’appendice A.
Il est possible de simplifier le résultat de l’équation 10.48 et d’exprimer les changements de la réflectivité comme une fonction explicite du changement de la partie de
la fonction diélectrique lié à la polarisation, des températures des électrons et du
réseau et du déplacement des atomes due aux oscillations harmoniques
∆R/R0 = A · ∆pol + B · ∆Te + C · ∆Tl + D · ∆q ,
(10.49)
où les coefficients A, B, C, et D peuvent être calculés en utilisant les équations
4.59 et 4.60, ainsi que les valeurs non-perturbées de la densité électronique et de la
fréquence de collision électron-phonon.
11 Résultats expérimentaux
11.1 Dispositifs expérimentaux
Dans ce paragraphe, les dispositifs expérimentaux qui ont été utilisés pour obtenir
les résultats de ce travail de thèse sont introduits. Ici, les principes généraux des
expériences sont décrits, une description en détail se trouve dans le chapitre 5.
Les mesures optiques présentées dans la suite sont effectuées avec la technique
pompe sonde, qui est une façon de profiter des impulsions laser courtes et de contourner les limitations des détecteurs en ce qui concerne leur résolution temporelle.
Dans une expérience pompe sonde, dont le principe de fonctionnement est présenté
dans la figure 5.1, une impulsion laser, dite l’impulsion de pompe, excite les processus
qui changent les propriétés optiques de l’échantillon. Après un certain délai ∆t, une
deuxième impulsion laser, l’impulsion de sonde arrive à la surface de l’échantillon.
La lumière de sonde réfléchie est détectée avec une photodiode. En variant le retard
entre la pompe et la sonde, la dynamique de la réflectivité peut être suivie pour une
période de temps choisie.
Les expériences ont étés faites avec un système laser qui est basé sur un cristal
Ti:Sa et qui produit des impulsions laser d’un énergie de 10 mJ à 800 nm avec
une durée de 35 fs en utilisant la technique CPA (chirped pulse amplification). La
cadence de répétition est 1 kHz. Le système est présenté dans la figure 5.2 et décrit
dans le paragraphe 5.2. Grâce à la haute énergie des impulsion femtosecondes, ce
laser puissant peut être utilisé pour des applications différentes, dans notre cas, il
suffit d’utiliser moins d’un pour cent de l’énergie pour effectuer les expériences.
Les résultats présentés dans les paragraphes suivants ont été obtenus en utilisant
trois dispositifs différents. Le premier, le dispositif pompe–sonde simple, consiste
d’un faisceau de pompe, qui contient 80% de l’énergie, et un faisceau de sonde
qui contient 20% de l’énergie. Avant la séparation du faisceau, une lame de verre
est utilisée afin de dévier une partie de la lumière pour l’utiliser comme signal de
référence. Le retard entre la pompe et la sonde peut être varié en modifiant le chemin
optique du faisceau de pompe avec en utilisant deux miroirs qui sont déplacés avec
un moteur à pas . Le faisceau de sonde, est focalisé sur une tache de 40 µm FWHM
sur l’échantillon avec une lentille de 100 mm de focale. Le faisceau de pompe est
focalisé avec une lentille de 500 mm de focale, ce qui produit une tache de 125 µm
157
158
11. Résultats expérimentaux
FWHM sur l’échantillon, afin d’assurer que la tache du faisceau de sonde peut être
alignée telle qu’elle sonde une zone de l’échantillon excitée d’une façon homogène.
Le faisceau réfléchi est détecté avec une photodiode, pour avoir un rapport signal
sur bruit plus élévé, une détection synchrone est utilisée. La cadence de pompe est
modulée en utilisant un chopper, et le signal, la différence des signaux de la photodiode de sonde et la photodiode de référence, est détecté à 500 Hz. Cette détection
permet de mesurer des changements de réflectivité ∆R/R0 , où R0 est la réflectivité
avant l’arrivée de l’impulsion de pompe de 10−5 . L’acquisition des données et le
contrôle de l’expérience sont faits avec une programme utilisant LabVIEW.
Le deuxième dispositif expérimental utilisé est le dispositif double sonde. En
principe, il s’agit du dispositif pompe sonde simple auquel une deuxième voix de
sonde à été rajouté. Ce dispositif, qui est présenté dans la figure 5.3, permet
de mesurer la réflectivité à deux angles d’incidence différents. Il est possible de
récupérer la fonction diélectrique à partir d’une telle mesure en faisant l’inversion
des équations de Fresnel. Une déscription détaillée de la procédure de calcul de la
partie réelle et imaginaire de la fonction diélectrique se trouve dans le paragraphe 5.4.
Le troisième dispositif expérimental utilisé dans ce travail de thèse est le dispositif
double pompe, tracé dans la figure 5.5.
Ce dispositif peut être vu comme le
dispositif pompe sonde simple avec une deuxième voix de pompe. Il permet de
mesurer la dynamique de la réflectivité en excitant l’échantillon avec deux impulsions
de pompe successives.
Tous les trois dispositifs expérimentaux ont été réalisés en polarisation croisée. Les
faisceaux de pompe ont une polarisation p et les faisceaux de sonde une polarisation
s. Ceci permet de discriminer la lumière de pompe des autres signaux en utilisant
des polariseurs et ainsi d’augmenter le rapport signal sur bruit de l’expérience.
11.2 Mesures de réflectivité après excitation optique
simple
Dans ce paragraphe, les résultats obtenus avec le dispositif pompe sonde simple
sont présentés. D’abord, les caractéristiques de la dynamique de la réflectivité sont
discutées à l’aide de deux mesures de réflectivité effectuées à deux fluences de pompe
différentes. Puis, deux séries de mesures sont présentées, une série en fonction de la
11.2. Mesures de réflectivité après excitation optique simple
159
fluence de pompe et une série en fonction de la température de l’échantillon.
11.2.1 Mesures pompe–sonde simple
Résultats expérimentaux
Dans la figure 6.1 il y a deux mesures de la réflectivité à des fluences de 6.7 mJ/cm2
et 2.7 mJ/cm2 . Les graphes qui montrent les changements de réflectivité ∆R(t)/R0 ,
où R0 est la réflectivité non-perturbée avant l’arrivée de l’impulsion de pompe,
font partie d’une série de mesures effectuées avec une durée d’impulsion de 35 fs
et un faisceau de pompe proche d’incidence normale (10◦ ). L’échantillon était un
cristal de bismuth dont l’axe (111), l’axe optique, était perpendiculaire à la surface.
L’expérience a été réalisée à température ambiante. Les courbes contiennent des
caractéristiques qui ont été observés pour des fluences de 1.5 mJ/cm2 à 15.0 mJ/cm2 .
Au début de la dynamique, pendant l’excitation, la réflectivité décroı̂t pour un
instant très bref, puis elle accroı̂t et atteint un maximum à t = 200 fs environ. La
largeur de la chute ultrabrève est comparable à la durée d’impulsion, elle est de
45 fs FWHM pour une fluence de 6.7 mJ/cm2 . Après le maximum, la dynamique
de la réflectivité consiste en une partie oscillatoire et une partie non-oscillatoire. La
première est caractérisée par une oscillation sinusoı̈dale amortie dont la fréquence
dépend de la fluence, et la deuxième par un décroissement lent. La fréquence de
l’oscillation est 2.90 THz dans le cas de 2.7 mJ/cm2 , ce qui coı̈ncide avec la fréquence
propre du mode A1g de 2.92 THz [84]. Pour la fluence de 6.7 mJ/cm2 , la fréquence
est décalée vers le rouge, et a une valeur de 2.86 THz. Ce décalage augmente avec
la fluence, ce qui est en accord avec les résultats préliminaires [48]. De même,
l’évolution de la fréquence montre une dépendance temporelle: le décalage est maximal au début des oscillations est se rapproche de la valeur propre du mode A1g de
2.92 THz avec le temps. Environ 10 ps après l’excitation, le signal ∆R/R0 change
de signe et atteint une valeur minimale après environ 20 ps. Puis il retourne vers la
valeur non–perturbée (∆R/R0 = 0) après 4 ns, indiquant que l’échantillon n’est pas
endommagé et retourne à l’état initial.
Analyse et discussion
La chute ultrabrève initiale et le changement négatif de la réflectivité après 10 ps
sont des observations nouvelles qui ne peuvent pas être expliquées dans le cadre
des théories DECP et ISRS présentées dans le chapitre 3. Dans la suite, les caractéristiques complexes de la dynamique de la réflectivité seront liées aux considérations théoriques du chapitre 4.
160
11. Résultats expérimentaux
Les courbes rouges dans la figure 6.1 représentent des fits qui ont été obtenus
en utilisant le résultat principal du chapitre 4, l’équation
∆R/R0 = A · ∆pol + B · ∆Te + C · ∆Tl + D · ∆q ,
(11.1)
qui prend en compte les contributions au changement de la réflectivité qui sont
traitées dans le cadre de l’approche théorique présentée dans ce chapitre. Les coefficients A, B, C et D sont des paramètres ajustables qui ont les mêmes signes que les
coefficients correspondants dans l’équation 4.58. A, C, et D sont négatifs et B est
positif. La procédure d’ajustement est décrite en détail dans le paragraphe 6.1.2.
Il est possible de comprendre tous les aspects de la dynamique de la réflectivité
à l’aide de l’équation 11.1: la chute initiale de la réflectivité est le résultat d’une
compétition entre les deux premières termes pendant l’impulsion de pompe. La polarisation due au champ du laser de pompe crée un changement négatif, qui, après un
certain temps, est compensé par la contribution positive des électrons excitées. La
contribution électronique augmente pendant l’impulsion de pompe. Une fois le maximum atteint, elle décroı̂t selon l’évolution temporelle de la température électronique.
Les électrons transfèrent leur énergie au réseau et induisent une augmentation de
la température du réseau, ce qui fait augmenter la troisième contribution selon
l’évolution temporelle de Tl . Après 10 ps, la contribution négative du réseau est
plus importante que la contribution électronique positive et le changement de la
réflectivité devient négatif. Quand le transfert de l’énergie des électrons au réseau
est terminé, la réflectivité atteint le minimum. Le dernier terme de l’équation 11.1
décrit la contribution due aux vibrations atomiques, dont l’évolution temporelle de
la fréquence est modélisée avec l’équation 4.47, ω 2 ≈ ω02 (1 − (Te /|b,0 |)).
L’accord entre les courbes théoriques et expérimentales est excellent. Pour des
retards plus grand que 30 ps, une déviation légère est observée due au fait que la
théorie ne prend pas en compte la diffusion de la chaleur plus en profondeur dans
les cristal. En conclusion, on peut constater que notre théorie décrit les processus
qui mènent aux changements complexes de la réflectivité d’une façon satisfaisante.
11.2.2 Mesures en fonction de la fluence
Les mesures présentées dans le dernier paragraphe ont été effectué en dessous du
seuil d’endommagement. Néanmoins, nous avons vu que l’impulsion de pompe a
créé un état fortement hors équilibre. Le fait que l’échantillon n’a pas montré de
signe d’endommagement interroge sur l’évolution de la dynamique de la réflectivité
proche du niveau d’excitation menant à la création du désordre dans le cristal. Dans
ce paragraphe, une série de mesures qui couvre une gamme plus élevée des fluences
jusqu’au seuil d’endommagement sera présentée.
11.2. Mesures de réflectivité après excitation optique simple
161
La série de mesures en fonction de la fluence d’excitation est présentée dans la
figure 6.3. Le cristal est la géométrie expérimentale étaient les mêmes que ceux
décrits dans le paragraphe précédent. Le graphe en haut de la figure montre
30 ps de la dynamique de la réflectivité pour quatre fluences entre 3.7 mJ/cm2 et
21.0 mJ/cm2 . Le deuxième graphe montre les premières oscillations de ces mêmes
courbes pour comparer directement l’évolution des amplitudes et de la fréquence
d’oscillation. L’énergie de laser appliquée à l’échantillon était en dessous du seuil
d’endommagement qui est discuté dans le paragraphe 6.2.3. La dynamique de la
réflectivité peut être décrite de la même façon pour toutes les fluences. La chute
ultrabrève de la réflectivité est observée juste après l’excitation. Cette chute est
moins prononcée dans cette série que dans les mesures présentées dans le paragraphe précédent, ce qui est due à la résolution temporelle de l’expérience qui
était légèrement inférieure pendant l’acquisition de ces données (∼ 50 ps). Puis,
la réflectivité accroı̂t, décroı̂t, le changement ∆R/R0 change de signe et atteint un
minimum après ∼ 20 ps avant d’atteindre le niveau non–perturbée sur une échelle
de ns.
Les graphes montrent que toutes les caractéristiques de la dynamique de la réflectivité dépendent de la fluence: l’amplitude et l’amortissement des oscillations,
le décalage vers le rouge de la fréquence A1g , l’instant où le changement de la
réflectivité change de signe et la valeur du minimum. Une analyse détaillée des
ces caractéristiques et une comparaison à notre approche théorique se trouvent dans
le paragraphe 6.2.2. Les résultats les plus importants de cette analyse sont les suivants:
• l’amplitude des oscillations augmente avec la fluence d’une façon linéaire.
• le décalage de la fréquence vers le rouge accroı̂t en augmentant la fluence.
Notre modèle théorique permet de reproduire le décalage d’une façon satisfaisante pour des fluences inférieures à ∼ 10 mJ/cm2 sans utiliser des paramètres
ajustables pour le calcul.
• Notre théorie permet également de reproduire l’évolution temporelle de la
fréquence pour cette gamme de fluences sans l’utilisation des paramètres ajustables.
• La constante d’amortissement augmente avec la fluence d’une façon linéaire.
• La valeur minimale du changement de la réflectivité décroı̂t d’une façon linéaire
avec la fluence.
L’ensemble des mesures montre qu’il n’y a aucun signe de transformation du
cristal. Cette observation est importante car les calculs effectués avec le modèle à
162
11. Résultats expérimentaux
deux températures suggèrent que les températures du réseau maximales atteintes
quelques ps après l’excitation sont bien supérieures à la température de fusion
pour des fluences de plusieurs mJ/cm2 (c.f. aussi la discussion à la fin du paragraphe 6.1.2).
11.2.3 Mesures en fonction de la température
Les mesures du paragraphe précédent ont montré la dépendance des caractéristiques
de la dynamique de la réflectivité avec la fluence d’excitation. Afin d’étudier soigneusement le couplage des sous–systèmes électrons et réseau, il est également nécessaire
de étudier l’influence de la température initiale du cristal. Pour cette raison, des
mesures effectuées à plusieurs températures différentes ont été réalisées. Ces mesures
sont présentées dans ce paragraphe.
La figure 6.10 montre cinq courbes différentes qui ont été obtenues à des températures entre 50 K et 510 K. Le flux d’excitation était 6.7 mJ/cm2 et le cristal ainsi
que la géométrie expérimentale sont les mêmes que ceux décrits dans les paragraphes
précédents. Le graphe en haut de la figure présente 30 ps de dynamique de la
réflectivité pour les cinq températures. Le graphe en bas montre les premières oscillations. La dynamique de la réflectivité consiste en une chute ultrabrève, une dimitution de la réflectivité, des oscillations harmoniques amorties, et un changement de la
réflectivité négatif qui apparaı̂t après quelques picosecondes et un minimum que est
atteint après environ 20 ps. Comme pour les expériences en fonction de la fluence,
il y a des différences considérables entre les mesures. Par exemple, l’amortissement
des oscillations est beaucoup plus faible à basse température, ce qui mène à un
signal de réflectivité avec des oscillations visibles jusqu’à ∼ 35 ps (non-présentées
dans la figure). Les fréquences d’oscillation et les composantes non–oscillatoires sont
différentes à chaque température. Le signal mesuré à 50 K montre une particularité:
les oscillations de la réflectivité sont modulées avec une fréquence qui est différente
de la fréquence A1g . Une analyse de Fourier montre que cette fréquence peut être
associée au mode Eg , ce qui est en accord avec des observations préliminaires [46].
Une analyse des caractéristiques des mesures se trouve dans le paragraphe 6.3.2.
Les résultats importants de cette analyse sont:
• La fréquence des oscillations décroı̂t avec la fluence d’excitation. Cette dépendance est très proche d’une dépendance linéaire. Le changement de ce décalage
vers le rouge peut être compris dans le cadre du modèle théorique que nous
avons dévellopé.
• La constante d’amortissement augmente avec la température. Il est possible d’expliquer cette dépendance pour des températures bien supérieure à la
11.3. Mesures de réflectivité après deux impulsions de pompe
163
température de Debye (TD = 119 K). L’amortissement peut être associé à la
désintégration d’un phonon optique en deux phonons acoustiques. L’analyse
indique que ce processus est couplé avec des processus d’ordre plus élevés.
• L’amplitude d’oscillations décroı̂t d’une façon presque linéaire pour des températures entre 50 K et 290 K. Pour les températures plus élevées, un changement
d’amplitude n’a pas été observé.
En mesurant le signal à 510 K aucun dommage de l’échantillon a pu être observé. A
cette température, le transfert d’énergie pourrait mener à une transition vers l’état
liquide. Le changement négatif de la réflectivité qui est maximale après ∼ 20 ps peut
indiquer cette transition (la réflectivité du bismuth à l’état liquide est inférieure à
celle de l’état solide). Une expérience pour conclure sur cette hypothèse est présentée
dans le paragraphe suivant.
11.3 Mesures de réflectivité après deux impulsions de
pompe
Les mesures présentées dans les paragraphes précédents ont permis l’analyse des
propriétés de la dynamique de la réflectivité du bismuth en fonction de la fluence
d’excitation et la température de l’échantillon. Néanmoins, l’ambiguı̈té de certains
résultats ainsi que la différence entre la température maximale du réseau estimée
avec le calcul TTM et celle qui a été estimée expérimentalement montrent qu’il est
nécessaire d’avoir des informations supplémentaires sur l’état de l’échantillon après
équilibration électrons-réseau. Pour cette raison, nous avons conduit une expérience
double–pompe, qui permet d’étudier comment l’échantillon réagi à une deuxième
excitation qui suit la première après que les électrons ont transféré leur énergie au
réseau.
L’expérience a été effectuée avec le dispositif double–pompe. L’angle d’incidence
de la sonde était 22◦ , ceux des faisceaux de pompe étaient 15◦ et 30◦ . Les fluences d’excitation étaient de 6.9 mJ/cm2 et 8.0 mJ/cm2 , respectivement, afin de
compenser la différence d’absorption aux angles différents. La partie en haute de la
figure 6.15 montre le signal mesuré lors de cette expérience pour une période de 50 ps.
La première impulsion de pompe induit la dynamique décrit dans les paragraphes
précédent. Après 25 ps, une deuxième impulsion de pompe est appliquée. Cette
impulsion déclenche une dynamique qui, au premier regard, est très similaire à celle
observée après la première impulsion de pompe. Afin de comparer les deux traces,
elles sont superposées graphiquement dans les graphes en bas de la figure 6.15. Ces
deux graphes montrent qu’il y a des différences entre les deux courbes en ce qui
164
11. Résultats expérimentaux
concerne la fréquence d’oscillation, l’amortissement des oscillations et le minimum
de ∆R/R0 atteint ∼ 25 ps après la première (la deuxième) impulsion de pompe.
Les oscillations déclenchées par la première impulsion de pompe sont caractérisées
par une fréquence de 2.82 THz et une constante d’amortissement de 0.43 ps−1 en
accord avec les mesures préliminaires. Après la deuxième impulsion de pompe, la
fréquence est de 2.75 THz et la constante d’amortissement est 0.48 ps−1 . Le fait
qu’il est possible de générer des vibrations atomiques avec une fréquence A1g à
t = 25 ps montre que l’échantillon est toujours dans l’état cristallin après que les
électrons ont transféré l’énergie de la première impulson de pompe au réseau. En
comparant ces résultats à ceux qui ont été observés en augmentant la température
de l’échantillon, on voit que le décalage vers le rouge ainsi que l’augmentation de
la constante d’amortissement peuvent être dus à un échauffement du cristal. Si le
cristal était simplement chauffé à l’équilibre, ces changements correspondraient à une
température du cristal de ∼ 330 K (cf. figures 6.12 and 6.13), donc une température
bien inférieure à la température de fusion. Ce changement de température ne correspond pas au résultat du calcul TTM, avec lequel une température de réseau
maximale supérieure à 1000 K a été calculée pour une fluence comparable.
En conclusion, cette expérience montre que le processus de transert d’énergie entre
les électrons et le réseau ne mène pas à une transition vers l’état liquide. Il montre
également que la différence entre la température du réseau estimée théoriquement et
celle estimée expérimentalement ne peut pas être expliquée uniquement sur la base
d’une expérience pompe–sonde simple.
11.4 Dynamique ultra-rapide de la fonction
diélectrique du bismuth
Les mesures présentées dans les paragraphes précédents ont indiqué qu’il est nécessaire
d’obtenir plus d’informations sur l’échantillon qu’il n’est accessible avec les mesures
pompe-sonde simples afin d’identifier l’état transitoire atteint environ 20 ps après
l’excitation. Pour cette raison, nous avons effectué une mesure de la dynamique
de la fonction diélectrique du bismuth à 800 nm. L’avantage d’une mesure de la
fonction diélectrique par rapport à une mesure de la réflectivité est que la fonction diélectrique dépend directement de la configuration électronique, et permet de
déterminer sans ambiguı̈té l’état de l’échantillon (solide ou liquide, conducteur ou
semi-conducteur).
11.4. Dynamique ultra-rapide de la fonction diélectrique du bismuth
165
11.4.1 Mesure résolue en temps de la fonction diélectrique à
800 nm
Pour l’étude de la fonction diélectrique à 800 nm, des mesures avec le dispositif
double-sonde ont été réalisées à température ambiante. Comme il est détaillé dans
le paragraphe 10.1.3, un cristal uniaxial comme le bismuth est caractérisé par une
fonction diélectrique qui est un tenseur avec deux éléments indépendants, la fonction
diélectrique ordinaire o et la fonction diélectrique extraordinaire e . La géométrie
expérimentale utilisée ici permet de mesurer la fonction diélectrique ordinaire o (cf.
paragraphe 7.2). La fonction diélectrique ordinaire sera apellée fonction diélectrique
dans la suite.
La figure 7.2 présente la dynamique de la fonction diélectrique pour une fluence
d’excitation de 6.7 mJ/cm2 . Les mesures de réflectivité à deux angles d’incidence
différents (α1 = 19.5◦ and α2 = 34.5◦ ) sont présentées en 7.2 a). Les deux mesures
sont similaires aux mesures présentées dans les paragraphes précédents. Cependant,
des différences sur l’amplitude d’oscillation et le minimum de ∆R/R0 qui dépendent
de l’angle sont visibles. Les fréquences ainsi que l’amortissement des deux courbes
sont identiques ainsi que l’instant pour lequel le signal change de signe.
Les parties réelles et imaginaires de la fonction diélectrique qui ont été calculées
en utilisant l’algorithme décrit dans le paragraphe 5.4 sont indiquées en 7.2 b).
Pour t < 0, les valeurs de la fonction diélectrique correspondent aux valeurs nonperturbées (re = −16.25 and im = 15.40, cf. paragraphe 7.1). Le comportement des
deux parties après l’arrivée de l’impulsion de pompe peut être décrit comme la superposition d’une composante oscillatoire et d’une composante non-oscillatoire. La
fréquence des oscillations et l’amortissement sont les mêmes que dans les mesures
de la réflectivité. La partie non-oscillatoire de la partie réelle décroı̂t et elle est
minimale à l’instant où la réflectivité est maximale. Après environ 4 ps, la partie
réelle croise la valeur non-perturbée et arrive à un maximum à ∼ 20 ps qui est
trans
= −13.80. La partie non-oscillatoire de la dynamique de la partie imaginaire
re
accroı̂t après l’arrivée de l’impulsion de pompe et atteint un maximum à l’instant
où la réflectivité est maximale. Puis, elle décroı̂t, croise la valeur non-perturbée à
∼ 4 ps et atteint une valeur minimale de trans
= 11.30 après ∼ 20 ps. Les changeim
ments de la valeur réelle et imaginaire ont des signes différents pendant toute la
période d’observation. Les valeurs atteintes après ∼ 20 ps restent constantes pendant les dernières 10 ps de la dynamique. Des résultats similaires ont été observés
pour des fluences d’excitation entre 1.5 mJ/cm2 et 14 mJ/cm2 .
166
11. Résultats expérimentaux
11.4.2 Analyse et discussion
Avant d’analyser et discuter les mesures de la fonction diélectrique dans le bismuth,
nous avons quantifié les sources d’erreurs sur les mesures expérimentales. Une analyse détaillée des erreurs se trouve dans le paragraphe 7.3.
La fonction diélectrique étant calculée à partir de deux mesures de réflectivité,
l’erreur sur les mesures de réflectivité influe sur la détermination de la fonction
diélectrique. Deux autres sources d’erreur ont été identifiées: la mesure d’angle
d’incidence et l’incertitude sur la superposition temporelle des deux faisceaux de
sonde. Ces erreurs ont été calculées et sont présentées dans la figure 7.3. L’estimation
des erreurs montre que la fonction diélectrique du bismuth à 800 nm a été déterminée
avec une exactitude satisfaisante. L’erreur sur la détermination de la partie réelle est
inférieure à 2%, celle sur la partie imaginaire est inférieure à 5%. Pour les premiers
extrema, l’erreur sur la partie réelle est d’environ 10%, celle sur la partie imaginaire
est de 40%. Afin de valider les résultats de l’inversion des équations de Fresnel,
nous avons effectué une mesure à deux paires d’angles d’incidence différentes, à
(14, 5◦ , 29, 5◦ ) et (19, 5◦ , 34, 5◦ ). Les résultats sont comparés dans la figure 7.4.
Les deux mesures présentent un accord satisfaisant et valident notre procédure de
détermination de la fonction diélectrique.
Les changements de la fonction diélectrique induits par le laser sont une manifestation du déplacement atomique que le cristal de bismuth subit après une excitation
laser. La fonction diélectrique change avec la fréquence du phonon. Un résultat
remarquable est que les parties réelles et imaginaires changent dans des directions
opposées pendant toute la periode d’observation. Au début de la dynamique, la
partie réelle évolue vers des valeurs inférieures, indiquant un état plus métallique.
Simultanément, la partie imaginaire croı̂t et croise la valeur de la partie imaginaire
de l’état liquide. Après plusieurs picosecondes, la partie réelle évolue vers la valeur
de l’état liquide sans l’atteindre. La partie imaginaire atteint un minimum qui est
inférieur à la valeur de la fonction diélectrique de l’état solide.
Sur tout l’intervale d’observation, au moins une des deux parties n’est pas comprise entre la valeur de l’état liquide et l’état solide. Nous pouvons donc exclure
une fusion jusqu’à 30 ps après l’excitation. Ce résultat corrobore le résultat de
l’expérience double-pompe, qui a également indiqué que le bismuth reste solide après
l’excitation optique. Nous avons observé que les valeurs des plateaux (les valeurs
après ∼ 20 ps) changent avec la fluence d’excitation. Les résultats sont présentés
dans la figure 7.5. Cette figure montre que plus fort est l’excitation, le plus grande
est la différence entre les valeurs non-perturbées et les valeurs des plateaux.
Afin de déterminer si les changements de la fonction diélectrique après ∼ 20 ps
peuvent être dus à un échauffement du réseau, nous avons effectué un mesure de la
11.4. Dynamique ultra-rapide de la fonction diélectrique du bismuth
167
fonction diélectrique en fonction de la température. Le résultat de cette expérience,
qui a déterminé les valeurs non-perturbées des parties réelles et imaginaires de la
fonction diélectrique en ellispométrie, sont présentés dans la figure 7.6. Cette figure
montre qu’en chauffant le cristal, la partie réelle ne change pas tandis que la partie
imaginaire présente des changements positifs. Ces changements sont différents aux
changements induits par le laser qui sont présentés dans la figure7.5. Nous pouvons
donc conclure que le cristal de bismuth se trouvre dans un état transitoire ∼ 20 ps
après l’excitation optique. Cet état ne correspond pas à un cristal chauffé, ni au
bismuth en état liquide, ni à un mélange des deux phases. Nos mesures étant
limitées à une longueur d’onde, il n’est pas possible d’obtenir des conclusions fiables
sur la configuration électronique du système. La connaissance de la dynamique de la
fonction diélectrique pour une gamme spectrale plus élévée (par exemple le visible
et l’infrarouge) permettrait de déterminer l’évolution de la structure de bande et
l’influence des déplacements atomiques. Il n’est donc pas possible de déterminer la
nature de l’état transitoire avec les informations disponibles à partir des résultats
présentés ici.
168
11. Résultats expérimentaux
12 Conclusions et perspectives
L’étude des phonons optiques cohérents présentée dans ce travail est basée sur un
dispositif expérimental qui permet de mesurer des changements de la réflectivité
d’un échantillon photo-excité avec une résolution temporelle de 35 fs et avec une
précision de ∆R/R0 = 10−5 . Ceci a permis d’effectuer une analyse détailée de la
dynamique de la réflectivité dans le bismuth après une excitation ultra-rapide et a
rendu possible la détermination de la fonction diélectrique à 800 nm.
L’étude de la dynamique de la réflectivité a mis à jour deux nouveaux effets: un
changement négatif ultra-bref pendant l’impulsion de pompe excitant l’échantillon
et un changement négatif qui est maximal après ∼ 20 ps. Les changements de
réflectivité ont été interprétés à l’aide des considérations théoriques qui prennent
en compte l’excitation des électrons, les forces induites par le laser et l’interaction
électrons-réseau. Le changement négatif initiale a été attribué à un déplacement
rapide des atomes pendant l’impulsion de pompe créé par polarisation. Dans le cadre
de la théorie, le changement négatif à l’échelle ps a été identifié comme le résultat
d’une compétition entre une contribution positive due à l’excitation des électrons
et une contribution négative qui est liée à l’échauffement du réseau par interaction
électron-phonon. La théorie a également permis de lier le décalage vers le rouge
de la fréquence A1g , qui dépend de la température et de la fluence d’excitation,
au potentiel inter-atomique perturbé par l’excitation optique. La dépendance de
l’amortissement de la température a été associée à la probabilité de relaxation
d’un phonon optique en deux phonons acoustiques. L’évolution temporelle de la
température des électrons et de la température du réseau ont été calculées avec le
modèle à deux températures. D’après ces calculs, la température électronique peut
monter à plusieurs milliers de Kelvin directement après l’excitation. La température
du réseau qui augmente due à l’interaction électron-phonon atteint des valeurs maximales qui sont supérieures à la température de fusion pour des fluences en dessous
du seuil d’endommagement. Néanmoins, aucun signe de transition vers la phase
liquide n’a été observé.
Afin d’étudier l’état excité qui est atteint ∼ 20 ps après l’excitation optique, nous
avons effectué une expérience double pompe. Les résultats de ces mesures ont montré
que la dynamique de la réflectivité après le transfert d’énergie électron-réseau est
similaire à la dynamique excitée par une seule impulsion. Ces mesures suggèrent
169
170
12. Conclusions et perspectives
que l’échantillon reste dans l’état solide pendant les premieres dizaines de picosecondes après l’excitation optique. La température du réseau ∼ 20 ps après l’arrivée
de l’impulsion de pompe a été estimée en comparant des résultats de l’expérience
double-pompe à la série de mesures effectuées à des températures différentes. La
comparaison indique que le changement de la température du réseau est de quelques
dizaines de Kelvin. Ce changement est moins important que le changement de la
température calculé avec le modèle à deux températures.
Pour valider notre théorie et comprendre l’origine du changement négatif de la
réflectivité, nous avons déterminé la fonction diélectrique à 800 nm à partir de deux
mesures de réflectivité. Les résultats montrent que les changements des parties
réelles et imaginaires ont des signes opposés pendant toute la période d’observation
de 30 ps, et que les partie réelles et imaginaires ne coı̈ncident pas avec les valeurs de
l’état liquide. La dynamique de la fonction diélectrique consiste en une contribution
non-oscillatoire et une contribution oscillatoire, dont cette dernière peut être décrite
par des oscillations harmoniques amorties avec la fréquence A1g . Similaire à la
dynamique de la réflectivité, les deux parties de la fonction diélectrique atteignent
des valeurs maximales environ 20 ps après l’excitation. La valeur de la partie réelle
à cet instant est entre les valeurs de l’état liquide et l’état solide, alors que la valeur
de la partie imaginaire est inférieure à la valeur de l’état solide. Afin de comparer
la fonction diélectrique de cet état transitoire avec celle solide chauffé dans des
conditions d’équilibre, nous avons effectué une mesure d’éllipsométrie en fonction
de la température du cristal, qui nous a permis de déterminer les changements
des parties réelles et imaginaires avec la température. Cette comparaison montre
clairement que l’état transitoire caractérisé par un accroissement de la partie réelle
et un décroissement de la partie imaginaire ne peut pas être décrit par un solide
chauffé. La mesure d’éllipsométrie montre que l’augmentation de la température ne
change pas la partie réelle de la fonction diélectrique d’une façon signifiante alors
que la partie imaginaire augmente. Une interprétation possible de l’état transitoire
du bismuth qui est atteint ∼ 20 ps après l’excitation est la signature d’un état moins
métallique caractérisé par une densité de porteurs moins importante.
Les résultats présentés suscitent la question de la caractérisation de cet état
transitoire. Il est possible que l’échantillon subisse une transition semi-métal–semiconducteur, mais la formation d’un état moins métallique ne peut pas être exclus.
Afin d’obtenir des informations supplémentaires sur la configuration électronique de
l’échantillon, il serait nécessaire de connaı̂tre la fonction diélectrique pour une large
gamme spectrale, par exemple le visible et l’infrarouge, et pour une large gamme
de température. Il faudrait également étudier précisément le processus de diffusion des électrons après l’absorption de l’impulsion laser de pompe. L’utilisation de
couches minces de bismuth d’épaisseurs différentes permettrait d’estimer l’effet du
171
confinement des électrons dans un volume limité.
Il serait également souhaitable de disposer d’impulsions laser encore plus brèves
que 35 fs, ce qui permettrait d’étudier le déplacement cohérent des atomes pendant
la durée de l’impulsion de pompe. Une configuration idéale serait un dispositif permettant de choisir la durée d’impulsion de la pompe et la sonde séparément. Cela
permettrait d’étudier la réponse ultra-rapide des atomes en analysant les changements de la réflectivité sans convolution avec les profils temporels.
Une possibilité prometteuse d’étude de la dynamique du réseau cohérente pourrait
être obtenue avec les sources X femtoseconde. Des publications récentes suggèrent
qu’il est possible de générer des impulsion X avec une durée de l’ordre d’une femtoseconde avec le laser à électrons libre [100]. En combinaison avec des impulsions
de pompe de quelques femtosecondes, les chercheurs disposeraient d’un outil permettant de résoudre des mouvements atomiques avec une résolution temporelle bien
inférieure à la période typique des oscillations atomiques. Ceci permettrait d’étudier
le déplacement cohérent ultrabref qui a été observé pour la première fois dans nos
expériences avec une résolution spatiale donnée par les changements de l’intensité
des pics de diffraction de Bragg.
172
12. Conclusions et perspectives
Part VI
Appendix
173
A Derivatives of the Drude dielectric
function and reflectivity
∂re
= re − 1 ,
∂ ln ne
∂re
ω2
= 2 · 22im ,
∂ ln νe−ph
ωp
∂im
= im ,
∂ ln ne
∂im
1 − (ν/ω)2
= im
,
∂ ln νe−ph
1 + (ν/ωp )2
∂R
im
=
∂im
||
∂R
=
∂re
(A.1)
(2 || + 4re − 2)
n
o2
1/2
1/2
1/2
1/2
2 (|| + re )
|| + 1 + 2 (|| + re )
21/2 {re (2re − 1 + ||) − || (|| + 1)}
o2
n
1/2
1/2
1/2
|| · (|| + re )
|| + 1 + 2 (|| + re )
(A.2)
175
176
A. Derivatives of the Drude dielectric function and reflectivity
Solid Bi
Liquid Bi
∂re
∂ ln ne
∂re
∂ ln νe−ph
∂im
∂ ln ne
∂im
∂ ln νe−ph
dR
dre
dR
dim
-17.25
-12
0.032
0.012
15.4
28.9
0.87
-11.94
−1.2 · 10−2
−7.5 · 10−3
−6 · 10−3
1.58 · 10−3
Table A.1: Numerical values for dielectric function derivatives for solid and liquid bismuth.
Solid Bi
Liquid Bi
Cne
Cve−ph
+0.1146 −5.6 · 10−3
+0.1357 −1.9 · 10−2
Table A.2: Coefficients in the reflectivity variations.
B Material properties of of bismuth
Property
Dielectric function (ordinary)
Complex refractive index
Plasma frequency
Electron density
Fermi-energy
Fermi velocity
Density
Atomic mass
Atomic density
Inter-atomic distance
Debye temperature
Debye frequency
Melting temperature
Value
Reference
re = −15.4; im = 16.25; || = 22.39 [EM]1
η = 1.87; κ = 4.35
[EM]
15
ωp = 1.31 · 10 s
[EM]
22
−3
ne = 5.39 · 10 cm
[EM]
F = 5.2 eV
[EM]
6
vF = 1.36 · 10 m/s
[EM]
3
ρ = 9.8 g/cm
[62]
−25
M = 209 u = 33.47 · 10
g
[62]
22
−3
na = 2.83 · 10 cm
[62]
−8
d = 3.3 · 10 cm
[62]2
TD = 119 K
[62]
13 −1
ωD = 1.56 · 10 s
[62]
Tmelt = 544.5 K
[39]
1
Obtained from ellispometry measurement presented in chapter 7. The other properties with
reference [EM] are calculated from re and im .
2
Calculated from na as d = na
− 13
177
178
B. Material properties of of bismuth
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