Time-resolved X-ray diffraction studies of phonons and phase transitions

Time-resolved X-ray diffraction studies

of phonons and phase transitions

Ola Synnergren

Department of Physics, Lund Institute of Technology Technology and Society, Malm¨o University

Doctoral Thesis

Atomic Physics Division

Department of Physics Technology and Society Lund Institute of Technology Malm¨o University P.O. Box 118

SE-221 00 Lund SE-205 06 Malm¨o

Sweden Sweden

Lund Reports on Atomic Physics, ISSN 0281-2762 LRAP-350

ISBN 91-628-6662-1 Printed by KFS i Lund AB

Abstract

This thesis summarizes work in which time-resolved X-ray diffraction has been used to probe crystalline materials, thereby revealing the dynamics of phonons and phase transitions.

X-ray diffraction is the standard tool in investigations of structure on the atomic scale. It has been used for a long time, and has successfully helped scientists to find the structure of a wide range of materials. The use of ultrafast time-resolved X-ray diffraction is a strongly emerging field which is still under development. Impulsive strain pulses, or coherent acoustic phonons, have been probed using optical techniques for at least two decades. Yet, optical pulses can only probe the surface of a semiconductor. X-rays penetrate deeper and can follow the phonons as they propagate into the sample.

Real time studies of phase transitions have also been conducted using optical methods. These measurements are indirect in the sense that they probe the susceptibility change of the sample rather than the positions of the atoms. Again, time-resolved X-ray diffraction can give new insights into the field by probing the structural changes directly.

This thesis focuses mainly on experimental work in which time-resolved X-ray diffraction has been used to probe phonons or samples undergoing phase transi-tions. A brief theoretical background will also be given, as well as a description of beamline D611 at MAX-lab, a synchrotron beamline for time-resolved X-ray diffraction measurements which has been developed during the work for this thesis.

Sammanfattning

Denna avhandling sammanfattar arbete d¨ar tidsuppl¨ost r¨ontgendiffraktion har anv¨ants f¨or att unders¨oka kristallina material och studera dynamiken hos fononer och fas¨overg˚angar.

R¨ontgendiffraktion ¨ar standardverktyget f¨or stukturbest¨amning p˚a en atom¨ar skala. Det har anv¨ants l¨ange och har framg˚angsrikt hj¨alp vetenskapsm¨an att best¨amma strukturen hos en stor m¨angd material. Anv¨andningen av ultra-snabb tidsuppl¨ost r¨ontgendiffration ¨ar ett starkt v¨axande omr˚ade som fort-farande utvecklas.

Akustiska impulser, eller koherenta akustiska fononer, har studerats med hj¨alp av optiska tekniker i ˚atminstonde tv˚a ˚artionden. Optiska pulser kan dock endast under¨oka en halvledares yta. R¨ontgen penetrerar djupare in i proven och kan f¨olja fononer d˚a de f¨ardas in i proven.

Realtidsstudier av fas¨overg˚angar har ocks˚a utf¨orts med hj¨alp av optiska metoder. Dessa m¨atningar ¨ar indirekta d˚a de m¨ater ¨andringar i susceptibiliteten hos provet ist¨allet f¨or atomernas positioner. ˚Aterigen kan tidsuppl¨ost r¨ontgendiffraktion ge en ny insyn i omr˚adet genom att den g¨or en direkt m¨atning av de strukturella ¨

andringarna.

Den h¨ar avhandlingen fokuserar huvudsaklingen p˚a experimentellt arbete d¨ar tidsuppl¨ost r¨ontgendiffraktion anv¨ants f¨or att studera fononer eller prover som genomg˚ar en fas¨overg˚ang. En kort teoretisk bakgrund finns med, liksom en beskrivning av D611, ett synkrotronljusstr˚alr¨or f¨or tidsuppl¨ost r¨ontgendiffraktion som utvecklats under arbetet med den h¨ar avhandlingen.

Acknowledgements

This thesis is a result of a collaboration between Malm¨o University and Lund University. I have been employed by Malm¨o University and registered as PhD student at Lund University. The research has been performed in the Ultra-fast X-ray Science group at the Atomic Physics division at Lund Institute of Technology. Many people have been very helpful, and I would like to thank. . . . . . my supervisor, J¨orgen Larsson, for giving me the opportunity to work in this exciting field, for guiding me through the work for this thesis, and for providing a rich working environment.

. . . Per J¨onsson, my co-supervisor, for good support and Johan Helgesson, my boss in Malm¨o, for being helpful and supportive. And the rest of my collegues in Malm¨o.

. . . people in the ultrafast X-ray science group who have provided a lot of use-ful help, interresting discussions and a pleasant working environment. Special thanks go to Peter Sondhauss for helping me with theoretical questions and Tue Hansen for lots of help on the experimental part. I would also like to thank all other former and present members of the group, including Thomas Mißalla, Michael Harbst, Vladimir Tenishev, Eva Danielsson, Alok Srivastava, Henrik Enquist, Hengameh Navirian and M˚ans Schultz.

. . . collaborators, especially Ben Lings, Katar´ına Rosolankov´a and Justin Wark from Oxford University, and Aaron Lindenberg, Klaus Sokolowski-Tinten, Chris-tian Bl¨ume, Kelley Gaffney, Carl Calleman, Jon Sheppard and others from the SPPS collaboration. Sophie Canton from the Department of Chemistry has been very helpful at MAX-lab.

. . . all fellow PhD-students at the atomic physics division and at MAX-lab. . . . my family for being supportive during the time I worked on my PhD, and my friends for providing other things to think about. And all the people in the scuba diving club for helping me to spend my free time.

. . . and, last but not least, Sara for always being supportive, and for being there throughout my PhD.

Contents

I

Overview

1

1 Introduction 3

2 Time-resolved X-ray diffraction 5

2.1 Introduction . . . 5

2.2 Kinematic X-ray diffraction theory . . . 6

2.3 Thermal effects . . . 8

2.4 Effects of phonons . . . 8

2.5 Absorption and extinction . . . 10

2.6 Asymmetrically cut crystals . . . 10

2.7 Rocking curves . . . 11

2.8 Probing depth . . . 12

2.9 Total external reflection . . . 12

3 Coherent acoustic phonons 13 3.1 Introduction . . . 13

3.2 Lattice vibrations in crystalline matter . . . 13

3.3 Detection of coherent acoustic phonons . . . 15

3.4 Generation of coherent acoustic phonons . . . 16

3.5 Phonons probed by laser produced X-rays . . . 17

3.6 Coherent phonons in heterostructures . . . 20

3.7 Coherent control of acoustic phonons . . . 21

3.8 Free-carrier effects in phonon generation . . . 22

4 Phase transitions 23 4.1 Introduction . . . 23

4.2 Structural phase transitions . . . 24

4.3 The ferroelectric phase transition in KDP . . . 25

4.4 Regrowth of InSb . . . 27

4.5 Non-thermal melting of InSb . . . 28

5 Beamline D611 31 5.1 Introduction . . . 31

5.2 X-ray source and optics . . . 31

5.6 Streak camera . . . 38

6 Ultrafast X-ray sources 41 6.1 Introduction . . . 41

6.2 Laser-produced plasma radiation . . . 42

6.3 Other laser-based X-ray sources . . . 42

6.4 Right angle Thompson scattering . . . 43

6.5 Synchrotron radiation . . . 44

6.6 Sliced synchrotron radiation . . . 45

6.7 Rotated-bunch synchrotron radiation . . . 46

6.8 LINAC-based light sources . . . 47

6.9 Recirculation LINAC-based sources . . . 47

6.10 The free electron laser . . . 48

7 Outlook 51

Summary of papers 53

List of papers

Paper I

O. Synnergren, M. Harbst, T. Misalla, J. Larsson, G. Kantona, R. Neutze, and R. Wouts, “Projecting picosecond lattice dynamics through X-ray topography,”

Appl. Phys. Lett., vol. 80, pp. 3727–3729, 2002.

Paper II

J. Larsson, P. Sondhauss, O. Synnergren, M. Harbst, P. A. Heimann, A. M. Lindenberg, and J. S. Wark, “Time-resolved X-ray diffraction study of the ferro-electric phase-transition in DKDP,” Chem. Phys., vol. 299, pp. 157–161, 2004.

Paper III

P. Soundhauss, J. Larsson, M. Harbst, G. A. Naylor, A. Plech, K. Scheidt,

O. Synnergren, M. Wulff, and J. S. Wark, “Picosecond X-ray studies of co-herent folded acoustic phonons in a multiple quantum well,” Phys. Rev. Lett., vol. 94, p. 125509, 2005.

Paper IV

A. M. Lindenberg, J. Larsson, K. Sokolowski-Tinten, K. J. Gaffney, C. Blome,

O. Synnergren, J. Sheppard, C. Caleman, A. B. MacPhee, D. Weinstein, D. P. Lowney, T. K. Allison, T. Matthews, R. W. Falcone, A. L. Cavalieri, D. M. Fritz, S. H. Lee, P. H. Bucksbaum, D. A. Reis, J. Rudati, P. H. Fuoss, C. C. Kao, D. P. Siddons, R. Pahl, J. Als-Nielsen, S. Duesterer, R. Ischbeck, H. Schlarb, H. Schulte-Schrepping, T. Tschentscher, J. Schneider, D. von der Linde, O. Hignette, F. Sette, H. N. Chapman, R. W. Lee, T. N. Hansen, S. Techert, J. S. Wark, M. Bergh, G. Huldt, D. van der Spool, N. Timneanu, J. Hajdu, R. A. Akre, E. Bong, P. Krejcik, J. Artur, S. Brennan, K. Luening, and J. B. Hastings, “Atomic-scale visualization of inertial dynamics,” Science, vol. 308, pp. 392–395, 2005.

M. Harbst, T. N. Hansen, C. Caleman, W. K. Fullagar, P. Jnsson, P. Sondhauss,

O. Synnergren, and J. Larsson, “Studies of resolidification of non-thermally molten InSb using time-resolved X-ray diffraction,” Appl. Phys. A, vol. 81, pp. 893–900, 2005.

Paper VI

K. J. Gaffney, A. M. Lindenberg, J. Larsson, K. Sokolowski-Tinten, C. Blome,

O. Synnergren, J. Sheppard, C. Caleman, A. B. MacPhee, D. Weinstein, D. P. Lowney, T. K. Allison, T. Matthews, R. W. Falcone, A. L. Cavalieri, D. M. Fritz, S. H. Lee, P. H. Bucksbaum, D. A. Reis, J. Rudati, P. H. Fuoss, C. C. Kao, D. P. Siddons, R. Pahl, J. Als-Nielsen, S. Duesterer, R. Ischbeck, H. Schlarb, H. Schulte-Schrepping, T. Tschentscher, J. Schneider, D. von der Linde, O. Hignette, F. Sette, H. N. Chapman, R. W. Lee, T. N. Hansen, S. Techert, J. S. Wark, M. Bergh, G. Huldt, D. van der Spool, N. Timneanu, J. Hajdu, R. A. Akre, E. Bong, P. Krejcik, J. Artur, S. Brennan, K. Luening, and J. B. Hastings, “Atomic-scale visualization of inertial dynamics,” Phys. Rev.

Lett., vol. 95, p. 125701, 2005.

Paper VII

O. Synnergren, P. Sondhauss, T. Hansen, S. Canton, H. Enquist, A. Srivas-tava, and J. Larsson, “Transient metal-like heat conduction in a semiconductor,”

Part I

Chapter 1

Introduction

When a short laser pulse hits a crystalline sample many different things can happen. If the bandgap of the crystal is smaller than the energy of the laser photons, the photons are absorbed. The valance electrons of the crystal then get excited to the conduction band. If there are only a small number of excited electrons, they will diffuse away into the crystal, but as they interact with the atoms in the crystal, energy is transferred from the electrons to the lattice. This will lead to a heating of the surface layer which will expand. As a reaction to this, according to Newton’s third law, an impulsive strain wave will be launched into the crystal.

If instead there is a large number of excited electrons, there will not be sufficient binding electrons left to keep the crystal together. On a very rapid time scale, the atoms will move away from their equilibrium positions. The crystal will lose its structure and melt. As time passes, the electrons give their energy to the atoms and the sample returns to thermal equilibrium. Later, the heat is transferred away from the interaction area and the sample solidifies again. Depending on how fast this happens, it may turn into a crystalline structure again, or it may become amorphous.

Even if the bandgap is larger than the photon energy, the photons can still interact with the sample, for example via Raman scattering. The photon is then scattered inside the crystal and part of its energy is transferred into vibrational modes in the crystal. Some of these vibrational modes may drive a structural phase transition (i.e. a soft phonon mode).

The scope of this thesis has been to study laser-induced processes in crystalline matter using time-resolved X-ray diffraction. The goal has been to learn more about how to generate and detect phonons, as well as to get more insight into laser-induced phase transitions.

The work performed includes a number of experiments performed at the Lund Laser Centre (LLC), MAX-lab in Lund, the European Synchrotron Radiation

Facility in Grenoble (ESRF), the Advanced Photon Source in Berkeley (ALS) and the Sub Picosecond Pulse Source in Stanford (SPPS), as well as the con-struction of beamline D611 at MAX-lab.

In chapter 2, I will give an overview of time-resolved X-ray diffraction (TRXD). I will also briefly describe some simple theoretical aspects of X-ray diffraction that will be needed to explain the measurements described later in the thesis. Chapter 3 contains information about coherent acoustic phonons. I will describe how phonons are created and detected. I will also give an overview of the experiments performed in this area. More information can be found in the bundled papers.

Phase transitions are described in chapter 4. This chapter also contains an overview of the field as well as an overview of the experiments performed. Again, more information can be found in the bundled papers.

Much of the work done was concerned with the construction of the D611 beam-line at MAX-lab, and this will be described in chapter 5. This chapter also contains a detailed description of the performance of and techniques used at the beamline.

The last chapter of the thesis is a summary and an outlook of what we can expect in the future. This is followed by the second part of the thesis which consists of published and submitted papers.

Chapter 2

Time-resolved X-ray

diffraction

2.1

Introduction

Time-resolved X-ray diffraction (TXRD) is a tool that makes studies of the temporal evolution of the structure of matter on the atomic scale possible. The range of time scales studied varies over twenty orders of magnitude or more. On the slow time scale, the aging of material can be detected and measured over tens of years. At the other end of the time scale, the non-thermal melting of semiconductors, which happens on a timescale of hundreds of femtoseconds, can be studied. This thesis is focused on fast and ultrafast TXRD, on time scales ranging from nanoseconds to femtoseconds.

A number of successful TXRD studies using X-rays from laser-based sources have been carried out. In the laser based X-ray sources, a high power laser is focused on a target, often a solid material. The laser-pulse creates a hot plasma from which electrons are accelerated into the target. This creates X-ray radiation in the same way as in a conventional X-ray tube. The pulse duration is very short, comparable to that of the generating laser pulse. Studies of both coherent acoustic phonons1 and non-thermal phase transitions2have been made. The integration of synchronized lasers and streak cameras at synchrotron radi-ation facilities has opened new possibilities in the field3. With a streak camera, only a single reflection can be measured at a time, sufficient to study, for ex-ample, coherent acoustic phonons4,5. Time resolutions of less than 1 ps can be achieved. If lower time resolution is acceptable, CCD cameras can be used instead of streak cameras, and many reflections can be recorded simultaneously. The time resolution will be limited to the duration of the X-ray pulse from the synchrotron light source, usually 50-100 ps. From this a molecular movie show-ing the motion of the atoms can be reconstructed6,7. Diffuse scattering from

simple molecules makes the study of chemical reactions8possible.

Some studies have been made using combined laser and accelerator sources9,10, and recently studies from LINAC-based sources have been presented.

Time-resolved X-ray diffraction enables us to study many phenomena that are hidden to other methods. To understand what can be studied it is essential to know some X-ray diffraction theory. Most often perfect crystals are studied, and to make quantitative calculations of the X-ray diffraction from these crystals it is necessary to use the dynamic theory of X-ray diffraction. However, many of the effects that occur in time-resolved X-ray diffraction can be understood qual-itatively using the more simple kinematic X-ray diffraction theory. This chapter includes the basics of the kinematic X-ray diffraction theory, and explanations of the time-resolved effects that can be studied.

2.2

Kinematic X-ray diffraction theory

Kinematic theory does not take into account effects that may occur in highly perfect crystals, such as multiple scattering and extinction. Yet, this theory often provides a sufficiently good description of the phenomena that may occur in time-resolved X-ray diffraction.

The X-ray photons mainly interact with the electrons in the diffracting material. This is because the energy of the photons are too low to interfere with the nucleus. It is known that the scattering by a single electron is given by (see figure 2.1a) Ie= I0 e 4 m2c4R2  1 + cos2φ 2 

for non-polarized light11. To calculate the scattering from an atom, the charge from each electron is assumed to be spread into a cloud with charge density ρ. For spherically symmetric electrons, the electronic scattering factor is given by

fe=  _∞ 0 4πr2ρ(r)sin kr kr dr where k = 4π sin θ λ

For an atom with several electrons, these complex quantities can simply be added

f₀=

n fen

Generally not all electron shells are spherically symmetric, and there is also absorption, thus the scattering factor is slightly modified

2.2. KINEMATIC X-RAY DIFFRACTION THEORY φ R (a) _Δk k0 k 2θ (b)

Figure 2.1: The coordinate system used in the derivation.

The values of these (energy dependent) parameters can be found in tables. To get the scattering from a unit cell, one simply adds the scattering from the different atoms in the unit cell (with the phase). Assume that there is an incoming X-ray beam with wave vector k₀ and a reflected X-ray beam with wave vector k and define Δk = k− k₀(see figure 2.1b). The scattering factor from the unit cell is then

F = n

fneiΔk·rn (2.1)

where rn is the position of atom n relative to the origin of the unit cell.

For a small crystal consisting of N₁× N2× N3unit cells, the scattered intensity is given by Ip = Ie|F |2sin 2_{[Δk/2· N} 1a1] sin2[Δk/2· a1] sin2[Δk/2· N₂a₂] sin2[Δk/2· a2] sin2[Δk/2· N₃a₃] sin2[Δk/2· a3] For large values of N this will vanish unless

Δk· a1= h, Δk· a2= k, Δk· a3= l

Where h, k and l are integer numbers. These are called the 3 Laue equations. By introducing the reciprocal lattice

b₁= 2π a2× a3

a₁· (a₂× a₃)

(b₂ and b₃are defined by cyclic permutation) and the vector



Hhkl= hb1+ kb2+ lb3

where h, k and l are integer numbers, the Laue equations can be rewritten as Δk = Hhkl

This can be used to calculate the stucture factor. Equation 2.1 becomes

Fhkl=



n

where (xn, yn, zn) is the relative position of the atoms within the unit cell.

The diffracted intensity is given by

I = IeFhkl∗ Fhkl (2.2)

2.3

Thermal effects

If the diffracting crystal has a non-zero temperature, the atoms will vibrate about an equilibrium position rn. Let δn denote the instantaneous distance to

the equilibrium position for atom n. The absolute position can then be written as rn+ δn. By inserting the position into equation 2.1 and then inserting the

result into equation 2.2 we get

I = Ie  n fneiΔk·(rn+δn)  n fn∗e−iΔk·(rn+δn)

By rearranging and taking the time average this becomes

I = Ie  n  n fnfn∗eiΔk·(rn−rn)eiΔk·(δn−δn)

Let un be the component of δn along Δk. If x is small or follows a Gaussian

distributioneix = e−x2/2. The intensity then becomes

I = Ie  n  n fne−Mnfn∗e−MneiΔk·(rn−rn)+ + Ie  n  n fne−Mnfn∗e−MneiΔk·(rn−rn)(ek 2_u nun− 1) _(2.3)

where Mn = k2u2n/2. The second term gives rise to thermal diffuse scattering

which will be discussed later. The first term can be included in equation 2.2 by introducing

FT =



n

fne−Mne2πi(hxn+kyn+lzn)

If there is only one kind of atom

|FT|2=|F |2e−2M

where e−2M is known as the Debye temperature factor.

2.4

Effects of phonons

An acoustic phonon is an elastic wave in the crystal (see chapter 3 for a more detailed description). A phonon with amplitude agj, polarization ej, wave vector g, frequency ωgj and phase δgj will displace atom n by



2.4. EFFECTS OF PHONONS

All phonons together will then displace atom n by

 un=



gj

agjegjcos(ωgjt− g · rn− δgj)

By inserting this into equation 2.3 and expanding the exponential, the second term becomes (assuming there is only one kind of atom)

|f|2_e−2M_I e  n  n eiΔk·(rn−rn) gj Ggjcos(g· (rm− rn))+ |f|2_e−2M_Ie n  n eiΔk·(rn−rn)  gjGgjcos(g· (rm− rn)) 2 2 + . . . where Ggj= 1 2(Δk· egj) 2_{< a}2 gj >

These terms describe the first and second order thermal diffuse scattering (TDS). Equation 2.3 can be abbreviated as

I(Δk) =|f|2e−2M(I₀(Δk) + I₁(Δk) + I₂(Δk) + ...) where I₀(Δk) = n  n eiΔk·(rn−rn) and I₁(Δk) =|f|2e−2M n  n  gj eiΔk·(rn−rn)_Ggjcos(g· (r_m− r_n))

By writing the cosine as exponentials this becomes 1 2|f| 2_e−2M gj Ggj  n  n ei(Δk+g)·(rn−rn)_{+ e}−i(Δk−g)·(rn−rn)

By comparing with the expressions above, this can be written

I₁(Δk) = 1 2|f| 2_e−2M gj Ggj  I₀(Δk + g) + I₀(Δk− g) 

I₀has sharp maximum where the argument is close to a reciprocal lattice vector. As a result, when a phonon is present in the X-ray diffraction process, the Laue equation will be modified to

2.5

Absorption and extinction

There are two processes determining how deep into a crystal the X-ray beam penetrates. The first process is absorption. Since the X-rays are absorbed by the inner shell electrons, the chemical bonds to other atoms can be neglected and the absorption coefficient for an arbitrary chemical compound can be calculated as

μ = n

ρnμmn

where the sum is over the different elements in the sample, and ρn the densities

of the elements. The mass absorption coefficients μmncan be found in tables.

Extinction is important if the reflectivity of the crystal is high, and means that the incident intensity is weakened by scattering into the diffracted beam. In the wings of the rocking curve, extinction is negligible. The extinction depth can be calculated using dynamic X-ray diffraction theory.

2.6

Asymmetrically cut crystals

Cubic crystals are mostly cut so that the surface of the crystal is parallel to the 111 or 100 lattice planes. They are then said to be symmetrically cut. They are said to be asymmetrically cut if, for example, they are cut at an angle to the 111 planes. The asymmetry angle will here be referred to as φ. When an X-ray beam is reflected by an asymmetrically cut crystal, the size of the beam

100 111

Figure 2.2: A simple cubic crystal. The bottom plane shows the 100-planes, the top plane shows the 111-planes and the plane in the middle shows how a crystal with a small asymmetry angle is often cut.

2.7. ROCKING CURVES

will change. The asymmetry parameter is given by

b = sin(θB+ φ)

sin(θB− φ)

If the size of the beam in the reflection plane is x before the reflection, it will be

x× b after the reflection.

If the incidence angle is smaller than the exit angle (φ is positive, shallow inci-dence), b will be smaller than one. The beam will therefore become bigger. In the other case, i.e. steep incidence, the beam will become smaller.

2.7

Rocking curves

A rocking curve of a crystal is the X-ray reflectivity of the crystal as a function of angle, given that the incident X-ray beam is monochromatic and non-divergent. Rocking curves can be calculated using the dynamic X-ray diffraction theory. An example of a rocking curve is shown in figure 2.3a.

If an X-ray beam is reflected from two identical and parallel crystals (as common in monochromators), the resulting rocking curve will be

R(θ) = _∞

−∞ R(φ)R(θ− φ)dφ ∞

−∞R(φ)dφ

The reflectivity of a single and double reflection in silicon is shown in figure 2.3b.

−200 0 200 400 600 0 0.5 1 Δθ / μrad R (a) −200 0 200 400 600 0 0.5 1 Δθ / μrad R (b)

Figure 2.3: (a) Rocking curves for InSb at 3600 eV (solid line) and 4600 eV (dashed line). (b) Rocking curves for a single (solid line) and double (dashed line) reflection in Si(111) at 4000 eV.

2.8

Probing depth

When designing an experiment it is important to match the depth of the X-ray probe to the absorption depth of the laser triggering the effect to be studied. To observe coherent acoustic phonons, a large penetration depth is often desired, whereas studies of phase transitions, such as melting, often require a small prob-ing depth. The probprob-ing depth of the X-rays in an asymmetrically cut crystal is given by

d = l

2

cos2φ− cos2θ

cos φ sin θ

where l is the absorption/extinction length, θ the Bragg angle and φ the asymme-try angle. At the peak of the rocking curve of a strong reflection, the extinction is often the dominant factor. For a weak reflection, and in the wings of a rocking curve, absorption is the dominant factor.

2.9

Total external reflection

The refractive index of a material depends on the absorption. For photon ener-gies below an absorption edge the refractive index is higher than one, and above the absorption edge it is below one. X-ray photons have a high energy and will experience a refractive index below one. This means that total reflection may occur at the surface in some cases. This is called total external reflection (TER) since the X-rays are outside the sample. Since the photon energy is far above the edge, the refractive index will be close to one. As a result, very small glancing angles are required for TER (see figure 2.4).

0 0.2 0.4 0.6 0.8

10−3 10−2 10−1 100

incidence angle / degrees

reflectivity

Figure 2.4: Reflectivity for specular reflection in InSb. Data from Sergey Stepanov’s X-ray Server12.

Chapter 3

Coherent acoustic phonons

3.1

Introduction

Impulsive strain pulses can be generated in crystals by using a femtosecond laser pulse. The strain pulses can be described as a set of coherent acoustic phonons, and they have been studied extensively using optical techniques13–15. By using a second femtosecond light pulse, the set of coherent acoustic phonons can be modified. This has also been studied using optical techniques16,17.

For an opaque crystal, optical techniques are limited to the study of phonons in the surface layer. X-rays provide a possibility to study phonons in the bulk of the crystal. In a number of studies, coherent acoustic phonons in semiconduc-tors have been probed using X-ray diffraction1,4,5,18,19. Femtosecond temporal resolution has made the study of optical phonons20 possible.

3.2

Lattice vibrations in crystalline matter

The interatomic forces for small atomic displacements in crystals are close to linear, just like the force from a spring. If a disturbance moves an atom out of its equilibrium position the atom will start vibrating. The energy in the vibration will be transferred to neighboring atoms. This can be described by a traveling wave and is referred to as a phonon. Due to the lattice structure of the crystal, there are limitations to the possible wave vectors for the phonons. For a one-dimensional chain of atoms, it is easy to realize that a phonon with a wavelength shorter than twice the distance between the atoms will be equivalent to a longer wavelength phonon, according to the Nyquist theorem (see figure 3.1). Hence, the wavenumber is limited to π/a, where a is the interatomic distance. The reciprocal lattice is defined such that the reciprocal lattice distance is b = 2π/a, and the wavenumber is limited to b/2.

Figure 3.1: Waves in a one-dimensional lattice, with a wavelength of 3a (solid line) and 1.5a (dashed line). They give rise to the same atomic displacements (dots).

For a simple cubic lattice, the same limitation will hold for the magnitude of all three components of the wave vector. If the wave vector originates in the origin (000), it will end in a volume around the origin enclosed by the planes that bisect the vectors from 000 to the 001, 010 and 100 reciprocal lattice points (see figure 3.2). This is referred to as the first Brillouin zone. This zone is defined in similar ways for other lattices.

The phonon dispersion for a one-dimensional chain of atoms is given by21

ω =4C/M| sin(ka/2)| (3.1)

where ω is the angular frequency of the phonon, C the restoring force constant,

M the atomic mass, k the wave vector of the phonon and a the distance

be-tween the atoms (see figure 3.3). The dispersion will show a similar behavior for phonons in a three-dimensional lattice for any given direction. For long wave-lengths, i.e. small k, ω is approximately proportional to k, which means that the speed of sound is independent of the wavelength.

100 010 001 111 110 101 000 011

Figure 3.2: The reciprocal lattice for a simple cubic crystal, with some reciprocal lattice points marked. The gray volume shows the first Brillouin zone.

3.3. DETECTION OF COHERENT ACOUSTIC PHONONS −1 −0.5 0 0.5 1 0 0.5 1 ω (4C/M) −1/2 ka/π

Figure 3.3: The relationship between frequency and wave vector of the phonons in a one-dimensional chain of atoms. The relationship is similar for phonons in the three-dimensional case, but π/a is replaced by the size of the first Brillouin zone.

3.3

Detection of coherent acoustic phonons

As seen earlier (equation 2.4), phonons modify the Laue condition so that the change in wave vector is equal to a reciprocal lattice vector, plus or minus the wave vector of the phonon. The phonons are therefore detected at the wings of the rocking curve. For a given distance from the peak, the frequency of the phonons is5

ω = vΔθ | GH|

tan θBcos φ + sin φ

where v is the speed of sound in the sample and φ the asymmetry angle. Instead of changing the angle of the sample (and having to change the position of the detector), the energy of the X-rays can be changed. By differentiating the Bragg equation one can easily see that these are equivalent unless E is close to an absorption edge.

ΔE

E =

Δθ tan θB

The wavelength of the phonons typically studied by time-resolved X-ray diffrac-tion is about 80 to 400 nm (corresponding to a period of 20-100 ps). The probing depth of the X-rays in the crystal is typically less than a few micrometers. As the wavelength and probe depth are comparable, the X-ray probe will only see a small number of wavelengths of the phonons. Depending on the phase of the phonon, the probe will either see more compression or more expansion (or an equal amount of each).

If the incidence angle of the X-rays is smaller than the Bragg angle, or the energy is lower, the lattice spacing would have to increase to fulfill the Bragg equation. Hence, this is called the expansion side. Similarly, if the incidence angle or energy is larger, a decrease of the lattice spacing is required, and this side is called the compression side.

Now, assume that the experiment is set up so that the expansion side is probed. At some point in time, the probe sees more expansion than compression, which leads to an increase in signal. The phonons propagate into the sample, and after a while the probe will see more compression than expansion, thus the signal will decrease. Later, the signal increases again due to the propagation of the phonons. Therefore, the signal will oscillate with the frequency of the phonons.

3.4

Generation of coherent acoustic phonons

When a laser pulse is absorbed by a semiconductor, the surface layer is rapidly heated. It will expand very quickly, and as a reaction to this, the atomic layers just below the surface are compressed. This creates an impulsive strain wave that travels into the bulk of the crystal.

The profile of the strain wave traveling into the crystal has been derived by Thomsen et al.13, and is given by

η₃₃(z, t) = (1− R) Qβ AζC 1 + ν 1− ν ×  e−z/ζ−1 2  e−(z+vt)/ζ+ e−|z−vt|/ζsgn(z− vt)  (3.2) where R is the reflectivity of the laser at the surface, Q is the energy in the laser pulse, β is the thermal expansion coefficient of the sample, A is the area of the laser spot, ζ is the absorption length of the laser, C is the specific heat of the sample and ν is the Poisson ratio. The first term on the second line is the static thermal strain, and the second and third terms are waves traveling out from and into the crystal. This model neglects thermal diffusion, carrier diffusion, electron-phonon coupling and electronic strain. Yet it provides good quantitative agreement in many cases. A refined model has been presented by DeCamp et al.22. They use a numerical solution of the carrier diffusion and heat conduction, where the energy of the electrons is turned into heat by Auger recombination. They then solve the equations of elasticity numerically, assuming a deformation given by the temperature and carrier concentration.

In some cases in which Bragg geometry is used, the heat diffusion is the only effect that needs to be taken into account. The heat diffusion equation is solved numerically. The equation is

∂T ∂t = K ∂2T ∂z2 + 1 Cq(t, z) T(t, 0) = 0 T(t, d) = 0 T (0, z) = T₀

3.5. PHONONS PROBED BY LASER PRODUCED X-RAYS

where K is the heat diffusion constant, q is the absorbed energy density and

d is the thickness of the sample. If the thickness of the sample prevents the

heat from reaching the other end of the sample on the timescale for which the equation is solved, a smaller thickness can be used as long as the heat does not reach this end. The solutions of this equation can then be put into the equation for the Thomsen strain (see equation 3.2)

η₃₃(z, t) = β C 1 + ν 1− ν  T (z, t)− (1 − R) × Q 2Aζ  e−(z+vt)/ζ+ e−|z−vt|/ζsgn(z− vt)  (3.3)

3.5

Phonons probed by laser produced X-rays

A laser plasma source is comparable to a synchrotron light source concerning the peak number of hard X-ray photons that are produced. The X-ray pulse and laser pulse are always synchronized and the pulse duration is a few hundreds of femtoseconds, which makes possible studies of ultrafast processes without using fast detectors. The main disadvantage is the high divergence of the X-ray burst (2π, extending over the half space that is not covered by the target). Other disadvantages are that even small variations in power, divergence and direction of the laser change the X-ray yield considerably. To overcome this problem, one often allows the laser to cover only half the X-ray spot on the target, and record both the time-resolved diffracted signal and a reference diffracted signal simultaneously.

If the source is relatively close to the sample and an imaging detector is used, it is possible to record the reflectivity at a range of angles of incidence simultaneously. Neutze et al. suggested that it would also be possible to record all time steps simultaneously by letting the laser sweep across the sample and excite different points at different times23. Since all time steps are recorded within the same X-ray burst, the fluctuation in the X-ray intensity is the same for all time steps. The proposed method was used to follow the strain generated by a femtosec-ond laser pulse in InSb. For more details see Paper I. In order to increase the probed time window, a grating was used to tilt the laser wave front. The laser was reflected on a grating, and the m = −1 reflection was used. A plane im-mediately after the grating was imaged onto the sample using a cylindrical lens. By changing the timing between the X-rays and the laser, the temporal effects were shifted along the time-axis in the image. Agreements with simulations were good. The data were also in good agreement with previous measurements1. Several studies of coherent acoustic phonons in semiconductors have been pub-lished1,I. These studies have in common that while they have shown clear proof of a fast evolving strain, the oscillations in the reflectivity caused by the phonons have not been seen. These studies use the natural bandwidth of the Kα lines in

detector

sample

x-ray source

laser

strain

Figure 3.4: The principle of X-ray topography. The laser hits the sample from above, and creates a temporal trace from top to bottom of the sample.

the source. If the bandwidth of the X-rays is too large, the X-rays will probe not only a single phonon mode but a range of phonons. The natural bandwidth of the Kα lines is so large that it smears out the oscillations sufficiently to make it very difficult to detect them (see figure 3.5).

The use of a toroidally bent crystal to focus and monochromatize the X-rays re-duces the bandwidth sufficiently to reveal the phonon oscillations. This was done in an experiment performed at the Lund Laser Centre. A schematic drawing of the setup is shown in figure 3.6. In this experiment, the X-rays were produced by focusing a high power laser onto a rotating titanium disc in a vacuum chamber. The X-rays escaped from the vacuum chamber through a beryllium window.

(a) Δθ / mrad t / ps −2 0 2 0 200 400 600 (b) Δθ / mrad t / ps −2 0 2 0 200 400 600

Figure 3.5: Reflectivity as a function of angle and time for a perfectly mono-chromatic source (a) and for a Ti Kα source (b).

3.5. PHONONS PROBED BY LASER PRODUCED X-RAYS polarizing beamsplitter x-ray source focusing crystal sample CCD camera

Figure 3.6: The setup used to detect coherent acoustic phonons in InSb.

They were then focused onto the sample by a toroidally bent germanium crystal using the 400 reflection in a Rowland circle geometry.

The results of the measurements are shown in figure 3.7. The periodic nature of the oscillation is revealed, as the X-ray intensity can be followed over a time corresponding to 1.5 periods. 0 100 200 300 400 500 0 1 2 3 4 t / ps I / I 0

Figure 3.7: Oscillations in the X-ray reflectivity caused by coherent acoustic phonons, probed using X-rays from a laser-based source.

3.6

Coherent phonons in heterostructures

A multilayer is a stack of repeated layers of two or more different materials. If the lattice is ordered over a long range reaching over several layers, the mul-tilayer is said to be a heterostructure. If the layers are so thin that the wave functions of the electrons can reach over several layers, the structure is said to be a superlattice. Semiconductor heterostructures have many important appli-cations, such as for diode lasers. They also have interesting phonon properties. A new periodicity is introduced by the repeated layers. The typical length of one set of layers is of the order of 10-100 nm, and the periodicity of the lattice is usually less than 1 nm. This will reduce the size of the first Brillouin zone considerably. This results in a phenomenon called branch folding, which means that the phonon branch from the original Brillouin zone is folded back into the new Brillouin zone (see figure 3.8), with only small modifications close to the zone boundaries. This creates new phonon branches with higher frequencies. Such folded acoustic phonons have been studied using optical techniques24–28. In a heterostructure there is a new periodicity of length dhs and an associated

reciprocal vector ghs. As in the case of phonons, the new periodicity will change

the Laue condition and introduce side-bands to the main rocking curve peak of each material according to

Δk = Ghkl± ghs

assuming that hkl are chosen along the direction in which the heterostructure is grown.

Bargheer et al. presented X-ray measurements of standing waves (k = 0) in a superlattice29. The use of X-ray diffraction revealed details about the nature of the excitation process which were hidden to optical measurements. In Paper III

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ω (4C/M) −1/2 ka/π

Figure 3.8: As a new periodicity is introduced into the sample, the size of the first Brillouin zone is reduced and the curve for the dispersion relation (dashed line) is folded back (solid line) into the new, reduced Brillouin zone.

3.7. COHERENT CONTROL OF ACOUSTIC PHONONS

the X-ray reflectivity is measured for several X-ray energies, thereby probing different wave vectors. The measurements showed that phonons are generated not only in the lowest acoustic phonon branch, but also in the upper branches generated in the folding process. The ability to probe heterostructures with phonons and X-rays can reveal information about the layers buried more deeply in the heterostructure.

3.7

Coherent control of acoustic phonons

The ability to control matter on the atomic scale is very intriguing. This topic has been addressed in a few studies in which the phonons have been probed either by visible light16,17 or X-rays30. Common to these studies is that they all use two laser pulses, one to create phonons and one to control them. This provides a good way to cancel certain phonon modes and enhance others, but the use of more pulses can increase the amplitude of the phonon modes further. In Paper VII, we have utilized four pulses to prove that the amplitude of certain phonon modes can be enhanced considerably.

The laser pulse train was generated by sending the laser through four Michelson interferometer-like stages (see figure 3.9). Each stage consisted of a polarizing beam splitter, splitting the incoming laser pulse(s) into two parts of different polarization. Each pulse was sent to an arm that consisted of a quarter wave plate and a back reflecting mirror. As the pulse passed through the quarter wave plate twice, the polarization was turned 90 degrees. The p-polarized beam that was initially reflected was turned into an s-polarized beam which was therefore transmitted by the polarizer, and vice versa. Due to this, the loss of 50% of the light that normally occurs in Michelson interferometers was avoided. Before entering the next stage, the polarization was turned by a half wave plate so that both pulses consisted of equal amounts of p- and s-polarized light. After the last stage, each of the pulses was split into two beams by a polarizer. The p-polarized beam was tripled and focussed onto the photo cathode of the streak camera, providing a time reference of each laser pulse. The s-polarized beam was focussed onto the sample using a two-lens telescope, providing a focus large enough to guarantee that the laser excitation is homogeneous across the probed area.

The ability to cancel phonon modes has also provided the possibility to study the temporal evolution of the surface strain, unobstructed by the effect of the phonons. This has given new exciting insights into the carrier diffusion, thermal diffusion and electronic deformation potential on short time- and length scales by comparing the experimental data with computer simulations. This has shown that the effects of the electron-hole plasma created by the laser have a large influence in this kind of measurements.

λ/2 λ/2

λ/4

λ/4

Figure 3.9: One of the four stages used to create the laser pulse train.

3.8

Free-carrier effects in phonon generation

Generation of phonons in semiconductors was described in section 3.4. In this section, the heating of the semiconductor was assumed to be instantaneous. Sev-eral experiments have shown that the absorptions process is more complex4,22. The laser pulse is absorbed by the electrons in the sample. Immediately after the pulse is absorbed, the electrons will have a non-thermal distribution defined by the laser spectrum. The electrons will rapidly (in a few tens of femtosec-onds) thermalize via carrier-carrier scattering. This will leave a hot electron-hole plasma in the sample, which will transfer its excess energy to the lattice by emitting optical phonons, a process which takes place within a picosecond. The phonons emitted will correspond to an increased temperature of the lat-tice. The dense electron-hole plasma, which is left after this process, will start diffusing into the sample. The electron-hole pairs will eventually recombine via Auger recombination. The Auger electron which receives the excess energy in this process loses its excess energy via optical phonon emission and thus heats the lattice.

Chapter 4

Phase transitions

4.1

Introduction

Well-known phase transitions are the phenomena of melting in which a solid turns into a liquid and boiling in which a liquid turns into a gas. However, many materials show a more complex behavior. In the solid phase, the material can be either crystalline (meaning that there is a long-range order) or amor-phous (meaning that there is only a short-range order), or have several different crystalline phases.

Phase transitions can be categorized in various ways, for instance, as first and second order phase transitions, where a first order phase transition involves latent heat. Transitions between solid, liquid and gaseous phases are all first order phase transitions. A second order phase transition does not involve latent heat. An example of this are some transitions from one crystalline structure to another in a solid. This is called a structural phase transition, and can be divided into displacive and order-disorder phase transitions (see section 4.2). Another way to categorize phase transitions is to make a distinction between thermal and non-thermal transitions. When a sample is slowly heated it under-goes a thermal phase transition. If a femtosecond laser hits a sample it excites a non-thermal distribution of electrons and phonons. This may lead to a phase transition before the sample has time to thermalize. In this case, the transi-tion is said to be non-thermal. Such phase transitransi-tions are usually faster and require less energy than thermal phase transitions. The sample will thermalize after the transition, and may then undergo a thermal phase transition. Different non-thermal phase transitions will be discussed later in this chapter.

−1 −0.5 0 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0 k BT position / a.u. energy / a.u.

Figure 4.1: A double well potential of a ferroelectric crystal showing an order-disorder-type phase transition. The straight line indicates the energy level of the ion.

4.2

Structural phase transitions

One kind of crystals that has a structural phase transition are the ferroelectric crystals. In such a crystals, the positive and negative charge in the unit cell have different centers of mass, implying that the crystal will have a permanent dipole moment. Most of these crystal are ferroelectric only below a certain temperature. As the temperature is increased above this value, the structure of the crystal changes such that the centers of mass of positive and negative charge coincide. This temperature is called the Curie temperature and is denoted Tc.

The phase transition usually does not involve any latent heat and is therefore a second order phase transition.

The two types of structural phase transitions, displacive, and order-disorder, are defined by the behavior of the lowest frequency (soft) optical phonons in the material. If a soft mode can propagate in the material at the phase transition, the phase transition is displacive. If the soft mode is non-propagating, the phase transition is of the order-disorder type. Many materials exhibit a phase transition that is a mixed order-disorder and displacive transition.

Displacive phase transitions can be explained by small anharmonicities of the inter-atomic potentials. Below the Curie temperature, one lattice structure is en-ergetically favored, while above this temperature another structure is preferred. At the transition temperature, the ions in the sample can move between the two positions with only a very small restoring force, i.e. the crystal is “soft” with respect to this motion. A small restoring force leads to a very low oscillation frequency, and the optical phonon mode describing this motion is softened. In order-disorder type phase transitions, the ions involved in the transition are located in a multiple well structure. Above the Curie temperature, the ions

4.3. THE FERROELECTRIC PHASE TRANSITION IN KDP

have enough energy to move across the entire structure, while below the Curie temperature, the ions will be trapped in a subset of wells, which leads to a displacement.

Laser-induced structural phase transitions have been studied using both optical and X-ray methods10,31–33.

4.3

The ferroelectric phase transition in KDP

In KDP (Potassium Dihydrogen Phosphate, KH₂PO₄), there are four KH₂PO₄ groups in each unit cell of the high temperature paraelectric phase. These are connected via hydrogen bonds between the phosphate groups. The proton is situated in a double-well potential. Below the Curie temperature, the proton will be trapped in one of the two wells. As the proton is fixed at one end of the bond, the other ions in the unit cell will be displaced, and thus the transition couples to an optical phonon mode. There is also a change in lattice as it goes from tetragonal to orthorhombic symmetry, and this couples the phase transition to an acoustic phonon mode.

One interesting feature of the phase transition is that when the hydrogen is re-placed by deuterium, the Curie temperature is increased from 123 K to 213 K. While replacing the hydrogen with deuterium almost doubles the Curie temper-ature, a change of isotope in the other atoms does not change the Curie tem-perature markedly34. This is believed to be an effect of the mass dependence of the de Broglie wavelength. The heavier deuterium has a shorter de Broglie wavelength, and as a result more energy is needed to tunnel between the two wells21. K K K K K K K K P P P P P P P P (a) (b)

Figure 4.2: The unit cell for KDP in the (a) ferroelectric phase and the (b) paraelectric phase.

001 100 010 110 110 paraelectic ferroelectric ferroelectric

Figure 4.3: The lattice of KDP. The domain walls are running either in the [110] or [1¯10] direction.

In the paraelectric phase, the crystal has a tetragonal unit cell (all angles are right angles, the a- and b-axis are equal, but the c-axis is different). In the ferroelectric phase, the unit cell becomes orthorhombic (the a- and b-axis are longer and shorter, respectively, while the c-axis retains its length).

The polarization of a unit cell affects the neighboring cells in such a way that the parallel alignment of the polarization vector is energetically favorable, leading to domains of equal polarization. Since there are many nucleation sites, domains of different polarization will be created. The walls between these domains will always run in either the [110] or the [1¯10] direction35. Depending on the ori-entation of the domain walls, the orthorhombic lattice is slightly rotated in a positive or a negative direction about the c-axis35 (see figure 4.3).

In Paper II we present two measurements on KDP, in which we probe the (233) and (323) reflections of the two polarizations, respectively. Due to the small change in the a- and b-axis of the crystal, one of the peaks will get a smaller and the other a larger Bragg angle compared to the Bragg angle of the reflection in the paraelectric phase.

In the first measurement, the reflectivity in the paraelectric phase was recorded during laser excitation, and it was shown to increase by about a factor of two, and then decay back again in a few nanoseconds. This indicates that the paraelectric phase exists for a short time. The most likely cause would be heating by the laser, but the absorbed energy is not sufficient. We believe that the optical phonon mode involved in the transition is excited via impulsive Raman scattering. In the second measurement, the reflectivity in the two peaks that correspond to different polarizations was measured over a long time of laser irradiation. One of the peaks grows at the expense of the other peak. This indicates that domains are reoriented by the laser pulses. Fahy et al. suggested that this would be possible36.

4.4. REGROWTH OF INSB

4.4

Regrowth of InSb

After the initial non-thermal melting, the electron and the lattice will thermal-ize, keeping the sample in the liquid phase. The sample will cool by thermal conduction, and will eventually resolidify. During this process, it will regrow from the bottom to the top. The speed of the boundary between the solid and liquid phase will determine whether the ions will have time to rearrange to the crystalline phase, or whether they will freeze in the amorphous phase.

The first study made at beamline D611 at MAX-lab was to measure the resolid-ification speed of InSb following non-thermal melting. In this study, the molten layer was assumed to be an equilibrium liquid which does not diffract the X-rays, but rather works as an attenuating layer on the surface of the diffracting crystal.

In Paper V, the results of measurements and simulations of regrowth of InSb are reported. In this case, the standard thermal diffusion equation gave good agree-ment with the experiagree-mental data (see figure 4.4), showing that carrier effects are not present on the nanosecond time scale. In the paper, contributions to the recorded signal, other than melting, are discussed. The Debye-Waller factor would only reduce the signal by 5-10%, much less than the recorded reduction. Other effects that contribute to the reduction in signal are for example amor-phization and multi-shot damage. These effects can be separated from the signal. In conclusion, this paper shows that it is possible to conduct these experiments in the multi-shot regime if laser power is kept sufficiently low.

Figure 4.4: Measured (dots) and simulated (line) melting depth as a function of time for InSb.

4.5

Non-thermal melting of InSb

If a femtosecond laser pulse is used to heat a sample, it will still take in the order of a few picoseconds for the phase transition to take place, since the energy is initially absorbed by the electrons and must be transferred to the lattice. Optical studies have been made, which shows that the susceptibility changes from that of the solid phase to that of the liquid phase on a much faster time scale37–45. This has been explained by the change in the energy surface on which the ions move, as a large number of binding electrons in the valance band are excited to the non-binding conduction band46. Some studies have also been reported in which time-resolved X-ray diffraction has shown a rapid disordering of the sample under intense laser radiation, which supports the statement that the samples are going into the liquid phase2,9,47–49.

In Paper IV, we demonstrate support for the theory of bond-breaking due to massive electronic excitation. The measurements for this paper were conducted at the Sub Picosecond Pulse Source (SPPS), a novel, short-pulse, X-ray source. To overcome the jitter between the exciting laser and the probing X-rays, the experiment was performed in a single shot in a crossed beam geometry (see chapter 3.5).

To match the probing depth and the melting depth, an incidence angle of 0.4 degrees was used. To find this angle accurately, we used a camera that could see the direct beam and the beam reflected from the crystal surface through total external reflection (TER, see section 2.9) simultaneously.

Data were recorded for two X-ray reflections, (111) and (220). The measured X-ray intensity fell more quickly for the (220)-reflection than for the (111)-reflection. This is explained by the fact that the (220)-reflection is more sensitive to atomic displacements since it is of higher order. The RMS atomic displace-ment calculated from the Debye model (see section 2.3) was the same for both reflections, and the slope of the curve, corresponding to the speed of the atoms, matched that of the speed of the atoms in InSb at room temperature.

Further analysis of the data was presented in Paper VI. This analysis showed that initially the atoms move from their initial positions without colliding, but after about 500 fs the atoms will start to collide. Even in this case there is some anisotropy in the system, indicating that the electron hole plasma modifies the potential in a non-isotropic way.

4.5. NON-THERMAL MELTING OF INSB -250 0 250 500 1.2 0.9 0.6 0.3 0.0 Time (fs) ] s m or t s g n A[ t n e m e c al p si d s mr ni e g n a h C 111 220

Figure 4.5: RMS displacements of the atoms in InSb extracted from the dif-fracted intensity using the Debye-Waller model.

t (fs) ( t ne me ca l ps i d s mr ) Α 0 500 1000 1500 2000 2500 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 _(a) -500 0 500 1000 1500 0.2 0.4 0.6 0.8 1.0 1.2 (b) t (fs) ( t ne me ca l ps i d s mr ) Α (111) (220)

Figure 4.6: RMS displacements of the atoms in InSb for a laser fluence of (a) 130 mJ/cm2 and (b) 50 mJ/cm2.

Chapter 5

Beamline D611

5.1

Introduction

D611 is a beamline at the MAX-II storage ring, located at MAX-lab in Lund, Sweden. The beamline is dedicated to time-resolved x-ray studies. In this chap-ter an overview of the beamline is given as well as detailed descriptions of some important parts of the beamline.

In the experiments a laser is used to start a process in the sample and the x-rays are used to probe the process. In order to do this, the laser light and the x-rays must not only arrive at the same point in space but also at the same time. A schematic view of the beamline is shown in figure 5.1. The x-rays that are used as a probe come from a bending magnet at an electron storage ring, and the laser light comes from a chirped pulse amplification, titanium sapphire laser. The laser is phase-locked to the electron storage ring radio frequency. The most commonly used detectors are avalanche photo diodes and streak cameras.

5.2

X-ray source and optics

The MAX-II electron storage ring is a third generation synchrotron light source, running at 1.5 GeV. The size is quite moderate, with a circumference of only 90 m. There are ten straight sections, of which one is used for the injection of electrons and one is used for the accelerating cavity. On the remaining eight straight sections there are three undulators and three multi-pole wigglers (of which two are superconducting). A new undulator will soon be installed, leav-ing only one straight section free. Some of the insertion devices have multiple beamlines, and in addition to this there are also three bending magnet beamlines. The total number of beamlines is 14 (not all are in operation yet). Important characteristics of MAX II is given below.

synchronization MAX-II Ti:sapphire laser streak camera 3-8 keV 400 ps 1000 ph/pulse 100 MHz 800 nm 35 fs ~1 mJ/pulse 4-8 kHz

Figure 5.1: Overview of the beamline.

Current 200 mA Life-time 24 h RF frequency 100 MHz Circumference 90 m Number of buckets 30 Bucket spacing 10 ns Bunch length 400 ps Magnetic field in bending magnets 1.5 T

D611 is located at the sixth bend and is a bending magnet beamline. Due to the relatively moderate size of the storage ring, the bending radius of the electrons in the bending magnet is relatively short, and the critical energy is comparable to other larger electron storage rings. The number of photons per second is shown in figure 5.3a.

A schematic layout of the beamline is shown in figure 5.2. A number of valves and windows protect the vacuum in the beamline and the ring. The total thickness of the beryllium windows is 45 μm.

The X-rays are focused by a toroidal mirror, which is made out of gold-coated, single-crystal silicon, polished to a cylindrical shape in one dimension and bent in the other. The bending radius can be controlled by a stepping motor. Char-acteristics of the mirror are presented below. The reflectivity of the mirror is shown in figure 5.3b.

Substrate material silicon Coating material gold Coating thickness 50 nm Top surface roughness 0.5 nm Horizontal bending radius 44.8 mm Vertical bending radius 400-600 m Incidence angle 0.5 deg

5.2. X-RAY SOURCE AND OPTICS

mirror _{monochromator} sample chamber

optical table MAX II streak camera x-rays laser 3ω Be window Be window D611 x-ray hutch chopper mask diodes streak camera Be window (removable)

Figure 5.2: The layout of the first part of the beamline.

The beam is monochromatized using a non-dispersive, double-crystal monochro-mator. The monochromator is equipped with both silicon crystals and multilayer mirrors. To change between them takes only a minute and makes it possible to chose the bandwidth to fit the experiment. The throughput of the monochro-mator is shown in figure 5.3c.

Since the repetition rate of the electron storage ring is 100 MHz and the laser runs at 4-8 kHz, there are a lot of X-ray pulses that are not useful. These pulses may be a problem since the average power may be too high for the detector. In order to reduce the average X-ray power without decreasing the peak power, a chopper is used. The chopper blade has a diameter of 10 cm and has 12 slits with a width of 500 μm. The slit can be this small since the chopper is mounted very close to the sample where the X-ray beam is small. The average power is decreased by a factor of 50 by the chopper. The chopper and the laser are synchronized, see chapter 5.4 below.

To get a more well defined X-ray profile, a mask is available. The mask has a set of rectangular holes with sizes ranging from 0.20×1.00 mm2to 0.55×2.75 mm2. The mask can be moved around using two stepper motors.

For diagnostics, a set of diodes is available. There are two big Si PIN diodes to measure the flux in the X-ray beam. One of them is covered with three different foils (Ti, Cu, Cl) mounted so that the beam can go through none, one, two or all three of them. There are also two small diodes with high time-resolution

0 2 4 6 8 10 0 2 4 6 8x 10 12 E / keV photons/second/mrad 2/0.1% BW

(a) bending magnet

0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 E / keV transmission / reflectivity

(b) Be windows and mirror

0 2 4 6 8 100 0.5 1 1.5 2 2.5 x 10−4 relative bandwidth 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 E / keV throughput (c) monochromator 0 2 4 6 8 10 0 2000 4000 6000 E / keV photons/pulse on sample (d) sample

Figure 5.3: (a) Output from the bending magnet. (b) The transmission of the beryllium windows (dashed line) and the reflectivity of the mirror (solid line). (c) The throughput of the monochromator (solid line) and the relative bandwidth of the monochromator (dashed line, right-hand scale). (d) Number of photons per pulse on the sample.

that can be used to check the timing between the laser and X-ray beam (see chapter 5.4 below).

The sample is mounted on a rotation stage so that the incidence angle can be adjusted. There are also two translation stages to move the crystal along its surface.

To calculate the number of photons per pulse that is accepted by the sample some factors have to be taken into account. First, the divergence is defined to 4.00*0.34 mrad2 by a beryllium window. Secondly, the repetition rate of the synchrotron light source is 100 MHz. Also, the throughput of the different optical elements have to be taken into account. The total number of photons per pulse is given by

Φsample(E) = ΦBM(E)× 4.00 · 0.34 _{ }

divergence

× 1 · 10−8

  rep. rate

×TBe(E)×

×Rmirror(E)× Tmono(E)× BWmono(E)/0.1%

5.3. LASER SYSTEM

the number of photons/second/mrad2/0.1% BW from the bending magnet (see figure 5.3d).

5.3

Laser system

When choosing a laser for integration at a synchrotron light source, several aspects must be taken into account. To cover the X-ray spot on the sample, the laser spot has to be 0.5×0.9 mm2. To get a laser fluence of 50 mJ/cm2, a laser pulse energy of 0.25 mJ is sufficient. However, the laser is also used to trigger the streak camera and to work as a time reference for the streak camera, and although small, there are always losses on mirrors in the beam transport system. Thus, a pulse energy of up to 1 mJ may be needed.

As mentioned in the previous section, the synchrotron light source runs at 100 MHz. There is no laser that is capable of giving 1 mJ pulse energy at this repetition rate. The sample also has to recover between the excitations, a process that almost always takes more than 10 ns. At very high repetition rates, the average laser power absorbed by the sample may cause thermal melting of the sample.

Laser systems which has an average output power higher than 5 watts are gen-erally relatively large. For D611 we found that a laser for which the repetition rate is adjustable between 4 kHz and 8 kHz, and with an average power of 6 W, was the best choice.

The oscillator at D611 is manufactured by KM-labs. It is pumped by a Coherent Verdi CW laser. The repetition rate is 100 MHz and the average output power is about 450 mW, hence the energy per pulse is 4.5 nJ. The bandwidth of the oscillator is 40 nm and the pulse duration is 20 fs.

The amplifier is also manufactured by KM-labs. It is pumped by a Coherent Corona pulsed laser. The amplifier uses the CPA technique which means that

cryogenically cooled high avarage power amplifier oscillator pump to X-ray hutch diagnostics delay lines