Experimental Setup of High Harmonic Generation Based Angle ...

Master of Science ThesisStockholm, Sweden 2013

TRITA-ICT-EX-2013:277

M D S A B B I R A H S A N

Experimental Setup of High HarmonicGeneration Based Angle Resolved

Photoemission Spectroscopy (HHG-ARPES) and Test Measurement on

Tungsten (W) [110] Surface

K T H I n f o r m a t i o n a n d

C o m m u n i c a t i o n T e c h n o l o g y

Experimental Setup of High Harmonic Generation Based Angle Resolved

Photoemission Spectroscopy

(HHG-ARPES) And Test Measurement on Tungsten (W) [110] Surface

M D S A B B I R A H S A N

Master of Science Thesis Stockholm, Sweden 2013


A Thesis submitted for the degree of Master of Science

Experimental Setup of High Harmonic

Generation Based Angle Resolved Photoemission

Spectroscopy and test measurement on Tungsten

W(110) surface

Md Sabbir AhsanDecember 20, 2013

PerformedUltrafast X-ray Physics group

Faculty of PhysicsLudwig Maximilians University(LMU)

Munich,Germany

SupervisorProf.Dr.Ulf Kleineberg

LMU,GermanyExaminer

Docent. Jonas WeissenriederKTH,Sweden

School of Information and communication TechnologyRoyal Institute of Technology (KTH)

Stockholm,Sweden

Contents

Acknowledgement vii

abstract viii

1. Introduction 11.1. Significance of studying the surface state: . . . . . . . . . . . . . . . . . . 21.2. Goal of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1. Familiar with Ulrashort pulse Generation and application to gen-erate attosecond pulse: . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3. Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3. Thesis structure: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

I. Theoretical Background: 5

2. Bandstructure and Fermi surface of solid 62.1. Free electrons in solid: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2. Energy Bands in a solid: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1. Electron in a weak periodic potential: . . . . . . . . . . . . . . . . 9

3. Spectroscopic Methods to study surface state 153.1. Photoelectron Spectroscopy: . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1. Kinematic of photoemission . . . . . . . . . . . . . . . . . . . . . . 163.1.2. Linear response in external field: . . . . . . . . . . . . . . . . . . . 183.1.3. Dipole approximation and selection rule: . . . . . . . . . . . . . . . 193.1.4. Model for Photoemission process . . . . . . . . . . . . . . . . . . . 213.1.5. Surface and bulk sensitive photoemission . . . . . . . . . . . . . . 23

3.2. ARPES tools: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.1. Hemispherical electron analyzer: . . . . . . . . . . . . . . . . . . . 243.2.2. Time of flight analyser (TOF) . . . . . . . . . . . . . . . . . . . . . 24

3.3. Auger Spectroscopy: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1. Principle: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4. Generation of Ultra-short pulses: From femtosecond laser to attosecond XUVpulse 274.1. Mathematical Idea of an ultra-short pulse: . . . . . . . . . . . . . . . . . . 27

i

Contents

4.2. Basic element for generating ultrashort pulse . . . . . . . . . . . . . . . . 294.2.1. Gain Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.2. Mode locking technique . . . . . . . . . . . . . . . . . . . . . . . . 294.2.3. Dispersion control and pulse compression . . . . . . . . . . . . . . 304.2.4. Principle of Dispersion compensation . . . . . . . . . . . . . . . . . 324.2.5. Mirror and output Coupler . . . . . . . . . . . . . . . . . . . . . . 34

4.3. Ionization process in strong laser field . . . . . . . . . . . . . . . . . . . . 344.4. Generation of attosecond pulse: High Harmonic Generation . . . . . . . . 364.5. Measuring the pulse duration: . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.5.1. Autocorrelation Technique . . . . . . . . . . . . . . . . . . . . . . . 384.5.2. Stereo ATI: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5.3. Attosecond streaking experiment . . . . . . . . . . . . . . . . . . . 40

II. Experimental setup,results and discussion 42

5. Experimental setup of HHG Based ARPES 435.1. Generation of attosecond XUV pulse . . . . . . . . . . . . . . . . . . . . . 43

5.1.1. Generation of few cycle pulse . . . . . . . . . . . . . . . . . . . . . 435.1.2. HHG setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2. Sample preparation chamber . . . . . . . . . . . . . . . . . . . . . . . . . 475.2.1. evaporation chamber . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2.2. cleaning chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2.3. Conditioning filament . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3. ARPES Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3.1. sample heater and sputter gun . . . . . . . . . . . . . . . . . . . . 505.3.2. Double Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3.3. Magnetic field compensation . . . . . . . . . . . . . . . . . . . . . 515.3.4. Ultra high vacuum in ARPES . . . . . . . . . . . . . . . . . . . . . 535.3.5. Connection: Beamline, ARPES and Sample preparation . . . . . . 56

5.4. Characterization of experimental setup . . . . . . . . . . . . . . . . . . . . 565.4.1. Determining laser spot size after HCF . . . . . . . . . . . . . . . . 565.4.2. Finding the focus point . . . . . . . . . . . . . . . . . . . . . . . . 575.4.3. Energy and time resolution . . . . . . . . . . . . . . . . . . . . . . 585.4.4. Determining the time offset . . . . . . . . . . . . . . . . . . . . . . 59

6. Experimental study on Tungsten W(110) surface 606.1. Auger spectroscopy on Tungsten(110) . . . . . . . . . . . . . . . . . . . . 60

6.1.1. Detection of surface contamination . . . . . . . . . . . . . . . . . . 606.1.2. Sample cleaning: Method and verification . . . . . . . . . . . . . . 62

6.2. Crystal property of Tungsten(W) . . . . . . . . . . . . . . . . . . . . . . . 646.3. Photoemission Spectroscopy on Tungsten (W) . . . . . . . . . . . . . . . . 65

6.3.1. Observation of Conduction band Spectrum . . . . . . . . . . . . . 666.3.2. Fermi surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

ii

Contents

6.3.3. Bandstructure observation . . . . . . . . . . . . . . . . . . . . . . . 70

III. Future work: Improvement, conclusion 72

7. future work 737.1. Area of improvements: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.2. Prospect for time resolved (TR) study:(TR-ARPES) . . . . . . . . . . . . 75

8. Conclusion 77

A. Calibration of evaporation chamber: 78

B. Helmholtz compensation 79

iii

List of Figures

2.1. Electron in an uniform potential . . . . . . . . . . . . . . . . . . . . . . . 62.2. Fermi surface in free electron model . . . . . . . . . . . . . . . . . . . . . 72.3. Formation of energy bands in a solid . . . . . . . . . . . . . . . . . . . . . 92.4. Bandstructure representation of a solid . . . . . . . . . . . . . . . . . . . . 112.5. Magnitude of energy gap at zone boundary . . . . . . . . . . . . . . . . . 132.6. Variation in potential energy . . . . . . . . . . . . . . . . . . . . . . . . . 132.7. Band gap of a solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8. Effect of weak periodic potential on Fermi surface . . . . . . . . . . . . . . 14

3.1. Photoelectron spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2. Photoemission Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3. Photoemission process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4. Photoemission model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5. Energy dependent mean free path . . . . . . . . . . . . . . . . . . . . . . . 233.6. ARPES electron analyzers . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.7. Auger process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1. Diagram of an ultrashort pulse . . . . . . . . . . . . . . . . . . . . . . . . 274.2. Mode locking mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3. Kerr-lens mode-locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4. self-phase modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5. Ionization field in strong Laser field . . . . . . . . . . . . . . . . . . . . . . 344.6. Regimes of Nonlinear optics . . . . . . . . . . . . . . . . . . . . . . . . . . 354.7. higher order harmonic generation process. . . . . . . . . . . . . . . . . . . 374.8. HHG Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.9. Autocorrelation technique to measure pulse duration . . . . . . . . . . . . 384.10. Measuring few fs pulse duration . . . . . . . . . . . . . . . . . . . . . . . . 404.11. Attosecond streaking experiment . . . . . . . . . . . . . . . . . . . . . . . 41

5.1. Chirped pulse amplification to generate femtosecond pulse . . . . . . . . . 445.2. Generation of ultrashort pulse . . . . . . . . . . . . . . . . . . . . . . . . . 455.3. Generation of few cycle laser pulse . . . . . . . . . . . . . . . . . . . . . . 455.4. Higher order harmonic spectrum . . . . . . . . . . . . . . . . . . . . . . . 465.5. Sample preparation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.6. tr-ARPES experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . 495.7. Sample stage in ARPES chamber . . . . . . . . . . . . . . . . . . . . . . . 50

iv

List of Figures

5.8. Reflectivity of double mirror . . . . . . . . . . . . . . . . . . . . . . . . . . 515.9. Effect of magnetic field in PES . . . . . . . . . . . . . . . . . . . . . . . . 525.10. Helmholtz like compensated ARPES chamber . . . . . . . . . . . . . . . . 525.11. Ultra High Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . 555.12. 3D view of HHG based ARPES experimental setup . . . . . . . . . . . . . 565.13. Determing spot size after HCF . . . . . . . . . . . . . . . . . . . . . . . . 575.14. Harmonic recorded in CCD camera . . . . . . . . . . . . . . . . . . . . . . 585.15. Determing Energy and time resolution . . . . . . . . . . . . . . . . . . . . 585.16. Determing time offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.1. Diagram for Auger and Photoemission process . . . . . . . . . . . . . . . 606.2. focusing electron beam on the sample surface . . . . . . . . . . . . . . . . 616.3. Auger Process in Tungsten surface . . . . . . . . . . . . . . . . . . . . . . 616.4. Auger Spectroscopy to verify surface cleanliness . . . . . . . . . . . . . . . 636.5. Auger peaks from clean W surface . . . . . . . . . . . . . . . . . . . . . . 646.6. direction and detector position . . . . . . . . . . . . . . . . . . . . . . . . 656.7. Exciting sample surface with XUV photons . . . . . . . . . . . . . . . . . 656.8. photoelectron Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . 666.9. W Photoelectron cross section . . . . . . . . . . . . . . . . . . . . . . . . . 676.10. photoelectron energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . 686.11. Fermi Surface of Tungsten(110) . . . . . . . . . . . . . . . . . . . . . . . . 696.12. Fraction of Brillouin zone using current experimental setup . . . . . . . . 696.13. Tungsten Bandstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.14. Theoretical Bandstructure of Tungsten . . . . . . . . . . . . . . . . . . . . 71

7.1. Separation of harmonics through grating . . . . . . . . . . . . . . . . . . . 737.2. Time-resolved ARPES study . . . . . . . . . . . . . . . . . . . . . . . . . 76

A.1. Calibration of dual electron beam evaporator . . . . . . . . . . . . . . . . 78

B.1. Magnetic field compensation by Helmholtz coil pair . . . . . . . . . . . . . 80

v

List of Tables

5.1. Magnetic Field compensation . . . . . . . . . . . . . . . . . . . . . . . . . 535.2. mean free path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

vi

Acknowledgement

First of all, I am extremely grateful to my GOD ALLAH for giving me the mental,physical strength as well as my enthusiasm to acquire knowledge far from my countryin Sweden and Germany.

I would like give special thanks and express my gratefulness to Prof. Dr.Ulf Kleinebergacting as my supervisor at Ludwig Maximilians University (LMU), Munich and Docent.Jonas Weissenrieder as my Examiner in Royal Institute of technology (KTH), Sweden.Prof. Kleineberg give me the opportunity to do my master thesis in his Ultrafast X-ray Physics Research group as an exchange student at LMU, Munich that enables meto be familiar with state-of-the-art attosecond science and technology. Based on mycareer goal and personal interest doing my master thesis in his research lab was the bestdecision for me. Also, I am very happy to get Dr. Jonas as my examiner. His nicesupervision from KTH through Phone conversation and frequent emails also helped meto find my goals and direction during my thesis work in abroad. Now, I understand verynicely, I would not get better supervisor, examiner, colleagues, office mates and betterenvironment to carry on my master thesis.

I am also grateful to Juergen Schmidt to whom I worked most of the time during mythesis work. To be honest, I never had experience to work with such complex instru-mental setup. Initially, Juergen helped me a lot to be familiar with every Instrumentand how to operate. Although, it took several times for him to describe me the samethings, but he always seems to be enthusiastic to explain me every setup. Moreover, Igot better understanding to handle such high power laser from him. Juergen helped mea lot to finish my thesis work at the right time. His hard working effort make it possibleto generate HHG based XUV attosecond pulse after severe laser breakdown for severalmonths. I greatly admire his friendly attitude towards me during my thesis work.

My Special thanks also goes to Soo Hoon Chew to whom I contacted first and expressmy interest to work in their Ultrafast X-ray research lab. She helped me a lot to comein Munich for doing my thesis work. I learned many things from her advice. I alsowanted to express my thanks to Alexander Guggenmos, for giving me access and to useDEKTEK tool in his well equipped clean room lab at MPQ (Max-Plank Instituteof Quantum Optics). Finally, I want to give thanks every group members Kellie,Christian, Jeryl, Huaihai, Clemens,Sebastian in our research group.

vii

abstract

Experimental set up of High harmonic Generation based angle resolved photoemissionspectroscopy (HHG based ARPES) has been developed and characterized. The setupis intended for ultrafast time resolved measurement of bandstructure dynamics on theattosecond time scale. As a first proof of principle experiment HHG based ARPESmeasurement on a tungsten W (110) sample was done in a static mode.

Initially 27 femtosecond (fs) laser pulse was generated by chirped pulse amplification(CPA) technique from a femtosecond laser source and a hollow core fiber was used toproduce few cycle (∼ 4fs) laser pulse. This Few cycle pulse ionize neon atoms andgenerates ultrafast attosecond (as) extreme ultra violet (XUV) pulses via HHG pro-cess that propagates towards the ARPES chamber in a ultrahigh vacuum condition(< 10−8mbar). Helmholtz like magnetic field compensated HHG based ARPES cham-ber was designed in combination with a multilayer broadband (5 eV) XUV mirror atphoton energy 65 eV with a 10 kHz repetition rate. A time-of-flight (TOF) electronanalyzer was installed in the UHV ARPES chamber that can collect and analyze thephotoemitted electrons within an acceptance angle of ±13 degrees. Besides the HHGbased ARPES chamber, a sample preparation chamber was developed and calibratedthat can be used either to grow thin-film or clean the sample. Performing photoemissionusing highly surface sensitive 65 eV XUV photon source demands an extremely cleansample surface. As a result sample surface cleaning was performed by high temperatureheating in a cleaning section of the preparation chamber before photo-exciting with theXUV pulse. Auger spectroscopy was employed by a 50 kHz pulsed electron source toverify the surface cleanliness. Finally, the attosecond XUV pulse has been focused ontothe tungsten sample using the multilayer XUV mirror to investigate the conduction bandphotoemission spectra, Fermi edge and Fermi surface as well as energy bands of the Wsample surface. Pronounced conduction band features in the photoelectron energy dis-tribution spectra were observed. The W energy bands are characterized by high densityof states near the Fermi level where Fermi surface is characterized by symmetrical lobesin vertical and horizontal direction. Moreover, the available setup will be prepared fortime-resolved (pump-probe) experiments using an appropriate double mirror delay unit.Following the excitation of the sample surface with a few cycle laser (pump) pulse, thetime resolved electron dynamics can be monitored using XUV attosecond (probe) pulsesin attosecond timescale.

Keywords: Attosecond XUV pulse, Magnetic compensation, E-beam evaporation, An-gle resolved Photoemission Spectroscopy, Auger Spectroscopy, Tungsten(110).

viii

Chapter 1.

Introduction

Our science and technology is getting smarter day by day. Such smartness is markedby considering how fast the devices can work and how small they may be build. Ourcomputers are getting smaller and faster but we are not satisfied. Our demand is quan-tum computer. With the remarkable advancement in the field of nanotechnology weare about at the limit in spatial dimension but still many things to be understood inshorter timescale limit. In that respect, ultrashort light sources can be used to study thenanostructure or few nanometer surface layer for better understanding about those sys-tem that can respond to the lower time scale i.e near attosecond region. High HarmonicGeneration (HHG) based Angle resolved photoemission Spectroscopy (ARPES) canbe such a tool where one can study the physics on the attosecond scale.

ARPES has already been proven to be a standard Spectroscopic technique to studysolid state system. In a solid systems, ARPES can provide many information about thesurface state, electronic bands, density of states etc. The observation of the electrondynamics in a solid state system requires proper integration of a pulsed light sourceswith ARPES. The shorter the pulse the shorter the event that can be observed.

With the development of ultrashort few cycle pulses and novel attosecond light sourcesthrough HHG process, an approach was taken to combine the novel light sources withthe state-of-the-art ARPES tool. At present, ARPES is well established using eitherany Synchroton or HHG sources driven by solid state laser sources that deliver pulsesin picosecond (10−12 s) or > 100 fs (10−15 s) range. but the tool is not developed usinga light source that deliver pulses of few fs or as(10−18 s) range. When the pulse getsshorter, the spectral shape of the pulse gets broader and vice versa1. That means aphysical phenomena can be sampled on a timescale with better time resolution thanany other sources. But On the other hand, one has to compromise with poor energyresolution.

Following the development of HHG based ARPES, first test measurement was per-formed in static mode to study the Tungsten (W) surface. Having novel light sources inhand, one can study the static as well as the dynamic process on the surface state.

1Space charge effect can dominate during photoemission when using high repetition rate laser pulsebut the current setup has no such effect due to low repetition rate of 10 kHz.

1

Chapter 1. Introduction

1.1. Significance of studying the surface state:

Surface state has drawn lot of interest to the scientific community since last decade [1–4].There are many interesting phenomena going on in the surface state i.e solid-vacuum,solid-liquid, solid-gas etc. One can explore light matter interaction, quasi particle in-teraction, motion of electrons in the surface electronic bands [5]. The fundamentalproperties of a solid is encoded in the energy bands or bandstructure. As a result newstate of matter can be discovered by studying the surface states or surface bands as wellas the bulk bands. Recently discovered Topological Insulators [6] (TI) is the resultof such study. TI surface states can offer persistent current flow [7] that can be used forquantum computation.

New device application can also be found by examining a modified surface layer bygrowing a thin layer on top of the substrate. Such work has been published [8] tounderstand about ultrafast magnetisation. An anti-ferromagnetic few nanometer thicklayer has been grown between two ferromagnetic layers. Exciting the top layer with aultashort laser pulse the magnetization has been transferred to the third layer within atimescale of few femtosecond.

Moreover, to understand the current flow one needs to understand the change ofoccupancy of the states near the topmost filled energy level called ”Fermi surface” and theshape of the Fermi surface determines the response due to electric, magnetic or thermalgradient. One can also find the superconductivity and magnetism simultaneously at theinterface of two band insulator [9]. In general, studying surface phenomena has greatimpact on understanding the behavior of fundamental properties i.e. magnetization,superconductivity, electronic motion and correleation in complex system etc.

W (110) was chosen for surface study for two reasons. It is relatively easy to prepareand has high density of states near the Fermi edge which is perfect for the first timedoing experiments on the newly build HHG based ARPES setup. Fermi level in W lieson the conduction band. Following the excitation on W surface using HHG beamline onecan examine bandstructure, conduction band photoelectron spectrum as well as Fermisurface to understand the fundamental properties of W.

1.2. Goal of the Thesis

My initial plan was to study the ultrafast electron dynamics in solid surface, but dueto instrumental breakdown as well as time constraint it has not been possible at thispoint. The aim of the thesis work was to get access into the emergent state-of-the-artAttosecond science and technology and apply such technological advancement inARPES experiment. The goals can be divided into few steps.

2


1.2.1. Familiar with Ulrashort pulse Generation and application to generateattosecond pulse:

The Preliminary stage of my thesis work was to familiar with the theoretical concept ofattosecond science and technology through literature study. Moreover, I develop someintuitive knowledge to study ultrafast phenomena such as studying cooper-pairs, band-structure dynamics etc. As a result, Special effort and concentration was to be familiarwith the generation of ultrashort broad bandwidth few cycle laser pulses, experimentaloptics, HHG process and apply such coherent pulses to study physical phenomena. Dur-ing thesis work a summer school visit to DESY(Deutsches Electron Synchroton)also enrich my knowledge to become familiar with other coherent light sources such asSynchroton radiation and FEL (Free Electron Laser).

1.2.2. Experimental setup

HHG-ARPES chamber: Attosecond extreme ultra-violet (XUV) pulses were gener-ated from HHG process by ionizing Ne atoms from a few cycle Infrared (IR) laser pulseand both of the pulses propagate collinearly to the experimental chamber. The connec-tion between beam line and ARPES chamber has been established. Two ultra thin metalfilters have been used to separate the XUV beam from IR beam for our experiment.

A complete setup of HHG based ARPES chamber has been developed that includesdouble mirror2, magnetic field compensation system, sample heater3, installing electrongun and sputter gun, maintaining ultra-high vacuum (UHV) condition etc. A time-of-flight (TOF) spectrometer has been installed to analyze the electrons from the samplesurface (E(kx, ky)). After development subsequent calibration of the TOF spectrome-ter was done to find energy and time resolution, time zero offset for both Auger andphotoemission spectroscopy.

Sample preparation chamber: A sample preparation chamber was developed duringthesis work that contains sample cleaning chamber and electron beam evaporation forthin film fabrication. This chamber has been attached with the ARPES chamber. Theevaporation chamber was also calibrated using four different materials.

1.2.3. Experimental Study

Study of Tungsten(110) surface: Final part of my thesis was to perform experimentalstudy using newly developed ARPES chamber using 65 eV XUV pulse through HHGprocess. The goal behind the experimental study was to clean the W sample surface and

2For pump and probe beam in time resolved experiment3For Instantaneous heating to clean the sample

3


investigate photoemission spectra, bandstructure as well as Fermi surface of the cleanW sample surface.

1.3. Thesis structure:

The Thesis work has been divided into several chapters that describes the theoreticalas well as the experimental observation on W(110) surface. Following Introduction inChapter.1 the report has been divided into three parts. Part.I contains a theoreticaldescription regarding bandstructure and Fermi surface chapter.2, spectroscopic tech-niques chapter.3 and ultrashort pulse generation chapter.4. Part.II of the reportdescribes the experimental setup for HHG based ARPES chapter.5 and experimentalstudy on W(110) surface chapter.6 respectively. Finally, Part.III is based on futurework chapter.7 and conclusion chapter.8.

4

Part I.

Theoretical Background:

5

Chapter 2.

Bandstructure and Fermi surface of solid

2.1. Free electrons in solid:

The behavior of the free electron was described classically by P. Drude [10] that treatedthe valence electrons in a metal as classical particles like molecules or an ideal gas.Crystal lattice and electron-electron interaction was completely neglected. Later thequantum mechanical version of the free electron model was extended by Arnold Som-merfeld incorporating the Fermi-Dirac distribution and uniform crystal potential. Themotion of the electron can be described by the quantum mechanical Schrodinger equa-tion. According to the model, electrons in three dimensional potential can be describedby infinite square-well potential.

V (x, y, z) =

0, if 0 ≤ x, y, z ≤ L∝, otherwise

Figure 2.1.: Electron in an uniform potential [11]

In 3D one can think the N electrons are confined in a cube box of edge L and solvethe Schrodinger equation in 3D with a boundary condition Ψk(x + L, y + L, z + L) =Ψk(x, y, z)

− h

2m(∂2Ψk

∂x2+∂2Ψk

∂y2+∂2Ψk

∂z2) = EkΨk (2.1)

6

Chapter 2. Bandstructure and Fermi surface of solid

Figure 2.2.: Free electron like Fermi surface in 3D space [12] (left) and 2D space [11](right)

The solution is a plain wave same

Ψk(r) = Ak(1

L)3/2eikr (2.2)

where ki = 0,±2πL , ......±

ni2πL and nx, ny, nz ∈ Z

Finally the energy eigenvalue is given by

Ek =h2k2

2m=

h2

2m(kx

2 + ky2 + kz

2) (2.3)

EF =h2kF

2

2m(2.4)

This equation 2.3 represent the occupied states with wavevector k ≤ kF , kF is theFermi wave-vector and a constant energy sphere can be defined for each k vector with

radius k =√

2mEh . System with N electrons, the orbitals can be represented by 3D k-

space. The energy of the occupied orbitals are included in a sphere and energy at thesurface sphere is defined as the Fermi energy. The topmost filled energy level defines theFermi level with a constant energy surface of sphere called ”Fermi Surface”.

Fermi wave-vector can be given by

kF = (3π2N

V)1/3 (2.5)

7


2.2. Energy Bands in a solid:

A solid can be considered as the symmetric collection of the atoms. In one dimensionalspace one can put N hydrogen atoms in a linear chain and examine how the molecular1s orbital look like in that chain. The total wavefunction can be written according tothe LCAO1.

|Ψ >=N∑j=1

cj |j > (2.6)

|j > defines the s band electron wavefunction in j’th atom. Schrodinger equation(HΨ = EΨ) can be applied to find the corresponding molecular coefficient |j > andenergy. The previous equation becomes

N∑j=1

cjH|j >= EN∑j=1

cj |j > (2.7)

solving the Hamiltonian matrix elements

N∑j=1

cj < p|H|j >= EN∑j=1

cj < p|j > (2.8)

p is defined as the wavefunction in one site. The overlap matrix element can beconstructed such as < p|j >= δp,j

δp,j =

α, if j = p

β, ifj = p± 1

0 otherwise

using the above condition one can get a system of N equation

αc1 + βc2 = Ec1

βc1 + αc2 + βc3 = Ec2

..

..

..

βcN−2 + αcN−1 + βcN = EcN−1

βcN−1 + αcN = EcN

1Linear Combination of Atomic Orbital

8


Figure 2.3.: Formation of bands by broadening the energy levels in a solid [13] (a) Nonde-generate energy levels in atomic potential (b) Broadening the energy levelswhen atoms are closely spaced with each others

After solving the linear equations the allowed energy levels can be found as

Em = α+ 2βcosmπ

N + 1(2.9)

where m= 0,1,2,3....N. Plotting the energy levels as a function of the length of thelinear chain of hydrogen atom, one can see the variation in energy levels. For smallervalue of N the energy levels are discreet, but the levels become continuous as N becomelarger and larger. In a practical solid, N is taken as infinite and the energy level spectrumis continuous. This continuum levels are called the energy band.

2.2.1. Electron in a weak periodic potential:

In a crystal lattice, atoms are arranged in a periodic manner. The electrons interactwith the massive neuclei and the other electrons. A effective periodic potential V(r) istreated that takes into account all the interactions for an electron in a periodic latticesite [13]. Electrons in an energy band can be treated to be perturbed by the periodiceffective potential of the ion cores.

A crystal lattice with periodic potential obeys translational symmetry,

V (r + rn) = V (r), rn = n1a1 + n2a2 + n3a3 (2.10)

9


Taking the Fourier expansion

V (r) =∑G

VGeiG.r,G = hg1 + kg2 + lg3 (2.11)

Writing Schrodinger equation using the wavefunction

Ψ(r) =∑k

Ck.eik.r (2.12)

HΨ(r) = [− h2

2m∇2 + V (r)]Ψ = EΨ (2.13)

Putting the value Ψ(r) and V(r)

(2.14)∑k

h2k2

2mCke

ik.r +∑k’G

Ck’VGeik’+G.r = E

∑k

Ckeik.r

where k’→k-G, after rearranging the equation one can get for any r

(h2k2

2m− E)Ck +

∑G

VGCk-G = 0 (2.15)

The potential couple each coefficient Ck with its reciprocal space translation Ck+G.Moreover, for each k in the first Brillouin Zone2 (BZ) one can get N independent prob-lem. Each of them will give a solution which is a sum over plane waves and wavevectorswill differ only by G. After summing over the lattice sites k,k+G,.... the wavefinctionat wavevector k is given by

Ψk(r) =∑G

Ck-Geik-G.r = (

∑G

Ck-Gei-G.r)eik.r (2.16)

The above equation can be written as

Ψk(r) = Uk(r)eik.r (2.17)

where Uk(r) = Uk(r + rn). Uk(r) is a periodic function and this periodic function ismodulated by the free-electron like plain wave. The wave function in a periodic potentialis defined as the periodic function times the plane wave. The above equation is known asthe Bloch equation. The wave function in a Bloch equation is called the Bloch states.

2Unit cell in reciprocal space

10


Figure 2.4.: Bandstructure a of solid (a)reduced (b)periodic(c)extended zone [11]

Due to the Bloch theorem one can expressed crystal potential, eigenfunction and energyeigenvalue as follows

uk(r + a) = uk(r) =∑G

Ck+GeiGr (2.18)

Ψn,k+G(r) = Ψn,k(r) (2.19)

εn,k+G = εn,k (2.20)

As both eigenstate and eigernvalue are periodic in reciprocal space, instead of findingthe wavefunction of the entire space in an infinite crystal Bloch wavefunction allows usto confine only in first Brillouin zone. Outside the first BZ, the value of the wavefunctionis simply identical due to translational invariance.

Taking into account the periodic (energy) eigenvalue and periodic weak potential,the electronic states will not be the same like a single parabola in free electron picture,but equally spaced parabolas shifted by any G vector. This is called periodic zonestructure. On the other hand, one can simply add or subtract the crystal potential Gto reach outside the single cell or return inside first BZ. This called the reduced zonerepresentation.

Figure 2.4 shows different Bandstructure representation of a solid. In reduced zonescheme all the bands are shown in the first Brillouin zone, every Band is shown in everyBrillouin zone in periodic zone scheme and different bands are shown in different zonesin extended zone scheme.

Now rearranging the equation 2.15 after shifting the eigenvalue by G and take thesummation over G’

11


Ck-G(E − h2(k-G)2

2m) =

∑G’

VG’Ck-G-G’ (2.21)

again writing G’ = G’−G

Ck-G(E − h2(k-G)2

2m) =

∑G’

VG’-GCk-G’ (2.22)

Ck-G =

∑G’

VG’-GCk-G’

(E − h2(k-G)2

2m )(2.23)

Setting E = h2k2

2m using first approximation and considering the highest coefficient forCk-G denominator of the above equation will vanish at

k2 = |k-G|2 (2.24)

which will give

k = ±πa

(2.25)

Which gives the Laue condition that is also equivalent to Bragg condition. One canalso examine how is the electronic band structure at the zone boundary.

Writing G’=G1 in the equation 2.21 for the first two coefficients Ck for G=0 andCk−G1 for G= G1

Ck(E − h2k2

2m) = VG1Ck-G1 (2.26)

Ck-G1(E − h2k-G12

2m) = V−G1Ck (2.27)

solving the above equations one can get

E± =1

2(E0

(k−G1) + E0k)± [

1

4(E0

(k−G1) + E0k)2 + |VG|2]

12 (2.28)

where E0k = h2k2

2m . At the zone boundary one can take E0k−G1

= E0k which will give

E+ − E− = 4E = 2|VG1| (2.29)

12


Figure 2.5.: Magnitude of energy gap at zone boundary [13]

That means at the zone boundary there is a gap in the electronic energy levels andbands are look like figure 2.5.

We have seen that at the zone boundary Bragg condition is satisfied and there isno electronic state. In other words, one get the Bragg reflection of electron waves in acrystal due to the energy gaps [14]. A question may arise how the gap occurs at thezone boundary?. The wavelike solution of Schrodinger equation does not exists, ratherforms a standing wave when the Bragg condition is satisfied (k = ±π

a ). At that pointthe wave traveling towards right is Bragg-reflected towards left and vice versa.

Figure 2.6.: Variation in potential energy [14]

The Eigenstates corresponding the eigenvalues E+ and E− can be represented asΨ(+) = eiπx/a + e−iπx/a = 2 cos(πx/a)

Ψ(−) = eiπx/a − e−iπx/a = i2 sin(πx/a)

13


Figure 2.7.: Band gap at zone boundary [14]

Now, calculating the probability density of the two states and plotting along with theion core potential, one can also understand the difference in potential energies betweenΨ(+) and Ψ(−) that is responsible for band gap.

Potential energy of an electron near a positive ion core is attractive. As a resultpotential energy due to Ψ(+) state gets lower where potential energy due to Ψ(−) statesgets higher and band gap arises. The wavefunction Ψ(+) situated just bellow the gapat A and Ψ(−) is located just above the gap at B.

Figure 2.8.: Effect of weak periodic potential on Fermi surface (a)free electron model(b)Weak periodic potential [15]

In the same way due to weak perturbation periodic potential Fermi Surface is alsoget distorted as the edge of Brillouin zone approaches. The surface is no more as aspherical as the free electron model suggest. Figure 2.8 shows the variation.

14

Chapter 3.

Spectroscopic Methods to study surfacestate

3.1. Photoelectron Spectroscopy:

When a solid, liquid or gas is illuminated with a energetic radiation having photon energyhigher than their work function, electrons are emitted from them. The emitted electronsare known as the photoelectrons and the process is known as the photoelectric effect. Theeffect was first discovered in 1987 by Heinrich Hertz [16] and later explained1 based onthe quantization of light by Albert Einstein in 1905 [17]. Photoelectron Spectroscopy(PES) is the technique that is based on the photoelectric effect and is a method toobserve the band structure of a material. The PES can give useful information to studythe electronic structure of metal, semiconductors, insulator and adsorbate molecules bymeasuring the energy and momentum distribution of the emitted photoelectrons fromthe non-insulating solid sample. Now-a-days it become a standard tool for solid stateand material science research2.

Figure 3.1 shows a schematic photoelectron spectrum. A beam having photon energyhν exceeding the work function of the sample Φ is used to excite the electrons. Thekinetic energy of the photoemitted electrons are given by

Ekin = hν − Φ− EB (3.1)

Where EB = EF − Ei, Ei is the initial state of the electron and EB is the bindingenergy of the electrons with respect to the Fermi level EF . If the energy of the photonis larger than (EB + Φ) then the photoemitted electron can leave the sample surfaceafter overcoming the surface barrier. So, binding energy of the photoelectrons can bedetermined by recording the energy spectrum I(Ekin) of the outgoing photoelectrons.

1Nobel Prize in Physics 1921 was awarded to Albert Einstein ”for his services to Theoretical Physics,and especially for his discovery of the law of the photoelectric effect”.

2The Nobel prize 1981 was awarded to Kai M. Siegbahn ”for his contributions to the development ofhigh-resolution electron spectroscopy”

15

Chapter 3. Spectroscopic Methods to study surface state

Figure 3.1.: Photoelectron spectrum as a function of the kinetic energy N(Ekin) of thephotoelectron [18]

PES measures the energy and momentum3 of the photoelectrons emitted from the non-insulating solid due to the photoelectric effect.

3.1.1. Kinematic of photoemission

A beam of radiation in Figure 3.2 either from laser or synchrotron radiation or gas dis-charge lamp is incident on the sample surface. Electrons are emitted due to photoelectriceffect, escape in vacuum and an electron analyzer is collecting the photoelectrons withina finite angle. One can measure the kinetic energy Ekin for a given emission directionand calculate the wavevector or momentum K in vacuum is given by

K =

√2meEkinh

(3.2)

The momentum has two component one is parallel K‖= Kx + Ky and another isperpendicular component K⊥= Kz.

3momentum is measured in angle-resolving mode

16


Figure 3.2.: Photoemission Geometry [18]. Photoelectrons are emitted from the soliddue to photoexcitation. A analyzer records the energy by measuring timeof flight and angle distribution of the incoming electrons.

Kx =1

h

√2meEkin sin ν cosφ (3.3)

Ky =1

h

√2meEkin sin ν sinφ (3.4)

Kz =1

h

√2meEkin cos ν (3.5)

To map the electronic dispersion relation E(k) inside the solid, one has to find out thewavevectors inside the soild by matching the free electron plane wave in vacuum (outsidethe solid) and Bloch wave inside the solid [18]. Parallel component of the wavevector isconserved under periodic boundary condition of the crystal lattice and given by

k‖ = K‖ =1

h

√2meEkinsinν (3.6)

k‖ is the surface parallel momentum component of the electron crystal in extended-zone scheme4.

The perpendicular component K⊥ alters during the photoemission process and is notconserved across the sample surface due to abrupt change of potential in z axis. It isnecessary to determine K⊥ to map the electronic dispersion E(k) with repect to totalcrystal momentum k. Assuming the final state momentum component using the band

4One can probe the higher order Brillouin zone by accessing the large angle ν and upon subtractingthe reciprocal-lattice vector G‖ can return to the reduced-zone scheme or in the first Brillouin zone

17


structure component or nearly free electron final block state description may enable usto calculate K⊥.

Ef (k) =h2

2m(k‖

2 + k⊥2)− | E0 | (3.7)

Where in equation 3.7 |E0|= V0 − Φ is defined as the bottom of the valence bandfrom the Fermi level that is connected with the inner potential V0 and work function Φ(Figure3.8). Using Equation 3.6 and 3.5 one can modify the equation 3.7 as follows.

k⊥ =1

h

√2me(Ekin cos2 ν + V0) (3.8)

Figure 3.3.: Three step Photoemission process [18]. Direct optical excitation (a), excitedelectrons in a free electron like final state (b), photoelectron spectrum (c).

After determining V0 the perpendicular momentum component k⊥ can be easily de-termined5. Several methods are mentioned in [18] to determine the inner potential.Inner potential V0 can be determined either by optimizing the theoretical and experi-mental band mapping for the occupied states or observing the periodicity by collectingthe photoelectrons along the surface normal while varying the photon energy and in turnEkin.

3.1.2. Linear response in external field:

Let us consider one electron system is subjected to a potential V(r). An external electro-magnetic field is applied to the system. The resultant Hamiltonian can be described [19]as

5A negligible dispersion along z is assumed for low dimensional anisotropic electronic structure andhence uncertainty in k⊥ is less relevant

18


(3.9)H =

1

2m(p− e

cA(r))2 + V (r)

=p2

2m+ V (r)− e

2mc[A(r).p+ p.A(r)] +

e2

2mc2[A(r)]2

= H0 +Hint

One can use the first order perturbation theory to find the interaction of the electro-magnetic field to the system as long as the external field intensity is low enough. Thephoto-current due to such external perturbation electromagnetic field can be find usingthe Fermi-Golden rule

I(f) = |Mif |2= |< Ψf |Hint|Ψi > |2 (3.10)

where Ψi and Ψf6 is defined as the eigenfunction of the Hamiltonian H0. The matrix

element can be be approximate by the common choice of Coulomb Gauge

∇.A(r) = 0 (3.11)

using the following commutation relation7 and equation 3.11 the interaction Hamil-tonian Hint can be written as

Hint(r) = − e

mc[A(r).p] +

e2

2mc2[A(r)]2 (3.12)

The quadratic term of A can be neglected in linear optical regime or low intensityexternal field. using p = −ih∇ The resultant matrix element can be expressed as usingequations 3.10 and 3.12

Mif =< Ψf |Hint|Ψi >=ieh

mc< Ψf |A(r).∇|Ψi > (3.13)

3.1.3. Dipole approximation and selection rule:

A periodic external electromagnetic field can be expressed [19] as

(3.14)A(r) = A0e.ei.k.r

= A0e(1 + ik.r + ....)

6Ψf is considered outgoing wave that extends to infinity7[p,A] = −ih∇.A

= (p.A(r)−A(r).p = −ih∇.A(r) = 0

19


Where A0 is the complex Amplitude of the external field in scalar form, the direction oflight polarization is defined by a unitary vector e and k vector represent the propagationdirection of the external electromagnetic field. In the dipole approximation it is assumed|k.r << 1|. Now the matrix element can be written as

Mif =ieh

mcA0 < Ψf |e∇|Ψi > (3.15)

using the commutation between position and Hamiltonian operator the momentumoperator can be expressed as

(3.16)[r, H0] = −(

h

i)p

m

p = − imh

[r, H0] = −ih∇ (3.17)

This relation gives the matrix element

Mif = − iehcA0(Ef − Ei) < Ψf |e.r|Ψi > (3.18)

The above dipole approximation easily provide certain selection rules in the symmetryof the photoemitted electron wavefunction. This selection rule is valid for considering asingle atom. The wavefunction can be expanded in the basis set of Spherical harmonicsYlm(Ωr). Now consider the photoemitted electron located initially in the core level withquantum numbers (li,mi) and electron initial wavefunction Ψi(r) can be written as

Ψi(r) = Rili,mi(r)Yli,mi

(Ωr) (3.19)

Final electron wavefunction Ψf (r)

Ψi(r) =∑lf ,mf

Rflf ,mf(r)Ylf ,mf

(Ωr) (3.20)

Consider the light is linearly polarized and e is parallel to the z axis. The dipoleoperator of the incoming linearly polarized light can be expressed as

e.r = (4π

3)1/2rY10(Ωr) (3.21)

Using Equations 3.19,3.20,3.21 the matrix element is

20


(3.22)Mif = − ie

hcA0(Ef − Ei)(

4π

3)1/2

∑lf ,mf

∫drr3[Rflf ,mf

]∗Rili,mi

×∫dΩr[Y

∗lf ,mf

(Ωr)]rY10(Ωr)Yli,mi(Ωr)

Using the general properties of the spherical harmonics, the integral is zero exceptfor two case lf = li ± 1 and mf = mi

For the case of circularly polarized incoming light, the plane of polarization of theincoming light will be perpendicular to the z axis and the dipole operator can be writtenas

e.r = (8π

3)1/2rY1m(Ωr) (3.23)

where m=1 or m=-1 for right and left circularly polarized light respectively. Usingequations 3.22 and 3.23 the selection rule for angular quantum number l will be the same(lf = li ± 1), but magnetic quantum number m will change as mf = mi ± 1.

3.1.4. Model for Photoemission process

The photoemission process can be described either Classically or Quantum mechanicallyusing 3-step and 1- step model [18] respectively.

Optical transition to the bulk (transition probability): Electrons are excited into theunoccupied bands above the vacuum level from the localized occupied bands . In thereduced zone scheme this transition is vertical as long as the photon momentum isnegligible and contains all the information about intrinsic electronic structure.

The optical transition probability from initial state |Ψi > to final state |Ψf > is givenby the Fermi’s Golden rule

Wfi =2Π

h|〈Ψf

N |Hint|ΨiN 〉|2δ(EfN − EiN − hv) (3.24)

Under Coulomb Gauge and linear approximation, interaction between the electronand incoming photon is described by the interaction Hamiltonian as follows,

Hint =e

mc~A.~p (3.25)

21


Where EiN = Ei

N−1 − EkB and EfN = Ef

N−1 + Ekin are the initial state and finalstate energy respectively. EkB is the binding energy of the photoelectron and , Ekin isthe kinetic energy with momentum k. Furthermore, to describe the complete photo-electron process in one step model it is required to conserve the energy and momentumconservation for the impinging photon and N electron system.

EfN − EiN = hv (3.26)

(3.27)kfN − kiN = khv

Travel of the excited electron to the surface (Scattering probability): various scatter-ing events (such as e-e, defect induced) can modify the electron energy and momentum sothat it contributes to the background photoemission spectrum in the form of secondaryelectrons. This is usually described by the effective mean free path proportional to theprobability that the excited electron will reach the surface without scattering [20].

Escape of the electron into vacuum (transmission probability): Depends on the elec-tron kinetic energy and material work function Φ. During transmission to the vacuum,Parallel component (K‖) of the momentum is conserved, but perpendicular component(K⊥) is altered due to work function as described in 3.1.1.

Figure 3.4.: Photoemission model [18], Three step model in (left) is characterized bytransition, scattering and transmission where in one step model is describedby the optical coupling between waves inside (Bloch) and outside (plane)the solid.

22


On the other hand, 1-step model of photoemission process (quantum mechanicalview) is described by the optical coupling between the Bloch waves inside the crystal tothe plane wave of the vacuum. Figure 3.4 shows the optical coupling decays exponentiallyinside the bulk crystal. In the one step model, many body wavefunction (both initial andfinal state) should overlap as shown in equation 3.24 and obey the boundary conditionat the solid interface.

3.1.5. Surface and bulk sensitive photoemission

Figure 3.5.: Energy dependent mean free path [20, 21]. Inelastic mean free paths ofphotoemitted electrons are plotted as a function of kinetic energy.

The photoelectrons detected in the PES experiment are from the uppermost layersof the solid. Photoemission can be used to probe just the first few monolayers at thesurface of the solid by the proper choice of experimental parameters. There is stronginteraction between the electron and matter. This interaction is responsible for the sur-face sensitivity. An electron travelling through the solid will have certain characteristiclength (known as mean free path) that it can travel without suffering the energy loss. Sothere are two kinds of electrons that are ejected due to photoelectric effect. One, inelas-tically scattered electron (suffered energy loss) and the other one is elastically scatteredelectrons (never lost their energy). An electron in the energy range (5-2000 eV) passingthe solid surface may lose their energy by a number of process such as electron-electronscattering, interband transition, Auger electron process etc. The net effect of this pro-cess is the mean free path of the electron in a solid is strongly dependent on the kineticenergy. The universal curve in figure shows that the mean free path is fairly independentof material and strongly dependent on the kinetic energy of the photoelectrons.

At low kinetic energy the electrons do not have enough energy to excite. On theother hand, at high kinetic energy the electrons are less likely to suffer energy loss asthey spend less time to pass through a given thickness of solid. Again the mean freepath is quite long. But between the two region, mean free path become minimum

23


in the energy range (∼ 20-150 eV) that lies in the XUV and soft X-ray zones in theelectromagnetic spectrum. The minimum mean free path is about∼ 0.4nm. This resultmeans that the incident photon source may have penetration depth on the order of fewmicrons, but electrons will escape from the solid surface due to their mean free path onlyfrom the top few layers. This makes crucial for surface sensitive tool using photoelectronspectroscopy. Outside the range (∼ 20-150 eV) the bulk sensitivity will increase.

3.2. ARPES tools:

As explained before initial state of an electron can be determined by measuring thekinetic energy and emission angle of the photoelectrons. Moreover, one can calculatethe binding energy as well as the band structure of a the material by determining theFermi level EF. Angle resolved photoemission spectroscopy uses this principle to mapthe band structure. Electrons are emitted due to photoemission from the sample surfacein all direction. There are two major ways to work with ARPES depending on how thekinetic energy of the electrons are determined.

3.2.1. Hemispherical electron analyzer:

This electron analyzer consists of a half-sphere extending into the vacuum away fromthe surface [22]. In this method, the photoemitted electron are collected within a givensolid angle by using an electron-static lens and passed between two capacitor plates witha applied voltage V. These capacitor plates are separated by a distance d and serve asthe energy filter. The electrons passing through the energy filter follow a circular path.The radius of curvature is determined by the applied voltage and kinetic energy of theelectrons. Only electrons with kinetic energy that are selected by the slit of the electro-static lens are able to pass through the hemisphere without hitting the capacitor plates.At the end of the hemisphere a detector is placed to measure their distribution alongperpendicular and radial directions. Along the radial direction electrons are distributedaccording their energies and along the perpendicular direction electrons are distributedaccording to their emission angle or the position along the entrance slit. By tilt androtation of the sample one can read the energy and angle distribution from the detector.Combining the whole emission angle will give the image of the band structure.

3.2.2. Time of flight analyser (TOF)

For non-relativistic particles, the kinetic energy is given by EK = 12mv

2. As a resultinstead of using the energy filter to determine the kinetic energy one can determine thekinetic energy from the velocity of the electron. One can determine the velocity (v = l/t)of the electrons using the distance (flight path,l) of the detector from the sample andflight time (t) to cover the distance. An electro-static lens same as the hemispherical

24


(a) (b)

Figure 3.6.: ARPES electron analyzers [22] (a)Hemispherical Analyzer (b)Time of Flightanalyzer

analyzer is used to focus the beam in a certain angle and image the beam in the detector.Due to different emission angle (θ 6= 0), flight path are not simply straight ( shown in3.6b) for different electrons. But the photoelectrons flight path will not differ so muchif one collects them in a very narrow acceptance angle.

In this method, a pulsed source that is synchronized with the detector and deliversshort pulse is used to measure the flight time. At the same time sufficient time betweenthe light pulses is needed to ensure the electrons with one pulse are not associated withthe next pulse. One can record the entire emitted electrons by simply tilting and rotatingthe sample same as the Hemispherical analyzer. Using TOF analyzer one can record theenergy and momentum data for a complete area E(kx, ky) of the BZ rather collectingthe data along the line E(kx) or E(ky) using hemispherical analyzer.

3.3. Auger Spectroscopy:

Auger Electron Spectroscopy (AES) was developed in the late 1960’s. Although, Theeffect was first observed by by Pierre Auger, a French Physicist (mid-1920). It’s a methodthat utilize the emission of low energetic electrons from a surface in the Auger Processand used to determine the surface contamination or surface composition.

3.3.1. Principle:

Auger Process can be divided into two basic steps,

Atomic ionization: A beam of high energetic electrons (typical energy 1-10keV) is usedto initiate the Auger process by creating the core holes in a particular atom. The beam

25


is energetic enough to ionize all the levels for lighter element and higher core levels forheavier elements. In figure the ionization is shown by removal of the k-shell electrons.

Relaxation and emission: The ionized atoms remains in the very excited state due toionization of core hole electrons. Then the atom relax back using any of the process.one, X-ray fluorescence two, Auger emission. As shown in figure, after the removal ofthe k-shell electrons one electron from the higher state (L1) fall down to fill the vacantstate. Energy liberated in this process transferred to other electrons (L23). Part of theliberated energy is used to overcome the binding energy the electron and rest of theenergy is gained as a kinetic energy of the emitted Auger electron. One can calculatethe kinetic energy of the Auger electrons using the binding energy information of thelevels associated with the Auger process.

Figure 3.7.: Auger transition, (a) the atom become ionized by exciting with a beam ofenergetic electron, (b) create vacancy in the core level (c) a electron fromrelaxed by filling the vacancy and transferred energy to the other electronto emit as a Auger electron

K.E = (EK − EL1)− EL23 (3.28)

Where EK is the binding energy of the K-shell electron that removed by the excitationsource, EL1 is the binding energy of the electron that fall from upper state to fill thevacancy and EL23 is the binding energy Auger electron. The transition is named asKL1L23 Auger transition.

26

Chapter 4.

Generation of Ultra-short pulses: Fromfemtosecond laser to attosecond XUV pulse

4.1. Mathematical Idea of an ultra-short pulse:

Consider an ultrashort pulse traveling in the z direction, which can be described as

E(z, t) = A(z, t)ei(ωt−kz+φ0) (4.1)

Figure 4.1.: Diagram of an ultrashort pulse(8 fs at 800nm) indicating carrier, CEP andenvelope [23]

Where, A(z,t) is the pulse envelope is assumed to be Gaussian, ω is the carrierfrequency, k is the wavenumber and is the carrier-envelope phase (CEP) defined as thetime difference between maxima of carrier and its envelope (figure 4.1). This equation4.1 can be written

27

Chapter 4. Generation of Ultra-short pulses: From femtosecond laser to attosecond XUV pulse

E(z, t) = A(z, t)ei(φ(z,t)) (4.2)

Where the temporal phase is

φ(z, t) = ωt− kz + φ0 (4.3)

Differentiating the temporal phase will give the instantaneous frequency

ω(z, t) =dφ(z, t)

dx(4.4)

For better understanding the behavior and manipulation of ultrashort pulse, it isuseful to describe it in the spectral domain. The temporal domain and spectral do-main description is related by the mathematical technique Fourier and inverse Fouriertransformation.

E(t) =1

2π

∞∫−∞

E(ω)e−iωtdω (4.5)

E(ω) =

∞∫−∞

E(t)eiωtdt (4.6)

From this relationship, the spectral width of the pulse ∆ω and pulse duration ∆t isrelated by an inequality

∆ωpulse∆tpulse ≥1

2(4.7)

The product ∆ω∆t is known as the time bandwidth product TBP and its valuedepends on the pulse shape (Gaussian= 0.441, sec2 h = 0.315,square=0.886). If theinequality is satisfied i.e. the pulse duration is as short as spectral profile will allow, thanthe pulse is said to be transform limited. Equation 4.1 describe the pulse in temporaldomain, similarly it can be written in spectral domain as

E(z, ω) = A(z, ω)eiφ(z,ω) (4.8)

where the spectral amplitude A(z, ω) ∝ e− ln 16ω2

∆ω2 and φ(z, ω) is the spectral phase.

28


4.2. Basic element for generating ultrashort pulse

4.2.1. Gain Medium

The gain medium should be as broad as possible to generate a ultrashort pulse, whichis governed by the relationship in equation 4.7. We can easily get output of ∼ 27fsfrom the commercial lasers. Now-a-days Titanium Sapphire Ti:Al2O3 having a largegain bandwidth [24] is used as a gain medium, where a Sapphire (Al2O3) crystal dopedwith titanium so that Ti3+ ions replace some of the Al3+ions. For the experimentalwork of this thesis, Titanium Sapphire lasers are also used to generate ultrashort pulses.It has several features. Firstly, broad absorption bandwidth ranging from 450 nm-600nm with peak at around 500 nm when irradiate with visible light. Secondly, broademission band of the medium with a width of 200 nm and peak at 750 nm. These twocriteria are necessary to generate ultrashort pulses. Thirdly, Sapphire has good thermalconductivity in Titanium Sapphire gain medium, so thermal effects are not present atthe high peak power of the pump laser used [25].

4.2.2. Mode locking technique

Mode-locking is a technique to generate extremely short pulses of light in the range offemtosecond (10−15 s). Laser produces light over a natural bandwidth or over a range offrequencies. The bandwidth of the laser depends on the gain medium that the laser isconstructed from. When different frequency component in a laser spectrum have randomrelative phase then the net intensity is constant in time, but when the modes are lockedin phase a series of short pulses can be obtained whenever the peaks are match up(Figure 4.2). Mode locking has two different kinds.

Figure 4.2.: (a) Random phase with constant intensity. (b) Modes are locked to give ashort pulse [23]

.

29


Active Mode Locking: One is active mode locking means putting a shutter in thecavity and opening it briefly once every cavity round trip time which determines thepulse duration. In this method the fastest shutters either acousto-optical or electro-optical shutters are transparent to light as short as several nanosecond. Other one ispassive mode locking one can generate pulses in sub-nanosecond, where pulse itself actas a gate.

Passive Mode Locking: One passive method of mode-locking named as Kerr LensMode-locking (KLM) (nonlinear response of the laser crystal). Now-a-days a crystallinematerial Ti:Sa having large value of χ3 is used as a gain medium which change the linearreflective index of the medium. The effective reflective index results

neff = n+ n2I (4.9)

Due to the effective reflective index of the gain medium, central most intense part ofthe Gaussian beam profile experience a different refractive index to the outer part of thebeam creating a lensing effect. The lensing effect is known as gradient index (GRIN)lens for itself and focusses in the material. The effect is called self-focusing. This effectallows the passive mode-locking. In hard aperture approach, a simple aperture blocksthe less focused beam paths corresponding to lower laser intensity. On the other hand insoft aperture approach, the overlap of the laser pulse and the pump pulse within the gainmedium is greatest for the highest intensity, more focused pulses thereby leading to theincreased gain until eventually over many complete trips around the laser cavity for onlyone peak, the shortest and the intense, will remain. Figure 4.3 shows the cavity beamself-focuses to the center of the spatial beam profile as a result the gain is greatest at thepeak of the temporal profile of the pulse whereas gain is less at the wings. Mode-lockingoccurs due to the highest gain at the center of the pulse, generate a pulse. Finallynarrowness of the pulse depends on the gain medium bandwidth and dispersion of thecavity.

4.2.3. Dispersion control and pulse compression

As we see from the time and bandwidth product equation 4.7, an ultrashort pulse hasa broad spectral bandwidth and different frequency components. We know the velocityof light in a medium depends on the frequency. The phase velocity of a monochromaticwave is defined as

v(ω) =c

n(ω)(4.10)

Different frequency component encounter different phase delays in passing through alength L of a material. As a result great care should be taken when choosing a suitable

30


Figure 4.3.: Kerr-lens mode-locking(KLM)(a)KLM causes the beam to see greater gainwhen light intensity is very high, (b) gain and loss dynamics at the centerof the beam [26]

.

optics for the propagation of the short pulse in a medium. Otherwise, dispersion canlead to change the pulse profile and duration.

∆φ(ω,L) = kω =L

cn(ω) (4.11)

Let us consider the central frequency of the pulse spectrum is at ω0. We can applythe Taylor expansion around its central frequency to find the phase delay at x=L

∆φ(ω,L) = ∆φ(ω0, L) +d∆φ(ω,L)

dx (ω=ω0)(ω − ω0) +

d2∆φ(ω,L)

dx2 (ω=ω0)(ω − ω0)2 + .....

(4.12)

First term in the expansion is the carrier envelope phase (CEP), which is importantfor few cycle pulses and it does not make any change to the pulse shape. Second term inthe expansion is the group delay indicates the time for the pulse to propagate throughthe material. Final term in the expansion is called the group delay dispersion (GDD)means the rate at which the pulse is stretched during propagation through the mediumdue to different spectral components propagate with different velocities.

As defined bellow,Phase delay

∆φ(ω0, L) (4.13)

Phase velocity

v(ω0) =L

∆φ(ω0, L)(4.14)

31


Now define,Group delay

∆φg(ω0, L) =d∆φ(ω,L)

dx (ω=ω0)(4.15)

Group Velocity

vg(ω0) =L

d∆φ(ω,L)dx (ω=ω0)

(4.16)

As we see from the equations group delay is defined as the first derivative of phasedelay. So if the phase delay is not linear in frequency, then group delay is not constantand we cannot ignore the higher order term of Taylor expansion. These terms areresponsible to change the shape of the pulse as it propagates. In particular they describethe variation of group velocity with frequency or group velocity dispersion (GVD).

For most optical materials, refractive index determines how fast the light can travelthrough the medium and gradually increases with the frequency. As a result group delayand phase delay increases with increasing frequency. Higher frequency (Blue component)in the pulse lags the lower frequency (Red) component. This is called normal GVD orpositive GVD. Otherwise it is anomalous or negative GVD.

4.2.4. Principle of Dispersion compensation

For IR pulses travelling through an optical medium such as silica GDD is a positive value.If the instantaneous frequency of a pulse (derivative of phase with respect to time) variesas a linear function of time then the pulse is described as being chirped where the positiveGDD will introduce an up-chirp meaning that the instantaneous frequency increases withtime [23]. Now, one has to apply negative GDD technique to the pulse using opticalcomponent such as prism pairs, grating pairs and chirped mirrors. This will compressit in time. GDD control is the underlying mechanism to produce the ultrashort laserpulse using chirped pulse amplification. Figure 4.4 shows the schematic representationof pulse compression of the transform limited pulse by exploiting the effect of dispersion.

One can compensate the dispersion or variation of group delay of a pulse by creatingdifferent paths for different frequencies through special optical arrangement. Now-a-days Prism/grating compressor, chirpped mirrors are used to compensate the dispersion.Consider the Taylor expansion (with higher order terms) of phase delay around thecentral frequency ω0 written in equation 4.17.

32


Figure 4.4.: Initially transform limited pulse is broadened by Self Phase modulation(SPM) and then introduction of negative GDD makes the pulse compressedin time domain [25]

(4.17)

∆φ(ω,L) = ∆φ(ω0, L) +d∆φ(ω,L)

dx (ω=ω0)(ω − ω0)

+d2∆φ(ω,L)

dx2 (ω=ω0)(ω − ω0)2 +

d3∆φ(ω,L)

dx3 (ω=ω0)(ω − ω0)3

+d4∆φ(ω,L)

dx4 (ω=ω0)(ω − ω0)4 +

d5∆φ(ω,L)

dx5 (ω=ω0)(ω − ω0)5.....

∆φ(ω,L) = 0 + 0(ω − ω0) +d2∆φ(ω,L)

dx2 (ω=ω0)(ω − ω0)2 +

d3∆φ(ω,L)

dx3 (ω=ω0)(ω − ω0)3

+d4∆φ(ω,L)

dx4 (ω=ω0)(ω − ω0)4 +

d5∆φ(ω,L)

dx5 (ω=ω0)(ω − ω0)5.....

(4.18)

The first two terms in the Taylor expansion is zero as they have no effect on thepulse duration. Each term in the expansion has smaller effect than the previous terms.Normally we do not need to compensate all the terms of the expansion. The quality ofdispersion depends on how many terms in the expansion should vanish through dispersioncompensation-called the orders of dispersion. One prerequisite to achieve shorter pulseswith greater bandwidth, Phase has to be flatter over a broader frequency range. Thehigher the orders of dispersion compensation imply the phase will remain flat over thegreater the range of frequencies (bandwidth).

33


4.2.5. Mirror and output Coupler

Mirrors should be highly reflecting at the laser frequency. The percentage of light trans-mitted by the output coupler determines the ratio of output power to the input powerwithin the cavity.

4.3. Ionization process in strong laser field

Depending on the intensity of the laser the ionization process will different in an atomicsystem. Multiphoton process will dominate at intensities around 1013 W/cm2 to 1013

W/cm2. In this process a bound atom is ionize with more than the required number ofphoton and the atomic potential remains the same. In the case of ATI, the peaks shouldappear at

Es = (n+ s)hω −W (4.19)

Where n and s represent the minimum and excess number of photons absorb in theprocess. W is the ionization energy and Es is the kinetic energy of the peak correspondingto s (excess photon) being absorbed. So, a system having same initial state is excitedwith different multiples of the number of photons then the photoelectron spectrum ofthe system contains repeated feature which is separated by the energy of the photonknown as Above Threshold Ionization (ATI) peaks.

Figure 4.5.: Ionization field in strong Laser field [25]

At higher laser intensities of around ∼ 1014W/cm2to1015W/cm2 in an atomic system,the tunnel ionization will dominate. These electric field will cause distortion of theatomic potential (shown in figure) until a potential barrier is formed and some of thewave packet to tunnel out into the continuum. At even higher intensities the atomicpotential will distort severely. As that moment, there is no longer a barrier present andelectrons may escape, also known as ATI. In contrast to the spectrum from multiphotonionization the spectrum from tunnel ionization is expected to be smoother [27].

34


To distinguish the two regions Keldysh parameter γ [28] is used which is derived fromthe ratio of the binding potential 1 and quiver energy 2. This is given by the followingequation,

γ =ω

E

√2meIpe

(4.20)

Where Ip the ionization is potential, me is the electron mass and ω is the laserfrequency. A perturbative approach [29] is valid if quiver energy is much less than thebinding potential, a value of γ << 1 or γ >> 1 indicate the transition lies in thetunneling or multiphoton respectively. Recently, it has been shown two regimes mayoverlap and are not mutually exclusive [25]. Equation shows γ is dependent on the pulseduration it is likely that tunneling regime can be reached with ultrashort pulse even thebelow the damage threshold. Figure 4.6 shows different regimes according to the laserintensities used.

Figure 4.6.: Regimes of Nonlinear optics: The defined cut-off boundaries are not sharpand illustrate different nonlinear effect according to laser intensities. Instrong field regime, intensities correspond to visible and near-infrared light[25]

.

1work function of a solid and ionization potential2energy gained by the E-field

35


4.4. Generation of attosecond pulse: High HarmonicGeneration

There are two methods to break the femtosecond barrier. One is stimulated Ramanscattering and the other is to generate attosecond pulse via higher order harmonic gen-eration (HHG). The process which involves the generation of higher order harmonicstheir wavefunction lies in the XUV region by focusing of a few cycle laser pulse onto atarget. The process was suggested by Paul Corkum in 1933 [30] using semiclassical threestep model.

HHG is a process where light of a particular wavelength taken from a solid statelaser is converted into a shorter wavelength through non-linear interaction medium.Normally an atomic gas can act as a medium. One can create polarization by displacingthe positive and negative charges by oscillating the electric field of the laser pulse. As aresult polarization of the medium also oscillates in time and can be expressed as

P (t) = ε0(χ1E1 + χ2E2 + χ3E3 + ......) (4.21)

Here χ1, χ2, χ3 are the linear, quadratic and cubic terms of susceptibilities. In a wealelectric field the linear terms dominate but in strong electric field one cannot neglect thehigher order terms. Consequently, polarization will contain higher order terms and canbe viewed as oscillating dipole that oscillates at integer values n of fundamental frequencyω0. One can achieve extremely short pulse in attosecond regime using such nonlinearprocess when the pulse laser is tightly focused through the atomic gas. However, forsuch HHG process one needs to have high power densities on the order of 1013 − 1015

Watt/cm2 that only can provided by a short pulse laser source (femtosecond regime).On the other hand system that provides such short pulses usually have repetition rate10 KHz or lower.

Figure 4.7 shows the classical view of the high harmonic generation process withtheir quantum mechanical wavefunction at different steps. in In the first step, stronglaser field add a potential (marked red) to the unperturbed electron state of coulombpotential. The superposition of the two potential bends down shown . As a resultcoulomb potential of the valence band become distorted under the influence of the stronglaser field (a, d) and electrons can tunnel out of the potential barrier. The free electronbecome accelerated under the influence of the oscillating laser field (b , e). Finally theelectron recombine when the laser field (linearly polarized) changes the direction (c ,f).At the time of recombination an XUV photon pulse is released due to energy gain duringthe acceleration. Electrons kinetic energy is transferred to the photon as the recollisionhappens.

Figure 4.8 shows the typical HHG spectrum. Since the pulse is symmetric and electroncan release around each peak amplitude, the HHG process will repeat itself at every halfcycle. This gives rise to a spectrum of attosecond pulse separated by 2ω. Initially the

36


Figure 4.7.: Classical view is shown in the upper (a,b,c) part, where lower part (d,e,f)shows the quantum mechanical wavefunction. Electrons tunneling throughthe barrier (a+d). Accelerated to the laser field (b+e) and very broadbandlight spectrum due electron recollision with the parent atom (c+f). [31]

Figure 4.8.: HHG Spectrum. Only odd harmonic has been produced that is separatedby twice the frequency [32]

intensity of the harmonic falls, later it stays constant with increasing the harmonic orderuntil a cut-off is reached.

Ecut−off = IPtarget + 3.17Up (4.22)

The electron can attain a maximum kinetic energy equal to 3.2 times the pondero-motive potential [33] given by,

Up =e2E2

4meω2laser

(4.23)

37


Where E is the laser electric field, e electron charge, me electron mass, ωlaser is thefrequency of laser. The ionization and recollision process happens every half cycle of thelaser, adds coherently and emission spectrum looks having odd harmonics. As a result,it is called high harmonic generation.

4.5. Measuring the pulse duration:

Several techniques are available to measure the pulse duration. These are autocorrelationtechnique, stereo ATI streaking technique etc. Autocorrelation works well when the pulseduration is 7-8 fs or above, but bellow the limit it does not work well. To measure thepulse duration bellow 7 fs,stereo ATI phase meter is used to find the pulse durationand phase stability. Furthermore, In the case of attosecond region one has to perform”Attosecond streaking experiment” to confirm the pulse duration.

4.5.1. Autocorrelation Technique

This technique is primarily used to measure the pulse duration. In this technique, a pulseis splitted into two identical pulses through a beamsplitter (BS). One of them follow thesame path, but other is moved back and forth by a mirror to create the time delay τbetween the pulses. A lense is used to focus both beam into common point where anonlinear crystal is placed to generate second harmonic (SH) light. Plotting SH signalas a function of time it is possible to estimate the pulse duration or pulse width.

Figure 4.9.: Autocorrelation technique to measure pulse duration [26]. BS= beamsplitter

The fundamental field is the sum of two pulses delayed by τ . On the other hand SHfield is proportional to the square of the fundamental field. Hence here the fig 4.9 shows

E(2ω, t) = [E(ω, t) + E(ω, t+ τ)e−iωτ ]2 (4.24)

The time integrated intensity is given by

38


(4.25)

S(τ) =

∞∫−∞

I(2ω, t)dt

∝∞∫−∞

[E(2ω, t)]2dt

∝∞∫−∞

[E2(ω, t) + E2(ω, t+ τ)e−i2ωτ + 2E(ω, t)E(ω, t+ τ)e−iωτ ]2dt

The first two terms due to the two beams those are delayed. Third term is due tosum-frequency generation (SH signal). As noted and showed in figure 4.9, the SH signaltravel along the direction of the incident beam and the detector detect the signal withoutinterference from the other two.

S(τ) = constant

∞∫−∞

I(ω)I(ω, t+ τ)dt (4.26)

The width of autocorrelation trace S(τ) is related to the actual pulse width. Assume apulse is Gaussian and it will also produce a Gaussian autocorrelation signal and duration(1/e or 1

2 width) is given by 1√2

times the corresponding width of the autocorrelationtrace.

S(τ) = S0e−( t

∆t)2

(4.27)

Many femtosecond oscillator produce pulses according to

I(t) = I0 sec2(t

∆t) (4.28)

Finally the autocorrelation width is 1.763 times the pulse-width [26]. This techniquehas one disadvantage that without knowing the pulse shape exactly, we cannot determinethe exact pulse duration. Recently, a new technique has been developed to determine thepulse shape and duration simultaneously known as FROG (Frequency Resolved OpticalGrating).

4.5.2. Stereo ATI:

Stereo ATI (Above threshold ionization) phase meter is capable to provide informationabout Carrier envelop phase(CEP) and pulse duration bellow 7fs. This technique was

39


(a) (b)

Figure 4.10.: Measuring few fs pulse duration(a)Phase potato (b)Extracting pulse dura-tion measuring the average radius

.

discovered by G.G Paulus [34] In a stereo like time of flight setup a horizontally polarizedfew cycle pulse is focused into the gas target to trigger the ATI process. The Stereo ATIPhase meter exploits the left-right asymmetry in the ATI Time of flight (TOF) spectrum.Two different energy region from the TOF spectrum is selected to plot the asymmetryparameters. Such asymmetry plot will give a ”Phase potato”. The measured asym-metry plot is plotted 100000 shots3 (figure 4.10a) and each dot represent one shot. Theradius of the phase potato (figure 4.10b) will give the pulse duration where the polarangle will give the Carrier Envelop Phase (CEP). The bigger the asymmetry parameterthe larger the potato and shorter the pulse duration of the laser pulse. Moreover, thisdevice does not suffer from the phase matching bandwidth limitation like SHG basedcorrelation.

4.5.3. Attosecond streaking experiment

When the light pulse is in the attosecond regime then one can not easily measure thepulse duration. A method discovered in MPQ, Munich named as attosecond streakingspectroscopy shown in Figure 4.11. Here the attosecond XUV pulse is generated bythe HHG process using femtosecond IR laser pulse. Both of them propagate in thesame direction where attosecond XUV pulse is used to generate photoelectrons andfemtosecond IR pulse is used to probe the photoemitted electrons. Electron released bythe XUV pulse parallel to the direction of electric field suffer a change in their initialmomenta. This momentum change is proportional to the vector potential of the electricfield of IR laser at the instant of release. The final momentum and energy distribution

3Measured in collaboration with the group of Matthias Kling at LMU

40


(a)

(b)

Figure 4.11.: Attosecond streaking Experiment (a) Conventinal streaking camera (b)light field streaking camera [25,35]

of the emitted photoelectrons maps the duration and evolution of the electron emissionthat imprints the pulse duration of attosecond XUV pulse.

41

Part II.

Experimental setup,results anddiscussion

42

Chapter 5.

Experimental setup of HHG Based ARPES

5.1. Generation of attosecond XUV pulse

The beamline is derived from Titanium: Sapphire (Ti: Sa) chirped pulse amplifier (CPA)system. The system produces 27 fs IR pulse with a central wavelength 800 nm at 10 kHzrepetition rate. This IR pulse can be used to perform experiments or compressed usinghollow core fiber to produce few cycle 4 fs IR pulse. For the photoemission spectroscopyexperiment an attosecond XUV pulse is generated from the 4 fs linearly polarized pulsethrough high harmonic generation (HHG).

5.1.1. Generation of few cycle pulse

Few Cycle pulse is generated using two step. In the first step commercial Ti:Sa lasersystem produce ∼27 fs laser pulse and in the second step the laser pulse is compressedto ∼4 fs.

Chirped pulse Amplification

Our laser system is the commercial Titanium: Sapphire laser (Femtopower compact pro.Femtolasers GmbH) delivers 420 µJ pulses with 27 fs pulse duration. The compact procomprises a KLM Ti:Sa oscillator followed by a multipass chirped pulse amplification(CPA) stage. In the KLM approach extremely self-focusing may lead to laser intensitiesabove the damage threshold of the gain medium thereby imposing a limit on the peakpower of the laser pulse. This problem can be overcome by CPA method which wasinvented by Donna Strickland and Gerard Mourou in 1980 [36]. It is a technique foramplifying the peak of an ultrashort pulse to the high energy level.

The oscillator generates femtosecond pulses of 2-3 nJ centered at 800 nm with 80 MHzrepetition rate. The pulse is stretched and amplified by 9 consecutive passes through aTi:Sa amplifier crystal which is pumped by Q-switched frequency doubled, DPSS Nd:Yag laser. A pockels- cell based pulse picker thereby reduces the rate to 10 kHz. At thatrate, the seed pulse deposit a lot of heat into the crystal and amplification in the mJ

43

Chapter 5. Experimental setup of HHG Based ARPES

level which requires to be cooled to maintain the pump energy slightly below the gainsaturation level. Otherwise, heat deposition on the crystal may lead to optical damage,nonlinear thermal effect, intensity fluctuation, pulse distortion etc. A combination of aset of prism pair and a pair of dielectric TOD mirrors for advanced third order dispersionis used to compress the pulse after amplification. The transmitted FWHM spectralbandwidth is ∼ 40 nm shown in figure 5.3.

Figure 5.1.: Chirped pulse amplification to generate femtosecond laser pulse. Block di-agram shows oscillator, stretcher, amplifier and compressor.

pulse compression by hollow core fiber

27 fs femtosecond output laser pulse can be compressed up to 4 fs pulse using hollowfiber pulse compression. It was first demonstrated by Nisoli in 1996 [37]. The fiber isfilled with the high pressure noble gas forms a waveguide along which the pulse propa-gates and spectrum of the incident pulse is broadened by Self-Phase Modulation (SPM).Reflections at the inner surface generate significant losses that discriminate against thehigher order modes and a long fiber (70 cm-1 m) is used so that only the fundamentalmode can propagate. 1 m long silica fiber with 250 µm core diameter has been usedin our experimental set up. We achieved 35% throughput having 140-150 µJ pulse en-ergy after the fiber. A GRENOUILLE-FROG1 and a CCD camera were employed toinvestigate the spatio temporal pulse distortion and inspecting the focus into the fiberrespectively. Beam pointing instability (beam direction and beam position) before thefiber can lead to intensity fluctuation like double the rms pulse to pulse intensity afterthe fiber. Two fast piezo-driven mirrors were mounted in our set up to avoid beampointing instability and kept the intensity fluctuation at the 5% level after the fiber.

Finally the pulses are de-chirped using 8 consecutive reflections on 4 balanced pairsof low ripple negative dielectric mirrors designed with enhanced TOD compensation,

1http://www.swampoptics.com/products˙grenoverview.htm

44


Figure 5.2.: Generation of ultrashort pulse, fs laser pulse compressed by HCF passedthrough the TOD mirror for dispersion compensation and focused on theHHG chamber

broad spectral bandwidth (450-1050 nm) and high reflectivity. Dispersion fine tuning isdone by a pair of thin glass wedges as shown in Figure 5.2.

Figure 5.3.: Generation of few cycle laser pulse. Monitoring pulse spectrum at differentstage during generation of few cycle fs laser pulse

45


5.1.2. HHG setup

A free gas jet configuration (shown in figure 5.2) was used to generate the high harmonicsfrom linearly polarization of fs few-cycle laser pulse. Neon (Ne) was used as the targetgas. The beam was folded and focused into the gas nozzle (upper end sealed and madeof thin-walled Nickel) by focusing mirror inside the vacuum chamber. In order to createhigher order of harmonics, the beam was focused with the focal length of 25 cm thatproduces a beam spot of 30 µm and intensity around 1015 watt/cm2 range. This intensityis able to drill a tiny gas outlet into the tube. The nozzle is fully movable in 3D spaceand focusing mirror is motorized to maintain precisely adjusting the beam and correctingthe beam direction respectively. The higher order harmonic is optimized by a number ofparameters and these parameters affect the phase matching condition of the conversionprocess of spectrally selective enhancement of the harmonic yield. Firstly, moving thenozzle along the beam direction, i.e. relative to the focus to where preferable trajectoryharmonics are generated. Secondly, proper adjustment of the backing pressure of thegas. Thirdly, the chirp (wedges) and the intensity (iris) of the fundamental beam needto be adjust properly.

Figure 5.4.: Higher order harmonic spectrum. Harmonics using (a) Al filter (c) Zr filter,where (b) and (d) shows the generated harmonics in energy (eV) scale

.

The optical breadboard is firmly attached to the optical table whereas the HHGchamber is connected to the optical table via vibration dampers to avoid vibrationtransmission from vacuum pump to optical setup. Both residual fundamental and har-monic XUV beam then propagate collinearly towards experimental chamber through adifferentially pumping section (from 10−3 mbar to 10−10 mbar) with a small interme-diate chamber in between. But the beam divergence of XUV beam is about 1√

n, n is

46


the harmonic order. We need UHV condition to do photoemission experiments on thesolid target since the short free mean path of electrons in the XUV energy makes mea-surement highly surface sensitive. The intermediate chamber connects HHG chamberwith the experimental chambers with two thin tubes. This chamber serves as auxiliarypumping point between pressure gap, XUV filter and diagnostic. The thin tubes containa set of apertures controlling gas flow conductivity and easily moveable. Retractableoutcoupling mirrors are installed after each tube in order to align them to the beam axisto avoid clipping.

The generated harmonics has been shown in figure 5.4. A XUV spectrometer wasplaced along the beamline to characterize the XUV beam. A set of metal (Zirconium(150 nm) and Aluminium (200 nm)) filters have been set up for a number of reasons.Firstly, blocking the fundamental IR beam. Secondly, the harmonics are bandpassedspectrally. Thirdly, enable to calibrate energy using the Aluminium sharp absorptionedge around 70 eV. At the back-end of the differential pumping line an iris is used forbeam attenuation and a set of three insertable XUV mirrors that reflect the harmonicsonto a XUV diode. These mirrors are designed for three spectrally distinct, adjacentreflection regions between 50 and 90 eV.

5.2. Sample preparation chamber

Sample preparation chamber is divided into two parts, one is for cleaning and other isfor growing a layer on top of the substrate. figure 5.5 shows the setup of our Samplepreparation chamber.

5.2.1. evaporation chamber

A dual ion beam evaporation chamber is installed in the Sample preparation section(lower portion) as shown in figure 5.5. A sample is kept on the sample holder and thematerial which will be used to grow on top of sample substrate is kept on one of the twotube. Usually a wire or piece of material on crucible is used for the evaporation purpose.A high voltage 1 kv is applied on the wire or in the crucible and the filament currentwas increased upto 8 A. As a result electrons from the filament accelerated towardthe wire or crucible and evaporate the material. Finally the atoms of the evaporatedmaterial proceed towards the sample and create a layer on top of substrate. During theevaporation process some ions are also created that gives the ion current. The thicknessis dependent on the ion current, time and type of material to be grown. A shutter isused to cover the two tube during adjustment of emission/filament current.

During the evaporation procedure excess heat can be generated. Some parts in theevaporation chamber cannot withstand the higher temperature due to excess heating.Moreover, such excess heating can heat up the whole chamber and as a result the vacuum

47


(a) (b)

Figure 5.5.: Sample preparation setup, (a) Dual electron beam evaporation chamber andsample heater (b) evaporation process for thin film fabrication

condition will destroy. A continuous flow of water is maintained to cool the chamberduring the evaporation procedure. The flow of water is monitored in PC.

It is very important to maintain the high vacuum condition in the evaporation cham-ber. The maximum threshold limit of pressure to work with the dual e-beam evaporationchamber is 10−5 mbar. Above the pressure in the chamber will cause random and highcurrent arc discharges around the high voltage parts of the evaporator that will lead tosevere damage. For our experimental condition to use sample preparation chamber thepressure was bellow 10−7 mbar. Calibration (Appendix.A) was done to find out theion current and time to grow a certain thin layer on top of a substrate. Silicon waferused as our substrate.

5.2.2. cleaning chamber

To study the surface of a particular material it is important to have extremely cleansample. Especially Carbon or Oxide contamination layer on top of the sample easilyhinders the actual surface state that makes difficult to study the surface state. Figure5.5a shows a cleaning chamber is placed on top of the sample preparation chamber.Although there is another sample heater in the ARPES chamber sometime one needsto clean a material requires much higher temperature as above 20000 C. Especiallyto remove the oxide layers. It is not safe to generate such higher temperature in theARPES chamber. Electron spectrometer, double mirror, lenses, chamber window etc.in the ARPES chamber cannot withstand above 2000 C and cleaning process cannotbe done in the ARPES chamber. A cleaning chamber is installed in the preparationchamber to prepare a clean sample for surface sensitive photoemission experiment.

This chamber also has one tungsten filament on top of the sample holder. A positive

48


voltage with respect to the tungsten filament is applied to the sample and the electronsfrom the tungsten filament are accelerated towards the sample and heats the sample.A temperature sensor Type C thermo-couple is used to monitor the temperature of thesample.

5.2.3. Conditioning filament

During opening the chamber or during construction process, dust particles in air mayenter into the chamber and one need to remove the dust particle gradually. Otherwiseapplying higher filament current might cause to flow of electricity between the dustparticles and create some unwanted effect like sparking, arching etc.

As a result the filament was conditioned properly each time we obtain the vacuumcondition in the sample preparation chamber. This was done by increasing the filamentcurrent in step by step rather increasing to the maximum value at a time. Initially, thefilament current was set to a lower value,wait 15 s and decrease to zero. Again after 15s later the current was increased in a step of 0.5 A , waited 15 s and set to zero. Theprocedure was maintained until the maximum filament current was obtained. Duringdeposition a thin layer on top of substrate and sample cleaning procedure maximumfilament current was 8 A and 4 A respectively.

5.3. ARPES Chamber

Figure 5.6.: ARPES experimental setup for both static and time resolved experiment,two beam is propagating collinearly and hit the sample by reflecting fromthe double mirror

.

49


5.3.1. sample heater and sputter gun

A Tetra sputter ion gun was installed in the ARPES chamber. A high mass and energeticAr ions or atoms from the gun will remove the surface contamination. But due to thehigh mass and high kinetic energy of the Ar atoms, this procedure can make hole orchange the lattice structure of the sample. As a result a sample heater in the samplestage was installed to heat the crystal after cleaning the surface contamination using thesputter gun. This heat will provide thermal energy to the lattice to rebuild the originallattice structure.

(a)

(b)

Figure 5.7.: Sample stage in ARPES chamber (a) Sample cleaning (b) Double mirror

5.3.2. Double Mirrors

Two mirrors (shown in 5.7b) in the ARPES chamber was installed. These are calleddouble mirror. The mirror having smaller diameter located inner part and other havinglarger diameter was placed in the outer part. Inner mirror is Mo/Si multilayer mirrorand designed for reflecting XUV radiation over bandwidth (figure 5.8) ∼ 5 eV (FWHM)centered at 65 eV, where outer part of the double mirror reflects the Infrared laser pulse.The double mirrors are suitable for time resolved experiment but they offer poor energyresolution.

Mirrors are controlled by three stepper motors. One can move the mirrors back andforth, left and right using the motors. By moving the inner mirror back and forth withrespect to the outer one, time delay between two pulses can be created.

50


Figure 5.8.: Reflectivity of double mirror

5.3.3. Magnetic field compensation

A charge (electron) moving in a magnetic field experience a force named as magneticforce. This magnetic force always is perpendicular to both the velocity of the electronsand direction of magnetic field. As a results electron are deflected from their straightpath. The force can be calculated from the charge times the vector product of velocityand field.

F = qvBsinθ (5.1)

where θ is the angle between magnetic field and the velocity of the electron. In ourARPES chamber photoemitted electrons are also experience magnetic field. The sourcesof magnetic fields are earth magnetic field, pressure gauge magnet, Sputter gun, currentcarrying wires in the lab etc. Due to such magnetic field effect photoemitted electronsare deflected. With such uncompensated magnetic field in the ARPES chamber, it isvery difficult to find the focus of the laser beam. Figure 5.9 shows the photoemissionspectrum using picosecond laser. The photoemitted electrons are spread away from thecenter.

To compensate the magnetic field for ARPES experiment, a Tesla-meter was placed indifferent place in the chamber in order to find the magnetic field in different points. Theearth magnetic field pointing north-south direction was found 60 µ Tesla. The magnetin the ion gauge pump was removed from the chamber during measurement. Finally theremaining field was compensated by the Helmholtz like compensation system.

51


Figure 5.9.: Effect of magnetic field in photoemission Experiment Tungsten

Figure 5.10.: Helmholtz like compensation to compensate the magnetic field in ARPESchamber

Helmholtz like compensation in ARPES chamber

Although the Ideal Helmholtz(Appendix.B) circular coil pairs can create maximumhomogeneity of magnetic field but for some application it is difficult to design rather rect-angular shape is preferred. Furthermore time constraint of doing research also proceedus to construct coil pairs in a rectangular shape aluminum profile. For such non-Idealcase it is not possible to get maximum homogeneity. As a result, the coil dimension wasset to large enough compared to the region where maximum compensation was required.

52


In our design three coils in 3-D space is placed in our ARPES chamber. Each coilsconsists of 12 wires and carrying same amount of current in any direction. Our targetedarea2 for such compensation was between the sample holder and the window of the spec-trometer. A Tesla-meter was placed in one direction and current was set to compensatethat field in that direction. The same procedure was repeated in every x,y,z direction.The magnetic field in the center of the rectangular coil pair is given by [38]

B =1.6nIab√a2 + b2 + c2

1

(a2 + c2)+

1

(b2 + c2)(5.2)

Table 5.1 shows the resultant magnetic field in ARPES chamber after such compen-sation. During such compensation pre-pump of the UHV system was switched on. Asthere is no vacuum sensor in the compensating system, it’s difficult to tell exactly theresultant field when every pump or other installed device is working. These device havecirculating current that can also effect the magnetic field condition. Although the effectwill not be more than few µ Tesla but it is not completely suitable to do photoemissionexperiment that generate electrons having lower kinetic energy. The current system ismore suitable to do photoemission experiment with photon source in the UV range ormore can generate more energetic electrons without deflecting too much.

Direction Magnetic field (µ Tesla) Current(mA) After compensation (µ Tesla)

X -24 0.89 0.001

Y -5 0.25 0.001

Z 66 1.84 0.001

Table 5.1.: Magnetic Field compensation

5.3.4. Ultra high vacuum in ARPES

Ultra high vacuum is characterized by the gas pressure 10−9 mbar or lower. For sur-face science application it is important to protect the surface from contamination or toprevent formation of oxide. UHV is prerequisite for certain experiments such as X-rayphotoelectron spectroscopy (XPS), Angle resolved photoelectron spectroscopy (ARPES),Auger electron spectroscopy (AES), Secondary ion mass spectroscopy (SIMS). Not onlythat UHV is also very important to grow thin film layer techniques like Molecular beamepitaxy (MBE), chemical vapor deposition (CVD).

UHV for photoemission experiment:

There may be some residual gas such as Hydrogen, Carbon monoxide, water in the pho-toemission experimental chamber. It is important to ensure that photoemitted electrons

253.3 mm×10 mm

53


do not collide with other residual gas molecules, otherwise kinetic energy of the pho-toemitted electrons will change. Now one can avoid the collision by increasing the meanfree path of the contaminated gas molecule. There is relation [39] between mean freepath and pressure of the gas chamber given in equation 5.3

λ =kBT√2Πξ2P

(5.3)

where P is the pressure of the gas, ξ is the molecular diameter, λ is the mean freepath. By decreasing the pressure one can get the mean free path of the gas moleculein the order of km, where in ambient condition the mean free path is nearly nanometerrange. Table5.2 shows the mean free path at different vacuum condition. That meansin the extremely low pressure condition molecules are likely to hit the walls of the UHVchamber rather colliding each other.

Vacuum Range Pressure hPa(mbar) Mean free path

Ambient pressure 1013 68nm

Low Vacuum 300 - 1 0.1-100 µm

Medium Vacuum 1-10−3 0.1-100mm

High Vacuum 10−3 − 10−7 10cm-1km

Ultra High Vacuum 10−7 − 10−12 1km-105km

Extremely High Vacuum < 10−12 > 105km

Table 5.2.: mean free path in vacuum condition [40]

That means mean free path of gas molecules is important effect during photoemission.As shown in the universal curve to study the surface state using XUV, we have very lowmean free path of the photoemitted electrons. Lower mean free path means highercollision rate. As a result one can decrease the vacuum pressure and the mean free pathof the gas molecules gets longer which reduces the probability collision. By decreasingthe pressure inside the chamber upto 10−8 mbar one can expect long mean free path ofthe contaminated gas molecule in the experimental chamber.

Vacuum setup

The gas flow under ambient condition is proportional to the pressure difference called”viscous”, but under UHV condition it is molecular that means one has to take intoaccount individual gas molecules. Achieving UHV is not an easy task. It requiresdifferent pumping stages. The prepared experimental chamber is leak-tight vessel. Theparts which we mounted was cleaned properly 3.

3Aceton was used for cleaning outside the screw and ultrasonic bath is used for cleaning the bog hole

54


(a)

(b)

Figure 5.11.: Ultra High Vacuum system(a)Setup for obtaining UHV (b)Baking of ourARPES chamber

Initially a prepump is used4 to evacuate the chamber. But this is not enough toobtain UHV as the molecular mean free path is very long. In the next stage a turbo-molecular pump is used to obtain the UHV condition and it supplies exhaust gas to thepre-pump to remove from the chamber. Figure 5.11a shows the vacuum setup where thepressure gauge monitor the vacuum pressure.

Baking:

Sometimes the gas molecule is adsorbed to the wall of the experimental chamber and topull the gas molecule (ex.water) by desorbtion is a very slow as well as time consumingprocess. One can accelerate the process by heating the chamber to a high temperaturenearly 1000 C or above5. Our experimental chamber (Figure 5.11b) was baked for 72 hourat 850C temperature. Sensor at different place of the chamber was placed to monitorthe temperature. After the bake-out the pressure was obtained < 10−8 mbar.

4works upto 10−2 mbar5depending on the system, different parts can withstand different temperature

55


Choice of materials Materials that create high vapor pressure 6 was avoided in theUHV chamber. At the same time stainless steel are used instead of steel to protect fromoxidation. As the chamber was baked out, low melting point metal 7 was also avoided.Threads that can trap the gas are modified by making a small hole in it.

5.3.5. Connection: Beamline, ARPES and Sample preparation

Figure 5.12.: Overall experimental setup for HHG based ARPES experiment

Finally the beamline is connected to the angle resolved TOF electron spectrometer(ARPES) for static and time resolved band structure analysis. The UHV condition ismaintained throughout the beamline8. Figure 5.12 shows the complete experimentalsetup for time and Angle resolved photoemission setup using HHG beamline. ARPESchamber is connected to the sample preparation chamber either for thin film fabricationor sample cleaning. As mentioned earlier for studying the surface using XUV pulseone has to take care about the surface cleanliness. Connecting the sample preparationchamber one can ensure the surface cleanliness or thin film fabrication before doingphotoemission experiment. There is a load-lock to transfer the sample from preparationchamber to ARPES chamber.

5.4. Characterization of experimental setup

5.4.1. Determining laser spot size after HCF

A CCD camera was employed to find out the beam shape after the hollow core fiber.The beam shape is important otherwise it will not hit the gas nozzle properly. Improper

6Zinc, Cadmium7Indium8Although IR beam can be passed through the air but XUV will be absorbed passing through in the

air

56


hitting of the gas nozzle will make a tiny hole due to high intense laser beam. Thistiny hole is unwanted because gas can leak through the hole and increase the pressureof HHG chamber. Figure 5.13 shows the beam spot (85µm × 100µm) after the hollowcore fiber. The shape of the beam is almost circular.

(a)

(b) (c)

Figure 5.13.: Determing spot size after HCF

5.4.2. Finding the focus point

To examine the Auger process or photoemission process, one need to find the exactfocusing point of the beam.

using XUV photons: A CCD camera is used to find the exact focus of the beam. Asthe XUV beam cannot be seen by normal eye, an alignment HeNe laser was used toget the focus. Moreover, improper focusing to the sample stage will not be enough toget sufficient photoemitted electron signal. After focusing the beam from an alignmentlaser, the IR pulse was used to generate the harmonics. A slit was placed before theexperimental chamber in such a way that passing through the slit will reflect from theXUV mirror and finally hit the sample.

57


(a) (b)

Figure 5.14.: Harmonics are passed through a slit and captured in a CCD camera

Figure 5.14 shows the harmonics are passed through a slit and recorded by the CCDcamera. The acquisition time of the camera was very short than the response time of thememory chip. As a result figure 5.14a shows the slit (marked by circle) and harmonicsare not really centered at the middle. As we increase the acquisition time shown in figure5.14b the generated harmonics are almost at the center of the image.

5.4.3. Energy and time resolution

(a) Determining Energy resolution (b) Determining time resolution

Figure 5.15.: Determing Energy and time resolution

Energy resolution was determine by taking the convolution between the experimentalFermi edge and theoretical Fermi-Dirac integral. The Energy resolution using our setupis very poor (8.01 eV). The resolution is limited in our case as the XUV mirror hasthe reflective bandwidth about 5 eV as well as low acquisition time for taking oneimage. Our experimental setup has good time resolution. As discussed before broadbandreflective XUV mirrors offer good time resolution but poor energy resolution. One canalso estimate mathematically from the energy and time uncertainty that indicate energyresolution is limited by time resolution and vice versa. Time resolution of our setup isdetermined by the Gaussian fit with the light peak of the recorded signal in TOF. Inour case we got the time resolution is 0.1 ns.

58


5.4.4. Determining the time offset

Determining time offset is the precondition to analyze the data that has been takenusing Time of flight(TOF) spectrometer.

XUV pulse: As one can see from the figure 5.15b that light peak is at 156.4 ns. Thetime that needed a photon to travel from the sample to the detector has been subtractedto get the right time offset.

Electron gun The time offset from the electron gun was determined by the elasticallyscattered electron peak. The theoretical value9 was calculated and subtract it from theinelastic electron peak.

Ttheoretical =1486.6√

Ens (5.4)

(a) (b)

Figure 5.16.: Determing time offset (a)Using XUV pulse (b)Using electron gun

9Provided by company SPECS

59

Chapter 6.

Experimental study on Tungsten W(110)surface

Figure 6.1.: Diagram for Auger and Photoemission process

Figure 6.1 shows the design to study the surface of a tungsten crystal by Augerand photoemission spectroscopy. Auger spectroscopy was used to find out the surfacecontamination level and photoemission spectroscopy was used to study the Fermi surface,bandstructure and photoelectrons energy spectrum.

6.1. Auger spectroscopy on Tungsten(110)

6.1.1. Detection of surface contamination

Auger spectroscopy can be used to detect the elements that present on a sample surface.Before starting the photoemission experiment, Auger spectroscopy was employed to findthe surface contamination on the tungsten surface.

A electron gun shown in Figure 6.1 was used that can generate high energetic andpulsed mode electrons at a repetition rate of 50 kHz. The sample was excited by a 1keV electron source to see the Auger process in a ultrahigh vacuum condition (8×10−8

60

Chapter 6. Experimental study on Tungsten W(110) surface

Figure 6.2.: focusing electron beam on the sample surface

Figure 6.3.: Auger Process in Tungsten surface

mbar). Figure 6.3 shows the Auger process in the Tungsten surface, where the upperone is plotted Count (N)vs Electron Kinetic energy (E). The spectrum was taken for

1000 s. To make the Auger intensity visible the lower figure was plotted abs(log(dN

dE))

61


vs E.

Several peaks have been identified in the Auger Spectra from W surface. At electronenergy 180 and 350 eV peaks are coming from Tungsten W [41]. According to theCoster-Kronig theoretical model [42] these peaks are coming from N4N6O3 and N3N6,7Xtransition. The peak at approximately at 280eV due to the Carbon(C) [41, 43, 44] KLLtransition [45]. Similar KLL transition in Oxygen (O) [41, 43, 46] has been found atapproximately 510 eV electron kinetic energy. Finally, the peak near 770 eV electronkinetic energy is due to the Cobalt (Co) [47] atoms. During the construction process inthe ARPES chamber some stainless steel screws were used that may have Co compound.

Especially Carbon and Oxygen are highly reactive with the W surface and henceproduce some strongly adherent top layers. As shown in universal figure 3.5 (Energyvs mean free path), when performing photoemission using 65 eV XUV photon sourcethen the mean free path of the photoemitted electrons from the tungsten surface is verysmall just below 1 nm. Moreover, XUV photons have penetration depth of few nm. Asa result few oxide monolayers in the W surface will change the expected photoemissionoutcome.

6.1.2. Sample cleaning: Method and verification

Before growing epitaxial Gadolinium layer on the tungsten, it is necessary to clean thesample surface. For the experimental purpose the tungsten surface was cleaned in ahighly efficient way [48]. A small contamination can change the whole physical property.The most familiar contamination elements in tungsten sample are Carbon and Oxygen.

Carbon compounds in tungsten sample can be segregated from the bulk to the surfaceby annealing. Now if the annealing is done at high temperature above 12000 C in Oxygenenvironment then it will react and form volatile CO (Carbon monoxide) compound. Thateasily remove the Carbon layers, but long time annealing will lead bad vacuum conditionand form a layer of Tungsten Oxide. Again such atomic oxygen contents and tungstenoxide can be removed at elevated temperature like 22000 C. The substrate was heatedby the electron bombardment. A constant dc voltage 1 kv was applied to the target anda dc current running to the filament results emission current from filament to the target.In order to maintain good UHV condition in our chamber against heat dissipation, ashort term heating (flashing) was applied to the system. The cleaning was done in twosteps as described in [48].

1.Formation of oxide layer: the heating is done in Oxygen atmosphere for a veryshort periodic basis. The heater is on and off for 20 s and 3 min subsequently withthe Oxygen partial pressure about 10−7 mbar. During the time surface contaminatedcarbon can react with the Oxygen and form carbon monoxide. During the heating athermocouple attached in the sample cleaning chamber records the temperature. Theemission current was obtained 100 mA along with 1.5 kV voltage during heating. Asa result Carbon contents are removed from the surface. The number of heating cycle

62


depends on the cleanliness of the Tungsten surface. This cycle is known as low powerheating or lowpower flashing at 12000 C temperature. 10-12 complete cycle was done toclean the sample.

2.Remove the oxide layer: The procedure is the same as mentioned before inlow power flashing, but the difference is that instead of low power heating a cycle ofhigh power flashing (HPF) (1.5 kV with emission current 250 mA) was applied withconsecutive turning on and off the heater. During the time temperature was obtainedabout 22000 C at vacuum condition (4.6×10−8 mbar) to remove the oxide layer on theTungsten surface. 5-8 complete HPF was done to remove the oxide layer

Figure 6.4.: Auger Spectroscopy to verify surface cleanliness

Figure 6.4 shows the Auger spectroscopy that has been performed in UHV condi-tion (4.6×10−8)on tungsten(110) surface in order to check the cleanliness of the samplesurface. Our main focus during cleaning process was to remove the carbon and oxidecomponents in the sample surface. The upper part of the figure 6.4 shows the Augerpeaks before cleaning. The Auger spectra detects C and O in our sample surface. Asmentioned in [44] several Auger peaks from C 1s elctrons will appear between 280-302eV. Moreover, experimental study report [43, 46] that the Auger peak of O will appearabove 500 eV. Before cleaning the spectra contains a broad peak both from carbon thatspans from 273-300 eV and oxygen at approximately 502-520 eV. Both of them are due toKLL Auger transition. After few high temperature flashings the C and O were removed

63


considerably. The sample was cleaned until O and C were completely removed.1

Figure 6.5.: Auger peaks from clean W surface

To see the Auger peaks from a clean W surface the Spectrum shown in figure 6.5has been taken in wide angle resolving mode with a narrow window in order to increasethe energy resolution. Three dominant peaks [42] are identified in the spectrum fromclean W in the range of 160-170 eV electron kinetic energy. Peaks in the electron kineticenergy at approximately 164 eV and 169 eV are identified as N5N6N7 and N4N7O3 Augertransition of W. The peak that has been identified near 160.6 eV was also theoreticallypredicted as due to N5N6N7 transition.

6.2. Crystal property of Tungsten(W)

Tungsten(W) is a prototypical transition metal that grows much interest to study fun-damental process. In real space Tungsten lattice is a body centered cubic lattice (BCC)structure having two atoms per unit cell, where in reciprocal space it becomes facecentered cubic (FCC) and vice versa.

The real space cell2 shown in figure 6.6a, the intersection point with the [100] directionis called X (H).The line Γ−X is called ∆.The intersection point with the [110] directionis called K (N).The line Γ−K is called Σ.The intersection point with the [111] direction

1For time constraint and Instrumental problem, the Auger spectra from clean tungsten surface has notbeen taken.

2Real space unit cell called Wigner-seitz unit cell

64


(a) (b)

Figure 6.6.: (a) real space (b) reciprocal space [110] direction and detector position [49]

is called L (P).The line Γ−L is called Λ. The corresponding reciprocal image Brillouinzone3 shown in 6.6b as well as the detector position and XUV beamline.

6.3. Photoemission Spectroscopy on Tungsten (W)

Figure 6.7.: Exciting sample surface with XUV photons

The sample has been excited by a 65eV XUV attosecond photon source. The at-tosecond XUV spectrum was generated from HHG process. The XUV pulse is focused

3Reciprocal of Brillouin zone is Wigner-seitz cell

65


to the sample surface after reflecting from multilayer Mo/si coated XUV mirrors. Thespectrum was taken at count rate 1500 count/sec for 10 min in a ultrahigh vacuumcondition (6×10−8mbar) as well as Helmholtz like magnetic field compensation system.Figure 6.7 shows raw image recorded by Themis 1000 Spectrometer.

6.3.1. Observation of Conduction band Spectrum

(a)

(b)

Figure 6.8.: photoelectron Energy spectrum (a) using HHG-ARPES (b)comparison be-tween synchroton and HHG spectrum [32]

Figure 6.8 shows the photoelectron spectrum after excitation with 65 eV XUV photonbeamline at 10 kHz repetition rate. Figure 6.8a shows the spectrum that was takenin wide angle resolving mode. Three regions have been indicated in the figure 6.8a

66


secondary electrons, conduction band electrons and Fermi edge. As mentioned dueto poor magnetic field shielding, low energetic (< 35 eV) electrons experience magneticfield. Moreover, the spin-orbit coupled 4f doublet from Tungsten will be located between25-28 eV. As a result the photoelectron energy spectrum shown in 6.8a has two differentcontribution one from secondary and other from 4f core electron states below 35 eV.Furthermore, due to poor energy resolution using broadband XUV multilayer mirror 4fdoublet peaks are not resolved in figure 6.8a. The result is also consistent with the figure6.8b. That shows a comparison [32] of photoelectron binding energy distribution between118 eV HHG based Photon source (reflected from 4.2 eV XUV multilayer mirror) andsynchroton source. One can also observe that the 4f peaks are not resolved by usingthe HHG source due to the broadband mirror where they are completely visible usingsynchroton source. In addition, 4f photoelectron cross section from W are very smallcompared to the total photoelectron cross section using 65 eV photon source, which alsocan suppress the 4f peaks from others.

Figure 6.9.: Total and 4f photoelectron cross section from W [50]

To see the exact shape how is the electron distribution above 35 eV electron kineticenergy the Figure 6.8a was filtered and shown in figure 6.10a. The filtered image wasalso fitted with the 10th order polynomial function which reflects the conduction bandfeature.

Furthermore, exciting a nicely cleaned W sample surface will give a pronouncedconduction band feature. Because if the sample surface has few oxide or carbon mono-layers, then due to low penetration depth of XUV photon and low mean free path ofthe surface electrons photoelectron excitation from the Tungsten conduction band would

67


(a) (b)

Figure 6.10.: photoelectron energy distribution (a)(35-67 eV) after filtering (b)Nearfermi edge spectrum to see conduction band spectrum, CB: conductionband, FE: Fermi edge

not be possible. To examine the conduction band near the Fermi edge more deeply, aspectrum shown in figure 6.10b has been taken in wide angle resolving mode with avery narrower window near Fermi level. Arrow indicates the conduction band and theFermi edge. One can see a conduction band hill is followed by a valley of the electrondistribution just below the Fermi edge. This feature also support the theoretical resultshown in [51]. Such distinct conduction band feature also confirm us that our sample isnicely cleaned using our cleaning method.

6.3.2. Fermi surface

Figure 6.11 shows the Fermi surface of W. Figure 6.11a shows the experimental mappingand figure 6.11b gives the theoretical Fermi surface. One can see the difference of thetheoretical and experimental Fermi surface mapping.

The reason for such disagreement in Fermi surface mapping was examined closely.There is a relation among k‖, detection angle φ and photon energy hν [53]. Figure 6.12shows the plot of equation 6.1 that explains the fraction of brillioun zone one can seewith ±130 angluar range.

hk‖ =√

2m(hν − ΦW ) sinφ (6.1)

The sample stage in the current setup is not rotate able in any direction, only itcan move back and forth, left or right. The experimental Fermi surface in figure 6.11ais mapped by the spectrometer that extends from -130 to 130 degree. Above ±130 the

68


(a) (b)

Figure 6.11.: Tungsten(110) Fermi Surface mapping using 65eV XUV pulse(a) Experi-mental, (b) Theoretical [52]

spectrometer can not detect the electrons. As a result, calculation using above equation6.1 shows the available photon source 65 eV XUV can map 51.53 % of the BZ (figure6.12) and so the full size of the lobes are not observable, one can just see their onset.The experimental plot also give us the hint that four lobes(left, right, up and down)4 areextending in four direction. In practice six different lobes extends in 3D space (figure6.11b) but due to 2D mapping lobes that are extending front and back side of the surfacecan not be seen due positioning of the [110] direction (figure 6.11a). Comparing betweenexperimental and theoretical Fermi surface one can easily understand that the (110)plane is oriented 450 with the major axis.

Figure 6.12.: Fraction of Brillouin zone using current experimental setup

4lobes are due to the BCC structure, In BCC horizontal direction has four-fold symmetry where verticaldirection has two-fold symmetry

69


6.3.3. Bandstructure observation

The Bandstructure of Tungsten(W) was recorded (shown in figure.6.13) in wide-angleresolving mode in ARPES experiment using 65 eV XUV photon source. Figure.6.13aplots the energy bands near the Fermi edge to see the conduction bands in Tungsten.Figure.6.13b was taken in a broad window in wide angle resolving mode to see boththe conduction band and core level electronic bands in W. As discussed before due tomomentum conservation under periodic boundary condition in ARPES experiment, onegets the direct information about the parallel momentum of the photoemitted electrons.Due to the sample surface orientation (110) one can only detect electron momenta inΓ-N-H direction, i.e. parallel to the surface.

Theoretical band structure shown in figure 6.14 has been plotted using linear muffin-tin orbital method with plane wave basis (LMTO-PLW) in different symmetry directionwith respect to k‖ space. It has been highlighted in the plot the maximum limit of k‖in bandstructure diagram one can see using our setup. The theoretical plot shows alongthe N-Γ-P direction have flat bands near the Fermi edge and theses flat bands have highdensity of electronic states. Exciting the surface with a photon source having photonenergy above the W work function will give rise more photoelectrons near Fermi edge dueto high density of states. As a result one can see from the experimental bandstructureplot in figure 6.13a more photoelectrons are located near the Fermi edge.

(a) (b)

Figure 6.13.: Tungsten Bandstructure in Angle resolving mode(a)small window (b)Broadwindow

Moreover, as one goes few eV below the Fermi level in W, dispersion due sp elec-tronic bands starts to appear (figure 6.14). Closely examine the bandstructure recorded

70


experimentally (figure 6.13a) one can get the hints that the parabolic feature starts tobe appear few eV below the Fermi edge. Figure 6.13b also shows a bandstructure thatextends far below to the Fermi edge to the core level states. Due to low acquisition timethe dataset is quite pixelized and so detailed features are not fully resolved.

Figure 6.14.: Theoretical Bandstructure N-Γ-H direction, FL: Fermi level

71

Part III.

Future work: Improvement,conclusion

72

Chapter 7.

future work

From the experimental results on Tungsten surface, it is quite obvious that the HHGbased ARPES setup has some difficulties. One needs to overcome the difficulties toinvestigate a material surface more accurately.

7.1. Area of improvements:

Lower Bandwidth Mirror or using grating: During our experimental study XUV mir-rors with 5 eV Bandwidth offers very poor energy resolution which is not good to studythe surface phenomena. As one can see from experimental photoelectron spectrum 6.8athat the doublet feature of the 4f electrons is not obvious. Using lower bandwidth (bel-low 1 eV) reflective mirror one can get good energy resolution to see better Fermi edge,steeper conduction band features, 4f doublets etc. One has to bear in mind that usinglower BW mirror is not good for time resolved study. Moreover, one can also use gratingto separate a particular harmonic instead of using lower BW mirror. Figure 7.1 showsthe process where the two different wavelength of light has been separated by usinggrating and an aperture.

(a) (b)

Figure 7.1.: A laminar periodic grating can be used to diffract a particular wavelengthλ of light (a), Different wavelength of light experience maxima at differentangle after diffracted from the grating and an aperture is used to filter [22]

73

Chapter 7. future work

Longer acquisition time: During our experiment the acquisition time was not enoughto resolve very good features in the recorded image. It was not possible to record imagefor longer time due to instrumental instability. Specially, I should point out this is veryimportant to see better Auger Spectra. In our recorded Auger spectra shown in figure6.3 and 6.4, the peaks from C and O are not so sharp (except W figure 6.5) rather broad.This is mainly due to the lower acquisition time as well as the poor resolution of ourspectrometer. The poor bandstructure features are also due to lower acquisition time.

Degree of rotation in sample stage: Rotational degree of freedom in the sample stageis extremely important for our setup. As we have seen from figure that BZ of the Fermisurface is mapped within ±130. Within this angle a part of the Fermi surface near thecenter of the Brillouin zone has been seen. Using our available light source inclusion ofrotational stage in sample holder can help one to record the entire part of the brilliounzone image by rotating the sample.

Increasing the photon energy and photon flux: Increasing the photon energy is an-other important criteria to see the whole brillioun zone. Using our setup without rota-tional degree of freedom one needs about 300 eV photon energy (figure 6.12) to see thewhole brillioun zone. On the other hand, increasing intensity or the photon flux one canextract more electrons and record some better features to study a physical phenomena.

XUV Pulse duration: The exact pulse duration of the XUV attosecond1 pulse thathad been used in ARPES experiment is not known. For studying the static processthis pulse duration is not so important but studying the dynamics process exact pulseis very important. A setup can be made to see the pulse duration that is familiar as”attoscond streaking experiment” shown in figure. 4.11.

Better Magnetic shielding: Our experimental setup is good to study the electronswith higher energy, but not good to study low energetic electrons. The ARPES cham-ber can be shielded by the µ metal sheets to study the low energetic electrons. Becausepresent magnetic field compensation system cannot generate homogeneous magnetic fieldthroughout our chamber. Moreover, there was no vacuum sensor that can give the read-ing when every pump was running or any current wire was crossed in our experimentalchamber. One can also understand from the photoelectron spectrum in figure 6.8a thatsecondary electrons those are affected by the magnetic field overlapped with the 4f coreelectrons.

1attosecond can be argued theoretically after seeing the pulse spectrum, but this will not give the exactduration

74


Introducing phosphor plate: Introducing a phosphor plate would reduce some effortduring experiment to find the exact focus of the XUV beam in the sample stage. Thesetup can be designed to introduce one phosphor plate in the other side of the doublemirror stage. The beam reflected from the XUV mirror will illuminate the phosphorplate and one can find the beam position. As XUV beam cannot be seen in normal eyeor eye-goggles, every time to find the beam position we close the HHG beamline and useIR laser.

Better Vacuum condition: Better vacuum condition means better experimental out-come. At present, the vacuum pressure is 8.6 × 10−8 mbar. Baking the UHV chamberfor longer time, this vacuum condition can be further improved to bellow 10−9 mbar.Especially for surface science experiment, the sample has to be clean enough due toshort mean free path. One theoretical study suggest that to grow one monolayer ofcontamination it’s enough to have only 10−6 mbar for one second,using 10−7 mbar itincrease to 10 s, using 10−8 mbar it increase to 100 s. Where few monolayer of oxidelayers are enough to destroy the whole photoemission process specially to study surface.But we are careful clean the sample (as mentioned in chapter.5) every time before anyphotoemission experiment or Auger process.

Increasing the repetition rate: With a better vacuum condition, better shielding onecan increase the repetition rate of the IR pulse. Currently, it is 10 kHz and one has towait for longer time to record one image which is very time consuming. On the otherhand, repetition rate of IR laser have an impact on the HHG process. The cutoff ofthe harmonics will decrease as the repetition rate goes higher. As a result, dependingon the experimental requirement one has to decide carefully regarding to increase therepetition rate of the IR laser.

7.2. Prospect for time resolved (TR) study:(TR-ARPES)

One limitation using static ARPES experiment is that it can give us spectroscopic in-formation about the occupied electronic state bellow Fermi level (E < EF ). Completeunderstanding of unoccupied electronic state above the Fermi level is demanding as itdetermines the material properties (i.e optical absorption). To explore the unoccupiedstates, two photon photoemission (2-PPE) was established where two pulses (pump andprobe) are used. Ultra-short pump pulse is used to excite the occupied electrons abovethe Fermi level. A probe laser pulse is used with a finite time delay to photoemit theelectrons and allow us to study the time evolution of transient population of unoccu-pied states by varying the delay of the two pulses with sub-fs time resolution. 2-PPEtechnique is a powerful tool to study the dynamics of hot electron distributions. In thisscheme, both photon energies are below the metal work function with low excitation

75


density to suppress the strong signal of direct photoemission and study small perturba-tion of the system [55]. Combining the 2-PPE technique with ARPES tool using novellight sources will lead towards Time and Angle resolved Photoemission spectroscopy(Tr-ARPES). One can access to the occupied and unoccupied electronic states usingTr-ARPES. A pump femtosecond few cycle IR pulse (∼ 1.5 eV) is used to excite thesystem and a probe attosecond XUV pulse (> 50 eV higher than the work function ofmetal)(through HHG process) can be used to observe the transient evaluation of theexcited electronic state. Tr-ARPES has been developed in the 1980s [54] but the pulseduration was not so short as now. The advantage of our setup is that our pulse durationis very short near sub-femtosecond or attosecond range.

Tr-ARPES can provide temporal evolution of the transient spectral function A(ω, k, t)that may give unique access to the non-equilibrium dynamics of the occupied band struc-ture. Tr-ARPES has several advantages over other time domain technique such as opticalspectroscopy or THz spectroscopy. First, it has direct access to the electronic states inthe energy domain. Secondly, along with the momentum resolution of static ARPES,Tr-ARPES can give access to the dynamical process in electronically anisotropic sys-tem [55]. Figure 7.2 shows the Banstructure electron dynamics in topological Insulatorusing Tr-ARPES.

Figure 7.2.: Time Resolved ARPES to study Topological Insulator [56], the bandstruc-ture of TI has been mapped in different time delay

As mentioned before our set up using XUV mirror and broad band pulses are moreperfect for time resolved experiment than any static experiment where one need betterenergy resolution. Static measurement on Tungsten(110) surface can be further extendedto pump-probe approach to investigate the bandstructure electron dynamics in the sur-face state. A experiment [57] has been performed on Tungsten surface using attosecondpulses through HHG process to find out the relative group delay of quantum mechan-ical wave packets those are originating from core state and conduction band. Usingpump-probe technique (Tr-ARPES) few IR pulse can be used to create non-equilibriumdynamics in Tungsten surface and XUV pulse can be used to study the transient eval-uation of the electron dynamics in bandstructure. That will provide a rich informationabout the unoccupied electronic state of the bandstructure to determine the materialproperties.

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Chapter 8.

Conclusion

The report describes the contribution in developing the scientific instrumentation as wellas experimental study using novel attosecond XUV beamline. Moreover, the technicalchallenges one has to consider to setup a HHG based ARPES experimental chamber forboth static and time-resolved experiment has been figured out and discussed briefly. Myexperimental thesis work can be summarized as follows:

Setup and calibration: Broadband few cycle IR laser pulse was successfully generatedat a repetition rate of 10 kHz from a commercial femtosecond laser that subsequentlywas used to generate attosecond XUV pulses having photon energy above 100 eV usingHHG process via ionizing the Ne atom. Two filters (Al and Zr) were used to calibratethe higher order harmonics by means of a XUV spectrometer. ARPES chamber wasmagnetically shielded by Helmholtz like compensation coils. The setup was completedby connecting the chamber to the HHG beamline, by installing double mirror stage, TOFspectrometer etc. to study the solid state sample surface. On the other hand, samplepreparation chamber was installed and connected to ARPES chamber in order to cleanthe sample surface before studying it by ARPES or to fabricate thin films on top of thesubstrate. Moreover, the calibration of TOF spectrometer and thin-film fabrication byelectron beam evaporation has been performed.

Experimental study and observation: The current HHG-based ARPES setup havingbroadband multilayer XUV mirror is also capable to do time resolved experiment. Beforestudying the dynamics, ARPES study was performed on a prototypical transition metalTungsten(W) [110] plane using highly surface sensitive 65 eV attosecond XUV pulses ina short term basis. The sample surface was successfully cleaned by high temperatureflashing and verified by Auger spectroscopy before ARPES study. Distinct conductionband features as well as the Fermi edge was observed by examining the photoelectronenergy distribution. The Fermi surface was mapped having symmetrical lobes along thefirst BZ. Finally, the conduction band bandstructure of W has been recorded that showsthe high density of states d band electronic states near the Fermi edge.

Overcoming every difficulties that we have in our present setup, this newly buildHHG based ARPES will add a new dimension in solid state research in coming days.

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Appendix A.

Calibration of evaporation chamber:

Four different kind of materials Silver(Ag), Gold(Au), Chromium(Cr), Gadolinium(Gd)was used to calibrate evaporation chamber. Figure A.1 shows the thickness of the layerwith different ion current.

Figure A.1.: Calibration of dual electron beam evaporator

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Appendix B.

Helmholtz compensation

Helmholtz coils are the 3D axial coils with a separation distance equal to their radiusthat are designed to provide homogeneous magnetic field as well as to compensate themagnetic field. Using such arrangement one can also create unidirectional magneticfield by changing the current ratio between coil pairs. This design was first proposed byGerman Physicist Herman von Helmholtz.The resultant magnetic field depends onthe winding, geometry and amount of current that flows in the coil pairs.

Whenever a charge is moving around the space or spinning around its orbit, a mag-netic field is created. As a result A current carrying wire (as shown in fig B.1a) or loopcan generate the magnetic field around their path. Magnetic field at the centerline ofa current carrying loop can be calculated using the Biot-Savart Law. The magneticfield due to the current carrying element is given by

dB =µ0I ~dL× ~1r

4πr2(B.1)

Where ~dL is the infinitesimal length of the current carrying conductor and ~1r isthe unit vector with distance r from the current to the field point. Each Infinitesimalelement in the figure B.1a makes a contribution at point p to generate a magnetic fieldperpendicular to the current carrying wire.

Now consider one pair of Helmholtz coils (1D) shown in figure B.1b each of themcarrying same amount of current and separated with a distance equal to their radiusR. Considering Biot-Savart law and number of current carrying wires in a coil themagnetic field is given by

dB =µ0nIR

2

(R2 + x2)3/2(B.2)

In Helmholtz coil pair system, x=R/2 a halfway point between two loop. after suchsubstitution one can get the modified formula

dB = (4

5)

32µ0nI

R(B.3)

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(a)

(b)

Figure B.1.: Magnetic field compensation by Helmholtz coil pair (a) current carryingloop generate magnetic field (b)Helmholtz compensation [58]

Resultant field using Helmholtz coils Suppose a pair of coils has been placed alongthe East-West direction. The earth magnetic field ( ~Bearth) point in the North-Southdirection. As a result a compass always point in the N-S direction when the currentthrough the loop is zero. However, electrical current through the loop produces thehomogeneous field ( ~Bc) in the halfway distance from the loops. The compass will pointto resultant field that is given by vector sum of the two field. The compass will graduallypoints towards the E-W direction as the current through the loop is increased.

~Bsum = ~Bearth + ~Bc (B.4)

| ~Bsum |=√| ~Bearth |+

√| ~Bc | (B.5)

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