Analysis of the electronic structure of LiCu_2O_2 and LiCu_3O_3 by angle resolved photoemission spectroscopy

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POLITECNICO DI MILANO

School of Industrial and Information Engineering

Master of Science’s Degree in

Engineering Physics

ANALYSIS OF THE ELECTRONIC STRUCTURE OF

LICU

2

O

2

AND LICU

3

O

3

BY ANGLE-RESOLVED

PHOTOEMISSION SPECTROSCOPY

Thesis Advisor: Prof. Claudia DALLERA

Research Supervisor: Prof. Marco GRIONI

Candidate:

Gianmarco GATTI

Matr. 817689

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List of Figures

1.1 The Hertz’s experiment setup . . . 12

1.2 The photoexcitation process . . . 14

1.3 A rough classification of vacuum pumps based on their pressure operating regime[17] . . . 18

1.4 Sections of four vacuum pumps: a) rotary pump, b) turbomolecu-lar pump, c) ion-getter pump, d) cryopump. . . 19

1.5 Structure of the Scienta VUV 5000 He-lamp. . . 20

1.6 The Scienta Phoibos 150 hemispherical analyser. . . 22

1.7 A typical wide energy Au XPS spectrum[5] . . . 27

1.8 This set of AIPES spectra at normal emission and various pho-ton energies from a LiCu2O2 sample shows several intensity peaks k⊥ dispersing at binding energy bigger than 3 eV and others not within 3 eV from Fermi level. [7] . . . 28

1.9 The typical photoemission geometry[9] . . . 28

1.10 LiCu2O2 constant energy maps . . . 29

1.11 LiCu3O3 ΓMXΓ path and a series of EDCs along the same contour 30 1.12 The solid line is a photoemission spectrum of the H2 molecule, as a model of electron-correlated system. Each peak corresponds to a different vibrational state. The dashed line represents a photoe-mission spectrum taken from a solid, within an interacting-electron model. It is made of a broad incoherent part, composed of several satellite peaks, and of a narrow coherent peak.[33] . . . 35

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1.14 A schematic representation of the escaping phase: only photoelec-trons with k⊥ > k⊥,min =

√ 2m ~

√

V0 may escape from the surface

of the solid. The thick solid line shows the internal and external escape cone. . . 37 1.15 Photo-excitation for a)a free-electron and b)a Bloch electron . . . 39 1.16 3-step Model and 1-step-Model . . . 41 2.1 MRAMs are among the most useful technologic applications

in-volving strongly-correlated materials. . . 44 2.2 The dispersion relation for an uncorrelated system . . . 50 2.3 The Hubbard model applied to the case of the H2 molecule:

op-posite spin configuration (AFM) allows hopping between the two sites, while like spin cannot delocalize without violating the Pauli principle. . . 55 2.4 The 1D-chain model for an extended system: a) represents the

ground state and b) stands for an excited state where an electron hops from the ith _{site to the j}th _{site. . . .} ₅₆

2.5 The Mott-Hubbard description of the lower energy states: the figure shows the lattice parameter dependence of the energy at-tributed to the states a) and b) in fig. 2.4 . As the lattice parame-ter a decreases, the bandwidth B of each state increases until the system undergoes a metal-insulator transition, when U = B.[16] . 57 2.6 The copper oxide(II) lattice structure: tenorite structure. . . 58 2.7 Localized electrons in transition metal compounds behave as in

molecules: metal d orbitals overlap with ligand oxygen p orbitals, leading to two mixed states, the bonding 3dn_{L and the antibonding}

state (3dn_{L)∗. The last one could be seen as a situation where the}

oxygen donates an electron to the metal.[35] . . . 59 2.8 A schematic illustration of Mott-Hubbard and Charge-transfer

in-sulators [13] . . . 60 2.9 Crystal structure of LSCO (at left) and the CuO2 plane

geom-etry(at right): coppers in blue, oxygens in pink. Each copper atom brings a hole which is coupled antiferromagnetically with the neighbouring one. . . 61 2.10 Orbitals scheme showing the simmetry of the Zhang-Rice singlet. 62

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2.11 The cuprates corner-sharing CuO2plane and the edge-sharing CuO2

plane: coppers in red, oxygens in blue. . . 63

3.1 Crystal structure of the LiCu2O2 and its projection onto the a − b plane to show chains structure. . . 66

3.2 On the left, Al Kα XPS spectrum of the valence band; on the right, a scheme of the density of states coherent with a Zaanen-Sawatsky-Allen model. . . 68

3.3 Papagno’s(left) and Cheng’s (right) Angle-Resolved spectra along the ΓY path at 22 eV photon energy. . . 68

3.4 The first two figures show a set of EDCs collected at 47 eV and 73 eV, respectively, along the b-axis (and a-axis); the third one is taken at 47 eV along 45◦ direction. . . 70

3.5 A LEED image showing diffraction spots corresponding to twin domains oriented along two perpendicular chain directions . . . . 71

3.6 The photoemission geometry of Berkeley beamline. . . 72

3.7 Integrated ARPES data taken at a photon energy of 118 eV . . . 73

3.8 On the left an ARPES map taken at a binding energy EB = 0.5 eV and on the right a representation of the a − b plane. High-symmetry points are shown into the ARPES image as well as the unit cells in both the direct and reciprocal space. The black one correspond to the primitive Brillouin zone while the red dotted one is linked to the oxygen sublattice. . . 74

3.9 ARPES cuts along kx . . . 75

3.10 ARPES cuts along ky . . . 76

3.11 An ARPES cut along the path ΓY'YΓ . . . 77

3.12 DFT calculations representing the ΓY'YΓ path: on the left a cal-culation that does not include correlation energy and on the right one that includes it. . . 78 3.13 Wannier-projected partial Density of States: different colours are

used to different orbital projections. Inert coppers are the Cu+ atoms placed between the coupled LiCuO2 planes, while O1 and

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3.14 This figure defines the hopping parameters that will be used in the TB model. Lithium atoms are grey, coppers are light red and oxygens are blue; the O-O distance changes by a distance ∆ along the a-axis between the two plaquettes, so that two inequivalent oxygens sites A and B are defined. The figure shows the nearest neighbours hopping parameters (t+, t−and tb), as well as the

next-nearest neighbours (tb+ and tb−). . . 81

3.15 The profile of the photoemission intensity (on top) and of the en-ergy dispersion of the two bands ϕ+ _{/ ϕ}− _{(at the bottom), for the}

case of a perfectly squared oxygen lattice (∆ = 0). Red and blue lines are related to the bands defined by the ϕ− or ϕ+ state, re-spectively. The thick black feature in the second picture combines bands dispersion and the intensity profile, representing how the photoemission signal should be. . . 83 3.16 The profile of the photoemission intensity (on top) and of the

en-ergy dispersion of the two bands ϕ+ _{/ ϕ}− _{(at the bottom), for}

the case of a perfectly squared oxygen lattice (∆ = 0.3˚A). Red and blue lines are related to the bands defined by the ϕ− or ϕ+

state, respectively. The thick black feature in the second picture combines bands dispersion and the intensity profile, representing how the photoemission signal should be. . . 84 3.17 ARPES cut along the ΓY'YΓ path. The bands coming from our

2-bands TB model are tracked by white dashed lines. . . 85 3.18 Experimental ARPES cut along the kx-axis at ky = π/b (top) and

ky = 0 (bottom). . . 88

3.19 ARPES cut along the path ΓY'YΓ and its relative color scale. The white dashed line shows the band dispersion extracted from the TB model. . . 89 3.20 ARPES Constant energy maps for LiCu2O2 at a binding energy

(from the top to the bottom) of 0.5 eV, 1 eV and 1.75 eV. On the left column, there is experimental data; on the right, one finds the results of Tight Binding calculations. . . 90 3.21 Representation of Wannnier functions |A > and |B > in the Cu+

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4.1 A simple scheme of LiCu3O3 lattice structure. [15] . . . 95

4.2 A set of ARPES constant energy maps taken at increasing binding energy; the valence band of a polycristalline gold sample has been probed to measure the Fermi energy and use it as the reference for the values of binding energy. . . 97 4.3 A portion of the surface of LiCu3O3 (001) cleaved between two

CuO2 planes (on the left): red atoms represent coppers while blu

atoms are oxygens. The black square is 2D unit cell and the yellow one is the antiferromagnetic cell of one corner-sharing sublattice. A constant energy map at E=0.7 eV where the 2D Brillouin Zone (in black) and the magnetic unit cell (in yellow) are shown (on the right). . . 98 4.4 Corner sharing CuO2 (a) and edge-sharing CuO2 (b) layers.

Cop-per d and oxygen p orbitals are respectively red and blue. The CuO2 edge-sharing layer is seen as a superposition of two

corner-sharing sublattices (I) and (II). . . 99

4.5 ARPES cut along the ΓMXΓ path where the undoped α band

appears. . . 100 4.6 ARPES cut along the ΓMXΓ path where the doped α band

ap-pears. A faint replica of this band is present at higher binding energies: this is the undoped α band. . . 100 4.7 ARPES cut along the ΓMXΓ path whose EDCs have been fitted

through an initial guess made up of two Gauss functions. Different colors of the markers identify the energy position of each peak, while energy width and signal intensity are represented by bars’ and markers’ dimension. . . 102 4.8 Double derivatives versus energy and momentum, respectively, of

raw data shown in fig.4.6. . . 103 4.9 ARPES cut along the ΓMXΓ path: raw data and double derivative

versus energy. Red markers are used to keep track of α band previously digitalized. Markers of different colors are used to follow displayed bands. . . 104

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4.10 ARPES energy cut along the ΓMΓ path: raw data and double derivative over energy. Yellow markers are used to put in evidence the ZRS dispersion. . . 105 4.11 ARPES energy cut along the ΓXΓ path: raw data and double

derivative over energy. Yellow markers are used to put in evidence the ZRS dispersion. . . 105 4.12 ARPES cut along the ΓMXΓ path: raw data and double derivative

versus energy are put in sequence. Yellow markers are set to follow a low dispersing state at the top of the valence band. . . 106 4.13 ARPES cut along the ΓMXΓ path: raw data (top) and raw data

fitted between X and Γ (bot). Markers make use of the same con-vention about the meaning of markers’ size, colours and position, as well as error bars. . . 107 4.14 ARPES Constant Energy map at a Binding Energy of 0.3 eV,

shown for LiCu2O2 (left) and LiCu3O3 (right). High-simmetry

points are recognized in the k-space and marked on the spectra. . 108 4.15 ARPES cut along the ΓY’YΓ path (LiCu2O2, top) and along the

ΓMXΓ path (LiCu3O3, bot). Notice that LiCu3O3datas have been

taken at a pass energy equal to 60 eV while LiCu2O2 one at a pass

energy of 50 eV. . . 109 4.16 ARPES constant energy map taken at the top of the valence band

for T-CuO (left) [27] and LiCu3O3 (right). T-CuO map contains

also marked high-simmetry points and an inset describing direct space lattice, where the lattice parameter a is shown. . . 110 4.17 Comparison between cut c) (see fig. 4.16) extracted from T-CuO

(top) and LiCu3O3 (bottom). . . 111

4.18 Comparison between ARPES cut along the AΓ path, zoomed close to the top of the valence band, taken from Ca2CuOCl2 (top-left),

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List of Tables

3.1 Fitting parameters for the TB model with Nearest-Neighbours . . 85 3.2 Fitting parameters for the TB model with Next-Nearest-Neighbours 86

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Acknowledgements

My experience at the Spectroscopy group at EPFL at Lausanne was a bit uncon-ventional. I joined the group in a quite difficult moment since a recent defection had affected the team, while another member was about to graduate and leave the academic world. In the reaction to this, the group leader (and my local su-pervisor, prof. Grioni) decided to get a new project started by the beginning of the new year, taking advantage of a new light source in the chemistry depart-ment. Thus, all the instrumentation of the lab was waiting to be dismounted and transferred to the new location. If from one side this prevents me to imme-diately get in touch with the electron analyser and the turbomolecular pumps, from the other side I had to the opportunity to test other important tasks of the researcher: the study of the literature and the data analysis. It is exactly in this frame that I have to say a HUGE thanks to Sara Fatale, for essentially adopting me and taking care of most of my difficulties. By the end of this experience I think it could be very much different, if she was not be there.

An acknowledgement should be given also to Simon Moser, for sharing his data with me and for his consultance coming from the other side of the Earth. I hope there will be the opportunity to meet each other sooner or later.

The begun of the new year was accompanied by the purpose of setting up the new lab trough the teamwork of three plus one brave guys. Assisting (when pos-sible) to this work has been a worthwhile experience and an intensive workout. I learned practically the working principles of each component and how to deal with them thanks to Alberto Crepaldi and Silvan Roth.

Special thanks go to prof. Marco Grioni, who kindly hosts me and always had the time to give an answer to my doubts. Beyond his undiscussed skills and knowledge, I have really admired his incredible vitality and curiosity, typical of a young mind.

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Abstract

A wide area of the condensed matter physics focuses on the so-called strongly-correlated materials. The physical properties of these solids are deeply influenced by the Coulomb interaction between electrons, which is not taken into account in the frame of the band theory. A first effective description of the electronic structure of these materials is provided by the empirical model developed by Mott and Hubbard, which is able to explain the unexpected isolant nature of numerous transition metal oxides.

Among them, two lithium copper oxides (LiCu2O2 and LiCu3O3) are analysed

through Angle-resolved PhotoEmission Spectroscopy (ARPES). In the case of the first compound, experimental data sided by Density Functional Theory (DFT) calculations, suggest that the origin of the first ionization state may be addressed to the simple overlap of oxygen pzorbitals, hybridized with copper 1+ dz2 orbitals.

This consideration, which excludes any electrons correlation, is strengthened by the result of a Tight Binding model built on the same wavefunctions.

The second compound, LiCu3O3, has also been probed by ARPES and the origin

of its states is discussed by comparison with the previous LiCu2O2 and with

the tetragonal CuO. This comparison, based on structural similarities between the compounds, points out that LiCu3O3 has the same first ionization state of

LiCu2O2 and shows another feature whose characteristics seem to be the one of

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Sommario

Una vasta area della fisica della materia condensata dedicata allo studio dei cosid-detti materiali fortemente correlati. Le propriet fisiche di questi solidi sono pro-fondamente influenzate dall’interazione di tipo Coulombiano elettrone-elettrone, di cui la comune teoria a bande non tiene conto. Attraverso il modello empirico sviluppato da Mott e Hubbard viene fornita una prima efficace descrizione della struttura elettronica di questi materiali, in grado di spiegare l’inaspettata natura isolante di numerosi ossidi di metalli di transizione.

Fra questi, due ossidi di litio e rame, LiCu2O2 e LiCu3O3, sono analizzati

at-traverso la tecnica della spettroscopia elettronica risolta in angolo (ARPES). Nel caso del primo composto, le misurazioni sperimentali affiancate da simulazioni basate sulla Teoria del Funzionale Densit´a (DFT) suggeriscono che l’origine del primo stato di ionizzazione sia dovuto alla semplice sovrapposizione di orbitali pz

dell’ossigeno, ibridizzati con orbitali d_z2 di rame 1+. Questa ipotesi, che vede la

totale assenza di correlazioni elettroniche, ´e rafforzata dai risultati di un modello Tight Binding costruito sulle medesime funzioni d’onda.

Il secondo composto, LiCu3O3, stato anch’esso esplorato tramite ARPES e

l’origine dei suoi stati discussa per comparazione dei dati con quelli del prece-dente LiCu2O2 e quelli del CuO tetragonale, ottenuti da letteratura. Il confronto,

originato da somiglianze strutturali tra i due composti, ha indicato la presenza dello stesso stato di prima ionizazzione riscontrato in LiCu2O2, affiancato da una

seconda banda le cui caratteristiche sembrano essere quelle di un singoletto di Zhang-Rice (ZRS).

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Chapter 1 Angle-Resolved PhotoEmission

Spectroscopy

In the far 1887, Heinrich Hertz performed a series of experiments by which he discovered that a material irradiated with UV light begins to conduct current. His source of electromagnetic waves was a spintherometer, which is an instrument able to generate sparks between two spherical conductors, while the receiver was a coil interrupted at a certain point. The experience showed that when sparks are created by the transmitter, a little version of those is seen between the two extremities of the open spire. He noticed also that by placing different materials between the transmitter and the receiver, the brightness of the electric arcs var-ied. Nevertheless, he was not able to explain why.

The full explanation of this question was given only at the beginning of the

Figure 1.1: The Hertz’s experiment setup

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rewarded with a Nobel Prize in 1921 [1]. He figured out that a ray of light yielding a high enough frequency is able to excite the electrons of the shone material and to make them escape from the solid. Einstein named this process the photoelectric effect and found out also that a classical description of it could not explain what experiments were showing. For this reason his discovery promoted a new way to do physics and to adopt a quantum description of the microscopic world.

These were the first steps of a long on-going walk that have led to important discoveries both in fundamental and experimental physics and then in technol-ogy. Among them, the photoelectric effect has been smartly used to develop a technique able to probe the electronic structure of materials and explain their most important physical properties. This technique is called Photoemission Spec-troscopy (PES) and it will be presented in this chapter, focusing on the angle-resolved version (ARPES).

1.1 General features

Before showing the mathematical description of PES, it is useful to introduce it with a description of its most generic characteristics. [5, 6]

Photoemission is a process where light hits the surface of a selected material in order to eject electrons from it. So the first feature to remember is that it is a photon-in electron-out technique. The target is usually a solid object several µm2 _{wide whose properties are under investigation. The physical phenomenon}

that stays behind PES is exactly the photoelectric effect explained by Einstein. Namely, a photon transporting an energy proportional to his frequency is ab-sorbed inside a solid giving it the same amount of energy. In this way, if the electron receives enough energy to overcome the ions’ attractive potential, it can get out the solid and travel as a free electron. These particles are then called photoelectrons. The photoexcitation process is governed by the equation:

Ekin= ~ω − Φ − |Eb| (1.1)

where Ekin is the kinetic energy of the photoelectron, ω is the frequency of the

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energy of the hit electron (relative to the Fermi Energy). The work-function is the minimum energy needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. It represents the attractive force exerted by the solid onto the photoelectron and it depends thus on its chemi-cal composition. From this perspective, the work-function is the analogous of the atomic ionization energy, applied to the case of a solid system. Thus, dif-ferent materials present difdif-ferent electronic and geometric structures so that the Coulomb interaction is also different from case to case.

Equation (1.1) tells us that having chosen the target and the photon energy,

Figure 1.2: The photoexcitation process

there is a relation between the kinetic energy of the photoelectron and its orig-inary binding energy. This means that if one is able to measure the speed of the emitted electrons, one can also predict which was their energy when they were bonded within the solid. This is the main purpose of the Photoemission Spectroscopy. A photoemission experiment takes place with an electron detector placed in front of the sample whose task is to measure the kinetic energy of the electrons. Probing a certain range of this energy is equivalent of scanning the energy distribution of the electronic levels of the target. The result of the mea-sure is a plot displaying Intensity (i.e. the number of photoelectrons detected) vs Kinetic energy and it is commonly called spectrum.

Depending on which states one would like to analyse, photons of different energy may be employed: X-ray photons have an energy higher than 100 eV, while UV photons have energies between 10 and 100 eV. The formers are used to excite core

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electrons and their related intensity peaks are tipically narrow, so that they act as a fingerprint for a selected element; thus X-ray PES is used to investigate the chemical composition of the sample’s surface. UV light is preferred to probe low energy states, the valence states, that form a continuum (bands). These states are the most interesting to observe because they determine the physical properties of materials. As an example, a conventional metal will show states filled up to the Fermi level, where insulators do not. At the same time, valence band states are not easy to treat and they deserve a deeper analysis to be correctly described. To go further, a theoretical model of PES is necessary to give a clear interpretation of the measured data: this is going to be the topic of the second part of this chapter.

I would like to end this short introduction discussing the more practical aspect of electron spectroscopies. High-energy photons can easily go through regular matter, while electrons do not. They scatter with high probability with atoms and other particles and this makes them lose the information we are interested in, i.e. their kinetic energy. So the only condition to obtain a PE spectrum is to place the entire experimental setup in vacuum, actually a Ultra High Vacuum (UHV). UHV chambers can also be made of materials with very high magnetic permeability (as µ-metal) able to screen magnetic fields and to withstand high difference of pressure. A differential pumping system is used to reach an internal value of pressure of around 10−10mbar, which ensures a minimum quantity of contaminants. Each chamber is connected to another one by the use steel flanges and can be isolated by closing valves set between them. In this way, it is possible to maintain the pressure in one chamber, while working on another one.

It is a common practice to attribute different roles to each chamber. A small one, called loadlock, is employed to introduce samples from the outside and to place them into an initial vacuum. A preparation chamber is tipically placed right after the loadlock and it is used to make all the preliminary operations (e.g. depositions) and analysis at a pressure of 10−10mbar. Finally, the measurement chamber is kept separated from the others to maintain the optimal vacuum con-ditions. Inside the latter, one finds the source of photons, the energy analyzer and the electron detector. Helium gas-discharge lamps are frequently employed to produce monochromatic UV light while solid Mg or Al anodes are tipically used to emit X rays. The electron analyzer fullfills the task to measure the

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ki-netic energy of the escaping electrons. Different types of analyser could be found in commerce working:

1. by monitoring the time of flight

2. by analyzing the deflected trajectory of the particles in presence of a con-trolled electromagnetic field.

The principles of these tools will be explained in detail at the end of the chapter.

1.2 Instrumentation

In this section, it would be briefly shown the principal instruments necessary to perform an Angle-Resolved PhotoEmission Spectrum. Each subsection is dedi-cated to a different component of a typical photoemission laboratory, presenting its main technical characteristics and key parameters.

1.2.1 Vacuum instrumentation

Since a photoemission process is extremely surface-sensitive, a correct measure-ment of the sample’s electronic structure needs the cleanest environmeasure-ment possi-ble. This means that the sample and thus the entire experimental set-up must be placed in vacuum. Different degrees of vacuum may be obtained combining a completely leak-tight chamber attached to a pump operating at constant vol-ume flow. A rough vacuum is produced when the internal pressure reachs values around 10−3 mbar, high vacuum is characterised by a pressure higher than 10−6 mbar and finally the ultra high vacuum (UHV) goes beyond 10−8 mbar. While in industry a high vacuum instrumentation is usually more than enough, UHV is a priority for research purposes. Therefore, this subsection will describe the three components responsible for the development of a stable ultra high vacuum environment: the UHV chambers, the vacuum pumps and the vacuum gauge[14].

1.2.1.1 UHV chambers

UHV chambers may be thought as big boxes having several openings in order to plug other instruments in, e.g. pumps, gauge, manipulators. They are made

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of a metal with a high magnetic permittivity, to better screen the inner parts from external electromagnetic fields, and with a high resistance to differences of pressure. The UHV chambers used for photoemission experiments are made of µ-metal, which is a nickel-iron alloy. Connections between two chambers or the attachment of an instrument to one of them are realised through the use of stain-less flanges with a copper gasket in between. This helps to prevent the entrance of air from the outside. The use of valves allows to work with different pressures in different chambers, which is particularly useful when introducing a new sample from the atmosphere.

The typical configuration of a vacuum system is composed by a little room, called loadlock, where samples are loaded into the vacuum environment. Then a prepa-ration chamber is used to make the sample undergo to several process, as metal depositions and electron diffraction. Finally, one finds the measurement chamber where the vacuum is particularly good since this is the place where photoemission experiments take place.

1.2.1.2 Vacuum pumps

The pumping stage is an important factor to take care of while considering to reach a high degree of vacuum. If all the molecules were contained in the volume of the vacuum chambers, it would be easy to pump all of them outside. Never-theless, a great amount of particles sticks to the inner surface of the chambers or further, they can also cross this surface and get bonded with atoms of the bulk (these phoenomena are called adsorption and absorption, respectively). On the other side, a contaminated surface naturally release particles sticked on it, i.e. outgases with a certain rate, which can be increased through physical process, e.g. heating the chamber (degassing). These are the principal phoenomena that occur at the surface of any material.

Vacuum pumps may operate only within a certain range of pressure. Some of them are suitable to pump out air in order to reach a rough vacuum, while others, if put in series with the previous ones, may create the condition of a high vacuum. Fig. 1.3 gives a classification of vacuum pumps based on their operating range.

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Figure 1.3: A rough classification of vacuum pumps based on their pressure operating regime[17]

The goal of the first vacuum is accomplished by using primary pumps such as the oil-sealed rotary pump. The rotary pump is made up of a rotor rotating inside a larger and non concentric circular cavity. On the inlet side of the pump the space close to the inlet increases because of the motion of the rotor and is filled with gas from the chamber. On the outlet side of the pump the volume adjacent to the outlet is decreased because of the motion of the rotor, which forces out the gas extracted from the chamber. After this stage a pressure of 10−3 mbar is obtained.

High vacuum is achieved through the use of turbomolecular pumps, directly con-nected to the primary pumps. This pump is a kinetic one composed of bladed rotor disks alternated with bladed stator disks. The rotor disks are whirling reaching a rotational speed beyond 1000 Hz and they hit the gas molecules forc-ing them out of the vacuum chamber in the direction of a primary pump. The blades of the stator disks are inversely inclined with respect to the ones of the rotor disks, preventing the gas molecules to flow towards the vacuum chamber.

In order to reach the UHV regime, a necessary step is the bake-out. The walls of a chamber previously exposed to air are tipically filled with water molecules that easily stick to the inner surface of the chamber itself. When the high vacuum regime is attained, the desorbing water constitutes the 99% of the gas load in the

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Figure 1.4: Sections of four vacuum pumps: a) rotary pump, b) turbomolecular pump, c) ion-getter pump, d) cryopump.

vacuum system, and the rate at which the pressure decreases starts to be ruled substantially by the desorption rate of the water. This is essentially controlled by the heat transferred from the walls of the chamber to the water molecules. This thermal energy allows the water molecules to overcome the energy of their bonds to the walls. Every time that the vacuum system is opened and exposed to air, after the high vacuum regime is reached, a bake-out oven is switched on. The temperature of the chambers is gradually raised to almost 150◦C and then kept constant for several days. This makes possible to get rid of the excess of water molecules adsorbed by the walls of the chambers.

A further improvement and stabilization of the UHV regime is made possible by means of entrapment pumps such as the ion getter pump and the cryopump. The ion getter pump ionizes the residual gas molecules and, by means of an electrical potential of several thousand volts, accelerates these ions towards a titanium cathode which entraps them. The cryopump traps the gas molecules thanks to a cooled surface upon which the gas molecules condense.

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1.2.2 Gas-discharge lamp

A gas-discharge lamp is made of two electrods in the middle of which a gas is injected and ionized. This process is responsible for the emission of light at en-ergies equal to the ionization enen-ergies of the gas inside the lamp. Therefore, the spectrum of the light emitted is usually made of peaks at tyipical values of energy depending on which gas is in. The narrower these peaks are, the more monochro-matic are the photons emitted. Helium-dicharge lamps are characterized by an emission spectrum with two main lines at 21.2 eV and and 40.8 eV. This makes them useful to be the light sources for PE experiment, with the goal to make a first description of the sample.

Figure 1.5: Structure of the Scienta VUV 5000 He-lamp.

The ionization process that takes place inside direct current discharge lamps could be described as follows[10]. The few electrons emitted due to the cosmic radiation are strongly accelerated by the high voltage supplied to the electrodes. These hit He atoms and transfer to them a certain amount of energy through inelastic collisions; if this energy is enough high, gas atoms relax themselves emitting one electron. The couple ion-electron is then accelerated under the difference of voltage and it is responsible for new ionization processes. This re-actions sustain the formation of a low-temperature plasma, i.e. an ionized gas, which naturally would tend to vanish under the neutralization mechanism be-tween ions and free electrons. The latter process is the cause of the emission of monochromatic light used as a source in photoemission experiments. On the other side, the plasma has to be sustained to keep emitting light.

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Different technologic solutions may be employed to the creation and stabi-lization of a plasma inside a helium-discharge lamp. Our lamp uses an electron cyclotron resonance plasma to generate UV light. The plasma generation pro-cess is here optimized through the interaction between a uniform magnetic field B and a radiofrequency electric field. Charged particles start preceding around magnetic field lines, greatly increasing their path length to the anode, with an angular cyclotron frequency ωc = eB/m. At the sime time, a linear

radiofre-quency (rf) electric field is guided to the lamp and it is directed in the same way as the magnetic one. A linear polarized field could be decomposed as a right-handed circularly polarized wave (RHP) and a left-right-handed circularly polarized one (LHP). Hence, when the frequency of the RHP reaches exactly the value of ωc, the travelling electron feels a direct current electric field that continuosly

accelerates it. The free electrons absorb the rf field and the remaining part of the injected electric power is measured and used as an index of the lamp emission efficiency.

1.2.3 Photoelectron analyser

The detection of photoelectrons is certainly the most important step of the PE spectroscopy. Once the sample is excited through the injection of photons, the material begins to emit photoelectrons in each direction. The photoemission intensity profile reflects the electronic structure of the material but it is also modulated by the degree of coupling between light and the electron distribution in the solid. Then, due to the low electron mean free path, only a small part of these particles may escape the solid, i.e. only those excited in a portion of the sample close to the surface.

Once an electron reaches the vacuum, it travels as a free particle in the same direction from which he escaped. Since an ARPES experiment wants to measure simultaneously its energy and its direction of emission, one needs an instrument that discriminates both of them. These instruments, called electrons analysers, may reveal the particle’s energy and momentum either using or not electromag-netic fields. While in the former class one finds analysers that measure the electron’s time of flight, the latter are instruments that use a potential barrier in order to discriminate electrons with different energies. Among these,

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hemispher-Figure 1.6: The Scienta Phoibos 150 hemispherical analyser. ical energy analysers will be described.

Hemispherical analysers are usually composed of three main parts: 1. the lens system

2. the 180◦ deflecting system 3. the electron detector.

The following paragraphs will introduce each of these components, explaining their working principles and parameters[34].

1.2.3.1 The lens system

The lens system is the first element photoelectrons encounter along their path towards the analysers. Its main duties are:

• make an image of the sample plane on the analyser entrance plane;

• define the analyzed sample area and the accepted solid angle on a sample; • accelerate or decelerate the particles with the observed energy Ekin to the

pass energy Epass

In the lens stage, the electron coming from the sample are imaged onto the entrance slit S1, with the sample being in the focal plane of the lens system. If

S1 has the dimension D1, then by theory the imaged area of the sample has the

dimension DS with

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where M is the lens magnification. Its value is electrically selectable by connecting appropriate voltages to the lens electrodes. The voltages are a function of the spectrometer voltage U0 through the following equation:

U0 = Ekin− Epass+ Φ (1.3)

where Φ is the analyser workfunction.

In the lens stage, the particles passing through an intermediate image and will focused onto the input slit S1 of the hemispherical analyser. At S1 the particles

have been slowed by the energy difference between the nominal particle kinetic energy Ekin and the nominal pass energy Epass.

The actual size of the analyzer sampling area DS is in principle given by

equa-tion 1.2. Due to spherical aberraequa-tion, however, the image in plane of the entrance slit is diffused. The degree of diffusion increases, for fixed magnification M , with the input lens acceptance angle ϑ1, as described by the following equation:

ϑ2 =

pVin/Vout

M ϑ1 (1.4)

where Vin and Vout are the electric potential of the inner and outer shell,

re-spectively, and ϑ2 is the acceptance angle at the entrance of the analyser. This

means that also the viewed area in the focal planes of the input lens system is broaded with increasing angle, resulting in larger sampling dimensions than given by eq.1.2. Thus, the lens acceptance angle could be managed by the magnifica-tion modes to keep the spherical aberramagnifica-tion at an acceptable value.

1.2.3.2 Hemispherical Analyser

The hemispherical analyzer (HA) with a mean radius R0 performs the

spectro-scopic energy measurement, due to energy dispersion. Charged particles entering the HA through the entrance slit S1 are deflected by the radial electrical field between the inner hemisphere Rin and the outer hemisphere Rout. The entrance

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a fixed electrical field gradient, only particles with kinetic energies in a certain energy interval are able to pass the full deflection angle from the entrance slit S1 to the exit plane S2. Particles with higher kinetic energy approach the outer

hemisphere, whereas particles with lower kinetic energy are deflected toward the inner hemisphere. Those particles which enter the HA normal to S1 and are able

to reach the exit slits have the nominal pass energy Epass:

Epass = (−q)k∆V (1.5)

where q is the charge of the electron, the potential difference ∆V = Vout− Vin is

applied to the hemispheres, k is the calibration constant. It is defined as: k = RinRout

2R0(Rout− Rin)

(1.6)

These particles reach S2 at the nominal radial position R0. If the HA accepts the

half angle α in the dispersion direction, the HA resolution or FWHM (full width at half maximum) of the transmitted line ∆Ean is given by:

∆Ean Epass = S 2R0 +α 2 4 (1.7)

where S = (S1 + S2)/2. This value is an analyzer constant.

There are additional contributions to the line width observed in the spectrum. The main are:

• intrinsic line width of the atomic level

• natural line width of the radiation used for excitation

The observed total FWHM total is given by the convolution of the single FWHMs. Its values are tipically between 0.5 and 1 meV, depending mostly on the degree of monochromatization of the light. At the same time, the signal intensity is described by the following:

I ∝ ∆EanΩSAS = ∆EanΩ0A0 Epass Ekin ∝ E 2 pass Ekin (1.8) where Ω and A are the accepted solid angle and the accepted sample area, respectively. The indexes S and 0 specify if the observable is considered at the

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entrance of the analyser or at the entrance slit, respectively. The last equation makes use of the Liouville’s theorem which states the conservation of the emit-tance.

The analyzer can operate in two different modes:

• Constant Retardation Ratio, where the retardation ratio B is defined as B = Ekin/Epass

• Constant Pass Energy Epass

In the former mode all particles are decelerated with the same fixed factor B, which means that Epass is dinamically changed with changing Ekin. This mode

makes signal intensity increase with the kinetic energy, while the energy resolu-tion decreases.

The latter mode keeps Epass fixed and so the analyser resolution. However, the

intensity of the signal is now decreased for increasing Ekin.

1.2.3.3 Electron detector

The detection of the electrons that go through the analyser is obtained employ-ing a system composed by a micro-channel plate (MCP), a phosphorous screen, and a CCD camera. The MCP is utilized to multiply the signal coming from a single incoming electron, while maintaining the spatial information encoded in the position at which the electrons arrive at the exit plane. The MCP consists of a plate of highly resistive material realized by several millions of parallel capillary tubes (microchannels). When the device is operating the set of tiny tubes con-stitutes a 2D network of electron multipliers. An incoming electron that enters a microchannel hits the wall of the capillary tube that in turn emits secondary electrons. A difference of potential applied between the two extremes of the MCP accelerates all the particles towards the phosphorous screen. Due to the several collisions of the electrons with the walls of the microchannels, a cloud of electrons is generated out of a single incoming electron. The ratio between the number of outgoing electrons and the number of electrons that enter the MCP is the MCP gain: its value is around 104_{. The collision of electrons with the phosphorous}

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of the incoming electron. In this way, the flux of incoming electrons is converted into a flux of photons. These photons are detected with 2D spatial resolution by the CCD camera. When the lens system is operated in angular dispersive mode, the use of a CCD camera with 2D spatial resolution allows one to exploit at the same time the energy and angular dispersion of the hemispherical ana-lyzer. Therefore, a parallel acquisition of the energy and angle dispersion of the electrons photoemitted by the sample can be performed.

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1.3 PES and ARPES data

In this section the results of a photoemission measurement are shown, focusing mostly on the angle-resolved spectra.

The data obtained from a photoemission experiment consist of spectra where the photoemission intensity is plotted against the measured kinetic energy of pho-toelectrons. Since the angular distribution of those is not revealed, but is rather averaged, this measures represent the analogous of Angle-Integrated PhotoEmis-sion spectra (AIPES).

Figure 1.7: A typical wide energy Au XPS spectrum[5]

As previously written, X-ray spectra are useful especially for chemical analysis of the sample’s surface or, if combined with a tunable light source, to see whether a certain intensity peak changes its energy position while varying the photon energy, i.e. if it disperses along kz . This information is valuable to conclude that

the state related to that specific peak expresses a localized character within the surface. This can be explained considering that the photoelectron momentum component perpendicular to the sample surface k⊥ is proportional to the square

root of the photon energy (see next section); then, remembering that at normal emission the peak energy is dependent only on k⊥, if its position does not change

at varying photon energy, it means that its energy depends only on k// and

therefore that it is a surface state.

When the photoelectron angular distribution is considered an Angle-Resolved photoemission spectrum is taken. These experiments measure photoemission in-tensity I depending on the polar and azimuthal angles (ϑ,φ) and on the pho-toelectron kinetic energy Ekin. The result is an ensemble of 4D data, which

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Figure 1.8: This set of AIPES spectra at normal emission and various photon energies from a LiCu2O2 sample shows several intensity peaks k⊥ dispersing at

binding energy bigger than 3 eV and others not within 3 eV from Fermi level. [7]

can be converted as I = f (Ekin, kx, ky), where kx and ky are the two in-plane

photoelectron momentum component obtained from (ϑ,φ).

Figure 1.9: The typical photoemission geometry[9]

Since 4D data cannot be visualized all at once it is necessary to make projec-tions onto one of the variables. When the data sets do not show any dispersion along kz, a projection is made on the 3D space of variables (E, kx, ky). Then,

multiple constant energy maps are investigated to see whether the electronic fea-tures follow a certain periodicity. Comparing it with the lattice periodicity, it is possible to recognize surface high-simmetry points. These correspond to the projection of the full 3D Brillouin zone on the chosen k-plane projections and they are used as a reference during the analysis of the whole spectrum. The electronic structure is periodic in k-space and each band shows its energy

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max-ima and minmax-ima at the same high-simmetry points. Therefore, the entire band dispersion can be resumed simply projecting the 3D data onto a high-symmetry path, which connects the high-simmetry points where the band extremes sit. Fig. 1.10 shows several constant energy maps extracted from angle-resolved data for a LiCu2O2 sample: bandstructure periodicity and high simmetry points are

high-lighted. Then an energy cut along one ΓY’YΓ path is showed.

Figure 1.10: LiCu2O2 constant energy maps

Another way to represent angle resolved spectra along a momentum contour is to show a set of Energy Distribution Curves (EDCs) or Momentum Distribution Curves (MDCs). Instead of displaying an image where the colour represents the value of intensity, several cuts at constant momentum (for EDCs) or at constant energy (for MDCs) are stacked to visually show the band dispersion (see fig.1.11).

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1.4 Photoemission Theory

A theoretical description of the whole photoemission process is necessary to help in the understanding of the measured spectra. Several contributions coming both from the characteristics of the used light source and from how this light interacts with the sample’s electrons determine the intensity profile measured by the analyser. Therefore, two models are here presented which account for the photoexcitation process, considered as interaction between a photon and a system of N bonded electrons.

1.4.1 Three-step Model

As a starting point, photoemission is considered only as a one-particle theory, namely the system of (N-1) electrons other than the photoelectron remains unper-turbed while the process occurs. Furthermore, the entire phenomenon is divided into three parts: photoexcitation within the solid, propagation of the photoelec-tron towards the surface and subsequent emission into vacuum[25, 16].

When one desires to evaluate the probability that an injected photon excites an electron inside a solid, perturbation theory comes to help if one considers the radiation-matter interaction as very small in comparison to the atomic potential. We can then use the Fermi Golden Rule to find the transition probability from the N-electron initial state and the free electron state:

w = 2π h | < Ψf|Hint|Ψi > | 2 _δ(E f − Ei− ~ω) (1.9) where Hint = e 2mc(A · p + p · A) − eφ (1.10)

with A and φ the vector and scalar electromagnetic potentials, respectively. Set-ting the EM field gauge so that φ = 0 and considering the translation invariance of the solid, one writes divA = 0. This hypothesis allows us to simplify what is inside the rounded parenthesis, commonly rewritten as 2A · p + i~ divA and then only 2A · p.

Taking into account the previous consideration, one can rewrite the matrix el-ement of eq. (1.9) in the following form, which contains interesting physical

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information about the PE process: MKk = e mc < Ψf|A · p|Ψi > = −i~e mc < Ψf| N X i=1 eikph·ri_{· ∇} i|Ψi > (1.11)

where is the polarization vector, kph is the momentum vector of the incoming

light and K and k are the wavevector of the electron before and after the excita-tion. The scalar product between these two vectors makes this 1-electron matrix element depend largely on geometrical factors about how light is coupled with electrons’ wavefunctions in the solid.

Within our hypothesis, a transition between an initial state of N electrons inside the solid and a final state of (N-1) electrons in the solid plus a photoemit-ted electron is now considered. This process may include the effect of electronic correlations. They behave like a glue that does not allow to consider each par-ticle as independent. It is indeed very difficult to mathematically describe this situation, thus it is necessary to make an early approximation; a very common one is the sudden approximation. It states that the photoelectron exits the solid and it is istantaneously decoupled from the remaining (N-1) electron-state. This assumption cannot be justified theoretically but it holds for sufficiently high pho-ton energy and could be verified a posteriori.

Thus, the initial state could be described as an antisymmetrized product of single-Bloch states inside the crystal

|Ψi > = ˆA |ϕ1 > |ϕ2 > . . . |ϕN > = |N > ,

where ˆA is an operator that guarantees the antisymmetrization of the following function. The final state is written as a tensor product of a free-electron state and an (N-1) electrons Bloch state

|Ψf > = |k > ⊗|(N − 1) > = |k > ⊗|ϕ1 > |ϕ2 > . . . |ϕN −1> .

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depends on the value of his wavevector k; similarly |Ψi > = |K > ⊗|(N − 1) >,

where K is the Bloch wave-vector of the electron in the solid. This allows to split up the matrix element as:

MKk = < Ψf|Hint|Ψi >=< k|Hint|K > < (N − 1)f|(N − 1)i > (1.12)

When no correlations are considered, both the initial and final |(N − 1) > states are eigenstates of the same Hamiltonian. The PE probability (1.2) thus takes the expression:

w = 2π_h| < k|Hint|K > |2 | < (N − 1)f|(N − 1)i > |2 δ(Ef − Ei− ~ω)

= 2π_h|MKk|2 δ(Ek− EK − ~ω) (1.13)

The energy spectrum is therefore made of ’spikes’ at energies Ek = EK + ~ω,

whose intensity is modulated by the dipole matrix elements MKk. It should

be noticed that the matrix element MKk depends on the angular momentum of

the initial state: in experiments where different angular momenta are involved, each momentum contribution will appear with different strength. However, in the frame of an XPS experiment of a policrystalline sample - where the k-dependence of the states is blurred - the distribution of photoemitted electrons shows exactly the Density of Occupied States. But in general this is not true when trying to describe a valence band spectrum, especially if one introduces also electron-correlations. In order to take them into account, one should slightly modify the mathematical approach just explained.

As done before, one can still rewrite the initial state as the antisymmetrized product of the orbital that is going to be excited and the remaining (N-1) electron single function state and similarly the final state with the difference that only his kinetic energy now describes the chosen electron. Now everything is determined by how one computes the second overlap integral, that involves all the remaining passive orbitals. In a frozen-orbital approximation the (N-1) electron system remains unperturbed and both the systems without or with hole are eigenstates of the unperturbed Hamiltonian. Thus the only measured energy is the one of the excited orbital.

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If we think more realistically that the final state has s different excited states, then there will be a certain probability |Cs|2 to reach these states after the PE;

then the passive orbitals integral becomes:

< (N − 1)f,s|(N − 1)i >= Cs (1.14)

The intensity of PES core-peaks results in:

I ∝ w =X f,i | < kf|Hint|Ki > |2 X s |Cs|2δ(Ef+ E(N −1),s− E(N −1),0− ~ω) (1.15)

For what concerns photoemission from the valence band, one should add the wave vector dependence due to the fact that electrons are travelling inside a periodic potential (they are Bloch waves). Furthermore, one can neglect the photon momentum for UV photons and thus k = K, i.e. only vertical transitions are considered (∆k =0). Now, the photoemission intensity reads:

I ∝X f,i | < kf|Hint|Ki > |2 X s |Cs|2 δ(Ef + E(N −1),s− E(N −1),0− ~ω) δ(E − Ef − Φ) δ(k//− K//+ G//) δ(k − K + G) (1.16)

This equation shows that the photoelectron spectrum would display now peaks for each reachable final state, depending on the probability |Cs|2. This situation

is shown in fig 1.12. It is clear that if one s-excited state is perfectly equal to the unperturbed one, < (N − 1)f,s|(N − 1)i > will be unity and all the other overlap

integrals will be zero: the PES spectrum will correspond again to delta functions at the Koopmans energies.

The second delta function in eq. 1.16 expresses energy conservation and under-lines that the phomoemitted electron needs to have an energy bigger than the material’s workfunction Φ. Finally, the wavevector’s conservation in the solid is made evident by the last term: this contains the information that k is conserved apart from a reciprocal lattice vector G, as Bloch theory suggests. However, when the excited electron is crossing the solid surface, only the momentum par-allel component is conserved and therefore this last term should be rewritten as δ(k//− K//+ G//)1.

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Figure 1.12: The solid line is a photoemission spectrum of the H2 molecule, as a

model of electron-correlated system. Each peak corresponds to a different vibra-tional state. The dashed line represents a photoemission spectrum taken from a solid, within an interacting-electron model. It is made of a broad incoherent part, composed of several satellite peaks, and of a narrow coherent peak.[33]

Before describing the other two steps of PES, the Spectral function A will be defined as: A(k, E) :=X s < Ψf,s(N − 1)|Ψi(N − 1) > 2 =X s |Cs|2 (1.17)

also written like:

A(k, E) :=X s < N − 1|ˆckN > 2 (1.18) in the second quantization formalism where ˆck is an operator that creates a hole

in am n-particle system. The last writing has the explicit meaning of the prob-ability with which an electron can be removed from an electron system in its ground state and placed to the s excited state. These definitions would be useful later when a many-body treatment of the photoemission will be introduced.

The second step of this model describes the transport of the electron through the surface. It is characterized mainly by scattering events with the Bravais lattice. This contributes to the finite lifetime of the PE final state τ that can be written: τ = λ(E, k) vg = λ(E, k)(1 ~ dE dk) −1 (1.19) where λ is the Inelastic Mean Free Path, which is assumed to be isotropic and dependent only on the electron’s energy, and vg is the group velocity of the

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energy follows roughly an universal curve (indipendent on which materials the electron is travelling through). The finite lifetime of photoemitted electrons

con-Figure 1.13: The inelastic mean free path of several elements

tributes to the energy-width of each intensity peak present inside a PE spectrum.

The third step, i.e. the trasmission through the surface could be seen using the escape-cone argument. Only electrons whose kinetic energy is higher than a surface potential V0 are able to escape the solid, otherwise they will be reflected

back to the bulk. Energy conservation during the process requires that: ~2k2f

2m =

p2

2m + V0 (1.20)

At the same time, the in-plane momentum ~kf // is conserved across the surface,

due to translational symmetry, so only the out-of-plane momentum is subjected to the effect of the surface potential:

~2kf ⊥2

2m =

p2_⊥

2m + V0 (1.21)

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Figure 1.14: A schematic representation of the escaping phase: only photoelec-trons with k⊥> k⊥,min=

√ 2m ~

√

V0 may escape from the surface of the solid. The

thick solid line shows the internal and external escape cone. the photoelectron momenta as

px =

p

2mEkinsin θkcos φk

py =

p

2mEkinsin θksin φk

pz =

p

2mEkincos θk

So one has

~2k2f // = p 2

//= 2mEkinsin2θk

~2k2f ⊥ = p2⊥+ 2mV0 = 2m(Ekincos2θk+ V0)

and finally

kf // =

1 ~

p

2mEkinsin θk (1.22)

kf,⊥ =

1 ~

p

2m(Ekincos2θk+ V0) (1.23)

(1.24) where the kinetic energy is further related to the binding energy by eq. (1.1):

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Ekin = hν − EB − Φ. From the first equation, remembering that the escaping

photoelectron kinetic energy is connected to the inner energy as Ekin= Ef − V0,

one can obtain the equation that describes the maximum angle of incidence θ0, simply rewriting (1.22) to explicit sin θ0:

kf // =

1 ~

p

2mEkinsin θk=

1 ~

q

2m(Ef − E0) sin θ0k (1.25)

so that, for the case limit θk = 90 :

sin θ_k,max0 = ( Ekin Ef − E0

)1/2. (1.26)

For each internal angle θ0 smaller than θ_k,max0 the electron would cross the surface, therefore photoemitted electrons would be those travelling through the crystal inside a cone, described by the angle θ0_k,max (see fig.1.14). Because of this, the higher Ekin, the bigger the region of the k-space that would be probed. In the

synchrotron data presented in the following chapters, the photon energy allows 3-4 Brillouin Zones to be probed.

It is interesting to underline that the previous discussion has been made within the free-electron approximation where the final state is a free travelling wave. But if one wants to draw the excitation process in the energy-momentum plane as a vertical transition, one will see that this process cannot happen, because there are no free final states to fill(see fig. 1.15.a). Only by considering the periodicity of the crystal, the simultaneous energy and momentum conservation can be assured with the intervention of a reciprocal lattice vector G(see fig. 1.15.b).

1.4.2 Many-Body Treatment

In the context of photoemission from solids, in particular when discussing the case of correlated systems, in which several |Cs| are different from zero, an approach

based on the Green’s function formalism may be alternatively used.

The one-particle Green’s function describes the electron propagation from r1

to r2 in a time t. It can be interpreted as the probability that an electron added

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Figure 1.15: Photo-excitation for a)a free-electron and b)a Bloch electron the same state after a time t. Switching to the k-space by Fourier transforming the Green’s function, one can explicitly write:

G(E, k) = Z

(A(k0, E)

(E − E0_{− iδ)}dE (1.27)

whose imaginary part can be written as: ImG(k, E) = π Z dEA(k, E)δ(E − E0) (1.28) or A(k, E) = 1 π|ImG(E, k)| (1.29)

It could be shown that poles in the Greens function correspond exactly to the energies of the peaks of PE spectrum. Intuitively, one can see that at the pole energy, G, the probability that the Bloch electron will stay in the same state diverges. Therefore the electron escapes the solid with exactly those energy and momentum until it would be detected.

For a non-interacting electron system one has: G0 = 1 (E − E0(k) − iδ) (1.30) A0(k, E) = 1 πδ(E − E0(k)) (1.31)

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the one-particle theory.

For an interacting electron system, one adds a complex Self-energy Σ term, where the real and imaginary part yield all the information about the ’renormalized’ energy and lifetime. This renormalization accounts for all electrons-electrons interactions, electrons-phonons interactions or scattering with impurities that involve each photoemission process. This leads to

G = 1 (E − E0(k) − Σ(k, E)) (1.32) A(k, E) = 1 π ImΣ (E − E0 − ReΣ)2+ (ImΣ)2 (1.33) Since the non-interacting electrons result must be recovered for small values of Σ, again the poles of G have to be equals to the energy position of PE spectrum. These may be found by solving:

E − E0(k) − Σ(k, E) = 0 (1.34)

If the self-energy Σ is small, G could be usefully rewritten as the sum of a coher-ent and an incohercoher-ent part. The first one accounts for the position and width of the poles, in the same way as G0 describe the non-interacting system. The

second incoherent part describes an ensemble of satellite lines at higher binding energy (see dashed line in fig.1.12). This approximation can justify the physical picture of a photoelectron travelling through the solid surrounded by a cloud of virtual excitations. The conjunction of the two entities, the free-electron and all these excitations, could be seen as a ’quasi-particle’.

Taking all the previous into consideration, the complete equation for the pho-tocurrent from a crystalline solid is :

I ∝X

f,i

ImΣ(ki)

(E − E0 − ReΣ(ki))2+ (ImΣ(ki))2

δ(E(kf) − E(ki) − ~ω)

δ(E − E(kf) − Φ) δ(ki//− kf //+ G//) δ(ki− kf + G) (1.35)

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1.4.3 One-Step Model

Although the three-step model in its one-electron expression can explain a great number of PE spectra, it remains an approximate approach to the problem. We should consider the entire process happening straightforwardly, in only one-step in order to better describe this phenomenon.

Figure 1.16: 3-step Model and 1-step-Model

The more rigorous 1-step model consists of applying the Fermi Golden Rule between the initial Bloch state in the solid and the final state describing the elec-tron travelling through the vacuum. It is different from the previous treatment that gives a quantistic description only of the photoexcitation in the solid, while introducing empirically the penetration through the surface and the subsequent escape from the solid.

In this frame, the final state is a so-called Inverse LEED wavefunction2, consisting of a free-electron wavefunction outside the solid and a combination of attenuated Bloch states inside the solid. Such a wavefunction is the time-reversal of a wave-function that describes the scattering of a free electron at the surface of a solid in a LEED (low energy electron diffraction) experiment. This is generally written

2_{LEED stands for Low Energy Electron Diffraction: it is a technique commonly used to}

investigate the cristallinity of the surface of a sample. A LEED experiment is performed by sending electrons towards the sample and measuring the diffraction pattern on a phosphorous screen. Thus, the LEED initial state - or the Inverse LEED final state - corresponds to the one of an electron which is going to enter the sample.

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as sums of propagating and evanescent waves expanded in terms of 2-D Bloch functions:

ψLEED ∝ exp(ikf //· %)

X

m

tmexp(ik⊥mz)um(r, kf ||, E). (1.36)

where % is the position vector on the sample surface, tm is the transmission

coefficient for the m Bloch function and where the wavevector of the mth Bloch function is a complex number :

k⊥,m= k0⊥,m+ ik 00

⊥,m. (1.37)

The magnitude of k0_⊥,m and k00_⊥,m determines the character of the LEED state, if more propagative-like or evanescent-like, respectively. The latter one could be physically explained by the scattering events inside the solid, but also if the energy of the considered state lies in a gap of the solid. Thus, it represents a state which is confined to the surface. The former is instead related to the bulk part of the wavefunction that is commonly more relevant in PE experiments. It could be shown that the matrix element MKk is now split in the surface and

the bulk part, where the latter corresponds to the one found within a three-step model and the first comes from the evanescent part of ψLEED. (k_⊥m00 )−1 is defined to be equal to λm, which describes the average distance from which an electron

can travel without scattering and tm contains the probability of escaping through

surface. Thus, the three-step is model is generally contained into the one-step model, which at the same time allows to further explore the photoemission from surface states through the analysis of the surface matrix element.

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Chapter 2 Strongly correlated materials

Modern condensed matter physics may explain the physical behaviour of a great number of materials, as simple metals, insulators and semiconductors. However, the description of materials with open d or f electron shells is a harder task and it needs a deeper understanding of the topic. Transition metals, for example, belong to this family and their valence electrons experience a strong Coulomb re-pulsion (or correlation). The influence of one electron to the others is too strong to treat them as independent particles and it determines deep changements to their physical properties.

The entire zoology of strongly correlated materials, where electronic correla-tions are the cause of weird effects, is quite broad[21]. Vanadium oxides show huge changes in the resistivity across the metal-insulator transition (see section. 2.4.1), while actinides and lanthanides experience a considerable change of volume across phase transitions. Copper oxides become superconductive at a relatively high temperature (above liquid-nitrogen temperature) and manganites display a great sensitivity to changes in an applied magnetic field (an effect called colossal magnetoresistance). These are only some examples of strongly-correlated mate-rials.

Even if the interest on studying them should be self-motivated by all these unusual physical behaviours, some of them have been used to develop technologic applications. Magnetic Random Access Memories (MRAMs) are probably one of the most interesting results obtained after the discovery of the giant

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magne-Figure 2.1: MRAMs are among the most useful technologic applications involving strongly-correlated materials.

toresistance effect. These are nothing but electronic devices which discriminate the high or low informatic state (bit) depending on the spin polarization (up or down) along a certain axes. The advantages of these mechanisms, instead of storing charge on a transistor, are that these memories are more resistant to external noise and reach an access time of around 10 ns, still remaining perfectly compatible with silicon-based processors. Another field of interest concerns su-perconducting magnets. Bulk MgB2 [36] is a strongly-correlated material which

becomes superconductive at 20 K. It produces a stable 3 T magnetic field that is used for several purposes, e.g. Magnetic Resonance Imaging. Finally, the last materials that deserve to be quoted for their hypothetical application as electronic devices or sensors are the so-called multiferroics. These objects show simultane-ously a ferroelectric behaviour and magnetic properties. Thus, the possibility to control spins through an electric field or to drive charges with a magnetic one makes them a good candidate for multifunctional devices. Among this class, one can find BiMnO3 or BiFeO3, whose magnetism arises from Fe3+ or Mn3+ ions, as

well as RMnO3 where an exotic spin spiral state arises at low temperatures.

Trying to mathematically describe strongly-correlated materials is something that goes beyond the simpler picture of metals. The most relevant physical properties of the latter may be caught by means of an effective lattice Hamilto-nian that takes into account electrons interaction with the crystal. Calculations based on the Tight Binding approximation are an example of this approach.[24] Nevertheless, this is not enough to explain the behaviour of strongly-correlated materials, where electron-electron repulsion plays a key role. Therefore, the Tight Binding model might be improved introducing a term that keeps track of them, resulting in what is known as the Mott-Hubbard model. An alternative theory

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has been successfully developed by Heisenberg, using electron’s spin as the mean of interaction between conduction electrons. Finally, both approaches have been related to usefully describe electron’s repulsion from two complementary points of view and create a connection between exchange interaction and the electrons delocalization.

2.1 The Tight binding model

The easiest way to describe a metal is to think about a periodic structure made by ions, surrounded by a cloud of free electrons travelling in every direction. If we consider this picture from the point of view of quantum physics, we are consider-ing an ensemble of fermions placed inside a finite volume, each one described by a free-electron wavefunction characterized by a particular value of the wave-vector k. However this kind of description is not enough to model what actually hap-pens in most solid materials. To go beyond this simple view, one should wonder how ions could interact with electrons. Among all mathematical approaches, two of them are useful to describe how the crystal lattice influences the electronic behaviour in a solid. The first one is to consider ionic Coulomb interaction as a periodic perturbative potential; the second one wants to build empirically the crystal wavefunction as a linear combination of atomic orbitals.

The first interpretation leads to the Bloch description of the electronic states, whose dispersion is periodic with a period inversely proportional to the lattice parameter. The second view is what is usually known as the tight-binding approx-imation and it is generally used when the atomic orbitals of close atoms interact enough to modify the electrons wave-function, but not that much to completely lose the concept of ‘atomic orbital’. This model is particularly efficient when trying to describe the behaviour of partially-filled d-bands, as in transition metal atoms, or the electronic structure of insulators.

2.1.1 Tight binding - Mathematical formulation

The Tight Binding model is based on the following assumptions:

Analysis of the electronic structure of LiCu_2O_2 and LiCu_3O_3 by angle resolved photoemission spectroscopy

POLITECNICO DI MILANO

School of Industrial and Information Engineering

Master of Science’s Degree in

Engineering Physics

ANALYSIS OF THE ELECTRONIC STRUCTURE OF

LICU

O

AND LICU

O

BY ANGLE-RESOLVED

PHOTOEMISSION SPECTROSCOPY

Thesis Advisor: Prof. Claudia DALLERA

Research Supervisor: Prof. Marco GRIONI

Candidate:

Gianmarco GATTI

Matr. 817689

Contents

List of Figures

List of Tables

Acknowledgements

Abstract

Sommario

Chapter 1

Angle-Resolved PhotoEmission

Spectroscopy

1.1

General features

1.2

Instrumentation

1.2.1

Vacuum instrumentation

1.2.2

Gas-discharge lamp

1.2.3

Photoelectron analyser

1.3

PES and ARPES data

1.4

Photoemission Theory

1.4.1

Three-step Model

1.4.2

Many-Body Treatment

1.4.3

One-Step Model

Chapter 2

Strongly correlated materials

2.1

The Tight binding model

2.1.1

Tight binding - Mathematical formulation