As a direct probe of the energy-momentum dispersion of charge carriers in a crystal, angle resolved photoemission spectroscopy (ARPES) has provided significant insights into both the fundamentals of solid state physics [57, 58] and the emergent physics of strongly correlated materials such as the cuprates high temperature superconductors [59–61]. It has also been pivotal in the discovery of new phases of matter such as topological insulators [62, 63] and more recently is playing a prominent role in the rapidly evolving understanding of Weyl semi-metals such as TaAs and (Mo/W)Te2 [64–66].
2.1 Theoretical description
All photoemission techniques are based on the photoelectric effect, the emis-sion of an electron from a material following absorption of a photon. Angle-resolved photoemission spectroscopy probes the energy and momentum dis-tribution of these photoelectrons. Very generally the photocurrent intensity I(k,!)can be written
I(k,!)/|M|2f(!)A(k,!) (2.1) where!is energy referenced to the Fermi level andkis the crystal momentum.
Direct photoemission probes only the occupied states which is described by the Fermi functionf(!).Mis a matrix element that determines the intensity modulation of the signal andA(k,!)is the one-electron spectral function
which describes the energy and momentum distribution of the interacting system with bare band dispersion⇠(k).A(k,!)includes many body effects through the electron self-energy⌃(k,!) =⌃0(k,!) +i⌃00(k,!)and is given by
A(k,!) = 1
⇡
⌃00(k,!)
(! ⇠(k) ⌃0(k,!))2+⌃00(k,!)2 (2.2) In the non interacting case ⌃(k,!) = 0 and the spectral function reduces to the delta function (! ⇠(k)at the energies and momenta defined by the bare band. For the case of weakly interacting electrons, it is clear from Eq. (2.2) that for a self-energy that depends only weakly on momentum and energy the spectral function for a linear bare band is approximately Lorentzian. The width of the Lorentzian is defined by the imaginary part of the self-energy ⌃00 and the pole of the spectral function position is offset from the bare band dispersion by the real part of the self-energy⌃0.
No complete theoretical description of the many-body photoemission process exists and there are different approaches for derivingMandA(k,!).
In linear response theory where the photocurrent is described as the steady response of the system to an applied electric field [67, 68] and is a so called one-step model in which the photoemission is treated as a single coherent process. Another example is the Fermi’s Golden rule approach which describes the photocurrent as the transition probability from an N particle initial state to an N particle final state. An advantage of linear response theory over the Fermi’s Goleden rule approach is that it is not necessary to know, or make assumptions about the full many body wave functions of the initial and final states and it naturally incorporates many-body behaviour. However, when combined with the common phenomenological three-step model of the photoemission process, the Fermi’s Golden rule approach is often more useful.
The three-step model artificially separates the single coherent photoemission process into the following three steps:
1. The bulk initial state is excited by a photon to a bulk final state inside the crystal.
2. The excited electron travels to the surface.
3. The photoelectron is transmitted through the surface into a free elec-tron vacuum final state.
In order to associate the raw photocurrent measured by ARPES with the energy-momentum dispersion of electrons in the crystal, we must map the measured quantities of the photoelectron emission angle#(see Fig. 2.1) and kinetic energyEkin to initial state crystal momentum kand binding energyEB respectively. Conservation of energy for a single coherent process requires that the kinetic energy of the final state photoelectron in vacuum is
2.1 Angle-Resolved Photoemission Spectroscopy
given by
Ekin =h⌫ EB (2.3)
where his the Planck constant,⌫ the frequency of the exciting radiation and the work function of the material. The free electron final state in the vacuum has total momentum ~ 1p2meEkin (here~is the reduced Planck’s constant andme is the free electron mass). Therefore the wave vector of the free electron final state in the vacuum can be completely determined by measuring the kinetic energyEkin and the angle at which the photoelectron is ejected from the crystal surface. The crystal momentum of the initial state kis given by the sum of the in-plane and out-of-plane componentskk and k?. Neglecting the photon momentum, conservation of momentum between the initial and final states and the translational symmetry of the crystal surface imply that the in-plane component of the crystal momentum of the initial state can be written as
kk =~ 1p
2meEkin·sin# (2.4)
where#is the angle at which the photoelectron is emitted measured from the surface normalnˆ as shown in Fig. 2.1. The surface breaks the translational symmetry of the crystal in the direction perpendicular to the surface therefore this component of the initial state crystal momentumk? is not conserved across the surface. Approximating the bulk final state inside the crystal as free-electron-like we can write
Ekin= ~2 2me
(kk2+k?2) +Vin (2.5) where the inner potential Vinis a free parameter that can be determined experimentally and kis expressed in the extended zone scheme. This ap-proximation is better for final states of high kinetic energy, however a large body of experimental work has shown that it is usually applicable down to kinetic energies around 20 eV [69]. Using Eq. (2.4) and Eq. (2.5), within the free electron final state approximation we can now write the perpendicular component of the crystal momentum as
k?=~ 1p
2meEkincos2#+Vin (2.6) Eq. (2.4) and Eq. (2.6) define the curved surface in reciprocal space probed at a given photon energy. Therefore, in order to probe the dispersion of states as a function of k? over an entire bulk Brillouin zone for a given k||we must vary the photon energyh⌫ over a large range. From such ak? dispersion it is possible to determineVin by matching the periodicity of the signal to the periodicity of the Brillouin zone. Photon energy dependent measurements allow us to determine if the initial state disperses alongk? which provides insight into the dimensionality of the crystals electronic structure.
The intensity of the energy and momentum resolved photocurrentI(!, k) is governed by the matrix element M of the electron-photon interaction.
Adopting thesudden approximation, which assumes that the photoelectron does not interact with the photohole after excitation, within the three step model we can factorize the N particle final state into a one-particle photoelectron state|fiand anN 1particle state of the system left behind.
Further assuming that the initial state can be similarly factorized into a single particle state |iiand an N-1 particle state, the matrix element takes the form
M/ hf|A·p|ii (2.7)
whereAis the vector potential of the exciting radiation andpis the electron momentum operator. Realistic calculations of the matrix elements for a given material require detailed knowledge of both the initial and final states.
However using Eq. (2.7) simple symmetry considerations (which are discussed more extensively in Sect. 3.1.1) allow the orbital character of the initial state to be deduced from polarization dependent measurements using different orientations of the vector potentialA.
The finite width of the pole of the spectral function is indicative of the finite lifetime of the photohole due to interactions described by the imaginary part of the self-energy. However when extracting experimental self-energies from data, the contributions to the total lineshape of the photocurrent from both resolution and matrix element effects must be considered. As the photoelectron travels to the surface it undergoes inelastic scattering processes which lead to a finite kinetic energy dependent photoelectron escape depth [70]. This is responsible for the surface sensitivity of the technique at VUV excitation energies. The finite inelastic mean free path in turn leads to uncertainty ink? of the final state. ARPES probes the initial state at the k? of the final state. Therefore the range ofk?values of the final state leads to a finite k? resolution through the matrix element M in Eq. (2.7). In materials with three dimensional electronic dispersion this broadens direct-transition peaks in the photocurrent. For a two-dimensional system this does not effect the measured line-width, but similar considerations can explain strong matrix element variations that reflect the spatial extent of the initial state wave function.
2.1 Experimental Principle
Angular and energy resolution of the photoelectrons is achieved with a cylindrical electron lens and hemispherical analyser respectively as sketched in Fig. 2.1. The lens and hemispheres image the photoelectron distribution onto a two-dimensional detector where single photoelectrons are amplified by a multi channel plate (MCP) and the resulting avalanche of electrons is accelerated to a phosphor screen which in-turn is monitored by a CCD camera.
2.1 Angle-Resolved Photoemission Spectroscopy
UV source
Hemispherical Analyzer
2D detector
Sample Lens
Lens axis
ˆn
ϑ,k
ϑ
Ekin
Figure 2.1: Schematic of an angle resolved electron spectrometer.#is defined as the angle between the surface normalnˆand the lens axis and encodes information about the momentumkkof the intial state.
The electron lens maps the angle at which electrons enter the lens onto positions on the entrance slit of the analyser and accelerates/decelerates the photoelectrons to the working energy of the hemispheres. The two concentric hemispheres are held at a certain voltage difference and the trajectories of the photoelectrons through the hemispheres is defined by their kinetic energy.
The hemispheres map the entrance slit (and therefore emission angle) on to the tangential coordinate of the MCP and disperses them in energy along the radial coordinate.
Spin- and Angle-Resolved Photoemission Spectroscopy
The spin polarization vector of the photoelectron emitted from a crystal surface can be measured by coupling the electron lens and hemispherical analyser to a Mott Polarimeter. Photoelectrons of the pre-selected energy and emission angle are accelerated to voltages on the order of a few 10 keV before scattering from a thin film target of highZ elements such as Au. The angular distribution of the back-scattered electrons depends on their spin vector due to a strong spin-orbit coupling near the heavy nuclei of the target which is known as Mott scattering. The spin polarization of the incoming electron beam along a given axis results in an asymmetric distribution of the back scattered electrons which is detected as an asymmetry in the photo-currents of symmetrically placed channeltrons. The detected asymmetry of a fully polarized electron beam is given by the effective Sherman function
of the spin-polarimeter Se↵, which is determined experimentally and for realistic Mott polarimeters takes values of ⇠15%[71]. Within this thesis I used such a setup the the Bessy II synchrotron in Berlin, Germany in order to investigate the spin structure of the STO (001) surface 2DEL.
Geneva Lab
During my doctoral studies I contributed to the set-up and commissioning of the ARPES lab at the University of Geneva. This system has three light sources: a Helium discharge lamp from MBS with a plane grating toroidal mirror monochromator from SPECS, a frequency converted CW diode laser fromLEOS Solutions providing photons of 6.05 eV and aLumeras source generating 113 nm (11 eV) radiation by mixing the fourth harmonic and the fundamental of a pulsed, narrow bandwidth pump laser with 1024 nm wavelength. It uses an MBS analyser with two deflectors in the lens which allows Fermi surface maps to be measured without moving the sample.
Combined with the small focus of the 6.05 eV laser this system is optimised for measuring very small samples or samples with very inhomogeneous surfaces. We have a home-built 6-axis cryo-manipulator with a sample base
Figure 2.2:Left: data from the Geneva lab on Sr3Ru2O7 at 4.5 K. Right: EDC on gold at 2K integrated across the entire detector demonstrating the ultimate resolution of the system with the 6.05 eV CW laser is 0.85 meV.
temperature of ⇡4.5K. The ultimate resolution of our system is ⇡0.85 meV as shown in Fig. 2.2 for a Fermi edge taken on polycrystalline gold at 2 K. This resolution was achieved in part by reducing noise on the sample position originating from poor high frequency electrical contact between the chamber and the cryostat at the rotary seal of thexyz-stage, to below 0.2 meV (see Fig. 2.3). An example of high resolution ARPES data from the Geneva lab is shown in Fig. 2.2(a).
2.1 Angle-Resolved Photoemission Spectroscopy
The data presented in this thesis were all acquired at Synchrotron beam-lines: I05 beamline of the Diamond Light Source, Surface and Interface Spec-troscopy (SIS) beamline of the Swiss Light Source (SLS) and the PHOENEX endstation on the UE112-PGM1 beamline of BESSY II. An advantage of synchrotron-based ARPES is the availability of higher and variable photon energies. Additionally, the small spot size of the synchrotron beam (compared to a He discharge lamp) reduces the detrimental effect of spatial averaging over inhomogeneous sample surfaces. These advantages were particularly important for the samples measured during the course of my thesis for several reasons:
• The cubic structure of the crystals studied in this thesis do not possess a natural cleavage plane. This leads to rather inhomogeneous surfaces being exposed by fracturing.
• The matrix elements of the STO 2DEL are more intense in the second Brillouin zone which is not accessible at laser energies.
• Variable polarization and photon energy were important for demon-strating the orbital nature and dimensionality of the investigated states respectively.
• Creation of 2DELs at the surfaces of STO, KTO and TiO2requires light induced oxygen vacancies. These vacancies are created more efficiently at the photon energies and intensities provided by a synchrotron.
1.0 0.8 0.6 0.4 0.2 Normalized Counts 0.0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Noise Amplitude (meV)
∆=0.2 meV ∆=0.9 meV
Figure 2.3:Histogram of the potential differences between sample and UHV chamber with (black) and without (red) good high frequency electrical contact between the system and the manipulator at the rotary seal. Full width half maximum is indicated by .
Surface Preparation
The extreme surface sensitivity of VUV photoemission experiments demands atomically clean sample surfaces, which imposes the need for ultra high
vacuum (UHV) conditions. To achieve an atomically clean surface the most common approach is to cleave the samplein situ. This is achieved by glueing a post to the sample surface (see figure) and hitting this in vacuum to break the crystal. This works well in samples with natural cleavage planes.
However in the cubic or quasi-cubic crystals studied in this thesis there is no natural mirror plane which makes them difficult to break. In order to facilitate cleavage in a preferred plane, we cut deep notches parallel to the plane using a diamond wire saw (as sketched in Fig. 2.4).
≈ 2 mm
ˆn
(a) (b)
Figure 2.4:(a) A sample mounted with a top-post prior to being loaded into the UHV chamber (upper) and a sketch of a cubic sample that has been notched with a wire saw perpendicular to the surface normalnˆ (lower). (b) CAD assembly (left) and photograph (right) of the UHV resistive heater in the Geneva lab.
Alternatively, clean surfaces can be obtained by sputtering and annealing.
During my studies I designed and built the resistive heater stage shown in Fig. 2.4 which can reach800 C and provides rotation around two axes and which is used forin situ annealing.