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Quantum Optics and Light-Matter Interactions: a Look Into ARPES

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Quantum Optics and Light-Matter Interactions: a Look Into ARPES Quantum Optics and Light-Matter Interactions: A Look Into ARPES Robin Erdakos (Dated: PHYS 24300 - Winter 2021) CONTENTS I. Introduction 2 II. Quantum Theories of Light and Matter 2 III. Solid-State and Condensed Matter Physics 4 IV. Applications 5 A. ARPES 5 B. trARPES 8 V. Conclusion 9 References 9 2 I. INTRODUCTION Quantum Optics and Nanophotonics are branches of physics concerned with study- ing the behavior of light and its interactions with physical materials on a small scale. In- teractions between photons and nanometer-scale materials play important roles in studying the intrinsic and extrinsic characteristics of a material, such as its electronic, magnetic, and topological properties. These interactions are deeply involved in spectroscopy, laser systems, sensing, and quantum information processing, and photons have been extremely useful in advancing studies in quantum mechanics and engineering. This paper is concerned with understanding the importance of quantum mechanically governed systems in nanoscale research and materials science and engineering. First, I will discuss relevant topics in quan- tum mechanics and introduce important concepts from solid-state and condensed matter physics. I will then discuss one of its important applications in modern research and current experimental studies, which is angle-resolved photoemission spectroscopy (ARPES). II. QUANTUM THEORIES OF LIGHT AND MATTER Theories of quantum mechanics are directly connected to how we study light-matter interactions. For example, the most fundamental observation relevant to interactions be- tween light and matter is the photoelectric effect. This phenomenon is directly concerned with the interactions between light and atomic particles, where electromagnetic radiation in the form of light causes the emission of electrons from a material. Historically, experimental results conflicted with classically defined theoretical predictions, which led to propositions that light was made up of discrete energy quanta, known as photons. This led to the even- tual theories of wave-particle duality that define light, however, here we are more concerned with its applications. Specifically, photoemission spectroscopy, which we will discuss in more detail later, allows us to determine the binding energy of electrons in a material. This is done using conservation laws of energy, with measured photoelectron energies and known incident photon energies. The importance in studying the nature of electrons within a ma- terial is that the atomic scale characteristics of a material often govern their larger scale physical properties. Electronic properties such as conductivity and resistivity, as well as other optical and topological properties are some of the examples of structurally dependent 3 characteristics within a material. Furthermore, the structural characteristics on the atomic scale are governed by quantum mechanics, which explains the importance of this section. Another relevant phenomena to consider from quantum mechanics is the stimulated emission of photons. Lasers specifically are fascinating quantum-mechanical devices that are based on this phenomena, and are extremely important for the field of quantum optics. Again being concerned with light-matter interactions, the stimulated emission of photons is a form of optical amplification to a coherent light source. To give a quick overview of how lasers work, they involve a material that acts as an amplifying "gain" medium, as well as the absorption and spontaneous emission of photons of specific energies. In the figure below we see how stimulated emission works, which is clearly different to spontaneous emission. Light amplification occurs when the rate of stimulated emission exceeds the rate of light absorption in the material. FIG. 1. Process of stimulated emission of photons for electrons in discrete energy levels, taken from Wikipedia page on Lasers The great importance and usefulness of these lasers is in the ability to control this amplification of photons. Optical resonators are used to further amplify stimulated emission, as well as polarize the photons to be in the same direction. This is a complicated process that I will not delve into in this paper, however, it is important that we recognize the importance of lasers in studying light-matter interactions. Lasing system are an important aspect of ARPES, and they are used to directly control and study interactions between incident photons from a laser and nanoscale substances. 4 III. SOLID-STATE AND CONDENSED MATTER PHYSICS In this section I want to give some information about relevant fields of research that can help us understand what is going on in light-matter experiments. Fundamentally, light- matter interactions are governed by quantum electrodynamics, and are often described as quantum transitions of electrons. Electron transitions are accompanied by the absorption or emission of electromagnetic radiation in the form of quantized photons. In solid-state physics, physical properties of materials such as resistivity and conductivity can be explained by electronic band structure. Band structure describes the range of energy levels that an electron may or may not populate, in bands and band gaps respectively. These bands form due to the overlap of atomic orbitals with discrete energy levels, which occurs when a large number of identical atoms form a solid. Due to the large number of atoms in a solid, the energy of adjacent levels are so close together that they create these energy bands. An important part of the electron band diagram is the Fermi level/energy, which is often defined as the highest energy that electrons assume at a temperature of 0 kelvin. In insulators and semiconductors there is a band gap that is directly surrounded by the conductance band above, and the valence band below. Metals and semimetals, for example, have one or more bands overlapping the Fermi level. These bands are the most important for determining the electronic and optical properties of a material. As the band gap refers to the energy difference between the valence and conductance bands, it also represents the energy required to move an electron from one band to the other. When the valence band of electrons is full and the conductance band is empty, charge does not flow. However, when electrons jump to the conductance band, the movement of charge carriers conducts an electric current. This helps us to realize why large band gaps are characteristics of insulators, because there is too much energy required to move electrons from the valence band. Furthermore, as conductors have very small or no band gap at all, electrons can more easily transition between levels, allowing for electrical current to be conducted. In the experiments we will discuss later, the electronic band structure is measured and these properties are observed. In understanding more about the materials that exhibit these properties, we can use them for future technologies as well as study new materials for interesting properties. 5 IV. APPLICATIONS A. ARPES An important application of these concepts is the technique of Angle-resolved Pho- toemission Spectroscopy, or ARPES. ARPES is a powerful tool for studying the intrinsic properties of a material, and it allows for the direct measurement of the electronic band structure of a material. This tool is based on the photoelectric effect, where the emit- ted photo-electrons are collected by a hemispherical analyzer, which resolves their emission angle and kinetic energy. This analyzer uses an electric field between two concentric hemi- spherical electrodes to manipulate the trajectory of incoming electrons, dependent on their kinetic energies. Using laws of conservation of energy and momentum, the electronic band structure can be mapped with energy and momentum distributions. This is all done in an ultrahigh vacuum environment of less than 10−11 Torr, which is necessary to reduce the scattering of electrons with gas molecules that would be present. Modern instruments for ARPES allow for precise measurements of energies and angles, up to about 1 meV and 0:1◦ respectively. Furthermore, ARPES provides relatively clean measurements with powerful resolutions sufficient to make very precise evaluations. Photons with frequency ν have energy E = hν where h is Planck's constant. Once stimulated, electrons will be emitted with a characteristic energy Ek at an angle θ to the normal of the surface, with the energy given as Ek = hν − EB, where EB is the electrons 6 p binding energy. The electrons momentum has a magnitude of jpj = 2meEk, which is linked 1 to the Bloch wave vector k by the expression kk = ¯h p. Using the emission angle, and the geometry of the apparatus, the components of the electrons momentum are given below. 1 p kx = ¯h 2meEk(sin(θ)cos(α)) 1 p ky = ¯h 2meEk(cos(θ)sin(η)cos(α)) ± sin(α)cos(η) FIG. 2. (a) Definition of angles used in ARPES measurements. (b) Configuration of photon incident on sample. [1] An important theorem that is used here from quantum mechanics is Bloch's the- orem, which states that the solutions to the Schr¨odinger equation in a periodic potential take the form of a plane wave with a periodic function modulating it [2]. This occurs in cases where the kinetic energy and crystal potential govern the behavior of the solid-state system, which is the case in ARPES samples. The most common example of this theorem is in describing electrons in a crystal, which contain periodically ordered structural properties. In ARPES it is used to characterize electronic properties, namely the band structure we are concerned with. ik·r (r) = e uk(r) [2] Mapping the band structure over this two-dimensional momentum space involves rotating the sample in the above geometries while keeping the incident laser at a fixed spot on the surface. Depending on the material and where the spot is located, different band structures can be generated. For instance, some samples are made using layers of different materials, such as MnBi2Te4. When these samples are prepared and cleaved for use in 7 ARPES, different cross-sections will have different materials on the surface.
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