Optical and Electrical Properties of Nanoscale Materials

In this chapter, we cover the basic principles involved in the interaction of light with crystals, thin films, and nanoscale materials necessary for discussing optical characterization. We discuss Fresnel’s equations for bulk materials and thin films on substrates. The Fresnel equations for isotropic, uniaxial, and biaxial materials are all presented in terms of the complex refractive index. This chapter introduces ellipsometric characterization of the dielectric function of nanoscale materials, and it also discusses Raman spectroscopy and photoluminescence of 2D materials.

The basics of electronic band structure are reviewed in this chapter. The chapter emphasizes the tight binding model. Spin orbit interactions are also introduced in this chapter using a semiclassical approach. The \(k \cdot p\) method of calculating electronic band structure is also briefly introduced. The \(k \cdot p\) method provides a useful method for determining the effective mass of electrons and holes from band curvature. The second quantization formalism is introduced. Spatial symmetry is discussed at the end of the chapter including point group and space group symmetry.

In this chapter we provide a brief overview of the instrumentation used for three key optical characterization methods: spectroscopic ellipsometry, Raman spectroscopy, and photoluminescence. We also discuss the relationship between the optical properties determined by each method and the data obtained by the measurement equipment.

In this chapter, we derive the dielectric function for direct gap transitions in the electronic band structure using Fermi’s Golden Rule. We discuss the relationship between the electronic band structure and locations in the Brillouin zone when there is a high density of transitions with the same energy resulting in a critical point. We show the analysis of the dielectric function at critical points to determine the transition energy and peak broadening. A number of examples of the dielectric functions of semiconductors over the visible wavelength range are presented. We also discuss the effect of doping on the dielectric function.

In this chapter, excitons and excitonic effects during optical transitions are discussed. The transition energy for 1D, 2D, and 3D materials which include the exciton binding energy provide a focal point for the chapter. The well-known Elliott description of optical absorption for 3D, 2D, and 1D materials and the Sommerfeld factor for Coulomb enhancement are discussed. The effect of quantum confinement is described. A quantum mechanical derivation of excitonic effects on direct gap optical transitions which alters the energy dependence of optical absorption and thus the dielectric function is presented. Photoluminescence spectra from semiconductors and quantum wells are also presented.

In this chapter we present the classical, quantum, and topological descriptions of electron transport measurements. Hall measurements are introduced using classical physics. The quantization of the electronic levels due to a magnetic field known as the Landau levels is shown. The observation of the quantization of the conductivity in a 2D electron gas at low temperature and high magnetic field due to the pioneering research of von Klitzing is presented. This leads to the introduction of the Berry phase and topological explanation of the quantized conductance. The relationship between the Kubo formula for conductance and topological quantification due to the research of Thouless, Kohmoto, Nightingale, and den Nijs is presented. The Hall characterization of single layer graphene and the observation of the Berry phase confirming the presence of Dirac carriers is used to demonstrate the topological properties of graphene. The family of Hall effects is also presented.

This chapter discusses the unique optical and electrical properties of graphene, bilayer graphene, and boron nitride. The tight binding electronic band structure of the \(\pi\) and \(\pi ^{*}\) bands is presented for both single layer graphene and bilayer graphene. This leads to a discussion of the low energy Dirac carriers present in Dirac cones in graphene. Ellipsometric characterization data for graphene is presented. The Dirac equation is used to show how the optical absorption for the zero mass, low energy carriers in the Dirac cones is directly related to the fine structure constant. The quantization of the Landau levels of the Dirac carriers in graphene is derived. The optical properties and Hall characterization of bilayer graphene and twisted bilayer graphene is discussed. The bands that result from the periodic potential of the moiré lattice due to the twisted alignment of the graphene layers are described. The electronic band structure of boron nitride is presented and heterolayers of graphene and boron nitride are discussed.

In this chapter, we discuss the electronic band structure, electrical, and optical properties of transition metal dichalcogenides. The different crystallographic structures for transition metal dichalcogenides are presented along with a discussion of the chemical bonding. Many of the transition metal dichalcogenides consist of van der Waals bonded monolayers where the monolayers consist of trilayers with a transition metal atom layer between a top and bottom chalcogenide layer. Often these monolayers have a trigonal prismatic arrangement of chalcogenide atoms around the metal atoms. A tight binding model for three of the \(d\) orbitals of the transition metal atoms provides a useful description of the highest energy valence band and lowest energy conduction bands of trigonal prismatic monolayer transition metal dichalcogenide. The impact of spin orbit coupling on the band structure is shown. We discuss how the electronic band structure due to the honeycomb lattice of many transition metal dichalcogenides monolayers interacts with spin orbit coupling resulting in differences in optical transitions between the \(K\) and \(K^{\prime}\) locations in the Brillouin zone. We present photoluminescence spectra demonstrating these differences. We also show theoretical and experimental dielectric function data for a variety of monolayer, multilayer, and bulk transition metal dichalcogenides. We show how Raman spectroscopy is sensitive to the layer structure. We also discuss the observation of superconductivity of TMD materials. A summary of the point group and space group symmetry and Raman Tensors of transition metal dichalcogenides is provided.

In this chapter, we present an overview of the structure, optical and electrical properties of materials that exhibit, or are predicted to exhibit, topological properties. We note that many of these materials consist of layers that are bonded by van der Waals forces and many have a local hexagonal structure. The materials are divided into those with a band gap and those without a band gap. We present the definitions of many of the key terms used in the topological description of materials as well as a description of the various types of topological materials such as topological insulators, Weyl semimetals, and Dirac semimetals. A useful and important part of this discussion is topological classification. For example, we describe the topological invariant \(\mathbb{Z}_{2}\). We present a brief overview of a tight binding, second quantization Hamiltonian that includes spin orbit and electron–electron interactions. Then we discuss materials with a band gap including the well-known tetradymites such as Bi₂Se₃ starting with a discussion of the crystal and electronic structure and resulting \(\mathbb{Z}_{2}\) classification. Next we present the optical and electrical properties of these materials. Whenever possible, experimental data for the dielectric functions are shown. Photoluminescence and Raman spectra are also shown, and the layer number dependence of the Raman spectra are discussed. This is followed by a similar discussion of gapped materials including Weyl semimetals, Dirac semimetals, and nodal line materials. When possible, experimental data is discussed in terms of whether or not topological properties are observed.