Optical and Electrical Properties of Nanoscale Materials

Preface
Acknowledgements
Contents
1 The Interaction of Light with Solids: An Overview of Optical Characterization
	1.1 The Wave Nature of Light
	1.2 Dielectric Tensor of Bulk Crystals
	1.3 Spectroscopic Ellipsometry
	1.4 Fresnel Equations for the Reflection of Light
		1.4.1 Fresnel Description of the Reflection of Light from an Isotropic Material
		1.4.2 Isotropic Bulk Materials
		1.4.3 Isotropic Thin Film on Isotropic Bulk Substrate
		1.4.4 Ultra-Thin Dielectric Film Ellipsometry
		1.4.5 Thin 2D Film on Transparent Solid
		1.4.6 Effective Medium Approximation for Surface Roughness
		1.4.7 Anisotropic Uniaxial Solid with Uniaxial Optical Axis Normal to the Surface
		1.4.8 Anisotropic Uniaxial Solid with Uniaxial Optical Axis Parallel to the Surface
		1.4.9 Anisotropic Uniaxial Thin Film with the Optical Axis Normal to the Surface of an Isotropic Substrate (or Anisotropic Uniaxial Thin Film with the Optical Axis Normal to a Uniaxial Substrate with Optical Axis also Normal to the Surface)
		1.4.10 Anisotropic Biaxial Solid with One Optical Axis Normal to the Surface and the Second Normal to the Plane of Incidence
		1.4.11 Anisotropic Biaxial Film on an Isotropic Substrate with One Optical Axis Normal to the Surface and the Second Normal to the Plane of Incidence
	1.5 Examples of Reflectance and Ellipsometry of 2D Films
		1.5.1 Graphene
		1.5.2 Monolayer TMD’s (Trilayers of Chalcogenide—Transition Metal—Chalcogenide)
		1.5.3 Topological Insulators
		1.5.4 2D Slab and Surface Current Models for the Optical Conductivity of 2D Films
	1.6 Generalized Ellipsometry: Optical Transition Matrix Approach for Crystals and Thin Films with Arbitrarily Oriented Optical Axes
	1.7 Optical Properties of Materials (Dielectric Function/Complex Refractive Index)
	1.8 The Particle Nature of Light
	1.9 Raman Spectroscopy
		1.9.1 Theory of Raman Scattering
		1.9.2 Diamond and Zinc Blende Crystals
		1.9.3 Wurtzite and other Uniaxial Crystals
		1.9.4 Van Der Waals (Layered) Materials
	1.10 Photoluminescence
	References
2 Introduction to the Band Structure of Solids
	2.1 Band Structure and Optical Properties
	2.2 Block Theorem
	2.3 First Brillouin Zone
	2.4 Block Function Wave Vector 2"0245k
	2.5 A Simple s Level Conduction Band for a Semiconductor Using a Tight Binding Approximation
	2.6 A Simple p Level Valence Band Using a Tight Binding Approximation
	2.7 Hybrid  sp3  Bonding in Semiconductors Versus the Band Picture
	2.8 Spin–Orbit Coupling (A Semiclassical Approach)
	2.9  k cdotp  Theory
	2.10 Effective Mass
	2.11 Tight Binding Model in the Second Quantization Formalism
	2.12 Crystal Structure Symmetry - Definitions of Point Groups and Space Groups
	References
3 Instrumentation
	3.1 Spectroscopic Ellipsometry
	3.2 Raman Spectroscopy
	3.3 Photoluminescence Spectroscopy
	References
4 Microscopic Theory of the Dielectric Function
	4.1 Relationship Between Dielectric Function and Optical Absorption
	4.2 Semiclassical Derivation of the Dielectric Function
	4.3 The Energy Dependence of the Dielectric Function for Parabolic Bands
	4.4 Joint Density of States, Critical Points, and Van Hove Singularities
	4.5 The Naming and Energy Dependence of the Critical Points
	4.6 Determining the Critical Point Energy Using Experimental Data
	4.7 Critical Points in Semiconductors (E1, E2, etc.) Review of Si, Ge, GaAs and Other Group IV and III-V Materials
		4.7.1 Brillouin Zone of Silicon, Germanium, Tin, and Diamond
		4.7.2 Critical Points of Silicon
		4.7.3 Critical Points of Germanium and Diamond
		4.7.4 Comments on Spin Orbit Splitting and CP Energies for Ge
		4.7.5 Critical Points of Sn
		4.7.6 Critical Points of GaAs and GaSb
		4.7.7 Critical Points of GaN
		4.7.8 Critical Points of CdSe
		4.7.9 Critical Points of Si1-xGex Alloys
		4.7.10 Critical Points of Ge1-xSnx Alloys
	4.8 The Effect of Doping on the Dielectric Function
	References
5 Excitons and Excitonic Effects During Optical Transitions
	5.1 Description of Excitons in 3D, 2D, and 1D
	5.2 Energy of Excitons in 3D, 2D, and 1D
		5.2.1 3D (Bulk Materials)
		5.2.2 2D (Nanofilms)
		5.2.3 1D (Nanowire)
		5.2.4 0D (Nanodots)
	5.3 Exciton Binding Energy in Semiconductor Dielectric Quantum Wells
	5.4 The Impact of Nanolayer Thickness on Band Gap and Photoluminescence Determination of Exciton Binding Energy
	5.5 Derivation of Dielectric Function Including Excitons and Excitonic Effects
		5.5.1 Quantum Mechanical Derivation of Excitonic Effects for a Direct Gap Transition
		5.5.2 Elliott Description of Absorption for 3D, 2D, and 1D and the Sommerfeld Factor for Coulomb Enhancement
	5.6 The Effect of Nanoscale Dimensions on the Band Gap, Band Structure and Exciton Energies of Semiconductors
		5.6.1 The Bandgap of Semiconductor Nanodots
		5.6.2 Thickness Dependence of Exciton Binding Energies in III-V Quantum Wells
		5.6.3 Electron–Phonon Interactions in Nanoscale SiO2-Si-SiO2 Quantum Wells
	5.7 Comments on Photoluminescence Lineshape
	References
6 Hall Effect Characterization of the Electrical Properties of 2D and Topologically Protected Materials
	6.1 Classical Hall Effect (HE)
		6.1.1 Classical Picture of Edge States
		6.1.2 Classical Picture of Magneto-Conductivity Tensor
	6.2 Integer Quantum Hall Effect (IQHE)
		6.2.1 Landau Levels—The Quantization of 3D and 2D Carrier Motion in a Magnetic Field
		6.2.2 Integer Quantized Transport
		6.2.3 Experimental Microscopic Observation of Carrier Transport and Chemical Potential for the IQHE
		6.2.4 Summary for experimental imaging of IQHE
	6.3 Topological Explanation of the Integer Quantum Hall Effect (IQHE)
		6.3.1 Berry Phase, Berry Curvature, and Berry Potential
		6.3.2 The Kubo Formula for the Conductivity and the TKNN Theory of the IQHE
		6.3.3 Why Topological
		6.3.4 Quantization of the Hall Conductance and the TKNN (Chern) Number
		6.3.5 Winding Number and Edge State Quantization in IQHE
		6.3.6 Brief Introduction to the Topological Description of Electronic Band Structure
	6.4 Integer Quantum Hall Effect for Graphene
	6.5 Fractional Quantum Hall Effect (FQHE): Many Body Physics in Action
	6.6 Anomalous Hall Effect (AHE)
	6.7 Quantum Anomalous Hall Effect (QAHE)
	6.8 Spin Hall Effect (SHE) and Quantum Spin Hall Effect (QSHE)
	6.9 Optical Measurement of Spin and Pseudospin Conductance
	6.10 Thermal (Nernst) Spin Hall Effect
	6.11 Skyrmion Hall Effect
	6.12 Summary
	References
7 Optical and Electrical Properties of Graphene, Few Layer Graphene, and Boron Nitride
	7.1 Hexagonal Graphene
		7.1.1 Bravais Lattice of Graphene
	7.2 Tight Binding Approximation for the π Bands of Graphene
		7.2.1 Another Look at the Reciprocal Lattice of Graphene
		7.2.2 Graphene’s Π Electronic Band Structure
		7.2.3 Comparing Nearest Neighbor Graphene Energy Bands to Ab Initio Results
		7.2.4 Sub-lattice PseudoSpin (Valley) and the Graphene Band Structure
		7.2.5 Dirac Points and Dirac Cones
		7.2.6 Dirac Cone Shape for Graphene with NNN (Next Nearest Neighbor) Hopping
		7.2.7 Hexagonal 2D Lattices with Different Atoms at A and B Positions (E.G., Hexagonal Boron Nitride, h-BN)
	7.3 The Importance of Understanding the Optical and Electrical Properties of Graphene: Proof of Dirac Carriers
		7.3.1 Electrical Test Structures for Graphene and Graphene Multilayers
	7.4 Introduction to Relativistic Quantum Mechanics for 2D Materials
		7.4.1 Sub-lattice Pseudospin, Valley Pseudospin, and Chirality for Dirac Fermions in Graphene
		7.4.2 Berry Phase of an Electron in the π Bands of Graphene
	7.5 The Berry Phase Correction for the Quantum Hall Effect and Shubnikov De Hass Oscillations in Graphene
	7.6 Electronic Structure of Bilayer Graphene
		7.6.1 Massive Dirac Fermions in Bilayer Graphene
		7.6.2 The Berry Phase Correction for the Quantum Hall Effect and Shubnikov De Hass Oscillations in Bilayer Graphene
	7.7 The Electronic Structure of TriLayer and TetraLayer Graphene
		7.7.1 The Berry Phase Correction for the Quantum Hall Effect and Shubnikov De Hass Oscillations in Trilayer Graphene
	7.8 Optical Characterization of Graphene and Multilayer Graphene
	7.9 Effect of Rotational Orietation Between Layers on Bilayer Graphene (Twisted Bilayer Graphene), Monolayer—Bilayer Graphene, and Bilayer-Bilayer Graphene Properties
		7.9.1 Twisted Bilayer Graphene
		7.9.2 Monolayer—Bilayer Graphene, Middle Layer—Twist Angle Trilayer Graphene, and Bilayer-Bilayer Graphene
	7.10 The Electronic Band Structure and Optical Properties of Hexagonal Boron Nitride (h-BN) and Graphene—h-BN
		7.10.1 Graphene—BN Heterostructures
	References
8 Optical and Electrical Properties of Transition Metal Dichalcogenides (Monolayer and Bulk)
	8.1 Structure and Bonding for TMD Materials
	8.2 Tight Binding Model for Highest Energy Valence Band and Lowest Energy Conduction Bands of Trigonal Prismatic Monolayer TMD
		8.2.1 Band Splitting Due to Spin Orbit Coupling
	8.3 Direct Observation of Monolayer TMD Valley Pseudospin and Valence Band Spin Splitting
	8.4 Massive Dirac Fermions: Physics and Optical Transitions at the K and K Points in the Brillouin Zone
	8.5 Band Gap Renormalization and Photoluminescence Lineshape
	8.6 The Complex Refractive Index (Dielectric Function) and Optical Conductivity of Monolayer TMD
		8.6.1 Optical Conductivity of Monolayer TMD
	8.7 Structure, Electronic Band Structure, and Optical Properties of Bilayer Trigonal Prismatic TMD
	8.8 Twisted Bilayer TMD
	8.9 The Complex Refractive Index (Dielectric Function) of Multilayer and Bulk TMD
	8.10 The Layer Number Dependence of Raman Scattering from Trigonal Prismatic TMD
	8.11 Transition-Metal Dichalcogenide Haeckelites (A Theoretical Material)
	8.12 Twisted and Hetero-Bilayers of Transition Metal Dichalcogenides with graphene and h-BN
	8.13 ReS2 and ReSe2 with the 12T Structure
	8.14 Practical Aspects of Characterization of TMD Materials Using Spectroscopic Ellipsometry
	8.15 Symmetry and Space Group Summary for Transition Metal Dichalcogenides
	References
9 Optical and Electrical Properties Topological Materials
	9.1 Overview of Topological (Dirac) Materials
		9.1.1 Topological Surface States on 3D Topological Insulators
		9.1.2 Weyl Semimetals and Dirac Semimetals
		9.1.3 Large Gap Quantum Spin Hall Insulator
		9.1.4 Axion and Axion Insulator
		9.1.5 Mott Insulator
		9.1.6 Chern Insulator
		9.1.7 Topological Superconductors
	9.2 Tight Binding Hamiltonian with Spin–Orbit and On-Site Coulomb (Hubbard) Interactions and a 3D Dirac Equation
	9.3 Optical and Electronic Properties of Topological Materials
	9.4 3D Topological Insulators
		9.4.1 Crystal and Electronic Band Structure of 3D Topological Insulators and Large Gap Quantum Spin Hall Insulators
		9.4.2 Optical Properties of 3D Topological Insulators and Large Gap Quantum Spin Hall Insulators
		9.4.3 Electrical Properties of 3D Topological Insulators and Large Gap Quantum Spin Hall Insulators
	9.5 Weyl, Dirac Semimetals, and Related Materials
		9.5.1 Structure, Bonding, and Electronic Band Structure of Weyl, Dirac Semimetals, and Related Materials
		9.5.2 Optical Properties of Weyl, Dirac Semimetals, and Related Materials
		9.5.3 Electrical Properties of Weyl, Dirac Semimetals, and Related Materials
	References
Appendix A Mueller Matrix Spectroscopic Ellipsometry
References
Appendix B Kramers–Kronig Relationships for the Complex Refractive Index and Dielectric Function
References
Appendix C Topological Periodic Tables
References
Index