Advanced Solid State Physics

Outline

• Fourier transforms
• Fourier series
• Fourier series in 1-D
• Reciprocal lattices
• Fourier series in 2-D
• Fourier series in 3-D
• Plane waves and reciprocal space
• Fourier transforms
• Plotting Fourier transforms
• Electrons in crystals
• Band structure calculations
• Free electrons
• Free electron model in 1-D
• Free electron model in 2-D
• Free electron model in 3-D
• Fermi function
• Empty lattice approximation: simple cubic, fcc, bcc, hexagonal, tetragonal, orthorhomic, simple monoclinic
• Brillouin zones of 2d Bravais lattices
• Fermi surface of a two-dimensional square lattice
• Fermi surfaces of some three-dimensional lattices
• Bloch waves in one dimension
• Band structure of one-dimensional potentials
• Kronig Penney Model
• Tight binding
• Table of tight binding band structure calculations
• Plane wave method
• Density Functional Theory
• Band structure calculations: Cr bcc, Li bcc, GaAs, GaN, GaP, Ge, InAs, 6H SiC,V bcc
• Calculated electron density of states
• Al fcc, Au fcc, Cu fcc, Cr bcc, Li bcc, Na bcc, Pt fcc, W bcc, Si diamond, Fe bcc, Ni fcc, Co fcc, Mn bcc, Cr bcc, Gd hcp, Pd fcc, Pd₃Cr, Pd₃Mn, PdCr, PdMn , GaN, 6H SiC, GaAs, GaP, Ge, InAs, V bcc
• The NOMAD Repository holds millions of band structure calculations
• Thermodynamic properties of non-interacting fermions
• Grand potential φ
• Chemical potential μ(T)
• Helmholtz free energy f
• Entropy s
• Energy spectral density u(E,T)
• Internal energy density u(T)
• Specific heat c_v(T)
• Fermi liquid theory     W
• Experimental methods
• Photoemission Spectroscopy (UPS XPS, ARPES, PEEM)
• Scanning tunneling spectroscopy      W
• Inverse photoemission spectroscopy
• Semiconductors
• Simplified band structures from the NSM Archive
  • Direct band gap: InAs, InP, GaAs, InN, GaN (zincblende), GaN (wurtzite), AlN
  • Indirect band gap: Ge, Si, GaP, Ga₂O₃
• Boltzmann approximation
• Intrinsic semiconductors     W
• Effective density of states N_c, N_v
• The density of electrons in the conduction band n = N_cexp((μ - E_c)/k_BT)
• The density of holes in the valence band p = N_vexp((E_v - μ)/k_BT)
• Law of mass action: np = N_cN_vexp(-E_g/k_BT)
• The intrinsic carrier density n_i =(N_cN_v)^1/2exp(-E_g/2k_BT)
• Chemical potential of intrinsic semiconductors
• Thermodynamic properties of intrinsic semiconductors
• Intrinsic semiconductors with a split-off band
• Table summarizing the thermodynamic properties of semiconductors in the Boltzmann approximation
• Extrinsic semiconductors
• Doping     W
• Extrinsic carrier densities
• Chemical potential in extrinsic semiconductors     W
• Determining chemical potential from the charge neutrality condition
• Semiconductor devices
• pn junctions
• diodes
• Current-voltage characteristics
• Light emitting diodes
• Solar cells
• Laser diodes
• Transistors: JFET, MOSFET, Bipolar
• Phonons
• Normal modes and phonons
• Using complex numbers to represent sinusoidal oscillations
• Linear chain
• Linear chain with two different masses
• fcc with linear springs to nearest neighbors
• bcc with linear springs to nearest neighbors and next nearest neighbors
• simple cubic with linear springs to nearest neighbors and next nearest neighbors
• Jupyter python notebook for calculating phonon dispersion equations
• Animations of some optical modes
• Thermodynamic properties of non-interacting bosons
• Density of states
• simple cubic, bcc, fcc
• Ag-fcc, Al-fcc, AlN, Au-fcc, Fe-bcc, GaN, Mg-hcp, Mo-bcc, Si-diamond, α-Sn, β-Sn, Ta-bcc, Tb-hcp, Ti-hcp, W-bcc, ZnO (rocksalt), ZnO (zincblende), ZnO (wurtzite), Zr-hcp
• Energy spectral density u(ω,T)
• Internal energy density u(T)
• Specific heat c_v(T)
• Helmholtz free energy density f(T)
• Entropy density s(T)
• Table summarizing the thermodynamic properties of phonons
• Magnetic effects and Fermi surfaces
• Schrödinger equation for electrons in a magnetic field
• Solutions to the Schrödinger equation for a charged particle in a magnetic field, Landau levels
• Thermodynamic properties of free electrons in two dimensions in a magnetic field at zero temperature
• de Haas - van Alphen oscillations      W
• Shubnikov - de Haas oscillations
• Drift and diffusion of electrons in electric and magnetic fields
• Quantum Hall effect      W
• Thermodynamic properties of free electrons in a magnetic field
• Measurements of the Fermi surface
• Magnetism
• Introduction
• Diamagnetism      W
• Paramagnetism
• Magnetic type for the elements
• Pauli paramagnetism / Landau diamagnetism
• Ferromagnetism     W
• Mean field theory
• Curie temperature     W
• Curie-Weiss law     W
• Exchange
• Ferrimagnetism and Antiferromagnetism
• Magnetite      W
• Spinel crystal structure     W
• Neel Temperature     W
• Magnetic domains     W
• Anisotropy energy
• Hard axis, easy axis
• Bloch walls     W
• Remnant magnetization
• Coersive field
• Hard magnets and soft magnets
• Superparamagnetism     W
• Magnons     W
• Giant magnetoresistance     W
• Magnetoresistive Random Access Memory     W
• Magnetic force microscope      W
• Superconductivity      W
• Meissner effect      W
• London equations      W
• Macroscopic quantum model
• BCS theory      W
• Ginzburg Landau theory      W
• Josephson effects      W
• SQUID      W
• Linear response theory
• Causality and the Kramers-Kronig relations     W
• Impulse response function, Green's function
• Generalized susceptibility
• Causality     W
• Fluctuation-dissipation theorem
• Dielectric function
• Insulators and semiconductors
• Breakdown field     W
• Low-k dielectrics     W
• High-k dielectrics     W
• Loss tangent     W
• Polarizability     W
• Inter- and Intraband transitions
• Joint density of states
• Metals
• Collisionless metals
• Diffusive metals
• Microwave engineering
• Experimental techniques
• Reflection electron energy loss spectroscopy (REELS)
• Ellipsometry
• Optical properties of selected materials
• Fresnel equations
• Transport
• Thermoelectric effects
• Crystal momentum
• Boltzmann equation
• Relaxation time approximation
• Current densities
• Electrochemical potential
• Electrical conductivity
• Graphite
• Thermoelectric currents
• Seebeck effect
• Free-electron model
• Electrical conductivity
• Electrical contribution to the thermal conductivity
• Thermoelectric coefficient, Seebeck coefficient
• Wiedemann Franz law
• Transport in 1-D crystals
• Current of Bloch waves in 1-D
• Thermionic-emission current in a Schottky diode
• Phonon contribution to thermal conductivity
• Crystal physics
• Calculating the equilibrium properties of crystals from the Gibbs free energy
• Nonlinear optics
• Symmetries
• Point groups     W
• Space groups     W
• Symmetry transformations of tensor quantities
• Table of the 32 crystal classes
• Space group → Bravais lattice
• Symmetric and asymmetric tensors
• Flowchart to determine the point group of a crystal
• SGTE data for pure elements - The Gibbs energy as a function of temperature for many elements.
• Structural phase transitions
• Tin
• Iron
• Piezoelectricity     W
• Ferroelectricity     W
• Paraelectricity     W
• Pyroelectricity     W
• Antiferroelectricity     W
• Verwey transition     W
• Perovskites     W
• Landau theory of phase transitions
• Broken symmetry
• Order parameter
• Landau theory of second order phase transitions
• Landau theory of first order phase transitions
• Examples
• Superconductivity
• Superfluidity
• Ferromagnetism
• Peierls transition
• Ferroelectricity
• Ginzburg-Landau theory     W
• Quasiparticles
• Particle-like quasiparticles
• Fermi liquid theory     W
• Electrons and holes
• Bogoliubov quasiparticles (superconducting quasiparticles)
• Polarons     W
• Bipolarons
• Mott-Wannier Excitons
• Frenkel Excitons
• Collective modes
• Phonons
• Polaritons     W
• Magnons
• Table of magnon properties
• Plasmons     W
• Surface Plasmons     W
• Surface plasmon simulation
• Experimental techniques
• Raman spectroscopy      W    5:38 
• Electron energy loss spectroscopy (EELS)
• Inelastic scattering
• Photoemission
• Electron-electron interactions
• Electron screening
• Single electron effects
• Tunneling transmission probability
• Coulomb blockade in single-electron transistors
• Current - voltage characteristics of a symmetrically biased SET transistor
• Current - voltage characteristics of an asymmetrically biased SET transistor
• Current - gate voltage characteristics of a SET transistor
• Electronic metal-insulator phase transitions
• Mott Transition
• Peierls transition     W

Appendices
• Quantization
• Recipe for quantization
• Construct the Lagrangian by inspection
• Differentiate the Lagragian to find the conjugate variables
• Use the conjugate variables to perform a Legendre transformation from the Lagrangian to the Hamiltonian
• Examples
• LC circuit
• Transmission line
• Dissipation in quantum mechanics
• Wave equation      W
• Electrons in a magnetic field
• Photons
• Photons in vacuum
• Review of Maxwell's unification of electromagnetism with optics      W
• Quantization of the normal mode solutions to the wave equation      W
• Thermodynamic properties of light
• Thermodynamic properties of non-interacting bosons
• Particle density n
• Grand potential φ
• Helmholtz free energy f(T)
• Entropy s(T)
• Energy spectral u(ω,T)
• Internal energy u(T)
• Specific heat c_v(T)
• Table summarizing the results of the quantization of light in vacuum
• Photonic crystals
• Light in one-dimensional layered material
• Empty lattice approximation: simple cubic, fcc, bcc, hexagonal, tetragonal, orthorhomic, simple monoclinic
• Plane wave method
• Table of photonic crystals