Course outline

PHY.K02UF Molecular and Solid State Physics

Course outline

Introduction

Review of atomic physics (pdf v.2018)

The solutions to the Schrödinger equation for the hydrogen atom

Atomic orbitals

Slater's rules

Helium
Many-electron wavefunctions
Slater determinants W
Singlet and triplet states
Exchange W
The intractability of the Schrödinger equation
Many-electron atoms

Molecules (pdf v.2018)

Molecular orbital theory W

A quantum mechanical description of molecules W

Harmonic oscillator

Solving the molecular orbital Hamiltonian

Valence bond theory

Heitler-London theory

Numerically calculated molecular orbitals

Carbon monoxide CO, Carbon dioxide CO₂, Water H₂O, Hydrogen sulfide H₂S, Nitrogen N₂, Ammonia NH₃, Methane CH₄, Ethene C₂H₄, Ethane C₂H₆, Butadiene C₄H₆, Benzene C₆H₆, Hexatriene C₆H₈

Programs to calculate molecular orbitals

GAMESS, Gaussian, Quantum Espresso, FHI-aims

Rotational and vibrational energy levels of some diatomic molecules
Chemical bonds

Covalent bond W
σ-bonds W
π-bonds W
sp, sp², sp³ orbitals
Double bond W
Triple bond W
Ionic bond W
Polar bond W
Metallic bond W
Van der Waals bond
Hydrogen bond W

Visualization tools

Crystal structure

Asymmetric unit
Symmetries
- Point groups W
- Space groups W
- Space Group → Bravais Lattice, Point Group
Examples of crystal structures

Crystal binding

Molecular crystals W
Ionic crystals W

Madelung constant W
Bulk modulus W

Crystal diffraction

Periodic functions

Interference of scattered waves
Brillouin zones W

Brillouin zones of 2D Bravais lattices
Brillouin zones of 3D Bravais lattices

Drawing 3D Brillouin zones

Symmetry points and lines: Simple Cubic, Face Centered Cubic, Body Centered Cubic, Hexagonal, Rhombohedral, Simple Tetragonal, Body Centered Tetragonal, Simple Orthorhombic, Base Centered Orthorhombic, Face Centered Orthorhombic, Body Centered Orthorhombic, Simple Monoclinic, Base Centered Monoclinic, Triclinic
Symmetry points of 2D lattices: Square, Hexagonal, Rectangular, Centered Rectangular, Oblique

Atomic form factors

Structure factor W
Bragg diffraction W

The reciprocal lattice vector G_hkl is orthogonal to the (hkl) plane.

Laue condition W

Applications of diffraction

powder diffraction W

American Mineralogist Crystal Structure Database (contains diffraction data)

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

Planck's radiation law W

Planck curve for black body radiation

Wien's displacement law, λ_max = 2.8977685 × 10^-3/T m K W
Stefan-Boltzmann law, I = σT⁴ W/m² W
Internal energy density W
Specific heat W
Radiation pressure W
Entropy W
Table summarizing the results of the quantization of the wave equation

Light in one-dimensional layered material
Photonic crystals W

Empty lattice approximation: simple cubic, fcc, bcc, hexagonal, tetragonal, body centered tetragonal, orthorhomic, simple monoclinic
Plane wave method
Table of photonic crystals

Lattice Vibrations and Phonons

Normal modes and phonons

Phonon contribution to the thermodynamic properties of solids

Density of states

linear chain
Einstein model
Debye model
linear chain with two masses
Ag-fcc, Al-fcc, AlN, 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

Kinetic theory

Thermal conductivity

Thermodynamic properties of free, noninteracting electrons

Free electron model in 1-D
Free electron model in 2-D
Free electron model in 3-D

Fermi function
Internal energy density
Specific heat
Helmholtz free energy density
Entropy
Table of thermodynamic properties of free electrons

Sommerfeld expansion

Energy bands

Table of tight binding band structure calculations

Some band structure calculations: Cr, Li bcc, GaAs, GaN, GaP, Ge, InAs, 6H SiC, V

Photoemission Spectroscopy (UPS XPS, ARPES, PEEM)
Metals, semimetals, semiconductors, insulators

Numerical determination of the thermodynamic properties of metals

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
Separable square wave potentials
Materials Project

Kinetic theory

Ballistic transport
Diffusive transport
Drift and diffusion simulation
Ohm's law
Mattheissen's rule
Hall effect
Thermal conductivity
Wiedermann-Franz law
Lorentz number

Summary: Electron Band Model

Crystal physics

Stress and strain

Einstein notation for tensors W
Review of statistical physics
Intrinsic symmetries
Maxwell relations W
Thermodynamic properties

Pyroelectricty W
Pyromagnetism
Piezoelectricty W
Piezomagnetism W
Electrocaloric effect W
Electrostriction W
Magnetostriction W
Thermal expansion W

Groups and symmetry
Table of crystal classes and their associated point groups
Flowchart to determine the point group of a crystal
Symmetric and asymmetric tensors
SGTE data for pure elements - The Gibbs energy as a function of temperature for many elements.

Semiconductors

Role of semiconductors in technology

Band structure of semiconductors

Conduction band E_c, valence band E_v, band gap E_g

Direct and indirect band gaps W
Absorption and emission of photons and phonons
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

Electrons and holes

Effective mass W
Holes W
Crystal momentum W
Ohm's law

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 materials

Silicon, SiC 4H

Semiconductor devices

pn junctions
diodes

Light emitting diodes
Solar cells
Laser diodes

Transistors

Transport in semiconductors

Ohm's law: σ = (neμ_n + peμ_p)
Mobility W