ECE407 Physics of Semiconductors and Nanostructures




Lecture notes

PART I: The Free Electron Gas in 1D, 2D, and 3D

Handout 1

Drude model for metals, DC and high frequency conductivity of electrons in metals, Drude expression for dielectric constant of metals, plasma frequency, plasma oscillations with and without electron scattering.

Handout 2

Sommerfeld model for electrons in metals, free Fermion gas, density of states in k-space and in energy, Fermi energy and Fermi surface, electron statistics, current flow under uniform electric field, conductivity of a free Fermion gas.

Handout 3

Electrons in low dimensional metals, 1D and 2D free Fermi gases, density of states in k-space and in energy, Fermi energy, Fermi circle and Fermi point, electron statistics, current flow under uniform electric field, conductivity of Fermion gases in 1D and 2D.

PART II: Crystal Lattices and the Reciprocal Lattice

Handout 4

Crystal lattices in 1D, 2D, and 3D, Bravais lattices, primitive cells and unit cells, Wigner-Seitz cell,  primitive lattice vectors, lattice with a basis, lattices of important metals and semiconductors in 3D and 2D.  

Handout 5 

Fourier transforms of lattices, reciprocal lattice in 1D, 2D, and 3D,  reciprocal lattice vectors, Brillouin zones, Fourier transforms of lattice periodic functions, lattice planes, X-ray diffraction, Bragg condition and Bragg planes. 

PART III: Electron States and Energy Bands in Molecules and Solids

Review Handout

First and second order perturbation theory in quantum mechanics.

Handout 6          

Electrons in a periodic potential, Bloch's theorem and Bloch functions, electron Bragg scattering, nearly free electron approach and perturbation theory, finite basis expansion, physical origin of bandgaps, location of bandgaps, zone folding, introduction to energy bands in solids.

Handout 7

Properties of Bloch functions, periodic boundary conditions, density of states in k-space, number of states per band, band filling, Fermi contours (2D) and Fermi surfaces (3D) in energy bands, energy bands and band filling in semiconductors.

Handout 8

Energy levels in atoms and molecules, covalent bonding in molecules, introduction to LCAO (linear combination of atomic orbitals), energy level splitting in molecules, finite basis expansions, sp3 and sp2 hybridizations, sigma and pi bonds, and examples.  

Handout 9

Tight binding method for energy bands in solids, atomic orbitals and Bloch functions, bandwidths and energy matrix elements, examples.

Handout 10

Tight binding method applied to lattices with more than one atom in the primitive cell, examples in 1D and 2D, pi-energy bands in conjugated hydrocarbons, energy bands in polyacetylene, energy bands of Germanium.   

Handout 11

Crystal structure and energy bands in graphene, tight-binding approach and comparison with the nearly-free-electron approach

Handout 12

Computation of energy bands in group IV (Silicon, Germanium, and Diamond) and group III-V (GaAs, InP) semiconductors with cubic symmetry using tight binding method, Bloch functions at the G-point, improved techniques for the calculation of energy bands in semiconductors.   

Advance Topic Handout: Symmetries, Energy Bands, and Electronic States

Relationship between crystal symmetries and energy bands in solids in the absence and in the presence of spin-orbit coupling effects.

Handout 13

Properties of Bloch functions, orthogonality and completeness, average momentum and velocity of electrons in energy bands, average momentum and crystal momentum, energy band dispersion near band extrema, effective mass tensor.

Handout 14

Electron statistics in energy bands, effective density of states and effective density of states mass, isotropic and anisotropic effective mass tensors, introduction to holes, electron and hole pockets, case of silicon, germanium, and gallium arsenide.

PART IV: Electron Dynamics in Energy Bands and Electron Transport

Handout 15

Dynamics of electrons in energy bands, the dynamical equation for the crystal momentum in the presence of external fields, applicability of Ehrenfest's theorem to electrons in energy bands, effective mass tensor and inertia of electrons, acceleration of electrons.

Advance Topic Handout: Berry's Phase and Electron Dynamics in Solids

Guage invariance and electron dynamics in solids, Berry's phase correction to electron dynamics, Berry's curvature.

Handout 16

Inversion symmetry of energy bands, current density in energy bands, electron-hole transformation, conductivity of electrons in conduction bands and holes in valence bands, conductivity tensor of semiconductors, conductivity effective mass, Bloch oscillations.  

PART V: Phonons in Solids

Handout 17

Lattice vibrations and phonons in 1D, phonons in 1D crystals with monoatomic basis, phonon in 1D crystals with diatomic basis, force constants and the dynamical matrix, optical and acoustic phonons, phonon phase and group velocities, phonon dispersion and eigenvectors.   

Handout 18

Phonons in 2D with monoatomic and diatomic basis, force constants and dynamical matrices, longitudinal acoustic and transverse acoustic phonon modes, longitudinal optical and transverse optical phonon modes, phonon dispersion and eigenvectors.

Handout 19

Phonons in 3D with monoatomic and diatomic basis, phonons bands in group IV and group III-V semiconductors, macroscopic models of acoustic phonons in solids, stress and strain tensors, Hooke's law, phonons in cubic semiconductors.

 Handout 20

Quantum mechanical description of lattice waves in solids, commutation relations and quantization of lattice waves, phonons, energy  of phonons.

Handout 21

Phonon statistics, Bose-Einstein distribution, phonon density of states in 1D, 2D and 3D, Debye and Einstein models, thermal energy, heat capacity of solids.

PART VI: Low Dimensional Quantum Confined Nanostructures

Handout 24

Electron states in crystals with external perturbations, the effective mass theorem and the effective mass Schrodinger equation, envelope function, donor and acceptor impurities in semiconductors, donor and acceptor energy levels and states (defect states in crystals), donor and acceptor statistics.  

Handout 25

Semiconductor heterostructures and heterojunctions, band diagrams in real space, electron affinity and work function, heterojunctions in equilibrium, band alignment and band offsets.

Handout 26

Electrons in quantum confined 2D nanostructures, semiconductor quantum wells, effective mass equation, energy quantization, reduced density of states, and device applications

Handout 27

Electrons in quantum confined 1D and 0D nanostructures, semiconductor quantum wires and dots, effective mass equation, energy quantization, reduced density of states, and device applications

Handout 31

Electron states and energy bands in carbon nanotubes, zigzag and armchair nanotubes, semiconductor and metallic behavior of nanotubes, device applications.  

PART VII: Optical Processes in Solids

Handout 29

Optical transitions in solids, Fermi's golden rule, electron-photon Hamiltonian, intraband and interband optical transitions, optical matrix elements, crystal momentum selection rule, transition rates, loss coefficients of semiconductors, stimulated absorption and stimulated emission of photons, population inversion and optical gain, semiconductor heterostructure lasers. 

Handout 30

Optical processes and the dielectric constant of solids, linear response functions and Kramers-Kronig relations, interband and intraband contributions to the dielectric constant of solids, dielectric constant of conductive solids and the plasma frequency.  

Advance Topic Handout: Phonon-Polaritons and Plasmon-Polaritons

Polaritons, longitudinal and transverse polaritons, phonon-polaritons and plasmon polaritons.

Advance Topic Handout: Excitons

Optical absorption and emission in the presence of electron-electron interactions, exciton states, interaction of excitons with light, stationary and moving excitons, exciton-polaritons.

PART VII: Electron Transport, Boltzmann Equation, Thermoelectric Effects, and Ballistic Transport

Handout 22

Non-equilibrium distribution functions, Liouville equation, Boltzmann equation, relaxation time approximation, conductivity, scattering beyond the relaxation time approximation.

Handout 23

Position dependent non-equilibrium distribution functions, Boltzmann equation, local equilibrium distribution, transport equations, transport via drift and diffusion, Einstein relations.

Handout 32

Transport of thermal energy by electrons, thermoelectric effects, Seebeck effect and thermopower, Peltier effect, thermoelectric cooling, thermoelectric energy conversion.   

Handout 28

Ballistic electron transport, ballistic electron transport in quantum wires, conductance and resistance quantization, quantum of conductance, scattering and transmission, conductance as transmission, Landauer's formula.