ECE407 Physics of Semiconductors and
Nanostructures

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

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.

**PART II: Crystal Lattices and the Reciprocal Lattice**

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

First and second order perturbation theory in quantum mechanics.

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.

Energy levels in atoms and molecules, covalent
bonding in molecules, i

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

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.

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**

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.

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**

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.

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.

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.

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

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**

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.

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

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

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

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**

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.

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 **

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

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

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

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.