The theory of semiconductors permits a reduction of quantum systems with a large number of interacting electrons to a small number of non-interacting coherent and/or collective states. These are nonlinear quantum states that cannot be explicitly computed in most instances, but are effectively simulated by Hartree and Hartree-Fock simulations. We will explain the underlying theory of these states and develop the mathematical and numerical methods to find them. Then driving the quantum systems with an AC or a DC bias these states bifurcate to new coherent and collective states, making a large number of quantum devices designable.

Syllabus

I. Introduction to Crystals and Semiconductors

1. A Crystal Lattice, Electrons, Ions and Holes

2. Semiconductors, the Conduction and the Valance Band

3. Examples of Superconductors, GaAs, AlGAAs, InAs

II. The Theory of Quantum Wells

1. Local Density Approximation, Homogeneous Quantum Wells

2. The Fermi Distribution and Ignoring the Exchange Correlation

3. Numerical Methods: The Hartree Iteration

4. Damping, Intersuband Absorption and L-O and Optical Phonons

5. Experimental Results, the Stark and the Blue Shift

III. Introducing the Lasers Drive

1. The Terahertz Laser and the Dipole Interaction

2. Numerical Methods: The Time-dependent Hartree Iteration

IV. Coherent Electron States and their Bifurcations

1. Time-periodic Plasmons and their Bifurcations

2. Photon Emisson and the Period-doubling Bifurcation

3. The Period-doubling Cascade to a Strange Attractor

4. The Experimental Search for the Period-doubling

V. The Case of Quantum Dots

1. Numerical Method: The Hartree-Fock Iteration

2. Limits of the Theory: Single Electron Dynamics

VI. The Theory of Semiconductor Superlattices

1. The Resonant Sequential Tunneling (RST) Model

2. Noise and Disorder and the Number of Wafers of Quantum Wells

3. The Design of the Superlattice

VII. Collective States in a Superlattice

1. The Gunn Oscillation and its Period-doubling Bifurcation

2. A Cascade of Period-doubling Bifurcations to a Strange Attractor

3. Can we get a Second Hopf Bifurcation and a Different Cascade?

4. The Numerical Method: A Fast ODE Solver of RST

5. Possible Device Design

VIII. The Theory of Strongly-Coupled Superlattices

1. What are the Equations, PDEs?

2. Do Coherent Collective States Exist?

3. Possible Devices in the Terahertz Range