Title: Solution of the Schrodinger equation in semiconductor device modelling


Supervisor: Dr E A B Cole



The electric currents which are controlled by certain nanoscale semiconductor devices are carried mainly by electrons. These electrons are quantum particles, and are subject to the rules of quantum mechanics which govern, among other quantities, their energies. These energies take up only discrete values, and are found as eigenvalues of the quantum mechanical Schrodinger equation.


When different layers of semiconducting material are brought together to form a HEMT (High Electron Mobility Transistor), then quantum wells form at the interfaces. The Schrodinger equation must be solved in these quantum wells. This project will be concerned with the solution of the Schrodinger equation for various physical profiles. Although these devices are three-dimensional structures, it turns out that we can very often get away with solving the Schrodinger equation only in one dimension.


Topics to be covered:

Some or all of the following topics will be covered:

Brief description of semiconductor devices and the equations describing current and energy transport through such devices.

Physical basis and "derivation" of the Schrodinger equation.

Time - independent Schrodinger equation (TISE).

Boundary and continuity conditions on the solution of the TISE.

Current density.

One-dimensional motion: solution of the TISE for step potentials, finite and infinite square well potentials, periodic arrays of potential barrier, reflection and transmission, tunnelling, the harmonic oscillator, the WKB approximation.

Quantum wells in semiconductor devices: self-consistent description of the coupled Poisson and Schrodinger equations.


Partial reading list:

1. A.I.M. Rae, Quantum Mechanics (IOP publication, 1992).

2. R.E.Miles, Introduction to Quantum Modelling (chapter in Compound Semiconductor Device Modelling, edited by Snowden and Miles, Springer-Verlag, 1993).

3. E.A.B.Cole, C.M. Snowden and T. Boettcher, Solution of the coupled Poisson-Schrodinger equations using multigrid methods, Int. J. Num. Mod. vol 10, 121-136 (1997).