4MM
PROJECT

Title:
**Solution of the Schrodinger equation in
semiconductor device modelling**

Supervisor:
Dr E A B Cole

**Outline**:

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).