Problem: Two-Dimensional Electron Gas

     The dimensionality of a system can be reduced by confining the electrons in certain directions. Consider an electron gas in an external potential: V=0 for and for . What is the density of states as a function of energy for ? (Discuss what happens at low and high energies.)

Assume d=100 Å. Up to what temperatures can we consider the electrons to be two-dimensional? If we can produce a potential of 100 meV and reach a temperature of 20 mK, what is the range of thicknesses feasible for the study of such a two-dimensional electron gas? (Note: A two-dimensional electron gas is produced in semiconductor devices and is used for the investigation of the quantum Hall effect as well as other phenomena.)