Temp

Problem 2

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 $z<d/2$ and $V=V_{0}$ for $|z|>d/2$. What is the density of states as a function of energy for $V_0 \rightarrow \infty$ (discuss what happens at low and high energies)? Assume $d = 100$\AA{}. 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 two-dimensional electron gas? (NOTE: A two-dimensional electron gas is produced in semiconductor heterostructures and is used for the investigation of the quantum Hall effect as well as other phenomena).