Temp

Problem 2

The dimensionality of a system can be reduced by confining the electrons in certain directions. A two-dimensional electron gas (2DEG) is produced in semiconductor heterostructures and is used for the investigation of the quantum Hall effect, creation of semiconductor quantum dots, quantum point contacts, nanowires, etc.

Consider a simplified model of a 2DEG where electron gas is in external potential $V=0$ for $z<d/2$ and $V=V_{0}$ for $|z|>d/2$.

(a) What is the density of states (DOS) as a function of energy for $V_0 \rightarrow \infty$? Discuss what happens at low energies and how 
DOS behaves in the limit of high energies (e.g., does it converge to the know DOS of 3D free electron gas?). 

(b) Assume $V_0 \rightarrow \infty$ and $d=100$ \AA{}. Up to what temperatures can we consider the electrons to be two-dimensional? (HINT: The electrons will behave two-dimensionally if $k_{B}T$ is less then the difference between the ground and first excited energy level in the confining potential).

(c) In real systems we can only produce a finite potential well. This puts a lower limit on $d$ since the ground state must be a bound state in the z direction with a clear energy gap up to the first excited state. If we can produce a potential of $V_{0}=100$ meV and reach a temperature of 20 mK, what is the range of thicknesses feasible for the study of such two-dimensional electron gas?