1. (20 points) Consider a metal with a simple cubic structure and a lattice parameter of 0.309 nm.

a) Assuming free electrons, if the Fermi surface just touches the centre of the faces of the 1st BZ boundary, what is the electron density? Give your answer in electrons per atom and electrons per unit volume.

b) What is the minimum electron density that would result in the Fermi surface being be completely outside of the 1st BZ? Give your answer in electrons per atom and electrons per unit volume.

c) Find the Fermi energy (Ef ) corresponding to the case in (b), as well as the Fermi speed (vf) and temperature (Tf) corresponding to electrons at Ef.

d) What are the lowest and highest energies that an electron on the 1st BZ boundary can have? Give the energy as well as k.

e) In the nearly free electron model, we see that turning on the ionic potential [U(r)] opens up energy gaps at the BZ boundaries. Assuming the SC structure above with two electrons per unit cell, describe (in terms of the energy gaps at the BZ boundary) when (if ever) you would expect the material to be metallic and when (if ever) you would expect it to be insulating. Incorporate a plot of E(k) (plotted along appropriate directions in k-space) in your discussion.

2. (16 points) On Assignment 4 we considered a h =12 nm thick metal layer, which we will assume was deposited on an insulating substrate. In this problem, you will model that structure as a quantum well, where the electrons are free to move in the plan of the layer but confined to remain in the layer. An analogous situation is that of conduction electrons in a semiconductor quantum well, where a thin layer (thickness of several nm) of one semiconductor material is sandwiched between a different semiconductor material, such that conduction electrons are confined within the thin layer. Such semiconductor quantum wells form the active region of semiconductor LEDs and lasers (as we will discuss later in the course).

For this problem, assume that the in-plane extension of the layer is a square with Ly ´ Lz = L ´ L

= (0.4 mm) ´ (0.4 mm). We want to solve the Schrӧdinger equation for free electrons using periodic boundary conditions in the y-z plane, and requiring the wave function to be zero at the boundaries of the quantum well (x=0, x=h). We can write the wave function as a product of 1D wave functions in x, y and z. In other words:

Y(���, ���, ���) = y���(���) ∙ y���(���) ∙ y���(���)

This allows us to use separation of variables and solve the 1D Schrӧdinger equation for each of the three directions. The overall energies will be given by E = Ex + Ey + Ez

a) Find the allowed energies of a free electron in the quantum well. The energy levels will be described by three quantum numbers: nx (an integer related to the quantization in the x-direction), as well as ky and kz. We say that the states associated with a particular nx are in a “sub-band”, with each sub-band behaving as a 2D electron gas. In fact, experiments are often done on thin samples where only the lowest sub-band is occupied and the electrons behave as a purely 2D system!