Problem 1
Cadmium selenide CdSe is an inorganic semiconductor that will form three different crystal structures wurzite (hexagonal), zincblende (cubic), and rock-salt (cubic).
(a) What can you say about the possible existence of pyroelectricity in CdSe?
(b) Describe how you could calculate the Seebeck coefficients for CdSe. What rank tensor are the Seebeck coefficients?
(c) How could you calculate which of the three crystal structures would be observed for a given temperature and pressure? What would be a good way to determine the crystal structure experimentally?
(d) Sketch the dielectric function for CdSe. How could you determine the real part and the imaginary part of the dielectric function at zero frequency experimentally?
(e) How would you calculate the absorption coefficient of CdSe? The absorption coefficient \(\alpha\) describes how the intensity of light decays as a function of distance \(I=I_0\exp (-\alpha x)\). Sketch the frequency dependence of \(\alpha\) for CdSe.
Problem 2
MXenes are two-dimensional inorganic compounds consisting of thin layers of transition metal carbides, nitrides, or carbon nitrides such as Nb2C and Ti3CN. They are metals with a hexagonal crystal structure. There is a large electron density of states at the Fermi energy.
(a) What would the Fermi surface look like?
(b) How could you calculate the bandstructure of these materials? How could you measure the bandstructure experimentally?
(c) Draw the Fermi surface of graphene and compare it to the metallic MXenes.
(d) How can you estimate the electron density of a metal using an optical measurement?
Problem 3
(a) How does the electron density of states change in the free electron model when a magnetic field is applied?
(b) How does the specific heat in the free electron model change when an magnetic field is applied?
(c) What are Shubnikov-de Haas oscillations?
(d) Why does the longitudinal resistivity $\rho_{xx}$ go to zero in the Quantum Hall effect?
Problem 4
Barium titanate is a ferroelectric that undergoes a first order structural phase transition from a rhombohedral phase to a monoclinic phase at about -80 C.
(a) What could be used as the order parameter? What symmetry is broken?
(b) Explain briefly what the Landau theory of first order phase transitions is.
(c) Sketch the specific heat and the electric susceptibility of barium titanate as a function of temperature. Where does the latent heat appear in these plots?