Problem 1

(a) Quartz is a transparent crystal that is often found in the mountains. What can you deduce about the electronic band structure of quartz from knowing it is transparent?

(b) Quartz makes a structural phase transition from α-quartz to β-quartz at a high temperature. How would you calculate the temperature at which this transition takes place? Your answer should discuss the electron contribution and the phonon contribution.

(c) A metal is well described by the free electron model with an effective mass equal to the free electron mass. The chemical potential is 1 eV above the bottom of the band. What is the Fermi wave vector $k_F$ of this metal?

(d) If the metal described by a free electron model is compressed, will the chemical potential increase, decrease, or stay the same? Explain your reasoning.

Solution

Problem 2

The element Sn is a metal above 13° C and a semiconductor below 13° C.

(a) The metallic phase has a body-centered tetragonal Bravais lattice with two atoms in the primitive unit cell. The conventional unit cell of the metallic phase is shown on the right where the black points are the Bravais lattice points. How many atoms are there in the conventional unit cells? Explain your reasoning.

(b) The semiconducting phase has an fcc Bravais lattice with two atoms in the primitive unit cell. The Miller indices are given in terms of the simple cubic conventional unit cell. Give the Miller indices of a direction that points from one Bravais lattice point to a nearest neighbor Bravais lattice point.

(c) The semiconducting phase has a very small direct band gap at $\Gamma$. Draw the electron dispersion relation $E\text{ vs. }k$ for the semiconducting phase showing the valence band, the conduction band, and the chemical potential.

(d) Which has more phonon modes, 10 grams of the metallic phase or 10 grams of the semiconducting phase? Explain your reasoning.

(e) Which has a higher specific heat at room temperature, the metallic phase or the semiconducting phase? Explain your reasoning.

Solution

Problem 3

(a) A cross-section of the Ewald sphere is shown below. What is the diffraction angle $2\theta$ for the Green reciprocal lattice vector?

(b) What is the largest $|\vec{G}|$ that could be measured in this experiment?

(c) What is the volume of the primitive unit cell in real space?

(d) What is the general form for a Fourier series in three dimensions? Define the terms in the expression you provide.

Solution

Problem 4

The phonon dispersion relation of GaN is shown below. (a) How many atoms are there in the basis of GaN?

(b) What is the frequency and the wavelength of the optical mode with the lowest frequency at the Γ point?

Solution