MAS.020 Introduction to Solid State Physics - 07.04.2026

Problem 1

Crystalographers typically specify a crystal structure using a CIF file.

(a) What information is contained in a CIF file?

(b) What information about the crystal structure do you need to calculate the reciprocal lattice?

(d) What information about the crystal structure do you need to calculate the electronic band structure?

(e) How can you use electronic band structure calculations to get the force constants you need for a phonon band structure calculation?

Solution

(a) A CIF file must specify the lattice constants, the space group, and the asymmetric unit. It may optionally a citation to the publication in which the crystal structure was reported, and the elements of the space group.

(b) The reciprocal lattice is calculated from the Bravais lattice. Knowing the primitive lattice vectors in real space is sufficient. The atoms in the basis are irrelevant for calculating the reciprocal lattice.

(c) For the intensities, you need to know the Bravais lattice and the atoms in the basis. It will be necessary to look up the atomic form factors of the atoms in the basis to calculate the intensities. The intensities are the squares of the <a href="https://lampz.tugraz.at/~hadley/ss1/crystaldiffraction/atomicformfactors/structurefactor.php">structure factors</a>. It is only possible to calculate the relative intensities of the different diffraction peaks.

(d) For a band structure calculation, you must know where all of the atoms are in a crystal. This can be specified by the space group and the asymmetric unit found in a CIF file.

(e) By varying the lattice constant of a crystal and performing a band structure for each value of the lattice constant, you can calculate the total energy as a function of the lattice constant. The minimum of the energy will indicate the equilibrium lattice constant, and the curvature at the minimum is the force constant needed for the phonon calculations.

Problem 2

Diamond has an fcc Bravais lattice. There are two atoms in the basis.

(a) How many $\vec{k}$ vectors satisfy periodic boundary conditions for a diamond crystal 1 cm³? (The lattice constant of the conventional unit cell is $a=3.567$ Å.)

(b) The lattice vibrations can be described in terms of normal modes. How many normal modes does a diamond crystal 1 cm³ have?

(c) Use the empty lattice approximation to sketch approximately the phonon dispersion relation for diamond along $L-\Gamma -X$. Label the energy axis.

(d) How many optical phonon branches does this crystal have?

Solution

(a) There are as many $\vec{k}$-vectors as there are primitive unit cells in a crystal. The number of conventional unit cells is 1 cm³/($a=3.567$ Å)³ = $2.2\times 10^{22}$. The number of primitive unit cells = the number of $\vec{k}$-vectors is four times this value, $8.8\times 10^{22}$.

(b) The number of normal modes is three times the number of atoms in the crystal. There are two atoms per primitive unit cell and four primitive unit cells per conventional unit cells so the number of phonon modes is,

$$3\times 2\times 4\times 22\times 10^{22} = 5.29\times 10^{23}.$$

(c) At low energies, th phonon dispersion relation is linear, $E=\hbar ck$, where $c$ is the speed of sound for that phonon branch. $L$ is closer to $\Gamma$ than $X$. There are three acoustic branches and three optical branches but the transverse phonon modes are degenerate.

Here $c_{\text{TA}}$ is the speed of sound of the transverse acoustic phonons.

(d) There are three optical phonon branches because there are two atoms in the basis.

Problem 3
The electron density of states for a particular material is given in the following figure. The total electron density is $17 \times 10^{27}$ electrons/m³. The form of the density of states was chosen to make it easy to integrate (by counting squares). The band edges are at integer or half-integer values in eV.

(a) Is this material a metal, an insulator, or a semiconductor?

(b) What is the limiting value of the Fermi energy as the temperature approaches zero?

Solution

Problem 4

(a) The electrical conductivity of a crystal is different in the $z-$direction than it is in the $x-$ and $y-$directions. What does this tell you about the crystal structure?

(b) In an electronic band structure diagram, how many electron states are there in each band?

(c) What are the Miller indices of a vector pointing from one bcc Bravais lattice point to another bcc Bravais lattice point? (There is more than one correct answer.)

(d) What is the necessary condition that the Boltzmann approximation is valid for semiconductors?

(e) What is an indirect band gap semiconductor?

Solution