MAS.020 Introduction to Solid State Physics - 20.11.2025

Problem 1

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 cell? 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. Label 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 2

Below is the phonon dispersion relation for a crystal with an fcc Bravais lattice. The lattice constant is a = 0.2 nm.

(a) How many atoms are there in the primitive unit cell?

(b) Choose a direction and a polarization (longitudinal or transverse) and estimate the speed of sound in this direction for long-wavelength sound waves.

(d) What is the mean number of phonons in the phonon mode with the highest energy at 300 K?

Solution

(a) There are three acoustic modes and three optical modes in the dispersion relation, so there are 2 atoms in the primitive unit cell.

(b) Between $\Gamma$ and $X$, there is one longitudinal acoustic mode and two degenerate acoustic transverse modes. The dispersion relation for all these modes is linear near $\Gamma$. If a straight line is drawn from $\Gamma$ with the initial slope of the longitudinal mode, it intercepts $X$ at about 14 THz $\rightarrow \omega = 2\pi \times 14\times 10^{12} = 8.7965 \times 10^{13} \text{ rad/sec}$. The transverse modes intercept $X$ at about 8 THz $\rightarrow \omega = 2\pi \times 8\times 10^{12} = 5.0265 \times 10^{13}\text{ rad/sec}$. At $X$, $k = \frac{2\pi}{a} = \pi\times 10^{10}$ 1/m. A linear dispersion relation has the form $\omega = ck$, where $c$ is the speed of sound.

For the longitudinal mode, $c_L=\omega /k = 1600\quad\text{m/s}$.

For the transverse mode, $c_T=\omega /k = 2800\quad\text{m/s}$.

(c) The shortest wavelength corresponds to the largest $k$ vector. From the image of the Brillouin zone, we see that the largest $k$ vector is at $W$ where $k = \frac{\sqrt{5}\pi}{a} = 3.5124\times 10^{10}$

$\lambda = \frac{2\pi}{k} = 1.789$ Å.
(d) The mean number of phonons in a mode of energy $\hbar \omega$ is given by the Bose-Einstein factor,
$$f_{BE} = \frac{1}{\exp\left(\frac{\hbar\omega}{k_BT}\right)-1}$$
The highest energy phonon has a frequency of about 15.5 THz. At 300 K, the mean number of phonons in this state is 0.091.

Problem 3

(a) What information do you get in a single crystal x-ray diffraction experiment that cannot be obtained from a powder diffraction experiment?

(b) In a single crystal x-ray diffraction experiment, how do you determine the Bravais lattice of a crystal?

(d) What is an Ewald sphere? What is the radius of an Ewald sphere?

Solution

(a) In a single crystal diffraction measurement, you measure the reciprocal lattice vectors and can fit them to $\vec{G}_{hkl} = h\vec{b}_1 + k\vec{b}_2 + l\vec{b}_3$ to find the primitive lattice vectors in reciprocal space. From the $\vec{b}_i$s you can calculate the primitive lattice vectors in real space, the volume of the primitive unit cell in real space, and the Bravais lattice. From the intensities of the diffraction peaks, you can deduce the atoms of the basis, and from the Bravais lattice and the basis, you can determine the space group. None of this is possible with powder diffraction, which only provides a list of the netplane spacings, $d_{hkl}$.

(b) As already stated in part (a), you measure the reciprocal lattice vectors and can fit them to $\vec{G}_{hkl} = h\vec{b}_1 + k\vec{b}_2 + l\vec{b}_3$ to find the primitive lattice vectors in reciprocal space. From the $\vec{b}_i$s you can calculate the primitive lattice vectors in real space $\vec{a}_i\cdot\vec{b}_j = 2\pi\delta_{ij}$, and thereby the Bravais lattice.

(c) All of the Bravais lattice points are equivalent. Choose one to be the origin and draw vectors from the origin to the other reciprocal lattice points. Bisect these vectors by planes. These planes are the Brillouin zone boundaries. If a $\vec{k}$-vector falls on a Brillouin zone boundary, there will be diffraction.

(d) The <a href="https://lampz.tugraz.at/~hadley/ss1/book/diffraction/ewald.php">Ewald sphere</a> is a sphere centered around the origin in $\vec{k}$ space. If a crystal is rotated in all directions relative to the incoming beam, diffraction will occur for all reciprocal lattice points within the Ewald sphere. The radius of the sphere is $2|\vec{k}|$ where $\vec{k}$ is the wave vector of the incoming beam in the diffraction experiment.

Problem 4

State whether the following materials are metals, semiconductors, or insulators. Briefly explain your reasoning in each case. It can happen that you cannot tell if a material is a semiconductor or an insulator. Write semiconductor/insulator in this case, and explain why you can't tell.

(a) The specific heat at low temperatures is proportional to temperature. There is an electron contribution to the specific heat and a phonon contribution to the specific heat to consider.

(b) The material is transparent to visible light. Photons are not absorbed if the photon cannot lift an electron from an occupied state to an empty state.

(c) The electrical conductivity improves dramatically when infrared light shines on this material. The electrical conductivity depends on the electron density.

(d) The electron density of states at the chemical potential is zero.

(e) A one-dimensional crystal with three valence electrons per unit cell. Consider a 1D band structure and fill the bands.

(f) The electrical conductivity increases as temperature increases. The electrical conductivity depends on the electron density.

(g) The phonon contribution to the internal energy density dominates at room temperature.

Solution

(a) Metal - Only metals have a linear specific heat at low temperatures.
(b) Insulator - Visible photons have energies up to about 3.5 eV. If these photons can't take an electron from the valence band to the conduction band, the band gap must be larger than this.
(c) Semiconductor - The conductivity of semiconductors and insulators increases dramatically when electrons are added to the conduction band. This is infrared light, which does not have enough energy to bring electrons into the conduction band of an insulator, so it must be a semiconductor.
(d) Semiconductor/Insulator - The density of states is zero the chemical potential for both insulators and semiconductors.
(e) Metal - In a one-dimensional crystal, two electrons go in the lowest band. Then there is a band gap. The third electron will go in the next band, which will be half-filled, so the crystal will be a metal.
(f) Semiconductor - For a metal, the electron density does not change as the temperature increases, but for a semiconductor, there are more electrons in the conduction band as the temperature increases so the electrical conductivity also increases.

(g) Semiconductor/Insulator - If the phonon contribution dominates, the electron contribution must be negligible. This is true for semiconductors and insulators.