## Notes for TU Graz Students

This website accompanies a course on solid state physics that is taught at Graz University of Technology in Graz, Austria. The course is taught in English. A web page has been made for every major section of the course. The sections generally correspond to chapters in the text we are using, Introduction to Solid State Physics, by Kittel. On each page, the topics that are covered in that section are listed. Often the topics are linked to the appropriate Wikipedia entry. Wikipedia provides useful definitions and it has the advantage that information is available in various languages.

The reading for each section is usually chapters (or parts of chapters) from Kittel. Sometimes other material is provided in electronic format. It is expected that the students participating in this course have read the material listed in the reading list.

Underneath the reading list, references are listed. These are not required reading. These references should be referred to if some topic in the reading list is not clear. The references might also be useful when solving problems.

Every student must contribute something that will improve the course for students in the future. Usually this will be making a multimedia presentation of about 3 minutes. The presentation should explain one of the topics The presentation will be posted on the course website. See making presentations for more information.

There will be a one hour writen exam. Once you have passed this exam, you must make an appointment to take an oral exam. Possible exam questions will be discussed during the lectures. Half of the written exam questions will be based on the exam questions listed on the website.

Possible projects:

• Use the plane wave method to calculate the photon dispersion relation and density of states for a photonic crystal. A description of this calculation is given here. We could start with the 1-D case create a webpage that explains the calculation and includes a program that calculates the dispersion relation. Programs that calculate the 2-D and 3-D dispersion relations would follow. An example with the associate matlab files can be found here.
• Help complete the table of the empty lattice approximation. There are 5 Bravais lattices in two dimensions and 14 Bravais lattices in three dimensions.
• Make a program to calculate the entropy, the grand potential, the pressure, the bulk modulus, the enthalpy, the Gibbs energy, or the specific heat at constant pressure from the density of states for photons like this one for the Helmholtz free energy.
• The photonic bandstructure and density of states can be determined numerically in much the same way that it is done for electrons. See: Band structure in 1-D. Make the corresponding program for 1-D photonic crystals.
• Make plots comparing the results of the Sommerfeld model to the thermodynamic properties determined by using the density of states for the free electron model in 1-D, 2-D, and 3-D. These density of states can be input the programs for the chemical potential, the energy spectral density, the internal energy density, and the specific heat.
• Throughout the course there are examples of numerical calculations that usually use JavaScript to perform the calculation. Javascript is suitable for calculations for problems where the execution time is less than a secont but it is less suitable for numerically intensive problems. In those cases it would be better to have a Jupyter notebook (python) or Pluto notebook (Julia) that explains the numerical calculation. Problems that would benefit from a Jupyter or Pluto notebook are: phonon band structure calculations in two or three dimensions, electronic band structure calculations using the plane wave method or the tight binding method, the calculation of the density of states, or calculating thermodynamic properties from the density of states.
• Make a program to calculate the entropy, the pressure, the bulk modulus, the enthalpy, the Gibbs energy, or the specific heat at constant pressure from the density of states for electrons like this one for the internal energy.
• Add a column to the table of phonon properties for a crystal with two atoms per unit like NaCl, GaAs, or CsCl. (2 students)
• Calculate the band structure and density of states of some material using a software package like Quantum Espresso or Wien2k. The density of states should be put in a form like this: Al fcc or V bcc.
• There is a DFT band structure calculation for lithium at here. Perform a band structure calculation for Li using tight binding or the plane wave method.
• In the tight binding table the density of states for simple cubic and bcc is symmetric around the middle of the band but this is not so for fcc. It would be interesting to calculate the effective mass of the holes at the tops of the bands in these three cases and see if this can explain the difference in the form of the density of states.
• Improve the table of properties of free electrons in a magnetic field by including the spin splitting to the density of states.
• Analytic expressions for the thermodynamic properties of a free electron gas in a magnetic field are found here. Plot the analytic solutions and compare them to the numerially produced solutions found here.
• A student did a project on Pauli paramagnetism that follows an discussion that is often presented in textbooks. The density of states of the spin up and spin down electrons are shown as growing like a square root. The problem is that the Landau splitting is about the same as the Zeeman splitting so these densities of states should exhibit the van Hove singularities (shown in the density of states here). These singularities are responsible for the de Haas - van Alphen oscillations. The project is to redo this calculation including the Landau splitting.
• Include the light hole band, the heavy hole band, the anisotropy of the effective mass, and multiple conduction band minima in the calculation of the density of states so that the table of semiconducting properties can be used for real materials. Right now the table assumes that there is a single isotropic conduction band minimum and valence band maximum.
• Make a simulation of an ellipsometer that shows the amplitudes and phases of the s- and p-polarizations before and after they strike the interfaces. Sliders should be used to adjust the angle and two polarizers.
• Make a page similar to the on on optical properties of diffusive metals for collisionless metals. For this case $g(t) = -eH(t)/m$ where the Fourier transform of $H(t)$ is given in the table of Fourier transforms.
• Calculate the joint density of states (Ibach and Lüth section 11.10) from the bandstructure of some material. Use the Kramers-Kronig relation calculate the dielectric function.
• At the beginning of the chapter on transport in the International Tables for Crystallography (2006) the following two equations are given, $$\vec{J} = \sigma(\vec{E}-S\nabla T)\\ \vec{J}_Q = \vec{J}TS - K\nabla T$$ where $\vec{J}$ is the electrical current density, $\vec{J}_Q$ is the thermal current density, $T$ is the temperature, $\vec{E}$ is the electric field, $S$ is the Seebeck coefficient, and $K$ is the thermal conductivity. Use the definition of $\vec{J}$ and $\vec{J}_Q$ to derive these relations.
• Calculate some transport properties (Seebeck effect, Nernst effect, etc.) for free electrons in 1-D, 2-D, or 3-D. Students have tried this and did not find a simple analytic expression. Write a program to calculate these effects if no simple expression can be found.
• The band structure $E(k)$ for any 1-D potential can be determined numerically. Write a program that numerically calculates a transport property.
• Make a page that gives the solutions to the Helmholtz equation. In three dimensions, the Green's function solution describes a screened Coulomb potential. In two dimensions, the solutions are modified Bessel functions. These are used to describe the magnetic field and current density around a superconducting vortex.
• Add a column to the magnon table.
• Calculate the polariton dispersion relation for some material. Since the material must have optical phonons, the dispersion relation for the phonons must be at least a 6×6 matrix. This must be coupled to the 3 components of the electric field.
• The 'Resources' sections for the different topics could be much better. Choose a topic like 'Linear response theory' or 'Quasiparticles' and search for the most useful books and websites on this topic for students studying this course.
• For some projects it is necessary to apply computing routines that were described in the numerical methods course. To help people prepare webpages that use these routines, it would be useful to make a page that discusses a numerical routine (like Gauss-Jordan elimination or interpolation using a cubic spline) and provide the Javascript version of this routine along with a web interface that illustrates the use of the routine. We are trying to convert the Sormann script on Numerical Methods to this format. Numerical methods online.
• Quasiparticles are low lying energy excitations from the ground state. One example of this are Bogoliubov excitations from a superconducting ground state. Two students could give a brief description of the BCS ground state and the quasiparticle excitations from the ground state. This is probably too much work if you don't already know something about the superconducting ground state.
• Determine α0 and β for some second order phase transition.
• Determine α0, β and γ for some first order phase transition. Equation 10 on the page, Landau theory of a first order phase transition expresses the difference in free energy in terms of α0, β and γ. Use the data at SGTE thermodynamic data table to get the free energy (the free energies are defined as functions in the source code.)