CMB Example SECTION A-type questions.
Band theory of solids, disorder, low dimensional systems, deffects
1. Show that the Bloch functions 𝝍(𝑟) = 𝑢(𝒓)𝑒 !𝒌∙𝒓 are not eigenstates of the momentum
ℏ
operator ! ∇ (and hence ℏk is not a free electron momentum p = ℏk).
2. A weak periodic potential 𝑉 𝑥 = 𝑉! + 𝑉! cos 2𝜋𝑥/𝑎 + 𝑉! cos 4𝜋𝑥/𝑎 + … is applied to a
one dimensional system of electrons (of mass 𝑚! ). Under what conditions can the nearly
free electron approximation be applied to this system?
3. Sketch the resistivity as a function of temperature down to 𝑇 = 0 K for a perfectly pure
metal (ignore electron-electron scattering), an impure (real) metal and for a
superconductor. Provide reasoning for your sketches.
4. What is the electron drift velocity
5. Give definition of the energy gap.
6. What is the difference in terms of density of states at Fermi energy in a metal and in a
semiconductor.
7. Explain in your own words the origin of the concept of effective electron mass in solid
state physics.
8. In your own words describe what an exciton is. State the key parameter used to
describe the size of an exciton (e.g. a Mott-Wannier exciton).
9. Describe the effect of disorder in a semiconductor lattice on the density of electron
states near the band gap. Give the definition of the mobility edge.
10. Explain why the density of states rather than dispersion relashionship is used to
describe ectronic properties in disordered materials.
11. Describe in your own words the origins of the Anderson localization. State the
localisation conditions for a hopping potential 𝑉!" (where 𝑖 and 𝑗 are the nearest neighbour
indices) and an electron energy distribution of characteristic width 𝑊.
12. Explain in your own words the difference between intrinsic and extrinsic
semiconductors.
13. Describe factors that affect the charge carrier mobility in semiconductors.
14. What is a p-n junction?
15. What is a Schottky barrier?
16. Describe the physical origins of quantum confinement.
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17. Derive an expression for the equilibrium number of vacancies, 𝑛, in a lattice of 𝑁
atoms, given that the probability of a lattice site being vacant is proportional to the
Boltzmann factor. Take the energy required to take an atom from the lattice site to be 𝐸! .
18. What are the Schottky and Frenkel defects?
19. Give a definition of the Burgers vector.
Magnetism
20. Describe how paramagnetism and diamagnetism are related to the electronic structure
of ions. Give definitions of these effects in terms of the value of the magnetic
susceptibility.
21. What is the Bohr magneton?
22. Describe the origins of exchange interaction that result in long range magnetic order.
Describe the differences between direct exchange, indirect exchange and superexchange.
23. Sketch 1D chain of periodically arranged atoms and show how atomic magnetic
moment should align in ferromag- netic, antiferromagnetic and ferrimagnetic material.
Write down the value of the exchange integral J in each of the cases.
24. What is the significance of the Curie temperature?
25. In your own words describe what a magnon is. Describe the effect of magnons on the
magnetisation of a sample.
26. Explain which experimental technique is normally used to determine the magnon
dispersion (𝝎(𝑸)). Show how the magnon dispersion can be obtained using this
technique.
Superconductivity
27. What is a Cooper pair? What is the origin of the effect of Cooper pairing?
28. What is the Meisnner effect?
Tensors
29. In an isotropic material the relationship between an external electric field and an
induced polarisation field is described by a simple scalar – susceptibility χ (𝑷 = ε! 𝜒𝑬)).
How does this change in an anisotropic material and why?