Latest update: March 2, 2004
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# Condensed Matter Physics: review questions

1. Introduction
1. Describe the different binding mechanisms in solids.
2. What distinguishes metals from other solids?
3. Experimentally, noble gas solids have either hexagonal closed packing (in the case of He) or face-centered cubic structures (all others). Why?
4. What is a Wigner crystal?
2. Free electron gas
1. Consider a free electron gas. Derive expressions for the Fermi wave vector, Fermi energy, and Fermi temperature in terms of the electron density.
2. Sketch the Fermi distribution for T = 0 and at room temperature at typical metallic densities.
3. Derive the density of levels for a free electron gas. Derive the electron pressure P = -(dE/dV)_N and (the electronic contribution to) the bulk modulus B = -V(dP/dV)_N of a metal at zero temperature.
4. Show that the electron specific heat is a linear function of T at low temperatures.
5. Describe some of the successes and shortcomings of free electron models for metals.
6. What is the main difference between the Drude and Sommerfeld models for metals?
3. Crystals
1. Define Bravais lattice, primitive vector, coordination number, primitive (unit) cell, (conventional) unit cell, lattice constant, Wigner-Seitz cell, crystal structure, basis.
2. Define reciprocal lattice and derive the reciprocal lattice to the simple cubic lattice. What is a Brillouin zone?
3. Show by explicit construction that the reciprocal of a reciprocal lattice is the corresponding direct lattice.
4. How are the Wigner-Seitz cell and the 1BZ related?
5. For a two-dimensional hexagonal lattice, construct the reciprocal lattice and the first Brillouin zone.
4. Independent electrons in a lattice 1: periodic potentials
1. Prove Bloch's theorem!
2. What is crystal momentum? How is it related to ordinary momentum?
3. What is the relation between the velocity of an electron and its energy?
4. What are van Hove singularities? Why are they important?
5. Independent electrons in a lattice 2: band structure
1. What is the basic assumption behind the "nearly-free-electron approximation"?
2. Which electron states are most influenced by a weak periodic potential? Why?
3. Sketch the energy bands in a 1D metal in the extended, reduced, and repeated zone scheme according to (i) the free electron approximation, and (ii) the nearly free electron approximation.
4. Consider the Fermi surface of a two-dimensional square lattice with lattice constant a and 3 electrons per unit cell. Assume that the periodic potential is weak and sketch the Fermi surface both in reduced and extended zone schemes. How many bands cross the Fermi surface?
5. What will happen to an incident beam of electrons if the beam energy lies within a band gap?
6. What is the basic idea behind the tight-binding method, and what can it be used for?
7. Sketch how the energy levels change as atoms condense to form a crystalline solid.
8. What is a metal-insulator transition? What is a Mott insulator? Why does the existence of Mott insulators indicate that there is something important missing from band theory? What is actually missing?
9. What does spin-splitting of energy bands mean? What is the origin of the splitting?
6. Methods for calculating band structures
1. What is the muffin-tin potential, and how/when is it used?
2. Discuss: Hartree-Fock equations, exchange.
3. What is jellium? Describe the Hartree-Fock approximation for jellium, and comment on its shortcomings.
4. What is a pseudopotential? When is is useful?
5. What is the advantage of the "one-electron approximation"? Which interactions must be included in the effective potential?
7. Semiclassical electron dynamics 1: electrons and holes as elementary excitations in a crystal
1. Derive and discuss the Drude formula for DC conductivity.
2. What kind of information can be obtained from measuring the Hall effect?
3. Derive the "classical" expression for the AC conductivity of a metal, and show that it reduces to the Drude DC formula in the limit of zero frequency.
4. What are Bloch oscillations and Wannier-Stark ladders? How large an electric field is needed to observe them?
5. What are the sources of electrical resistance in metals?
6. Define electrical breakthrough, magnetic breakthrough, effective mass tensor, magnetoresistance.
7. How can one distinguish between metals, semiconductors and insulators. Illustrate with some examples from the periodic table.
8. What is a hole? Properties?
9. Why is the effective mass of an electron different from its rest mass? Plot the dispersion curve for an electron with a small and a large effective mass, respectively.
10. What are de Haas-van Alphen and Shubnikov-de Haas effects? What is their origin?
8. Semiclassical electron dynamics 2: relaxation time approximation and physics on the Fermi surface
1. Explain: distribution function, relaxation time approximation, local equilibrium.
2. Derive the formula for DC conductivity in the relaxation time approximation. Show that the free-electron case follows as a special case. Also show that the full conductivity tensor for a crystal with sc-structure may be replaced by a scalar.
3. Discuss different definitions of effective mass.
4. Show that filled bands don't contribute to electrical conductivity. Also show that the electron- and hole descriptions are equivalent.
5. What is the de Haas-van Alphen effect? How can it be used to map out the Fermi surface of a metal? Discuss the basic ideas going into the explanation of the effect!
6. Explain: collision mechanism, scattering probability (per unit time), the Boltzmann equation, drift term, collision term, impurity scattering, elastic scattering, detailed balance.
9. Semiclassical dynamics 3 and beyond independent electrons
1. What are the dominating scattering mechanisms in real crystals at low temperatures?
2. What assumptions underlie the relaxation time approximation?
3. Outline the basic ideas of Landau Fermi Liquid Theory! What is a "quasiparticle"?
4. Give examples of cases when an interacting fermion system cannot be described using the Landau Fermi liquid theory. What is common to the examples?
5. What is the zero sound? Why is it an interesting property of a Fermi liquid?
10. Microscopic view on conductivity
1. How does the transport relaxation time differ from the ordinary relaxation time and why?
2. Discuss the possible origins of metal-insulator transitions.
3. What is localization?
4. Show that classically the conductivity G varies as a function of the sample size L as lnG/lnL = (d-2) where d is the dimensionality of the sample. What are the most important quantum mechanical corrections to this behavior?
5. What is weak localization? How can it be observed experimentally?
11. Review 1: independent electrons
1. Discuss the successes and failures of the free electron model.
2. Discuss band structure, its origins and consequences.
3. Discuss the successes and failures of the semiclassical approximation to electron dynamics in metals.
12. Electron-electron interactions: screening, plasmons
1. Why are metals usually not transparent? Under which conditions can they become transparent?
2. What is a charge density oscillation? Explain why the plasma frequency defines the "onset of optical transparency".
3. How is the speed of propagation of electromagnetic waves related to the dielectric constant of a metal?
4. Describe a method to detect plasmons experimentally, and to determine the plasmon dispersion relation omegap(q).
5. Define the dielectricity function epsilon(q,omega). Derive an approximation for epsilon(q,omega) at small wavevectors and large frequencies.
6. Sketch the real and imaginary parts of epsilon(q=0,omega) for a insulator as a function of omega.
7. Explain why the range of the Coulomb potential is strongly reduced in a metal. Consequences?
8. Describe the Thomas-Fermi theory of screening.
9. Discuss the optical response of metals at different frequencies.
13. Phonons 1: classical normal modes and quantum oscillators
1. List the main deficiencies of the static lattice model!
2. Discuss: harmonic approximation, anharmonicity, the adiabatic approximation, normal modes, dispersion curve, acoustic branch, optical branch, phonon, polarization vector, dynamical matrix!
3. Sketch the dispersion relations for acoustic and optical phonons for a lattice with two atoms per unit cell. Why must the frequency of an acoustic phonon vanish as the wavevector tends to zero?
4. Why is in general not correct to express the potential energy of a crystal as a sum of pairwise potentials?
5. When is the harmonic approximation OK, and when is it not? Why is the harmonic approximation used at all?
6. Derive the Dulong-Petit law for the specific heat due to lattice vibrations. When does it apply? Why do the experimentally measured specific heats deviate from the Dulong-Petit law?
7. What is the characteristic energy scale of phonons in a crystal? Compare to other processes in the crystal.
8. How does the phonon contribution to the specific heat depend on T at low temperatures?
14. Phonons 2: approximations and consequences
1. What are the Debye and Einstein models for phonons? When are they applicable?
2. What is Debye temperature? What is its typical value for metals?
3. Sketch the phonon density of states in the Debye and Einstein approximations.
4. Estimate the speed of acoustic phonons in metals.
5. How do phonons influence the interaction between conduction electrons in a metal? Why?
6. Why does the resistivity of a metal increase as T5 at low temperatures?
7. What are polarons, polaritons, and excitons?
8. What are Kohn anomalies?
9. If you were to choose a material to fill cavities in teeth, what kind of Debye temperature would you prefer? Why?
10. Describe an experimental way of determining phonon dispersion relations.
15. Magnetism 1: free electrons and ions
1. What is the origin of the magnetic moment of an atom?
2. Explain and discuss: magnetization, magnetic susceptibility, Bohr magneton, g-factor, diamagnetism, Larmor susceptibility, Hunds' rules, paramagnetism, van Vleck paramagnetism, Pauli paramagnetism, Curie's law, NMR, Landau diamagnetism.
3. How are Pauli and Landau susceptibilities related to each other for free electrons, and for electrons in metals?
16. Magnetism 2: interaction effects
1. Discuss: magnetic interaction, ferromagnet, antiferromagnet, ferrimagnet, spontaneous magnetization, exchange interaction
2. What is the Quantum Hall Effect? Integer Effect? Fractional?
3. Compare the Zeeman energy g muB B and the cyclotron energy. Which is larger for free electrons? What about electrons in GaAs where g=0.4 and the effective mass is meff = 0.067m0?
4. What is a Landau level? What is the maximal density of electrons in a Landau level?
5. Sketch the Hall resistance as a function of the magnetic field.
6. What are edge states? Why are they important in the quantum Hall regime?
7. Why does Laughlin's explanation of the fractional quantum Hall effect not allow for a special state at a half-filled Landau level?
17. Magnetism 3: collective behavior
1. What is the most important source for magnetic interactions?
2. Estimate the magnetic ordering temperature for a gas of non-interacting atoms by treating them as elementary magnetic dipoles.
3. What is the Lieb-Mattis theorem?
4. What are the ground states of a Heisenberg ferromagnet and antiferromagnet?
5. What is frustration? Give an example!
6. The Bloch T3/2 law states that the magnetization of a ferromagnet at temperature T deviates from the zero temperature magnetization by an amount that is proportional to T3/2. Explain the origin of this behavior!
7. What is the origin of magnetic domains?
8. What are magnons? What is their dispersion relation? Why must the energy of (at least one branch of) magnons go to zero at long wavelengths?
9. How does the magnon contribution to the specific heats of Heisenberg ferromagnets and antiferromagnets depend on temperature at low temperatures?
10. Why are multi-magnon states not eigenstates of the Heisenberg Hamiltonian?
11. How do magnons influence the magnetization of two- and one-dimensional Heisenberg ferromagnets at non-zero temperatures?
18. Superconductivity
1. What are the two distinct properties that a material should exhibit in order to be considered a superconductor?
2. Discuss: the Meissner effect, penetration depth, type I and II superconductors, Abrikosov flux lattice, vortex line, energy gap, isotope effect, Cooper pair, coherence length, critical fields and currents, flux quantization, Ginzburg-Landau parameter.
3. Sketch the main ideas that go into the BCS theory. Write down the BCS groundstate and discuss its properties.
4. Discuss the Josephson effects.
5. What is the London equation? Discuss its derivation and implications.
6. In type 2 superconductors, for Hc1 < H < Hc2, the magnetic field penetrates the superconductor in small separated regions, vortices. At low temperatures the vortices are organized as an ordered lattice, but at temperatures exceeding the ``flux lattice melting temperature'' Tf the vortices may move around. Discuss the consequences of this motion in materials for which Tf is lower than the superconducting transition temperature.
7. How can one see that the superconducting behavior cannot be obtained by treating the electron-electron interactions by means of a perturbation theory?
19. Review 2: beyond independent electrons
1. List some of the possible ground states for the ionic, electronic, and magnetic subsystems in solids, and the elementary excitations supported by them!
2. What is an elementary excitation?
3. Discuss some of the shortcomings of the independent electron approximation.