PHYS 4050.03/PHYS 5100.03 - Concepts Review (part 2).
- What is the electron wavefunction in the free electron model of metals? Use the periodic boundary condition to obtain possible values of the wavevector k for the electron wavefunction.
- Derive the expression for the energy using the free electron model of metals.
- Define the Fermi energy. Show that the Fermi energy is proportional to the concentration of electrons.
- Explain the meaning of the Fermi-Dirac distribution function. Sketch this function at T = 0 K,
T = 300 K and T = 5000 K.
- Under what conditions is the Fermi-Dirac distribution function well approximated by the classical Maxwell-Boltzmann distribution function?
- Use the free electron model to show that the density of states function D(ε) is proportional
to ε1/2 . Explain qualitatively why D(ε) increases with energy ε.
- Define the Fermi energy. What is the typical value of the Fermi energy in metals? How does it compare with the thermal energy at room temperature?
- Define the Fermi temperature, Fermi momentum and Fermi velocity and give typical values of these quantities (order of magnitude in convenient units).
- What is the Fermi sphere?
- Calculate the ratio of the Fermi radius to the shortest distance across the Brillouin zone for copper (fcc lattice), which has one valence electron per atom.
- Based on the classical statistical mechanics, the heat capacity of metals is (TF/T) times larger than the value predicted by the free electron model of metals. Explain why. (TF is the Fermi temperature).
- According to the free electron model the electrical conductivity σ = ne2τ/m.
Explain the meaning of τ.
- State the Wiedemann-Franz law.
-What is meant by “Pauli paramagnetism”?
- Why alkali metals are “colourless”?
- Explain why is the resistivity of metals at high temperature is proportional to the temperature.
- Define π ("flipp over") scattering processes of electrons?
- The Hall coefficient RH does not agree for many metals with predictions based on the free electron model. Why?
- Derive the central equation for a one-dimensional crystal.
- Use the central equation to show that energy gaps develop at the Brillouin zone boundary. Assume that the potential energy contains only one Fourier component.
- What is the Bloch function? What does it represent?
- Show that Bloch function is periodic in crystal lattice.
- Explain the difference between metals, semimetals, semiconductors and insulators.
-Energy gap at the Brillouin zone boundary “shrinks” the free Fermi surface. Explain why.
- How is the electron wave function constructed in the Tight Binding Approximation model of metals?
-Use the Tight Binding Approximation to calculate the energy for a simple cubic crystal. Assume that the atomic orbitals overlap only between the nearest neighbours.
- Explain why d-bands in transition metals are narrow.
- Explain the difference between electron and hole orbits.
- What is a typical energy gap in semiconductors? How does it compare with the thermal energy at room temperature?
- Describe at least two methods to measure the energy gap in semiconductors.
-Explain the difference between direct and indirect absorption processes in semiconductors. Which processes are more desirable in optoelectronic devices?
-What are holes? Do they contribute to the electrical conductivity?
-Define the effective mass. Can it be negative? If so, what is the meaning of the negative effective mass?
-Set up the integral that would allow you to evaluate the concentration of electron (holes) in the valence band.