Silicon's conduction band minimum lies near and its indirect gap is at low temperature
PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure · Dielectric and optical properties of solids
Solid State Physics, Ch Tight binding and the band structures of real materials
Problem
Silicon's conduction band minimum lies near X and its indirect gap is Eg=1.12eV at low temperature. The phonon branch that supplies the required crystal momentum has an energy of ℏωph=18meV at the relevant wavevector (a value obtained from inelastic neutron scattering, of the kind described for measuring phonon dispersion). Recall that the number of phonons present in a mode of energy ℏω at temperature T is set by its thermal occupation, which vanishes as T→0, whereas a phonon can always be emitted regardless of the initial occupation.
((a)) At T=0K only phonon emission is possible. Write the energy-conservation condition for the threshold photon energy of this process and evaluate it.
((b)) At room temperature, phonon absorption also becomes possible because the 18meV mode is thermally populated (kBT=25meV at 300K). Evaluate the threshold photon energy for this second process.
((c)) Silicon's valence band is 12.5eV wide, about the width a free-electron gas of four electrons per atom at silicon's density would occupy. Explain why the bandwidth criterion alone cannot decide whether silicon's bands should be started from tight-binding orbitals or from nearly-free electrons, and state which test does decide it, and how.