P461 - Conductors1 Properties of Energy Bands Band width, gaps, density of states depend on the properties of the lattice (spacing, structure) First approximation is Fermi Gas (like 460). But if wavelength becomes too small, not overlapping ---> sets width of band next approximation adds in periodic structure of potential can cause interference of “traveling” waves (reflection/transmission). Essentially vibrational modes of the solid destructive interference causes energy gaps which are related to dimensions of lattice Note often the “band energy” is measured from the bottom of the band (which is the electronic energy level)

P461 - Conductors2 Fermi Gas Model From 460: Ex What are the number of conduction electrons excited to E > E F for given T ? EFEF T=0 T>0 n*D

P461 - Conductors3 Fermi Gas Model II Solids have energy bands and gaps Can calculate density of states D(E) from lattice using Fourier Transform like techniques (going from position to wavelength space) can change D(E) by changing lattice - adding additional atoms during fabrication - pressure/temperature changes -----> PHYS 566, 480, 690A techniques E ideal real D

P461 - Conductors4 Fermi Gas Model III 1D model. N levels and min/max energy For 2D/3D look at density of states. Grows as E.5 until circle in k-space “fills up” then density falls (can’t have wavelength ~ smaller than spacing) E D 2D/3D Na=L a ky kx Ef

P461 - Conductors5 Interactions with Lattice Study electron wavefunction interactions with the lattice by assuming a model for the potential Kronig-Penney has semi-square well and barrier penetration will sort of look at 1D --> really 3D and dependent on type of crystal which gives inter-atom separation which can vary in different directions solve assuming periodic solutions Bragg conditions give destructive interference but different fs “sin” or “cosine” due to actual potential variation. will have different points where wavefunction=0 V0V0

P461 - Conductors6 Interactions with Lattice Get destrctive interference at leads to gaps near those wavenumbers once have energy bands, can relate to conductivity materials science often uses the concept of effective mass. Electron mass not changing but “inertia” (ability to be accelerated/move) is. So high m* like being in viscous fluid-> larger m* means larger interaction with lattice, poor conductor m* ~ m in middle of unfilled band m* > m near top of almost filled band m* < m near bottom of unfilled band always dealing with highest energy electron (usually near Fermi energy)