Band Theory of Electronic Structure in Solids Continuing with Chapter 11 (semiconductors) Sections 11.4, 11.6, 11.7

Band Theory: “Bound” Electron Approach For the total number N of atoms in a solid (1023 cm–3), N energy levels split apart within a width E. Leads to a band of energies for each initial atomic energy level (e.g. 1s energy band for 1s energy level). Two atoms Six atoms Solid of N atoms Electrons must occupy different energies due to Pauli Exclusion principle. Phys 320 - Baski

Conductors, Insulators, Semiconductors NaCl is an insulator, with a band gap of 2 eV, which is much larger than the thermal energy at T=300K Therefore, only a tiny fraction of electrons are in the conduction band

Conductors, Insulators, Semiconductors Silicon and germanium have band gaps of 1 eV and 0.7 eV, respectively. At room temperature, a small fraction of the electrons are in the conduction band. Si and Ge are intrinsic semiconductors

Intro to Semiconductors and p-n junction devices

Band Diagram: Insulator Conduction band (Empty) T > 0 EC Egap EF EV Valence band (Filled) At T = 0, the lower valence band is filled with electrons and the upper conduction band is empty, leading to zero conductivity. Fermi energy: EF is at midpoint of large energy gap (2-10 eV) between conduction and valence bands.

Band Diagram: Intrinsic Semiconductor Conduction band (Partially Filled) EC EF EV Valence band (Partially Empty) At T = 0, lower valence band is filled with electrons and upper conduction band is empty, leading to zero conductivity. Fermi energy EF is at midpoint of small energy gap (<1 eV) between conduction and valence bands.

Donor Dopant in a Semiconductor For group IV Si, add a group V element to “donate” an electron and make n-type Si (more negative electrons!). “Extra” electron is weakly bound, with donor energy level ED just below conduction band EC. Dopant electrons easily promoted to conduction band, increasing electrical conductivity by increasing carrier density n. Fermi level EF moves up towards EC. EC EV EF ED Egap~ 1 eV n-type Si

Band Diagram: Acceptor Dopant in Semiconductor For Si, add a group III element to “accept” an electron and make p-type Si (more positive “holes”). “Missing” electron results in an extra “hole”, with an acceptor energy level EA just above the valence band EV. Holes easily formed in valence band, greatly increasing the electrical conductivity. Fermi level EF moves down towards EV. EA EC EV EF p-type Si

Dopant Density via Hall Effect Why Useful? Determines carrier type (electron vs. hole) and carrier density n for a semiconductor. How? Place semiconductor into external B field, push current along one axis, and measure induced Hall voltage VH along perpendicular axis. Derived from Lorentz equation FE (qE) = FB (qvB). Carrier density n = (current I) (magnetic field B) (carrier charge q) (thickness t)(Hall voltage VH) Hole Electron + charge – charge Solid-State Physics

pn Junction: Band Diagram pn regions “touch” & free carriers move Due to diffusion, electrons move from n to p-side and holes from p to n-side. Causes depletion zone at junction where immobile charged ion cores remain. Results in a built-in electric field (103 to 105 V/cm), which opposes further diffusion. Note: EF levels are aligned across pn junction under equilibrium. n-type electrons EC EF EF EV holes p-type pn regions in equilibrium – – EC – – + – + – EF + + – – – + + + – – + + – + + + EV Depletion Zone