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, lower valence band is filled with electrons and 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

Forward Bias and Reverse Bias Forward Bias : Connect positive of the positive end to positive of supply…negative of the junction to negative of supply Reverse Bias: Connect positive of the junction to negative of supply…negative of junction to positive of supply.

PN Junction: Under Bias Forward Bias: negative voltage on n-side promotes diffusion of electrons by decreasing built-in junction potential  higher current. Reverse Bias: positive voltage on n-side inhibits diffusion of electrons by increasing built-in junction potential  lower current. Equilibrium Forward Bias Reverse Bias p-type n-type p-type n-type p-type n-type –V +V e– e– e– Majority Carriers Minority Carriers

pn Junction: IV Characteristics Current-Voltage Relationship Forward Bias: current exponentially increases. Reverse Bias: low leakage current equal to ~Io. Ability of pn junction to pass current in only one direction is known as “rectifying” behavior. Forward Bias Reverse Bias Manifestly not a resistor: V=IR Not Ohm’s law

Devices: Light-related Absorption: incoming photon creates electron-hole pair (solar cell). Spontaneous Emission: electron-hole pair spontaneously decays to eject photon (LED). Stimulated Emission: incoming photon stimulates electron-hole pair to decay and eject another photon, i.e. one photon in  two photons out (LASER). Energy E2 2 1 1 1 2 2 3 E1 Absorption Spontaneous Emission Stimulated Emission

LED Solar Cell Light-emitting diode (LED) Converts electrical input to light output: electron in  photon out Light source with long life, low power, compact design. Applications: traffic and car lights, large displays. Solar Cell Converts light input to electrical output: photon in  electron out (generated electrons are “swept away” by E field of pn junction) Renewable energy source! LED Solar Cell Phys 320 - Baski Solid-State Physics

LED-Electrical Properties-Hetero junctions

Laser Diodes Similar to LED structure with degenerate doping. Degenerate doping- where fermi level is ( ) on P-side is in the valence band (VB)and on the N-side is in the conduction band (CB). When there is no applied voltage the fermi level is continuous across the diode. .

Laser Diode (Population Inversion) More electrons in the CB at energies near Ec than electrons in VB near Ev. This is the result of a Population Inversion in energies near EC and EV. The region where the population inversion occurs develops a layer along the junction called an inversion layer or active region.