Semiconductor Physics: Band Theory and PN Junction

Semiconductors like silicon and germanium conduct electricity halfway between conductors and insulators. Their secret is the energy band structure: electrons live in allowed bands separated by a forbidden gap. A modest energy kick can push electrons across the small gap into conduction, which is why their conductivity can be tuned so precisely.

They matter because every chip, solar panel, and LED is built from them. By adding tiny amounts of impurities, engineers create n-type and p-type regions; joining them makes the pn junction, the building block of diodes, transistors, and integrated circuits.

Key Concepts

• Valence band: the highest energy band filled with electrons at absolute zero.
• Conduction band: the next band up, where electrons move freely and carry current.
• Band gap (Eg): the forbidden energy between them: about 1.1 eV for silicon, 0.66 eV for germanium, 0 eV for metals.
• Intrinsic semiconductor: pure crystal; electrons and holes appear only in equal pairs by thermal excitation.
• Extrinsic semiconductor: doped crystal. Donor impurities (phosphorus) add free electrons making n-type; acceptor impurities (boron) add holes making p-type.
• Hole: a missing electron that behaves like a positive charge carrier moving through the valence band.
• Depletion region: the charge-free zone at a pn junction where electrons and holes have recombined, leaving uncovered ions.
• Biasing: forward bias shrinks the depletion region and conducts; reverse bias widens it and blocks current.

Formulas

• Conductivity: sigma = n e mu_n plus p e mu_p, where n, p are electron and hole concentrations per cubic metre, e the elementary charge, and mu the mobilities.
• Mass action law: n times p = ni squared, where ni is the intrinsic carrier concentration.
• Built-in potential: V0 = Vt times ln((NA times ND) divided by ni squared), with thermal voltage Vt about 26 mV at room temperature.
• Diode current: I = Is times (e^(V divided by (eta Vt)) minus 1), where Is is the tiny reverse saturation current and eta is about 1 to 2.
• Photon threshold: a photon needs energy above Eg to create an electron-hole pair, so lambda_max = h c / Eg.

Worked Example

A silicon diode has Is = 1 pA (10^(-12) A) and eta = 1. Estimate its forward current at V = 0.7 V and room temperature (Vt = 26 mV).

1. V divided by Vt = 0.7 / 0.026 = 26.9.
2. e^26.9 is about 4.9 times 10^11.
3. I = 10^(-12) times (4.9 times 10^11 minus 1), about 0.49 A: raising V by just 60 mV multiplies the current roughly tenfold.
4. Band-gap check: silicon Eg = 1.1 eV, so photons need wavelength below 1240/1.1 = 1127 nm (infrared) to generate carriers.

Common Misconceptions

• Doping makes the semiconductor electrically charged. No: donors add an electron but also a positive ion core, so the crystal stays neutral. Doping changes carrier type, not net charge.
• Holes are just missing electrons with nothing real about them. In the mathematics of bands, a hole moves and carries current exactly like a positive particle, and Hall measurements confirm it.
• Current flows through the depletion region easily. The depletion region has almost no mobile carriers; that is precisely why the junction can block reverse current.

Real-World Applications

• Diodes: rectifiers convert AC to DC in every phone charger.
• Transistors: billions of MOSFETs switch and amplify inside processors and memory.
• Solar cells: pn junctions separate photo-generated carriers into useful current.
• LEDs and laser diodes: forward bias drives electron-hole recombination that emits light.

From this single junction grow diodes, transistors, and solar cells.