The Forbidden Energy Gap
In a crystalline semiconductor, quantum mechanics dictates that electrons can only occupy specific energy ranges called bands. Between the valence band (filled with electrons at absolute zero) and the conduction band (empty at absolute zero) lies the band gap — a range of energies no electron can possess. This gap is what makes a semiconductor a semiconductor: small enough that thermal energy or light can kick electrons across it, large enough that conductivity can be precisely controlled.
Temperature and the Varshni Equation
The band gap is not fixed — it shrinks as temperature rises. The Varshni empirical formula E_g(T) = E_g(0) − αT²/(T + β) captures this behavior accurately. For silicon, E_g decreases from 1.17 eV at 0 K to 1.12 eV at 300 K. This temperature dependence has cascading effects: it increases intrinsic carrier concentration exponentially, shifts LED emission wavelengths, reduces solar cell voltages, and increases transistor leakage currents.
Doping: Engineering the Fermi Level
Pure silicon at room temperature has only about 1.5 × 10¹⁰ free electrons per cubic centimeter — orders of magnitude too few for practical devices. Doping introduces controlled impurities: phosphorus (5 valence electrons) donates an extra electron per atom, while boron (3 valence electrons) creates a hole. At 10¹⁷ cm⁻³ doping, the majority carrier concentration exceeds the intrinsic value by seven orders of magnitude, and the Fermi level shifts from mid-gap toward the relevant band edge.
The Mass-Action Law
One of the most elegant relationships in semiconductor physics is the mass-action law: n × p = n_i², where n is the electron concentration, p is the hole concentration, and n_i is the intrinsic concentration. Doping increases one carrier type while the other decreases reciprocally. At 10¹⁷ cm⁻³ N-type doping in silicon, electrons number 10¹⁷ per cm³ while holes drop to just 2,250 per cm³. This enormous asymmetry is what makes PN junctions, transistors, and every semiconductor device possible.