Semiconductor Physics

Every chip starts with the physics of silicon. Learn energy bands and the band gap, intrinsic carriers, doping with donors and acceptors, electron and hole concentrations (n·p = nᵢ²), the Fermi level, drift and diffusion currents, mobility and the Einstein relation, conductivity, and how temperature changes it all — with a calculator for doped silicon.

Metals conduct because their electrons are free; insulators don't because theirs are tightly bound. Semiconductors sit in between — and crucially, we can control how well they conduct by adding tiny amounts of impurities (doping) and by applying electric fields. That controllability is what makes transistors possible.

Silicon has four valence electrons forming covalent bonds in a crystal. At room temperature a few bonds break, creating a free electron and a 'hole' (a missing electron that behaves like a positive carrier). Doping with phosphorus adds extra electrons (n-type); doping with boron adds holes (p-type). Everything in VLSI — diodes, MOSFETs, CMOS logic — builds on these two ideas.

An analogy

Imagine a car park (valence band) full of cars and an empty motorway above it (conduction band). Traffic flows only on the motorway. The band gap is the ramp height: silicon's ramp is moderate, so a few cars get up by heat alone. Doping is like adding a feeder road that parks extra cars right next to the motorway (n-type) or opens empty spaces in the full car park so cars can shuffle (p-type holes).

Energy bands and the band gap

Electrons in a crystal can only have energies inside allowed bands. The valence band is (almost) full; the conduction band is (almost) empty. The gap between them, E_g, is 1.12 eV for silicon. An electron needs at least E_g to jump up and become free to conduct — heat, light or electric fields can provide it.

Electrons, holes and doping

When a bond breaks, both the freed electron and the hole it leaves behind can move and carry current. In pure (intrinsic) silicon n = p = nᵢ, which is tiny — pure silicon is a poor conductor. Adding 1 phosphorus atom per 10⁷ silicon atoms (≈10¹⁶ cm⁻³) raises the electron density a million-fold: that is n-type silicon. Boron creates p-type.

The mass-action law n·p = nᵢ² holds in equilibrium: adding majority carriers suppresses minority carriers.

Drift and diffusion

Carriers move in two ways. Drift: an electric field pushes them (velocity = μE). Diffusion: carriers spread from high to low concentration, like perfume in a room. The Einstein relation D = μV_T links the two. A diode's current is diffusion; a MOSFET channel's current is (mostly) drift.

The PN junction

Joining p- and n-type silicon makes carriers diffuse across, leaving behind charged ions: a depletion region with a built-in field and potential V_bi ≈ 0.6–0.9 V. Forward bias lowers the barrier (exponential current); reverse bias widens the depletion region (tiny leakage). Every MOSFET contains PN junctions at its source and drain.

Temperature, high fields and beyond silicon

nᵢ roughly doubles every 11 °C in silicon, so leakage currents rise exponentially with temperature — a key reason chips throttle when hot. At high fields, carrier velocity saturates (~10⁷ cm/s in Si), which limits modern short-channel transistors. Wide-bandgap materials (SiC 3.3 eV, GaN 3.4 eV) tolerate higher voltages and temperatures, powering EV inverters and fast chargers.

Formulas

Worked example: Minority carriers

Given: Si, N_D = 10¹⁶ cm⁻³, nᵢ = 1.5×10¹⁰ cm⁻³

Find: n and p.

Answer: p = 2.25×10⁴ cm⁻³ (a trillion times fewer holes than electrons)

Worked example: Resistivity of a wafer

Given: N_D = 10¹⁶ cm⁻³, μₙ = 1350 cm²/V·s

Find: ρ.

Answer: ρ ≈ 0.46 Ω·cm

Worked example: Built-in potential

Given: N_A = 10¹⁷, N_D = 10¹⁶ cm⁻³

Find: V_bi.

Answer: V_bi ≈ 0.75 V

Carrier concentrations, resistivity and Fermi level of doped silicon

import math

q, VT, ni = 1.602e-19, 0.02585, 1.5e10          # C, V (300 K), cm^-3 (silicon)
mu_n, mu_p = 1350, 480                          # cm^2/V.s

for ND in (1e14, 1e16, 1e18):                   # donor doping, cm^-3
    n, p = ND, ni**2 / ND                       # majority / minority carriers
    sigma = q * (n * mu_n + p * mu_p)           # conductivity, S/cm
    ef = VT * math.log(ND / ni)                 # E_F - E_i, eV
    print(f"ND={ND:.0e}: p={p:.2e} cm^-3  rho={1/sigma:.3g} ohm.cm  EF-Ei={ef:.3f} eV")

NA, ND = 1e17, 1e16
Vbi = VT * math.log(NA * ND / ni**2)
print(f"PN junction built-in potential Vbi = {Vbi:.3f} V")
# ND=1e+14: p=2.25e+06 cm^-3  rho=46.2 ohm.cm  EF-Ei=0.228 eV
# ND=1e+16: p=2.25e+04 cm^-3  rho=0.462 ohm.cm  EF-Ei=0.347 eV
# ND=1e+18: p=2.25e+02 cm^-3  rho=0.00462 ohm.cm  EF-Ei=0.466 eV
# PN junction built-in potential Vbi = 0.753 V

Each 100× increase in doping cuts resistivity 100× and moves the Fermi level up by kT·ln(100) ≈ 0.119 eV. Real mobility falls at high doping (impurity scattering), so heavily doped silicon is a little more resistive than this simple model predicts.

Common mistakes

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Key takeaways

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