Semiconductor Physics for Circuit Designers

Why silicon conducts a little — and how we control it. Learn conductors, insulators and semiconductors, silicon, germanium and compound/wide-bandgap materials (GaAs, GaN, SiC), atomic structure and covalent bonding, energy bands (valence, conduction, bandgap), electrons and holes, the Fermi level, intrinsic semiconductors, doping (donors/acceptors, N- and P-type), majority and minority carriers, the mass-action law, drift and mobility, diffusion and the Einstein relation, generation and recombination, and temperature effects — with Python computing intrinsic concentration vs temperature, carrier densities, Fermi level and resistivity.

Every diode, transistor and integrated circuit relies on precisely controlling how many charge carriers exist in a semiconductor and where they flow. Adding one phosphorus atom per ten million silicon atoms raises conductivity by about a million times.

Circuit designers don't need to solve Schrödinger's equation, but understanding bands, doping, drift and diffusion explains diode voltages, leakage that doubles every 10 °C, transistor gain and why GaN and SiC dominate modern power electronics.

An analogy

Picture a two-storey car park: the lower floor (valence band) is full so no car can move; the upper floor (conduction band) is empty. Promote a car upstairs (an electron) and it can drive freely — and the empty space left downstairs (a hole) lets cars below shuffle too. Doping adds extra cars upstairs (N-type) or extra empty spaces downstairs (P-type).

Conductors, insulators and semiconductors

Metals have overlapping bands and huge free-electron densities (~10²² cm⁻³). Insulators have wide bandgaps (> 4 eV), so almost no electrons are thermally excited. Semiconductors (E_g ≈ 0.5–3.5 eV) sit in between, and — uniquely — their conductivity can be set precisely by doping and changed by light, heat and electric fields.

Doping

Controlling carriers deliberately:

Drift, diffusion and recombination

Drift: carriers accelerate in an electric field; mobility µ describes how easily (electrons ≈ 2.8× more mobile than holes in Si, which is why NMOS/NPN devices are faster). Diffusion: carriers spread from high to low concentration; D = µkT/q links the two. Generation creates electron–hole pairs (heat, light); recombination destroys them. Minority-carrier lifetime and diffusion length set diode reverse recovery and BJT base design.

Materials beyond silicon

Germanium (small gap, low forward voltage, high leakage — historic transistors). GaAs and InP (direct bandgap, high electron mobility — RF, LEDs, lasers). SiC (3.26 eV) and GaN (3.4 eV) wide-bandgap materials withstand higher fields and temperatures — compact, efficient power converters, EV inverters and fast chargers. Mobility falls at high doping and high temperature; velocity saturates at high fields (~10⁷ cm/s in Si), limiting short-channel transistors.

Formulas

Worked example: Minority carriers

Given: P-type N_A = 10¹⁷ cm⁻³

Find: n.

Answer: 10³ cm⁻³ electrons

Worked example: Resistivity of a doped layer

Given: N_D 10¹⁸, µ_n 250 cm²/V·s (high doping)

Find: ρ.

Answer: 0.025 Ω·cm

Worked example: Einstein relation

Given: µ_p = 480 cm²/V·s at 300 K

Find: D_p.

Answer: 12.4 cm²/s

Energy-band picture

  Energy
    ▲   ─────────────── E_C  conduction band (empty at 0 K)
    │        ‥ E_D  donor level (N-type, just below E_C)
    │   E_g  - - - - - - E_F (N-type: moves up toward E_C)
    │   ‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ E_i intrinsic level (mid-gap)
    │        ‥ E_A  acceptor level (P-type, just above E_V)
    │   ─────────────── E_V  valence band (full at 0 K)
  Si: E_g = 1.12 eV   Ge: 0.66 eV   GaAs: 1.42 eV   SiC: 3.26 eV   GaN: 3.4 eV

Electrons need at least E_g of energy to jump into the conduction band. Donors sit just below E_C and give up electrons easily; acceptors sit just above E_V and capture electrons, creating holes. The Fermi level shows which way the material is doped.

Carrier concentrations, Fermi level and resistivity

import math
# Semiconductor physics: intrinsic carriers vs temperature, doping, Fermi level and conductivity
k = 8.617e-5                                   # eV/K
def Eg(T): return 1.17 - 4.73e-4 * T**2 / (T + 636)       # silicon band gap (Varshni), eV
def ni_si(T):                                  # intrinsic concentration, normalised to 1e10 cm^-3 at 300 K
    return 1.0e10 * (T / 300) ** 1.5 * math.exp(-Eg(T) / (2 * k * T) + Eg(300) / (2 * k * 300))
for T in (250, 300, 400):
    print(f"T = {T} K: Eg = {Eg(T):.3f} eV, ni = {ni_si(T):.2e} cm^-3")
ni = 1.0e10
Nd = 1e16                                      # phosphorus donors
n, p = Nd, ni**2 / Nd
print(f"N-type Nd = 1e16: n = {n:.1e}, p = {p:.1e} cm^-3 (minority)")
Ef = k * 300 * math.log(Nd / ni)
print(f"Fermi level {Ef:.3f} eV above intrinsic level")
q, mu_n, mu_p = 1.602e-19, 1350, 480           # cm^2/Vs
sigma = q * (n * mu_n + p * mu_p)
print(f"conductivity {sigma:.2f} S/cm, resistivity {1/sigma:.3f} ohm*cm (intrinsic Si: {1/(q*ni*(mu_n+mu_p)):.2e} ohm*cm)")
print(f"diffusion coefficient Dn = mu*kT/q = {mu_n*k*300:.1f} cm^2/s (Einstein relation)")
# T = 250 K: Eg = 1.137 eV, ni = 7.41e+07 cm^-3
# T = 300 K: Eg = 1.125 eV, ni = 1.00e+10 cm^-3
# T = 400 K: Eg = 1.097 eV, ni = 5.28e+12 cm^-3
# N-type Nd = 1e16: n = 1.0e+16, p = 1.0e+04 cm^-3 (minority)
# Fermi level 0.357 eV above intrinsic level
# conductivity 2.16 S/cm, resistivity 0.462 ohm*cm (intrinsic Si: 3.41e+05 ohm*cm)
# diffusion coefficient Dn = mu*kT/q = 34.9 cm^2/s (Einstein relation)

Heating silicon from 300 K to 400 K multiplies intrinsic carriers by ~500 — that's why leakage currents and minority-carrier effects grow so fast with temperature, and why silicon devices stop working properly above ~150–200 °C (wide-bandgap SiC and GaN survive much hotter). Doping at 10¹⁶ cm⁻³ cuts resistivity by nearly a million times compared with pure silicon.

Common mistakes

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Key takeaways

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