Band Theory and Doping

Why Band Theory is Needed

Classical physics cannot explain the conductive behavior of semiconductors. Band theory uses quantum mechanics to describe the allowed energy states of electrons in solids, forming the foundation for understanding all semiconductor devices.

Band Structure

Three Basic Bands

Conductors / Semiconductors / Insulators

Fermi Level

Fermi Level Ef: The energy level where the probability of electron occupation is 50%

Intrinsic Semiconductor: Ef is in the middle of the band gap
N-type Semiconductor: Ef is close to the conduction band (more electrons)
P-type Semiconductor: Ef is close to the valence band (more holes)

Fermi-Dirac Distribution:
f(E) = 1 / (1 + e^((E-Ef)/kT))

T=0K: Step function
T>0K: Transition region width ≈ kT (room temp ≈ 26meV)

Semiconductor Materials

Silicon (Si) — Absolute Mainstream

Atomic Number: 14
Crystal Structure: Diamond structure (each atom has 4 covalent bonds with neighbors)
Eg = 1.12 eV
Intrinsic Carrier Concentration ni ≈ 1.5×10¹⁰ cm⁻³ (300K)

Advantages: Cheap, natural SiO₂ insulating layer, mature process technology

Other Materials

Doping

Problems with Intrinsic Semiconductors

Pure silicon has very weak conductivity at room temperature (ni is too low), so impurities must be intentionally added to alter the conductivity.

N-type Doping (Adding Group 5 Elements)

     Si       Si       Si
      │        │        │
  Si ─ P ─ Si      Si ─ As ─ Si
      │        │        │
     Si       Si       Si
      │                  
      e⁻  ← Excess "free electrons"

Donors: P, As, Sb (Group 5)
Majority Carriers: Electrons
Majority Carrier Concentration n ≈ Nd (Donor Concentration)

P-type Doping (Adding Group 3 Elements)

     Si       Si       Si
      │        │        │
  Si ─ B ─ Si      Si ─ Al ─ Si
      │        │        │
     Si       Si       Si
      │
      h⁺  ← Missing electron = "Hole"

Acceptors: B, Al, Ga (Group 3)
Majority Carriers: Holes
Majority Carrier Concentration p ≈ Na (Acceptor Concentration)

Doping Concentration Range

Light Doping: 10¹⁴ ~ 10¹⁶ cm⁻³  (Substrate, high-resistance regions)
Medium Doping: 10¹⁶ ~ 10¹⁸ cm⁻³  (Channel, Base region)
Heavy Doping: 10¹⁸ ~ 10²⁰ cm⁻³  (Source/Drain, Emitter region)
Degenerate:   > 10²⁰ cm⁻³       (Ohmic contacts)

Carriers

Two Types of Carriers

Electron: Free electrons in the conduction band, negatively charged
Hole:     Electron vacancy in the valence band, behaves like a positively charged particle

Intrinsic Semiconductor: n = p = ni
N-type Semiconductor: n >> p  (Electrons are majority carriers)
P-type Semiconductor: p >> n  (Holes are majority carriers)

Law of Mass Action: n × p = ni²  (Holds true under thermal equilibrium)

Mobility and Conductivity

Drift Velocity: v = μ × E
  μ: Mobility (cm²/V·s) — Electrons are about 3 times faster than holes

Conductivity: σ = q × (nμn + pμp)
  q: Electron charge

Resistivity: ρ = 1/σ

Diffusion

Concentration gradient drives carriers to diffuse from high to low concentration
Diffusion Current ∝ Concentration Gradient (dC/dx)

Einstein Relation: D/μ = kT/q = VT ≈ 26mV (300K)

Temperature Effects

Overall: Temperature↑ → Semiconductor Resistance↓ (NTC characteristic, note this is opposite to metals!)

Keywords: Band, Conduction Band, Valence Band, Band Gap, Fermi Level, Doping, N-type, P-type, Carrier, Mobility, Diffusion