--- title: Band Theory and Doping url: https://doc.liz6.com/en/hardware/03-semiconductor/01-band-theory-and-doping locale: en area: hardware tags: - hardware - semiconductor date: 2026-06-30 modified: 2026-07-16 description: Band Theory and Doping Why Band Theory is Needed Classical physics cannot explain the conductive behavior of semiconductors. Band theory uses quantum mechanics … --- # 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 Three Basic Bands: Conduction, Band Gap, Valence Energy Conduction Band Electrons can move freely (conductive) Band Gap: Eg Energy range where electrons cannot exist Valence Band Electrons are bound (non-conductive) Core Conclusion: The bandgap width Eg determines conductivity—electrons must cross Eg to jump from the valence band to the conduction band to conduct, the wider the Eg, the harder it is for electrons to jump, and the more insulating the material. ### Conductors / Semiconductors / Insulators Conductors / Semiconductors / Insulators: Bandgap Width Eg Determines Conductivity Conductor (Cu, Al) Semiconductor (Si, Ge) Insulator (SiO₂) Conduction Band ≈ Valence Band (Overlapping / Partially filled) Eg = 0 Conduction Band Eg ≈ 1.12eV Valence Band Conduction Band Eg > 5eV Valence Band Key Rule: Eg increases from 0 (conductor) → ~1eV (semiconductor) → >5eV (insulator), while conductivity decreases. The unique feature of semiconductors is their moderate Eg—conductivity can be artificially controlled via doping, temperature, etc., which is the physical basis for semiconductor devices. --- ## 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 | Material | Eg (eV) | Characteristics | Applications | |------|---------|------|------| | Ge (Germanium) | 0.67 | High mobility, high leakage | Early transistors | | GaAs (Gallium Arsenide) | 1.43 | Direct bandgap, high speed | RF / Optoelectronic devices | | SiC (Silicon Carbide) | 3.26 | Wide bandgap, high voltage/temperature resistance | Power devices | | GaN (Gallium Nitride) | 3.4 | Wide bandgap, high frequency/efficiency | Fast chargers / 5G base stations | --- ## 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 | Effect | Description | |------|------| | Intrinsic Carrier ni ↑ | Doubles for every ~11°C rise | | Mobility μ ↓ | Increased lattice vibration leads to more scattering | | Eg ↓ | Bandgap width slightly shrinks | | PN Junction Vf ↓ | Decreases by about 2mV per 1°C | ``` 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*