1. Introduction to Semiconductors
The key to understanding semiconductors lies in their band structure – the arrangement of allowed and forbidden energy levels for electrons in the solid.
| Property | Conductor | Semiconductor | Insulator |
|---|---|---|---|
| Band Gap (Eg) | 0 or overlapping | 0.5 - 3 eV | > 5 eV |
| Conductivity (σ) | 10⁶ - 10⁸ S/m | 10⁻⁶ - 10⁴ S/m | < 10⁻¹⁰ S/m |
| Resistivity (ρ) | 10⁻⁸ - 10⁻⁶ Ω·m | 10⁻⁴ - 10⁶ Ω·m | > 10¹⁰ Ω·m |
| Temp. dependence | σ decreases with T | σ increases with T | σ slightly increases |
| Examples | Cu, Ag, Al | Si, Ge, GaAs | Diamond, Glass, Rubber |
2. Direct and Indirect Band Gap Semiconductors
The band gap is not just characterized by its energy but also by the momentum relationship between the valence band maximum and conduction band minimum. This determines how electrons transition between bands.
The conduction band minimum and valence band maximum occur at the same momentum (k-value).
- Electron can transition directly by absorbing/emitting a photon
- No change in momentum required
- Efficient light emission → used in LEDs, lasers
- Examples: GaAs, InP, GaN, CdS
The conduction band minimum and valence band maximum occur at different momentum values.
- Transition requires both photon AND phonon (lattice vibration)
- Momentum change needed (provided by phonon)
- Inefficient light emission → not good for LEDs
- Examples: Si, Ge
- Optoelectronics: Direct band gap materials (GaAs, GaN) are used for LEDs and laser diodes because electron-hole recombination efficiently produces photons
- Solar Cells: Both types work, but direct band gap absorbs light more efficiently for thinner cells
- Silicon dominates electronics: Despite being indirect, Si is abundant and has excellent oxide (SiO₂) for transistors
| Property | Direct Band Gap | Indirect Band Gap |
|---|---|---|
| Transition | Photon only | Photon + Phonon |
| Light Emission | Efficient (radiative) | Inefficient (mostly non-radiative) |
| Light Absorption | High absorption coefficient | Lower absorption coefficient |
| LED/Laser use | Excellent | Poor |
| Examples | GaAs, InP, GaN, CdTe | Si, Ge |
3. Electrical Conductivity of Semiconductors
In semiconductors, electrical conduction occurs through two types of charge carriers: electrons in the conduction band and holes in the valence band.
Where:
- n = concentration of electrons (per m³)
- p = concentration of holes (per m³)
- e = electron charge (1.6 × 10⁻¹⁹ C)
- μₑ = electron mobility (m²/V·s)
- μₕ = hole mobility (m²/V·s)
A pure semiconductor with no impurities. At T > 0 K, thermal energy excites some electrons from valence to conduction band, creating equal numbers of electrons and holes:
4. Drift Velocity, Mobility, and Conductivity
4.1 Drift Velocity
4.2 Mobility
Therefore: vd = μE
- Temperature: Higher T → more lattice vibrations → more scattering → lower mobility
- Impurity concentration: More impurities → more scattering → lower mobility
- Effective mass: Lighter carriers have higher mobility
- Crystal structure: Determines scattering mechanisms
4.3 Relation Between Conductivity and Mobility
Current density J = charge × number density × drift velocity
Using vd = μE:
Since J = σE (Ohm's law):
| Material | Electron Mobility μₑ (m²/V·s) | Hole Mobility μₕ (m²/V·s) |
|---|---|---|
| Silicon (Si) | 0.135 | 0.048 |
| Germanium (Ge) | 0.39 | 0.19 |
| Gallium Arsenide (GaAs) | 0.85 | 0.04 |
5. Fermi-Dirac Distribution Function
Electrons in solids follow Fermi-Dirac statistics because they are fermions (particles with half-integer spin that obey the Pauli exclusion principle). The Fermi-Dirac distribution gives the probability that an energy state E is occupied by an electron.
Where:
- f(E) = probability of occupation of state with energy E
- Eғ = Fermi energy (or Fermi level)
- k = Boltzmann constant (1.38 × 10⁻²³ J/K)
- T = absolute temperature (K)
- At T = 0 K: f(E) = 1 for E < Eғ, f(E) = 0 for E > Eғ (step function)
- At E = Eғ: f(Eғ) = 0.5 at all temperatures
- For E >> Eғ: f(E) → 0 (states mostly empty)
- For E << Eғ: f(E) → 1 (states mostly filled)
- Higher T: Distribution becomes more gradual (thermal smearing)
At T = 0, electrons fill all states from the lowest energy up to Eғ. As temperature increases, some electrons gain thermal energy and can occupy states above Eғ, leaving holes below Eғ. The Fermi-Dirac function describes this thermal excitation probability.
6. Position of Fermi Level in Semiconductors
6.1 Intrinsic Semiconductor
In a pure (intrinsic) semiconductor, the number of electrons in the conduction band equals the number of holes in the valence band: n = p = nᵢ.
If mₕ* ≈ mₑ* (effective masses approximately equal):
6.2 Extrinsic Semiconductor (n-type)
When a semiconductor is doped with donor impurities (e.g., P, As, Sb in Si), extra electrons are introduced. The Fermi level shifts toward the conduction band.
6.3 Extrinsic Semiconductor (p-type)
When doped with acceptor impurities (e.g., B, Al, Ga in Si), holes are introduced. The Fermi level shifts toward the valence band.
| Type | Dopant | Majority Carrier | Fermi Level Position |
|---|---|---|---|
| Intrinsic | None (pure) | n = p = nᵢ | Middle of band gap |
| n-type | Donor (P, As, Sb) | Electrons (n >> p) | Closer to Ec |
| p-type | Acceptor (B, Al, Ga) | Holes (p >> n) | Closer to Ev |
- At low T: Fermi level is close to the dopant level (Ed or Ea)
- At moderate T: All dopants ionized, Eғ between dopant level and intrinsic position
- At high T: Intrinsic carriers dominate, Eғ moves toward middle of band gap
Summary: Key Formulas
- Direct vs Indirect Band Gap: Direct allows efficient light emission; Indirect requires phonon assistance
- Conductivity: Depends on carrier concentration and mobility
- Mobility: Drift velocity per unit field; affected by scattering
- Fermi-Dirac: Probability distribution for electron occupancy
- Fermi Level: 50% occupancy level; shifts with doping
- n-type: Donor doping, Eғ near Ec, electrons are majority carriers
- p-type: Acceptor doping, Eғ near Ev, holes are majority carriers