Semiconductor Physics
Part of The Transistor
Semiconductor physics explains why certain materials conduct electricity only under specific conditions — the band theory of solids, carrier generation, and the behavior that makes transistors and diodes possible.
Why This Matters
You can build working transistor circuits without understanding the quantum mechanics behind them — but you cannot fabricate transistors, diagnose unusual failures, or extend your knowledge to novel situations without understanding why semiconductors behave the way they do.
Semiconductor physics explains what a “hole” actually is, why silicon conducts at 25°C but not at −200°C, why doping with a few parts per million of phosphorus changes resistivity by a factor of a million, and why the collector-base junction of a transistor creates the electric field that amplifies signals. These are not intuitive results — they follow from the quantum mechanics of electrons in crystalline solids.
The good news: you do not need to understand the full mathematics to have a working model. The band theory provides the framework, and the rest follows from a handful of physical principles. This article builds that framework.
The Band Theory of Solids
In an isolated atom, electrons occupy discrete energy levels. In a crystal lattice with billions of atoms in close proximity, these discrete levels merge into continuous bands of allowed energies, separated by forbidden gaps.
Three types of materials result:
Conductors (metals): The highest occupied band (the conduction band) is partially filled. Electrons can absorb even tiny amounts of energy and jump to slightly higher states within the same band, becoming mobile. Even at absolute zero, metals conduct.
Insulators: The valence band (highest occupied band) is completely filled. The next allowed states are in the conduction band, separated by a large energy gap (>4 eV for typical insulators). At room temperature, thermal energy (~0.026 eV) is insufficient to bridge this gap. Almost no electrons reach the conduction band — the material does not conduct.
Semiconductors: Like insulators, but with a smaller bandgap:
| Material | Bandgap (eV) | Classification |
|---|---|---|
| Silicon (Si) | 1.12 | Semiconductor |
| Germanium (Ge) | 0.67 | Semiconductor |
| Gallium arsenide (GaAs) | 1.43 | Semiconductor |
| Diamond (C) | 5.5 | Insulator |
| Copper (Cu) | 0 (overlapping bands) | Conductor |
A semiconductor’s bandgap is small enough that thermal energy at room temperature breaks a fraction of bonds, generating a small but useful number of free carriers.
Carrier Generation and Recombination
In a pure semiconductor at room temperature, thermal energy breaks covalent bonds, freeing electron-hole pairs:
Generation: An electron absorbs thermal energy → breaks free from its bond → enters conduction band → leaves behind a hole in the valence band.
Recombination: A free electron encounters a hole → falls back into the valence band → releases energy (as heat or light) → both carriers disappear.
At equilibrium, generation rate equals recombination rate. This sets the intrinsic carrier concentration n_i:
| Material | n_i at 25°C (cm⁻³) |
|---|---|
| Silicon | 1.5 × 10¹⁰ |
| Germanium | 2.4 × 10¹³ |
| GaAs | 1.8 × 10⁶ |
For comparison, silicon has ~5 × 10²² atoms/cm³. Only about 1 in every 3 trillion silicon atoms contributes a free electron at room temperature — hence the near-insulating behavior of pure silicon.
Temperature dependence: n_i rises steeply with temperature because higher thermal energy breaks more bonds. For silicon, n_i doubles approximately every 11°C. This is why semiconductor devices have temperature limits — above ~150°C for silicon, thermally generated carriers begin to swamp the doped carrier concentration, and the material loses its controlled semiconductor behavior.
Carrier Transport: Drift and Diffusion
Free carriers move through the crystal by two mechanisms:
Drift
An electric field accelerates carriers: electrons drift in the direction opposite to the field, holes drift in the direction of the field. The velocity is proportional to field strength up to a maximum (saturation velocity):
Drift velocity: v = µ × E
Where µ is the carrier mobility and E is the electric field.
Mobility values:
| Material | Electron Mobility (cm²/V·s) | Hole Mobility (cm²/V·s) |
|---|---|---|
| Silicon | 1,400 | 450 |
| Germanium | 3,900 | 1,900 |
| GaAs | 8,500 | 400 |
Electrons are always faster than holes in the same material. Silicon electrons are 3× faster than silicon holes. This is why NPN transistors (electrons as active carriers) are faster than PNP transistors.
Germanium has 2.8× higher electron mobility than silicon — one reason early transistors used germanium (faster operation despite lower bandgap). GaAs has 6× higher electron mobility than silicon — why GaAs is used in microwave devices.
Diffusion
Carriers move from regions of high concentration to low concentration, driven by concentration gradients. Diffusion current:
J_diff = q × D × dn/dx
Where D is the diffusion coefficient (related to mobility by the Einstein relation: D = µ × k × T / q).
Diffusion is the mechanism by which electrons cross the base of a transistor. There is no electric field pushing them across the base — instead, the high electron concentration at the emitter side and low concentration at the collector side creates a gradient. Electrons diffuse from high to low concentration, arriving at the collector-base depletion region where they are swept into the collector by drift.
The Fermi Level
The Fermi level E_F is the energy at which the probability of finding an electron is exactly 50%. It is a thermodynamic quantity that describes the distribution of electrons among available energy states.
In intrinsic silicon: E_F sits near the middle of the bandgap.
In n-type silicon: Doping adds many electrons. The Fermi level rises toward the conduction band — more energy states near the conduction band are occupied.
In p-type silicon: Doping removes electrons (adds holes). The Fermi level drops toward the valence band.
At a PN junction: The Fermi level must be constant (flat) in equilibrium. But n-type and p-type materials have different Fermi level positions. When joined, the bands must bend to align the Fermi levels — this band bending creates the built-in electric field and built-in voltage of the junction.
Carrier Concentration and the Mass Action Law
In any semiconductor in equilibrium, the product of electron and hole concentrations is constant:
n × p = n_i²
Where n = electron concentration, p = hole concentration, n_i = intrinsic concentration.
Implications for doped material:
If you add N_D donor atoms/cm³ (n-type doping), the majority carrier electron concentration n ≈ N_D. From the mass action law: p = n_i² / N_D (minority carrier holes)
For silicon with N_D = 10¹⁶ /cm³:
- n ≈ 10¹⁶ electrons/cm³
- p = (1.5×10¹⁰)² / 10¹⁶ = 2.25×10⁴ holes/cm³
The minority carrier concentration (holes in n-type) is vanishingly small — 12 orders of magnitude less than the majority carriers. This is why n-type and p-type materials do not spontaneously recombine all their carriers — the concentration asymmetry prevents it.
Practical Implications
Choosing between Si and Ge:
- Silicon: higher operating temperature, lower leakage, better for power and digital circuits
- Germanium: higher mobility (faster for RF), lower forward voltage (~0.2V vs 0.6V for Si), better for sensitive detectors and early transistors
- Silicon dominates because it is abundant, has a convenient native oxide (SiO₂ for passivation), and refines to higher purity
Bandgap and light emission:
- Direct-bandgap semiconductors (GaAs) emit light when carriers recombine — basis for LEDs and laser diodes
- Indirect-bandgap semiconductors (Si, Ge) require a phonon to conserve momentum — extremely inefficient light emitters
- Silicon does not make practical LEDs; GaAs and related compounds do
Summary
Semiconductor Physics — At a Glance
- Band theory: semiconductors have a small bandgap (Si: 1.12 eV, Ge: 0.67 eV) that allows thermal carrier generation
- Intrinsic carrier concentration: only ~10¹⁰/cm³ in silicon at 25°C — nearly insulating
- Two transport mechanisms: drift (electric field) and diffusion (concentration gradient)
- Electrons are 3× faster than holes in silicon (mobility 1,400 vs 450 cm²/V·s)
- Mass action law: n × p = n_i² — doping one carrier type suppresses the other exponentially
- Band bending at PN junction creates built-in voltage — the origin of the ~0.6V silicon diode threshold
- Silicon preferred over germanium: higher temperature stability, less leakage, better passivation