Carrier Transport
Part of Semiconductors
How electrons and holes move through semiconductor material under electric fields and concentration gradients.
Why This Matters
Semiconductor devices work because charge carriers — electrons and holes — move in controlled ways through semiconductor material. The two fundamental transport mechanisms, drift and diffusion, underlie every device characteristic: the I-V curve of a diode, the frequency response of a transistor, the sensitivity of a photodetector. Without understanding carrier transport, device behavior is a collection of memorized rules rather than comprehended physics.
For a rebuilding civilization, transport concepts directly guide practical decisions. Why does base doping concentration affect transistor speed? Because minority carriers must diffuse across the base before they recombine — transport physics determines the time this takes. Why does reverse leakage current increase with temperature? Thermally generated carriers drift under the junction field — more thermal generation at higher temperature means more leakage. These are not mysteries to be accepted; they are consequences of transport physics.
The mathematics requires calculus to derive precisely, but the physical pictures are intuitive. Carriers in an electric field are like sand grains blown by wind (drift). Carriers at high concentration spreading to regions of low concentration are like perfume diffusing through air (diffusion). Both pictures are exactly right.
Drift: Transport in Electric Fields
When an electric field is applied to a semiconductor, free charge carriers accelerate. They do not accelerate indefinitely — they repeatedly collide with lattice vibrations (phonons) and impurity atoms, losing momentum. The average velocity reached in a given field is the drift velocity:
v_drift = µ × E
where µ is the carrier mobility (cm²/V·s) and E is the electric field (V/cm).
Mobility is a material property that quantifies how easily carriers move. Higher mobility means faster response to fields. Key values at room temperature:
- Germanium electrons: 3900 cm²/V·s
- Germanium holes: 1900 cm²/V·s
- Silicon electrons: 1350 cm²/V·s
- Silicon holes: 480 cm²/V·s
Germanium’s higher mobility is why early transistors used germanium despite its narrower band gap — faster carriers meant better high-frequency performance with 1950s fabrication capabilities.
Mobility decreases with temperature (more phonon scattering at higher temperatures) and with doping concentration (impurity atoms scatter carriers). This is why heavily doped semiconductor has lower carrier mobility and higher resistivity than expected from carrier count alone.
Drift current density: J_drift = q(nµ_e + pµ_h)E, where q is electron charge, n is electron concentration, p is hole concentration. Both electrons and holes contribute to current. In n-type material, electrons dominate; in p-type, holes dominate.
Ohm’s law emerges: For uniform doping and low fields, drift current is proportional to field — this is Ohm’s law. Resistivity ρ = 1/[q(nµ_e + pµ_h)]. Semiconductor resistivity ranges from ~0.001 Ω·cm (heavily doped) to ~50,000 Ω·cm (very pure silicon at room temperature).
Diffusion: Transport Down Concentration Gradients
Diffusion occurs whenever carrier concentration varies in space. Carriers in a region of high concentration move randomly in all directions; more of them move toward low-concentration regions than in the reverse direction. The net result is carrier flow from high to low concentration — diffusion current.
Diffusion current density: J_diff = qD_n (dn/dx) for electrons, where D_n is the electron diffusion coefficient and dn/dx is the concentration gradient.
Diffusion coefficients relate to mobility through the Einstein relation: D = µ × kT/q
where kT/q is the thermal voltage (~0.026 V at room temperature, often written V_T). At room temperature: D_n(Si) ≈ 35 cm²/s, D_p(Si) ≈ 12 cm²/s.
This is the critical mechanism in bipolar transistor operation. In a forward-biased emitter-base junction, the emitter injects minority carriers (electrons in NPN) into the base. These electrons exist at high concentration near the emitter and low concentration at the collector junction. The concentration gradient drives electron diffusion across the base. Since the collector is reverse-biased, it sweeps up arriving electrons. Current flows.
The total current across the base is determined by how many injected carriers make it across without recombining. The fraction that survives is the base transport factor — a key component of current gain.
The Continuity Equation
In real devices, carriers are simultaneously being generated (thermally or by injection), recombining, drifting, and diffusing. The continuity equation tracks all these processes:
∂n/∂t = G - R + (1/q) dJ_n/dx
where G is generation rate, R is recombination rate, and dJ_n/dx is the divergence of electron current density.
For steady-state operation (∂n/∂t = 0), the equation simplifies and can be solved for the carrier distribution in each region of a device.
Minority carrier lifetime (τ): Excess minority carriers recombine with majority carriers over a characteristic time τ. For germanium, τ can be hundreds of microseconds. For silicon, it ranges from microseconds (heavily doped) to milliseconds (pure). Lifetime determines how far minority carriers travel before recombining — the diffusion length L = √(Dτ).
Diffusion length: L = √(Dτ). For silicon minority electrons in p-type base: D_n ≈ 35 cm²/s, τ ≈ 1 µs gives L_n ≈ 60 µm. The base must be thinner than this for efficient transistor action. This is why base width is a critical fabrication parameter.
Recombination Mechanisms
Carriers recombine when an electron falls into a hole. The energy released can go to:
Radiative recombination: A photon is emitted. Dominant in direct band-gap semiconductors (GaAs, GaN, InP). Basis of LEDs and laser diodes. Rare in indirect band-gap materials like silicon and germanium.
Non-radiative (trap-assisted) recombination: Energy goes to lattice vibrations via intermediate trap states created by impurities or crystal defects. Dominant in silicon and germanium. Reducing trap density (reducing impurities and crystal defects) increases minority carrier lifetime and improves transistor performance. This is why semiconductor purification is critical.
Auger recombination: Three-particle process important in heavily doped material. An electron and hole recombine, transferring energy to a third carrier. Reduces lifetime in the emitter region.
For primitive semiconductor fabrication, trap-assisted recombination through metallic impurities is the primary enemy. Iron, copper, gold, and nickel create deep trap states in silicon and germanium even at part-per-billion concentrations. These cut minority carrier lifetime from microseconds to nanoseconds, destroying transistor gain. Zone refining and careful contamination control during fabrication target this problem directly.
Practical Implications for Device Design
Understanding transport guides device design decisions:
High-frequency transistors: Speed requires thin base (short transit time) and high carrier mobility. Use germanium for highest mobility; thin the base to minimum achievable thickness; minimize base resistance by doping appropriately (heavier base doping reduces resistance but also reduces mobility — optimization required).
Power handling: High current causes high carrier concentrations at junctions, changing mobility (high injection effects). Large-area junctions distribute current, reducing local carrier density and maintaining efficiency.
Temperature effects: Mobility decreases with temperature (phonon scattering), reducing speed. Intrinsic carrier concentration increases exponentially with temperature, increasing leakage. Both effects limit maximum operating temperature — germanium to ~70°C, silicon to ~150°C.
Contamination tolerance: Devices with thin, lightly-doped bases are most sensitive to recombination centers from metallic impurities. If fabrication cannot achieve very high purity, use designs with wider bases and lower gain expectations rather than attempting thin-base high-gain designs that will fail due to short minority carrier lifetime.