Band Gap

Part of The Transistor

How the energy gap between valence and conduction bands determines semiconductor properties and transistor performance limits.

Why This Matters

The band gap is a single number — measured in electron-volts — that predicts a semiconductor’s operating temperature range, forward junction voltage, leakage current, optical properties, and maximum operating frequency. Every semiconductor material has a characteristic band gap. Understanding it allows you to choose the right material for each application, predict device behavior at temperature extremes, and understand why certain devices fail in ways that others do not.

For transistors specifically, band gap determines two critical limits: the maximum operating temperature (above which thermal carrier generation overwhelms controlled doping) and the forward voltage drop across the base-emitter junction (which sets biasing requirements and minimum supply voltage). These are practical engineering constraints that affect every circuit designed around a transistor.

For a rebuilding civilization, band gap knowledge guides material selection strategy: start with germanium (0.67 eV, accessible, lower temperature requirements) to establish fabrication, then transition to silicon (1.1 eV, abundant, better temperature range) as process capability improves.

What the Band Gap Means Physically

In a solid, electron energies are organized into bands (ranges of allowed energy) separated by gaps (forbidden ranges). The valence band is the highest fully-occupied band at absolute zero — electrons here are bound in lattice bonds. The conduction band is the next higher band — electrons here are free to carry current.

The band gap energy E_g is the minimum energy an electron must gain to jump from the top of the valence band to the bottom of the conduction band. Once there, it can contribute to conductivity. It leaves behind a hole in the valence band, which also contributes to conductivity.

At absolute zero (0 K), no electrons have sufficient thermal energy to cross the gap: intrinsic semiconductor is a perfect insulator. At room temperature (300 K), thermal energy is kT = 0.026 eV. The probability of an electron having energy E above the band edge is proportional to exp(-E/kT).

Intrinsic carrier concentration (ni): The number of thermally generated electron-hole pairs per cubic centimeter at temperature T:

ni = A × T^(3/2) × exp(-Eg / 2kT)

where A is a material constant. This exponential dependence on band gap is the dominant factor:

• Silicon (Eg = 1.1 eV) at 300 K: ni ≈ 1.5 × 10^10 cm
• Germanium (Eg = 0.67 eV) at 300 K: ni ≈ 2.4 × 10^13 cm
• GaAs (Eg = 1.4 eV) at 300 K: ni ≈ 1.8 × 10^6 cm

Germanium has 1600× more thermally generated carriers than silicon at room temperature. This is why germanium transistors leak more and have lower maximum operating temperature.

Temperature Limits

A transistor works because doping controls carrier concentration to a level much higher than intrinsic. When temperature rises, ni rises exponentially. At the temperature where ni equals the doping concentration, the material reverts to near-intrinsic behavior — the transistor loses its designed properties.

For n-type germanium with ND = 10^15 cm^-3: ni equals ND at approximately 70°C. Above this temperature, the material is intrinsic — the transistor’s gain drops, leakage soars, and circuit behavior becomes unpredictable.

For n-type silicon with ND = 10^15 cm^-3: ni equals ND at approximately 180°C. Silicon transistors reliably operate to 150°C in most applications.

This is why silicon replaced germanium for nearly all applications despite germanium’s processing advantages: a device that fails at 70°C (summer ambient temperature in many locations) has limited practical use. Silicon’s wider band gap extends operating temperature to ranges compatible with automotive, industrial, and military use.

Leakage current and band gap: Reverse junction leakage current is proportional to ni. Since ni doubles approximately every 10°C for germanium and every 12°C for silicon, leakage doubles at these temperature intervals. Germanium diodes at 60°C have leakage ~60× higher than at 20°C. Silicon diodes at 60°C have leakage ~10× higher.

This temperature dependence creates a practical maximum for germanium: even if gain is acceptable, leakage current at the collector junction draws base current, changing the operating point in unpredictable ways. Careful circuit design can compensate with temperature-sensitive bias networks, but the fundamental limit exists.

Band Gap and Forward Junction Voltage

The built-in potential of a p-n junction is approximately: V_bi ≈ (kT/q) × ln(NA × ND / ni^2)

Since ni^2 = A^2 × T^3 × exp(-Eg/kT), the built-in potential scales approximately with the band gap. Higher band gap → higher built-in potential → higher forward voltage to overcome the junction.

At 300 K:

• Germanium junction: forward voltage ~0.3-0.4V at low currents, ~0.5V at moderate currents
• Silicon junction: forward voltage ~0.6-0.7V at moderate currents
• GaAs junction: forward voltage ~1.1-1.3V

For transistor circuit design, VBE ≈ 0.7V for silicon and ≈ 0.3V for germanium. These values must be supplied by the bias network. In low-voltage systems (battery-powered from 1-2 cells), germanium’s lower VBE is a practical advantage — it works from lower supply voltages than silicon.

Temperature coefficient of VBE: As temperature rises, VBE decreases. For silicon: approximately -2 mV/°C. For germanium: approximately -1.5 mV/°C. This temperature coefficient is used in temperature compensation circuits: a diode’s VBE tracks the transistor’s VBE, so connecting them in appropriate configurations cancels drift.

Direct vs. Indirect Band Gap

The band gap classification (direct vs. indirect) affects optical properties:

Direct band gap: The momentum of the valence band maximum and conduction band minimum are identical. An electron can transition across the gap by absorbing or emitting only a photon — no momentum change needed. GaAs, InP, GaN are direct band gap.

Indirect band gap: Valence and conduction band extrema occur at different momenta. Transition requires simultaneous photon and phonon (lattice vibration) — a two-particle process, much less probable. Silicon and germanium are indirect band gap.

Consequence: direct band gap semiconductors efficiently emit light (LEDs, lasers) when forward-biased. Indirect band gap semiconductors emit light extremely weakly — silicon LEDs exist but are inefficient.

For rebuilding: this means LEDs and laser diodes require GaAs or similar materials, not silicon or germanium. Silicon photodetectors work well (absorption is efficient even for indirect gap) but silicon light emitters are impractical.

Tailoring band gap: Binary and ternary compound semiconductors can be formulated with any desired band gap between those of their constituent materials. GaAs1-xPx adjusts band gap from GaAs (1.4 eV) to GaP (2.3 eV) — covering red through green wavelengths. This is the path to full-spectrum LED technology, a medium-term goal after mastering junction diodes and transistors.

Band Gap Narrowing at High Doping

At high doping levels (above ~10^18 cm^-3), the energy bands become slightly distorted — the effective band gap narrows. This “band gap narrowing” reduces VBE in heavily doped emitters, affecting the transistor’s gain at high injection.

For primitive transistor fabrication, this effect is a second-order correction. Understand it exists; note that heavily doped emitters have slightly lower VBE than expected from the undoped material band gap. Design emitter doping high (>10^18 cm^-3) but not so high that band gap narrowing significantly reduces injection efficiency.

Silicon emitters at 10^20 cm^-3 experience approximately 0.1 eV of band gap narrowing — a 10% reduction that increases ni^2 in the emitter by 50×, reducing emitter injection efficiency. For germanium alloy transistors, the alloy recrystallization gives emitter doping of ~10^19 cm^-3 (limited by indium solubility in germanium) — moderate band gap narrowing occurs but does not dominate device behavior.