Energy Band Theory

Energy band theory explains why silicon conducts electricity differently from copper or glass by describing the allowed energy levels of electrons in a crystal. Without this model, the operating principle of every diode, transistor, and solar cell would be impossible to explain.

Core Concept

When atoms bond together to form a crystal, their discrete energy levels split and broaden into continuous energy bands. The two most important bands are the valence band (highest filled band at 0K) and the conduction band (the next allowed band above it). Electrons can carry current only in the conduction band, where they are free to move through the crystal.

The energy gap between the top of the valence band and the bottom of the conduction band is called the bandgap energy Eg. Silicon has Eg = 1.12 eV at 300K. Germanium has Eg = 0.67 eV. Gallium arsenide (GaAs) has Eg = 1.42 eV. An insulator like SiO2 has Eg above 5 eV, which means thermal energy at room temperature cannot excite electrons across the gap, so no current flows.

In a conductor like copper, the conduction band and valence band overlap. Electrons sit right at the Fermi level, which lies inside the conduction band. Even a tiny electric field moves them. In a semiconductor at room temperature, a small but non-zero fraction of electrons have enough thermal energy (kT ≈ 0.026 eV at 300K) to cross the 1.12 eV gap in silicon, creating electron-hole pairs.

Key Equations

Thermal voltage: VT = kT/q = 26 mV at T = 300K

where k = 1.38e-23 J/K (Boltzmann constant), q = 1.6e-19 C (electron charge).

Bandgap of silicon: Eg(Si) = 1.12 eV at 300K

Temperature dependence of Eg: Eg decreases with increasing temperature (approximately -2.3 meV/K for silicon).

Fermi-Dirac probability: f(E) = 1 / (1 + exp((E - Ef) / kT))

At the Fermi level: f(Ef) = 0.5 (50% probability of electron occupancy at any temperature).

Given:
  Silicon at T = 400K
  k  = 1.38e-23 J/K
  q  = 1.6e-19 C
  Find: Thermal voltage VT at 400K

Why this formula:
  Thermal voltage scales directly with absolute temperature.

Formula:
  VT = kT / q

Substitution:
  VT = (1.38e-23 * 400) / (1.6e-19)
     = (5.52e-21) / (1.6e-19)

Calculation:
  VT = 5.52e-21 / 1.6e-19
     = 0.03450 V
     = 34.5 mV

Final Answer:
  VT = 34.5 mV at 400K
  (compared to 26 mV at 300K)
  Note: VT increases proportionally with absolute temperature.

Exam Tip: GATE uses VT = 26 mV as the standard value at 300K. Any problem specifying a different temperature requires you to scale: VT = (T/300) * 26 mV. Also, the Fermi level in an intrinsic semiconductor lies very close to the middle of the bandgap, not at the band edges. Students who place the Fermi level at the conduction band edge in an intrinsic semiconductor will get doping and carrier concentration problems wrong.

Key Properties

• Silicon bandgap Eg = 1.12 eV at 300K; Germanium Eg = 0.67 eV; GaAs Eg = 1.42 eV (direct bandgap, used in LEDs).
• Conductors have overlapping valence and conduction bands, so the Fermi level sits inside the conduction band and electrons are always free to move.
• Insulators have Eg above 5 eV; thermal energy at 300K (kT = 0.026 eV) is far too small to bridge the gap.
• The Fermi level in an intrinsic semiconductor lies near the middle of the bandgap at 300K, slightly offset due to the difference in effective density of states.
• Bandgap decreases with increasing temperature for all semiconductors, which is why diode forward voltage (proportional to Eg/q) decreases by about 2 mV/°C.
• Direct bandgap semiconductors (GaAs, GaN, InP) emit photons when electrons recombine with holes, enabling LEDs and laser diodes. Indirect bandgap materials (Si, Ge) cannot.

Quick Revision

• Valence band: highest filled band at 0K. Conduction band: next allowed band above it.
• Bandgap Eg: Si = 1.12 eV, Ge = 0.67 eV, GaAs = 1.42 eV, SiO2 > 5 eV.
• Thermal voltage VT = kT/q = 26 mV at 300K.
• Fermi-Dirac function: f(Ef) = 0.5 at the Fermi level always.
• Fermi level in intrinsic semiconductor is near midgap, not at band edge.
• Eg decreases with temperature; Vf of diodes decreases ~2 mV/°C.
• Exam trap: Students place the Fermi level of an intrinsic semiconductor at the conduction band edge rather than near the middle of the bandgap, leading to wrong answers on carrier concentration and doping problems.