Energy Bands, Bandgap & Band Diagrams

Why solids have bands and what the gap means. Learn how discrete atomic levels split into bands when atoms come together, the valence band, conduction band and forbidden gap, electrons and holes as carriers, the band pictures of metals, semiconductors and insulators, the Fermi level, the E–k diagram with direct and indirect gaps, effective mass, temperature dependence of the gap (Varshni), band bending near surfaces and junctions, and material data for Si, Ge, GaAs, GaN, SiC and InP — with Python computing E_g(T) and photon cut-off wavelengths.

Energy-band diagrams are the most important picture in device physics. Almost every device in this course — junctions, transistors, LEDs, solar cells, HEMTs — is explained by drawing its bands and watching how they bend under voltage or light.

The bandgap sets how many carriers exist, how much voltage a device can block, which light it absorbs or emits, and how hot it can run.

An analogy

Think of a two-storey car park: the lower floor (valence band) is completely full, so no car can move; the upper floor (conduction band) is empty. Lift a car upstairs (supply energy ≥ E_g) and it can drive freely — and the empty space left downstairs (a hole) lets cars below shuffle along too.

From levels to bands

An isolated atom has sharp energy levels. Bring N atoms together and each level splits into N closely spaced levels — effectively a continuous band. In silicon the bonding (valence) states form the valence band and the antibonding states form the conduction band, with no allowed states in between: the forbidden gap.

Electrons and holes

Two kinds of carrier:

E–k diagrams and effective mass

Plotting energy against crystal momentum k shows band shapes. Near a band edge E ≈ ħ²k²/2m*: the curvature defines an effective mass that describes how carriers accelerate in fields (Si electrons m* ≈ 0.26m₀ for conductivity, GaAs electrons 0.067m₀ → higher mobility). Direct gaps have both extrema at the same k; indirect gaps need a phonon to supply momentum.

Band bending and alignment

Charge near surfaces, interfaces and junctions creates fields that bend bands; in equilibrium E_F is constant throughout. Different materials align according to electron affinity and gap (type I straddling, type II staggered, type III broken) — the basis of heterojunctions, HEMTs, quantum wells and lasers. Band diagrams also include vacuum level, work function and electron affinity for metal–semiconductor contacts.

Formulas

Worked example: Absorption

Given: Si at 300 K (1.12 eV), 940 nm IR LED

Find: Absorbed?

Answer: Yes (weakly — indirect gap near the edge)

Worked example: Bandgap shift

Given: GaAs 300 → 400 K

Find: ΔE_g.

Answer: −46 meV

Worked example: Emission wavelength

Given: GaN LED, E_g 3.4 eV

Find: λ.

Answer: ≈ 365 nm (UV); InGaN alloys give blue

Band pictures

  Metal            Semiconductor           Insulator
  ▓▓▓▓▓ partly     ───── E_c (empty)        ───── E_c
  ▓▓▓▓▓ filled      ↕ E_g ≈ 1 eV            ↕ E_g ≈ 9 eV
  ───── E_F        ─ ─ E_F (mid-gap)
                   ▓▓▓▓▓ E_v (full)         ▓▓▓▓▓ E_v
  n-type: E_F near E_c        p-type: E_F near E_v
  Junction/surface: bands bend, E_F stays flat in equilibrium

A metal's Fermi level lies inside a band, so electrons move freely. In a semiconductor it sits in the gap; doping moves it towards E_c (n-type) or E_v (p-type). Electric fields tilt or bend the bands — electrons roll 'downhill' in E_c, holes float 'uphill' in E_v.

Bandgap vs temperature and cut-off wavelengths

# Bandgap vs temperature (Varshni) and photon cut-off wavelength for common semiconductors
params = {  # Eg(0) eV, alpha eV/K, beta K
    "Si":   (1.170, 4.73e-4, 636),
    "Ge":   (0.7437, 4.77e-4, 235),
    "GaAs": (1.519, 5.405e-4, 204),
    "GaN":  (3.507, 9.09e-4, 830),
}
for m, (e0, al, be) in params.items():
    eg = lambda T: e0 - al * T * T / (T + be)
    print(f"{m:4s}: Eg(300 K) = {eg(300):.3f} eV, Eg(400 K) = {eg(400):.3f} eV, cut-off {1240/eg(300):.0f} nm")
print(f"photon energy of a 650 nm red LED: {1240/650:.2f} eV; 1550 nm telecom: {1240/1550:.3f} eV")
print("a 1550 nm photon (0.80 eV) cannot be absorbed by Si (1.12 eV) -> InGaAs/Ge detectors are used")
# Si  : Eg(300 K) = 1.125 eV, Eg(400 K) = 1.097 eV, cut-off 1103 nm
# Ge  : Eg(300 K) = 0.663 eV, Eg(400 K) = 0.624 eV, cut-off 1869 nm
# GaAs: Eg(300 K) = 1.422 eV, Eg(400 K) = 1.376 eV, cut-off 872 nm
# GaN : Eg(300 K) = 3.435 eV, Eg(400 K) = 3.389 eV, cut-off 361 nm
# photon energy of a 650 nm red LED: 1.91 eV; 1550 nm telecom: 0.800 eV
# a 1550 nm photon (0.80 eV) cannot be absorbed by Si (1.12 eV) -> InGaAs/Ge detectors are used

Bandgaps shrink as temperature rises (lattice expansion and electron–phonon interaction): silicon loses ~28 meV between 300 and 400 K, which is one reason diode voltages fall with temperature. Photon energy must exceed E_g to be absorbed — silicon is transparent at 1550 nm, so telecom detectors use InGaAs or Ge.

Common mistakes

Applications

Key takeaways

Going further