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Degenerate semiconductors; Si vs GaAs E-k

Short answer

Non-degenerate: moderate doping, EF in band gap, Boltzmann statistics. Degenerate: heavy doping (~10^19 cm⁻³), EF inside band, Fermi-Dirac statistics, metallic behavior. Si has indirect gap (1.12 eV), CB minimum at k≠0, poor light emission. GaAs has direct gap (1.42 eV), both band edges at k=0, efficient photon emission, high electron mobility.

Differentiate between non-degenerate and degenerate semiconductor. Compare between Si and GaAs semiconductor with their respective E-K curves.

Non-degenerate vs degenerate semiconductor

A non-degenerate semiconductor is moderately doped (n≪Ncn \ll N_c), so EFE_F lies inside the band gap; a degenerate one is so heavily doped (about 101910^{19} cm−3^{-3} or more in Si) that EFE_F enters the conduction or valence band.

PointNon-degenerateDegenerate
DopingNd≪NcN_d \ll N_cNd≳NcN_d \gtrsim N_c
Fermi levelIn gap, over 3kT3kT from band edgesInside CB (n+^+) or VB (p+^+)
StatisticsBoltzmann; np=ni2np = n_i^2Fermi-Dirac; np≠ni2np \ne n_i^2
Impurity levelsDiscreteMerge into an impurity band
BehaviourSemiconductorMetal-like, weak T dependence
UsesNormal diodes, transistorsTunnel diodes, lasers, ohmic contacts

Comparison of Si and GaAs

PropertySiGaAs
Band gap1.12 eV, indirect1.42 eV, direct
Electron mobilityabout 1350 cm2^2/Vsabout 8500 cm2^2/Vs
Electron effective mass1.08me1.08m_e (dos)0.067me0.067m_e
nin_i at 300 Kabout 101010^{10} cm−3^{-3}about 2×1062\times10^{6} cm−3^{-3}
StructureDiamondZinc blende
Native oxideSiO2_2None useful
UseICs, power devices, solar cellsLEDs, lasers, microwave ICs

E-k curves

     Si (indirect)          GaAs (direct)
  E                       E
  |      \   /  CB        |    \   /  CB
  |       \_/             |     \_/
  |        ^ min at k0    |      | min at k=0
  |   Eg  /  (phonon)     |   Eg | photon
  |    /\                 |     /\
  |   /  \   VB           |    /  \  VB
  +---0----k0---- k       +-----0------ k
  • Si: CB minimum at k0≠0k_0 \ne 0, VB maximum at k=0k = 0. A transition needs a phonon for momentum, so light emission is poor.
  • GaAs: CB minimum and VB maximum both at k=0k = 0. Electrons drop directly and emit photons (λ≈1.24/1.42=0.87\lambda \approx 1.24/1.42 = 0.87 μ\mum). Sharp CB curvature gives small me∗=0.067mem_e^* = 0.067m_e and high mobility.

✓ Answer checked by Claude Opus 5.5 (high effort)

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