Direct vs indirect bandgap

Semiconductors are classified as "direct-bandgap" or "indirect-bandgap" based on whether the conduction-band minimum (lowest electron energy) and the valence-band maximum (highest hole energy) occur at the same crystal momentum $\mathbf{k}$. This single distinction has profound consequences for optoelectronic device performance: direct-bandgap semiconductors emit light efficiently; indirect-bandgap semiconductors do not.

Direct bandgap. The conduction-band minimum and valence-band maximum both occur at $\mathbf{k}=0$ (the $\mathrm{\Gamma }$ point in Brillouin zone notation). An electron-hole transition requires no change in momentum, so a photon (carrying negligible momentum) can mediate it directly. Radiative recombination is a fast, allowed process.

Direct-bandgap semiconductors include the III-V family used in all commercial semiconductor lasers and LEDs:

These materials have $B\sim {10}^{-10}$ cm³/s radiative coefficient and form the basis of nearly all commercial light-emitting devices.

Indirect bandgap. The conduction-band minimum and valence-band maximum occur at different $\mathbf{k}$ points. Optical transitions between them require simultaneous emission or absorption of a phonon to conserve momentum, making the radiative process slow.

Indirect-bandgap semiconductors:

For these, the radiative recombination coefficient is $\sim {10}^{-15}-{10}^{-13}$ cm³/s (Si $5\times {10}^{-15}$, Ge $5\times {10}^{-14}$), three to five orders of magnitude smaller than in direct-gap materials.

Why the distinction matters: photon vs phonon momentum. Photon wavevector at optical frequencies: $k_{\text{photon}}=2\pi n\mathrm{/}\lambda \sim {10}^7$ m⁻¹, negligible compared to the electron wavevector at the Brillouin zone edge: $k_{\text{electron}}=\pi \mathrm{/}a\sim {10}^{10}$ m⁻¹. In an indirect transition, the missing momentum must come from a phonon, a second-order process with much smaller matrix element.

Quantitatively: the radiative recombination rate in indirect-gap Si is suppressed by a factor of $\sim {10}^{-5}$ relative to direct-gap GaAs, almost entirely because of the phonon-coupling factor in the matrix element.

Absorption coefficients. Direct-gap materials have sharper, larger optical absorption near the band edge:

The shape of the absorption edge near $E_g$ also differs:

• Direct gap: $\alpha \propto (h\nu -E_g{)}^{1\mathrm{/}2}$ (sharp threshold)
• Indirect gap: $\alpha \propto (h\nu -E_g\pm h{\mathrm{\Omega }}_{\text{phonon}}{)}^2$ (softer threshold with phonon-mediated steps)

This is why photodetector materials are selected:

• Si detectors (450 – 1100 nm): use Si's wide-window indirect absorption; require thick absorbing layers in the near infrared (the absorption length is about 20 μm at 850 nm and over 100 μm at 1000 nm)
• InGaAs detectors (900 – 1700 nm): use InGaAs's direct absorption; thin layers (~1 μm) suffice
• Ge-on-Si detectors (1310/1550 nm): use Ge's near-direct absorption at $\mathrm{\Gamma }$ for tighter integration with Si photonics

Engineering indirect-to-direct conversion.

Several strategies make indirect-gap materials emit light:

1. Tensile strain in Ge: shifts the $\mathrm{\Gamma }$ valley below the $L$ valley, making Ge effectively direct
2. Ge-Sn alloys: $>7\mathrm{\%}$ Sn shifts to direct
3. SiGe quantum wells: confinement modifies energy hierarchy
4. Si nanocrystals: quantum confinement opens direct transitions
5. Si:Er (erbium doping): introduces a direct radiative transition at 1.54 μm independent of Si band structure
6. Heterogeneous integration: bond III-V direct-gap material onto Si chips (a widely used silicon photonics approach)

Quantum cascade lasers. QCLs sidestep the bandgap altogether. They emit on intersubband transitions between confined states within the conduction band of a multilayer III-V structure, and the photon energy is set by the layer thicknesses rather than by the gap. The transitions are direct in $\mathbf{k}$ and couple only to light polarized perpendicular to the layers (TM).

Why direct-gap III-Vs are also good detectors. GaAs, InP, InGaAs: all the materials that make great lasers also make great photodetectors. The same large absorption coefficient that enables efficient emission enables efficient absorption. This explains why InGaAs PIN photodiodes and InGaAs/InP avalanche photodiodes are the standard for telecom receivers.

Direct-indirect crossovers in alloys. Some alloy systems transition between direct and indirect as composition changes:

• AlₓGa₁₋ₓAs: direct for $x<0.45$; indirect ($X$-valley) above. Critical for AlGaAs/GaAs LEDs and lasers.
• InₓGa₁₋ₓP: direct for $x\gtrsim 0.3$ (Ga fraction below about 0.7); indirect below
• AlₓIn₁₋ₓAs: direct at the composition lattice-matched to InP ($x=0.48$); indirect at high aluminum fractions, toward AlAs

Cladding layers need a wider gap than the active region, not an indirect one: the Al₀.₃Ga₀.₇As cladding of a GaAs laser is direct. The crossover matters mostly as a limit, since an alloy near or beyond it emits light inefficiently, which sets the shortest wavelengths that AlGaAs and (Al)GaInP emitters reach.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 16 (semiconductor materials); Yu & Cardona, Fundamentals of Semiconductors (4th ed., 2010) for the canonical band-structure treatment; Coldren, Corzine & Mašanović, Diode Lasers and PICs (2nd ed., 2012), Ch. 2.