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 $\Gamma$ 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:

Material	$E_g$ (eV, 300 K)	$\lambda_g$
GaN	3.4	365 nm (UV)
GaAs	1.42	873 nm (NIR)
InP	1.35	918 nm (NIR)
In₀.₅₃Ga₀.₄₇As (LM to InP)	0.74	1675 nm (SWIR)
InGaAsP (varies)	0.74 – 1.35	918 – 1675 nm
InAs	0.36	3440 nm (mid-IR)
InSb	0.17	7300 nm (mid-IR)
HgCdTe (varies)	0 – 1.5	tunable LWIR

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:

Material	$E_g$ (eV, 300 K)	Conduction band min
Si	1.12	$X$ valley (along [100] direction)
Ge	0.66	$L$ valley (along [111] direction); $\Gamma$ valley only 140 meV higher
GaP	2.26	$X$ valley
AlAs	2.15	$X$ valley
AlSb	1.61	$X$ valley
SiC	2.4 – 3.3 (polytype-dependent)	indirect

For these, radiative recombination coefficient is $\sim 10^{-15} - 10^{-14}$ cm³/s — about 5 orders of magnitude smaller than direct-gap materials.

Why the distinction matters: photon vs phonon momentum. Photon momentum at optical frequencies: $\hbar k_\text{photon} \sim 10^4$ m⁻¹, completely negligible compared to electron crystal-momentum at the Brillouin zone edge: $\hbar k_\text{electron} \sim 10^{10}$ 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:

Material	$\alpha$ at $E_g$ + 50 meV	$\alpha$ at $E_g$ + 500 meV
GaAs (direct)	$\sim 10^4$ cm⁻¹	$\sim 5 \times 10^4$ cm⁻¹
InP (direct)	$\sim 10^4$ cm⁻¹	$\sim 5 \times 10^4$ cm⁻¹
InGaAs (direct)	$\sim 10^4$ cm⁻¹	$\sim 5 \times 10^4$ cm⁻¹
Si (indirect)	$\sim 10$ cm⁻¹	$\sim 10^3$ cm⁻¹
Ge (indirect)	$\sim 100$ cm⁻¹	$\sim 10^4$ cm⁻¹

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

• Direct gap: $\alpha \propto (h\nu - E_g)^{1/2}$ — sharp threshold
• Indirect gap: $\alpha \propto (h\nu - E_g \pm h\Omega_\text{phonon})^2$ — 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 (~10 μm)
• 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 $\Gamma$ 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 $\Gamma$ valley below the $L$ valley, making Ge effectively direct
2. Ge-Sn alloys: $> 7\%$ 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 (the standard silicon photonics approach)

Quantum cascade lasers: indirect-gap-like operation. QCLs operate via intersubband transitions within the conduction band of multi-layer III-V structures. The "vertical" intersubband transition is direct in $\mathbf{k}$, but the device design has many similarities to indirect-gap absorption (TM-polarization only, oscillator strength engineering, etc.).

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/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 $x > 0.55$; indirect below
• AlₓIn₁₋ₓAs: direct throughout (LM to InP at $x = 0.48$, indirect-like at higher $x$)

These crossovers are exploited to make heterostructures where the cladding is indirect (low-loss) and the active region is direct (efficient emission).

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.