In semiconductors, optical transitions are fundamentally governed by the electronic band structure and the interactions between electrons and lattice vibrations, known as phonons. While direct bandgap materials like GaAs exhibit efficient radiative recombination due to momentum-conserving transitions, indirect bandgap semiconductors such as silicon (Si) and gallium phosphide (GaP) require phonon participation to satisfy momentum conservation. This article explores the mechanisms of phonon-assisted optical transitions, their manifestation in photoluminescence spectra, and the role of phonon sidebands in indirect semiconductors.

Indirect bandgap semiconductors possess conduction band minima and valence band maxima at different points in the Brillouin zone. For example, in silicon, the conduction band minimum lies near the X-point, while the valence band maximum is at the Γ-point. A direct optical transition between these states is forbidden by momentum conservation, as photons carry negligible momentum compared to the crystal momentum of electrons. To bridge this momentum mismatch, phonons—quanta of lattice vibrations—participate in the transition process. The interaction between electrons, holes, and phonons enables indirect radiative recombination, albeit with lower probability than direct transitions.

The probability of phonon-assisted transitions depends on the electron-phonon coupling strength and the availability of phonons with the required momentum and energy. Two types of processes can occur: phonon emission or phonon absorption. In the case of phonon emission, an electron recombines with a hole while emitting a phonon to conserve momentum. Conversely, phonon absorption involves the absorption of a phonon during the transition. At finite temperatures, both processes contribute, but their relative importance varies with temperature due to the thermal population of phonons.

Photoluminescence (PL) spectroscopy reveals distinct signatures of phonon-assisted transitions in indirect semiconductors. The PL spectrum of silicon, for instance, exhibits a no-phonon (NP) line corresponding to transitions where momentum conservation is achieved through impurity scattering or other mechanisms. However, the dominant features are phonon replicas, or sidebands, shifted from the NP line by the energy of the participating phonons. In silicon, the transverse optical (TO) phonon near 58 meV and the transverse acoustic (TA) phonon near 18 meV are commonly observed in PL spectra. The intensity ratio of these replicas reflects the relative coupling strengths to different phonon modes.

Gallium phosphide (GaP) provides another illustrative example. Although GaP has an indirect bandgap, its PL spectrum shows strong phonon sidebands due to interactions with zone-boundary phonons. The no-phonon line in GaP is weak, while the phonon-assisted transitions involving LO phonons (around 50 meV) and TO phonons (around 45 meV) dominate the emission spectrum. The energy separation between the NP line and the phonon replicas provides direct evidence of the phonon energies involved in the transition.

The temperature dependence of phonon-assisted PL spectra offers further insights into the underlying physics. At low temperatures, phonon absorption processes are suppressed due to the low thermal population of phonons, leaving phonon emission as the dominant mechanism. As temperature increases, phonon absorption becomes more significant, leading to changes in the relative intensities of the phonon replicas. Additionally, higher temperatures enhance non-radiative recombination pathways, reducing the overall PL intensity but preserving the phonon sideband structure.

Quantitative analysis of phonon-assisted transitions requires consideration of the joint density of states, the phonon dispersion relation, and the electron-phonon matrix elements. The transition rate for an indirect process can be expressed as a product of electronic and phononic terms, reflecting the coupling between the two systems. Theoretical models based on perturbation theory, such as Fermi's golden rule, provide a framework for calculating these rates and predicting the PL spectrum's shape and intensity.

Experimental studies have measured the phonon-assisted PL spectra of silicon and GaP with high precision. For silicon at low temperatures, the TO phonon replica is typically the most intense feature, with an energy shift matching the known TO phonon energy at the X-point. In GaP, the LO phonon replica often dominates due to strong Fröhlich coupling between electrons and longitudinal optical phonons. These observations align with theoretical predictions and confirm the critical role of phonons in indirect optical transitions.

Beyond photoluminescence, phonon-assisted processes influence other optical phenomena in indirect semiconductors. Absorption edges in materials like silicon exhibit a characteristic Urbach tail at higher energies, where phonon participation broadens the absorption spectrum. Similarly, Raman scattering spectra reveal the same phonon modes involved in optical transitions, providing complementary information about electron-phonon interactions.

The study of phonon-assisted optical transitions has practical implications for optoelectronic devices based on indirect semiconductors. While these materials are less efficient light emitters than direct bandgap counterparts, their widespread use in electronics motivates efforts to enhance their optical properties. Strategies such as strain engineering, nanostructuring, and defect engineering can modify the electron-phonon coupling and improve radiative efficiency. For example, silicon nanocrystals exhibit enhanced luminescence due to quantum confinement effects that alter the momentum conservation requirements.

In summary, phonon-assisted optical transitions are a defining feature of indirect bandgap semiconductors like silicon and GaP. The participation of phonons enables radiative recombination despite the momentum mismatch between conduction and valence band extrema. Photoluminescence spectra reveal phonon sidebands that provide valuable information about the electron-phonon coupling and the material's vibrational properties. Understanding these processes is essential for optimizing the optical performance of indirect semiconductors and developing new applications in photonics and optoelectronics.