2.2 Direct and indirect bandgap semiconductors

Direct vs indirect bandgaps

Semiconductors fall into two categories based on their band structure: direct bandgap and indirect bandgap. The difference comes down to whether an electron can jump between the valence and conduction bands by absorbing or emitting just a photon, or whether it also needs a phonon (a quantum of lattice vibration) to conserve momentum. This single distinction has huge consequences for which materials work best in LEDs, lasers, solar cells, and transistors.

Valence and conduction bands

The valence band is the highest energy band that's fully occupied by electrons at absolute zero. The conduction band is the lowest energy band that's empty at absolute zero. The energy gap separating them is the bandgap energy, $E_g$.

When an electron absorbs energy greater than $E_g$, it gets promoted from the valence band into the conduction band. That excited electron, along with the hole it leaves behind in the valence band, can now move through the crystal and contribute to electrical conductivity.

Crystal momentum conservation

This is the core concept that separates direct from indirect bandgap materials. Every transition between bands must conserve both energy and crystal momentum (represented by the wave vector $k$).

• In a direct bandgap semiconductor, the conduction band minimum and the valence band maximum sit at the same $k$-value in the Brillouin zone. An electron can transition between bands by absorbing or emitting a single photon, since photons carry negligible momentum compared to the scale of the Brillouin zone.
• In an indirect bandgap semiconductor, the conduction band minimum and valence band maximum occur at different $k$-values. A transition now requires both a photon (to supply the energy) and a phonon (to supply or absorb the difference in crystal momentum). Because this is a two-particle process, it's far less probable.

This lower transition probability is why indirect bandgap materials are poor light emitters but can still function well in electronics and photovoltaics.

Photon absorption and emission

Direct bandgap semiconductors absorb and emit photons efficiently at energies near $E_g$. That efficiency is what makes them the go-to choice for LEDs and laser diodes.

Indirect bandgap semiconductors absorb light too, but the phonon requirement makes each absorption event less likely. To compensate, devices made from indirect materials (like silicon solar cells) need to be physically thicker so that enough photons get absorbed over a longer optical path.

Band structure and lattice

A semiconductor's band structure is the relationship between electron energy and crystal momentum throughout the material. It's set by the periodic potential of the crystal lattice and captures everything you need to know about allowed and forbidden electron states.

Brillouin zone

The Brillouin zone is the fundamental cell in reciprocal (momentum) space, analogous to the unit cell in real space. The first Brillouin zone contains all the unique $k$-values needed to fully describe the electronic properties of the crystal.

Key high-symmetry points label specific locations in the zone:

• Γ (gamma): the zone center, $k = 0$
• X, L, and others: zone-boundary points whose positions depend on the crystal structure

For example, in GaAs both the conduction band minimum and valence band maximum sit at Γ, making it a direct bandgap material. In silicon, the valence band maximum is at Γ but the conduction band minimum is near the X point, making it indirect.

Reciprocal lattice

The reciprocal lattice is the Fourier transform of the real-space crystal lattice. Each point in the reciprocal lattice corresponds to a set of crystal planes in real space. The Brillouin zone is constructed from this reciprocal lattice, and together they provide the framework for plotting and interpreting band structures.

E-k dispersion relation

The E-k diagram plots electron energy versus crystal momentum and is obtained by solving the Schrödinger equation for the crystal's periodic potential. Two things to read from it:

• Band edges: where the conduction band minimum and valence band maximum sit tells you whether the gap is direct or indirect.
• Curvature near band edges: the curvature of the E-k curve determines the effective mass $m^*$ of carriers. A sharper curvature means a smaller effective mass, which translates to higher carrier mobility. Mathematically, $m^* = \hbar^2 \left(\frac{d^2E}{dk^2}\right)^{-1}$.

Optical properties

How a semiconductor interacts with light follows directly from its band structure. The distinction between direct and indirect gaps shows up clearly in measurable optical quantities.

Absorption coefficient

The absorption coefficient $\alpha$ describes how quickly light intensity drops as it travels through the material. A high $\alpha$ means photons are absorbed over a short distance.

• Direct bandgap semiconductors have steep absorption edges and large $\alpha$ values near $E_g$. GaAs, for instance, absorbs most above-bandgap light within about 1 µm.
• Indirect bandgap semiconductors have much smaller $\alpha$ near the band edge because each absorption event requires a phonon. Silicon needs roughly 100 µm or more to absorb the same fraction of light.

This difference is why GaAs solar cells can be thin-film devices while silicon cells are typically hundreds of micrometers thick.

Luminescence efficiency

Luminescence efficiency (or internal quantum efficiency) is the fraction of electron-hole recombination events that produce a photon.

• In direct bandgap materials, radiative recombination dominates because the transition is a simple one-step process. Efficiencies can exceed 90% in high-quality material.
• In indirect bandgap materials, radiative recombination is slow (requiring phonon assistance), so non-radiative pathways tend to win. These include Shockley-Read-Hall (defect-mediated) recombination and Auger recombination, both of which waste energy as heat rather than light.

Exciton binding energy

An exciton is a bound electron-hole pair held together by Coulomb attraction. The exciton binding energy is the energy needed to pull the pair apart into free carriers.

In most common semiconductors at room temperature, thermal energy ($k_BT \approx 26$ meV) exceeds the exciton binding energy, so excitons dissociate quickly. In wide-bandgap materials like GaN (~26 meV binding energy) or ZnO (~60 meV), excitons can survive at room temperature and play a significant role in optical absorption and emission.

Carrier dynamics

Once electrons and holes are generated, their behavior (how they recombine, scatter, and move) determines device performance.

Electron-hole recombination

When a conduction-band electron drops back into the valence band and annihilates a hole, that's recombination. There are two broad categories:

• Radiative recombination: produces a photon with energy $\approx E_g$. This is the dominant mechanism in direct bandgap semiconductors and the basis for LEDs and lasers.
• Non-radiative recombination: energy is released as heat or transferred to another carrier instead of producing light. Shockley-Read-Hall (trap-assisted) and Auger recombination are the main non-radiative pathways.

Phonon-assisted transitions

In indirect bandgap materials, an electron transitioning between bands must exchange momentum with the lattice. This happens through phonon absorption or emission simultaneously with photon absorption or emission.

Because this is a second-order process (two particles must cooperate), the transition rate is much lower than for a direct transition. That's the fundamental reason indirect materials are inefficient light emitters and have weaker absorption near the band edge.

Auger recombination

Auger recombination involves three carriers. An electron and hole recombine, but instead of emitting a photon, the released energy kicks a third carrier (either an electron or a hole) to a higher energy state. That third carrier then thermalizes, losing its energy as heat.

Auger recombination scales with carrier density (it goes roughly as $n^3$ or $n^2p$), so it becomes a serious efficiency-limiting factor at high injection levels. This is one reason LED and laser diode efficiency drops at high drive currents, a phenomenon called efficiency droop in GaN-based LEDs.

Applications

The direct/indirect distinction maps neatly onto device categories: if you need light in or out, you generally want a direct bandgap material. If you need robust, cheap electronics, indirect materials (especially silicon) dominate.

Light-emitting diodes (LEDs)

LEDs convert electrical current into light through radiative recombination. Direct bandgap semiconductors are essential here because you need efficient photon emission.

• GaAs-based LEDs emit in the infrared and red.
• InGaN alloys cover blue through green, and when combined with phosphor coatings, produce white light for general illumination.
• The emitted wavelength is set by the bandgap: $\lambda \approx \frac{1240}{E_g \text{ (eV)}}$ nm.

Laser diodes

Laser diodes produce coherent light via stimulated emission. They require high optical gain, which demands efficient radiative recombination, so direct bandgap materials are used almost exclusively.

• GaAs and InGaAsP alloys are standard for telecom wavelengths (1.3 µm and 1.55 µm).
• The lasing wavelength depends on both $E_g$ and the design of the optical cavity (which provides feedback for stimulated emission).

Photovoltaic devices

Solar cells absorb photons and convert them to electrical energy. Here, indirect bandgap silicon dominates the market despite its weaker absorption, for practical reasons:

• Silicon is abundant and inexpensive.
• Its bandgap of 1.12 eV is close to the theoretical optimum (~1.34 eV) for single-junction solar cells under the Shockley-Queisser limit.
• The weaker absorption is compensated by using wafers ~150–200 µm thick, which is still practical for manufacturing.

Direct bandgap thin-film alternatives like GaAs achieve higher efficiencies per unit thickness but cost significantly more, so they're mainly used in space applications and concentrator systems.

Common semiconductors

Direct bandgap materials

These materials have their conduction band minimum and valence band maximum aligned at the same $k$-point (typically Γ), enabling efficient optical transitions.

Gallium arsenide (GaAs)

• Bandgap: 1.42 eV at room temperature (near-infrared, ~870 nm)
• Key properties: high electron mobility (~8500 cm²/V·s), efficient light emission
• Applications: LEDs, laser diodes, high-frequency RF electronics, and high-efficiency solar cells
• GaAs is the workhorse of III-V optoelectronics

Indium phosphide (InP)

• Bandgap: 1.34 eV at room temperature (~925 nm)
• Key properties: high electron mobility, serves as a substrate for the InGaAsP material system
• Applications: fiber-optic communication lasers and detectors (1.3–1.55 µm range), high-speed transistors (HEMTs, HBTs)

Indirect bandgap materials

These materials have their conduction band minimum and valence band maximum at different $k$-points, requiring phonon assistance for optical transitions.

Silicon (Si)

• Bandgap: 1.12 eV at room temperature (conduction band minimum near the X point)
• Key properties: earth-abundant, inexpensive, extremely mature fabrication technology
• Applications: integrated circuits, transistors, solar cells, MEMS
• Silicon's dominance in electronics comes from decades of manufacturing optimization, not from superior intrinsic properties

Germanium (Ge)

• Bandgap: 0.67 eV at room temperature (narrower gap, sensitive to longer-wavelength infrared)
• Key properties: higher electron and hole mobilities than silicon, but the narrow bandgap leads to higher leakage currents
• Applications: infrared photodetectors, thermophotovoltaics, SiGe alloys for high-speed transistors, and as a substrate for growing III-V materials like GaAs