10.1 Absorption and emission

Absorption and emission fundamentals

Absorption and emission describe how electrons in solids gain or lose energy by interacting with light. These two processes sit at the heart of every optoelectronic device, from solar cells to lasers, and understanding them means understanding why materials look, glow, and conduct light the way they do.

Three ideas control everything here: the energy levels available to electrons, the probability that a transition between levels actually happens, and the selection rules that decide which transitions are allowed in the first place.

Energy levels in solids

In an isolated atom, energy levels are sharp and discrete. In a solid, the periodic potential of the crystal lattice causes those levels to broaden into bands. The two most important bands are the valence band (filled states) and the conduction band (empty states), separated by the bandgap $E_g$.

The density of states (DOS) tells you how many energy states are available per unit energy per unit volume. Its shape depends on the dimensionality of the system:

• 3D bulk: DOS goes as $\sqrt{E}$
• 2D (quantum well): DOS is a step function
• 1D (quantum wire): DOS has van Hove singularities ($1/\sqrt{E}$ divergences)
• 0D (quantum dot): DOS is a series of delta functions (discrete levels)

The DOS matters because transitions can only happen where states actually exist on both sides of the gap.

Transition probabilities and Fermi's golden rule

Not every pair of energy levels leads to a transition with equal likelihood. Fermi's golden rule gives the transition rate $P_{if}$ from an initial state $i$ to a final state $f$:

$P_{if} = \frac{2\pi}{\hbar} |M_{if}|^2 \delta(E_f - E_i - \hbar\omega)$

Here's what each piece means:

• $M_{if}$ is the matrix element of the perturbation (the light-matter interaction Hamiltonian) between the initial and final states. A larger matrix element means a stronger coupling and a faster transition.
• The Dirac delta function $\delta(E_f - E_i - \hbar\omega)$ enforces energy conservation: the photon energy $\hbar\omega$ must exactly match the energy difference between the two states.
• $\hbar$ is the reduced Planck constant.

In practice, you also sum over all available final states weighted by the DOS, so both the matrix element and the availability of states determine the overall absorption or emission rate.

Selection rules for transitions

Selection rules come from symmetry. When you evaluate the matrix element $M_{if}$, certain combinations of initial and final states give zero, making those transitions forbidden.

• Electric dipole transitions (the most common type) require a change in orbital angular momentum quantum number: $\Delta l = \pm 1$. The initial and final states must have opposite parity.
• Magnetic dipole transitions are much weaker and follow $\Delta m_s = 0, \pm 1$.
• Phonon-assisted transitions can relax these rules by supplying extra momentum or changing the symmetry of the process. This is why indirect-gap semiconductors can still absorb light, just less efficiently.

Absorption processes

When a solid absorbs a photon, an electron gets promoted to a higher energy state. Several distinct mechanisms exist, and which one dominates depends on the photon energy, the band structure, and the material quality.

Direct absorption in semiconductors

Direct absorption is the simplest case. A photon with energy $\hbar\omega \geq E_g$ excites an electron from the valence band to the conduction band at the same point in k-space. Crystal momentum is automatically conserved because the photon carries negligible momentum compared to the electron.

Direct bandgap semiconductors like GaAs ($E_g \approx 1.42$ eV) and InP ($E_g \approx 1.34$ eV) have their valence band maximum and conduction band minimum at the same k-point (usually $\Gamma$). This gives them strong absorption near the band edge and makes them ideal for LEDs and laser diodes.

Indirect absorption via phonons

In indirect bandgap semiconductors like Si ($E_g \approx 1.12$ eV) and Ge ($E_g \approx 0.66$ eV), the conduction band minimum sits at a different k-point than the valence band maximum. A photon alone can't bridge the gap because it can't supply the required crystal momentum.

The solution: a phonon (quantized lattice vibration) participates in the transition, providing the missing momentum. This makes indirect absorption a second-order process involving two particles (photon + phonon), so its probability is significantly lower than direct absorption. That's why silicon absorbs light much more weakly near its band edge than GaAs, and why Si solar cells need to be much thicker to capture the same fraction of sunlight.

Free carrier absorption

Once carriers are already in a band (free electrons in the conduction band or free holes in the valence band), they can absorb photons and move to higher states within the same band. This is free carrier absorption (FCA).

FCA scales with carrier concentration and increases at longer wavelengths (infrared), roughly following a $\lambda^2$ or $\lambda^3$ dependence depending on the scattering mechanism. In heavily doped semiconductors, FCA can become a significant source of optical loss, which is a real concern in laser design and waveguide engineering.

Exciton absorption

An exciton is a bound electron-hole pair held together by Coulomb attraction. Think of it like a hydrogen atom, but inside a crystal. The binding energy is typically a few meV in bulk semiconductors (e.g., ~4 meV in GaAs) but can reach tens of meV in wide-gap materials like GaN or ZnO.

Exciton absorption shows up as sharp peaks in the absorption spectrum at energies just below the bandgap, offset by the exciton binding energy. These features are most visible in high-purity samples at low temperatures, where thermal broadening doesn't wash them out. In quantum wells and 2D materials, exciton binding energies are much larger, and excitonic effects can dominate even at room temperature.

Urbach tail absorption

Below the bandgap, absorption doesn't drop to zero instantly. Instead, it falls off exponentially in what's called the Urbach tail:

$\alpha(E) \propto \exp\left(\frac{E - E_g}{E_U}\right)$

The Urbach energy $E_U$ characterizes the width of this tail. It arises from disorder, defects, and electron-phonon coupling that create localized states near the band edges. A smaller $E_U$ indicates a more ordered, higher-quality material. Typical values range from ~10 meV in crystalline GaAs to ~50 meV or more in amorphous silicon.

Emission processes

Emission is the reverse of absorption: an electron in a higher energy state relaxes to a lower one, and the energy difference can be released as a photon (radiative) or as heat (non-radiative).

Spontaneous emission

In spontaneous emission, an excited electron drops to a lower state on its own, emitting a photon with energy equal to the transition energy. The timing is random, and the emitted photons have random phases and directions.

The spontaneous emission rate depends on the matrix element and the density of final states, just like absorption. This is the process behind LEDs, which is why LED light is incoherent and has a relatively broad spectrum (typically tens of nm wide).

Stimulated emission

In stimulated emission, an incoming photon triggers an excited electron to drop, producing a second photon that is identical to the first: same energy, phase, polarization, and direction. This is the mechanism that makes lasers possible.

For stimulated emission to dominate over absorption, you need population inversion, meaning more electrons in the excited state than in the ground state. This is a non-equilibrium condition that requires external pumping (electrical or optical). Without population inversion, the material absorbs more than it emits, and you don't get amplification.

Luminescence: fluorescence vs. phosphorescence

Luminescence is light emission that isn't caused by thermal radiation (it's sometimes called "cold light").

• Fluorescence: Emission happens fast, typically on nanosecond timescales. The transitions involved are spin-allowed (e.g., singlet-to-singlet). Once you remove the excitation source, fluorescence stops almost immediately. Examples include organic dyes and quantum dots.
• Phosphorescence: Emission is delayed, from milliseconds to hours. The transitions are spin-forbidden (e.g., triplet-to-singlet), so the excited state is long-lived. This is what makes glow-in-the-dark materials work. Examples include some transition metal complexes and rare-earth-doped phosphors.

The key distinction is the spin state involved: allowed transitions give fluorescence, forbidden transitions give phosphorescence.

Radiative vs. non-radiative recombination

When an electron-hole pair recombines, the energy can go two ways:

• Radiative recombination: Energy is released as a photon. This is what you want in LEDs and lasers.
• Non-radiative recombination: Energy is released as phonons (heat) or transferred to another carrier. The main non-radiative mechanisms are:
  • Shockley-Read-Hall (SRH) recombination: Carriers recombine through defect or trap states in the bandgap.
  • Auger recombination: The recombination energy is transferred to a third carrier, which then thermalizes. This becomes significant at high carrier densities.

The competition between these two pathways determines how much light you actually get out of a material.

Quantum efficiency of emission

Quantum efficiency (QE) quantifies how effectively absorbed energy is converted to emitted photons.

Internal quantum efficiency (IQE) looks only at what happens inside the material:

$IQE = \frac{R_{rad}}{R_{rad} + R_{non-rad}}$

where $R_{rad}$ is the radiative recombination rate and $R_{non-rad}$ is the non-radiative recombination rate.

External quantum efficiency (EQE) accounts for how many of those photons actually escape the device:

$EQE = IQE \times \eta_{extraction}$

The extraction efficiency $\eta_{extraction}$ is often surprisingly low because of total internal reflection at the semiconductor-air interface (high refractive index materials trap most of the light). For example, a flat GaAs surface only lets out about 2% of internally generated photons without special extraction structures.

Optical properties of solids

The optical properties of a solid describe how it responds to electromagnetic radiation across the spectrum. These properties connect directly to the electronic structure and are measured using techniques like ellipsometry and reflectance spectroscopy.

Complex refractive index

The optical response of a material is captured by the complex refractive index:

$\tilde{n} = n + i\kappa$

• $n$ (real part) is the ordinary refractive index. It determines the phase velocity of light in the material and governs refraction.
• $\kappa$ (imaginary part) is the extinction coefficient. It describes how quickly light is attenuated as it propagates through the material.

Both $n$ and $\kappa$ depend on wavelength. A material can be transparent at one wavelength ($\kappa \approx 0$) and strongly absorbing at another (large $\kappa$).

Kramers-Kronig relations

The real and imaginary parts of the refractive index are not independent. The Kramers-Kronig relations connect them through integral transforms:

$n(\omega) - 1 = \frac{2}{\pi} \mathcal{P} \int_0^{\infty} \frac{\omega' \kappa(\omega')}{\omega'^2 - \omega^2} d\omega'$

These relations follow from causality: the material's response cannot come before the excitation. The practical payoff is that if you measure $\kappa(\omega)$ over a broad frequency range, you can calculate $n(\omega)$ without a separate measurement (and vice versa).

Absorption coefficient and penetration depth

The absorption coefficient $\alpha$ describes how rapidly light intensity decays inside a material. It follows the Beer-Lambert law:

$I(z) = I_0 e^{-\alpha z}$

where $I_0$ is the initial intensity and $z$ is the propagation distance.

The penetration depth $\delta = 1/\alpha$ is the distance at which the intensity drops to $1/e$ (about 37%) of its initial value. The absorption coefficient relates to the extinction coefficient by:

$\alpha = \frac{4\pi\kappa}{\lambda}$

For context, silicon at 500 nm has $\alpha \approx 10^4$ cm$^{-1}$, giving a penetration depth of about 1 $\mu$m. At 1000 nm (near its indirect gap), $\alpha$ drops to ~100 cm$^{-1}$, and the penetration depth grows to ~100 $\mu$m.

Reflectivity and transmittivity

Reflectivity $R$ is the fraction of incident light reflected at an interface. For normal incidence between two media:

$R = \left|\frac{n_1 - n_2}{n_1 + n_2}\right|^2$

For a material with a complex refractive index, you use $\tilde{n}$ in this formula, which means absorption affects reflectivity too.

Transmittivity $T$ through a slab of thickness $d$ (ignoring multiple reflections) is:

$T = (1-R)^2 e^{-\alpha d}$

The $(1-R)^2$ factor accounts for reflection losses at both surfaces. These quantities are central to designing anti-reflection coatings for solar cells, optical filters, and mirrors.

Optical conductivity

The optical conductivity $\sigma(\omega)$ describes how a material conducts in response to an oscillating electric field. It connects to the complex dielectric function $\tilde{\varepsilon}(\omega)$ through:

$\tilde{\varepsilon}(\omega) = 1 + \frac{i\sigma(\omega)}{\varepsilon_0 \omega}$

• The real part $\sigma_1(\omega)$ represents dissipation (energy absorbed from the field).
• The imaginary part $\sigma_2(\omega)$ represents the reactive (out-of-phase) response.

Optical conductivity is particularly useful for metals and strongly correlated systems, where it reveals information about carrier dynamics, interband transitions, and collective excitations that complement what you learn from the refractive index alone.

Applications of absorption and emission

The physics of absorption and emission underpins a wide range of technologies. Here are the major ones you should know.

Photovoltaic devices and solar cells

Solar cells convert photon energy to electrical energy through the photovoltaic effect:

1. A photon with $\hbar\omega \geq E_g$ is absorbed, creating an electron-hole pair.
2. The built-in electric field at a p-n junction separates the electron and hole.
3. Carriers are collected at the electrodes, driving current through an external circuit.

The efficiency depends on matching the bandgap to the solar spectrum. The Shockley-Queisser limit sets the maximum efficiency for a single-junction cell at about 33% (for $E_g \approx 1.34$ eV). Common materials include Si, GaAs, CdTe, and perovskites (e.g., methylammonium lead iodide).

Light-emitting diodes (LEDs)

LEDs generate light through electroluminescence: electrons and holes are injected from opposite sides of a p-n junction, and their radiative recombination produces photons.

The emission wavelength is set by the bandgap:

• GaN/InGaN: blue and green ($\lambda \approx$ 450–530 nm)
• AlGaInP: red and amber ($\lambda \approx$ 590–650 nm)
• OLEDs: organic emitters covering the full visible range

White LEDs typically use a blue GaN chip coated with a yellow phosphor (Ce:YAG), and the combination of blue + yellow appears white. Quantum wells are used in the active region to increase carrier confinement and radiative efficiency.

Lasers and optical amplifiers

A laser requires three ingredients:

1. A gain medium that can be pumped to achieve population inversion.
2. An optical cavity (usually two mirrors) that provides feedback by bouncing photons back through the gain medium.
3. A pumping mechanism (electrical current, optical excitation, or chemical reaction) to maintain population inversion.

Stimulated emission in the gain medium amplifies light at the lasing wavelength, producing coherent, monochromatic, and highly directional output. Optical amplifiers (e.g., erbium-doped fiber amplifiers used in telecom) use the same stimulated emission process but without a cavity, amplifying a signal in a single pass.

Examples: semiconductor diode lasers (telecom, optical storage), Ti:sapphire lasers (ultrafast spectroscopy), fiber lasers (industrial cutting).

Photodetectors and optical sensors

Photodetectors convert light back into electrical signals. The basic mechanism is photon absorption followed by carrier collection, similar to a solar cell but optimized for speed and sensitivity rather than power generation.

Key performance metrics:

• Responsivity: photocurrent per unit optical power (A/W)
• Dark current: current flowing without illumination (lower is better)
• Bandwidth: how fast the detector can respond to modulated light

Examples include Si photodiodes (visible), InGaAs photodiodes (near-IR telecom), and avalanche photodiodes (APDs) that provide internal gain. Optical sensors extend this concept by detecting changes in absorption or emission caused by external stimuli like temperature, pressure, or chemical species.

Spectroscopic techniques for material characterization

Spectroscopy uses absorption and emission to probe what's happening inside a material.

• UV-Vis absorption spectroscopy: Measures wavelength-dependent absorption to determine bandgaps, defect levels, and impurity concentrations.
• FTIR (Fourier-transform infrared spectroscopy): Probes vibrational modes and free carrier absorption in the infrared.
• Photoluminescence (PL) spectroscopy: Excites the material optically and measures the emission spectrum. Reveals radiative recombination channels, defect states, and material quality. Time-resolved PL gives carrier lifetimes.
• Raman spectroscopy: Measures inelastic light scattering to probe phonon modes and crystal structure.
• Ellipsometry: Measures changes in light polarization upon reflection to extract $n$ and $\kappa$ across a wide spectral range.

Each technique gives complementary information, and combining several of them builds a complete picture of a material's electronic and optical properties.