Electron-hole pair

An electron-hole pair is the basic excitation of a semiconductor: an electron promoted from the full valence band into the empty conduction band, and the vacancy it leaves in the valence band. The vacancy, called a hole, behaves as a particle of positive charge $+e$ with its own effective mass, and it moves through the crystal as neighboring valence electrons fill it in turn. The electron and hole are created together, so generation always adds equal numbers of each. Creating a pair takes at least the bandgap energy: 1.12 eV in silicon, 1.42 eV in GaAs and 0.75 eV in lattice-matched InGaAs at room temperature. Photodetectors, solar cells, LEDs and semiconductor lasers all work by generating, separating or recombining pairs.

Generation by light

A photon is absorbed across the gap when its energy $h\nu$, the photon energy, exceeds $E_g$. A photon at 1550 nm carries 0.800 eV, enough for InGaAs and too little for silicon or GaAs; the GaAs absorption edge lies near 870 nm. Excess energy above the gap goes into the kinetic energy of the carriers and is lost to the lattice as heat within a picosecond or so: a 532 nm photon (2.33 eV) absorbed in silicon leaves 1.21 eV to be thermalized. Whether absorption needs a phonon to conserve momentum depends on the band structure, covered under direct and indirect bandgaps; silicon's indirect gap is why it absorbs weakly near its edge, with an absorption length of about 20 µm at 850 nm.

Each absorbed photon creates at most one pair at ordinary photon energies. In a photodiode the junction field separates the pair before it recombines, and the electron and hole together deliver one electron charge to the external circuit. If a fraction $\eta$ of incident photons yields a collected pair, the quantum efficiency, the responsivity is

$$\mathcal{R} = \frac{\eta\, e}{h\nu} = \frac{\eta\, e\,\lambda}{hc}.$$

At 1550 nm with $\eta$ = 1 this is 1.25 A/W. One microwatt at that wavelength is $7.80 \times 10^{12}$ photons per second, and the same number of pairs per second gives 1.25 µA.

Ionizing radiation and avalanche gain

Charged particles and X-rays create many pairs, each through a cascade of secondary excitations. The average energy spent per pair is larger than the gap because part of it goes into phonons; in silicon it is about 3.6 eV, roughly three times the bandgap. A 5.9 keV X-ray absorbed in a silicon detector therefore produces about 1640 pairs, which is the basis of energy-resolving X-ray detectors.

In an avalanche photodiode, the high field accelerates carriers until they create additional pairs by impact ionization. The multiplication gain $M$ is a random process, so it raises the shot noise by the excess noise factor in addition to the gain.

Recombination

A pair disappears when the electron falls back into the hole. In radiative recombination the energy leaves as a photon near the gap energy, the process behind LEDs and laser diodes, and it is efficient only in direct-gap materials. Non-radiative recombination, through defect states (Shockley-Read-Hall) or by the three-carrier Auger process, converts the energy to heat. The average time a pair survives is the carrier lifetime: a few nanoseconds in the active region of a III-V laser at threshold, and up to milliseconds in high-purity silicon.

Excitons

The electron and hole attract each other through the Coulomb force, and they can form a bound, hydrogen-like state called an exciton, with energy slightly below the gap. Its binding energy is reduced from hydrogen's 13.6 eV by the small effective masses and the large dielectric constant of the crystal: about 4.2 meV in GaAs, about 15 meV in silicon and about 60 meV in ZnO. At 300 K the thermal energy $k_BT$ is 25.85 meV, so excitons in GaAs and silicon are mostly ionized into free pairs at room temperature, while in ZnO and in quantum wells, where confinement raises the binding energy, sharp excitonic absorption and emission lines persist.

Common questions

What is a hole in a semiconductor?

A hole is the empty state left in a nearly full valence band when an electron leaves it. The collective motion of the remaining electrons is described as the motion of a single positive charge, the hole, with its own effective mass and mobility.

Can one photon create more than one electron-hole pair?

At visible and infrared photon energies, no: the excess energy is lost as heat. Photons with several times the gap energy can create a second pair by impact ionization, a small effect in bulk silicon in the ultraviolet, and X-rays create thousands. Avalanche gain in a detector multiplies pairs after absorption.

What is the difference between an electron-hole pair and an exciton?

A free pair consists of an electron and a hole moving independently. An exciton is the pair bound by its Coulomb attraction, with an energy just below the bandgap; it carries no net charge and does not contribute to current until it is separated.

References: B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007); C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005); G. F. Knoll, Radiation Detection and Measurement, 4th ed. (Wiley, 2010).