Auger Recombination in Semiconductors: Explanation, Equation and Rate

What is Auger recombination?

Auger recombination is a non-radiative three-carrier recombination process in semiconductors. An electron and a hole recombine, but instead of releasing their energy as a photon, the energy is transferred to a third carrier, which may be another electron or a hole. Because three carriers are involved, the Auger recombination rate increases very rapidly with carrier density and can become an important loss mechanism under high-injection conditions.

To understand how Auger recombination differs from radiative processes, it is useful to first consider free-to-free recombination. As shown on the left-hand side of ??, an electron relaxes into a hole and emits a photon, thereby conserving energy. The recombination rate is typically written as \(R = k np\), since the process requires both an electron and a hole to be present.

However, light does not have to be emitted for energy to be conserved. In Auger recombination, as shown on the right-hand side of ??, an electron again relaxes into a hole, but no photon is produced. Instead, the recombination energy is transferred to a third carrier. In the electron-assisted Auger process, this energy is given to another electron, which is excited higher into the conduction band.

This excited (“hot”) electron does not remain at high energy. At some later time, it relaxes back towards the bottom of the conduction band by emitting phonons. In this two-step process-recombination followed by thermal relaxation-energy is conserved without the emission of light. This is why Auger recombination is referred to as a dark recombination mechanism.

Why is Auger recombination a cubic process?

The presence of this third carrier fundamentally changes the carrier-density dependence of the recombination rate. While the initial recombination event still requires an electron and a hole (giving an \(np\) dependence), an additional carrier must be present to absorb the energy. For the electron-assisted process, the rate is therefore proportional to \(n^{2}p\), and for the hole-assisted process it is proportional to \(np^{2}\). Auger recombination is thus a cubic process in carrier density.

Auger recombination equation

The Auger recombination rate contains two contributions. In the electron-assisted Auger process, a second electron receives the recombination energy, giving a contribution proportional to \(n^{2}p\):

\[ R_{eeh} = C^{AU}_{n} n^{2}p \]

In the hole-assisted Auger process, a second hole receives the energy, giving a contribution proportional to \(np^{2}\):

\[ R_{ehh} = C^{AU}_{p} np^{2} \]

Including both processes and subtracting the equilibrium contribution, the net Auger recombination rate used in OghmaNano is

\[R^{AU} = \big(C^{AU}_{n} n + C^{AU}_{p} p\big)\,(np - n_{0}p_{0}) \]

where \(n\) and \(p\) are the electron and hole densities, \(n_{0}\) and \(p_{0}\) are their equilibrium values, and \(C^{AU}_{n}\) and \(C^{AU}_{p}\) are the electron- and hole-assisted Auger coefficients with units of m⁶s⁻¹. These coefficients describe the probability that the recombination energy of an electron–hole pair is transferred to a third carrier rather than emitted as light.

Because of its cubic dependence on carrier density, Auger recombination is typically negligible at low injection, but becomes increasingly important at high carrier densities, where it can dominate recombination and act as a significant loss mechanism.

What is the Auger recombination coefficient?

The Auger recombination coefficients \(C^{AU}_{n}\) and \(C^{AU}_{p}\) determine the strength of the electron-assisted and hole-assisted Auger processes. Their units are m⁶s⁻¹, although values are also commonly reported in cm⁶s⁻¹. The coefficients depend on the semiconductor material, its band structure, temperature and the particular Auger transition involved.

A larger Auger coefficient means that Auger recombination becomes significant at a lower carrier density, whereas a smaller coefficient allows higher carrier densities to be reached before Auger losses become important.

Auger recombination in k-space: energy and momentum conservation

Above, we introduced recombination using the most common picture found in introductory physics texts, shown in ??. This description focuses on energy, but in all recombination processes both energy and momentum (the k-vector) must be conserved. A more complete representation is therefore given in energy–momentum (E–k) space, shown in ??, where energy is plotted along the vertical axis and momentum (k) along the horizontal axis.

On the left-hand side of ??, free-to-free recombination is shown. An electron relaxes directly into a hole at the same k-vector, so there is no change in momentum. The emitted photon carries negligible crystal momentum, and thus both energy and momentum are conserved in a direct transition.

On the right-hand side, Auger recombination is shown. In this case, an electron recombines with a hole, but the transition does not occur at a single k-vector. Instead, momentum is redistributed between carriers: the recombining electron changes its k-state, while a second electron is excited to a higher energy state and also shifts in k-space. In this way, both energy and momentum are conserved without the emission of a photon.

The excited (“hot”) electron produced in the Auger process subsequently relaxes back towards the bottom of the conduction band by emitting phonons. This delayed relaxation converts the recombination energy into heat, completing the non-radiative Auger process.

Auger vs radiative and SRH recombination

When is Auger recombination important?

Auger recombination becomes particularly important at high carrier densities. Because its rate increases approximately with the cube of carrier density, Auger recombination can rapidly overtake first-order trap-assisted and second-order radiative recombination processes as injection increases. This makes it a critical loss mechanism in III–V semiconductors (e.g. GaAs, InGaN), high-brightness LEDs, and semiconductor lasers, where carrier densities are intentionally driven to high levels during operation. It is also important in silicon solar cells under strong illumination (e.g. concentration) and has been identified as a limiting factor in some inorganic perovskites under high-injection conditions such as pulsed-laser excitation.

By contrast, Auger recombination is usually much weaker in devices operating at relatively low carrier density. In many organic semiconductor and conventional thin-film solar-cell devices under approximately 1-sun illumination, trap-assisted or bimolecular recombination is therefore much more important than Auger recombination. In these regimes, recombination is instead dominated by trap-assisted (SRH) or bimolecular (free-to-free) pathways.

In OghmaNano, the Auger coefficients \(C^{AU}_{n}\) and \(C^{AU}_{p}\) can be set directly in the Electrical parameter editor. This allows Auger recombination to be included when modelling devices in which high carrier densities are expected to impact efficiency and performance.