In solid-state physics of semiconductors, carrier generation and carrier recombination are processes by which mobile charge carriers (electrons and electron holes) are created and eliminated. Carrier generation and recombination processes are fundamental to the operation of many optoelectronic semiconductor devices, such as photodiodes, light-emitting diodes and laser diodes. They are also critical to a full analysis of p-n junction devices such as bipolar junction transistors and p-n junction diodes.
The electron–hole pair is the fundamental unit of generation and recombination in inorganic semiconductors, corresponding to an electron transitioning between the valence band and the conduction band where generation of an electron is a transition from the valence band to the conduction band and recombination leads to a reverse transition. Slow recombination rates (or equivalently, long carrier lifetimes) are a favorable attribute of the semiconductors that comprise solar cells.
Overview
Like other solids, semiconductor materials have an electronic band structure determined by the crystal properties of the material. Energy distribution among electrons is described by the Fermi level and the temperature of the electrons. At absolute zero temperature, all of the electrons have energy below the Fermi level; but at non-zero temperatures the energy levels are filled following a Fermi-Dirac distribution.
In undoped semiconductors the Fermi level lies in the middle of a forbidden band or band gap between two allowed bands called the valence band and the conduction band. The valence band, immediately below the forbidden band, is normally very nearly completely occupied. The conduction band, above the Fermi level, is normally nearly completely empty. Because the valence band is so nearly full, its electrons are not mobile, and cannot flow as electric current.
However, if an electron in the valence band acquires enough energy to reach the conduction band as a result of interaction with other electrons, holes, photons, or the vibrating crystal lattice itself, it can flow freely among the nearly empty conduction band energy states. Furthermore, it will also leave behind a hole that can flow like a physically charged particle.
Carrier generation describes processes by which electrons gain energy and move from the valence band to the conduction band, producing two mobile carriers; while recombination describes processes by which a conduction band electron loses energy and re-occupies the energy state of an electron hole in the valence band.
These processes must conserve quantized energy crystal momentum, and the vibrating lattice which plays a large role in conserving momentum as in collisions, photons can transfer very little momentum in relation to their energy.
Relation between generation and recombination
Recombination and generation are always happening in semiconductors, both optically and thermally. As predicted by thermodynamics, a material at thermal equilibrium will have generation and recombination rates that are balanced so that the net charge carrier density remains constant. The resulting probability of occupation of energy states in each energy band is given by Fermi–Dirac statistics.
The product of the electron and hole densities ( and ) is a constant at equilibrium, maintained by recombination and generation occurring at equal rates. When there is a surplus of carriers (i.e., ), the rate of recombination becomes greater than the rate of generation, driving the system back towards equilibrium. Likewise, when there is a deficit of carriers (i.e., ), the generation rate becomes greater than the recombination rate, again driving the system back towards equilibrium. As the electron moves from one energy band to another, the energy and momentum that it has lost or gained must go to or come from the other particles involved in the process (e.g. photons, electron, or the system of vibrating lattice atoms).
Carrier generation When light interacts with a material, it can either be absorbed (generating a pair of free carriers or an exciton) or it can stimulate a recombination event. The generated photon has similar properties to the one responsible for the event. Absorption is the active process in photodiodes, solar cells and other semiconductor photodetectors, while stimulated emission is the principle of operation in laser diodes.
Besides light excitation, carriers in semiconductors can also be generated by an external electric field, for example in light-emitting diodes and transistors.
When light with sufficient energy hits a semiconductor, it can excite electrons across the band gap. This generates additional charge carriers, temporarily lowering the electrical resistance of materials. This higher conductivity in the presence of light is known as photoconductivity. This conversion of light into electricity is widely used in photodiodes.
Recombination mechanisms
Carrier recombination can happen through multiple relaxation channels. The main ones are band-to-band recombination, Shockley–Read–Hall (SRH) trap-assisted recombination, Auger recombination and surface recombination. These decay channels can be separated into radiative and non-radiative. The latter occurs when the excess energy is converted into heat by phonon emission after the mean lifetime , whereas in the former at least part of the energy is released by light emission or luminescence after a radiative lifetime . The carrier lifetime is then obtained from the rate of both type of events according to:
From which we can also define the internal quantum efficiency or quantum yield, as:
However, distinguishing between the contributions of different recombination mechanisms with lateral or depth-dependent resolution remains challenging, and improved experimental and analytical methods for resolving these mechanisms are still under active development.Sakowski, K. et al. “On method of estimating recombination rates by analysis of time-resolved luminescence.” Journal of Luminescence 269 (2024): 120473. https://doi.org/10.1016/j.jlumin.2024.120473
Radiative recombination
Band-to-band radiative recombination Band-to-band recombination is the name for the process of electrons jumping down from the conduction band to the valence band in a radiative manner. During band-to-band recombination, a form of spontaneous emission, the energy absorbed by a material is released in the form of photons. Generally these photons contain the same or less energy than those initially absorbed. This effect is how LEDs create light. Because the photon carries relatively little momentum, radiative recombination is significant only in direct bandgap materials. This process is also known as bimolecular recombination.
This type of recombination depends on the density of electrons and holes in the excited state, denoted by and respectively. Let us represent the radiative recombination as and the carrier generation rate as G.
Total generation is the sum of thermal generation G0 and generation due to light shining on the semiconductor GL:
Here we will consider the case in which there is no illumination on the semiconductor. Therefore and , and we can express the change in carrier density as a function of time as
Because the rate of recombination is affected by both the concentration of free electrons and the concentration of holes that are available to them, we know that Rr should be proportional to np:
and we add a proportionality constant Br to eliminate the sign:
If the semiconductor is in thermal equilibrium, the rate at which electrons and holes recombine must be balanced by the rate at which they are generated by the spontaneous transition of an electron from the valence band to the conduction band. The recombination rate must be exactly balanced by the thermal generation rate .
Therefore：
where and are the equilibrium carrier densities. Using the mass action law ，with being the intrinsic carrier density, we can rewrite it as
The non-equilibrium carrier densities are given by
Then the new recombination rate becomes,
Because and , we can say that
In an n-type semiconductor,
and
thus
Net recombination is the rate at which excess holes disappear
Solve this differential equation to get a standard exponential decay
where pmax is the maximum excess hole concentration when t = 0. (It can be proved that , but here we will not discuss that).
When , many (63.21...%) of the excess holes will have disappeared. Therefore, we can define the lifetime of the excess holes in the material
So the lifetime of the minority carrier is dependent upon the majority carrier concentration.
Stimulated emission
Stimulated emission is a process where an incident photon interacts with an excited electron causing it to recombine and emit a photon with the same properties as the incident photon, in terms of phase, frequency, polarization, and direction of travel. Stimulated emission together with the principle of population inversion are at the heart of operation of lasers and masers. It has been shown by Einstein at the beginning of the twentieth century that if the excited and the ground level are non degenerate then the absorption rate and the stimulated emission rate are the same. Else if level 1 and level 2 are -fold and -fold degenerate respectively, the new relation is:
Trap emission Trap emission is a multistep process wherein a carrier falls into defect-related wave states in the middle of the bandgap. A trap is a defect capable of holding a carrier. The trap emission process recombines electrons with holes and emits photons to conserve energy. Due to the multistep nature of trap emission, a phonon is also often emitted. Trap emission can proceed by use of bulk defects or surface defects.
Non-radiative recombination Non-radiative recombination is a process in phosphors and semiconductors, whereby charge carriers recombine releasing phonons instead of photons. Non-radiative recombination in optoelectronics and phosphors is an unwanted process, lowering the light generation efficiency and increasing heat losses.
Non-radiative life time is the average time before an electron in the conduction band of a semiconductor recombines with a hole. It is an important parameter in optoelectronics where radiative recombination is required to produce a photon; if the non-radiative life time is shorter than the radiative, a carrier is more likely to recombine non-radiatively. This results in low internal quantum efficiency.
Shockley–Read–Hall (SRH) In Shockley-Read-Hall recombination (SRH), also called trap-assisted recombination, the electron in transition between bands passes through a new energy state (localized state) created within the band gap by a dopant or a defect in the crystal lattice; such energy states are called traps. Non-radiative recombination occurs primarily at such sites. The energy is exchanged in the form of lattice vibration, a phonon exchanging thermal energy with the material.
Since traps can absorb differences in momentum between the carriers, SRH is the dominant recombination process in silicon and other indirect bandgap materials. However, trap-assisted recombination can also dominate in direct bandgap materials under conditions of very low carrier densities (very low level injection) or in materials with high density of traps such as perovskites. The process is named after William Shockley, William Thornton Read and Robert N. Hall, who published it in 1952.
Types of traps
Electron traps vs. hole traps