Carrier generation and recombination
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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.
Overview
Like other solids, semiconductor materials have an
In undoped semiconductors the Fermi level lies in the middle of a forbidden band or
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 physicaly 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.[1] 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,
From which we can also define the internal quantum efficiency or quantum yield, as:
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
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:
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 .[4]
Therefore:
The non-equilibrium carrier densities are given by [5]
Then the new recombination rate becomes,[4][5]
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 , all 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
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[7] or surface defects.[8]
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
Shockley–Read–Hall (SRH)
In Shockley-Read-Hall recombination (SRH), also called trap-assisted recombination, the electron in transition between
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[9] and Robert N. Hall,[10] who published it in 1952.
Types of traps
Electron traps vs. hole traps
Even though all the recombination events can be described in terms of electron movements, it is common to visualize the different processes in terms of excited electron and the electron holes they leave behind. In this context, if trap levels are close to the
Shallow traps vs. deep traps
The distinction between shallow and deep traps is commonly made depending on how close electron traps are to the conduction band and how close hole traps are to the valence band. If the difference between trap and band is smaller than the thermal energy kBT it is often said that it is a shallow trap. Alternatively, if the difference is larger than the thermal energy, it is called a deep trap. This difference is useful because shallow traps can be emptied more easily and thus are often not as detrimental to the performance of optoelectronic devices.
SRH model
In the SRH model, four things can happen involving trap levels:[11]
- An electron in the conduction band can be trapped in an intragap state.
- An electron can be emitted into the conduction band from a trap level.
- A hole in the valence band can be captured by a trap. This is analogous to a filled trap releasing an electron into the valence band.
- A captured hole can be released into the valence band. Analogous to the capture of an electron from the valence band.
When carrier recombination occurs through traps, we can replace the valence density of states by that of the intragap state.[12] The term is replaced by the density of trapped electrons/holes .
Where is the density of trap states and is the probability of that occupied state. Considering a material containing both types of traps, we can define two trapping coefficients and two de-trapping coefficients . In equilibrium, both trapping and de-trapping should be balanced ( and ). Then, the four rates as a function of become:
Where and are the electron and hole densities when the quasi Fermi level matches the trap energy. In steady-state condition, the net recombination rate of electrons should match the net recombination rate for holes, in other words: . This eliminates the occupation probability and leads to the Shockley-Read-Hall expression for the trap-assisted recombination:
Where the average lifetime for electrons and holes are defined as:[12]
Auger recombination
In Auger recombination the energy is given to a third carrier which is excited to a higher energy level without moving to another energy band. After the interaction, the third carrier normally loses its excess energy to thermal vibrations. Since this process is a three-particle interaction, it is normally only significant in non-equilibrium conditions when the carrier density is very high. The Auger effect process is not easily produced, because the third particle would have to begin the process in the unstable high-energy state.
In thermal equilibrium the Auger recombination and thermal generation rate equal each other[13]
where are the Auger capture probabilities. The non-equilibrium Auger recombination rate and resulting net recombination rate under steady-state conditions are[13]
The Auger lifetime is given by[14]
The mechanism causing
Surface recombination
Trap-assisted recombination at the surface of a semiconductor is referred to as surface recombination. This occurs when traps at or near the surface or interface of the semiconductor form due to dangling bonds caused by the sudden discontinuation of the semiconductor crystal. Surface recombination is characterized by surface recombination velocity which depends on the density of surface defects.[17] In applications such as solar cells, surface recombination may be the dominant mechanism of recombination due to the collection and extraction of free carriers at the surface. In some applications of solar cells, a layer of transparent material with a large band gap, also known as a window layer, is used to minimize surface recombination. Passivation techniques are also employed to minimize surface recombination.[18]
Langevin recombination
For free carriers in low-mobility systems, the recombination rate is often described with the Langevin recombination rate.[19] The model is often used for disordered systems such as organic materials (and is hence relevant for organic solar cells[20]) and other such systems. The Langevin recombination strength is defined as .
See also
References
- S2CID 19785166.
- ISBN 9780199588336
- .
- ^ ISBN 978-0-387-28893-2.
- ^ OCLC 964380194.
- OCLC 249201544.
- .
- ISSN 1520-6106.
- .
- ^ Hall, R.N. (1951). "Germanium rectifier characteristics". Physical Review. 83 (1): 228.
- OCLC 964380194.
- ^ ISBN 9781782622932
- ^ ISBN 978-0-387-28893-2.
- ISBN 978-0-387-28893-2.
- ^ Stevenson, Richard (August 2009) "The LED's Dark Secret: Solid-state lighting won't supplant the lightbulb until it can overcome the mysterious malady known as droop". IEEE Spectrum
- ^ Justin Iveland; Lucio Martinelli; Jacques Peretti; James S. Speck; Claude Weisbuch. "Cause of LED Efficiency Droop Finally Revealed". Physical Review Letters, 2013. Science Daily. Retrieved 23 April 2013.
- ISBN 978-1-86094-340-9.
- ISSN 0021-8979.
- ^ "Recombination in low mobility semiconductors: Langevin theory". 4 April 2008.
- PMID 24423376.
Further reading
- N.W. Ashcroft and N.D. Mermin, Solid State Physics, Brooks Cole, 1976