Carrier generation and 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[3].

This type of recombination depends on the density of electrons and holes in the excited state, denoted by ${\displaystyle n(t)}$ and ${\displaystyle p(t)}$ respectively. Let us represent the radiative recombination as ${\displaystyle R_{r}}$ 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 G_L:

$${\displaystyle G=G_{0}+G_{L}}$$

Here we will consider the case in which there is no illumination on the semiconductor. Therefore ${\displaystyle G_{L}=0}$ and ${\displaystyle G=G_{0}}$, and we can express the change in carrier density as a function of time as

$${\displaystyle {dn \over dt}=G-R_{r}=G_{0}-R_{r}}$$

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 R_r should be proportional to np:

$${\displaystyle R_{r}\propto np}$$

and we add a proportionality constant B_r to eliminate the ${\displaystyle \propto }$ sign:

$${\displaystyle R_{r}=B_{r}np}$$

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 ${\displaystyle R_{0}}$ must be exactly balanced by the thermal generation rate ${\displaystyle G_{0}}$.[4]

Therefore：

$${\displaystyle R_{0}=G_{0}=B_{r}n_{0}p_{0}}$$

where ${\displaystyle n_{0}}$ and ${\displaystyle p_{0}}$ are the equilibrium carrier densities. Using the mass action law ${\displaystyle np=n_{i}^{2}}$，with ${\displaystyle n_{i}}$ being the intrinsic carrier density, we can rewrite it as

$${\displaystyle R_{0}=G_{0}=B_{r}n_{0}p_{0}=B_{r}n_{i}^{2}}$$

The non-equilibrium carrier densities are given by [5]

$${\displaystyle n=n_{0}+\Delta n,}$$

$${\displaystyle p=p_{0}+\Delta p}$$

Then the new recombination rate ${\displaystyle R_{\text{net}}}$ becomes,[4][5]

$${\displaystyle R_{\text{net}}=R_{r}-G_{0}=B_{r}np-G_{0}=B_{r}(n_{0}+\Delta n)(p_{0}+\Delta p)-G_{0}}$$

Because ${\displaystyle n_{0}\gg \Delta n}$ and ${\displaystyle p_{0}\gg \Delta p}$, we can say that ${\displaystyle \Delta n\Delta p\approx 0}$

${\displaystyle R_{\text{net}}=B_{r}(n_{0}+\Delta n)(p_{0}+\Delta p)-G_{0}=B_{r}(n_{0}p_{0}+\Delta np_{0}+\Delta pn_{0})-B_{r}n_{i}^{2}}$

${\displaystyle R_{\text{net}}=B_{r}(n_{i}^{2}+\Delta np_{0}+\Delta pn_{0}-n_{i}^{2})}$

${\displaystyle R_{\text{net}}=B_{r}(\Delta np_{0}+\Delta pn_{0})}$

In an n-type semiconductor,

${\displaystyle p_{0}\ll n_{0}}$ and ${\displaystyle \Delta p\ll n_{0}}$

thus

${\displaystyle R_{net}\approx B_{r}\Delta pn_{0}}$

Net recombination is the rate at which excess holes ${\displaystyle \Delta p}$ disappear

${\displaystyle {\frac {d\Delta p}{dt}}=-R_{\text{net}}\approx -B_{r}\Delta pn_{0}}$

Solve this differential equation to get a standard exponential decay

${\displaystyle \Delta p=p_{\max }e^{-B_{r}n_{0}t}}$

where p_max is the maximum excess hole concentration when t = 0. (It can be proved that ${\displaystyle p_{\max }={\frac {G_{L}}{Bn_{0}}}}$, but here we will not discuss that).

When ${\displaystyle t={\frac {1}{Bn_{0}}}}$, all of the excess holes will have disappeared. Therefore, we can define the lifetime of the excess holes in the material ${\displaystyle \tau _{p}={\frac {1}{Bn_{0}}}}$

So the lifetime of the minority carrier is dependent upon the majority carrier concentration.

Stimulated emission is a process in which an incident photon interacts with an excited electron causing it to recombine and emit a photon with the same properties of the incident one, 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 ${\displaystyle W_{12}}$and the stimulated emission rate ${\displaystyle W_{21}}$are the same.[6] Else if level 1 and level 2 are ${\displaystyle g_{1}}$-fold and ${\displaystyle g_{2}}$-fold degenerate respectively, the new relation is:

$${\displaystyle g_{1}W_{12}=g_{2}W_{21}.}$$

Trap emission is a multistep process wherein a carrier falls into defect-related wavestates 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]

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[9] and Robert N. Hall,[10] who published it in 1952.

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 conduction band, they can temporarily immobilize excited electrons or in other words, they are electron traps. On the other hand, if their energy lies close to the valence band they become hole 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 k_BT 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.

Electron and hole trapping in the Shockley-Read-Hall 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 ${\displaystyle p(n)}$is replaced by the density of trapped electrons/holes ${\displaystyle N_{t}(1-f_{t})}$.

$${\displaystyle R_{nt}=B_{n}nN_{t}(1-f_{t})}$$

Where ${\displaystyle N_{t}}$ is the density of trap states and ${\displaystyle f_{t}}$ is the probability of that occupied state. Considering a material containing both types of traps, we can define two trapping coefficients ${\displaystyle B_{n},B_{p}}$ and two de-trapping coefficients ${\displaystyle G_{n},G_{p}}$. In equilibrium, both trapping and de-trapping should be balanced (${\displaystyle R_{nt}=G_{n}}$ and ${\displaystyle R_{pt}=G_{p}}$). Then, the four rates as a function of ${\displaystyle f_{t}}$become:

$${\displaystyle {\begin{array}{l l}R_{nt}=B_{n}nN_{t}(1-f_{t})&G_{n}=B_{n}n_{t}N_{t}f_{t}\\R_{pt}=B_{p}pN_{t}f_{t}&G_{p}=B_{p}p_{t}N_{t}(1-f_{t})\end{array}}}$$

Where ${\displaystyle n_{t}}$ and ${\displaystyle p_{t}}$ 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: ${\displaystyle R_{nt}-G_{n}=R_{pt}-G_{p}}$. This eliminates the occupation probability ${\displaystyle f_{t}}$ and leads to the Shockley-Read-Hall expression for the trap-assisted recombination:

$${\displaystyle R={\frac {np}{\tau _{n}(p+p_{t})+\tau _{p}(n+n_{t})}}}$$

Where the average lifetime for electrons and holes are defined as:[12]

$${\displaystyle \tau _{n}={\frac {1}{B_{n}N_{t}}},\quad \tau _{p}={\frac {1}{B_{p}N_{t}}}.}$$

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 ${\displaystyle R_{A}}$ and thermal generation rate ${\displaystyle G_{0}}$ equal each other[13]

$${\displaystyle R_{A0}=G_{0}=C_{n}n_{0}^{2}p_{0}+C_{p}n_{0}p_{0}^{2}}$$

where ${\displaystyle C_{n},C_{p}}$ are the Auger capture probabilities. The non-equilibrium Auger recombination rate ${\displaystyle r_{A}}$ and resulting net recombination rate ${\displaystyle U_{A}}$ under steady-state conditions are[13]

$${\displaystyle r_{A}=C_{n}n^{2}p+C_{p}np^{2}\,,\quad R_{A}=r_{A}-G_{0}=C_{n}\left(n^{2}p-n_{0}^{2}p_{0}\right)+C_{p}\left(np^{2}-n_{0}p_{0}^{2}\right)\,.}$$

The Auger lifetime ${\displaystyle \tau _{A}}$ is given by[14]

$${\displaystyle \tau _{A}={\frac {\Delta n}{R_{A}}}={\frac {1}{n^{2}C_{n}+2n_{i}^{2}(C_{n}+C_{p})+p^{2}C_{p}}}\,.}$$

The mechanism causing LED efficiency drop was identified in 2007 as Auger recombination, which met with a mixed reaction.[15] In 2013, an experimental study claimed to have identified Auger recombination as the cause of efficiency droop.[16] However, it remains disputed whether the amount of Auger loss found in this study is sufficient to explain the droop. Other frequently quoted evidence against Auger as the main droop-causing mechanism is the low-temperature dependence of this mechanism, which is the opposite of that found for the drop.

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]

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 ${\displaystyle \gamma ={\tfrac {q}{\varepsilon }}\mu }$.