# Carrier Recombination in Semiconductors

This is a continuation from the previous tutorial - electron and hole concentrations.

In a semiconductor, electrons in the conduction bands and holes in the valence bands can be generated through many mechanisms, including thermal excitation, current injection, and optical excitation.

Meanwhile, an electron in a conduction band and a hole in a valence band can be eliminated together through a recombination process. In an equilibrium state, electron-hole generation is exactly balanced by electron-hole recombination.

Recombination Processes

There are many different electron-hole recombination processes. Based on the mechanisms responsible for these processes, they are classified into three general categories:

2. The bimolecular recombination processes
3. The Auger recombination processes

These basic mechanisms are schematically illustrated in Figure 12-4 below. A Shockley-Read process involves one carrier at a time; a bimolecular process takes place with an electron and a hole simultaneously; an Auger process is a three-body process with three participating carriers at the same time.

Notwithstanding these differences, any of the three types of recombination processes ends with the annihilation of one electron with one hole. Depending on whether or not electron-hole recombination in a particular process results in the emission of electromagnetic radiation, a process can also be classified as a radiative recombination process or as a nonradiative recombination process.

A Shockley-Read process is a recombination process that takes place through the capture of one carrier, either an electron or a hole, at a time by recombination centers, as illustrated in Figure 12-4(a).

A recombination center is created by an imperfection, such as a defect or an impurity, in a semiconductor. Its energy level lies somewhere in the bandgap of the semiconductor.

The density of the recombination centers in a given piece of semiconductor is determined by the density of the defects and impurities in the semiconductor and is independent of the electron or hole concentration.

There are four processes associated with the recombination centers: electron capture, hole capture, electron emission, and hole emission.

A recombination center is available for the capture of an electron only if it is not already occupied by an electron, whereas it is available for the capture of a hole only if it is occupied by an electron.

Two basic time constants can be defined: $$\tau_\text{e0}^{-1}=v_\text{th}\sigma_\text{e}N_\text{r}$$ for electrons and $$\tau_\text{h0}^{-1}=v_\text{th}\sigma_\text{h}N_\text{r}$$ for holes, where $$v_\text{th}$$ is the thermal velocity of an electron, $$\sigma_\text{e}$$ and $$\sigma_\text{h}$$ are the capture cross sections of the recombination center for an electron and a hole, respectively, and $$N_\text{r}$$ is the concentration of the recombination centers.

[Note that $$\sigma_\text{e}$$ here is not to be confused with the emission cross section of a laser transition.]

The physical meaning of $$\tau_\text{e0}$$ is that $$n/\tau_\text{e0}$$ is the electron capture rate if all recombination centers are empty and are thus able to trap electrons, whereas that of $$\tau_\text{h0}$$ is that $$p/\tau_\text{h0}$$ is the hole capture rate if all recombination centers are occupied and are thus able to trap holes.

In many situations, however, the recombination centers are neither all empty nor all occupied. In such situations, the Shockley-Read recombination lifetimes of electrons and holes are different from $$\tau_\text{e0}$$ and $$\tau_\text{h0}$$, respectively.

After a recombination center first captures a carrier, either an electron or a hole, a Shockley-Read electron-hole recombination process is completed only if the recombination center subsequently captures another carrier of opposite charge to result in the annihilation of an electron-hole pair.

A carrier can also be captured and then reemitted without completing the recombination process. The net Shockley-Read recombination rate for electrons is the difference of the electron capture rate and the electron emission rate by recombination centers, whereas that for holes is the difference of the hole capture rate and the hole emission rate.

Because only one carrier is captured at a time, the rates of electron capture and hole capture in the Shockley-Read recombination processes are linearly proportional to the total electron and hole concentrations, respectively. We therefore express the net electron recombination rate in the form of $$A_\text{e}(n-n_0)$$ and the net hole recombination rate in the form of $$A_\text{h}(p-p_0)$$.

Because a completed recombination process always results in the annihilation of an electron-hole pair, the net electron and hole recombination rates are necessarily equal: $$A_\text{e}(n-n_0)=A_\text{h}(p-p_0)$$ though $$A_\text{e}$$ and $$A_\text{h}$$ can be different if the excess electron concentration $$\Delta{n}=n-n_0$$ is different from the excess hole concentration $$\Delta{p}=p-p_0$$.

The net Shockley-Read recombination rate is found from balancing the capture and emission of electrons and holes in steady state to be

$\tag{12-48}R_\text{SR}=A_\text{e}(n-n_0)=A_\text{h}(p-p_0)=\frac{np-n_0p_0}{\tau_\text{h0}(n+n_1)+\tau_\text{e0}(p+p_1)}$

where $$n_1$$ and $$p_1$$ characterize the emission rates of the recombination centers for electrons and holes, respectively, and $$n_1p_1=n_0p_0=n_\text{i}^2(T)$$.

We see from this relation that the two coefficients $$A_\text{e}$$ and $$A_\text{h}$$ both vary with temperature, the doping condition of the semiconductor, and the excess carrier concentrations.

A Shockley-Read recombination process can be either radiative or nonradiative, depending on the type of recombination centers involved in the process.

Bimolecular process

A bimolecular recombination process always involves an electron an a hole at the same time. As illustrated in Figure 12-4(b), there are primarily two types of bimolecular electron-hole recombination processes: band-to-band recombination, which takes place between an electron in a conduction band and a hole in a valence band, and exciton recombination.

An electron-hole pair in a semiconductor can be held together by their Coulomb attraction to form an exciton like an electron-proton pair forming a hydrogen atom.

A free exciton is free to wander in the semiconductor; its energy is reduced by the energy needed to hold the electron and hole together and is slightly less than the bandgap of the semiconductor.

A bound exciton is localized and bound to an impurity center in the semiconductor; its energy depends on the properties of the impurity and is generally lower than that of a free exciton in the same semiconductor.

The bimolecular recombination processes have the same contribution to the rate of electron recombination and the rate of hole recombination. This rate is proportional to the product of the electron and hole concentrations and can be expressed as $$Bnp$$, where $$B$$ is the bimolecular recombination coefficient.

Bimolecular recombination processes are radiative processes.

Auger process

An Auger recombination process is a three-body process that requires the participation on each occasion of two electrons and one hole, with a rate of $$C_\text{e}n^2p$$, or one electron and two holes, with a rate of $$C_\text{h}np^2$$, where $$C_\text{e}$$ and $$C_\text{h}$$ are the Auger recombination coefficients.

The total Auger recombination rate for both electrons and holes is $$C_\text{e}n^2p+C_\text{h}np^2$$.

As illustrated in Figure 12-4(c), in an Auger recombination process, the energy released by band-to-band recombination of an electron and a hole is picked up by a third carrier, either another electron or another hole, as kinetic energy of the third carrier.

This energy is eventually converted to the thermal energy of the semiconductor lattice as the third carrier relaxes toward the band edge. Consequently, an Auger process is nonradiative.

Each recombination process has a corresponding inverse process for generating free electrons and holes. Irrespective of the absence or presence of an external excitation such as current injection or optical excitation, free electrons and holes are continuously generated by thermal excitation through these inverse processes.

The inverse processes of electron capture and hole capture in the Shockley-Read processes are electron emission and hole emission, respectively. An electron captured by a recombination center can be thermally reemitted back to the conduction band, and a hole captured by a recombination center can be thermally reemitted back to the valence band.

Similarly to the difference in electron and hole capture rates by the recombination centers, the processes of electron and hole emission by the recombination centers generally have different thermal generation rates for electrons and holes.

The thermal generation rates of electrons and holes in the Shockley-Read processes are already accounted for in the net Shockley-Read recombination rate given in (12-48).

The inverse process of band-to-band bimolecular recombination is the generation of an electron-hole pair by thermal excitation of a valence electron to the conduction band. This process has the same generation rate of $$Bn_0p_0$$ for electrons and holes as they are generated in pairs.

The inverse of an Auger process can also generate additional electrons and holes. This process also generates electrons and holes in pairs and has the same generation rate of $$C_\text{e}n_0^2p_0+C_\text{h}n_0p_0^2$$ for electrons and holes.

To summarize, the total thermal generation rates for electrons, $$G_\text{e}^0$$, and for holes, $$G_\text{h}^0$$, are generally different in the presence of recombination centers, but are the same if the density of the recombination centers is very small compared to the equilibrium electron and hole concentrations.

These thermal generation rates are a function of temperature and the properties of the semiconductor and its impurities.

From the above discussions, the net recombination rate for electrons and that for holes can be respectively expressed as

$\tag{12-49}R_\text{e}=A_\text{e}n+Bnp+C_\text{e}n^2p+C_\text{h}np^2-G_\text{e}^0$

and

$\tag{12-50}R_\text{h}=A_\text{h}p+Bnp+C_\text{e}n^2p+C_\text{h}np^2-G_\text{h}^0$

where $$G_\text{e}^0=A_\text{e}n_0+Bn_0p_0+C_\text{e}n_0^2p_0+C_\text{h}n_0p_0^2$$ and $$G_\text{h}^0=A_\text{h}p_0+Bn_0p_0+C_\text{e}n_0^2p_0+C_\text{h}n_0p_0^2$$.

Because electrons and holes always recombine in pairs, they must have the same net recombination rate:

$\tag{12-51}R=R_\text{e}=R_\text{h}$

The net bimolecular recombination rate is clearly the same for electrons and holes, and so is the net Auger recombination rate.

For the Shockley-Read processes, $$A_\text{e}n$$ and $$A_\text{h}p$$ might be different, but the difference is exactly balanced by the difference between $$A_\text{e}n_0$$ and $$A_\text{h}p_0$$. Therefore, as indicated in (12-48), the net Shockley-Read recombination rate is also the same for electrons and holes.

When a semiconductor is in thermal equilibrium with its environment, recombination of the carriers has to be exactly balanced by thermal generation of the carriers so that the electron and hole concentrations are maintained at their respective equilibrium values of $$n_0$$ and $$p_0$$. Therefore, the net recombination rate is zero, $$R=R_\text{e}=R_\text{h}=0$$, when a semiconductor is in thermal equilibrium.

In practice, the $$A$$, $$B$$, and $$C$$ coefficients that characterize the three basic recombination processes in (12-49) and (12-50) are not completely independent of the carrier concentrations.

Despite this fact, we can still see significant differences between the functional dependencies of the recombination rates on the carrier concentrations for the three different processes.

Besides, the $$A$$, $$B$$, and $$C$$ coefficients are generally different by many orders of magnitude, with $$A$$ often being the largest and $$C$$ being the smallest.

For these reasons, the significance of each of the three different recombination processes varies strongly with the concentrations of the carriers. Only the Shockley-Read process is important at low carrier concentrations, whereas the Auger process can be significant only at very high carrier concentrations. Between the two limits, the bimolecular recombination process can be the dominant recombination process.

The specific quantitative carrier concentrations for each process to be significant vary from one kind of semiconductor to another and from one specific sample to another, depending on many factors such as band structure, bandgap, type of impurity, doping concentrations, defect density, and temperature.

In general, however, the $$B$$ coefficients of direct-gap semiconductors such as GaAs and InP are much larger, by orders of magnitude, than those of indirect-gap semiconductors such as Si and Ge.

Therefore, carrier recombination in most indirect-gap semiconductors is predominantly nonradiative and is often completely characterized by the Shockley-Read process with a net recombination rate of $$R=R_\text{SR}$$ given in (12-48) for carrier concentrations up to a pretty high level.

When the electron and hole concentrations in a semiconductor are higher than their respective equilibrium concentrations, due to current injection or optical excitation for example, the excess carriers will relax toward their respective thermal equilibrium concentrations through recombination processes.

The relaxation time constant for excess electrons is the electron lifetime, defined as

$\tag{12-52}\tau_\text{e}=\frac{n-n_0}{R}$

and that of excess holes is the hole lifetime, defined as

$\tag{12-53}\tau_\text{h}=\frac{p-p_0}{R}$

From these relations, we find that

$\tag{12-54}\frac{\tau_\text{e}}{\tau_\text{h}}=\frac{\Delta{n}}{\Delta{p}}$

where $$\Delta{n}=n-n_0$$ and $$\Delta{p}=p-p_0$$ are the excess electron and hole concentrations, respectively.

The lifetime of the minority carriers in a semiconductor is called the minority carrier lifetime, and that of the majority carriers is called the majority carrier lifetime.

In an n-type semiconductor, $$\tau_\text{e}$$ is the majority carrier lifetime, and $$\tau_\text{h}$$ is the minority carrier lifetime. In a p-type semiconductor, $$\tau_\text{e}$$ becomes the minority carrier lifetime, and $$\tau_\text{h}$$ is the majority carrier lifetime.

When the density of the recombination centers is not small compared to the thermal-equilibrium carrier concentrations, the excess minority carrier density can be less than the excess majority carrier density. In this situation, $$n-n_0\ne{p}-p_0$$ and, consequently, $$\tau_\text{e}\ne\tau_\text{h}$$.

When the concentrations of free electrons and holes are much less than the density of the recombination centers, the majority carrier lifetime can be much greater than the minority carrier lifetime.

A sufficient condition for electrons and holes to have the same lifetime is that the electron and hole concentrations are both very large compared to the density of the recombination centers. When this condition is satisfied, $$n-n_0=p-p_0$$; then, we can define the excess carrier density as

$\tag{12-55}N=n-n_0=p-p_0$

which is also the density of excess free electron-hole pairs.

Then, the free electrons and the free holes have the same lifetime: $$\tau_\text{e}=\tau_\text{h}=\tau_\text{s}$$, which is the spontaneous carrier recombination lifetime of the excess electron-hole pairs given by

$\tag{12-56}\tau_\text{s}=\frac{N}{R}$

This relation is valid in practical operating conditions of semiconductor lasers and light-emitting diodes.

Under the condition of (12-55), we also find from (12-48) that $$A_\text{e}=A_\text{h}=A$$. By applying (12-55) to (12-49), (12-50), and (12-51) and then using (12-56), we find that

\tag{12-57}\begin{align}\frac{1}{\tau_\text{s}}=&A+B(N+n_0+p_0)+C_\text{e}[N^2+(2n_0+p_0)N+(n_0^2+2n_0p_0)]\\&+C_\text{h}[N^2+(2p_0+n_0)N+(p_0^2+2n_0p_0)]\end{align}

This spontaneous carrier recombination lifetime is the saturation lifetime of a semiconductor because it has the effect of the saturation lifetime $$\tau_\text{s}$$ defined in (10-74) [refer to the population inversion and optical gain tutorial] in defining the saturation intensity of a semiconductor.

Example 12-6

The n-type GaAs considered in Example 12-3 [refer to the electron and hole concentrations tutorial] under optical excitation with $$n-n_0=p-p_0=N$$ is found to have the following recombination coefficients:

$A=5.0\times10^5\text{ s}^{-1}$

$B=8.0\times10^{-11}\text{ cm}^3\text{ s}^{-1}=8.0\times10^{-17}\text{ m}^3\text{ s}^{-1}$

$C=C_\text{e}+C_\text{h}=5.0\times10^{-30}\text{ cm}^6\text{ s}^{-1}=5.0\times10^{-42}\text{ m}^6\text{ s}^{-1}$

[Note: In the literature, the coefficients $$B$$ and $$C$$ are commonly quoted in the units of $$\text{cm}^3\text{ s}^{-1}$$ and $$\text{ cm}^6\text{ s}^{-1}$$, respectively. Here we convert them to $$\text{m}^3\text{ s}^{-1}$$ and $$\text{ m}^6\text{ s}^{-1}$$, respectively, for the convenience of computation in SI units.]

Assume that $$C_\text{e}=C_\text{h}=C/2$$ for simplicity, as the ratio between $$C_\text{e}$$ and $$C_\text{h}$$ is not found. Find the ranges of the excess carrier concentration $$N$$ where each of the three different recombination processes dominates. Plot the spontaneous carrier lifetime $$\tau_\text{s}$$ as a function of the excess carrier concentration $$N$$ for $$N$$ in the range between $$10^{18}$$ and $$10^{26}\text{ m}^{-3}$$.

For $$n-n_0=p-p_0=N$$ and $$C_\text{e}=C_\text{h}=C/2$$ considered in this problem, we have, from (12-57),

$\frac{1}{\tau_\text{s}}=A+B(N+n_0+p_0)+C\left[N^2+\frac{3}{2}(n_0+p_0)N+\frac{1}{2}(n_0^2+p_0^2)+2n_0p_0\right]$

Because the values of $$A$$, $$B$$, and $$C$$ are different by many orders of magnitude, each term dominates over a certain range of $$N$$ values. To find these ranges, we only have to compare two neighboring terms at a time.

The Shockley-Read recombination process dominates when $$A\gt{B}(N+n_0+p_0)$$, thus

$N\lt\frac{A}{B}-n_0-p_0=6.25\times10^{21}\text{ m}^{-3}$

The Auger recombination process becomes important when

$C\left[N^2+\frac{3}{2}(n_0+p_0)N+\frac{1}{2}(n_0^2+p_0^2)+2n_0p_0\right]\gt{B}(N+n_0+p_0)$

Because $$N\gg{n}_0,p_0$$ when this condition is satisfied, we find that the Auger recombination process is important for

$N\gt\frac{B}{C}=1.6\times10^{25}\text{ m}^{-3}$

In the middle range, for $$6.25\times10^{21}\text{ m}^{-3}\lt{N}\lt1.6\times10^{25}\text{ m}^{-3}$$, bimolecular recombination is the dominant process.

The carrier lifetime as a function of excess carrier concentration is plotted in Figure 12-5 below. It can be seen from the change of slope in the curve that the Shockley-Read recombination process dominates for $$N\lt6.25\times10^{21}\text{ m}^{-3}$$, where $$\tau_\text{s}$$ is almost constant; the bimolecular recombination process dominates for $$6.25\times10^{21}\text{ m}^{-3}\lt{N}\lt1.6\times10^{25}\text{ m}^{-3}$$, where $$\tau_\text{s}$$ decreases approximately linearly with increasing $$N$$; and finally, the Auger process becomes significant for $$N\gt1.6\times10^{25}\text{ m}^{-3}$$, where $$\tau_\text{s}$$ decreases with increasing $$N$$ more than linearly.

The next tutorial covers the topic of current density in semiconductors.