# Radiative Recombination in Semiconductors

This is a continuation from the previous tutorial - **semiconductor junctions**.

The general characteristics of electron-hole recombination processes in a semiconductor are discussed in the carrier recombination tutorial.

The net result of any recombination process is the transition of an electron from an occupied state at a higher energy to an empty state at a lower energy, accompanied by the release of the energy that is the difference between these two states.

An electron-hole recombination process in a semiconductor can be either radiative or nonradiative. In a radiative recombination process, the released energy is emitted as electromagnetic radiation. In a nonradiative recombination process, no radiation is emitted, and the released energy is eventually converted to thermal energy in the form of lattice vibrations. Only radiative processes are useful to the function of semiconductor lasers and LEDs.

There are primarily three different radiative recombination processes:

- Band-to-band recombination
- Exciton recombination, through either a free exciton or a bound exciton
- Recombination through impurity states.

The most important radiative recombination process is the bimolecular band-to-band recombination process, the details of which are discussed in the following section.

Free exciton recombination is radiative, but it is not important for practical device applications at room temperature because free excitons can form only at very low temperatures due to their small ionization energies.

Radiative recombination of certain types of bound excitons can be useful. Certain radiative recombination processes associated with impurities in a semiconductor are important in the operation of some semiconductor lasers or LEDs.

A photon emitted by band-to-band recombination has an energy slightly higher than the bandgap, whereas one that is emitted through a process involving the impurities has an energy lower than the bandgap of the semiconductor.

Certain impurities in a semiconductor can form ** isoelectronic centers**. An isoelectronic center is normally neutral but introduces a local potential that can trap an electron or a hole, depending on the type of impurity that creates the isoelectronic center.

An isoelectronic center that traps an electron becomes negatively charged. The negatively charged center can then capture a hole from the valence band to form a bound exciton.

Similarly, an isoelectronic center that traps a hole becomes positively charged and is able to capture an electron from the conduction band to form a bound exciton.

Subsequent annihilation of the electron-hole pair in the bound exciton is a radiative process that results in the emission of a photon of an energy equal to the bandgap minus the binding energy of the center.

Because the momentum of a trapped, localized electron or hole is highly diffused according to the uncertainty principle of quantum mechanics, conservation of momentum can easily be satisfied in the radiative recombination process through an isoelectronic center no matter whether the host semiconductor is a direct-gap or an indirect-gap material. Consequently, this mechanism of radiative recombination is important in indirect-gap semiconductors, in which band-to-band radiative recombination probabilities are very low.

In particular, this process is responsible for improving the luminescence efficiency of the indirect-gap semiconductors GaP, GaAs_{x}P_{1-x}, and In_{x}Ga_{1-x}P for their applications as materials for LEDs.

As an example, the energy levels of the isoelectronic traps created by the impurities N and Zn,O in GaP are illustrated in Figure 13-1. The N and Zn,O impurities in GaP both act as electron traps. At room temperature, a photon emitted through a N center in GaP : N has an energy of about 2.20 eV, and that emitted through a Zn,O center in GaP : Zn,O has an energy of about 1.79 eV.

A high impurity concentration in a semiconductor can lead to the formation of conduction and valence ** bandtail states**, which in effect extend the conduction- and valence-band edges into the gap. Optical transitions associated with such bandtail states result in the absorption and emission of photons of energies less than the bandgap of a semiconductor. This bandtail effect is important only in a direct-gap semiconductor that is doped with a high concentration of impurities.

**Radiative Efficiency**

The total recombination rate for the excess carriers in a semiconductor can be expressed as the sum of radiative and nonradiative recombination rates:

\[\tag{13-1}R=R_\text{rad}+R_\text{nonrad}\]

The lifetime of an excess electron-hole pair associated with radiative recombination is called the ** radiative carrier lifetime**, \(\tau_\text{rad}\), and that associated with nonradiative recombination is called the

**, \(\tau_\text{nonrad}\). They are related to the total**

*nonradiative carrier lifetime***, \(\tau_\text{s}\), of the excess carriers by**

*spontaneous carrier recombination lifetime*\[\tag{13-2}\frac{1}{\tau_\text{s}}=\frac{1}{\tau_\text{rad}}+\frac{1}{\tau_\text{nonrad}}\]

The ** spontaneous carrier recombination rate**, \(\gamma_\text{s}\), is defined as

\[\tag{13-3}\gamma_\text{s}=\frac{1}{\tau_\text{s}}\]

This parameter is the total rate of carrier recombination including the contributions from all, radiative and nonradiative, spontaneous recombination processes but excluding the contribution from the stimulated recombination process. In the presence of stimulated emission, the effective recombination rate of the carriers can be much higher than that given by \(\gamma_\text{s}\) because of stimulated recombination.

The ** radiative efficiency**, or the

**, of a semiconductor is defined as**

*internal quantum efficiency*\[\tag{13-4}\eta_\text{i}=\frac{R_\text{rad}}{R}=\frac{\tau_\text{s}}{\tau_\text{rad}}\]

**Bimolecular Radiative Lifetime**

In a practical operating condition of a semiconductor laser or LED, the radiative recombination rate is almost entirely contributed by bimolecular recombination, including band-to-band and exciton recombination processes.

In thermal equilibrium, bimolecular recombination is balanced by bimolecular thermal generation. The bimolecular thermal generation rate, \(G_0\), is the same for the generation of electrons and holes.

Therefore, the net radiative recombination rate in the presence of excess electron-hole pairs is given by

\[\tag{13-5}R_\text{rad}=Bnp-G_0=Bnp-Bn_0p_0\]

where \(G_0\) is identified with \(Bn_0p_0\) because \(R_\text{rad}=0\) in the state of thermal equilibrium when \(n=n_0\) and \(p=p_0\).

In contrast to the bimolecular recombination rate, which depends on the total electron and hole concentrations, the thermal generation rate is largely independent of the carrier concentrations because the bound electrons in the valence bands and the empty states in the conduction bands that are available for thermal generation of free electrons and free holes are always much more numerous than the values of \(n\) and \(p\).

Consequently, even in a semiconductor that has a high concentration of excess carriers generated by external excitation, the bimolecular thermal generation rate remains \(G_0=Bn_0p_0\).

With the excess carrier density \(N=n-n_0=p-p_0\) as expressed in (12-55) [refer to the carrier recombination tutorial], the radiative lifetime of the excess carriers is then given by

\[\tag{13-6}\tau_\text{rad}=\frac{N}{R_\text{rad}}=\frac{1}{B(N+n_0+p_0)}\]

A short radiative lifetime corresponds to a high radiative recombination rate.

In the case when the excess carrier density is low so that \(N\ll{n}_0,p_0\), the radiative lifetime is a constant that is independent of the density of the excess carriers:

\[\tag{13-7}\tau_\text{rad}\approx\frac{1}{B(n_0+p_0)}\]

In the case when the excess carrier density is high so that \(N\gg{n}_0,p_0\), the radiative lifetime varies inversely with the excess carrier density:

\[\tag{13-8}\tau_\text{rad}\approx\frac{1}{BN}\]

**Example 13-1**

Find the radiative carrier lifetime and the internal quantum efficiency for the optically excited n-type GaAs considered in Example 12-6 [refer to the carrier recombination tutorial] if both the Shockley-Read and the Auger recombination processes in this semiconductor are nonradiative while the bimolecular process is purely radiative.

Plot them as a function of excess carrier concentration \(N\) for \(N\) in the range between \(10^{18}\) and \(10^{26}\text{ m}^{-3}\). In what range of carrier densities is high radiative efficiency found? What is the peak internal quantum efficiency?

From Example 12-6 [refer to the carrier recombination tutorial], we have the following spontaneous carrier lifetime:

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

Because the bimolecular process is purely radiative while the Shockley-Read and the Auger recombination processes are nonradiative, the radiative carrier lifetime is that given in (13-6):

\[\tau_\text{rad}=\frac{N}{R_\text{rad}}=\frac{1}{B(N+n_0+p_0)}\]

From Example 12-6, we have \(A=5.0\times10^5\text{ s}^{-1}\), \(B=8.0\times10^{-17}\text{ m}^3\text{ s}^{-1}\), and \(C=5.0\times10^{-42}\text{ m}^6\text{ s}^{-1}\).

From Example 12-3 [refer to the electron and hole concentrations tutorial], we have \(n_0=5.0\times10^{18}\text{ m}^{-3}\) and \(p_0=1.1\times10^6\text{ m}^{-3}\).

Using these parameters, we can find \(\tau_\text{s}\) and \(\tau_\text{rad}\) as a function of the carrier concentration \(N\). Then the internal quantum efficiency can be found by using (13-4) as \(\eta_\text{i}=\tau_\text{s}/\tau_\text{rad}\). The results are plotted in Figure 13-2 below.

We find from these results that \(\eta_\text{i}\gt0.5\) for carrier concentrations in the range of \(6.25\times10^{21}\text{ m}^{-3}\lt{N}\lt1.6\times10^{25}\text{ m}^{-3}\) where the bimolecular recombination process dominates, according to Example 12-6 [refer to the carrier recombination tutorial].

We also find from Figure 13-2 that the peak internal quantum efficiency is \(96.2\%\) for \(N=3.16\times10^{23}\text{ m}^{-3}\).

The next tutorial covers the topic of **band-to-band optical transitions in semiconductors**.