# Radiative Recombination in Semiconductors

This is a continuation from the previous tutorial - ** emission characteristics of semiconductor lasers**.

## Introduction

The next few tutorials discuss the electron-hole recombination mechanisms in a direct-band-gap semiconductor.

Recombination mechanisms can in general be classified into two groups, radiative and nonradiative.

** Radiative recombination** occurs when an electron in the conduction band recombines with a hole in the valence band and the excess energy is emitted in the form of a photon. Radiative recombination is thus the radiative transition of an electron in the conduction band to an empty state (hole) in the valence band.

The optical processes associated with radiative transitions are (i) spontaneous emission, (ii) absorption or gain, and (iii) stimulated emission.

Stimulated emission, in which the emitted photon has exactly the same energy and momentum as the incident photon, forms the basis for laser action. The concept of stimulated emission dates back to the work of Einstein in 1917.

In thermal equilibrium, a direct-band-gap semiconductor (e.g., GaAs, InP, or GaSb) has a few electrons in the conduction band and a few holes (empty electron states) in the valence band.

When a photon of energy greater than the band gap passes through such a semiconductor, the photon has a high probability of being absorbed, giving its energy to an electron in the valence band, thereby raising the electron to the conduction band.

In principle, such a photon could stimulated the emission of an identical photon with the transition of an electron from the conduction to the valence band. The emitted photo derives its energy from the energy lost by the electron.

In thermal equilibrium the number of electrons in the conduction band is very small (~\(10^6\text{ cm}^{-3}\) for GaAs), so the probability of stimulated emission is negligible compared to the probability for absorption.

However, external excitation, can sufficiently increase the number of electrons in the conduction band such that the probability of stimulated emission eventually becomes higher than the probability of absorption. This situation corresponds to population inversion in a laser medium and is necessary for optical gain.

The external excitation which generates a high density of electron-hole pairs in a semiconductor is usually provided by current injection. It can also be achieved by optical pumping (absorption of radiation higher in energy than the band gap). In this tutorial we discuss the absorption and emission rates. These are then used to study the dependence of optical gain on the injected current and the carrier density.

* Nonradiative recombination* of an electron-hole pair, as the name implies, is characterized by the absence of an emitted photon in the recombination process. This of course makes the experimental study, and hence identification, of such processes very difficult.

In indirect-band-gap semiconductors such as Ge or Si, the probability of nonradiative recombination dominates that of radiative recombination by several orders of magnitude.

The measurable quantities associated with nonradiative recombinations are the internal quantum efficiency and the carrier lifetime. The variation of these quantities with parameters like temperature, pressure and carrier concentration is, by and large, the only way to identify a particular nonradiative recombination process.

One of the effects of nonradiative recombination on the performance of injection lasers is to increase the threshold current.

The nonradiative recombination processes that affect the performance of long-wavelength semiconductor lasers are Auger recombination, surface recombination, and recombination at defects.

The Auger recombination mechanism involves four particle states (three electrons and one hole) and is believed to be important at high temperatures and for low band-gap semiconductors.

## Radiative Recombination

The processes associated with the radiative recombination of electron-hole pairs in semiconductors are spontaneous emission, optical absorption or gain, and stimulated emission.

The rates of these processes are related to each other by the Einstein relations. These relations were first derived for electronic transitions between two discrete energy levels (i.e., in a gaseous medium).

Similar relations also hold for a semiconductor where the recombining electrons and holes can occupy a continuous band of energy eigenstates.

Figure 3-1 shows a simplified energy-versus-wave-vector diagram for a direct-band-gap semiconductor.

An accurate description of the band structure requires sophisticated numerical techniques. A commonly used approximation of the exact band structure in a direct-gap semiconductor is the parabolic band model.

In this model the energy-versus-wave-vector (\(E\) versus \(\mathbf{k}\)) relation is assumed to be parabolic, that is,

\[\tag{3-2-1a}E_\text{c}=\frac{\hbar^2{k^2}}{2m_\text{c}}\qquad\text{for electrons}\]

\[\tag{3-2-1b}E_\text{v}=\frac{\hbar^2{k^2}}{2m_\text{v}}\qquad\text{for holes}\]

where \(m_\text{c}\) and \(m_\text{v}\) are the effective masses of electrons and holes, respectively, and \(k\) is the magnitude of the wave vector \(\mathbf{k}\).

In a direct-gap semiconductor, the minimum in the conduction band curve and the maximum in the valence band curve occur at the same value of the wave vector \(\mathbf{k}\) (\(\mathbf{k}=0\) in Figure 3-1). Since a photon carries negligible momentum compared with the carrier momentum \(\hbar\mathbf{k}\), radiative transitions occur between free electrons and free holes of essentially identical wave vectors.

At a given temperature \(T\), the available number of electrons and holes are distributed over a range of energies. The occupation probability \(f_\text{c}\) of an electron with energy \(E_\text{c}\) is given by the Fermi-Dirac statistics, and is

\[\tag{3-2-2}f_\text{c}(E_\text{c})=\frac{1}{\exp[(E_\text{c}-E_\text{fc})/k_\text{B}T]+1}\]

where \(E_\text{fc}\) is the quasi-Fermi level for the conduction band. Similarly for the valence-band holes, the occupation probability of a hole with energy \(E_\text{v}\) is

\[\tag{3-2-3}f_\text{v}(E_\text{v})=\frac{1}{\exp[(E_\text{v}-E_\text{fv})/k_\text{B}T]+1}\]

where \(E_\text{fv}\) is the quasi-Fermi level for the valence band.

The notations used here assume that \(E_\text{c}\) and \(E_\text{fc}\) are measured from the conduction-band edge and are positive into the conduction band. On the other hand, \(E_\text{v}\) and \(E_\text{fv}\) are measured from the valence-band edge and are positive into the valence band. Note that \(f_\text{v}\) represents the occupation probability of the hole and not of the electron.

Consider the transition shown in Figure 3-1 in the presence of an incident photon whose energy \(E=h\nu=E_\text{c}+E_\text{v}+E_\text{g}\), where \(E_\text{g}\) is the band gap. The photon can be absorbed creating an electron of energy \(E_\text{c}\) and a hole of energy \(E_\text{v}\).

The absorption rate is given by

\[\tag{3-2-4}R_\text{a}=B(1-f_\text{c})(1-f_\text{v})\rho(E)\]

where \(B\) is the transition probability, \(\rho(E)\) is the density of photons of energy \(E\), and the factors \(1-f_\text{c}\) and \(1-f_\text{v}\) represent the probabilities that the electron and hole states of energy \(E_\text{c}\) and \(E_\text{v}\) are not occupied.

The stimulated-emission rate of photons is given by

\[\tag{3-2-5}R_\text{e}=Bf_\text{c}f_\text{v}\rho(E)\]

where the Fermi factors \(f_\text{c}\) and \(f_\text{v}\) are the occupation probabilities of the electron and hole states of energy \(E_\text{c}\) and \(E_\text{v}\) respectively.

The stimulated-emission process is accompanied by the recombination of an electron-hole pair. The condition for net stimulated emission or optical gain is

\[\tag{3-2-6}R_\text{e}\gt{R}_\text{a}\]

Using Equations (3-2-4) and (3-2-5), this condition becomes

\[\tag{3-2-7}f_\text{c}+f_\text{v}\gt1\]

With the use of \(f_\text{c}\) and \(f_\text{v}\) given by Equations (3-2-2) and (3-2-3), this becomes

\[\tag{3-2-8}E_\text{fc}+E_\text{fv}\gt{E_\text{c}}+E_\text{v}\]

By adding \(E_\text{g}\) to both sides and noting that \(E_\text{c}+E_\text{v}+E_\text{g}=h\nu\), it follows that the separation of the quasi-Fermi levels must exceed the photon energy in order for the stimulated-emission rate to exceed the absorption rate. This is the necessary condition for net stimulated emission or optical gain.

### Absorption and Emission Rates for Discrete Levels

The spontaneous-emission rate and the absorption rate for photons in a semiconductor can be calculated using time-dependent perturbation theory and summing over the available electron and hole states.

We first consider the case of two discrete energy levels of energy \(E_\text{i}\) and \(E_\text{f}\). The electron makes a transition from an initial state of energy \(E_\text{i}\) to a final state of energy \(E_\text{f}\), and in the process a photon of energy \(E=E_\text{i}-E_\text{f}\) is emitted.

The transition probability or the emission rate \(W\) of such a process is given by Fermi's golden rule

\[\tag{3-2-9}W=\frac{2\pi}{\hbar}|H'_\text{if}|^2\rho(E_\text{f})\delta(E-E_\text{i}+E_\text{f})\]

where \(H'_\text{if}\) is the matrix element \(\langle\text{i}|H'|\text{f}\rangle\) of the time-independent part of the perturbation Hamiltonian \(H_\text{I}\) between the initial state \(|\text{i}\rangle\) and the final state \(|\text{f}\rangle\), and \(\rho(E_\text{f})\) is the density of the final states.

In obtaining Equation (3-2-9), the perturbation Hamiltonian is assumed to be of the form

\[\tag{3-2-10}H_\text{I}=2H'\sin(\omega{t})\]

The time-dependent sinusoidal part gives rise to the delta function of Equation (3-2-9) arising from the energy conservation during the emission process.

For photons interacting with free electrons of mass \(m_0\), the perturbation Hamiltonian is given by

\[\tag{3-2-11}H_\text{I}=-\frac{q}{m_0}\mathbf{A}\cdot\mathbf{p}\]

where \(\mathbf{A}\) is the vector potential of the electromagnetic field and \(\mathbf{p}\) is the electron-momentum vector.

The vector potential is related to the electric field \(\pmb{\mathscr{E}}\) by the equation

\[\tag{3-2-12}\pmb{\mathscr{E}}=-\frac{\partial\mathbf{A}}{\partial{t}}\]

For a traveling plane wave with the angular frequency \(\omega\), the electric field is written as

\[\tag{3-2-13}\pmb{\mathscr{E}}=E_0\hat{\boldsymbol{\epsilon}}\cos(\omega{t}-\mathbf{k}\cdot\mathbf{r})\]

where \(\hat{\boldsymbol{\epsilon}}\) is a unit vector determining the polarization of the field, \(E_0\) is its amplitude, and \(\mathbf{k}\) denotes the propagation vector such that \(\mathbf{k}\times\hat{\boldsymbol{\epsilon}}=0\).

The magnitude of the electric field \(E_0\) associated with one photon can be obtained as follows.

The magnitude of the Poynting vector that represents the energy flux is given by

\[\tag{3-2-14}S=|\langle\pmb{\mathscr{E}}\times\pmb{\mathscr{H}}\rangle|\]

where angular brackets denote time averaging and \(\pmb{\mathscr{H}}\) is the magnetic field, which can be obtained using Maxwell's equations [refer to the Maxwell's equations for semiconductor lasers tutorial]. Using them, we obtain

\[\tag{3-2-15}S=\frac{1}{2}E_0^2\bar{\mu}\epsilon_0c\]

where \(k=\bar{\mu}\omega/c\) and \(\bar{\mu}\) is the refractive index of the medium. The factor 1/2 arises from time averaging over one optical cycle.

The energy flux per photon is also given by the product of the photon energy \(\hbar\omega\) and the group velocity \(c/\bar{\mu}\). Hence it follows that

\[\tag{3-2-16}S=\hbar\omega{c}/\bar{\mu}\]

From Equations (3-2-12), (3-2-15), and (3-2-16), the vector potential \(\mathbf{A}\) is given by

\[\tag{3-2-17}\mathbf{A}=-\hat{\boldsymbol{\epsilon}}\left(\frac{2\hbar}{\epsilon_0\bar{\mu}^2\omega}\right)^{1/2}\sin(\omega{t}-\mathbf{k}\cdot\mathbf{r})\]

and using Equation (3-2-11) the perturbation Hamiltonian becomes

\[\tag{3-2-18}H_\text{I}=\frac{q}{m_0}\left(\frac{2\hbar}{\epsilon_0\bar{\mu}^2\omega}\right)^{1/2}(\hat{\boldsymbol{\epsilon}}\cdot\mathbf{p})\sin(\omega{t}-\mathbf{k}\cdot\mathbf{r})\]

The square of the transition matrix element \(H'_\text{if}\) is now given by

\[\tag{3-2-19}|H_\text{if}'|^2=\frac{q^2}{m_0^2}\frac{2\hbar}{\epsilon_0\bar{\mu}^2\omega}\frac{1}{4}|\langle\text{i}|\hat{\boldsymbol{\epsilon}}\cdot\mathbf{p}|f\rangle|^2\]

where we have made the usual assumption that the electron wave functions spread over a linear dimension shorter than the wavelength (\(r\ll\lambda\)) so that \(\mathbf{k}\cdot\mathbf{r}\ll1\).

For stimulated emission or absorption, \(\rho(E_\text{f})=1\), and the absorption or emission rate at a photon energy \(E=\hbar\omega\) for transitions between two discrete levels obtained from Equations (3-2-9) and (3-2-19) is given by

\[\tag{3-2-20}W=\frac{\pi{q}^2}{m_0^2\epsilon_0\bar{\mu}^2\omega}|M_\text{if}|^2\delta(E_\text{i}-E_\text{f}-\hbar\omega)\]

where \(M_\text{if}=\langle\text{i}|\hat{\boldsymbol{\epsilon}}\cdot\mathbf{p}|f\rangle\) is the momentum matrix element between the initial and final electron states.

In the case of absorption, the number of photons absorbed per second is \(W\). Since a photon travels a distance \(c/\bar{\mu}\) in 1 second, the number of photons absorbed per unit distance or the absorption coefficient \(\alpha=\bar{\mu}W/c\). Using (3-2-20), the absorption coefficient for photons of energy \(E=\hbar\omega\) becomes

\[\tag{3-2-21}\alpha(E)=\frac{q^2h}{2\epsilon_0m_0^2c\bar{\mu}E}|M_\text{if}|^2\delta(E_\text{i}-E_\text{f}-E)\]

For spontaneous emission, the quantity \(\rho(E_\text{f})\) in Equation (3-2-9) equals the number of states for photons of energy \(E\) per unit volume per unit energy. it is given by

\[\tag{3-2-22}\rho(E_\text{f})=(2)\frac{4\pi{k^2}}{(2\pi)^3}\frac{\text{d}k}{\text{d}E}\]

where the factor 2 arises from two possible polarizations of the electromagnetic field.

Since \(E=\hbar\omega\) and \(k=\bar{\mu}\omega/c\), Equation (3-2-22) reduces to

\[\tag{3-2-23}\rho(E_\text{f})=\frac{\bar{\mu}^3\omega^2}{\pi^2c^3\hbar}\]

Thus using Equations (3-2-9), (3-2-19), and (3-2-23), the spontaneous emission rate per unit volume at the photon energy \(E\) is given by

\[\tag{3-2-24}r_\text{sp}(E)=\frac{4\pi{q^2}\bar{\mu}E}{m_0^2\epsilon_0c^3h^2}|M_\text{if}|^2\delta(E_\text{i}-E_\text{f}-E)\]

### Absorption and Emission Rates in Semiconductors

We now calculate the spontaneous-emission rate and the absorption coefficient for direct-gap semiconductors. These quantities are obtained by integrating Equations (3-2-21) and (3-2-24) over the occupied electron and hole states in a semiconductor.

A realistic band model for a III-V direct-gap semiconductor is the four-band model shown in Figure 3-1. In this mode the valence band is represented by three subbands: the heavy-hole band, the light-hole band, and the spin-split-off-hole band.

The light- and heavy-hole bands are degenerate at \(\mathbf{k}=0\). Usually the split-off energy \(\Delta\) is large compared with the thermal energy \(k_\text{B}T\); hence the split-off band is full of electrons, that is, no holes are present.

The quasi-Fermi levels for the conduction and valence bands can be obtained from the known effective masses and the density of states in each band.

For the conduction band, \(E_\text{fc}\) is obtained using

\[\tag{3-2-25}n=\displaystyle\int\frac{\rho_\text{c}(E)\text{d}E}{1+\exp[(E-E_\text{fc})/k_\text{B}T]}\]

where \(\rho_\text{c}(E)\) is the conduction-band density of states and \(n\) is the number of electrons in the conduction band.

The density of states for electrons of energy \(E\) can be obtained using Equation (3-2-22) with the dispersion relation (3-2-1a) and becomes

\[\tag{3-2-26}\rho_\text{c}(E)=(2)\frac{4\pi{k^2}}{(2\pi)^3}\frac{\text{d}k}{\text{d}E}=4\pi\left(\frac{2m_\text{c}}{h^2}\right)^{3/2}E^{1/2}\]

where \(E=\hbar^2k^2/2m_\text{c}\), and \(m_\text{c}\) is the conduction-band effective mass. The factor 2 in Equation (3-2-26) arises from the two electronic spin states.

Equation (3-2-25) can be rewritten as

\[\tag{3-2-27}n=N_\text{c}\frac{2}{\pi^{1/2}}\displaystyle\int\frac{\epsilon^{1/2}\text{d}\epsilon}{1+\exp(\epsilon-\epsilon_\text{fc})}\]

where \(N_\text{c}=2(2\pi{m_\text{c}}k_\text{B}T/h^2)^{3/2}\) and \(\epsilon_\text{fc}=E_\text{fc}/k_\text{B}T\).

Equation (3-2-27) can be used to calculate \(\epsilon_\text{fc}\) for a given electron concentration.

A useful approximation for \(\epsilon_\text{fc}\) is given by Joyce and Dixon. They shows that \(\epsilon_\text{fc}\) can be represented by a convergent series of the form

\[\tag{3-2-28}\epsilon_\text{fc}=\ln\left(\frac{n}{N_\text{c}}\right)+\sum_{i=1}^\infty{A_i}\left(\frac{n}{N_\text{c}}\right)^i\]

where the first few coefficients are

\[\begin{align}A_1=\quad&3.53553\times10^{-1}\\A_2=-&4.95009\times10^{-3}\\A_3=\quad&1.48386\times10^{-4}\\A_4=\quad&4.42563\times10^{-6}\end{align}\]

Another approximation, although not as elegant but showing faster convergence, is the Nilson approximation.

For a nondegenerate electron gas (\(n\ll{N}_\text{c}\)), all terms except the first in Equation (3-2-28) can be neglected and the Fermi energy is given by

\[\tag{3-2-29}E_\text{fc}=k_\text{B}T\ln\left(\frac{n}{N_\text{c}}\right)\]

Using Equation (3-2-29) in Equation (3-2-2) and neglecting 1 in the denominator, the occupation probability becomes

\[\tag{3-2-30}f_\text{c}(E)\approx\frac{n}{N_\text{c}}\exp\left(\frac{-E}{k_\text{B}T}\right)\]

This is often referred to as the ** Boltzmann approximation**. The use of Equation (3-2-30) simplifies considerably the calculation of recombinations rates in semiconductors.

**The nondegenerate approximation \(n\ll{N}_\text{c}\), however, does not always hold for semiconductor lasers and should be used with caution.**The quasi-Fermi level for the holes in the valence band can be similarly calculated. The hole density is given by

\[\tag{3-2-31}p=\sum_{i=\text{l},\text{h}}\displaystyle\int\frac{\rho_{\text{v}i}(E)\text{d}E}{1+\exp[(E-E_\text{fv})/k_\text{B}T]}\]

where \(p\) is the density of holes and the summation is over the light- and heavy-hole bands.

Using an expression similar to Equation (3-2-26) for the density of states, we obtain

\[\tag{3-2-32}p=N_\text{v}\frac{2}{\pi^{1/2}}\displaystyle\int\frac{\epsilon^{1/2}\text{d}\epsilon}{1+\exp(\epsilon-\epsilon_\text{fv})}\]

where

\[N_\text{v}=2(2\pi{k_\text{B}}T/h^2)^{3/2}(m_\text{lh}^{3/2}+m_\text{hh}^{3/2})\]

and \(m_\text{lh}\) and \(m_\text{hh}\) are the effective masses of light and heavy holes, respectively.

For the nondegenerate case where \(p\ll{N_\text{v}}\), Equation (3-2-32) can be simplified to obtain the following expressions for the quasi-Fermi energy and the hole-occupation probability:

\[\tag{3-2-33a}E_\text{fv}=k_\text{B}T\ln\left(\frac{p}{N_\text{v}}\right)\]

\[\tag{3-2-33b}f_\text{v}(E)=\frac{p}{N_\text{v}}\exp\left(\frac{-E}{k_\text{B}T}\right)\]

So far we have considered the idealized case of parabolic bands with the density of states given by Equation (3-2-26). Under high injected carrier densities or for high doping levels, the density of states is modified.

In the case of heavy doping, this occurs because randomly distributed impurities create an additional continuum of states near the band edge. These are called ** band-tail states** and are schematically shown in Figure 3-2.

Several models of band-tail states exist in the literature. Principal among them are the Kane model and the Halperin-Lax model. Figure 3-3 shows a comparison of the calculated density of states for the two models.

The curves were calculated by Hwang for p-type GaAs with a hole concentration of \(3\times10^{18}\text{ cm}^{-3}\). The material was heavily compensated with acceptor and donor concentrations \(N_\text{A}\) and \(N_\text{D}\) of \(6\times10^{18}\text{ cm}^{-3}\) and \(3\times10^{18}\text{ cm}^{-3}\), respectively.

Figure 3-3 shows that for the Kane model the band tails extend much more into the band gap than for the Halperin-Lax model. The absorption measurements on heavily compensated samples indicate that the Halperin-Lax model is more realistic.

However, this model is not valid for states very close to the band edge. To remedy the situation, Stern approximated the density of states by fitting a density-of-state equation of the Kane form (which is Gaussian) to the Halperin-Lax model of band-tail states.

The fit is done at ** one** energy value in the band tail, which lies within the validity of the Halperin-Lax model. The Kane form of band tail merges with the parabolic band model above the band edge as seen in Figure 3-3.

The importance of band-tail states in the operation of semiconductor lasers arises from two considerations. First, band-tail states can contribute significantly to the total spontaneous-emission rate and hence to the injection current needed to reach threshold. Second, band-tail transitions can have significant optical gain so that the laser can lase on these transitions, especially for heavily doped semiconductors at low temperatures.

The matrix element for transitions between band-tail states differs from that involving free electron and hole states in one important aspect.

The former does not satisfy momentum conservation; i.e., since the electron and holes states are not states of definite momentum, the \(\mathbf{k}\)-selection rule does not apply.

By contrast, for transitions involving parabolic band states, the initial and final particle states obey the \(\mathbf{k}\)-selection rule.

Thus the exact matrix element for optical transitions in heavily compensated semiconductors must not invoke the \(\mathbf{k}\)-selection rule for transitions involving band-tail states and, at the same time, should extrapolate to that obeying the \(\mathbf{k}\)-selection rule for above-band-edge transitions involving parabolic band states.

The model for such a matrix element has been considered by Stern. Before discussing his model in the next section, we consider the simpler case where the \(\mathbf{k}\)-selection rule holds and obtain the absorption coefficient and the spontaneous emission rate.

**The \(\mathbf{k}\)-Selection Rule**

When the \(\mathbf{k}\)-selection rule is obeyed, \(|M_\text{if}|^2=0\) unless \(\mathbf{k}_\text{c}=\mathbf{k}_\text{v}\).

If we consider a volume \(V\) of the semiconductor, the matrix element \(|M_\text{if}|^2\) is given by

\[\tag{3-2-34}|M_\text{if}|^2=|M_\text{b}|^2\frac{(2\pi)^3}{V}\delta(\mathbf{k}_\text{c}-\mathbf{k}_\text{v})\]

The \(\delta\) function accounts for the momentum conservation between the conduction-band and valence-band states. The quantity \(|M_\text{b}|^2\) is an average matrix element for the Bloch states. Using the Kane model, \(|M_\text{b}|^2\) in bulk semiconductors is given by

\[\tag{3-2-35}|M_\text{b}|^2=\frac{m_0^2E_\text{g}(E_\text{g}+\Delta)}{12m_\text{c}(E_\text{g}+2\Delta/3)}=\xi{m_0}E_\text{g}\]

where \(m_0\) is the free-electron mass, \(E_\text{g}\) is the band gap, and \(\Delta\) is the spin-orbit splitting. For GaAs, using \(E_\text{g}=1.424\text{ eV}\), \(\Delta=0.33\text{ eV}\), \(m_\text{c}=0.067m_0\), we get \(\xi=1.3\).

We are now in a position to calculate the spontaneous-emission rate and the absorption coefficient for a bulk semiconductor.

Equations (3-2-24) and (3-2-34) can be used to obtain the total spontaneous-emission rate per unit volume. Summing over all states in the band, we obtain

\[\tag{3-2-36}\begin{align}r_\text{sp}(E)=&\frac{4\pi\bar{\mu}q^2E}{m_0^2\epsilon_0h^2c^3}|M_\text{b}|^2\frac{(2\pi)^3}{V}(2)\left(\frac{V}{(2\pi)^3}\right)^2\frac{1}{V}\\&\times\sum\displaystyle\int\ldots\int{f_\text{c}}(E_\text{c})f_\text{v}(E_\text{v})\text{d}^3\mathbf{k}_\text{c}\text{d}^3\mathbf{k}_\text{v}\delta(\mathbf{k}_\text{c}-\mathbf{k}_\text{v})\delta(E_\text{i}-E_\text{f}-E)\end{align}\]

where \(f_\text{c}\) and \(f_\text{v}\) are the Fermi factors for electrons and holes. The factor 2 arises from the two spin states. In Equation (3-2-36), \(\sum\) stands for the sum over the three valence bands (see Figure 3-1).

For definiteness, we first consider transitions involving electrons and heavy holes. The integrals in Equation (3-2-36) can be evaluated with the following result:

\[\tag{3-2-37}r_\text{sp}(E)=\frac{2\bar{\mu}q^2E|M_\text{b}|^2}{\pi{m_0}^2\epsilon_0h^2c^3}\left(\frac{2m_\text{r}}{\hbar^2}\right)^{3/2}(E-E_\text{g})^{1/2}f_\text{c}(E_\text{c})f_\text{v}(E_\text{v})\]

where

\[\begin{align}E_\text{c}&=\frac{m_\text{r}}{m_\text{c}}(E-E_\text{g})\\E_\text{v}&=\frac{m_\text{r}}{m_\text{hh}}(E-E_\text{g})\\m_\text{r}&=\frac{m_\text{c}m_\text{hh}}{m_\text{c}+m_\text{hh}}\end{align}\]

and \(m_\text{hh}\) is the effective mass of the heavy hole.

Equation (3-2-37) gives the spontaneous-emission rate at the photon energy \(E\). To obtain the total spontaneous-emission rate, a final integration should be carried out over all possible energies. Thus the total spontaneous-emission rate per unit volume due to electron-heavy-hole transitions is given by

\[\tag{3-2-38}R=\displaystyle\int\limits_{E_\text{g}}^\infty{r_\text{sp}}(E)\text{d}E=A|M_\text{b}|^2I\]

where

\[\tag{3-2-39}I=\displaystyle\int\limits_{E_\text{g}}^\infty(E-E_\text{g})^{1/2}f_\text{c}(E_\text{c})f_\text{v}(E_\text{v})\text{d}E\]

and \(A\) represents the remaining constants in Equation (3-2-37).

A similar equation holds for the electron-light-hole transitions if we replace \(m_\text{hh}\) by the effective light-hole mass \(m_\text{lh}\).

The absorption coefficient \(\alpha(E)\) can be obtained in a similar way using Equations (3-2-21) and (3-2-34) and integrating over the available states in the conduction and valence bands. The resulting expression is

\[\tag{3-2-40}\begin{align}\alpha(E)=&\frac{q^2h}{2\epsilon_0m_0^2c\bar{\mu}E}|M_\text{b}|^2\frac{(2\pi)^3}{V}(2)\left(\frac{V}{(2\pi)^3}\right)^2\left(\frac{1}{V}\right)\\&\times\displaystyle\int\ldots\int(1-f_\text{c}-f_\text{v})\text{d}^3\mathbf{k}_\text{c}\text{d}^3\mathbf{k}_\text{v}\delta(\mathbf{k}_\text{c}-\mathbf{k}_\text{v})\delta(E_\text{i}-E_\text{f}-E)\\=&\frac{q^2h|M_\text{b}|^2}{4\pi^2\epsilon_0m_0^2c\bar{\mu}E}\left(\frac{2m_\text{r}}{\hbar^2}\right)^{3/2}(E-E_\text{g})^{1/2}[1-f_\text{c}(E_\text{c})-f_\text{v}(E_\text{v})]\end{align}\]

For the case of a constant matrix element, the assumption is that \(|M_\text{if}|^2\) is independent of the energy and wave vector of the initial and final states. In such a case, the total spontaneous emission rate and absorption coefficient are obtained by summing over the initial and final states using Equations (3-2-21) and (3-2-24). The calculation is similar tot hat presented above; the only difference is that in the latter case the delta function representing momentum conservation is absent.

### Absorption Coefficient and Optical Gain

We now describe the calculation of the spontaneous-emission rate and the absorption coefficient using the Gaussian Halperin-Lax band-tail model and Stern's matrix element. The results in this section are presented for the quaternary InGaAsP alloy emitting near 1.3 μm or 1.55 μm.

The optical absorption (or gain) coefficient for transitions between the valence and conduction bands is obtained by integrating Equation (3-2-21) over all states, and is given by

\[\tag{3-2-41}\alpha(E)=\frac{q^2h}{2\epsilon_0m_0^2c\bar{\mu}E}\displaystyle\int\limits_{-\infty}^\infty\rho_\text{c}(E')\rho_\text{v}(E^")|M_\text{if}|^2[1-f_\text{v}(E^")-f_\text{c}(E')]\text{d}E'\]

where \(E^"=E'-E\) and \(\rho_\text{c}\) and \(\rho_\text{v}\) are, for the conduction and valence bands respectively, the densities of states per unit volume per unit energy.

The integral is evaluated numerically and summed for both light-hole and heavy-hole bands.

The matrix element \(M_\text{if}\) can be expressed as a product of two terms, \(M_\text{if}=M_\text{b}M_\text{env}\), where \(M_\text{b}\) is the previously defined contribution [Equation (3-2-35)] from the band-edge Bloch functions and \(M_\text{env}\) is the matrix element of the envelope wave functions representing the effect of band-tail states.

The envelope wave function is a plane wave above the band edge and takes the form of the ground state of a hydrogen atom for impurity states in the band tail.

The calculation described here uses the envelope matrix element of Stern. The square of the Bloch-function matrix element \(|M_\text{b}|^2\) is given by Equation (3-2-35) for the Kane model.

Correction to the value of \(|M_\text{b}|^2\) can arise from the contribution of other conduction bands. These corrections would simply scale the gain values calculated using Equation (3-2-41).

Figure 3-4 shows the calculated absorption curves for p-type InGaAsP (\(\lambda=1.3\) μm) with different doping levels. The parameters used in the calculation are listed in Table 3-1.

The heavy-hole mass, the light-hole mass, the dielectric constant, and the spin-orbit splitting are obtained from a linear extrapolation of the measured binary values.

The reported experimental results are in good agreement with the extrapolated light-hole mass.

The carrier-concentration-dependent band-gap reduction is assumed to be the same as for GaAs,

\[\tag{3-2-42}\Delta{E}_\text{g}(\text{in eV})=(-1.6\times10^{-8})(p^{1/3}+n^{1/3})\]

where \(n\) and \(p\) are in units of \(\text{ cm}^{-3}\).

The effect of \(\Delta{E}_\text{g}\) is only to shift the position of the absorption edge or the gain peak, and the magnitude of the maximum gain at a given current density remains unchanged.

The results shown in Figure 3-4 assume that acceptors are fully ionized, i.e. \(N_\text{A}-N_\text{D}=p\) where \(N_\text{A}\) and \(N_\text{D}\) are the acceptor and the donor concentrations, respectively and \(p\) is the majority carrier concentration. The ratio \(N_\text{A}/N_\text{D}=5\) was chosen on the basis of the experimental studies on Zn-doped GaAs.

The calculation shows that with an increase in the doping level, the effect of band-tail states becomes more pronounced and the absorption edge is less steep. For the same reason, the absorption on the high-energy side is reduced with an increase in the doping level.

The total spontaneous-emission rate per unit volume can be calculated in a similar manner. Integrating Equation (3-2-24) over all states, it is given by

\[\tag{3-2-43}R(E)=\frac{4\pi{q^2}\bar{\mu}E}{m_0^2\epsilon_0c^3h^2}\displaystyle\int\limits_{-\infty}^\infty\rho_\text{c}(E')\rho_\text{v}(E^")f_\text{c}(E')f_\text{v}(E^")|M_\text{if}|^2\text{d}E'\]

where \(E^"=E'-E\).

The integral is evaluated numerically. Equation (3-2-43) yields the spontaneous-emission spectrum if \(R\) is calculated as a function of the photon energy \(E\).

Figure 3-5 shows the calculated spontaneous-emission spectra for different hole concentrations. Note that with increasing carrier concentration the peak of the spontaneous emission shifts to lower energies while its height (maximum emission intensity) decreases. Also note that the width of the emission spectrum increases with increasing carrier concentration as a result of the band-tail states.

Figure 3-6 shows the effect of acceptor concentration on the width (FWHM) of the spontaneous emission spectrum. The measured data are also shown. The absorption edge shifts to lower energies due to the formation of band-tail states with increasing impurity concentration. The emission and absorption spectra of n-type semiconductors have essentially the same feature as those of p-type semiconductors.

We now calculate the gain in InGaAsP excited by external pumping. Under sufficient optical excitation or current injection, the number of electrons and holes may be large enough to satisfy Equation (3-2-8); the semiconductor material would then exhibit optical gain.

The rate of excitation can be expressed in terms of the injected carrier density or by the nominal current density defined below by Equation (3-2-44).

Figure 3-7 shows the gain spectra at various injected carrier densities. For injected carrier densities less than \(10^{18}\text{ cm}^{-3}\), there is no net gain at any photon energy. As the carrier density is further increased, the absorption coefficient \(\alpha\) becomes negative over a limit energy range and gain occurs.

The photon energy at which the maximum gain occurs shifts to higher energy with increasing injection while, at the same time, the gain appears at a lower energy with increasing injection. This effect is due to a shift in the band edge given by Equation (3-2-42).

Figure 3-8 shows the maximum gain (peak value in Figure 3-7) plotted as a function of the injected carrier density at different temperatures. The calculation is done for undoped material with a residual concentration of \(2\times10^{17}\text{ cm}^{-3}\) acceptors and donors.

Note that the gain is higher at a lower temperature for the same excitation. The temperature plays a role through the Fermi factors and affects the distribution of electrons and holes in the conduction and valence bands. At higher temperatures the carriers are distributed over a wider energy range, so the maximum gain is less.

A convenient way to express the excitation rate is to use the nominal current density \(J_\text{n}\), which is defined as the total spontaneous-emission rate per unit volume and equals the injected current density at unit quantum efficiency.

The nominal current density is usually expressed in \(\text{A}/(\text{cm}^2\cdot\text{ μm})\) and is given by

\[\tag{3-2-44}J_\text{n}=qR\]

where \(R\) is the total spontaneous-emission rate per unit volume given by Equation (3-2-43).

Figure 3-9 shows the calculated maximum gain as a function of nominal current density for InGaAsP at 1.3 μm. The nominal current density equals the current lost to radiative recombination in a double-heterostructure laser.

Figure 3-9 shows that to achieve the same gain, higher current density is needed at higher temperatures. Figures 3-8 and 3-9 show the basis of the increase in threshold current of injection lasers as the temperature is increased.

Results of a similar calculation for the variation of the maximum gain with the injected carrier density and the nominal current density at 1.55 μm are given in some literatures. The results are very similar to those of Figure 3-8 and 3-9 except that a smaller injected carrier density and a smaller nominal current density are needed at the longer wavelength to achieve the same gain. This arises from the smaller conduction-band effective electron mass at the longer wavelength, which allows the gain condition (3-2-8) to be satisfied at a lower carrier denstiy.

We next consider the effect of doping on the optical gain. Figure 3-10 shows the calculated maximum gain as a function of the injected electron density in p-type InGaAsP (\(\lambda=1.3\) μm) with different hole concentrations. The calculation assumes \(N_\text{A}/N_\text{D}=5\).

Figure 3-11 shows the result for n-type InGaAsP as a function of the injected hole density.

The two figures show that the excitation rate required for transparency (\(g=0\)) decreases with higher doping both for p- and n-type materials. The decrease is more rapid for n-type than for p-type materials.

The difference in behavior between the n- and p-type materials is due to the small effective mass of electrons compared with that of holes. This may be seen as follows.

The condition for net stimulated emission or gain is given by Equation (3-2-8). Increasing the doping level increases both \(E_\text{fc}\) and \(E_\text{fv}\); hence fewer additional injected carriers are needed to satisfy the condition for gain.

Since the effective mass of the conduction band is smaller than that of the valence band,

\[\left|\frac{\partial{E_\text{fc}}}{\partial{n}}\right|\gt\left|\frac{\partial{E_\text{fv}}}{\partial{p}}\right|\]

This explains why the effect of doping is more pronounced in n-type than in p-type material.

Figures 3-8 to 3-11 show that the optical gain varies almost linearly with the injected carrier density and can be approximately written as

\[\tag{3-2-45}g=a(n-n_0)\]

where the slope \(a\) is called the gain coefficient and \(n_0\) is the injected carrier density required to achieve transparency (i.e., \(g=0\) when \(n=n_0\)).

From Equation (3-2-8), the condition for transparency is given by

\[\tag{3-2-46}E_\text{fc}+E_\text{fv}=0\]

The quantity \(n_0\) can be calculated from Equation (3-2-27) using the approximation (3-2-28) for parabolic bands.

Figure 3-12 shows the calculated \(n_0\) for undoped InGaAsP lattice-matched to InP. The smaller \(n_0\) for low band gap is due to a smaller effective mass.

The variation of the gain with the carriers density is approximately linear. As such, it is useful for a simple but fairly accurate description of semiconductor lasers as was presented in the gain and stimulated emission for semiconductor laser tutorial. Equation (3-2-45) was used in the phenomenological model introduced in the gain and stimulated emission in semiconductor laser tutorial.

The gain parameters \(a\) and \(n_0\) can be determined either using numerical results presented in Figures 3-8 to 3-12 or deduced from the experimental data.

Typical values for InGaAsP lasers are in the range of \(1.2-2.5\times10^{-16}\text{ cm}^{-3}\) for \(a\) and \(0.9-1.5\times10^{18}\text{ cm}^{-3}\) for \(n_0\) depending on the laser wavelength and doping levels.

It should be stressed that because of uncertainties in the band-structure parameters, the numerical results are only accurate to within a factor of 2.

The total spontaneous emission rate \(R\) given by Equation (3-2-43) can be approximated by

\[\tag{3-2-47}R=Bnp\]

where \(B\) is the radiative recombination coefficient and \(n\) and \(p\) are the electron density and the hole density respectively. For undoped semiconductors, \(R=Bn^2\).

The calculated \(B=1.2\times10^{-10}\text{ cm}^3/\text{s}\) for InGaAsP (\(\lambda=1.3\) μm) at a carrier concentration of \(1\times10^{18}\text{ cm}^{-3}\).

Calculation of the radiative recombination rate shows that \(B\) decreases with increasing carrier density. This has been confirmed by Olshansky et. al using carrier-lifetime measurements.

The next tutorial covers the topic of ** nonradiative recombination in semiconductors**.