# Gain and Stimulated Emission for Semiconductor Lasers

This is a continuation from the previous tutorial - threshold condition and longitudinal modes of semiconductor lasers.

In the semiclassical laser theory the medium response is governed by the polarization $$\pmb{\mathscr{P}}$$ induced by the optical field $$\pmb{\mathscr{E}}$$ and leads to the susceptibility $$\boldsymbol{\chi}$$, as defined in Equation (2-2-17) [refer to the Maxwell's equations for semiconductor lasers tutorial].

In terms of the density-matrix operation $$\rho$$, the induced polarization is given by

$\tag{2-4-1}\pmb{\mathscr{P}}=\text{Tr}(\rho\mathbf{p})=\sum_{c,v}(\rho_{cv}\mathbf{p}_{vc}+\rho_{vc}\mathbf{p}_{cv})$

where $$\mathbf{p}$$ is the dipole-moment operator and the sum is over all the energy states per unit volume in the conduction and valence bands.

The dynamic evolution of the density-matrix operator is governed by

$\tag{2-4-2}\frac{\text{d}\rho}{\text{d}t}=\frac{1}{\text{i}\hbar}[H_0-\mathbf{p}\cdot\mathbf{E},\rho]-\frac{1}{2}[\gamma\rho+\rho\gamma]+\Lambda$

where $$H_0$$ is the unperturbed Hamiltonian of the semiconductor, $$\gamma$$ is the decay operator, and $$\Lambda$$ takes into account the carrier generation in the active region because of external pumping.

The decay operator $$\gamma$$ in general should include all decay mechanisms through which electrons in a given energy state can decay out of that state. These can be divided into two categories corresponding to the intraband and interband decay mechanisms.

Intraband processes in a semiconductor constitute electron-electron scattering and electron-phonon scattering and occur at a fast time scale of ~0.1 ps. By contrast, interband processes occur at a time scale of a few nanoseconds; they consist of radiative recombination leading to spontaneous and stimulated emissions as well as nonradiative recombination. In long-wavelength semiconductor lasers an important source of nonradiative recombination is due to the Auger process.

A first-principle approach based on Equations (2-4-1) and (2-4-2) has proved useful for gas and solid-state lasers. Such an approach has also been used for semiconductor lasers by considering both the many-body effects and the intraband scattering processes.

However, its usefulness for semiconductor lasers is limited because a realistic analysis is extremely complex. The reasons are severalfold. The inclusion of all the decay processes governed by $$\gamma$$ in Equation (2-4-2) is usually difficult since intraband scattering processes are not well understood. Further, the description of the unperturbed system through $$H_0$$ in Equation (2-4-2) requires knowledge of the band structure and the density of states in the conduction and the valence bands. A further complication arises in a semiconductor laser because the active region is often heavily doped and the band-tail effects become important.

An alternative approach has been used to calculate the optical gain. It is based on the generalization of Einstein's $$A$$ and $$B$$ coefficients, which relate the rate of spontaneous and stimulated emissions to the net gain or absorption coefficient. This method is discussed in a later tutorial. A shortcoming of this approach is that it provides only the small-signa gain, and the gain-saturation effects cannot be treated. It is, however, useful for studying the dependence of the gain on various parameters such as the doping levels and the carrier density.

In view of the above-mentioned difficulties, a phenomenological approach is generally used for describing a semiconductor laser and has proven to be extremely successful. It is based on the observation that the numerically calculated gain at the lasing frequency (corresponding to the value at which the gain spectrum peaks for a given current density $$J$$) varies almost linearly with the injected carrier density $$n$$ for all values of $$J$$. The gain $$g$$ can therefore be approximated by

$\tag{2-4-3}g(n)=a(n-n_0)$

where $$a$$ is the gain coefficient and $$n_0$$ is the carrier density required to achieve transparency (corresponding to the onset of population inversion). Both of the parameters can be estimated from the numerical calculations or determined experimentally.

Note that the product $$an_0$$ is just the absorption coefficient of the unpumped material. To complete the phenomenological description, the refractive index is also assumed to vary linearly with the carrier density; i.e., $$\Delta\mu_\text{p}$$ in Equation (2-3-3) [refer to the threshold condition and longitudinal modes of semiconductor lasers tutorial] is given by

$\tag{2-4-4}\Delta\mu_\text{p}=bn$

where $$b=\partial\mu/\partial{n}$$ and is often determined experimentally.

The parameters $$a$$, $$b$$, and $$n_0$$ are the three parameters of the phenomenological model.

The assumption that both $$g$$ and $$\Delta\mu_\text{p}$$ vary linearly with $$n$$ appears to be quite drastic at first sight. However, the carrier density $$n$$ changes little above threshold and the linear variation is a reasonable approximation for small changes in $$n$$.

On comparing Equations (2-4-3) and (2-4-4) with Equations (2-3-3) and (2-3-4) [refer to the threshold condition and longitudinal modes of semiconductor lasers tutorial], we can see that the present model assumes a linear variation of the complex susceptibility $$\boldsymbol{\chi}_\text{p}$$ with the carrier density $$n$$, i.e.,

$\tag{2-4-5}\boldsymbol{\chi}_\text{p}=\mu_\text{b}(2b-\text{i}a/k_0)n$

A parameter that is found to be quite useful is the ratio of the real to the imaginary parts of $$\boldsymbol{\chi}_\text{p}$$ and is given by

$\tag{2-4-6}\beta_\text{c}=\frac{\text{Re}(\boldsymbol{\chi}_\text{p})}{\text{Im}(\boldsymbol{\chi}_\text{p})}=-\frac{2k_0b}{a}=-2k_0\left(\frac{\partial\mu/\partial{n}}{\partial{g}/\partial{n}}\right)$

Since $$\beta_\text{c}\propto{b}$$, it is often used in place of $$b$$ in the phenomenological description of a semiconductor laser. Note, however, that since $$b$$ is negative, $$\beta_\text{c}$$ is a positive dimensionless number.

Depending on the context, different names and symbols have been attributed to $$\beta_\text{c}$$. It is commonly referred to as the antiguiding parameter or the line-width enhancement factor.

The phenomenological description is complete once the carrier density $$n$$ is related to the pump parameter, the current density $$J$$. This is accomplished through a rate equation that incorporates all the mechanisms by which the carriers are generated or lost inside the active region.

In general, the continuity equations for both electrons and holes should be considered. The two are interrelated because of charge neutrality, and it suffices to consider one rate equation for electrons. In its general form, the carrier-density rate equation is

$\tag{2-4-7}\frac{\partial{n}}{\partial{t}}=D(\nabla^2n)+\frac{J}{qd}-R(n)$

The first term accounts for carrier diffusion, and $$D$$ is the diffusion coefficient.

The second term governs the rate at which the carriers, electrons and holes, are injected into the active layer because of the external pumping. The electron and hole populations are assumed to be the same to maintain charge neutrality. In the second term, $$q$$ is the magnitude of the electron charge and $$d$$ is the active-layer thickness.

Finally, the last term $$R(n)$$ takes into account the carrier loss owning to various recombination processes, both radiative and nonradiative.

A rigorous derivation of (2-4-7) has to be based on the density-matrix approach. Yamada has carried out this procedure using Equation (2-4-2) with some simplifying assumptions and has shown that carrier diffusion is a consequence of intraband scattering.

Carrier diffusion in general plays an important role in semiconductor lasers and complicates significantly their analysis.

Depending on the device geometry, diffusion effects sometimes are of minor nature. This is the case, for example, for strongly index guided semiconductor lasers where the active-region dimensions (in the plane perpendicular to the cavity axis) are often small compared to the diffusion length.

Since the carrier density in that case does not vary significantly over the active-region dimensions, it can be assumed to be approximately constant and the diffusion term in Equation (2-4-7) can be neglected. In the steady state, $$\partial{n}/\partial{t}=0$$ and we obtain

$\tag{2-4-8}J=qdR(n)$

The situation is entirely different for gain-guided semiconductor lasers where the lateral variation (parallel to the heterojunction) of the carrier density arising from carrier diffusion makes the gain in Equation (2-4-3) spatially inhomogeneous and helps to confine the optical mode. The inclusion of the diffusion term is then a necessity. This case is considered in a later tutorial.

The threshold condition (2-3-9) [refer to the threshold condition and longitudinal modes in semiconductor lasers tutorial] together with Equations (2-4-3) and (2-4-8) can be used to model the light-current characteristics of semiconductor lasers.

To complete the description, we need a suitable form of the carrier-recombination rage $$R(n)$$ appearing in Equation (2-4-8). The charge carriers recombine through several radiative and nonradiative mechanisms described in detail in later tutorial. Radiative recombination may lead to either spontaneous or stimulated emission. A suitable form for $$R(n)$$ is

$\tag{2-4-9}R(n)=A_\text{nr}n+Bn^2+Cn^3+R_\text{st}N_\text{ph}$

where it is assumed that the doping level of the active layer is well below the injected carrier density.

The quadratic term $$Bn^2$$ is due to spontaneous radiative recombination wherein an electron in the conduction band recombines with a hole in the valence band and a photon is spontaneously emitted.

The cubic term $$Cn^3$$ is due to Auger recombination, and its inclusion is particularly important for long-wavelength semiconductor lasers ($$\lambda\gt1$$ μm). The nonradiative Auger process is described in a later tutorial.

The last term $$R_\text{st}N_\text{ph}$$ is due to stimulated recombination that leads to coherent emission of light. It is directly proportional to the intracavity photon density $$N_\text{ph}$$ and to the net rate of stimulated emission

$\tag{2-4-10}R_\text{st}=(c/\mu_g)g(n)$

where $$g(n)$$ is the optical gain given in Equation (2-4-3).

The present phenomenological modal assumes that the coefficients $$A_\text{nr}$$, $$B$$ and $$C$$ in Equation (2-4-9) are constants independent of the external pumping. However, the bimolecular radiative coefficient $$B$$ is known to depend on the carrier density and is often approximated by

$\tag{2-4-11}B\approx{B_0}-B_1n$

This effect can be incorporated in Equation (2-4-9) by using effective values of $$B$$ and $$C$$.

Similarly the effect of active-layer doping can be included by suitably modifying the coefficients $$A_\text{nr}$$, $$B$$, and $$C$$.

The numerical values of these coefficients and the method of calculation are descried in later tutorials.

The next tutorial covers the topic of waveguide modes in semiconductor lasers.