(p.339) A Trends with band gap
(p.339) A Trends with band gap

A.2 Gain and emission 340
It is instructive to step back from the detailed accounts of absorption, gain, and recombination and look at trends in these characteristics with the choice of material, specified by its direct gap, since these underpin the trends of device characteristics with wavelength.
The equations for emission and gain spectra depend upon the occupation factors of the initial and final states, the transition density, and, in the case of modal gain, upon the optical confinement. They also contain explicit dependence on photon energy: gain $\propto 1/(h\mathrm{\nu})$ and spontaneous emission $\propto (h\mathrm{\nu})$. To examine the broad trends with the band gap of the medium we consider fixed occupation factors, for convenience a fully inverted structure with $f({E}_{\text{c}})=1$ and $f({E}_{\text{v}})=0$. Since the gain medium interacts with light at a photon energy $h\mathrm{\nu}\approx {E}_{\text{g}}$, the trend with band gap is determined principally by three factors:
• the dependence of the matrix element on the band structure,

• the dependence of the transition density on effective mass, and

• the explicit dependence on photon energy.
The first expresses the strength of the fundamental light–matter interaction.
A.1 The matrix element
The basis function momentum matrix element (eqn 11.25) is related to the band structure (Section 11.3.2) by^{1}
where ${E}_{\text{g}}$ is the band gap, $\mathrm{\Delta}{E}_{\text{SO}}$ is the energy of the splitoff band relative to the valence band edge, and ${m}_{\text{c}}^{\ast}$ is the conduction band effective mass. These parameters are known from band structure calculations or measurements and, taking literature values for GaAs, InP, GaSb, InAs, and InSb spanning band gaps from 1.424 to 0.17 eV, the values for $M{}^{2}$ (Exercise 11.4) vary by only about 20%, as shown in Fig. A.1, so the general trend is for the basis function momentum matrix element to be roughly independent of band gap.
(p.340) The relation between the momentum matrix element and the dipole matrix element contains the optical frequency (eqn 7.73) due to the relation between electric field and vector potential (eqn 7.67). The dipole matrix element can be calculated from $M{}^{2}$ using eqn 7.73 with $\mathrm{\hslash}\mathrm{\omega}={E}_{\text{g}}$, producing the results for ${\mu}_{12}^{2}$ shown in Fig. A.1. The square of the dipole matrix element varies inversely with the square of the band gap.
There is a trend for the band gap to decrease with increasing lattice parameter (Fig. 3.3) and, as shown in Fig. A.2, the trend is for the dipole length ${\mu}_{\text{cv}}/e$ to increase linearly with lattice parameter.
A.2 Gain and emission
Gain and emission of a well are given by equations of the form of eqns 11.38 and 11.46. The basis function matrix element in the well is a property of the unit cell (eqn 11.25) and is characteristic of the well material. Although quantum confinement increases the subband separation, in examining trends we take $h\mathrm{\nu}\propto {E}_{\text{g}}$ of the well material.
(p.341) Transition density
Values for the transition density ${\mathrm{\rho}}_{\text{trans}}(h\mathrm{\nu})$ (eqn 11.18), calculated from literature data for the conduction and heavy hole effective masses of III–V compounds and shown in Fig. A.3 relative to GaAs, increase with band gap owing primarily to the trend for ${m}_{\text{c}}^{\ast}$ to increase with ${E}_{\text{g}}$.
Fractional absorption
Gain is, explicitly, inversely proportional^{2} to $h\mathrm{\nu}$; however, modal gain also depends on the optical coupling of the mode to the well. To focus on the well alone consider first the term in large [. . .] in eqn 11.38:
This is the fractional increase in energy for light propagating normal to the layer for full inversion and is the same quantity as the fraction absorbed, ${\mathrm{\gamma}}_{\text{well}}$ (eqn 11.34). Combining the photon energy $h\mathrm{\nu}={E}_{\text{g}}$ with values for $M{}^{2}$ and ${\mathrm{\rho}}_{\text{trans}}$, the relative variations in ${\mathrm{\gamma}}_{\text{well}}$ with band gap are shown in Fig. A.3. These changes are the same as for the material gain, the {. . .} term in eqn 11.40, with a well of fixed width. The trend is for ${\mathrm{\gamma}}_{\text{well}}$ to decrease with increasing band gap.^{3}
Modal gain and mode width
Generally speaking similar well widths are used for a wide range of well materials and, provided the values of L_{z} are all such that the amplitude of the optical field is uniform over the width of the well, the effective mode width defined by eqn 4.47 depends only on the properties of the waveguide at the laser wavelength, the latter determined by the well width.
(p.342) The waveguide can be designed to provide the greatest optical field at the well by optimising the width of the core as illustrated in Fig. 4.11. The trend is for the effective mode width to increase with increasing wavelength;^{4} consequently, as the material band gap increases, ${w}_{\text{mode}}$ decreases, so the variation of modal gain with band gap is weaker than that of the fractional gain and absorption shown in Fig. A.3 in structures with a waveguide optimised to each wavelength.
Spontaneous emission
The spontaneous emission rate is, explicitly, proportional to $h\mathrm{\nu}$ because the density of freespace modes available for spontaneous emission, ${\mathrm{\rho}}_{\text{mode}}\propto (h\mathrm{\nu}{)}^{2}$ (eqn 7.9). Combining this with the results for $M{}^{2}$ eqn 11.46 produces the relative variation with band gap shown in Fig. A.3.
Implications for lasers
At first sight it appears that narrowgap structures should produce their maximum gain with much smaller intrinsic recombination current because of their lower freespace mode density. However, Auger recombination is also an intrinsic process and its rate increases exponentially with decreasing band gap (eqn E.37), which more than masks the decrease in the radiative rate, as illustrated by Fig. E.8.
A.3 Optical cross section
The strength of the light–matter interaction as measured by the integrated optical cross section ${\mathrm{\sigma}}_{\text{OE}}$ is proportional to ω and ${\mu}_{\text{cv}}^{2}$ (eqn 7.58); (p.343) the proportionality to ω arises from the definition in terms of energy absorbed in eqn 7.56. Overall we therefore expect the integrated cross section to decrease in inverse proportion to the band gap energy, as shown in Fig. A.4, where it is plotted as a function of the band gap wavelength. The variation of modal gain (eqn 9.18) is weaker than this owing to the increase in ${w}_{\text{mode}}$ with wavelength.
Notes:
(^{1}) Coldren and Corzine 1995, eqn A8.14.
(^{2}) The origin of this can be seen from Section 11.4.1 for absorption. Both S and $\mathrm{\Delta}S$ are proportional to ${A}_{0}^{2}$; gain is the fractional change in energy, so $\mathrm{\Delta}S$ also contains $h\mathrm{\nu}=\mathrm{\hslash}\mathrm{\omega}$; S contains ω^{2}, so the ratio $\mathrm{\Delta}S/S\propto 1/h\mathrm{\nu}$.
(^{3}) The same trend applies to the modal gain for structures with the same mode width, ${w}_{\text{mode}}$. This requires that for each different well material the waveguide width d is different to keep ${w}_{\text{mode}}$ constant as the wavelength changes.