## Peter Blood

Print publication date: 2015

Print ISBN-13: 9780199644513

Published to Oxford Scholarship Online: November 2015

DOI: 10.1093/acprof:oso/9780199644513.001.0001

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# (p.339) A Trends with band gap

Source:
Quantum Confined Laser Devices
Publisher:
Oxford University Press

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 $∝1/(hν)$ and spontaneous emission $∝(hν)$. To examine the broad trends with the band gap of the medium we consider fixed occupation factors, for convenience a fully inverted structure with $f(Ec)=1$ and $f(Ev)=0$. Since the gain medium interacts with light at a photon energy $hν≈Eg$, 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) by1

(A.1)
$Display mathematics$

where $Eg$ is the band gap, $ΔESO$ is the energy of the split-off band relative to the valence band edge, and $mc∗$ 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.

Fig. A.1 (a) Basis function momentum matrix element $|M|2$, calculated from the electronic band structure using eqn A.1 as a function of band gap (units are ([M][L][T]−1)2). (b) Dipole matrix element calculated from $|M|2$ using eqn 7.73 as a log–log plot of $μ122$ versus 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 $ℏω=Eg$, producing the results for $μ122$ shown in Fig. A.1. The square of the dipole matrix element varies inversely with the square of the band gap.

Fig. A.2 Linear plot of the dipole length versus lattice parameter for III–V compounds, from the data in Fig. A.1.

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 $μcv/e$ 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 sub-band separation, in examining trends we take $hν∝Eg$ of the well material.

### (p.341) Transition density

Values for the transition density $ρtrans(hν)$ (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 $mc∗$ to increase with $Eg$.

### Fractional absorption

Gain is, explicitly, inversely proportional2 to $hν$; 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:

(A.2)
$Display mathematics$

Fig. A.3 Values for the transition density (open circles), fractional gain $γwell$, and spontaneous emission rate for a single sub-band pair of a quantum well calculated for C–HH transitions relative to GaAs as a function of band gap. The dashed lines are guides.

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, $γwell$ (eqn 11.34). Combining the photon energy $hν=Eg$ with values for $|M|2$ and $ρtrans$, the relative variations in $γwell$ 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 $γwell$ 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 Lz 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, $wmode$ 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 $hν$ because the density of free-space modes available for spontaneous emission, $ρmode∝(hν)2$ (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 narrow-gap structures should produce their maximum gain with much smaller intrinsic recombination current because of their lower free-space 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 $σOE$ is proportional to ω‎ and $μcv2$ (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 $wmode$ with wavelength.

Fig. A.4 Integrated optical cross section as a function of band gap expressed as a wavelength.

## Notes:

(2) The origin of this can be seen from Section 11.4.1 for absorption. Both S and $ΔS$ are proportional to $A02$; gain is the fractional change in energy, so $ΔS$ also contains $hν=ℏω$; S contains ω2, so the ratio $ΔS/S∝1/hν$.

(3) The same trend applies to the modal gain for structures with the same mode width, $wmode$. This requires that for each different well material the waveguide width d is different to keep $wmode$ constant as the wavelength changes.

(4) Equation 4.48: κ1 decreases with increasing wavelength, so $wmode$ increases.