## 4 Homogeneous Broadening
### Compliance Check against Experimental Spectra  
Both the JDOS softening and the charge carrier scattering originate from the lattice disorder, but the latter is very seldom included in the studies for optical spectra. However, these two effects are simultaneously present at all times, and the lifetime broadening exists even in spectra where the exponential Urbach tail spans several orders of intensity magnitude. Likewise, the softening of band edges is usually omitted in the carrier mobility studies in semiconductor materials.  
This demands for a compliance check using the momentum relaxation times available from charge carrier mobilities and scattering rate calculations. For gallium arsenide at 300 K, the mobilities are given as: \(\mu_{e}\) = 8500 and \(\mu_{h}\) = 400 cm\({}^{2}\)/(Vs) for the electrons and holes, respectively (light holes \(\mu_{lh}=1067\) and heavy holes \(\mu_{hh}=341\)) [27; 28; 29; 30; 31]. Using these values and the electron (0.063) and heavy hole (0.51) effective masses [32] for the \(m_{eff}\) in Equation (12) we get the first line in Table 1, where the heavy hole mass was applied for its higher density of states.  
\[\tau=\mu*m_{eff}/e \tag{12}\]  
At room temperature in GaAs, the relaxation times are determined mainly by the scattering from longitudinal optical (LO) phonons, with phonon energy, \(\hbar\omega_{LO}\approx 35\) meV [27; 28; 29; 30; 31]. Up to this energy, LO-phonon absorption dominates the relaxation for the electrons and holes above the conduction band minimum and below the valence band maximum, respectively. The corresponding scattering times are given on the second line in Table 1. Above the \(\hbar\omega_{LO}\) energy, instead of phonon absorption, hot carrier scattering arises from the emission of optical phonons and the associated relaxation times are on the last table line.  
Finally, to validate these values, the corresponding Lorentzian lineshapes (\(\tau_{r}\) = 84 fs, 181 fs) are convolved with the Inhomog. Urbach spectrum in Figure 4. It was evaluated that, in order to comply with the Urbach tails, for two or more intensity magnitude orders, [4; 5] approximately 2000 fs values would be needed for the \(\tau_{r}\). We conclude that the basic Lorentzian lineshape poorly fits the experimental emission and absorption spectra. It is also problematic for the LED quantum wells [33; 34].  
On the other hand, using the above \(\tau_{e}\) and \(\tau_{h}\) values in the product of two Lorentzians broadening function in Equation (10) it convolves into the plotted lineshapes L (271 fs) \(*\) L (13 fs) and L (540 fs) \(*\) L (27 fs) and the latter does not significantly diverge from the original Inhomog spectrum.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**E(e), E(h)** & \(\tau_{e}\) & \(\tau_{h}\) & \(\tau_{r}\) & \(\tau_{be}\) & \(\tau_{bh}\) \\ \hline
**from mobility** & 304 & 116 & 84 & 271 & 13 \\
**\textless{}**\#\omega_{LO}\)** & 600 & 260 & 181 & 540 & 27 \\
**\textgreater{}**\#\omega_{LO}\)** & 168 & 60 & 44 & 150 & 7 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Electron and hole relaxation times [fs] in GaAs at 300 K. The \(\tau_{r}\), \(\tau_{be}\) and \(\tau_{bh}\) columns were obtained from Equations (9) and (11) and used in Equations (2) and (10).