# Study of the Photo-Response of Doped GaAs with Aging
## 3 Results and Discussion
With the Eg values and the concentrations of the impurities, we draw the red dots in Figure 5. The fitted values are shown in Figure 5 (red points and line). The theoretical energy gap values based on the concentration given by Equation (2) [21] are plotted in black. \[E_{g\ 300\mathrm{K}}=1.424\ \mathrm{eV}-aN^{1/3} \tag{2}\]  
where \(N\) is the impurity concentration and \(a=9.8\times 10^{-9}\) eVcm for p-type GaAs and \(a=16.5\times 10^{-9}\) eVcm for n-type GaAs. In our case, doping with Sn generated an n-type semiconductor and a p-type semiconductor with Ge. As observed in Figure 5, the value of \(Eg\) obtained from PR exhibits a larger shift compared to the calculated value. This is because Equation (1) does not include the defects formed in GaAs over the 26 years. To incorporate the defects, the value of \(N\) in Equation (2) was changed to \(N+\mathrm{x}\), and \(\mathrm{x}\) was solved for the different experimental values of \(Eg\). This yielded an average value of \(\mathrm{x}\) or defect concentration of \(5\times 10^{17}\) atoms cm\({}^{-3}\). When we add this concentration of defects to \(N\) of Equation (1), the agreement is quite good. This value of the vacancy concentration is in line with the value obtained in another publication of ours using Raman spectroscopy [19], which indicates a value of \(7\times 10^{17}\) cm\({}^{-3}\). Figure 5: Values of the energy gap in GaAs according to Equation (1) and the energy gap values obtained from the PR fitting (shown in red). The stars include the correction explained in the main text. We reviewed the phenomenological broadening parameter in Table 1 because we can obtain information about the energy depth of the deep traps or defects. The total width squared is the sum of the squares of the instrumental, phononic or thermal, defects and impurities contributions, as follows:  
\[\Gamma_{obs}^{2}=\Gamma_{int}^{2}+\Gamma_{pho}^{2}+\Gamma_{def}^{2}+\Gamma_{ imp}^{2} \tag{3}\]  
For the width of defects, we will use the expression proposed by [PERSON] and [PERSON] [22]  
\[\Gamma_{def}=(\Gamma_{tr}(N_{T}+1)p)(exp(-2N_{T}S)) \tag{4}\]  
In Equation (4), \(\Gamma_{tr}\) is the internal width of the trap, \(N_{T}\) is the thermal distribution of the phonons, \(S\) is the Huang-Rhys factor or average of participating phonons and \(p\) is the ratio of the trap energy to the phonon energy. We already saw previously that the participating phonon is the LO phonon and that there are two phonons participating in the spectra in Figure 3, and therefore, \(S\) = 2. Calculation of the thermal distribution of the LO phonon at 300 K gives 0.46. In this way, Equation (4) becomes:  
\[\Gamma_{def}=\big{(}\Gamma_{tr}(1.46)^{p}\big{)}(0.025). \tag{5}\]  
From here we will do a rough calculation to estimate the value of the trap's energy. We took the observed PR widths from Equation (3) in 2021 (\(\Gamma_{21}^{2}\)) and 2001 (\(\Gamma_{01}^{2}\)) and subtract them. In this way, the instrumental width and the phononic width cancel out and we will assume that the width of the trap is the same since we have argued that the defect is the same only when its charge state changes. We will also consider that in low doping, the width due to impurities is negligible compared to the width of the defects. In this way, the calculations are:  
\[\Gamma_{21}^{2}-\Gamma_{01}^{2}=\Gamma_{tr}^{2}\ \Big{[}1.46^{2p_{21}}-1.46^{2p_{01}} \Big{]} \tag{6}\]  
[PERSON] [14] measured \(p_{01}\) = 3 and \(\Gamma_{tr}\) = 0.007 eV in 2001. From Table 1, we take the widths \(\Gamma_{21}\) and \(\Gamma_{01}\) equivalent to \(\Gamma_{1}\) for the 1 \(\times\) 10\({}^{16}\) cm\({}^{-3}\) sample in 2021 and 2001, respectively. That way, \(p_{21}\) will be 4.74. Since the energy of the trap is Et = 0.030p eV, then Et\({}_{01}\) = 0.090 eV and Et\({}_{21}\) = 0.142 eV. [PERSON] and [PERSON] [23] calculated the different charge states in both V\({}_{\rm Ga}\) and V\({}_{\rm As}\).