## 3 Theoretical Background
### Dielectric Properties in the NIR \(\rightarrow\) UV Region
#### 3.3.1 \(\mathrm{E_{0}}\) and \(\mathrm{E_{0}}+\Delta_{0}\) Transition  
The \(\mathrm{E_{0}}\) and \(\mathrm{E_{0}}+\Delta_{0}\) transitions are three-dimensional (3D) \(\mathrm{M_{0}}\) CPs and occur in InP (for instance) at photon energies \(\sim\)1.35 eV and \(\sim\)1.45 eV at RT (see Table 2). Assuming the bands to be parabolic, the contribution of these energy gaps to \(\stackrel{{\sim}}{{\varepsilon}}(\mathrm{E})\) can be calculated by using [37; 38]  
\[\stackrel{{\sim}}{{\varepsilon}}^{\mathrm{a}}(\mathrm{E})=\mathrm{ A_{0}^{-1.5}}\Bigg{\{}\mathrm{f}(\chi_{0})+\frac{1}{2}\bigg{[}\frac{\mathrm{E_{0}}}{ \mathrm{E_{0}}+\Delta_{0}}\bigg{]}^{1.5}\mathrm{f}(\chi_{\mathrm{s.o.}}) \Bigg{\}}, \tag{8}\]  
with  
\[\mathrm{f}(\chi_{0})=\chi_{0}^{-2}\Big{[}2-(1+\chi_{0})^{0.5}-(1-\chi_{0})^{0.5 }\Big{]}, \tag{9a}\]  
\[\mathrm{f}(\chi_{\mathrm{s.o.}})=\chi_{\mathrm{s.o.}}^{-2}\Big{[}2-(1+\chi_{ \mathrm{s.o.}})^{0.5}-(1-\chi_{\mathrm{s.o.}})^{0.5}\Big{]}, \tag{9b}\]  
\[\chi_{0}=(\hbar\omega+\mathrm{i}\Gamma)/\mathrm{E_{0}}, \tag{9c}\]  
\[\chi_{\mathrm{s.o.}}=\frac{\hbar\omega+\mathrm{i}\Gamma}{(\mathrm{E_{0}}+ \Delta_{0})}. \tag{9d}\]  
In Equations (8) and (9a)-(9d), the terms A and \(\Gamma\) are the strength and broadening parameters, respectively, for the \(\mathrm{E_{0}}\) and \(\mathrm{E_{0}}+\Delta_{0}\) energy transitions. The discrete series  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**MDF Parameters** & **InSb** & **InP** \\ \hline E\({}_{0}\) (eV) & 0.18 & 1.35 \\ E\({}_{0}+\Delta_{0}\) (eV) & 0.99 & 1.45 \\ \(\Gamma_{0}\) (eV) & 0.015 & 0.23 \\ A\({}_{0}\) (eV)\({}^{1.5}\) & 0.19 & 7.1 \\ A\({}_{0\mathrm{K}}\) (eV\({}^{-1}\)) & 0.007 & 0.009 \\ E\({}_{1}\) (eV) & 1.86 & 3.1 \\ E\({}_{1}+\Delta_{1}\) (eV) & 2.37 & 3.31 \\ \(\Gamma_{1}\) (eV) & 0.19 & 0.34 \\ B\({}_{1}\) & 4.10 & 2.94 \\ B\({}_{2}\) & 2.70 & 0.96 \\ B\({}_{1x}^{1}\) (eV) & 1.87 & 1.64 \\ B\({}_{1 sx}^{1}\) (eV) & 0.98 & 0.82 \\ E\({}_{2}\) (eV) & 3.85 & 4.70 \\ E\({}_{0}^{\prime}\) (eV) & 3.26 & 4.2 \\ E\({}_{0}^{\prime}+\Delta_{0}^{\prime}\) (eV) & 3.63 & 4.40 \\ E\({}_{1}^{\prime}\) (eV) & 5.11 & 5.5 \\ C\({}_{2}\) & 2.35 & 0.22 \\ C\({}_{0}^{\prime}\) & 0.50 & 0.50 \\ C\({}_{30}^{\prime}\) & 0.35 & 0.70 \\ C\({}_{1}^{\prime}\) & 0.34 & 0.36 \\ \(\Gamma_{2}\) (eV) & 0.17 & 0.12 \\ \(\Gamma_{0}^{\prime}\) (eV) & 0.18 & 0.18 \\ \(\Gamma_{0}^{\prime}\) (eV) & 0.18 & 0.18 \\ \(\Gamma_{1}^{\prime}\) (eV) & 0.18 & 0.18 \\ \(\varepsilon_{10}\) & 1.20 & 0.50 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The best-fitted MDF parameters are exploited in the calculations of optical constants for binary InSb and InP materials (see text).  
(n = 1, 2, 3, \(\ldots\)) of exciton lines at E\({}_{0}\) and E\({}_{0}+\Delta_{0}\) gaps can be written with the Lorentzian line shape by using [38]  
\[\widehat{\epsilon}^{\rm a1}(\rm E)=\sum_{n=1}^{\infty}\frac{A_{0\rm x}}{n^{3}} \left[\frac{1}{E_{0}-\frac{G_{0}}{n^{2}}-h\omega-i\Gamma}+\frac{1}{2}\left\{ \frac{1}{E_{0}+\Delta_{0}-\frac{G_{0}}{n^{2}}-h\omega-i\Gamma}\right\}\right], \tag{10}\]  
where the parameters A\({}_{0\rm x}\) and G\({}_{0}\) are the 3D exciton strength parameter and exciton binding [G\({}_{0}=34\) meV] energy, respectively.