## 3 Theoretical Background
### Dielectric Properties in the NIR \(\rightarrow\) UV Region
#### 3.3.2 E\({}_{1}\) and E\({}_{1}+\Delta_{1}\) Transition  
The E\({}_{1}\) and E\({}_{1}+\Delta_{1}\) transitions in the zb-type semiconductors can take place along the [111] directions (\(\Lambda\)) or at L point in the BZ. These transitions are of the 3D M\({}_{1}\) type and occur in the zb InP at energies \(\sim\)3.1 eV and \(\sim\)3.31 eV, respectively. However, as the M\({}_{1}\) CP longitudinal effective mass is much larger than its transverse counterparts, one can treat these 3D M\({}_{1}\) CPs as a two-dimensional (2D) minimum M\({}_{0}\). The contribution to \(\widetilde{\epsilon}(\rm E)\) of the 2D minimum is given by [38]  
\[\widehat{\epsilon}^{\rm b}(\rm E)=-B_{1}\chi_{1}^{-1}\ln\left(1-\chi_{1}^{2} \right)-B_{2}\chi_{1\rm s}^{-2}\ln\left(1-\chi_{1\rm s}^{2}\right), \tag{11}\]  
with  
\[\chi_{1}=(h\omega+i\Gamma)/E_{1}, \tag{12a}\] \[\chi_{1\rm s}=\frac{h\omega+i\Gamma}{(E_{1}+\Delta_{1})}, \tag{12b}\]  
where B\({}^{\prime}\)s and \(\Gamma\) are the strength and broadening parameters, respectively, of energy transitions. One must note that Equation (11) is a consequence of the one-electron approximation. Excitonic states should, in principle, exist at each type of CP, since the Coulomb-like interaction is always present between the electrons and holes. There may be only two analytical equations which enable us to treat the excitonic effects in the E\({}_{1}\) and E\({}_{1}+\Delta_{1}\) spectral region: (i) the effective mass (EM) approximation and (ii) the Koster-Slater (KS) method [38]. Both the EM approximation and the KS method can dramatically modify and sharpen the E\({}_{1}\) and E\({}_{1}+\Delta_{1}\) structures of semiconductors. We, however, find a better fit with experiments using the EM approximation rather than the KS method. In the case of 3D M\({}_{1}\) CPs (i.e., the saddle-point excitons or hyperbolic excitons), the EM equation is much more difficult to solve. However, in the approximation of 2D M\({}_{0}\) CP, the equation gives a series of 2D Wannier-type excitons (discrete excitons). The contribution of these excitons to \(\widetilde{\epsilon}(\rm E)\) can now be written with Lorentzian line shape as [38]  
\[\widehat{\epsilon}^{\rm b1}(\rm E)=\sum_{n=1}^{\infty}\frac{1}{\left(2n-1 \right)^{3}}\left[\frac{B_{1\rm x}^{1}}{E_{1}-\frac{G_{1}}{(2n-2)^{2}}-h\omega -i\Gamma_{1}}+\frac{B_{1\rm x}^{1}}{E_{1}+\Delta_{1}-\frac{G_{1}}{(2n-1)^{2} }-h\omega-i\Gamma_{2}}\right], \tag{13}\]  
where the \(B_{1\rm x}^{1}\) and \(B_{1\rm x}^{1}\) are the 2D exciton strength parameters at the E\({}_{1}\) and E\({}_{1}+\Delta_{1}\) saddle point excitons, respectively, and G\({}_{1}\) is the 2D exciton binding energy. The 2D EM approximation also gives a continuum part of the excitonic states. However, one can consider that the contribution of this part is like that of the one-electron approximation (i.e., Equation (10)). 3.3 E\({}_{0}^{\prime}\), E\({}_{0}^{\prime}+\Delta_{0}^{\prime}\), E\({}_{2}\) and E\({}_{1}^{\prime}\) Transitions  
In the optical spectra of InP (InSb), the more pronounced structures found in the region higher than E\({}_{1}\) and E\({}_{1}+\Delta_{1}\) energies [37; 38] can be labeled E\({}_{0}^{\prime}\), E\({}_{0}^{\prime}+\Delta_{0}^{\prime}\), E\({}_{2}\) and E\({}_{1}^{\prime}\). The E\({}_{0}^{\prime}\) and E\({}_{0}^{\prime}+\Delta_{0}^{\prime}\) energy transitions in the zb semiconductors are believed to take place either at the \(\Gamma\) point or in the \(\Delta\) direction near \(\Gamma\) point (Figure 1). One must note that the nature of E transition is more complicated as it does not correspond to a single, well-defined CP. The E\({}_{1}^{\prime}\) transitions may take place, however, from the top of the valence band (\(\Lambda_{3}^{V}\)) to the second-lowest conduction band (\(\Lambda_{3}^{C}\)) near the L point.