## 3 Results and Discussion
### PIN-Diodes Based on InGaN/GaN DHS with 5.5% In
The extrema of FKOs are given by [90; 94; 95]:  
\[Z(\omega)=R_{S}+\frac{1/R_{P}-i\omega C}{1/R_{P}^{2}+\omega^{2}C^{2}} \tag{2}\]2  
where \(m_{FKO}\) is the index of the \(m\)-th extremum, \(\varphi\) is an arbitrary phase factor, \(E_{m}\) is the photon energy of the \(m\)-th oscillation, \(E_{0}\) is the energy gap, and \(\hbar\Theta\) is the electro-optic energy given by:  
\[Z(\omega)=R_{S}+\frac{1/R_{P}-i\omega C}{1/R_{P}^{2}+\omega^{2}C^{2}} \tag{2}\]3  
\[Z(\omega)=R_{S}+\frac{1/R_{P}-i\omega C}{1/R_{P}^{2}+\omega^{2}C^{2}} \tag{2}\]4 ER spectra of PIN-diodes based on (**a**) the GaN reference sample and (**b**) the In\({}_{0.055}\)Ga\({}_{0.945}\)N/GaN DHS with three-split valence bands \(E_{a}\), \(E_{b}\), and \(E_{c}\) of the wurtzite crystal structure. with \(q_{e}\) the electron elementary charge and \(\mu_{//}\big{(}=m_{e}^{*}m_{hh}^{*}\big{/}\big{(}m_{e}^{*}+m_{hh}^{*}\big{)}\big{)}\) the reduced interband effective mass for the electron and heavy-hole pair in the direction of \(F\). The electric field \(F\) can, therefore, be obtained directly from the period of the FKOs, if \(\mu_{//}\) is known. A plot of the FKO extrema \(E_{m}\) as a function of redefined index \(m=\big{[}(3/4)m_{\text{FKO}}\pi\big{]}^{2/3}\) (Figure 5) yields a straight line with slope \(\hbar\Theta=0.01536\) eV and ordinate-intercept the \(E_{a}\) transition energy (fundamental gap energy \(E_{0}\)) of the In\({}_{0.055}\)Ga\({}_{0.945}\)N/GaN quantum well heterostructure, \(E_{0}=3.260\pm 0.002\) eV, averaged over several measurements, \(<\)\(E_{0}\)\(>\) = 3.248 \(\pm\) 0.006 eV. The bandgap energies of III-V ternary compounds have an approximately quadratic dependence on the alloy composition \(x\): \(E_{0}\)(A\({}_{\text{x}}\)B\({}_{\text{1-x}}\)C) = x \(E_{0}\)(AC) + (1 - x) \(E_{0}\)(BC) + \(p\)(A-B) x(1 - x) [7; 100]. According to recently published literature on the fundamental properties of III-V ternary and quaternary alloys [100], the band-gap energies of the related III-V binaries at 300 K and the bowing parameter \(p\)(A-B), in case of A\({}_{\text{x}}\)B\({}_{\text{1-x}}\)C = In\({}_{\text{x}}\)Ga\({}_{\text{1-x}}\)N, are: \(E_{0}\)(InN) = 1.1 eV, \(E_{0}\)(GaN) = 3.42 eV, and \(p\)(In-Ga) = -1.64. Considering that the (fundamental) energy band gap of bulk InGaN with 5.5% In-fraction is \(E_{g}\)(In\({}_{0.055}\)Ga\({}_{0.945}\)N) = 3.207 eV, the excess energy \(\Delta E\) = (3.248-3.207 eV =) 0.041 \(\pm\) 0.006 eV originates from the band gap broadening of the InGaN/GaN QW heterostructure as described by Equation (8). For the respective interband transition, rearrangement of the terms in Equation (8) yields the width \(L_{W}\) (n = 1) of the InGaN/GaN quantum well heterostructure:  
\[Z(\omega)=R_{S}+\frac{1/R_{P}-i\omega C}{1/R_{P}^{2}+\omega^{2}C^{2}} \tag{2}\]5  
where \(\Delta E\) = 0.041 eV (1 eV = 1.602 \(\times\) 10\({}^{-19}\) J) is the band broadening due to quantum confinement and \(\mu_{//}\) is the reduced effective mass in In\({}_{0.055}\)Ga\({}_{0.945}\)N set equal to \(\mu_{//}\) in GaN. For the calculation of the internal electric field \(F\) with Equation (10) and the QW width \(d_{\text{QW}}\) (\(=L_{W}\)) with Equation (11), the effective mass of electrons and holes in In\({}_{0.055}\)Ga\({}_{0.945}\)N was set equal to \(m_{e}^{*}=0.20m_{0}\) and \(m_{hh}^{*}=2.20m_{0}\) in GaN [101], with the free electron rest mass \(m_{0}\) = 9.109 \(\times\) 10\({}^{-31}\) kg, the elementary charge \(q_{e}\) = 1.602 \(\times\) 10\({}^{-19}\) C, and [PERSON]'s constant \(\hbar=h/2\pi\), \(h\) = 6.626 \(\times\) 10\({}^{-34}\) m\({}^{2}\) kg/s.