## 3 Results and Discussion
### PIN-Diodes Based on InGaN/GaN DHS with 5.5% In
The calculated QW width \(d_{\text{QW}}\) = \(d_{\text{InGaN}}\) = 7.09 \(\pm\) 1.04 nm converges to the value of nominal thickness (9.5 \(\pm\) 0.5 nm) striven  
\[Z(\omega)=R_{S}+\frac{1/R_{P}-i\omega C}{1/R_{P}^{2}+\omega^{2}C^{2}} \tag{2}\]6 Linear fitting of the Franz–Keldysh oscillation (FKO) extrema of the ER spectrum of the In\({}_{0.055}\)Ga\({}_{0.945}\)N/GaN DHS of Figure 4b (inset: electric field in the barrier and depletion region interface in dependence of bias-voltage). The calculated values of the In\({}_{0.055}\)Ga\({}_{0.945}\)N/GaN field \(F\) in the barrier interface region (\(F=F_{B}\)) are plotted in the inset of Figure 5 in dependence of the modulation voltage. FKOs are usually related to surface/interfaces, thus barrier regions of heterojunctions, and, in this case, to the field \(F_{B}\) existing at the InGaN/GaN interface barrier, which is expected to be weaker than the (total) field \(E_{pol}\). For a pin structure, there are three contributions to the FKOs [99]: one from the unintentionally doped QW barriers that have a uniform electric field of \(F_{B}\); the other two from the depletion regions, at the p- and n-sides, that have non-uniform electric fields with the maximum fields equal to \(F_{B}\). Under small AC modulation, the period of the FKOs originating in the depletion region is determined by the maximum electric field \(F_{B}\)[95; 99]. In general, if a quantum well (QW) structure is incorporated in the intrinsic region of a p-i-n diode, an external field can be applied, leading to an expression of the voltage drop across the DHS [72]:  
\[Z(\omega)=R_{S}+\frac{1/R_{P}-i\omega C}{1/R_{P}^{2}+\omega^{2}C^{2}} \tag{2}\]7  
where \(V_{appl}\) and \(V_{bi}\) are the externally applied bias and the built-in potential, \(L_{w}(L_{b})\) and \(E_{w}(E_{b})\) are the well (barrier) length and well (barrier) field, respectively. The QW-structure, however, has an internal envelope electric field, \(\overset{\rightarrow}{E}_{int}\), across the DHS that opposes the conventional built-in field [73]:  
\[Z(\omega)=R_{S}+\frac{1/R_{P}-i\omega C}{1/R_{P}^{2}+\omega^{2}C^{2}} \tag{2}\]8  
As a result, the energy minima and envelope functions of all sub-bands supported by the QW are strongly modified relative to the ideal flat-band case (\(\overset{\rightarrow}{E}_{w}=\overset{\rightarrow}{E}_{b}=0\)) through the quantum-confined Stark effect. The majority of approaches to determine the internal electric field have relied upon counterbalancing the quantum-confined Stark effect with an **externally applied reverse bias**\(V_{appl}\) and measuring properties of the quantum well as a function of this applied reverse bias. At low bias, the well is skewed due to the internal field. At a critical bias, the contributions from the applied bias and the internal field are equal and opposite [28]. The total field (\(\overset{\rightarrow}{F}=\overset{\rightarrow}{E}_{total}\)) in the DHS is approximated as follows:  
\[Z(\omega)=R_{S}+\frac{1/R_{P}-i\omega C}{1/R_{P}^{2}+\omega^{2}C^{2}} \tag{2}\]9  
Figure 2.0  
Figure 2.1  
where \(\overset{\rightarrow}{E}_{int}\) represents the fields due to the piezoelectric effect and the spontaneous polarization in the unintentionally doped (intrinsic) and depletion regions with spatial widths \(d_{u}\) and \(d_{d}\), respectively, and \(V_{bi}\) is the built-in voltage from the p-n junction. Considering that on (0001) substrates the electric fields in the wells and barriers are of equal strength (\(E_{w}=E_{b}=E_{pol}\)) [72], Equations (14)-(16) yield:  
Figure 2.2  
Hence, for a QW sandwiched inside a \(p\)-\(n\) junction (p-i(QW)-n), the electric field \(\overset{\rightarrow}{F}(=\overset{\rightarrow}{E}_{total}\)) is related to the polarization field \(\overset{\rightarrow}{E}_{pol}\) and bias voltage \(V_{appl}\) by Equation (17). The first term of Equation (17) is the background field written as the total voltage drop divided by the distance (\(d_{u}+d_{d}\)) over which that drop occurs. The externally applied voltage is negative for reverse bias.