## 3 Results and Discussion
### PIN-Diodes Based on InGaN/GaN DHS with 5.5% In
At a flat-barrier (FB) condition, the electric field in the barrier-interface region (\(\overset{\rightarrow}{F}=\overset{\rightarrow}{F}_{B}\)) is zero and the polarization field \(\overset{\rightarrow}{E}_{pol}\) can be determined by solving Equation (17) for \(F_{B}(=F)=0\)[99]:  
Figure 2.3  
where \(V_{FB}\) is the bias voltage at flat-barrier condition. In the vicinity of flat-barrier, \(F_{B}(=F)\) approaches a linear relation with bias voltage. For the In\({}_{0.055}\)Ga\({}_{0.945}\)N/GaN field studied in this work, the flat-barrier condition is sustained for voltages below the emission threshold voltage of the respective PIN-diode of 4 V AC, as demonstrated in Figure 2.4 The field is maintained in the voltage-plateau 4-6 V, and is overwhelmed above 8 V, where FKOs are almost buried in the superimposed UV emission. In the inset of Figure 5, a fit of the electric field in dependence of voltage \(F_{B}\)(V) with a straight line up to \(F_{B}=0\) intersects the \(x\)-axis at a flat-barrier voltage of \(V_{FB}=8.47\pm 0.58\) V. Using the built-in potential value selected at 0 V in capacitance-voltage measurements \(V_{bi}=3.28\pm 0.22\) V (part 3 Results and Discussion), since all equations were derived close to equilibrium [65], and \(d_{u}=23.0\pm 2.0\) nm (Figure 1b), the polarization field in the In\({}_{0.055}\)Ga\({}_{0.945}\)N/GaN DHS is found to be \(E_{pol}=-2.26\pm 0.33\) MV/cm in the same direction as the built-in field. The field \(F\) experienced by carriers with concentration \(N_{InGaN}\) and charge \(q_{e}\) across the InGaN/GaN DHS under application of an external voltage \(V_{AC}\) can be approximated by [90]:  
Figure 2.5  
where \(\varepsilon_{InGaN}=10.518\) is the relative permittivity of the intrinsic InGaN layer [26] and \(\varepsilon_{0}=8.854\times 10^{-12}\) F/m is the vacuum permittivity. The field \(F\) modulated with \(V_{AC}=2\) V is according to Equation (10) \(F=4.18\times 10^{6}\) V/m. Figure 2.6 ER spectra of PIN-diodes based on In\({}_{0.055}\)Ga\({}_{0.945}\)N/GaN DHS in dependence of AC modulation voltage. The carrier concentration determined by Equation (20) after term-reorder of Equation (19):  
Figure 2.7  
is \(N_{InGaN}\) = 4.04 \(\times\) 10\({}^{15}\) cm\({}^{-3}\), which is a realistic value considering that the InGaN layer is unintentionally doped with dopant concentration typically in the range 10\({}^{15}\)-10\({}^{16}\) cm\({}^{-3}\), and that FKOs are generally observed in this doping range [90]. Oscillations at energies below the band gap energy, as demonstrated in Figure 7, are thickness interferences [102; 103; 104; 105] and can be used to determine the thickness of the GaN layer, alternately, the thickness of the GaN/InGaN/GaN layer region penetrated by the incident beam:  
Figure 2.8  
where \(t\) is the thickness of the film, \(\lambda_{1}\) and \(\lambda_{2}\), \(\lambda_{1}>\lambda_{2}\), are the begin- and end-wavelength of the spectral region where the oscillations appear, \(N\) is the number of oscillations within this region, \(n_{f}\) is the refractive index of the film, and \(\theta\) is the angle between the incident beam and the normal to the surface [104]. For \(N\) = 10 oscillations in the range 2.583-3.107 eV (400-480 nm), at angle of incidence \(\theta\) = 45\({}^{\circ}\), a thickness of \(t\) = 4.5 \(\upmu\)m was calculated by inserting n\({}_{InGaN}\)\(\approx\) n\({}_{GaN}\) = 2.82 [89; 106; 107; 108; 109] in Equation (21), which accounts for both the GaN and the InGaN DHS based PIN-diode layer thickness in Figure 1a,b.