## 3 Results and Discussion  
### PIN-Diodes Based on InGaN/GaN DHS with 5.5% In  
The electric performance of the group-III nitride semiconductor PIN-diodes analyzed in the present work by electroreflectance (ER) and electroluminescence (EL) spectroscopy in view of performance relevant electro-optical field effects was initially approached by capacitance-voltage measurements (CVM) in [44]. A brief review of the CVM measurement principles applied in [44; 45; 46] on the PIN-diodes studied by ER and EL is given in the following, since it has delivered the built-in potential \(V_{bi}\) used to evaluate the polarization fields by ER. The capacitance \(C\) of a PIN-diode is directly related to the area and interspace of the equivalent capacitor (Equation (1)), in this case the (intrinsic) unintentionally doped (\(d_{u}\)) inclusive the depletion (\(d_{d}\)) region width \(d\) (=\(d_{u}+d_{d}\)) and the area \(A\) of the diode defined by the p-contact and the mesa area:  
\[C=\frac{\varepsilon_{0}\varepsilon_{r}A}{d} \tag{1}\]  
Various methods utilizing the equivalent circuit characteristics can be addressed for accurate determination of the capacitance and the herewith related material and device properties, as reported in previous and recent publications [44; 45; 46; 47; 48; 49; 50]. By application of CVM techniques, the capacitance \(C\) can be determined from the current in dependence of voltage I(V) characteristics using an equivalent circuit model to evaluate the resistances [46] or alternately fitted to the impedance complex function of a PIN-diode equivalent circuit [44]:  
\[Z(\omega)=R_{S}+\frac{1/R_{P}-i\omega C}{1/R_{P}^{2}+\omega^{2}C^{2}} \tag{2}\]  
where \(C\) is the measured capacitance, \(R_{S}\), \(R_{P}\) are the series and parallel resistance of the equivalent circuit, respectively, \(Z(\omega)\) is given by \(Im(Z(\omega))=\big{|}Z(\omega)\big{|}sin\theta\) and \(\big{|}Z(\omega)\big{|}=V_{AC}/I_{AC}\) with \(V_{AC}\), \(I_{AC}\) the rms-values of current and voltage, respectively, \(\theta\) is the phase-shift of current with respect to voltage, and \(\omega\) is the frequency. The main observable in CVM is the inverse of the capacitance square (1/C\({}^{2}\)), which depends linearly on the applied voltage [44; 45; 46]. The 1/C\({}^{2}\) dependence of the GaN reference on the applied bias-voltage is depicted in Figure 2. The slope of the 1/C\({}^{2}\)(V) curve is inversely proportional to the effective dopant concentration \(1/N_{A}+1/N_{D}\) of the PIN-diode, which in principle defines the width \(d\) of the unintentionally doped and the depletion region (\(d=d_{u}+d_{d}\)) [45]:  
\[d=\sqrt{d_{u}^{2}\frac{(N_{D}-N_{i})(N_{A}+N_{i})}{N_{D}N_{A}}+\frac{2 \varepsilon_{0}\varepsilon_{r}}{q}\bigg{(}\frac{N_{D}+N_{A}}{N_{D}N_{A}}\bigg{)} \bigg{(}V_{bi}-V-\frac{2 kT}{q}-E_{pol}d_{QW}\bigg{)}} \tag{3}\]  
Inserting Equation (3) in Equation (1) results in an explicit relationship of the squared inverse capacitance dependence on voltage 1/C\({}^{2}\)(V) described by Equation (4):  
\[\frac{1}{C^{2}}=\frac{d_{u}^{2}}{\varepsilon_{0}^{2}\varepsilon_{r}^{2}A^{2}} \bigg{(}\frac{(N_{D}-N_{i})(N_{A}+N_{i})}{N_{D}N_{A}}\bigg{)}+\frac{2}{ \varepsilon_{0}\varepsilon_{r}qA^{2}}\bigg{(}\frac{N_{D}+N_{A}}{N_{D}N_{A}} \bigg{)}\bigg{(}V_{bi}-V-\frac{2 kT}{q}-E_{pol}d_{QW}\bigg{)} \tag{4}\]  
where \(C\) is the capacitance, \(V\) is an externally applied bias voltage, \(V_{bi}\) is the built-in potential, and \(E_{pol}\) is the polarization field of group-III nitrides. \(N_{i}\), \(N_{A}\), and \(N_{D}\) are the intrinsic, the acceptor, and the donor dopant concentrations, respectively, \(d_{QW}\) is the thickness of the central ternary nitride (InGaN or InAlN) layer of the DHS, and \(d_{u}\) is the thickness of the unintentionally doped region in the binary nitride (GaN) reference and ternary nitride (InGaN or InAlN) DHS based PIN-diodes.