## 3 Results and Discussion
### PIN-Diodes Based on InGaN/GaN DHS with 5.5% In
\(A\) is the p-contact area that equals the diode surface area, \(q\) is the elementary charge, \(T\) is the temperature in Kelvin, and \(k\) is the Boltzmann constant; \(\varepsilon_{0}\), \(\varepsilon_{r}\) are the vacuum and relative permittivity, respectively. For a constant doping profile, the slope of 1/C\({}^{2}\)(V) is constant. The built-in potential \(V_{bi}\) and the polarization field \(E_{pol}\) of group-III nitrides can thus be accurately determined, as demonstrated in recent publications [44; 45; 46]:  
\[V_{bi}=V+\frac{2 kT}{q}+\left(\frac{\partial}{\partial V}\frac{1}{C_{ref}^{2}(V)} \right)^{-1}\left(\frac{d_{u,ref}^{2}}{e_{0}^{2}c_{r,ref}^{2}A_{ref}^{2}}\frac{ (N_{D}-N_{i})(N_{A}+N_{i})}{N_{A}N_{D}}-\frac{1}{C_{ref}^{2}}\right) \tag{5}\]  
\[E_{pol}=\frac{1}{d_{QW}}\left[V_{bi}-V-\frac{2 kT}{q}+\left(\frac{\partial}{ \partial V}\frac{1}{C^{2}(V)}\right)^{-1}\left(\frac{1}{C^{2}}-\frac{d_{u}^{2} }{e_{0}^{2}c_{r}^{2}A^{2}}\frac{(N_{D}-N_{i})(N_{A}+N_{i})}{N_{D}N_{A}}\right)\right] \tag{6}\]  
The built-in potential \(V_{bi}\) deduced by CVM from the 1/C\({}^{2}\)(V) characteristic curve of the GaN reference sample exhibited a relatively weak dependence on applied voltage with \(V_{bi}\) (1/C\({}^{2}\) (V = 0)) = 3.28 \(\pm\) 0.22 V, which is relevant for all processes close to equilibrium. This value is used in the determination of \(E_{pol}\) by CVM in [44; 45] and by ER in this work. The polarization field \(E_{pol}\), which is a strictly decreasing function of the indium content, promotes interface charge generation and, therefore, the increase of the capacitance. Dependent on the sign and strength of the polarization fields, the 1/C\({}^{2}\)(V) characteristics of the In\({}_{\rm x}\)Ga\({}_{\rm 1-x}\)N/GaN, x = 0.055, and In\({}_{\rm x}\)Al\({}_{\rm 1-x}\)N/GaN, x = 0.15, 0.18, 0.21, PIN-diodes are expected to be: the former higher and the latter lower than the 1/C\({}^{2}\)(V) of the GaN reference. Group-III nitride semiconductors possess direct band gaps over a large energy range. It is the ability to create direct gap ternary alloys such as In\({}_{\rm x}\)Ga\({}_{\rm 1-x}\)N, Al\({}_{\rm x}\)Ga\({}_{\rm 1-x}\)N or In\({}_{\rm x}\)Al\({}_{\rm 1-x}\)N with energies beyond those of the constituent binary III nitrides that makes them profitable for use in optoelectronic devices. A band diagram of the wurtzite (WZ) crystal structure [33] in a simplified \(E(k)\) representation at \(k\) = 0 (\(\Gamma\)-point) involves a non-degenerate conduction band (CB) and three separate valence split bands (VBs) upon degeneracy removal due to the hexagonal crystal field \(\Delta_{cr}\) and the spin-orbit interaction \(\Delta_{so}\), as demonstrated in Figure 3 (left side). In reality, the bands extend over a larger range of momentum with non-parabolic dispersions and there may also be more than one confined states for both the CB and VBs. The top of the valence band is split into twofold- and single-degenerate states originated by the hexagonal crystal field \(\Delta_{cr}\); the former is labeled \(\Gamma_{6}\) and the latter \(\Gamma_{1}\)[61; 62; 63]. In case of GaN, \(\Gamma_{6}\) is higher than \(\Gamma_{1}\), while in case of AlN, the \(\Gamma_{1}\) level is higher. The two-fold degenerate \(\Gamma_{6}\) state is split into \(\Gamma_{9}\) and \(\Gamma_{7}\) by the spin-orbit interaction \(\Delta_{so}\); the single degenerate state has also \(\Gamma_{7}\) symmetry. Hence, the two \(\Gamma_{7}\) states are mixed by the spin-orbit coupling. The hole masses of the three states: \(\Gamma_{9}\), \(\Gamma_{7+}\), and \(\Gamma_{7-}\) have a large k dependence [61; 62], and are labeled HH (heavy-holes), LH (light-holes), and CH (or SO) (crystal field split-off holes) [62; 64], as shown in Figure 3. Near the band  
Figure 2: Plot of 1/C\({}^{2}\)(V) dependence of the PIN-diode based on the GaN reference [44] (inset: V\({}_{\rm bi}\) deduced from 1/C\({}^{2}\)(V) of GaN reference).