## 3 Results and Discussion
### Comments on Formalism of Polarization Field Determination  
A basic property of nitride semiconductors, which directly affects the electronic, transport, and optical properties of heterostructures, is the presence of strong spontaneous \(\mathrm{P_{SP}}\) and piezoelectric \(\mathrm{P_{PE}}\) polarizations. The orientation of both \(\mathrm{P_{SP}}\) and \(\mathrm{P_{PE}}\) polarizations along the c-axis depends on the sequence of atomic layers in the crystal. The compositional dependence and large magnitude of \(\mathrm{P_{SP}}\) and \(\mathrm{P_{PE}}\) play an important role in the optoelectronic device physics of nitride semiconductors due to the associated polarization charge distributions. A surface charge distribution results in the formation of an electrostatic field perpendicular to the interface. In heterostructures, polarization charges are induced on each heterojunction and can produce large internal electric fields that strongly bend the conduction- and valence-band edges near the interface. The electric fields produced by the polarization charges in heterostructures can be evaluated using principles of electrodynamics. In case of multiple-quantum-well (MQW) structures, which are used routinely in most optoelectronic device applications, the well and barrier materials can be InGaN and GaN, AlGaN and GaN, or InAlN and GaN, as in the present case, and their properties are denoted by the subscripts w and b, respectively. In accordance with [PERSON]'s theory of the electromagnetic field, the electric displacement \(\overset{\rightarrow}{D}\) of a material of dielectric permittivity \(\varepsilon\) subjected to electrical polarization \(\overset{\rightarrow}{P}\) under the influence of an (external) electric field \(\overset{\rightarrow}{E}\) is given by:  
\[\overset{\rightarrow}{D}=\varepsilon\overset{\rightarrow}{E}=(1+\chi) \overset{\rightarrow}{E}=\overset{\rightarrow}{E}+\chi\overset{ \rightarrow}{E}=\overset{\rightarrow}{E}+\overset{\rightarrow}{P} \tag{22}\]  
Continuity of the displacement flux across each interface of a well-barrier pair requires that:  
\[\overset{\rightarrow}{E}_{w}+\overset{\rightarrow}{P}_{w}=\overset{ \rightarrow}{E}_{b}+\overset{\rightarrow}{P}_{b} \tag{23}\]  
Figure 9: Comparison of the polarization field values \(\mathrm{E_{pol}}\) of \(\mathrm{In_{0,055}Ga_{0.945}N}\) DHS (open triangle) and \(\mathrm{In_{0,21}Al_{0.79}N}\) DHS (open circle) determined by ER in this work with the field values of \(\mathrm{In_{0,055}Ga_{0.945}N}\)[44], \(\mathrm{In_{0,08}Ga_{0.92}N}\) DHS [46] (full triangles) and \(\mathrm{In_{0.15}Al_{0.85}N},\mathrm{In_{0.18}Al_{0.82}N},\mathrm{In_{0.21}Al_{ 0.79}N}\) DHS [45] (full circles) determined by CVM, and with theoretically predicted values [19; 38] (dashed line) and [26] (dot-dashed line). A simple analytical solution for the fields in well \(\vv{E}_{w}\) and barrier \(\vv{E}_{b}\) can be obtained in the limit of an infinitely periodic structure, where the intrinsic voltage drop across each well/barrier pair must be zero [110]:  
\[\vv{E}_{w}L_{w}+\vv{E}_{b}L_{b}=0 \tag{24}\]  
with \(L=L_{w}+L_{b}\) being the layer thickness. Combining Equations (23) and (24) yields the polarization-induced built-in electric field in the well layers:  
\[\vv{E}_{w}=\frac{\pqty(P_{b}-\vv{P}_{w})L_{b}}{L_{w}+L_{b}} \tag{25}\]  
Very large values of up to a few MV/cm are typically computed for these fields in c-plane nitride heterostructures. As a result, the energy minima and envelope functions of all sub-bands supported by the QWs are strongly modified relative to the ideal flat-band case (\(\vv{E}_{w}=\vv{E}_{b}=0\)) through the quantum-confined [PERSON] effect. The existence of strong piezoelectric fields results thus in a red-shift of the excitonic transitions within strained QWs. Application of an external bias can modulate these fields to obtain a blue-shift of the absorption edge.