## 3 Results and Discussion
### Comments on Formalism of Polarization Field Determination
If a multiple-quantum-well (MQW) structure is incorporated in the intrinsic region of a p-i-n diode, an external field can be applied leading to the expression [72]:  
\[E_{w}L_{w}+E_{b}L_{b}=-\pqty(V_{bi}+V_{appl}) \tag{26}\]  
where \(V_{appl}\) and \(V_{bi}\) are the applied and built-in potentials, respectively. Combining Equations (23) and (26) gives the polarization-induced built-in electric field in the well layers under the influence of an external field:  
\[E_{w}L_{w}+E_{w}L_{b}+(P_{w}-P_{b})L_{b}=-\pqty(V_{bi}+V_{appl}) \tag{27}\]  
On (100) substrates, the electric fields in the wells and barriers are of equal strength, while, for structures grown on substrates other than (100), the piezoelectric field \(\vv{E}_{p}\) can be expressed as [72]:  
\[\vv{E}_{p}\equiv\vv{E}_{w}-\vv{E}_{b} \tag{28}\]  
Taking into consideration Equation (23), it can also be expressed as:  
\[\vv{E}_{p}\equiv\vv{P}_{b}-\vv{P}_{w} \tag{29}\]  
By substitution of Equation (29) into Equation (27), the well and barrier fields for a p-i-n structure, grown so that the p-doped layer is uppermost, are:  
\[E_{w}=\frac{-\pqty(V_{bi}+V_{appl})+(P_{b}-P_{w})L_{b}}{L_{w}+L_{b}}=-\frac{(V _{bi}+V_{appl})}{L}+\frac{E_{p}L_{b}}{L} \tag{30}\]  
\[\overset{\rightarrow}{E}_{w}+\overset{\rightarrow}{P}_{w}=\overset{ \rightarrow}{E}_{b}+\overset{\rightarrow}{P}_{b} \tag{23}\]0  
Equations (30) and (31) indicate that for a given external bias \(V_{appl}\), the well \(\vv{E}_{w}\) and barrier \(\vv{E}_{b}\) fields are dependent on the i-region length and the lengths of the barrier and the well material, respectively. The magnitude of the overall well field is reduced as more wells are incorporated into an i-region of a p-i(MQW)-n diode structure of a given thickness; likewise the barrier fields must increase accordingly:  
\[\overset{\rightarrow}{E}_{w}+\overset{\rightarrow}{P}_{w}=\overset{ \rightarrow}{E}_{b}+\overset{\rightarrow}{P}_{b} \tag{23}\]1  
\[\overset{\rightarrow}{E}_{w}+\overset{\rightarrow}{P}_{w}=\overset{ \rightarrow}{E}_{b}+\overset{\rightarrow}{P}_{b} \tag{23}\]2  
and  
\[\overset{\rightarrow}{E}_{w}+\overset{\rightarrow}{P}_{w}=\overset{ \rightarrow}{E}_{b}+\overset{\rightarrow}{P}_{b} \tag{23}\]3  
where \(L_{w}\) is the length of a single QW and \(n\) is the number of wells. Using Equation (34), Equation (32) can be then rewritten as:  
\[\overset{\rightarrow}{E}_{w}+\overset{\rightarrow}{P}_{w}=\overset{ \rightarrow}{E}_{b}+\overset{\rightarrow}{P}_{b} \tag{23}\]4  
In the absence of illumination, the electric field \(\overset{\rightarrow}{E}_{w}\) within the wells amounts [73]:  
\[\overset{\rightarrow}{E}_{w}+\overset{\rightarrow}{P}_{w}=\overset{ \rightarrow}{E}_{b}+\overset{\rightarrow}{P}_{b} \tag{23}\]5  
The internal electric field is determined by counteracting the quantum-confined Stark effect (QCSE) with an externally applied reverse bias; the reverse bias \(V_{appl}\) acts to oppose the internal field and attenuate the induced QCSE. A decrease of the internal polarization field \(\overset{\rightarrow}{E}_{p}\) implies cancellation of the piezoelectric field \(\overset{\rightarrow}{E}_{piezo}\) by the reverse bias. The piezoelectric field is set, in principle, against the built-in \(V_{bi}\) and reverse bias field \(V_{appl}\). In [69], the total internal field (\(\overset{\rightarrow}{E}_{total}\)) in the well layer was approximated as follows:  
\[\overset{\rightarrow}{E}_{w}+\overset{\rightarrow}{P}_{w}=\overset{ \rightarrow}{E}_{b}+\overset{\rightarrow}{P}_{b} \tag{23}\]6  
\[\overset{\rightarrow}{E}_{w}+\overset{\rightarrow}{P}_{w}=\overset{ \rightarrow}{E}_{b}+\overset{\rightarrow}{P}_{b} \tag{23}\]7  
where \(V_{bi}\), \(V_{appl}\), \(\overset{\rightarrow}{E}_{piezo}\), and \(\overset{\rightarrow}{E}_{i}\) represent the built-in potential, the applied voltage, the piezoelectric field, and the internal field in the undoped and depletion regions, respectively.