# Study on Optical Properties and Internal Quantum Efficiency Measurement of GaN-based Green LEDs
## 3 Results
### Determination of IQE Values  
In TDPL experiment, IQE values at different temperature can be expressed as  
\[\text{IQE}_{\text{TDPL}}(T,P)=\frac{I(T,P)}{I(\text{LT},P)} \tag{1}\]  
where \(I(T,P)\) denotes integrated intensity at the temperature \(T\) and excitation power \(P\) and \(I(\text{LT},P)\) denotes integrated intensity at the lowest temperature, e.g., LT equals 10 K in this study. This expression reflects the proportions of IQE values at different temperatures correctly, which are relative values. However, this is based on the fact that there is no nonradiative recombination at LT, which is not always reasonable and needs further verification. To correct this latent mistake, PDPL experiment can be performed to calculate the IQE value at LT and modify the TDPL results. According to [PERSON] et al. [21], the relationship between carrier generation rate \(G\) and excitation power \(P\) can be expressed as  
\[G=P\frac{(1-R)\alpha}{A_{\text{spot}}h\
u} \tag{2}\]  
where \(R\) denotes the reflection ratio, \(\alpha\) denotes the absorption coefficient, \(A_{\text{spot}}\) denotes the area of the laser spot, and \(h\
u\) denotes photon energy. It is generally accepted that the non-equilibrium carriers generated are likely to recombine through three channels, that is, nonradiative recombination, bimolecular radiative recombination, and Auger nonradiative recombination. So, the rate equation can be expressed as  
\[G=An+Bn^{2}+Cn^{3} \tag{3}\]  
where \(A\), \(B\), and \(C\) are nonradiative recombination coefficient, bimolecular radiative recombination coefficient, and Auger nonradiative recombination coefficient, respectively, which is called the ABC model [22]. Whether Auger recombination can be neglected or not depends on carrier density, which equals the product of carrier generation rate and carrier lifetime. Here we should take Auger recombination into consideration, which can be proved by fitting the \(P-I\) curves with the contribution of Auger recombination. Assume that integrated intensity \(I\) is proportional to carriers involved in radiative recombination, namely \(I=\beta Bn^{2}\) where \(\beta\) is a scale factor related to light extraction efficiency. Thus, the relationship between \(P\) and \(I\) can be expressed as  
\[P=mG=\frac{mA}{\sqrt{B\beta}}\sqrt{I}+\frac{m}{\beta}I+\frac{mC}{(B\beta)^{ \frac{3}{2}}}I^{\frac{3}{2}}=Q_{1}\sqrt{I}+Q_{2}I+Q_{3}I^{\frac{3}{2}} \tag{4}\]  
Polynomial curve fit is used to extract fitting coefficients \(Q_{1}\), \(Q_{2}\), and \(Q_{3}\), which is shown in Figure 4. Rearrange this equation with these fitting coefficients and we can express \(P\) as a function of \(\sqrt{mBn}\). Solve this cubic equation using numerical approach and IQE can be derived as  
\[\text{IQE}_{\text{FDPL}}(T,P)=\frac{B(T)n(T,P)^{2}}{G(T,P)}=\frac{\left(\sqrt {m(T)B(T)}n(T,P)\right)^{2}}{P} \tag{5}\]  
Figure 4(a) shows the IQE values stemming from TDPL experiment and IQE values decrease monotonically with increasing temperature, which reflects the proportion of nonradiative recombination increases as the temperature rises. The IQE values calculated decrease slightly with increasing excitation power, which means nonradiative recombination centers (NRCs) are saturated even at relatively low excitation power and Auger recombination starts to dominate. Figure 5b shows the IQE values stemming from PDPL experiment at LT and room temperature (RT) and IQE values at LT are not 100%. Assuming that IQE values at LT measured by PDPL experiment \(\mathrm{IQE_{PDPL}(LT,P)}\) are reasonable, we can calculate the absolute IQE values at RT as  
\[\mathrm{IQE_{a}(T,P)=IQE_{PDPL}(LT,P)\times IQE_{TDPL}(T,P)} \tag{6}\]  
which should be close to IQE values at RT measured by PDPL experiment \(\mathrm{IQE_{PDPL}(RT,P)}\) to make sure PDPL results are trustworthy. For instance, \(\mathrm{IQE_{PDPL}(LT,250\ mW)}\) equals 92.68% and the proportion of IQE values at LT and RT we acquired by TDPL experiment \(\mathrm{IQE_{TDPL}(RT,250\ mW)}\) equals 39.26%.