## 3 Results and Discussions
### Model Calculations  
To reproduce the TR-PL signals and their temperature dependences observed in Section 3.2, we consider the energy transfer model shown in Figure 7 and solve the following rate equations:  
\[\frac{dn_{0}}{dt} = -\Gamma n_{0}+A_{\rm r}n_{1}-C_{41}n_{4}n_{0}+C_{14}n_{3}n_{1}, \tag{2}\] \[\frac{dn_{1}}{dt} = W_{21}n_{2}-\left(A_{\rm r}+W_{\rm nr}\right)n_{1}-\left(1-\frac {n_{\rm d}}{N_{\rm d}}\right)W_{\rm d}n_{1}+C_{41}n_{4}n_{0}-C_{14}n_{3}n_{1},\] (3) \[\frac{dn_{2}}{dt} = \Gamma n_{0}-W_{21}n_{2},\] (4) \[\frac{dn_{\rm d}}{dt} = \left(1-\frac{n_{\rm d}}{N_{\rm d}}\right)W_{\rm d}n_{1},\] (5) \[\frac{dn_{3}}{dt} = \frac{n_{4}}{\tau_{43}}+\frac{n_{5}}{\tau_{53}}+C_{41}n_{4}n_{0}- C_{14}n_{3}n_{1}+C_{45}n_{4}^{2},\] (6) \[\frac{dn_{4}}{dt} = -2C_{45}n_{4}^{2}-\frac{n_{4}}{\tau_{43}}+\frac{n_{5}}{\tau_{54}} -C_{41}n_{4}n_{0}+C_{14}n_{3}n_{1},\] (7) \[\frac{dn_{5}}{dt} = C_{45}n_{4}^{2}-\frac{n_{5}}{\tau_{53}}-\frac{n_{5}}{\tau_{54}}, \tag{8}\]  
where \(n_{i}\) is the population probability of the level labeled \(i\) (\(i=0\)\(-\)5, d), \(\Gamma\) is the pumping rate, \(C_{14}\) and \(C_{41}\) are the inter-site ET rate between the states 1 (\(Y_{1}^{\prime}\)) and 4 (\(Y_{1}\)), and \(C_{45}\) is the upconversion rate in \(C_{2}\) sites from 4 to 5 (UC levels) states. In this model, the cross-relaxation and Auger process are assumed for inter-site ET and UC, respectively. For state 1, the radiative relaxation rate \(A_{\rm r}\) is explicitly introduced and the radiative and non-radiative relaxation processes are definitely distinguished; by contrast, the lifetimes of states 4 and 5, \(\tau_{43}\), \(\tau_{54}\), and \(\tau_{53}\) are determined by both the radiative and  
Figure 6: Temperature dependences of the relaxation times, \(\tau_{\rm fast}\) (open circles) and \(\tau_{\rm slow}\) (solid circles), for the three samples; (**a**) A: \(x=1.000\), (**b**) B: \(x=0.054\), and (**c**) C: \(x=0.012\). The solid and dashed lines are results of model calculation. (**d**) Arrhenius plot of \(\tau_{\rm slow}\) in Sample A (circles), B (triangles), and C (squares). The solid lines are the fittings by using Equation (1). non-radiative relaxation processes (i.e., level lifetimes). The rapid internal relaxation from states 2 to \(1(Y_{3}^{\prime}\rightarrow Y_{1}^{\prime})\) with a rate \(W_{21}\) follows the optical pumping to state 2 with a rate \(\Gamma\). In our model, two kinds of non-radiative relaxation processes, the temperature-sensitive relaxation with a rate \(W_{\rm nr}\) and the temperature-insensitive relaxation with a rate \(W_{\rm d}\) are considered for state 1. Here, the rate \(W_{\rm nr}\) has the thermally-activation type as similar to the \(\tau_{\rm slow}\) process in Section 3.2, and it can be written as \(W_{\rm nr}=W_{\rm nr}^{0}\exp\left(-E_{\rm A}/k_{\rm B}T\right)\). Further, the temperature-insensitive non-radiation relaxation process with \(W_{\rm d}\) needs to have the density-of-state limitation property as shown in Equation (5) in order to reproduce the double exponential decay curve. Although the UC procces is one of the candidates to induce the double exponential decay, it can be negligible in our experimental condition with a low excitation power, especially for the low Er-concentration Samples B and C. The energies via these non-radiative relaxations are dissipated from the considered system. Additionally, the UC rate C\({}_{45}\) is set to zero because the TR-PL measurements were carried out under the low power excitation condition and the effect of the UC process was negligible.