## 3 Results and Discussions
### Time-Resolved Photoluminescence Measurements  
The population dynamics can be reflected more directly in TR-PL signals. In this section, we investigate the decay dynamics of the PL signal corresponding to the Y\({}_{1}^{\prime}\)-Z\({}_{1}^{\prime}\) transition and its temperature dependence (\(T\) = 4-60 K) under the Y\({}_{3}^{\prime}\) resonant excitation at \(P_{\rm exc}\) = 1 mW. Figure 5 shows the TR-PL signals of the Y\({}_{1}^{\prime}\)-Z\({}_{1}^{\prime}\) transition in the three samples at (a) 6 K and (b) 20 K in one instance. Though the signal in Sample A decays in one order faster than those in Samples B and C, all the TR-PL signals for each Er-concentration and each temperature could be decomposed into two components by the double exponential fitting: fast decay and slow decay components with characteristic times \(\tau_{\rm fast}\) and \(\tau_{\rm slow}\). Please note that the solid curves in the figures represent the calculated results based on the population dynamics model discussed in the next section. The relaxation times \(\tau_{\rm fast}\) and \(\tau_{\rm slow}\) deduced from the double exponential fitting are plotted as a function of the temperature for Samples A, B, and C in Figure 6a-c, respectively. As a whole, we note that both \(\tau_{\rm fast}\) and \(\tau_{\rm slow}\) get shorter with increasing Er-concentration. As clearly shown in the figures, while the \(\tau_{\rm fast}\) is insensitive to the change in temperature and is almost constant in the temperature range 4-60 K, the \(\tau_{\rm slow}\) shows a rapid reduction with increasing temperature; These behaviors are similarly observed in all three samples. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Sample**} & **Er\({}^{3+}\)** & **\(E_{\rm A}\)** & **\(A_{\rm r}\)** & **\(\tau_{\rm r}\)** & **\(W_{\rm nr}^{0}\)** & **\(\tau_{\rm nr}^{0}\)** \\  & **\(x\)** & **(meV)** & **(ms\({}^{-1}\))** & **(ms)** & **(ms\({}^{-1}\))** & **(ms)** \\ \hline A & 1.000 & 1.02 & 3.92 & 0.255 & 122 & 0.008 \\ B & 0.054 & 9.01 & 0.88 & 1.14 & 8.08 & 0.124 \\ C & 0.012 & 2.89 & 0.56 & 1.78 & 2.70 & 0.370 \\ \hline \end{tabular}
\end{table} 
Table 2: Summary of the fitting parameters to reproduce the temperature dependence of \(\tau_{\rm slow}\) depicted in Figure 6d. First, we focus on the \(\tau_{\text{slow}}\), which is assumed to include the two different decay components, the radiative and the non-radiative relaxations. Since the radiative relaxation rate is determined quantum-mechanically and is independent of temperature in principle, we consider that the temperature-sensitive property of \(\tau_{\text{slow}}\) component is caused by the thermally-activation type non-radiative relaxation process. The \(\tau_{\text{slow}}\) can be expressed as a sum of the rates of radiative and thermally-activation type non-radiative relaxations:  
\[\frac{1}{\tau_{\text{slow}}}=A_{\text{r}}+W_{\text{nr}}^{0}\exp\left(-\frac{E_ {\text{A}}}{k_{\text{B}}T}\right), \tag{1}\]  
where \(A_{\text{r}}\) is the radiative relaxation rate, \(W_{\text{nr}}^{0}\) is the non-radiative relaxation rate at 0 K, \(k_{\text{B}}\) is the Boltzmann constant, and \(E_{\text{A}}\) is the activation energy of the non-radiative center. Figure 6d shows the observed \(\tau_{\text{slow}}\) and the fitting curves as a function of the inverse temperature, and the fitting parameters are summarized in Table 2, where the radiative and non-radiative relaxation times, \(\tau_{\text{r}}=1/A_{\text{r}}\) and \(\tau_{\text{nr}}^{0}=1/W_{\text{nr}}^{0}\), are also listed. Remarkably, the characteristic parameter for the thermally-activation type non-radiative relaxation process, \(W_{\text{nr}}^{0}\) is strongly reduced to \(\sim\)1/30 by decreasing the Er-concentration from \(x=1.000\) to \(x=0.012\). This is because the longer inter-Er\({}^{3+}\) distance suppresses the inter-site and/or the intra-site ET process and prevents the photo-excited electrons from being captured in the quenching centers consequently.