## 3 Results and Discussions
### Time-Integrated Photoluminescence Measurements  
The UC-PL means the photon emission with shorter wavelength than that of the excitation laser, and it comes from the excited-state absorption and/or the cooperative UC (Auger UC) in which the excitation into a higher lying state and de-excitation to a lower lying state occur simultaneously as shown in Figure 1. The UC levels of Er\({}^{3+}\) in \(C_{2}\) sites were summarized in Figure 3a, and Figure 3b shows examples of the UC-PL spectra observed in Sample A: 4 K under \(Y_{3}^{\prime}\) resonant excitation, the excitation powers (\(P_{\rm exc}\)) of 1 mW (upper panel) and 30 mW (lower panel). While the observed UC-PLs were negligible at \(P_{\rm exc}\) = 1 mW, the UC-PLs from various manifolds such as \({}^{4}S_{3/2}\) (\(\sim\)550 nm), \({}^{4}F_{9/2}\) (\(\sim\)680 nm), \({}^{4}I_{9/2}\) (\(\sim\)850 nm), and \({}^{4}I_{11/2}\) (\(\sim\)1000 nm) were observed clearly at \(P_{\rm exc}\) = 30 mW. Especially, the UC-PLs from \({}^{4}F_{9/2}\) and \({}^{4}S_{3/2}\) manifolds were much stronger than other UC-PLs. Figure 3c shows the excitation power dependences of the integrated UC-PL intensities from \({}^{4}F_{9/2}\), \({}^{4}S_{3/2}\), and \({}^{4}I_{11/2}\) to \({}^{4}I_{15/2}\) manifolds. It clearly indicates that the appearance of the UC-PLs requires more than a critical excitation power, and the UC-PLs become remarkable in the high power excitation region. Additionally, we found a similar power dependence of the UC-PLs in Samples B and C, and the clear but weaker UC-PLs were observed at more than the similar critical excitation power with that in Sample A (not shown here). Therefore, the influence of the UC process is considered to be negligible in the low power excitation region of 0.1-a few mW. All the measurements in the following experiments were carried out under the low power excitation condition (\(P_{\rm exc}\) = 1 mW). Please note that the observed UC-PLs can be addressed to the transitions from the higher-lying manifolds in \(C_{2}\) sites although the excitation laser was tuned to the \(Y_{3}^{\prime}\) resonance in \(C_{3i}\) sites. The assignment of UC-PLs is also supported by the fact that the optical transitions between the higher-lying states and the ground states in \(C_{3i}\) sites are forbidden [21]. Thus, the appearance of UC-PLs indicates the presence of the inter-site ET process (\(C_{3i}\to C_{2}\)) in the \({}^{4}I_{13/2}\) manifold. Under the low power excitation condition (\(P_{\rm exc}\) = 1 mW), we study the inter-site ET process and the thermal quenching of PL spectra. Figures 4a-c show the TI-PL spectra at some different temperatures under the resonant excitation of the \(Y_{3}^{\prime}\) level for Samples A, B, and C, respectively. The strongest PL spectra at 1548 nm in Sample A corresponds to the transition between \(Y_{1}^{\prime}-Z_{1}^{\prime}\) (\(C_{3i}\) site). The transition energy of the peak shifts to \(\sim\)1551 nm in Samples B and C because the crystal field surrounding Er\({}^{3+}\) is changed by decreasing the Er concentration. Despite the resonant excitation to the \(Y_{3}^{\prime}-Z_{1}^{\prime}\) transition in \(C_{3i}\), the PL peaks that were assigned to be the transitions Y\({}_{1}-Z_{1}\) and Y\({}_{1}-Z_{2}\) in \(C_{2}\) sites appeared along with \(Y_{1}^{\prime}-Z_{1}^{\prime}\) transition in \(C_{3i}\) sites. The observation evidently indicates that the energy transfer from \(C_{3i}\) to \(C_{2}\) occurs. Please note that the PL intensities from \(C_{2}\) sites in the low Er-concentration samples B and C become weaker than those in Sample A. These results show that the interaction between Er ions and the resultant inter-site ET process can be reduced by the extension of the averaged inter-ionic distance through decreasing the Er-concentration.