## 2 Samples and Experimental Setup  
Er\({}_{2}\)O\({}_{3}\) crystals have a lattice constant (\(a=10.54\) A) which is almost twice one of Si(111) (\(a=5.43\) A). Thus, the material is lattice-matched to Si(111) and can be epitaxially grown on Si with very low density of defects [31]. The averaged nearest neighbor distances of Er\({}^{3+}\) between the \(C_{2}\)-\(C_{2}\), \(C_{2}\)-\(C_{3i}\), and \(C_{3i}\)-\(C_{3i}\) sites are 1.945, 2.656, and 5.268 A, respectively. (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) has similar properties and can also be epitaxially grown on a Si(111) substrate [18]. Sc\({}_{2}\)O\({}_{3}\) is completely transparent for photons in the visible-to-telecom-band range [27]. Some of the Er\({}^{3+}\) ions are replaced with Sc\({}^{3+}\) ions to control the distance between the Er\({}^{3+}\) ions as shown Figure 2a. (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) crystals with \(x=0.012-1.000\) and the thickness of about 50 nm were grown on Si (111) surfaces with \(7\times 7\) reconstruction by molecular beam epitaxy at a growth temperature of 715 \({}^{\circ}\)C, and were successfully to be single-crystalline [17; 18], which was confirmed by the streak pattern of reflection high-energy electron diffraction. The cross-sectional image obtained with a transmission electron microscope (TEM) also indicated that the (Er,Sc)\({}_{2}\)O\({}_{3}\) was epitaxially grown on the Si(111) surface in Figure 2b. The composition of the grown films was determined by Rutherford backscattering. An \(\omega-2\theta\) scan of the X-ray diffraction (XRD) measurements after growth (not shown here) also proved that single crystal (Er,Sc)\({}_{2}\)O\({}_{3}\) layers were grown, and the crystal quality was approximately equivalent in all Er compositions [20]. Moreover, we found that the Er concentration-dependency of the lattice constant of (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) satisfied [PERSON]'s law. Thus, we can assume macroscopic uniformity of Er\({}^{3+}\) distribution in the grown samples. The uniformity was supported also by the mapping of the PL spectra in the scanned area of 5\(\times\)5 mm\({}^{2}\) which indicated no change of the spectral properties (intensity, linewidth, and peak position) for all studied samples. In this work, three (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) samples with different Er-concentrations (\(x=1.000\), 0.054, and 0.012) were studied. Please note that a large number of Er ions (\(\sim\)3\(\times\)10\({}^{20}\) cm\({}^{-3}\)) are still contained even in a sample (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) with \(x=0.012\) (the lowest value in this work) compared to a commercial Er-doped fiber amplifier material (typically \(<\)10\({}^{19}\) cm\({}^{-3}\)). Hereafter, the samples with \(x=1.000\), 0.054, and 0.012 are referred to as Sample A, B, and C, respectively. The Er\({}^{3+}\) density, number of Er\({}^{3+}\) in unit cell, averaged inter-Er\({}^{3+}\), and intra (inter) site distance of the studied (Er\({}_{x}\)Sc\({}_{1-x}\))\({}_{2}\)O\({}_{3}\) samples are summarized in Table 1.  
As shown in Figure 3a, Er\({}^{3+}\) has the optical transition between the first excited state (\({}^{4}I_{13/2}\) manifold with Stark levels Y\({}_{i}^{(^{\prime})}\), \(i=1-7\), labeled from lower to higher energy level) and the ground state (\({}^{4}I_{15/2}\) manifold with Stark levels Z\({}_{j}^{(^{\prime})}\), \(j=1-8\)), which exhibits the photon emission and absorption at around 1.5 \(\upmu\)m-region. The (forced) electric-dipole transitions between the Stark levelsare allowed for \(\text{Er}^{3+}\) in \(\text{C}_{2}\) sites (\(\text{Y}_{i}-\text{Z}_{j}\)), while only magnetic-dipole transitions are possible for \(\text{Er}^{3+}\) in \(\text{C}_{3i}\) sites (\(\text{Y}_{i}^{\prime}-\text{Z}_{j}^{\prime}\)). Since \(\text{Sc}_{2}\text{O}_{3}\) is transparent in the visible to telecommunication-band region, the emission and absorption at around 1.5 \(\upmu\)m in (\(\text{Er}_{x}\text{Sc}_{1-x}\))\({}_{2}\text{O}_{3}\) originate from \(\text{Er}\) ions.