Investigation of Population Dynamics in 1.54-μm Telecom Transitions of Epitaxial (Er_xSc_1-x)₂O₃ Thin Layers for Coherent Population Manipulation: Weak Excitation Regime

Abstract

We have investigated the energy transfers in the 1.54-$\mathsf{\mu }$m region of (Er,Sc)${}_2$O${}_3$ epitaxial thin films grown on Si(111). The interplay of the energy transfers between Er ions in the different and the same symmetry sites makes the dynamics complicated. To suppress the energy transfer upconversion, low power and resonant excitation of the third crystal-field level (${}^4{I}_{13/2}:{\mathrm{Y}}_3^{\prime }$) of the Er${}^{3+}$ site with $C_{3i}$ symmetry was employed. The time-resolved photoluminescence measurements of the ${\mathrm{Y}}_1^{\prime }$-${\mathrm{Z}}_1^{\prime }$ transition indicate the existence of two decay components having fast (10–100 $\mathsf{\mu }$s) and slow (0.1–1 ms) relaxation times in the range of 4–60 K. The model calculation including the inter-site energy transfers, the temperature-sensitive and -insensitive non-radiative relaxations fits the experimental results well. Moreover, the long averaged inter-Er${}^{3+}$ distance obtained by decreasing Er concentration was found to reduce two kinds of non-radiative relaxation rates and the energy transfer rates between Er ions very effectively.

1. Introduction

Recently, there has been a great deal of interest in the subject of the population (and coherence) manipulation in quantum states, which provoked by the emergence of the quantum information paradigm [1,2]. The exciting researches are currently progressed in different solid-state systems such as semiconductor quantum dots [3,4,5,6] and nitrogen-vacancy centers in diamond [7,8] based on the interplay of the solid-state nature of the material engineering and the atom-like character of the sharp discrete energy levels. In those solid-state materials and structures, the techniques and concepts demonstrated previously in quantum optics of atoms and molecules [9,10,11] can be put to practical use.

Rare earth sesquioxides, R${}_2$O${}_3$ (R is rare earth), are alternative attractive candidates to exploit the coherent quantum phenomena because the intra-4f transitions in rare-earth ions are weakly perturbed by the crystalline environments, and they exhibit a resonance with a very narrow inhomogeneous linewidth [12,13,14,15]. In the various R${}_2$O${}_3$, we focus on the single crystalline (Er${}_x$Sc${}_{1-x}{)}_2$O${}_3$ including Er${}_2$O${}_3$ since the trivalent Er (Er${}^{3+}$) can interact with a telecommunication-band photon (∼1.5 $\mathsf{\mu }$m) and is a potential platform for the coherent population manipulation in the quantum information network using the well-developed infrastructure [15,16,17,18,19,20]. However, for this purpose, some obstacles have to be overcome: a relatively small oscillator strength of a single ion [21,22], a relatively large upconversion (UC) rate [23,24,25,26,27,28,29,30], and a large concentration quenching [20,22]. These issues complicate the physics of the energy transfer (ET) in Er${}^{3+}$-doped solids as well as in other R${}_2$O${}_3$ solids.

Figure 1 shows the various kinds of ET from one of the Stark levels in the ${}^4{I}_{13/2}$ manifold of Er${}^{3+}$ [30], which are observed in (Er${}_x$Sc${}_{1-x}{)}_2$O${}_3$ epitaxial thin films in this study. (Er${}_x$Sc${}_{1-x}{)}_2$O${}_3$ crystals have a cubic bixbyite structure (Figure 2a). A unit cell of the crystal contains 32 Er and/or Sc ions, 24 at sites with $C_2$ (non-inversion) symmetry and 8 at sites with $C_{3i}$ (inversion) symmetry (denoted by $C_2$ and $C_{3i}$ sites hereafter). In Er${}^{3+}$ in $C_2$ site, the (forced) electric-dipole transitions between the Stark levels are allowed. Since $C_{3i}$ is a higher symmetry than $C_2$, the electric-dipole transitions are forbidden strictly and only magnetic-dipole transitions between ${}^4{I}_{13/2}{-}^4{I}_{15/2}$ are allowed. Therefore, the UC occurs only in $C_2$ sites and gives rise to a severe population dissipation from a target state in the case of high power excitation [20]. Also, we observe mutual ET between Er${}^{3+}$ at $C_2$ and $C_{3i}$ sites and the ET between Er${}^{3+}$ with the same site symmetry ($C_2$ or $C_{3i}$). In this work, we will refer to the former ET as the inter-site ET and the latter ET as the intra-site ET as shown in Figure 1. Moreover, the non-radiative relaxation occurs due to the crystal imperfection and multiple phonon emission. The intra-site ET induces the energy migration to open the channels to the quenching centers generated by the crystal defects and Er-ion clustering. By using the single crystalline (Er${}_x$Sc${}_{1-x}{)}_2$O${}_3$ with various Er concentrations, we can investigate how the inter-Er distance affects the energy flow in a system. Also, in order to investigate the energy flow in detail, the initial energy injection into well-resolved point such as one of the Stark levels in the multiplet is very important.

In our previous study [17], we investigated the ETs in the Er${}_2$O${}_3$ single crystal under the resonant excitation of a Stark level (Y${}_2$) in a $C_2$ site and estimated the large inter-site ET rate (∼0.5 $\mathsf{\mu }$s${}^{-1}$) and intra-site ET rate (∼0.1 $\mathsf{\mu }$s${}^{-1}$) via the excitation power dependence of the time-integrated photoluminescence (TI-PL) intensities, where the non-radiative relaxation and UC process were not explicitly taken into account. However, the energy dissipation via non-radiative relaxation and the UC process is supposed to affect the population dynamics greatly, and therefore, further detailed studies are necessary to reveal the population dynamics in a whole system.

In this study, we investigate the population dynamics in a whole system which consists of not only the inter- and intra-site ETs but the non-radiative relaxation in Er${}_2$O${}_3$ and (Er, Sc)${}_2$O${}_3$ single crystals grown on Si(111). Since the UC process occurring in the $C_2$ site makes the population dynamics complicated, the low power resonant excitation of one of Stark levels (Y${}_3^{\prime }$) in the $C_{3i}$ site is employed in order to minimize the effect of UC process. After deciding the proper experimental conditions, the time-resolved PL (TR-PL) measurements and their temperature dependence have directly revealed the ETs in the same site ($C_{3i}$, one of intra-site ET) and between the different sites ($C_{3i}\leftrightarrow C_2$, inter-site ET) by using three samples with different Er-concentrations. The simple model requires the temperature sensitive and insensitive non-radiative relaxations to reproduce the TR-PL signals and their temperature dependence well.

2. Samples and Experimental Setup

Er${}_2$O${}_3$ crystals have a lattice constant (a = 10.54 Å) which is almost twice one of Si(111) (a = 5.43 Å). 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 Å, 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 Vegard’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 5×5 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 (∼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. Summary of the studied (Er${}_x$Sc${}_{1-x}{)}_2$O${}_3$ samples. The averaged distances are evaluated from the lattice constant and Er concentration.

Sample	x	Er${}^{3+}$ Density	Er${}^{3+}$	Averaged Inter-Er${}^{3+}$	Averaged Intra(inter)-Site Distance
		(cm${}^{-3}$)	in Unit Cell	Distance (Å)	C${}_2$-C${}_2$ (Å)	C${}_{3\mathit{i}}$-C${}_{3\mathit{i}}$ (Å)	C${}_2$-C${}_{3\mathit{i}}$ (Å)
A	1.000	2.7 $\times {10}^{22}$	32	3.3	1.95	5.27	2.66
B	0.054	1.4 $\times {10}^{21}$	1.6	8.8	5.15	13.94	7.03
C	0.012	3.4 $\times {10}^{20}$	0.4	14.5	8.50	23.01	11.60

.

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 $\mathsf{\mu }$m-region. The (forced) electric-dipole transitions between the Stark levels are allowed for Er${}^{3+}$ in $C_2$ sites (${\mathrm{Y}}_i-{\mathrm{Z}}_j$), while only magnetic-dipole transitions are possible for Er${}^{3+}$ in $C_{3i}$ sites (${\mathrm{Y}}_i^{\prime }-{\mathrm{Z}}_j^{\prime }$). Since Sc${}_2$O${}_3$ is transparent in the visible to telecommunication-band region, the emission and absorption at around 1.5 $\mathsf{\mu }$m in (Er${}_x$Sc${}_{1-x}$)${}_2$O${}_3$ originate from Er ions.

The as-grown samples were mounted in a continuous He-flow cryostat, and the temperature was controlled. A continuous-wave tunable laser (1470−1560 nm, spectral width of 400 kHz) with an erbium-doped fiber amplifier was used to excite the ${}^4{I}_{13/2}$ manifold. Since the bandwidth of the excitation laser was much narrower than the energy intervals in the ${}^4{I}_{13/2}$ manifold, the resonant excitation of one Stark level was possible.

In the TI-PL measurements, the PL peaks in the 1.54-$\mathsf{\mu }$m (visible) wavelength regions were detected by InGaAs-photodiode array (Si-CCD) through the monochromator under the ${}^4{I}_{13/2}$ manifold excitation by varying the sample temperature 4–150 K. The excitation laser was focused on a sample surface with the diameter of ∼15 $\mathsf{\mu }$m by an objective lens, and the incident angle was set to be 45${}^{\circ }$ from the sample growth axis. Avoiding a strong stray light, the PL signal was collected perpendicularly by an another objective lens with a numerical aperture of 0.4. The total spectral resolution was about 60 $\mathsf{\mu }$eV for the TI-PL measurements.

For the TR-PL measurements, a pulsed excitation formed from the continuous-wave laser light by an acousto-optic modulator and a streak camera (model C11293S, Hamamatsu Photonics, Hamamatsu, Japan) were used to observe the temporal evolution of PL signals from ${}^4{I}_{13/2}$ manifold, and the sample temperature was changed around 4–60 K. The spectral and temporal resolutions of the TR-PL measurement system were 500 $\mathsf{\mu }$eV and 20 ps (for a 1 ns time range), respectively.

3. Results and Discussions

One of the important ET processes for fundamental and application aspects is known as UC process in the Er-doped materials [30]. The UC process brings a severe reduction in the population of the target energy level, and it also makes the population dynamics in the considered system complicated. We seek the proper condition to avoid the UC process effectively from the excitation power dependence of TI-PL signals before studying the population dynamics in detail.

3.1. 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 ($P_{\mathrm{exc}}$) of 1 mW (upper panel) and 30 mW (lower panel). While the observed UC-PLs were negligible at $P_{\mathrm{exc}}$ = 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 $P_{\mathrm{exc}}$ = 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 ($P_{\mathrm{exc}}$ = 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 ($P_{\mathrm{exc}}$ = 1 mW), we study the inter-site ET process and the thermal quenching of PL spectra. Figure 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 ∼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. Actually, the averaged value of the inter-site distance of Er${}^{3+}$, $R_{C_2-C_{3i}}$ extends from 2.66 Å to 7.03 (11.6) Å as the Er-concentration is reduced from x = 1.000 to 0.054 (0.012) in Sample A to B (C) as shown in Table 1.

Other important information about the inter-site ET process is found from the photoluminescence excitation (PLE) measurement as shown in Figure 4d where the excitation wavelength was swept around ${}^4{I}_{13/2}$ manifold (1520–1555 nm). At the Y${}_1^{\prime }$ resonant excitation (${\lambda }_{\mathrm{exc}}$ = 1548 nm), the PL signal of Y${}_1$−Z${}_1$ transition (1535 nm) was observed. Accordingly, the Y${}_1^{\prime }\to$Y${}_1$ transition seems to be a dominant process relevant to the $C_{3i}\to C_2$ ET rather than the Y${}_3^{\prime }\to$Y${}_1$ transition, and the inter-site ET occurs after the rapid internal energy relaxation in $C_{3i}$ sites (i.e., Y${}_3^{\prime }\to$Y${}_1^{\prime }\to$Y${}_1$). Further, it should be noted that the strong PL of the Y${}_1^{\prime }$−Z${}_1^{\prime }$ transition (1548 nm) appeared at the Y${}_1$ and Y${}_2$ resonant excitation; it denotes the presence of the inter-site ET in the reverse direction ($C_2\to C_{3i}$).

For the energy transfer mechanisms between Er ions, the $\mathrm{F}\ddot{\mathrm{o}}\mathrm{rster}$ type (dipole-dipole interaction) and Dexter type (wave-function overlapping) are well known from the previous studies [32,33], and the former and the latter are proportional to $R^{-6}$ and $exp(-2R/L)$, respectively (R: the inter-ionic distance, L: the van der Waals radius). In our case, the remarkable suppression by extending $R_{C_2-C_{3i}}$ and the energy mismatch between ${\mathrm{Y}}_1^{\prime }$ and ${\mathrm{Y}}_1$ levels of ∼7 meV suggest that the Dexter-type mechanism, which is more sensitive to the change in R than the Förster-type mechanism, seems to be dominant in the inter-site ET process.

Figure 4e shows the temperature dependence of the TI-PL intensities of the ${\mathrm{Y}}_1^{\prime }$-${\mathrm{Z}}_1^{\prime }$ transition in Figure 4a–c for the three samples. In Sample A, a significant thermal quenching of the PL intensity ($I_{\mathrm{PL}}$) occurs till around 20 K, and the $I_{\mathrm{PL}}$ at 20 K is reduced approximately to the one-quarter of $I_{\mathrm{PL}}$ at 4 K. In contrast, the decrease in PL intensities in Samples B and C are more moderate; the $I_{\mathrm{PL}}$ at ∼20 K is about a half of $I_{\mathrm{PL}}$ at 4 K. These behaviors appear clearly in the values of non-radiative recombination rate $W_{\mathrm{nr}}^0$ (or time ${\tau }_{\mathrm{nr}}^0$) in

Table 2. Summary of the fitting parameters to reproduce the temperature dependence of ${\tau }_{\mathrm{slow}}$ depicted in Figure 6d.

Sample	Er${}^{3+}$	${\mathit{E}}_{\mathbf{A}}$	${\mathit{A}}_{\mathbf{r}}$	${\mathit{\tau }}_{\mathbf{r}}$	${\mathit{W}}_{\mathbf{nr}}^0$	${\mathit{\tau }}_{\mathbf{nr}}^0$
	x	(meV)	(ms${}^{-1}$)	(ms)	(ms${}^{-1}$)	(ms)
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

, which is deduced from the time-resolved PL measurements in the next subsection. The data suggests that the extension of the inter-ionic distance serves as a suppression of the thermal quenching as well as the reduction in the inter-site ET. The suppression of the thermal quenching in Samples B and C may come from the reduction of the possibility to reach non-radiative centers due to the reduction in the inter-site ET. The dilution of Er ions replacing by Sc ions induces the increase of the lattice mismatch since the lattice constant of Sc${}_2$O${}_3$ is 9.85 Å, and thus the increase of the defect density may be expected. However, the PL intensity in Sample C does not change and is even stronger than that in Sample B as shown in Figure 4b,c. This observation leads to the conclusion that the change of the defect density via the lattice mismatch due to the change of x does not have a significant effect. We consider that the extension of the inter-Er distance by replacing Er with Sc is more effective to suppression of the non-radiative recombination rather than the increase of non-radiative centers via the induced lattice mismatching.

In this section, we obtained the following findings about the energy transfers in (Er${}_x$Sc${}_{1-x}{)}_2$O${}_3$ crystals under the resonant excitation of the Y${}_3^{\prime }$ level:

(1)

The UC process occurring in $C_2$ sites is negligible under a low power excitation condition of $P_{\mathrm{exc}}$ < a few mW.

(2)

The inter-site ET occurs mutually between the lowest states of Y${}_1^{\prime }$ and Y${}_1$, and it follows the internal energy relaxation in $C_{3i}$ sites (Y${}_3^{\prime }\to$Y${}_1^{\prime }$).

(3)

The extension of the inter-Er${}^{3+}$ distance is very effective in suppressing the inter-site ET and thermal quenching.

These findings will help to construct the model of population dynamics in the target energy levels of Er-ions.

3.2. 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 $P_{\mathrm{exc}}$ = 1 mW.

Figure 5 shows the TR-PL signals of the ${\mathrm{Y}}_1^{\prime }$-${\mathrm{Z}}_1^{\prime }$ 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 ${\tau }_{\mathrm{fast}}$ and ${\tau }_{\mathrm{slow}}$. 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 ${\tau }_{\mathrm{fast}}$ and ${\tau }_{\mathrm{slow}}$ 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 ${\tau }_{\mathrm{fast}}$ and ${\tau }_{\mathrm{slow}}$ get shorter with increasing Er-concentration. As clearly shown in the figures, while the ${\tau }_{\mathrm{fast}}$ is insensitive to the change in temperature and is almost constant in the temperature range 4–60 K, the ${\tau }_{\mathrm{slow}}$ shows a rapid reduction with increasing temperature; These behaviors are similarly observed in all three samples.

First, we focus on the ${\tau }_{\mathrm{slow}}$, 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 ${\tau }_{\mathrm{slow}}$ component is caused by the thermally-activation type non-radiative relaxation process. The ${\tau }_{\mathrm{slow}}$ can be expressed as a sum of the rates of radiative and thermally-activation type non-radiative relaxations:

$$\begin{array}{c}\hfill \frac{1}{{\tau }_{\mathrm{slow}}}=A_{\mathrm{r}}+W_{\mathrm{nr}}^{0}exp\left(-\frac{E_{\mathrm{A}}}{k_{\mathrm{B}}T}\right),\end{array}$$

(1)

where $A_{\mathrm{r}}$ is the radiative relaxation rate, $W_{\mathrm{nr}}^0$ is the non-radiative relaxation rate at 0 K, $k_{\mathrm{B}}$ is the Boltzmann constant, and $E_{\mathrm{A}}$ is the activation energy of the non-radiative center. Figure 6d shows the observed ${\tau }_{\mathrm{slow}}$ 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 , ${\tau }_{\mathrm{r}}=1/A_{\mathrm{r}}$ and ${\tau }_{\mathrm{nr}}^0=1/W_{\mathrm{nr}}^0$, are also listed. Remarkably, the characteristic parameter for the thermally-activation type non-radiative relaxation process, $W_{\mathrm{nr}}^0$ is strongly reduced to ∼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.

In Table 2, there is also a difference in the radiative rate $A_{\mathrm{r}}$ between the low and high Er-concentration samples. In Samples B and C, the experimentally deduced $A_{\mathrm{r}}$ almost agrees with the theoretically estimated values [34,35] which was obtained by using the Judd-Ofelt parameters ${\Omega }_t$ (t = 2, 4, 6) in (Er${}_x$,Sc${}_{1-x}$)${}_2$O${}_3$ transparent ceramics with $x=0.003$ [36]. On the other hand, a larger $A_{\mathrm{r}}$ was observed in Sample A, and it suggests the increase in the effective transition dipole moment. The smaller inter-Er${}^{3+}$ distance achieved in high Er-concentration crystals enhances the Er-Er interaction and may induce a larger coherence volume [37] as one plausible candidate. It can lead to a larger effective dipole moment, and thus, to a larger radiative relaxation rate compared with those in low Er-concentration systems.

Next, we mention the origin of the fast, temperature-insensitive decay. Since the radiative relaxation process is included in the ${\tau }_{\mathrm{slow}}$, the ${\tau }_{\mathrm{fast}}$ is determined only by the non-radiative relaxation process. At this stage, we suppose that the non-radiative energy dissipation from the ${\mathrm{Y}}_1^{\prime }$ level due to the multi-phonon (MP) emission process to the quenching centers is one candidate for the ${\tau }_{\mathrm{fast}}$ component. The MP relaxation time ${\tau }_{\mathrm{MP}}$ ordinarily depends on the sample temperature, and it is proportional to the following factor, ${\left[1-\exp (-E_p/k_{\mathrm{B}}T)\right]}^{-m}$ in the single frequency model, where $E_{\mathrm{p}}$ is the phonon energy and m is the number of phonons, respectively [38]. Since the $E_{\mathrm{p}}$ in Sc${}_2$O${}_3$ is about 74 meV [18] and the corresponding one in (Er, Sc)${}_2$O${}_3$ crystal seems to be close with that, the MP relaxation rate for m = 1−15 is insensitive to the change in temperature below 100 K, and it agrees qualitatively with the behavior of the experimentally obtained ${\tau }_{\mathrm{fast}}$.

3.3. 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:

$$\begin{array}{c}{\displaystyle \frac{d{n}_0}{dt}=-\Gamma n_0+A_{\mathrm{r}}{n}_1-C_{41}{n}_4{n}_0+C_{14}{n}_3{n}_1,}\hfill \end{array}$$

(2)

$$\begin{array}{c}{\displaystyle \frac{d{n}_1}{dt}=W_{21}{n}_2-\left(A_{\mathrm{r}}+W_{\mathrm{nr}}\right)n_1-\left(1-\frac{n_{\mathrm{d}}}{N_{\mathrm{d}}}\right)W_{\mathrm{d}}{n}_1+C_{41}{n}_4{n}_0-C_{14}{n}_3{n}_1,}\hfill \end{array}$$

(3)

$$\begin{array}{c}{\displaystyle \frac{d{n}_2}{dt}=\Gamma n_0-W_{21}{n}_2,}\hfill \end{array}$$

(4)

$$\begin{array}{c}{\displaystyle \frac{d{n}_{\mathrm{d}}}{dt}=\left(1-\frac{n_{\mathrm{d}}}{N_{\mathrm{d}}}\right)W_{\mathrm{d}}{n}_1,}\hfill \end{array}$$

(5)

$$\begin{array}{c}{\displaystyle \frac{d{n}_3}{dt}=\frac{n_4}{{\tau }_{43}}+\frac{n_5}{{\tau }_{53}}+C_{41}{n}_4{n}_0-C_{14}{n}_3{n}_1+C_{45}{n}_4^2,}\hfill \end{array}$$

(6)

$$\begin{array}{c}{\displaystyle \frac{d{n}_4}{dt}=-2C_{45}{n}_4^2-\frac{n_4}{{\tau }_{43}}+\frac{n_5}{{\tau }_{54}}-C_{41}{n}_4{n}_0+C_{14}{n}_3{n}_1,}\hfill \end{array}$$

(7)

$$\begin{array}{c}{\displaystyle \frac{d{n}_5}{dt}=C_{45}{n}_4^2-\frac{n_5}{{\tau }_{53}}-\frac{n_5}{{\tau }_{54}},}\hfill \end{array}$$

(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 $A_{\mathrm{r}}$ 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 non-radiative relaxation processes (i.e., level lifetimes). The rapid internal relaxation from states 2 to 1(Y${}_3^{\prime }\to$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 $W_{\mathrm{nr}}$ and the temperature-insensitive relaxation with a rate $W_{\mathrm{d}}$ are considered for state 1. Here, the rate $W_{\mathrm{nr}}$ has the thermally-activation type as similar to the ${\tau }_{\mathrm{slow}}$ process in Section 3.2, and it can be written as $W_{\mathrm{nr}}=W_{\mathrm{nr}}^{0}exp\left(-E_{\mathrm{A}}/k_{\mathrm{B}}T\right)$. Further, the temperature-insensitive non-radiation relaxation process with $W_{\mathrm{d}}$ needs to have the density-of-state limitation property as shown in Equation () 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.

The calculated results deduced from the temporal profile of $n_1$ were depicted as solid curves in Figure 5. The parameters about $W_{\mathrm{nr}}$ and $A_{\mathrm{r}}$ were estimated from the corresponding ones listed in Table 2, and the set of parameters used in the rate equation analysis was summarized in

Table 3. The set of parameters used in the rate equation analysis.

Er${}^{3+}$		${\mathbf{Y}}_1\to {\mathbf{Z}}_1$		${\mathbf{Y}}_1^{\prime }\to {\mathbf{Z}}_1^{\prime }$		${\mathbf{Y}}_1^{\prime }\to {\mathbf{Y}}_1$		${\mathbf{Y}}_1\to {\mathbf{Y}}_1^{\prime }$		Non-Radiative Decays from ${\mathbf{Y}}_1^{\prime }$
x		${\mathit{\tau }}_{43}$ (ms)		${\mathit{\tau }}_{\mathbf{r}}$ (ms)		$1/{\mathit{C}}_{14}$ (ms)		$1/{\mathit{C}}_{41}$ (ms)		${\mathit{\tau }}_{\mathbf{nr}}^0$ (ms)	${\mathit{W}}_{\mathbf{d}}$ (s${}^{-1}$)	${\mathit{N}}_{\mathbf{d}}$
1.000		0.10		0.26		1.00		0.008		0.006	1.0 $\times {10}^5$	0.4
0.054		1.59		1.18		500		30.3		0.160	3.3 $\times {10}^3$	0.4
0.012		1.30		1.75		2 $\times {10}^3$		167		0.300	1.4 $\times {10}^3$	0.4

. As clearly shown, the calculations can reproduce the experimental results for all the samples, and their temperature dependences are described well in the framework of our model, that is, the assumed $W_{\mathrm{nr}}$ works adequately to describe the thermal quenching of the Y${}_1^{\prime }$-Z${}_1^{\prime }$ emissions. In addition, we extracted two relaxation times corresponding to the fast and slow decay components from the calculated curves of $n_1$, and we plotted them in Figure 6a–c as solid and dashed lines. The extracted relaxation times show the similar temperature dependences with the experimentally obtained ones. The agreements between the experimental and calculated results support the validities of our model and assumptions, and the model calculations successfully explain the energy transfers and its temperature dependence especially of the target energy level Y${}_1^{\prime }$.

Further, the other non-radiative relaxation rate, $W_{\mathrm{d}}$ can also estimated as listed in Table 3, and the $W_{\mathrm{d}}$ in Sample C is reduced to 1/70 of $W_{\mathrm{d}}$ in Sample A. It is found that the reduction of Er-concentration is effective to suppress the temperature-insensitive non-radiative rate as well as the temperature sensitive non-radiative rate.

Although the excellent agreement between the observations and model calculations by a simple model in Figure 7 was obtained, some assumptions and the parameters that have not been obtained experimentally were used. Therefore, further detail experimental investigation to specify the physical origins of those non-radiative relaxations and the analysis by the microscopic theory such as Inokuti-Hirayama model [39,40] are required.

4. Summary

In summary, we have investigated the temperature dependence of the PL intensity and lifetime in (Er${}_x$Sc${}_{1-x}$)${}_2$O${}_3$ (x = 1.0, 0.054, 0.012) epitaxial thin films grown on Si(111) under the excitation conditions that the energy transfer upconversion process from $C_2$ site is minimized; resonant excitation of the third Stark level (${}^4{I}_{13/2}$: Y${}_3^{\prime }$) with a low power was employed. The extension of the averaged inter-Er distance by replacing Er with Sc is more effective to suppression of the non-radiative recombination rather than the increase of non-radiative centers via the induced lattice mismatching. In addition, thermal quenching could be greatly suppressed. These observations lead to the conclusion that the extension of inter-Er distance suppresses the energy transfers in inter and intra Er sites and reduces the captured possibility by the quenching centers. The simple model calculation introducing the inter-site ET, the temperature-sensitive and -insensitive non-radiative relaxations explains well the temperature dependence of the relaxation times as well as the time-resolved PL signals. The obtained results contribute to the coherent population manipulation and development of highly efficient optical devices on Er-related materials on Si substrates.