Stochastic Model of Energy-Transfer Processes Among Rare-Earth

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Stochastic Model of Energy-Transfer Processes Among Rare-Earth Ions. The Example of AlO:Tm 2

3

3+

Pavel Loiko, and Markus Pollnau J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b09594 • Publication Date (Web): 03 Nov 2016 Downloaded from http://pubs.acs.org on November 8, 2016

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The Journal of Physical Chemistry

Stochastic Model of Energy-Transfer Processes Among Rare-Earth Ions. The Example of Al2O3:Tm3+ Pavel Loiko and Markus Pollnau* Department of Materials and Nano Physics, School of Information and Communication Technology, KTH – Royal Institute of Technology, Electrum 229, Isafjordsgatan 22–24, 16440 Kista, Sweden

ABSTRACT: Energy-transfer processes strongly affect the performance of lanthanide-doped photonic devices. In this work, we introduce a simple stochastic model of energy-transfer processes and successfully apply it to the example of cross-relaxation (CR) and energy-transfer upconversion (ETU) in amorphous Al2O3:Tm3+ waveguides on silicon intended for lasers operating at ~2 μm. The stochastic model is based on the rate-equation formalism and considers two spectroscopically distinct ion classes, namely single ions and ions with neighbours (pairs and clusters), with the corresponding ion fractions being dependent on the doping concentration. We prove that a more accurate description of the luminescence properties of amorphous Al2O3:Tm3+ is obtained when accounting for the presence of these distinct ion classes. Based on the developed model, we derive microscopic CR and ETU parameters of CCR = 5.83×10-38 cm6s-1, CETU1 = 0.93×10-40 cm6s-1, and CETU2 = 7.81×10-40 cm6s-1, and determine the laser quantum efficiency ηq of excitation of Tm3+ ions in the upper laser level. For the maximum Tm3+ concentration of 5.0×1020 cm-3 studied experimentally in this investigation, ηq reaches 1.73. Furthermore, the transition crosssections at the pump and laser wavelengths are determined. For the 3H6 → 3F4 transition, the maximum stimulated-emission cross-section is σe = 0.47 × 10-20 cm2 at 1808 nm.

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I. INTRODUCTION Most of the trivalent rare-earth ions (RE3+) possess a rich energy-level system1. Combined with the fact that electronic transitions within the 4f sub-shell are parity forbidden, leading to long intrinsic excited-state lifetimes in the µs to ms range, it facilitates various energy-transfer processes between closely located ions2-7. Such interionic processes exhibit a high probability, especially when the involved transitions are spectrally resonant with each other2, but the occurrence of phonon-assisted energy-transfer processes further extends the variety of possible interactions between RE3+ ions8. These processes appear in systems containing one type9-13 (single doping) or different types14-18 (co-doping) of RE3+ ions. Well-known processes are cross-relaxation (CR)19-24 and energy-transfer upconversion (ETU)2,3,9. Both processes can play a negative or positive role in a RE3+-doped material regarding its potential applications in luminescent or laser devices. For instance, CR can increase the excitation rate of the upper laser level in thulium (Tm3+) doped media, leading to an improved slope efficiency of Tm3+ 2-µm lasers22. ETU, in turn, can lead to the depopulation of the upper laser level, thus having a detrimental effect on the performance of lasers and amplifiers9. On the other hand, ETU plays a crucial role in the realization of upconversion lasers25 and is an underlying principle of near-IR-to-visible light-conversion materials (phosphors)26-30. Therefore, a proper description of interionic processes in RE3+-doped materials is of high importance for understanding and boosting the efficiency of various photonic systems. In particular, determination of the micro-parameters of energy-transfer processes is desirable for a quantitative description of the population dynamics in RE3+-doped devices. Tm3+ ions have attracted a lot of interest during the last two decades due to their applications in lasers emitting in the eye-safe spectral region around ~2 μm21,22,31-34. This emission is related to the 3 F4 → 3H6 transition. Depending on the host matrix, the corresponding emission band of the Tm3+ ions can span from ~1.8 up to ~2.1 μm where the absorption lines of several relevant molecules, e.g. H2O, CO2, etc., are located. This makes Tm3+ lasers highly attractive for applications in sensing and medicine. Direct excitation of the Tm3+ ions to the upper laser level (3F4) is complicated by the lack of commercially available and powerful diode pump sources. However, Tm3+ ions can be pumped at ~0.8 μm on the 3H6 → 3H4 transition using commercial AlGaAs laser diodes. Generally, this pump scheme suffers from a low Stokes efficiency, ηSt = λP/λL  0.4, where λL and λP are the laser and pump wavelengths, respectively. However, thanks to the energy-gap resonances between the ground state, the upper laser level, and the pump level, efficient CR between neighboring Tm3+ ions is possible21-23: Tm1(3H4) + Tm2(3H6) → Tm1(3F4) + Tm2(3F4). This process produces two excitations in the upper laser level from one absorbed pump photon, ideally leading to a pump quantum efficiency of ηq = 2. CR can become very efficient even at relatively low Tm3+ concentrations22. Thus, it is easily accessible in a broad variety of amorphous and crystalline Tm3+-doped materials. 2-μm Tm3+ lasers based on CR may exhibit very high laser slope efficiency22 approaching the theoretical limit set by the factor of ηStηq ~0.8. In the present paper, we introduce a new stochastic model addressing various energy-transfer processes (CR and ETU). The model considers a different ability of different classes of active ions to participate in energy-transfer processes. The model has high relevance for the understanding of luminescence decay and population dynamics in RE3+-doped materials, independent of the amorphous or crystalline nature of the host matrix, and for the development of efficient laser materials. As an example, we have selected amorphous Al2O3:Tm3+ waveguides on silicon. During recent years, amorphous Al2O3:RE3+ has been successfully exploited for waveguide amplifiers and lasers35-38 at ~1 μm and 1.5 μm based on activation with Yb3+, Nd3+, and Er3+ ions, respectively, 2 ACS Paragon Plus Environment

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showing exceptional properties with respect to laser linewidth and stability39,40. The waveguide geometry allows one to achieve excellent laser efficiency. Amorphous Al2O3 is particularly attractive because of its demonstrated integration with conventional silicon technology40. From the physical point of view, an amorphous material is a system where clustering of RE3+ ions may be more relevant12 as compared to crystalline matrices and, thus, the verification of the proposed stochastic model is more convincing.

II. MODEL OF ENERGY-TRANSFER PROCESSES IN Tm3+ IONS A. Stochastic Model and Ion Fractions. In our simple stochastic model, we consider two different ion classes, namely (i) single ions having a concentration Nsi and (ii) ions with active neighbors, including ion pairs, multimers, and clusters, having a concentration Nni. The sum of the two concentrations equals the dopant concentration Nd. All ions belonging to the same class of ions are considered spectroscopically identical. Single ions do not exhibit energy-transfer processes (CR and ETU), neither among their own class nor with ions from the other class, whereas ions with neighbors may participate in CR and ETU with other ions within their own class. Consequently, ions belonging to the different classes cannot communicate with each other. For each of these two ion classes a separate rate-equation system considering the spectroscopic processes relevant for this ion class is solved. This model simplifies the description of ions with active neighbors, because ions having different constellations of active neighbors usually exhibit different probabilities of energy transfer. For example, in pairs each ion has only one neighbor, whereas in multimers the number of neighbors is higher, thereby increasing the probability of energy transfer. The situation can easily become rather complex, because already for three neighboring ions two constellations are possible: ions arranged in a line such that ion 1 and ion 3 do not directly interact with each other, and ions arranged in a triangle such that any two of the three ions can interact with each other. With increasing number of neighboring ions the situation becomes further involved. In order not to introduce a large number of additional parameters (at least the concentration of ions belonging to each sub-class or sub-subclass would be such an additional parameter) to the model, we make a first-order approximation by considering only two ion classes, with and without active neighbors. Among all the possible levels of refinement, this first is the most important step, because it has the strongest impact toward a more precise description of the spectroscopy of the investigated system, while adding only one additional parameter, namely either Nsi or Nni, since their sum is known. Furthermore, we make the simplifying assumptions that all active ions form chemical bonds leading to the same coordination, e.g., tetrahedral, octahedral, or dodecahedral coordination, and that the active ions replace the relevant host cations such that they are statistically distributed, i.e., neither a uniform replacement nor enhanced clustering occurs. Although there is indication that in amorphous and glass materials enhanced clustering of rare-earth ions beyond a statistical distribution can occur, detailed quantitative information would be required in each individual case if one wanted to take this fact into account in the model. Under these conditions, the probability for the occurrence of active neighbor ions in the first coordination sphere of an active ion is given by Pm,n  Cnm p m q nm 

n! nm . p m 1  p  m! n  m !


(1)


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Mathematically, n is the number of trials, m is the number of successful trials, and p is the probability of success. Physically, n corresponds to the coordination number of the first coordination sphere of nearest-neighbor rare-earth sites, m represents the tested number of active ions in the first coordination sphere, where m = 0 corresponds to a single ion, m = 1 to an ion pair, and m > 1 to a multimer or cluster, and p = Nd/Nmax represents the relative dopant concentration, where Nd is the absolute dopant concentration (Nd = NTm in our case) and Nmax is the maximum possible dopant concentration, i.e., for 100% replacement of the relevant host-matrix ions by rare-earth ions. Thus, statistically the fraction of single ions can be derived as x



P0,n n P m 0 m,n

.

(2)

This simple model is generally applicable to all rare-earth-doped materials and, although not taking all details into account, improves the accuracy of the spectroscopic description significantly. In the following, we will apply this model to the case of CR and ETU in amorphous Al2O3: Tm3+. B. Spectroscopic Processes in Tm3+ Ions. A partial energy-level scheme of Tm3+ is presented in Fig. 1, including the electronic states from 3H6 to 1D2 which are consecutively numbered as 0−6. The Tm3+ ions are excited at ~0.8 μm on the ground-state-absorption (GSA) transition 3H6 → 3H4. Further excitation to higher-lying excited states may be achieved by pump excited-state absorption (ESA) after a fast non-radiative (NR) multi-phonon relaxation. The ESA channels 3H5 → 1G4 and 3 F2,3 → 1D2 are possible, albeit relatively weak in oxide materials, as each of the competing NR relaxations from the 3H5 and 3F2,3 states is very strong due to the low energy gap to the next lowerlying state. Luminescence from various excited states on the transitions to the 3H6 ground state and 3 F4 first excited state can be observed, as indicated in Fig. 1 by solid arrows and notations Lumij, where i is the emitting and j is the terminating multiplet. Also displayed are the potential energytransfer processes22,23,41-43. As mentioned above, CR is a relevant process for the depopulation of the 3H4 pump level of the Tm3+ ions. The energy of the first excited state of Tm3+ is ~5600 cm-1 which is more than four times larger than the maximum phonon energy hph of usual oxide and fluoride hosts, hence the NR multi-phonon relaxation from this state is rather weak and intrinsic lifetimes in the ms range are typically observed. The long lifetime of the 3F4 state, however, may facilitate its depopulation through the two ETU processes44 indicated in Fig. 1. The ETU1 process, Tm1(3F4) + Tm2(3F4) → Tm1(3H6) + Tm2(3H4), is the inverse process of the afore-mentioned CR process. It removes two Tm3+ ions from the upper laser level. The phonon-assisted ETU2 process, Tm1(3F4) + Tm2(3F4) → Tm1(3H6) + Tm2(3H5) + nphhph, followed by fast NR relaxation of the Tm1 ion from the 3H5 state back to the 3F4 state, effectively removes only one Tm3+ ion from the upper laser level.


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Figure 1. Partial energy-level scheme of Tm3+. GSA: ground-state absorption, ESA: excited-state absorption, NR: non-radiative (multi-phonon) relaxation, Lumij: luminescent transition from ith to jth state, CR: cross-relaxation, ETU1 and ETU2: energy-transfer upconversion. The wavelengths of GSA and Lumij, and the intrinsic lifetimes τ1 and τ3 are indicated for amorphous Al2O3:Tm3+.

C. Rate-Equation System for Tm3+ Ions. In the following, we will ignore energy levels lying above the pump level 3H4, because their weak excitation by ESA is in strong competition with fast NR relaxation. In addition, we will consider that the 3H5 level has a very short intrinsic lifetime τ2, which is a good approximation for the vast majority of oxide hosts, because the energy gap between the 3H5 state and the lower-lying 3F4 state is only ~2600 cm-1 and, thus, τ2 is typically a few μs. The population densities within each of the two ion classes are denoted as Nsi (singles) and Nni (ions with active neighbors), respectively, for i = 0 (3H6), 1 (3F4), and 3 (3H4). The ion densities are related to the Tm3+ doping concentration NTm as follows: Ns 0  Ns1  Ns3  xNTm ,

(3)

Nn0  Nn1  Nn3  (1  x) NTm .

(4)

Here, x and 1 – x are the fractions of single ions and ions with neighbors, respectively. The rate-equation system for the class of single ions is 5 ACS Paragon Plus Environment


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dN s 3 1   Ps N s 0  N s 3 , dt 3

(5)

dN s1 '31 1  N  N , dt  3 s 3  1 s1

(6)

whereas for the class of ions with neighbors it is dNn3 1  Pn Nn0  WCR Nn0 Nn3  WETU 1 Nn21  Nn3 , dt 3

(7)

dNn1 ' 1  2WCR Nn0 Nn3  31 Nn3  Nn1  2WETU 1 Nn21  WETU 2 Nn21 . dt 3 1

(8)

Here, τ1 and τ3 are the intrinsic lifetimes of the 3F4 upper laser level and the 3H4 pump level, respectively. β'31 is the combined branching ratio of radiative and non-radiative transitions on the 3H4 → 3 F4 and 3H4 → 3H5 → 3F4 relaxation channels, β'31 = ANR(32)τ3 + (β32 + β31)τ3/τ3rad, where ANR is the rate of non-radiative relaxation (see below), βij are the radiative branching ratios, and τ3rad is the radiative lifetime of the 3H4 state. WCR, WETU1, and WETU2 are the macroscopic parameters of the CR and two ETU processes, respectively, and ρPs/n are the pump-rate constants for the classes of single ions and ions with neighbors, respectively: Ps / n   Ps / n ( z ) 

Pas / n ( z ) . h P AP ( z )

(9)

Here, Pas/n is the absorbed pump power for the classes of single ions and ions with neighbors, respectively, h is the Planck constant, νP is the pump frequency, AP is the lateral cross-section of the pump beam, Δ is the length over which pump light is absorbed, and z is the axial coordinate. In order to extract the macroscopic parameters of CR and ETU and the fraction of single ions, Eqs. (3)−(9) are exploited to model the measured decay curves of luminescence from the 3H4 and 3 F4 states, which are the Lum30 and Lum10 transitions in Fig. 1, respectively. In accordance with our experiments, we consider the case of quasi-continuous-wave excitation. We assume that a steady-state excitation of the Tm3+ excited states is reached during the pump pulse so that in the moment when the pump power is switched off (t = 0), the condition dNsi/dt = dNni/dt = 0 holds. It is used to determine the ion densities Nsi(t = 0) and Nni(t = 0). With these values as a starting point, the rate-equation system is then solved for the time t > 0, while ρPs/n = 0 in this situation. The solution yields time-dependent total ion densities Ni(t) = Nsi(t) + Nni(t). The time-dependent intensity of luminescence from the ith state is proportional to Ni(t). If the luminescence decay curve is normalized to unity, its temporal shape can be directly fitted with the Ni(t)/Ni(t = 0) dependence. D. Pump Absorption. In this sub-section, we will describe the determination of the pump-rate constants ρPs/n. We take into account the geometry of the investigated Tm3+-doped planar waveguides. A similar approach can be applied to the description of a bulk material excited with a divergent pump beam. For a planar waveguide, the pump-beam propagating along the z-axis is confined in the vertical direction but diverges in the horizontal direction, therefore AP = AP(z). We consider a rectangular profile of the pump beam in the transverse plane. The pump power absorbed in a small longitudinal slice of material with a length of Δz located at a distance z from the entrance face is Pa ( z  z )  Pinc ( z )(1  eloss z )(1  e N0 ( z ) a z ) .


(10)

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Here, Pinc(z) is the local incident pump power, αloss is the coefficient of passive propagation losses in the device, σa is the GSA cross-section at the pump frequency P corresponding to the 3H6 → 3 H4 transition, and N0(z) is the local population density of Tm3+ ions in the ground state. At the entrance face of the waveguide, Plaunch = ηlaunch Pinc, where Plaunch is the pump power launched into the waveguide at the entrance face, ηlaunch is the launching efficiency to the waveguide, and Pinc is the pump power incident upon the waveguide entrance face. In our optical waveguides rather high light intensities can easily be reached due to the strong mode confinement, and an accordingly strong ground-state bleaching can be present. Thus, in general, N0(z) < NTm. Pa(z) can be determined with good accuracy by dividing the waveguide into a reasonably large number of longitudinal slices and solving the rate-equation system of Eqs. (3)−(9) under steady-state conditions, dNsi/dt = dNni/dt = 0, in conjunction with Eq. (10) for each slice. In this way, one can describe the power propagation of the pump beam through the waveguide.

III. EXPERIMENTAL RESULTS A. Preparation of Al2O3:Tm3+ Waveguides. Al2 O3:Tm3+ planar layers were deposited by radio-frequency (RF) reactive co-sputtering from Al and Tm metallic targets in an AJA ATC 1500 sputtering machine. The RF sputtering power on the Tm target was varied in the range 9.2-26 W to fabricate layers with different Tm3+ concentrations NTm. A controlled oxygen (O2) flow was added to the deposition process through a gas inlet in the deposition chamber wall. The pressure in the chamber was set at ~5 mTorr to reduce the OH- level. During the synthesis, the substrate holder was heated to ~600 °C which ensured densification of the layers. The substrate, a standard 1-inch silica wafer, was thermally oxidized, resulting in an 8-μm-thick SiO2 buffer layer that ensured waveguiding within the active layer. The deposited Al2O3:Tm3+ active layers had the thickness of ~1 μm. The amorphous nature of the grown layers was confirmed by X-ray diffraction (XRD) analysis. The Tm3+ concentration of each layer was determined by Energy-Dispersive X-ray (EDX) spectroscopy using a Scanning Electron Microscope (SEM, Zeiss, model ULTRA 55) equipped with an Inca micro-analyzer (Oxford Instruments). The seven fabricated samples had Tm3+ concentrations of NTm = 0.8, 1.5, 2.1, 2.7, 3.4, 4.1, and 5.0 ×1020 cm-3. The propagation losses in the samples were αloss ~0.21±0.05 dB/cm, as measured at 829 nm using a Metricon 2010/M machine. No dependence of αloss on NTm was detected. SEM and EDX studies also confirmed good uniformity of the deposited layers. B. Luminescence Measurements. For the spectroscopic studies, ~1×1 cm2 rectangular samples were diced from the as-prepared layers. All spectroscopic studies were performed at room temperature (293 K). The studied samples were passively cooled. The Tm3+ ions were excited at ~790 nm on the 3H6 → 3H4 transition by a continuous-wave (CW) Ti:Sapphire laser (Spectra Physics, model 3900S). The pump light was coupled into the waveguides by an antireflection (AR) coated 60 microscope objective lens (OLYMPUS, N.A. = 0.85). The luminescence was collected from the top of the waveguide by a low-OH Si multimode fiber (Z-light, core diameter: 600 μm). The center of the fiber was located at a distance of ~1 mm from the entrance face of the waveguide, see Fig. 2.



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Figure 2. Geometry of the collection of luminescence from the planar Al2O3:Tm3+ waveguide.

Luminescence spectra were measured with a 0.55 m grating monochromator (Horiba-Yvon, model iHR550) equipped with two visible and near-IR photomultiplier tubes (PMT, Hamamatsu) and an electrically-cooled PbS detector. The luminescence decay curves were measured under quasi-CW excitation. The pump beam was focused and re-collimated by a pair of AR-coated spherical lenses (focal length: 30 mm). A mechanical chopper was inserted in the focal spot between the two lenses. The duty cycle was 1:14 and the rotation rate was 45 Hz. The cut-off time for the pump beam was ~5 μs. The quasi-CW incident pump power was varied with a pair or rotatable Glan-Taylor prisms. The decay curves were detected with a 1 GHz digital oscilloscope (Lecroy). The luminescence decay curves were measured at 800 nm (luminescence from the 3H4 pump level) and at 1700 nm (luminescence from the 3 F4 upper laser level). In the latter case, the choice of wavelength was determined by the sensitivity of the fast near-IR PMT. C. Luminescence Spectra. No dependence of the shape of emission spectra on the excitation wavelength in the range 0.78−0.81 μm was detected. The spectra were also very similar for all studied Tm3+ concentrations. The full emission spectrum of Tm3+ ions in amorphous Al2O3 is shown in Fig. 3. In each of the parts (a-c), (d), and (e) of this figure, the spectra are normalized to unity. In the near-IR spectral region, a broad luminescence band spanning from 1.55 to 2.05 μm is related to the 3F4 → 3H6 transition (Lum10 in Fig. 1). Luminescence from the pump level is detected at 1.4−1.55 μm (3H4 → 3F4, Lum31) and at 0.76−0.84 μm (3H4 → 3H6, Lum30). Furthermore, in the visible and near-IR spectral region, several emission bands related to transitions from the upper excited states 1D2, 1G4, and 3F2,3, see Fig. 1, are observed.


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Figure 3. Luminescence spectra of Tm3+ in amorphous Al2O3 in the visible (a,b) and near-IR (c−e) spectral region. The spectra in (a-c), (d), and (e) are each normalized to unity. The excitation wavelength is 780 nm (marked by an arrow). In (c), the excitation peak caused by the detection of residual pump power is eliminated by information from an additional measurement at the excitation wavelength of 810 nm.

D. Luminescence Decay. Luminescence decay curves were measured for all seven Al2O3:Tm3+ samples at 800 nm (from the 3H4 state) and at 1700 nm (from the 3F4 state). For each sample, the measurements were performed for several quasi-CW launched pump powers, Plaunch, ranging from 35 to 525 mW. The results are shown in Fig. 4 for the maximum and minimum Plaunch. All decay curves are plotted in a semi-logarithmic scale.



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Figure 4. Decay curves of ground-state luminescence from the 3H4 pump level (a,c) and from the 3F4 upper laser level (b,d) in amorphous Al2O3:Tm3+ planar waveguides for various Tm3+ concentrations NTm which are given in panel (b). The excitation wavelength is 790 nm, the luminescence wavelength is 800 nm (a,c) or 1700 nm (b,d), and the launched pump power is 35 mW (a,b) or 525 mW (c,d).

The luminescence decay from the 3H4 pump level at low pump power and low NTm is singleexponential, see Fig. 4(a). With increasing NTm, the decay becomes faster, which is an evidence of CR. It also deviates from a single-exponential law, which can be understood as a result of ETU1 re-feeding the pump level from the 3F4 state, leading to a slow tail at long delay times. At low pump power where the excitation density is very weak, the effect of ETU is noticeable only for high Tm3+ concentrations. For luminescence from the 3F4 upper laser level at low pump power, the decay is single-exponential for all studied NTm, see Fig. 4(b). The decay becomes faster with increasing Tm3+ concentration, which is an evidence of enhanced energy migration within and subsequent de-excitation of the 3F4 level by, e.g., ETU1 among Tm3+ pairs or energy transfer to other impurity ions. Our current spectroscopic information is insufficient to determine the reason. With increasing pump power, the shape of the decay curves changes, see Fig. 4(c) and 4(d). For luminescence from the 3H4 pump level, the slow component in the decay curves due to the refeeding becomes more significant, because at high pump power more ions are excited to the upper laser level and, thus, the ETU1 rate of WETU1N1 is increased. For low NTm, however, the decay is single-exponential, which means that at low Tm3+ concentrations the effects of ETU are generally negligible. For luminescence from the 3F4 upper laser level at high NTm, the decay curves also slightly deviate from the single-exponential law due to the appearance of a fast component attributed to the two ETU processes which depopulate the 3F4 state. However, at long delay times the decay is again single-exponential, similarly to the case of low pump power, because the 3F4 population density finally becomes very small again and, thus, the ETU processes are diminished. For 10 ACS Paragon Plus Environment

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low NTm, the luminescence decay from the 3F4 state measured at high pump power is still singleexponential. This agrees with our previous conclusion about weak ETU for low NTm. E. Transition Cross-Sections. For the development of amorphous Al2O3:Tm3+ waveguide amplifiers and lasers, it is crucial to know the absorption and stimulated-emission cross-sections, σa and σe, respectively, for the relevant absorption transitions 3H6 → 3H4 and 3H6 → 3F4 (the two possible pump channels) and emission transition 3F4 → 3H6 (the laser transition). To determine the σe spectrum, we used the Füchtbauer-Ladenburg (F-L) equation45,

5 B( JJ ')W ( )  e ( )  , 2 8 n  rad c  W ( )d

(11)

where λ is the wavelength, n is the refractive index of the material, τrad is the radiative lifetime of the emitting state, c is the speed of light, W(λ) is the measured luminescence spectrum corrected for the spectrometer response, and B(JJ') is the luminescence branching ratio for the considered transition (B = 1 in our case). For ground-state transitions, the luminescence spectra may be affected by reabsorption and the σe spectrum calculated from the F-L equation may be incorrect. In our case, W(λ) was measured in the sample with the lowest NTm from the top of a 1-μm-thick active layer, hence the reabsorption effects are almost negligible. A separate issue is to determine τrad. It can be achieved by accounting for the NR relaxation46,47, 1

 exp



1

 rad

 ANR ,

ANR  CeE .

(12) (13)

Here, ANR is the non-radiative relaxation rate constant, τexp is the experimental luminescence lifetime measured at low doping concentration, ΔE is the energy gap to the next lower-lying state, and C and α are temperature-dependent parameters representing the NR relaxation in our material. For the 3F4 state in amorphous Al2O3:Tm3+, τexp = 3.1 ms, see later in Fig. 11(c), and ΔE ~5600 cm-1. The parameters in Eq. (13) for amorphous Al2O3 are C = 4.1±2×107 s-1 and α = 2.32±0.14×10-3 cm47. Thus, τrad(3F4) = 4.3 ms. The refractive index of amorphous Al2 O3 at ~1.9 μm is n = 1.64±0.01, as deduced from the Sellmeier curve based on the refractive index measurements (Metricon 2010/M). To determine the shape of the σa spectra, we used the reciprocity method48,

 hc  ,  kT  

 e ( ) ~  a ( )exp  

(14)

where h is the Planck constant, k is the Boltzmann constant, and T is the temperature (room temperature), together with direct absorption measurements performed at a certain wavelength corresponding to the peak absorption. For the 3H6 → 3F4 transition (at ~1.63 μm), such measurements were previously reported49 and for the 3H6 → 3H4 transition (at ~0.78 μm), they have been performed in the present work. The results on the σa and σe spectra for Tm3+ ions in amorphous Al2O3 are presented in Fig. 5. For the 3H6 → 3H4 transition, the maximum absorption cross-section is 0.45×10-20 cm2 at 784 nm and the full width at half maximum (FWHM) of the corresponding absorption peak is ~14 nm. For the 3H6 → 3F4 transition, the maximum σa = 0.28×10-20 cm2 at 1630 nm and the maximum σe = 0.47 × 10-20 cm2 at 1808 nm. 11 ACS Paragon Plus Environment


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Figure 5. Spectra of (a) absorption cross-section σa on the 3H6 → 3H4 transition and (b) absorption cross-section σa and stimulated-emission cross-section σe on the 3H6 ↔ 3F4 transition in amorphous Al2O3:Tm3+. The σe spectrum is calibrated with the Füchtbauer-Ladenburg Eq. (11) and the σa spectra are determined by a combination of the reciprocity method of Eq. (14) and direct absorption measurements.

IV. APPLICATION OF THE STOCHASTIC MODEL In this Section, we justify our choice of the model introduced in Section II.B for the description of energy-transfer processes in amorphous Al2O3:Tm3+. The evidence of spectroscopically distinct ion classes has been discussed earlier for the case of amorphous Al2O3:Er3+.9 It was shown that there is a fraction of Er3+ ions which are quenched and do not participate in radiative processes. A similar conclusion was drawn for Al2O3:Yb3+.50 In the present paper, we show that even for those ions which participate in the emission, it is relevant to consider the presence of their potential neighbors. With Eq. (1), we have proposed an analytical expression for the quantification of the ion fractions and we will verify this formula experimentally. A. Effect of Various Models on Fitting the Luminescence Decay Curves. We applied four different models for fitting the measured luminescence decay curves: (i) a model taking into account only CR, (ii) a model considering CR and only one ETU process (ETU1), (iii) a model considering CR, ETU1, and ETU2, and (iv) a model considering CR, ETU1, ETU2, and two distinct ion classes (single ions and ions with neighbors). For each model, we fitted the whole set of measured decay curves. WCR(NTm), WETU1(NTm), WETU2(NTm), τ3(NTm), τ1(NTm), and x(NTm) were used as free fit parameters (where applicable). For each model, a separate fit was performed, i.e., the parameters determined with the different models are different. In Fig. 6, we analyze the quality of the fit to the decay curve for luminescence from the 3H4 pump level measured for the highest NTm = 5.0×1020 cm-3 and the highest Plaunch = 525 mW, where the nonlinear effects of CR and ETU are strongest. We present the least-squares fit of the same experimental decay curve 1 (in black) with the four models, resulting in curves 1' to 1'''' (in color).


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Figure 6. Improvement by the stochastic model of interionic processes on the fit quality of the measured luminescence decay from the 3H4 pump state in amorphous Al2O3:Tm3+ planar waveguides. 1 = measured decay curve. Individually optimized fits including the following processes: (a) 1' = model with CR, (b) 1'' = model with CR and ETU1, (c) 1''' = model with CR, ETU1, and ETU2, and (d) 1'''' = model with CR, ETU1, ETU2, and two ions classes. The Tm3+ concentration is NTm = 5.0×1020 cm-3, the excitation wavelength is 790 nm, the luminescence wavelength is 800 nm, and the launched pump power is 525 mW.

From Fig. 6, one can see that model (i) can explain the faster decay for higher NTm as compared to the lowest NTm = 0.8×1020 cm-3, which is related to CR. However, this model cannot explain the existence of the slow component in the luminescence decay. The slow component can be explained by applying models (ii) or (iii), which is a confirmation of the physical mechanism of this effect, namely ETU. However, neither model (ii) nor (iii) allows for an accurate fit of the experimental decay curves. Only when applying model (iv) which takes into account the existence of two distinct ion classes, an excellent fit to the experimental luminescence decay curve is achieved, see Fig. 6(d). To date, several models of energy-transfer processes in RE3+-doped materials based on a statistical description of the nearest-neighbor shell have been developed51-56. Such models often require knowledge of the exact crystallographic structure of the material51,52, which is difficult to deduce and even varies locally for amorphous solids or is very complicated for low-symmetry or disordered materials. Some models use a fully statistical approach by considering a large number of possible donor-acceptor environments (either by their discretization or by considering distribution functions)53-56. However, due to the complexity of such derivations, normally only simplified ener13 ACS Paragon Plus Environment


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gy-transfer mechanisms can be described. For systems with several competing energy-transfer processes and an involved energy-level system to be considered, the rate-equation formalism is simpler and can be directly applied to fitting the experimentally observable pump-, concentration-, and time-dependent emission intensities. By adding to this formalism a simple set of distinct ion classes, we introduce to the rate-equation formalism the flexibility to describe the statistical nature of ion interactions, while keeping the number of free parameters at a minimum. B. Results on Ion Fractions. By applying model (iv), as described above, to fit all measured decay curves, we were able to extract the experimental values of the fraction x of single ions for each Tm3+ concentration. The results are shown in Fig. 7. For NTm increasing from 0.8 to 5.0×1020 cm-3, x decreases accordingly from 0.80±0.15 to 0.07±0.02. To accomplish the derived x values, we performed calculations using Eqs. (1) and (2). According to our studies on the variation of the refractive index of rare-earth-doped amorphous Al2O3 films (unpublished) and previous structural studies of this material57-59, it is very unlikely that the RE3+ ions (e.g., Tm3+) would replace the Al3+ ions which are predominantly IV-fold O2-coordinated. One possible mechanism for Tm3+ ions to enter amorphous Al2O3 is that they are located in interstitial positions, most probably corresponding to VI-fold O2--coordination. This naturally limits the maximum RE3+ doping concentration in amorphous Al2O3 thin films to Nmax  12±3 × 1020 cm-3, according to our studies. This allows us to evaluate the parameter p = NTm/Nmax in Eq. (1). The coordination number for the first coordination sphere of the nearestneighbor rare-earth sites in Al2O3:RE3+ is ~8. The results of the calculations are shown in Fig. 7 as solid curves. One can see that the theoretical curve for single ions matches pretty well with the points derived from fitting the decay curves with the stochastic model considering two distinct ion classes. According to Fig. 7, at the maximum NTm studied experimentally, a major fraction of Tm3+ ions is located within clusters which mostly contribute to the energy-transfer processes.

Figure 7. Fractions of spectroscopically distinct Tm3+ ions in amorphous Al2O3 versus Tm3+ concentration. Symbols: experimental data for single ions determined from the luminescence decay. Curves: calculated data for single ions and ions with neighbors (pairs, clusters) using Eqs. (1) and (2).

C. Fitting of the Luminescence Decay Curves. The model described in Section II.B allowed us to fit the whole set of decay curves of luminescence from both excited states (3H4 and 3F4) measured for all Tm3+ concentrations. In Fig. 8, both experimental and calculated decay curves for all studied samples at the lowest Plaunch = 35 mW are displayed.


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Figure 8. Measured (black) and calculated (red) decay curves of luminescence from (a) the 3H4 pump level and (b) the 3F4 upper laser level in amorphous Al2O3:Tm3+ planar waveguides for various Tm3+ concentrations of NTm = 0.8 (curves 1,1'), 1.5 (2,2'), 2.1 (3,3'), 2.7 (4,4'), 3.4 (5,5'), 4.1 (6,6'), and 5.0 (7,7') × 1020 cm-3. The excitation wavelength is 790 nm, the luminescence wavelength is (a) 800 nm or (b) 1700 nm, and the launched pump power is 35 mW.

Figure 9. Measured (black) and calculated (red) decay curves of luminescence from (a,c) the 3H4 pump level and (b,d) the 3F4 upper laser level in amorphous Al2O3:Tm3+ planar waveguides. The Tm3+ concentration is (a,b) NTm = 5.0×1020 cm-3 or (c,d) 0.8×1020 cm-3, the excitation wavelength is 790 nm, the luminescence wavelength is (a,c) 800 nm or (b,d) 1700 nm, the launched pump power is 35 mW (curves 1,2) or 525 mW (curves 1', 2'). For the graphs (c,d), the curves 2,2' are multiplied by a factor of 0.7 for the convenience of observation.

Moreover, by using the same set of parameters we were able to fit the decay curves measured at different pump powers, as illustrated in Fig. 9. For low NTm, the increase of pump power has an almost negligible effect on the luminescence decay. For high NTm, an increase of pump power 15 ACS Paragon Plus Environment


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slows the decay from the 3H4 pump level and fastens the decay from the 3F4 upper laser level. This is attributed to the effect of ETU, according to the mechanisms described above. D. Energy-Transfer Micro-Parameters and Laser Quantum Efficiency. The modeling of luminescence decay curves described above yields the set of macroscopic CR and ETU parameters, WCR, WETU1, and WETU2, respectively, for each of the studied Tm3+ concentrations. The resulting values of these parameters are shown in Fig. 10. By fitting their dependences on NTm with a linear law47, namely WCR = CCRNTm and WETU1/2 = CETU1/2NTm, one can extract the concentrationindependent micro-parameters of CR and ETU, denoted as CCR and CETU1/2, respectively, which equal CCR = 5.83±0.26×10-38 cm6s-1, CETU1 = 0.93±0.04×10-40 cm6s-1, and CETU2 = 7.81±0.25×10-40 cm6s-1. Thus, in amorphous Al2O3:Tm3+, CR is about two orders of magnitude stronger than ETU. In addition, we conclude that the process ETU2 may be stronger than ETU1, which is somewhat surprising, because for amorphous Al2O3, according to our measurements, the maximum phonon energy is hph  820 cm-1, which means that nph ~3 are required to bridge the energy mismatch for ETU2. In contrast, for YAG:Tm these processes were found to be competitive44,60. Therefore, one may expect domination of one of the ETU processes depending on the splitting of the multiplets and the phonon spectra which are determined by the host matrix.

Figure 10. Macroscopic parameters (a) WCR, (b) WETU1, and (c) WETU2 in amorphous Al2O3:Tm3+ versus Tm3+ concentration. Symbols: data determined from the luminescence decay. Lines: their linear fits for the determination of concentration-independent microscopic parameters (a) CCR, (b) CETU1, and (c) CETU2.

The determination of microscopic CR and ETU parameters allowed us to calculate the effective lifetime τ3,eff of the 3H4 pump level and the corresponding pump quantum efficiency ηq of population of the 3F4 level22: 16 ACS Paragon Plus Environment

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1 1   WCR ( NTm ) NTm ,  3,eff ( NTm )  3

q ( NTm )  1 

WCR ( NTm ) NTm . 1/  3  WCR ( NTm ) NTm

(15)

(16)

Here, τ3 is the intrinsic lifetime of the 3H4 level, which accounts for the intra-ionic radiative and NR transitions, τ3,eff(NTm) represents the effective decay time due to the radiative and NR transitions and CR, and ηq is the mean number of excitations to the 3F4 upper laser level created by one absorbed pump photon (one excitation to the 3H4 pump level). In Fig. 11(a) and 11(b), we present results on τ3,eff and ηq, respectively. Here, the points are plotted according to the parameters derived for each Tm3+ concentration individually (points in Fig. 10) and the solid lines are calculated according to the concentration-dependent micro-parameters CCR and CETU1/2. The intrinsic lifetime of the 3H4 pump level in Al2O3:Tm3+ is τ3 = 210 μs. For the maximum Tm3+ concentration of 5.0×1020 cm-3 studied experimentally, the laser quantum efficiency reaches ηq = 1.73. According to our model, we expect ηq > 1.9 for NTm > 9×1020 cm-3, a dopant concentration that is feasible for amorphous Al2O3:Tm3+ waveguides. In addition, in Fig. 11(c) we show the results for the derived lifetime τ1 of the 3F4 upper laser level. For NTm increasing from 0.8 to 5.0×1020 cm-3, τ1 decreases from 3.10 to 2.32 ms.

Figure 11. (a) Effective decay time τ3,eff of the 3H4 pump level, (b) quantum efficiency ηq of population of the 3F4 upper laser level, and (c) experimental lifetime τ1 of the 3F4 upper laser level in amorphous Al2O3:Tm3+ versus Tm3+ concentration. Symbols: data determined from the luminescence decay. Curves– calculation according to (a) Eq. (15) and (b) Eq. (16).



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E. Applications of Al2O3:Tm3+ waveguides. Amorphous Al2O3:Tm3+ waveguides are very promising for amplifiers and lasers operating around 2 μm. Due to the relatively broad emission band of this material, one can expect broad tunability of Al2O3:Tm3+ waveguide lasers in the spectral range of 1.8-2.05 μm. Such devices can be pumped into the 3H4 state by AlGaAs laser diodes or directly into the 3F4 upper laser level (resonant or in-band pumping). In the latter case, a high laser slope efficiency is naturally provided by the low quantum defect and relatively weak ETU. In the former case, one can also expect efficient laser operation, because in amorphous Al2O3:Tm3+ CR is almost two orders of magnitude stronger than ETU. In addition, relatively high laser quantum efficiency is reached at moderate Tm3+ doping levels. To date, laser operation in amorphous Al2O3:Tm3+ on silicon was demonstrated in a micro-ring resonator configuration under in-band pumping at 1608 nm49. This micro-ring laser generated ~240 μW of output power at 1.80−1.88 μm (multi-peak spectral behavior) with a slope efficiency of η  40% with respect to absorbed pump power; NTm was 2.5×1020 cm-3. We expect better power scaling capabilities and higher laser slope efficiency for channel waveguides with higher NTm and a distributed Bragg reflector (DBR) written on top of the waveguide. The development of such lasers can offer ultra-narrow linewidth of the laser emission, which is crucial for sensing applications. The laser emission wavelength in this case can be varied by changing the period of the DBR structure to fit precisely the absorption transitions of various gases and bio-molecules.

V. CONCLUSIONS Energy-transfer processes in lanthanide-doped materials, e.g. cross-relaxation and energy-transfer upconversion, can be quantified correctly only if considering the presence of spectroscopically distinct ion classes. We have presented a novel stochastic model which distinguishes between single ions and ions with neighbors (pairs, clusters). Single ions do not participate in energy-transfer processes. Their fraction drops with increasing doping concentration. An analytical expression for the ion fractions has been presented and experimentally verified. This simple stochastic model may prove very useful for the description of luminescence decay and the quantification of energytransfer processes in various lanthanide-doped amorphous and crystalline materials. Our stochastic model has been applied to the case of energy-transfer processes in amorphous Al2O3:Tm3+ waveguides on silicon excited to the 3H4 state. It accounts for competitive interaction of the CR and two ETU processes and yields the concentration-independent microscopic energytransfer parameters, CCR and CETU1/2, and the pump quantum efficiency ηq for the excitation of Tm3+ ions to the 3F4 upper level of the 2-µm laser transition. The determined microscopic CR and ETU parameters are CCR = 5.83 × 10-38 cm6s-1, CETU1 = 0.93 × 10-40 cm6s-1, and CETU2 = 7.81 × 10-40 cm6s-1. For the maximum studied Tm3+ concentration of 5.0 × 1020 cm-3, ηq reaches 1.73. We expect amorphous Al2O3:Tm3+ films to be highly suitable for the development of efficient DBR narrow-linewidth channel waveguide lasers for sensing applications operating in the spectral range of 1.8–2.05 μm. The slope efficiency of such lasers under pumping into the 3H4 state can be boosted by reaching high Tm3+ concentrations of 5–8 × 1020 cm-3.

AUTHOR INFORMATION Corresponding author *e-mail: [email protected] Notes The authors declare no competing financial interest. 18 ACS Paragon Plus Environment

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ACKNOWLEDGMENTS The authors acknowledge financial support by the ERC Advanced Grant “Optical Ultra-Sensor” No. 341206 from the European Research Council and the Swedish Research Council (VR) via the Linnaeus Center for Advanced Optics and Photonics (ADOPT). Kerstin Wörhoff is thanked for the deposition of the Al2O3:Tm3+ planar layers and Dimitri Geskus is thanked for help with the experimental setups.

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Stochastic Model of Energy-Transfer Processes Among Rare-Earth

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