Absolute quantum yield for understanding upconversion and downshift luminescence in PbF₂:Er³⁺,Yb³⁺ crystals

Abstract

The search for new materials capable of efficient upconversion continues to attract attention. In this work, a comprehensive study of the upconversion luminescence in PbF₂:Er³⁺,Yb³⁺ crystals with different concentrations of Yb³⁺ ions in the range of 2 to 7.5 mol% (Er³⁺ concentration was fixed at 2 mol%) was carried out. The highest value of upconversion quantum yield (ϕ_UC) 5.9% (at 350 W cm⁻²) was found in the PbF₂ crystal doped with 2 mol% Er³⁺ and 3 mol% Yb³⁺. Since it is not always easy to directly measure ϕ_UC and estimate the related key figure of merit parameter, saturated photoluminescence quantum yield (ϕ_UCsat), a method to reliably predict ϕ_UCsat can be useful. Judd–Ofelt theory provides a convenient way to determine the radiative lifetimes of the excited states of rare-earth ions based on absorption measurements. When the luminescence decay times after direct excitation of a level are also measured, ϕ_UCsat for that level can be calculated. This approach is tested on a series of PbF₂:Er³⁺,Yb³⁺ crystals. Good agreement between the estimates obtained as above and the directly experimentally measured ϕ_UCsat values is demonstrated. In addition, three methods of Judd–Ofelt calculations on powder samples were tested and the results were compared with Judd–Ofelt calculations on single crystals, which served as the source of the powder samples. Taken together, the results presented in our work for PbF₂:Er³⁺,Yb³⁺ crystals contribute to a better understanding of the UC phenomena and provide a reference data set for the use of UC materials in practical applications.

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Introduction

The synthesis of materials with efficient upconversion (UC) luminescence is a hot topic in materials science due to their use in a wide range of applications. These include potential applications in photovoltaics and solar energy harvesting,^1–3 security markers,^4–7 luminescent thermometry,^8,9 as well as tracers for advanced plastics sorting.¹⁰ Most of these applications are made possible by the unique optical properties of the trivalent ions of the lanthanides, which are defined by partially forbidden transitions within 4f electronic shells. UC processes in lanthanide doped materials can be realised by excited state absorption, photon avalanche or energy transfer up-conversion (ETU),¹¹ with ETU providing the highest photoluminescence quantum yield (ϕ_UC) at intensities that can be regarded as low (<40 W cm⁻²).^12–14

Recently, efficient ETU in crystalline materials based on MF₂ (M = Ca, Sr, Ba) hosts doped with Er³⁺ and Yb³⁺ has received increasing attention.^15–19 These hosts generally have a cubic unit cell, which is considered unfavourable for radiative transitions and efficient energy transfer between doping ions, which is crucial for the ETU process. However, when doped with trivalent lanthanide ions, the symmetry of the local environment is reduced, increasing the probability of the transitions. This is because trivalent ions replace divalent cations and require charge compensation via F⁻ ions taking interstitial positions in the lattice.^20–23 In addition, MF₂ hosts tend to have a lower maximum phonon energy than other fluoride hosts: CaF₂ – 320 cm⁻¹,²⁴ SrF₂ – 284 cm⁻¹,²⁴ BaF₂ – 240 cm^−1 24versus β-NaYF₄ – 360 cm⁻¹,²⁵ LaF₃ – 350 cm⁻¹,²⁶ and LiYF₄ – 460 cm⁻¹.^27,28 Low phonon energy hosts favour achieving high ϕ_UC values. For near-infrared (980 nm) to visible UC, the highest ϕ_UC values observed in MF₂ materials are 6.5% in SrF₂:Er³⁺,Yb^3+ 16 and 10.0% in BaF₂:Er³⁺,Yb³⁺,¹⁹ whereas the highest known ϕ_UC in a material doped with Er³⁺ and Yb³⁺ ions is 11% in β-NaYF₄.¹³ However, to date, there have been only a small number of publications on lanthanide-doped crystalline PbF₂ hosts. From the available data it can be inferred that PbF₂ doped with Er³⁺ and Yb³⁺ should perform similarly to the other materials mentioned above due to low phonon energy (257 cm⁻¹).²⁹ In addition, the observed ϕ_UC increases when moving from SrF₂ to BaF₂ thus suggesting that heavier cations help to achieve high ϕ_UC values, thus PbF₂ could offer even better UC performance.

Usually, the UC samples can be obtained either in a form of micro/nanometer size particles or in a form of single crystals. While the characterisation of crystalline materials is best performed when the samples are in the form of single crystals due to the convenient and reliable measurement of the absorption coefficient, luminescence decay time and ϕ_UC value as well as Judd–Ofelt (JO) calculations, the micro- or nanoparticles are often more feasible for applications and easier to synthesise. Thus, it is crucial to compare the results obtained with these two forms of the material.

This paper presents a comprehensive study of PbF₂ crystals doped with Er³⁺ and Yb³⁺ – the Er³⁺ concentration was set to either 2 or 1.5 mol% and the Yb³⁺ concentration was varied in the range 1.5–7.5 mol%. To gain insight the upconversion properties of the studied crystals, both the power density dependent ϕ_UC under 976 nm excitation and the down-shifting quantum yield (ϕ_DS) under 522 and 652 nm excitation were obtained. The analysis of ϕ_UC and ϕ_DS values and radiative lifetimes from JO calculations was used to understand the UC mechanism and to find factors limiting ϕ_UC in Yb³⁺/Er³⁺ codoped PbF₂ crystals. In addition, powder samples prepared by grinding the above crystals were studied. JO analysis on crystalline and powder samples provided a useful comparison of three different methods for calculating JO parameters on powder materials.

Experimental part

Synthesis procedure

The single crystals based on PbF₂ doped with Yb³⁺ and Er³⁺ were grown by the Bridgman technique in a vacuum using the CF₄ fluorination atmosphere in multichannel graphite crucibles with a temperature gradient (7 deg mm⁻¹). The growth rate (7 mm hour⁻¹) was estimated from the stability function of a flat crystallization front.³⁰ Based on the PbF₂–RF₃ (R = Yb, Er)³¹ phase diagrams, the single crystal growth temperature was chosen to be 870 °C. The crystalline samples were prepared in a shape of disks 10 mm in diameter and about 1.7 mm in thickness, cut perpendicular to the long axis of the crystal boule.

Characterization

To estimate maximum host phonon energy, the Raman spectrum of the undoped PbF₂ sample was recorded (Polytec i-Raman instrument) using 785 nm excitation and with a 3.5 cm⁻¹ resolution. It was not possible to obtain the Raman spectrum of a doped PbF₂ sample due to presence of emission from ⁴I_9/2 energy level of Er³⁺ under 785 nm excitation that makes the detection of the pure Raman bands complicated.

The crystalline structure of the samples was determined using the powder XRD patterns recorded with a diffractometer (Bruker, D2 PHASER) (CuKα radiation). A small part of the single crystal was ground into powder. The patterns were recorded in the 2 theta range from 10 to 70 degrees.

Absorption spectra of the crystals were recorded at a room temperature using a ultraviolet (UV)-visible (Vis)-NIR spectrophotometer (PerkinElmer Lambda 950) in absorbance mode. The instrument provided the absorbance data, which was then converted to the absorption coefficient using eqn (1):

where α is the absorption coefficient in cm⁻¹, A is the absorbance data obtained from the instrument, and d is sample thickness in centimetres.

The concentration of Er³⁺ and Yb³⁺ ions (Table S2, ESI†) was determined by wavelength dispersive X-ray fluorescence (WDXRF) spectroscopy (Pioneer S4, Bruker AXS).

Excitation spectra were recorded using a calibrated spectrophotometer (Varian Cary Eclipse). Diffuse reflectance spectra were recorded at room temperature using a spectrophotometer (PerkinElmer Lambda 950) in absorbance mode with the sample placed inside an integrating sphere.

The setup and the methodology for estimating ϕ_UC under 976 nm excitation have been described previously.^16,19,32 The setup is built around an integrating sphere (Labsphere, Ø6′′, 3 P-LPM-060-SL) and uses two calibrated spectrometers (Avantes, AvaSpec-ULS2048×64TEC, Thorlabs, CCS200/M) to register the intensity of the sample emission and the incident laser. A 976 nm laser diode (Roithner) driven by a laser diode controller (ITC4001, Thorlabs) is used as the excitation source and the incident intensity is varied with a variable filter wheel (Thorlabs, NDC-100C-2).

To record the ϕ_DS of the ⁴S_3/2 → ⁴I_15/2 and ⁴F_9/2 → ⁴I_15/2 transitions of the Er³⁺ ions, a tunable continuous wave (CW) laser (Solstis with EMM-Vis, M-Squared Lasers Ltd) pumped by 532 nm laser (Verdi-V18, Coherent) is used. The system is tuned to 522 nm for the direct excitation of the ⁴S_3/2 level and to 652 nm for the direct excitation of the ⁴F_9/2 level. For the measurement of ϕ_DS of the ⁴I_13/2 → ⁴I_15/2 transition under direct excitation the tunable laser kit (Thorlabs, TLK-L1550M) operating at 1495 nm was used as the excitation source. The rest of the setup was the same as described in our previous publication.^19,32

Luminescence lifetimes are measured using an optical system described previously.^17,19 525 nm, 976 nm, and 633 nm (Roithner) and 1550 nm (Thorlabs) laser diodes mounted in temperature stabilized mounts (TCLDM9, Thorlabs) and driven by a laser diode controller (ITC4001, Thorlabs) are used as the excitation sources. The luminescence wavelength is selected with a double monochromator (Bentham, DTMS300) and the signal is detected with a photomultiplier tube (R928P, Hamamatsu) mounted in a temperature-cooled housing (CoolOne, Horiba) in the UV-Vis region or with an infrared single-photon detector (ID Quantique, ID220) in IR region-both detectors are coupled to the multi-channel scaling card (TimeHarp 260, Picoquant).

Results and discussion

Crystal structure characterization

The measured powder XRD patterns are presented in Fig. 1(a). The data is in good agreement with JCPDS card # 76-1816. The XRD patterns show that the parameters of a unit cell change with doping concentration. To illustrate this the position of the [111] peak in samples with different Yb³⁺ contents is plotted in Fig. 1(b). The position of the [111] peak is shifted to the higher angles as the concentration of the dopant ions is increased. This is due to the fact that the ionic radius of the Er³⁺ and Yb³⁺ ions is smaller than that of the Pb²⁺ ions, which causes the shrinking of the unit cell in samples with higher doping concentration. The Raman spectrum for an undoped PbF₂ crystal is presented in Fig. 1(c). It consists of a broad band with a maximum at 260 cm⁻¹. This value agrees well with the previously reported phonon energy of a PbF₂ crystal equal to 257 cm⁻¹.²⁹

Fig. 1 (a) Powder XRD patterns of the PbF₂:Er³⁺,Yb³⁺ samples; (b) change in the [111] peak position with increasing Yb³⁺ concentration; (c) room temperature Raman spectrum of the undoped PbF₂ crystal.

Absorption spectra and JO calculations

The absorption spectra of the samples investigated are given in Fig. 2. The spectra contain absorption bands in the UV, Vis and NIR regions typical for Er³⁺ and Yb³⁺ ions. The transitions corresponding to the most intense bands are labelled in Fig. 2. The positions of all bands remain the same in all samples and are in agreement with the available literature data.^16,19 The absorption data can be used in the JO method to calculate key features of electron levels in luminescent materials, such as radiative lifetime and branching ratios. The comparison of these radiative lifetimes with experimentally obtained luminescence decays can additionally reveal the fraction of excitation energy emitted via radiative processes, which, in turn, can help to predict both the down-shifting quantum yield ϕ_DS and the upconversion quantum yield ϕ_UC.^33,34

Fig. 2 Absorption spectra of the PbF₂:Er³⁺,Yb³⁺ crystals.

A standard procedure was used to obtain JO parameters in our work.^35,36 The JO theory uses absorption cross-sections to determine oscillator strengths (Table S1, ESI†). These values were then used to calculate JO parameters Ω_t, which allow the description of a radiative transition between any two levels. To obtain the Ω_t values, some additional parameters, such as the barycentre wavelength, the doping concentration of Er³⁺ ions, the refractive index and the reduced matrix elements should be accurately estimated beforehand. These values are given in Table 1 (reduced matrix elements³⁷ and refractive index data³⁸ were taken from the literature and are universal for all samples) and Table S2, ESI† (concentrations of doping ions estimated by the WDXRF method).

Table 1 Barycentre wavelength (λ_b), along with literature values for the reduced matrix elements (U)³⁷ and refractive index (n)³⁸ of the host at the appropriate wavelengths

Excited state	λ b, nm	[U^(2)]	[U^(4)]	[U^(6)]	n
^4G11/2	378.5	0.9156	0.5263	0.1167	1.8314
^2H9/2	406.5	0	0.0243	0.2147	1.8149
^4F7/2	487.0	0	0.1465	0.6272	1.7857
^2H11/2	522.0	0.7158	0.4128	0.0927	1.7774
^4S3/2	542.0	0	0	0.2235	1.7740
^4F9/2	653.0	0	0.55	0.4621	1.7593
^4I13/2	1520.0	0.0195	0.1172	14.325	1.7335

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Briefly, the parameters used in the JO calculations are obtained as follows. The absorption cross sections were calculated from the absorption spectra shown in Fig. 2 and the doping concentration of the Er³⁺ ions. The barycentre energy (E_b) and then the barycentre wavelength were determined using the method described by Hehlen et al.³⁵ In general, when an absorption band spans over energies from E₀ to E₁ then E_b is defined as where α(λ) is absorption cross-section. The reduced matrix elements are taken from the work of Carnall et al.³⁷ as Hehlen et al.³⁵ demonstrated that the host has an insignificant effect on the values of the reduced matrix elements. The refractive indices n of the PbF₂ corresponding to the wavelength of each absorption transition in the Er³⁺ ions are taken from the work of Malitson and Dodge.³⁸ The root mean square (RMS) approach described by Hehlen et al.³⁵ was used to calculate the JO parameters Ω_t. This approach allows to reduce the influence of more intense absorption bands on the calculation result. The uncertainties in the JO parameters Ω_t were calculated using the matrix approach described by Zhang et al.³⁹ and by Görller–Walrand and Binnemans.⁴⁰ The uncertainty of the radiative lifetime could also be calculated from the known uncertainty in Ω_t (Table 2). The full results of the JO calculations are given in the Tables S2–S7 (ESI†), while Table 2 summarizes the values of radiative lifetimes (τ_rad) and branching ratios (β) that are important for the further discussion.

Table 2 Radiative lifetimes (τ_rad) and branching ratios (β) of some transitions of the PbF₂:Er³⁺,Yb³⁺ samples, and calculated Judd–Ofelt parameters

		Er1.5Yb15	Er2Yb2	Er2Yb3	Er2Yb5	Er2Yb7.5
^4S3/2–^4I15/2	τ rad, ms	1.12 ± 0.06	0.85 ± 0.05	0.81 ± 0.04	0.82 ± 0.04	0.63 ± 0.04
	β	0.67	0.67	0.67	0.67	0.67
^4F9/2–^4I15/2	τ rad, ms	1.47 ± 0.14	1.11 ± 0.12	1.06 ± 0.09	1.03 ± 0.11	0.86 ± 0.10
	β	0.91	0.91	0.91	0.91	0.92
^4I13/2–^4I15/2	τ rad, ms	9.71 ± 0.17	8.39 ± 0.20	8.12 ± 0.17	8.17 ± 0.19	7.00 ± 0.21
	β	1.00	1.00	1.00	1.00	1.00
Judd–Ofelt parameters, ×10^−20 cm^2
Ω 2		1.053 ± 0.105	1.342 ± 0.134	0.986 ± 0.099	0.952 ± 0.095	1.289 ± 0.129
Ω 4		0.836 ± 0.125	0.469 ± 0.070	1.163 ± 0.174	0.125 ± 0.019	1.381 ± 0.207
Ω 6		0.839 ± 0.042	1.795 ± 0.090	1.160 ± 0.058	1.140 ± 0.057	1.480 ± 0.074

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The following observations can be made from these results. Firstly, the increase in the doping concentration of Yb³⁺ leads to higher transition probabilities in Er³⁺ and thus shorter Er³⁺ radiative times. On the other hand, the branching ratios are similar in all samples. Unfortunately, in many cases there is no a simple relationship between the radiative lifetime and the number of emitted photons emitted, since non-radiative relaxation and various quenching processes must be taken into account. Nevertheless, the study of luminescence decay and knowledge of the radiative lifetime is a useful data set that can shed light on the quantum yield of luminescent materials.

Fig. 3(a) demonstrates the luminescence spectra of the PbF₂:Er³⁺,Yb³⁺ crystals excited at a wavelength of 375 nm. The emission spectra consist of several peaks corresponding to ⁴S_3/2–⁴I_15/2, ⁴F_1/2–⁴I_15/2, ⁴S_3/2–⁴I_13/2, {Er³⁺:⁴I_11/2 & Yb³⁺:²F_7/2}–⁴I_15/2 and ⁴I_13/2–⁴I_15/2 transitions. In order to correlate the results of the JO calculations with the experimental results, the luminescence decays of three states (⁴S_3/2, ⁴F_1/2 and ⁴I_13/2) were studied (Fig. 3(b)–(d)) using the direct excitation of the above states. The first interesting observation based on these decays is that the decay time of the ⁴I_13/2–⁴I_15/2 transition (Fig. 3(d) and Table S8, ESI†) is always longer than the radiative lifetime calculated using the JO model (Table 2). The uncertainties of the experimentally determined decay times were calculated using a method described in the work of Fišerová and Kubala.⁴¹ It can be assumed that such an extension of the decay time is due to the re-absorption of emitted photons within the crystal.^42,43 To test this hypothesis, two exemplary crystals (Er2Yb5 and Er2Yb2) were carefully ground and the powder was diluted with undoped PbF₂ powder (the experimental protocol is inspired by the works of de Mello Donegá et al.⁴² and Rabouw et al.⁴³) until the ratio between the undoped and doped fractions was 19 to 1. In such an experiment, the local concentration of Er³⁺ and Yb³⁺ remains unchanged, whereas the reabsorption can be significantly reduced by dilution. Indeed, the dilution led to a pronounced reduction in the decay time (Fig. 3(e)), which decreased from 8.8 ms to 5.4 ms for the powder with undoped/doped (Er2Yb5 crystal) ratios of 9/1 and 19/1 (Fig. 3(f)). The luminescence decays of the other two transitions ⁴S_3/2–⁴I_15/2 and ⁴F_9/2–⁴I_15/2 were also prolonged in the crystal (which may indicate an effect of re-absorption), but to a much lesser extent (Fig. S1, ESI†). Another example of data with a similar trend can be found in ESI† for the crystal Er2Yb2 (Fig. S2, ESI†).

Fig. 3 (a) Emission spectra under 375 nm excitation, (b) luminescence decay of the Er³⁺:⁴S_3/2–⁴I_15/2 transition under 525 nm excitation, (c) luminescence decay of the Er³⁺:⁴F_9/2–⁴I_15/2 transition under 640 nm excitation, (d) luminescence decay of the Er³⁺:⁴I_13/2–⁴I_15/2 transition under 1535 nm excitation in the PbF₂:Er³⁺,Yb³⁺ samples; (e) luminescence decay and (f) luminescence decay time of the Er³⁺:⁴I_13/2–⁴I_15/2 of the PbF₂:2 mol% Er³⁺,5 mol% Yb³⁺ crystal and diluted powders under 1535 nm excitation.

A second interesting phenomenon can be observed in Fig. 3(b) with multi-exponential behaviour of all decays. It is well known that the ⁴S_3/2 state of Er³⁺ undergoes cross-relaxation with the ground states of Er³⁺(⁴I_15/2) and Yb³⁺(²F_5/2).⁴⁴ Due to the distribution of the inter-ion distance (Er³⁺–Er³⁺ or Er³⁺–Yb³⁺) the cross-relaxation rate can take different values, resulting in a multi-exponential decay. For simplicity, the decays in Fig. 3(b) were fitted with a bi-exponential model, giving the results shown in Table S8 (ESI†). It is important to note that crystals with the lowest Er³⁺ concentration (Er1.5Yb1.5 and Er2Yb2) demonstrate the presence of the short-lived component with decay times of 0.18 and 0.08 ms and a significant contribution from the long-lived component with decay times of 1.08 and 0.94 ms, respectively. Simultaneously, the JO calculation predicts radiative lifetimes of 1.12 and 0.85 ms (Table 2), respectively. Thus, the close examination of the decays for Er1.5Yb1.5 and Er2Yb2 crystals shows that the long-lived component of the decay is close to or even exceeds the radiative lifetime obtained with the JO method, which cannot be explained by simple re-absorption (with almost no extension of the experimental decay times in Fig. S1 and S2, ESI†) and requires a reasonable explanation.

It can be assumed that due to the formation of clusters of lanthanide ions in MF2 (M = Ca, Sr, Ba, Pb),^20,45,46 which can affect the radiative transitions, the radiative decay time may have a distribution resulting in a multi-exponential decay. The long-lived component of the decay could correspond to Er³⁺ ions in a highly symmetric environment (cubic symmetry of the PbF₂ crystal structure), whereas the short-lived component corresponds to Er³⁺ ions in a less symmetric environment (corresponding to ion clusters). These two symmetries are distinguishable in luminescence decays due to the strong cross-relaxation of the ⁴S_3/2 state. The strong cross-relaxation is expected for the cluster environment (with a shorter interionic distance), which strongly reduces the lifetime of the short-lived component. The two different environments are more difficult to detect for the ⁴F_9/2 and ⁴I_13/2 states because these states do not participate in cross-relaxation and the difference between the radiative lifetimes is hardly noticeable in experimental decays.

Analysis of the amplitude ratio (A_i) for the short-lived and long-lived components (Table S8, ESI†) for decays of ⁴S_3/2 state allows us to understand which fraction of the Er³⁺ ions are distributed in the two crystalline environments. The crystals with the lowest doping concentration of Er1.5Yb1.5 and Er2Yb2 have comparable A₁ and A₂ values (amplitudes of the short-lived and long-lived components, respectively). In this case, the processing of the absorption spectra with the JO model probably gives an “average” value of the radiative lifetime, although the correctness of the application of the JO model for a material with two radiative lifetimes may be questionable. As the concentration of Yb³⁺ increases (Er2Yb3, Er2Yb5 and Er2Yb7.5 crystals), the number of Er³⁺ ions in the cluster fraction increases and becomes >95% for the Er2Yb5 crystal. Therefore, the Er2Yb5 crystal (with a dominant radiative lifetime) was chosen for further analysis of the correlations between JO calculations and experimental results.

Assuming that in the Er2Yb5 crystal (i) there is only one value of the radiative lifetime for each ⁴S_3/2, ⁴F_9/2 and ⁴I_13/2 state; (ii) there is a constant quenching rate for the ⁴S_3/2, ⁴F_9/2 and ⁴I_13/2 states, where the quenching mechanism of the ⁴S_3/2 state is cross-relaxation with the ground states (⁴I_15/2 or ²F_5/2); and (iii) the measurement artefact associated with the re-absorption of emitted photons is suppressed, the radiative lifetime, decay time, and quantum yield are bound by a simple equation:

where β is the branching ratio, τ_rad is the radiative decay time calculated from the JO analysis and τ is the experimentally determined decay time.

It is very likely that the conditions (i)–(iii) can be fulfilled for the ⁴S_3/2–⁴I_13/2 transition in the Er2Yb5 sample. Especially because this transition between two excited states is less affected by re-absorption. For example, Fig. S1c, and S3 (ESI†) indicate that there is a similar decay time for the ⁴S_3/2–⁴I_13/2 transition in the Er2Yb5 crystal and all powder mixtures prepared on the bases of this crystal. Taking into account the decay time (τ = 0.05 ± 0.01 ms), the radiative lifetime (τ_rad = 0.82 ± 0.01 ms) and the branching ratio of β = 0.28 (Table S6, ESI†), eqn (2) allows to calculate the quantum yield of the ⁴S_3/2–⁴I_13/2 luminescence as ϕ_DS = 1.4 ± 0.3%, which agrees very well with the measured value of ϕ_DS = 1.3% (Table 3). When the same equation is applied to the ⁴S_3/2–⁴I_15/2 and ⁴F_9/2–⁴I_15/2 transitions, the calculation gives values of quantum yield of ϕ_DS = 3.3 ± 0.07% and ϕ_DS = 29.2 ± 4.7% respectively. However, lower values of quantum yield of ϕ_DS = 3.1% and ϕ_DS = 24.7% (Table 3) have been measured experimentally using an integrating sphere and correction procedure in agreement to Wilson and Richards⁴⁷ (Fig. S4, ESI†). The values before the correction as measured in an integrating sphere are given in Table S9 (ESI†). It is assumed that the difference between the quantum yields predicted (based on lifetime) and experimentally measured quantum yields – 3.3% vs. 3.1% (for the ⁴S_3/2–⁴I_13/2 transition) and 29.2% vs. 24.7% (for the ⁴F_9/2–⁴I_15/2 transition) – illustrates the realistic degree of agreement that can be expected between the JO calculations and the experimental result.

Table 3 ϕ _DS (%) of some transitions in the PbF₂:Er³⁺,Yb³⁺ samples upon excitation of the ⁴S_3/2, ⁴F_9/2, Er³⁺:⁴I_11/2 & Yb³⁺:²F_7/2 and ⁴I_13/2 levels measured in an integrating sphere and corrected as described in Fig. S4 (ESI). Intensities of the 522 nm, 652 nm, 940 nm and 1495 nm lasers are 0.1 W cm⁻², 0.3 W cm⁻², 0.1 W cm⁻² and 0.1 W cm⁻², respectively

Excitation	Emission	Er1.5Yb1.5	Er2Yb2	Er2Yb3	Er2Yb5	Er2Yb7.5
522 nm	^4S3/2–^4I15/2	5.4	3.2	3.5	3.1	2.5
	^4S3/2–^4I13/2	2.0	1.2	1.5	1.3	0.9
	^4F9/2–^4I15/2	8.7	7.6	7.0	4.0	1.7
	{Er^3+:^4I11/2 & Yb^3+:^2F7/2}	46.7	51.9	57.0	48.3	42.0
652 nm	^4F9/2–^4I15/2	22.0	23.5	28.2	24.7	23.4
	{Er^3+:^4I11/2 & Yb^3+:^2F7/2}	29.5	31.9	31.5	29.4	18.9
940 nm	{Er^3+:^4I11/2 & Yb^3+:^2F7/2}	37.3	60.6	69.4	52.9	43.2
	^4I13/2–^4I15/2	3.0	4.4	4.5	2.7	3.0
1495 nm	^4I13/2–^4I15/2	85.1	74.8	69.2	68.6	69.5

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Given the results presented in Table 3, the probabilities of the transitions ⁴S_3/2–⁴F_9/2 and {Er³⁺:⁴I_11/2 & Yb³⁺:²F_7/2}–⁴I_13/2 can be also estimated from the values of ϕ_DS. Considering the model with three energy levels (0, 1 and 2) (Fig. S5, ESI†), the probability of transition (as the sum of radiative and non-radiative transitions) φ₂₋₁ can be calculated using the quantum yields of the radiative transition (ϕ₁₋₀) at excitations (0–1) and (0–2):

Since β for ⁴S_3/2–⁴F_9/2 is very small – 3 × 10⁻⁴, φ_2–1 for the ⁴S_3/2 state represents the fraction of all excited ⁴S_3/2 states that decay non-radiatively (via multi phonon relaxation) to the ⁴F_9/2 state. In the case of {Er³⁺:⁴I_11/2 & Yb³⁺:²F_7/2}–⁴I_13/2, β is unknown. Therefore, φ_2–1 represents the fraction of all {Er³⁺:⁴I_11/2 & Yb³⁺:²F_7/2} excited states that decay both radiatively and non-radiatively to the ⁴I_13/2 state.

The data in Table 4 demonstrates that φ_2–1 for the {Er³⁺:⁴I_11/2 & Yb³⁺:²F_7/2}–⁴I_13/2 transition is in the range of 0.04–0.07 for all Yb³⁺ concentrations and is slightly smaller than the probability for the ⁴I_11/2–⁴I_13/2 radiative transition, since it has been shown that the branching ratio for the ⁴I_11/2–⁴I_13/2 transition in Er³⁺ doped fluorides is in the range 0.11–0.15.^33,48,49 In contrast, the φ_2–1 value for the ⁴S_3/2–⁴F_9/2 transition decreases with increasing Yb³⁺ concentration, indicating the predominance of cross-relaxation over non-radiative relaxation in crystals with high Yb³⁺ doping concentration.

Table 4 Transition probabilities between certain Er³⁺ excited states in the PbF₂:Er³⁺,Yb³⁺ samples in the low excitation intensity range (0.1–0.3 W cm⁻²)

φ 2–1	Er1.5Yb1.5	Er2Yb2	Er2Yb3	Er2Yb5	Er2Yb7.5
^4S3/2–^4F9/2	0.40	0.32	0.25	0.14	0.07
{Er^3+:^4I11/2 & Yb^3+:^2F7/2}–^4I13/2	0.04	0.06	0.07	0.04	0.04

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Judd–Ofelt calculations for powder samples

JO calculations can help predicting parameters like radiative transition rates, branching ratios and even quantum efficiencies that can give an insight into the applicability of the materials. However, it is not always possible to obtain single crystals of the desired chemical composition for the JO analysis. In most cases, newly synthesized materials are obtained in a form of micro- or nanoparticles. For such samples, it is not easy to obtain the absorption cross-section for the JO method as it first requires determination of the absorption coefficient that in turn can be defined if the light propagation path is known. Instead, powder samples are usually treated using excitation⁵⁰ and diffuse reflection^51–53 spectra. We assumed that the accuracy of JO calculations for a powder sample can be evaluated if the result of JO calculations for the equivalent crystal is already known. To test this approach a part of the Er2Yb5 crystal was ground into microparticles, followed by measurements of excitation and diffuse reflectance spectra.

In the case of Er³⁺ ions, all methods of JO analysis of powder samples use the luminescence decay time of the ⁴I_13/2–⁴I_15/2 transition to calibrate the JO parameters Ω_t. It is assumed that the luminescence decay time of the ⁴I_13/2–⁴I_15/2 transition is equal to the radiative lifetime and that the ϕ_DS of the ⁴I_13/2–⁴I_15/2 transition is equal to 100% upon the excitation of the ⁴I_13/2 state. However, this assumption does not take into account the various quenching processes that can take place in the non-ideal sample. In order to quantify this effect, the ϕ_DS of the ⁴I_13/2–⁴I_15/2 transition was determined using 1495 nm excitation, giving a value of 68.8%. Furthermore, the results of the previous section clearly show that the decay times obtained for the crystal and for the powder with the same chemical composition differ significantly due to reabsorption effect.⁵⁴ The decay time of the ⁴I_13/2–⁴I_15/2 transition in the Er2Yb5 crystal is 8.8 ms whereas the same transition in the powder has a decay time of 7.4 ms (5.4 ms for the diluted powder (Fig. 2(f))). It can be also noted that the experimentally measured lifetime in the crystal exceeds the radiative lifetime obtained from the JO analysis, which is 8.4 ms.

Using eqn (1) (with β = 1) it is possible to determine experimentally that the radiative lifetime of the ⁴I_13/2–⁴I_15/2 transitions as 7.9 ms (given ϕ_DS = 68.6% and τ = 5.4 ms), which is in the good agreement with the value of 8.4 ms obtained from the JO calculations for the crystal. The results lead to a rather curious observation: quenching decreases the luminescence decay time, whereas reabsorption increases it. Thus, the experimental decay time for the powder (7.4 ms) is only coincidently close to the radiation lifetime predicted by JO theory (8.4 ms) for the single crystal. For the sake of simplicity, the value of 8.4 ms (calculated using JO theory for the crystal) was taken as the radiative lifetime in the further calculations for the powder sample.

With the radiative lifetime of the ⁴I_13/2–⁴I_15/2 transition is defined (8.4 ms), it is now possible to perform the JO calculations on powder samples. First, the Method A described in the paper⁵⁰ was tested. It uses the excitation spectrum recorder while monitoring the emission of the ⁴S_3/2–⁴I_15/2 transition. The spectrum obtained is given in Fig. 4(a). Using the transitions marked with arrows, the JO parameters Ω_t were calculated in arbitrary units and then recalculated using the radiative lifetime (8.4 ms) of the ⁴I_13/2–⁴I_15/2 transition established above. These parameters were then used to estimate the transition probabilities and radiative lifetimes of other transitions. The algorithm of this method is illustrated in Fig. S6a (ESI†).

Fig. 4 (a) The excitation spectrum of the ⁴S_3/2–⁴I_15/2 transition (detection at 540 nm); (b) the reflectance (R) spectrum of the Er2Yb5 powder sample. The arrows indicate the bands used to calculate the Judd–Ofelt parameters.

Secondly, the Method B described in the paper⁵¹ was evaluated. This approach uses the diffuse reflectance spectrum of the powder sample. The spectrum obtained is presented in Fig. 4(b). The band located around 800 nm was excluded from the calculation due to its extremely low signal to noise ratio. The band with a maximum at 980 nm was also not included in the calculations because the observed absorption band is an overlap of the ⁴I_11/2–⁴I_15/2 transition of Er³⁺ ions and the ²F_7/2–²F_5/2 transition of Yb³⁺ ions. To convert from arbitrary units of the diffuse reflectance spectrum to cm² of the absorption cross-section, the experimentally determined oscillator strength of the ⁴I_15/2–⁴I_13/2 transition is calibrated to the radiative lifetime of the ⁴I_13/2–⁴I_15/2 transition. The treated spectrum is then used to calculate the JO parameters Ω_t. The algorithm of this method is illustrated in Fig. S6b (ESI†).

The final approach that was tested was Method C described in ref. 52,53. Unlike the previous case, here the absorption in arbitrary optical density units is used to calculate the relative intensity parameters. The actual JO parameters Ω_t are calculated afterwards using the radiative lifetime of the ⁴I_13/2–⁴I_15/2 transition. In this case, a diffuse reflectance spectrum was used in combination with Kubelka–Munk theory to perform the calculations in arbitrary units. The algorithm of this method is illustrated in Fig. S6c (ESI†).

In order to compare the results obtained using these three different methods the RMS values of the radiative lifetimes and Ω_t values obtained for the crystalline and powder samples were calculated (Table 5). The results of lifetimes in Table 5 and the comparison branching ratios in Table S10 (ESI†) clearly show the difference between the three methods. Although none of these approaches provide a 100% consistency between transition probabilities, the Method A⁵⁰ as well as Method C^52,53 show adequate agreement between values calculated on a single crystal sample and results obtained on powder. If the difference in Ω_t values and branching ratios is also taken into account, then the Method C^52,53 is preferred. It should also be noted that the radiative lifetimes of the weak ⁴I_9/2–⁴I_15/2 transition show the greatest difference between the methods. This state has a lower absorption and emission intensity as well as a worse agreement between experimental and calculated transition probabilities compared to other emissive excited states of the Er³⁺.⁴⁹ However, excluding this transition from the calculation of RMS (Table 5) did not change the conclusion that the method described in Method C^52,53 better fits the parameters calculated for the single crystal.

Table 5 Comparison of the radiative lifetime of Er2Yb5 powder obtained with different approaches based on the JO theory

Emission band	Lifetime, ms
	Crystal	Method A^50	Method B^51	Method C^52,53
^4G9/2–^4I15/2	0.56	0.40	0.58	0.50
^2H11/2–^4I15/2	0.62	0.35	0.89	0.77
^4S3/2–^4I15/2	0.82	0.73	0.71	0.61
^4F9/2–^4I15/2	1.03	0.51	1.66	1.43
^4I9/2–^4I15/2	8.42	3.81	17.56	14.69
^4I11/2–^4I15/2	9.02	8.44	9.01	7.26
^4I13/2–^4I15/2	8.17	8.19	12.71	8.09
Relative RMS (with ^4 I 9/2 )		0.91	1.44	0.94
Relative RMS (w/o ^4 I 9/2 )		0.73	0.95	0.57
Judd–Ofelt parameters, × 10^−20 cm^2
Ω 2	0.95	1.08	0.68	0.73
Ω 4	0.12	1.79	0.31	0.34
Ω 6	1.14	0.95	1.00	1.08
Relative RMS		2.86	0.74	0.67

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As an additional proof, the same procedure was carried out with the Er2Yb2 crystal. Similarly, to the Er2Yb5 the JO calculations are first performed with the material in the form of a crystal, which was then ground to powder and the same three methods are tested. The results obtained are presented in Fig. S7 (ESI†), as well as Tables S11 and S12 (ESI†). The data allow the same conclusions to be drawn: Method C^52,53 gave better agreement between the values obtained on a crystal sample and on a powder sample.

Up-conversion luminescence in PbF₂:Er³⁺,Yb³⁺ crystals

Knowing the ϕ_DS of the ⁴S_3/2 and ⁴F_9/2 emitting states and the decay times of the {⁴I_11/2 & Yb³⁺:²F_7/2} and ⁴I_13/2 intermediate states, one can estimate the efficiency of the UC process based on excited state energy transfer. In general, ϕ_UC should be less than or equal to 0.5ϕ_DS as UC is a two- or three-photon process. In addition, the longer decay time of {⁴I_11/2 & Yb³⁺:²F_7/2} and ⁴I_13/2 states provides a greater probability of UC at lower excitation intensities, which is important for many applications.

Fig. 5(a) demonstrates the upconversion emission spectra under 976 nm excitation with an intensity of 350 W cm⁻², where all spectra are normalised to the maximum intensity of the ⁴F_9/2–⁴I_15/2 transition. First, the UC properties of the crystals were investigated experimentally using a direct approach: ϕ_UC was measured in the integrating sphere at different excitation intensities of the 976 nm laser. It is important to note that the results of measurements of ϕ_UC can be biased in a number of ways.⁵⁵ First, at high intensities the samples can heat up, reducing the emission intensity. Second, the emission can be reabsorbed by the sample as it propagates trough the integrating sphere, reducing the luminescence. The last major source of the error in the estimation of the ϕ_UC is related to the geometry of the sample. During propagation through the sample, the excitation intensity is being absorbed. This leads to a decrease in the incident intensity inside the sample, and due to the non-linear nature of the up-conversion process, a decrease in observed ϕ_UC.

Fig. 5 (a) UC emission spectra under the 976 nm excitation (intensity of 350 W cm⁻²); (b) ϕ_UC in the intensity range of 0.1–350 W cm⁻².

The first problem can be solved by estimating the temperature of the sample from the ratio of the ²H_11/2–⁴I_15/2 and ⁴S_3/2–⁴I_15/2 emission bands, which are thermally coupled. The temperature can be calculated as

where k_B is the Boltzmann constant, ΔE is the energy difference between the ²H_11/2 and ⁴S_3/2 levels, I₅₂₀ and I₅₅₀ is the emission intensities of the ²H_11/2 and ⁴S_3/2 levels respectively, T₀ is the initial temperature in the absence of excitation and and are the emission intensities at the initial temperature.

The procedure was described in detail in the literature⁵⁶ and the results of the calculation with eqn (4) are displayed in Fig. S8 (ESI†). It can be seen that only in the sample with the highest concentration of Yb³⁺ (Er2Yb7.5) there is noticeable change in the temperature of the sample.

The problem of re-absorption within the integrating sphere can be solved as described previously for ϕ_DS in Table 3 and Fig. S4 (ESI†). Finally, in order to account for the effect of sample size on the calculated ϕ_UC values, several assumptions should be made. A crystalline sample of a given thickness is considered to be a seamless stack of 100 layers. The ϕ_UC of each layer was assumed to be ϕ_UC ∝ Iⁿ where I is the incident intensity and n is varied from 0 to 1 in 0.1 steps to illustrate different power dependencies of ϕ_UC. It is then possible to calculate the number of incident and absorbed photons for each layer as well as the ϕ_UC of the layer. Combination of these two values (ϕ_UC and n) gives the number of photons emitted by each layer. By summing the emitted and absorbed photons in each layer, the ϕ_UC of the sample can be calculated based on measured ϕ_UC (Fig. S9, ESI†).

The ϕ_UC values after the above-mentioned corrections (similar to the detailed explanation given by Madirov et al.⁵⁵) are presented in the Fig. 5(b). The data in Fig. 5(b) indicate that ϕ_UC increases as the incident intensity increases, reaching its maximum value of 5.9% observed at 350 W cm⁻² for the Er2Yb3 crystal. It is known that the synthesis of PbF₂ single crystals by the Bridgman method can lead to a certain amount of defects in the crystal lattice,⁵⁷ which could quench the emission. As a result, it might lead to slightly lower observed ϕ_UC values compared to SrF₂:Er³⁺,Yb³⁺ (6.5%)¹⁶ and BaF₂:Er³⁺,Yb³⁺ (9.9%)¹⁹ crystals. As in BaF₂:Er³⁺,Yb³⁺ and SrF₂:Er³⁺,Yb³⁺, at low intensities (<10 W cm⁻²) the highest ϕ_UC is observed in the samples with the maximum amount of the Yb³⁺ ions (7.5%), whereas at higher intensities (>10 W cm⁻²) samples with a lower concentration (3%) of Yb³⁺ demonstrate the highest ϕ_UC.

The experimental dependence of ϕ_UC on excitation intensity can be examined using the approach proposed by Joseph et al. in order to estimate a single figure of merit parameter of the UC process – critical power density (CPD).³² Based on the CPD value, other important parameters of the UC process – the maximum value of the quantum yield (ϕ_UCsat) and the energy transfer rate between donor and acceptor ions (k₁₂) can be derived.³² It should be noted that the CPD concept was derived for a two-photon UC process (such as population and emission from the ⁴S_3/2 state) and not for a three-photon process (which sometimes refers to emission from the ⁴F_9/2 state). Table 6 displays experimental values of the UC quantum yield at a maximum intensity of 350 W cm⁻² (Max ϕ_UC), the values of the CPD, the values of ϕ_UC CPD (UC quantum yield at an intensity corresponding to the CPD), as well as ϕ_UCsat, and k₁₂ derived from the CPD.

Table 6 Maximum (at 350 W cm⁻²) ϕ_UC values for UC emission in the 400–900 nm range (Max ϕ_UC total) and for the ⁴S_3/2–⁴I_15/2 emission (Max ϕ_UC⁴S_3/2–⁴I_15/2); CPD; ϕ_UC at CPD (ϕ_UC CPD); and saturation ϕ_UC (ϕ_UCsat) as well as energy transfer rate (k₁₂) of the ⁴S_3/2–⁴I_15/2 transition

	Er1.5Yb1.5	Er2Yb2	Er2Yb3	Er2Yb5	Er2Yb7.5
Max ϕUC total, %	4.3	4.4	5.9	4.7	5.8
Max ϕUC^4S3/2–^4I15/2, %	1.0	0.9	1.2	1.1	1.3
CPD ^4S3/2–^4I15/2, W cm^−2	48.4	37.9	17.8	16.0	9.2
ϕ UC CPD, %	0.4	0.3	0.4	0.2	0.2
ϕ UCsat ^4S3/2–^4I15/2, %	2.7	1.3	2.6	1.2	1.3
k 12, ×10^−17 cm^3 s^−1	0.6	1.5	2.1	1.9	5.4

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The following conclusions can be drawn from the data in Table 6. The CPD value decreases as the concentration of the Yb³⁺ ions increases and reaches 9.2 W cm⁻² for the Er2Yb7.5 sample. Slightly lower CPD values (∼1 W cm⁻²) have previously been reported for the most efficient hosts (NaYF₄, YF₃, YCl₃, and La₂O₃) ³². However, these lower values were observed at a much higher concentration of Yb³⁺ (18%). Thus, a high concentration of Yb³⁺ is preferred to obtain a high quantum yield at a lower excitation intensity. In contrast, the highest value of ϕ_UCsat is expected for the sample with the lowest doping concentration (Er1.5Yb1.5), the sample with the lowest probability of the cross-relaxation. It is interesting to note that ϕ_UCsat (Table 6) is 0.5 ϕ_DS (Table 3) for most samples, with a slightly larger deviation for Er2Yb7.5. Furthermore, it could be assumed that in the case of the ⁴S_3/2 state, the relatively low ϕ_DS is a limiting factor for achieving high ϕ_UC.

While it is well known that the green UC emission from the ⁴S_3/2 state is a two-photon process, the red UC emission from the ⁴F_9/2 state may have a more complex origin. It is hypothesized that if the ⁴F_9/2 state is populated via relaxation of the ⁴S_3/2 state (and also a two-photon process), the red-to-green (R/G) ratio in the UC spectra should be similar to the R/G ratio observed for the direct excitation of the ⁴S_3/2 state. In the latter case, the R/G ratio can be estimated from the values of ϕ_DS given in Table 3. Therefore, a comparison of the R/G ratios for the UC and DS processes should shed light on the mechanism of the ⁴F_9/2 state population. Fig. 6(a) demonstrates how the R/G ratio changes as a function of excitation intensity in the UC process for the Er2Yb5 sample. At low excitation intensities (<1 W cm⁻²), the R/G ratio in the UC spectra corresponds well to the R/G value obtained by direct excitation of the ⁴S_3/2 state (R/G = 1.5). It is therefore likely that the ⁴F_9/2 state originates from the ⁴S_3/2 state via non-radiative relaxation. However, increasing the excitation intensity (>1 W cm⁻²) clearly leads to an increase in the R/G ratio. This rise in red emission can be explained either by the model proposed by Berry and May,⁵⁸ where the ⁴F_9/2 state results from the three-photon process of populating the ²H_9/2 state and the subsequent back energy transfer to Yb³⁺, or by the ETU process involving ⁴I_13/2 state.

Fig. 6 Er2Yb5 sample: (a) ratio of the intensities of the ⁴F_9/2–⁴I_15/2 (red) transition and the ⁴S_3/2–⁴I_15/2 (green) transition as a function of 976 nm excitation intensity in the 0.1–350 W cm⁻²; (b) ϕ_DS of the Er³⁺:⁴I_13/2–⁴I_15/2 transition as a function of intensity at 940 nm excitation.

In turn, the ⁴I_13/2 state can stem either by radiative transition from the ⁴I_11/2 or ²S_3/2 states, or by cross-relaxation of the ⁴S_3/2 state. The probability of the ⁴I_11/2–⁴I_13/2 transition (0.04) was calculated earlier in Table 4 as the contribution of both radiative and phonon-assisted relaxation processes. This probability does not depend on the excitation intensity and, thus ϕ_DS of the ⁴I_11/2–⁴I_15/2 radiative process should also remain constant. However, an interesting observation can be found in Fig. 6(b). The quantum yield of the ⁴I_11/2–⁴I_15/2 luminescence increases with increasing laser intensity, suggesting that the ⁴I_13/2 state is populated via a new pathway at high excitation intensity – mainly via cross-relaxation of the ²S_1/2 state and minimally via the ²S_3/2–⁴I_13/2 radiative transition. Although it is rather difficult to decide how the ⁴F_9/2 state is populated – either via the Berry and May model or via the ⁴I_13/2 state – it can be assumed that the sublinear increase in the number of ⁴I_13/2 states with excitation intensity leads to an increase in the number of ⁴F_9/2 states (by the ETU process: ⁴I_13/2 + ²F_5/2 → ⁴F_9/2 + ²F_7/2) and thus to an increase in the R/G ratio. Fig. S11 (ESI†) also demonstrates the similar trends in the R/G ratio and ϕ_DS for other investigated samples. Thus, the assumption made about additional population of the ⁴F_9/2 state via⁴I_13/2 state can be valid for a wide range of Yb³⁺ concentrations (1.5–7.5%).

Conclusions

The optical properties of a series of crystalline PbF₂:Er³⁺,Yb³⁺ samples (with fixed Er³⁺ concentration of 2 mol% and variable Yb³⁺) concentration of (2–7.5 mol%) were investigated. Since the luminescence properties of the ⁴S_3/2 and ⁴F_9/2 states of Er³⁺ are important for understanding of the UC in PbF₂:Er³⁺,Yb³⁺ crystals, they were investigated by applying JO analysis, luminescence decay measurements and determination of the absolute luminescence quantum yield at the direct excitation of the corresponding states. It was shown that the quantum yield of the ⁴S_3/2 and ⁴F_9/2 states can be predicted using the JO model and luminescence decays, and that these values are in good conformity with experimental values of quantum yield measured by an integrating sphere. In the case study for the Er2Yb5 crystal, ϕ_DS values of 3.3% (⁴S_3/2–⁴I_15/2 transition) and 29.2% (⁴F_9/2–⁴I_15/2 transition) were obtained from the JO model, while 3.1% and 24.7%, respectively, were measured using an integrating sphere. The proposed method can also be applied to other upconversion materials doped with rare earth ions (Tm³⁺ and Ho³⁺), as long as it is possible to perform Judd–Ofelt analysis for the emitting ions and obtain luminescence decay times. Upon excitation with a 976 nm laser, all PbF₂:Er³⁺,Yb³⁺ crystals exhibit bright UC emission. In the case study for the Er2Yb5 crystal, the value of ϕ_UCsat for the ⁴S_3/2–⁴I_15/2 transition was estimated to be 1.2%. This value follows an empirical rule ϕ_UCsat ≈ 0.5ϕ_DS, leading to the conclusion that the pure emission property of the ⁴S_3/2 state is the limiting factor for a UC quantum yield. Overall, ϕ_UC can be as high as 4.4% for the Er2Yb5 crystal (at intensity of 350 W cm⁻²) because it also includes the ⁴F_9/2–⁴I_15/2 transition with much higher ϕ_DS. The ⁴F_9/2 state originates from the ⁴S_3/2via multi phonon relaxation (with probability of 0.136 for the Er2Yb5 sample) or via the ETU from the ⁴I_13/2 state or via the ²H_9/2 state. This complex path reduces the ϕ_UC for the ⁴F_9/2–⁴I_15/2 transition, although the emission property of the ⁴F_9/2 state is good. The highest value of ϕ_UC 5.9% was measured for the Er2Yb3 crystal in the sample series.

UC materials are often not available in the form of transparent crystals, but are synthesized as microcrystalline or nanocrystalline powders. Several methods of JO parameters calculation for powder samples are available in the literature. In the present work, comparison of three methods (for powder samples) with results obtained for a single crystal of identical material was done. The RMS analysis reveals that method,^52,53 utilizing the reflection spectrum and giving Ω_t parameters measured in arbitrary units, gives better agreement between the results obtained for the crystal and the powder. The true values of Ω_t (in cm²) were then recovered by the known value of radiative lifetime for the ⁴I_13/2–⁴I_15/2 transition. However, it has been demonstrated that the radiative lifetime of the ⁴I_13/2–⁴I_15/2 transition cannot simply be considered equal to the measured decay time of the ⁴I_13/2–⁴I_15/2 transition, as very often due to the non-perfect crystalline materials and the presence of additional quenching channels ϕ_DS < 100%. More careful data evaluation of radiative rate for the ⁴I_13/2–⁴I_15/2 transition is therefore required before applying decay time of the ⁴I_13/2–⁴I_15/2 transition in the JO analysis.

Although ϕ_UC of PbF₂:Er³⁺,Yb³⁺ crystals does not outperform SrF₂:Er³⁺,Yb³⁺ (ϕ_UC = 6.5%) and BaF₂:Er³⁺,Yb³⁺ (ϕ_UC = 10.0%) crystal series, the comprehensive data set presented in our work for PbF₂:Er³⁺,Yb³⁺ crystals can contribute to a better understanding of UC phenomena and provide a reference data set for the use of UC materials in practical applications.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The reported study was funded DFG (project no. TU 487/8-1 for A. T. and B. S. R.). The financial support provided by the Helmholtz Association is gratefully acknowledged: (1) a Recruitment Initiative Fellowship for B. S. R.; (2) the funding of chemical synthesis equipment from the Helmholtz Materials Energy Foundry (HEMF); and (3) Research Field Energy—Program Materials and Technologies for the Energy Transition—Topic 1 Photovoltaics. The authors acknowledge T. Bergfeldt (IAM, KIT) for conducting WDXRF analysis.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3cp00936j

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