Understanding Yb³⁺-sensitized photon avalanche in Pr³⁺ co-doped nanocrystals: modelling and optimization

Abstract

Among different upconversion processes where the emitted photon has higher energy than the one absorbed, photon avalanche (PA) is unique, because the luminescence intensity increases by 2–3 orders of magnitude in response to a tiny increase in excitation intensity. Since its discovery in 1979, PA has been observed in bulk materials but until recently, obtaining it at the nanoscale has been a significant challenge. In the present work, the PA phenomenon in β-NaYF₄ colloidal nanocrystals co-doped with Pr³⁺ and Yb³⁺ ions was successfully observed at 482 nm (³P₀ → ³H₄) and 607 nm (³P₀ → ³H₆) under excitation at 852 nm. The impact of Pr³⁺ ion concentration and pump power dependence on PA behavior was investigated, i.e. PA non-linearity slopes of luminescence intensity curves as a function of pump power density as well as PA thresholds. The highest slopes, namely 8.6 and 9.0, and the smallest thresholds equal to 286 kW cm⁻² and 281 kW cm⁻², observed for emission bands at 607 nm and 482 nm, respectively, were obtained for NaYF₄:0.5%Pr³⁺,15%Yb³⁺@NaYF₄ colloidal nanocrystals. Besides experimental research, simulations of PA behavior in Pr³⁺, Yb³⁺ co-doped materials were performed based on differential rate equations describing the phenomena that contribute to the existence of PA. The influence of different processes leading to PA, e.g. the rates of nonradiative and radiative transitions as well as energy transfers, on PA performance was simulated aiming to understand their roles in this complex sensitized system.

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Introduction

Photon avalanche (PA) is a highly non-linear upconversion process during which the luminescence intensity increases rapidly with a minute pump power density increase. In consequence, the characteristic s-shape curve of luminescence intensity (I_L) as a function of pump power density (I_P) is obtained and could be described by the following power law:

I_L = I^S_P, (1)

where S signifies a slope of the curve and, next to the PA threshold (PA_TH) indicating the value above which PA exists, is a characteristic parameter used to describe the PA process. PA was first observed in 1979 by Jay S. Chivian in singly Pr³⁺-doped LaCl₃ and BaCl₃ single crystals, with a vision to count medium infrared photons at ca. 4.5 mm,¹ which were responsible for the intermediate ³H₅ level population and enabled efficient photoexcitation by a green laser source, leading to the ³H₅ → ³P₁ ESA transition. It was predicted that PA nanoparticles may open new possible fields of applications;² however, obtaining PA at the nanoscale has been a challenge. All features and characteristics of the PA phenomenon have only recently been clearly and undoubly evidenced in colloidal NaYF₄ and LiYF₄ nanocrystals doped with Tm³⁺ ions at the nanoscale.^3,4 Since then, the impacts of the local environment,⁵ host matrix⁶ and light induced photodarkening and recovery in singly Tm³⁺-doped⁷ ions have been studied, and a general multicolor PA emission thanks to the core–shell design and energy migration through a Yb³⁺ (ref. 8) or Gd³⁺ (ref. 9) migration network has been demonstrated. At the nanoscale, PA emission has also been demonstrated in NaYF₄ nanocrystals co-doped with Pr³⁺ and Yb³⁺ ions.⁸ In the latter case Yb³⁺ ions work as a sensitizer and establish an energy migration network, hosting the migrating PA mechanism (MPA)⁸ or sensitized PA (SPA).¹⁰ These studies confirmed the applicability of the Yb³⁺, Pr³⁺ PA system for sub-diffraction imaging⁸ and luminescence nanothermometry,¹⁰ as predicted previously.^2,11 However, the mechanisms of sensitization of Pr³⁺ avalanche emission by co-doping with Yb³⁺ have to be further studied and clarified.

There are some key conditions required to obtain PA. Firstly, the excitation wavelength has to be resonant with excited state absorption (ESA) and simultaneously far from resonance with ground state absorption (GSA). Additionally, the absorption cross section from the excited state (σ_ESA) has to be much higher than from the ground state (σ_GSA), which is described as and should be above 10⁴. In order to populate the metastable state, from which ESA can occur, cross-relaxation (CR) processes are necessary. CR relays on the exchange of energy between two ions, one of which is in a higher excited state and the other in a basic energy state. These two ions average their energies and thus produce two ions in an excited, metastable state. Consequently, the population of the metastable level increases in a loop, and absorption occurs more efficiently, which translates into PA emission. The characteristic features of PA are also slow luminescence rise times. Furthermore, to quantify the dynamics of relative changes of the luminescence intensity for the pump power changes, we introduce the parameter D_AV, which is the PA luminescence intensity enhancement in response to doubling the excitation power densities, i.e.:

The subject of the present work is SPA in β-NaYF₄ nanocrystals co-doped with Pr³⁺ and Yb³⁺ ions. Multicolor PA emission at 482 nm (³P₀ → ³H₄ transition) and 607 nm (³P₀ → ³H₆ transition) was successfully observed under excitation at 852 nm wavelength in samples co-doped with 15%Yb³⁺ ions and 0.1, 0.3, 0.5 or 0.7% of Pr³⁺ ions. The impacts of the concentration of Pr³⁺ ions and nanocrystal architecture on PA properties have been studied. In particular, the essential parameters describing the avalanching system, like S, PA_TH, rise times and D_AV were investigated in detail. Additionally, β-NaYF₄ nanocrystals singly doped with Pr³⁺, i.e. 0.1%, 0.5%, 1% and 3% ions, were synthesized and investigated, aiming to understand the role of CR and sensitization by Yb³⁺ ions. Moreover, another motivation behind our work was to critically evaluate the origin of red Pr³⁺ emission, as a discrepancy exists in the literature about the energy level (either ³P₀ or ¹D₂) responsible for this transition. Furthermore, simulations of the PA process in Pr³⁺, Yb³⁺ co-doped crystals were performed. Based on the supposed PA mechanisms and relatively low phonon tetrafluoride materials, differential rate equations (DREs) were rationalized as compared to previous reports. Additionally, we have adopted a ‘knock-out’-like approach in DRE modelling, aiming to understand the influence of different individual phenomenological parameters on the PA process, i.e. the role of the rates of all the important nonradiative and radiative transitions as well as energy transfers was investigated. This knowledge is essential to further optimize the performance (e.g. decrease the PA threshold, increase PA gain or PA slopes) of these avalanching materials for numerous future photonic applications, and likewise to qualitatively understand the generic mechanisms behind sensitization in photon avalanching by other co-dopants.

Results and discussion

The aim of the present work was to understand the origin and mechanism of PA emission of Pr³⁺ ions at the nanoscale. Therefore, a set of β-NaYF₄ colloidal nanocrystals singly doped with 0.1%, 0.5%, 1% and 3% of Pr³⁺ ions was synthesized. Additionally, based on earlier reported MPA in Pr³⁺, Yb³⁺ co-doped β-NaYF₄ nanoparticles,⁸ a set of samples co-doped with 15% of Yb³⁺ ions in a wide range of concentrations of Pr³⁺ ions, i.e. 0.1%, 0.3%, 0.5% and 0.7%, was synthesized. All materials were synthesized by thermal decomposition of lanthanide salts in a core and core–shell architecture. Morphology and structural characterization was carried out for all prepared nanocrystals (Fig. 1 and Fig. S1, S2†). X-ray powder diffraction (XRD) reflexes of the investigated samples are compatible with the XRD pattern of the hexagonal (β) structure of NaYF₄ crystals (Fig. 1o and Fig. S2†), which confirms a pure β-NaYF₄ phase of the synthesized nanomaterials. The sizes of the nanocrystals were estimated based on transmission electron microscopy (TEM) images. Exemplary TEM pictures and histograms with average dimensions of the synthesized nanocrystals are presented in Fig. 1 and panels (a–l) in Fig. S1.† The diameters of the core nanoparticles are in the range of 16–20 nm and the core–shell dimensions are within 18–31 nm. Most of the crystals feature narrow size distribution, except for core–shell nanocrystals doped with 0.5% of Pr³⁺ ions, where, next to bigger core–shell particles (21 nm), smaller ones (around 10 nm) are observed (Fig. S1b†). We suppose that the shell material, besides crystallization on core nanoparticles, crystallized also as smaller undoped NaYF₄ nanocrystals, as we discussed later on the basis of core and core–shell dimensions. An indirect proof of shell deposition is the bigger size of core–shell nanoparticles than that of core nanocrystals. Additionally, undoped shell deposition is directly evidenced by the exemplary energy dispersive X-ray spectroscopy (EDS) map and the profile of Yb³⁺ and Y³⁺ ions prepared for core–shell NaYF₄:0.5%Pr³⁺,15%Yb³⁺@NaYF₄ nanocrystals (Fig. 1m and n). The thicknesses of shells were calculated based on the average dimensions of nanocrystals. For most of the spherical core–shell nanoparticles, the shell thickness is around 4 nm (Table 1) with the exception of 0.5%Pr³⁺-doped ones, which have a shell thickness of around 2 nm (Table S1†). It is evidence that part of the shell material in this sample crystallized as smaller, undoped β-NaYF₄ particles. In the case of ellipsoidal core–shell nanocrystals, the shell thickness is around 2 nm in width and between 4 nm and 6 nm in length (Table S1†).

Fig. 1 Characteristic of β-NaYF₄ nanocrystals co-doped with x%Pr³⁺ and 15%Yb³⁺ ions showing PA: (a, d, g and j) transmission electron microscopy (TEM) images of core nanoparticles; (b, e, h and k) TEM images of core–shell nanocrystals; (c, f, i and l) histograms with average sizes of core and core–shell nanocrystals. (m) Energy dispersive X-ray spectroscopy (EDS) map and (n) EDS profile of Yb³⁺ and Y³⁺ ions in the core–shell NaYF₄:0.5%Pr³⁺,15%Yb³⁺@NaYF₄ nanocrystals; (o) X-ray powder diffractions of the β-NaYF₄ structure (template) and all the samples.

Table 1 Average dimensions and shell thicknesses of β-NaYF₄ nanocrystals co-doped with x%Pr³⁺ and 15%Yb³⁺ ions

Dopants x%Pr^3+, 15%Yb^3+	Average core diameter [nm]	Average core–shell diameter [nm]	Average shell thickness [nm]
0.1%Pr^3+	20.1 ± 1.3	28.1 ± 1.4	4.0 ± 1.4
0.3%Pr^3+	20.2 ± 1.0	27.7 ± 1.6	3.8 ± 1.3
0.5%Pr^3+	20.34 ± 0.81	30.9 ± 1.2	5.3 ± 1.0
0.7%Pr^3+	17.67 ± 0.73	25.3 ± 1.0	3.80 ± 0.86

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To determine the PA luminescence parameters, the 852 nm wavelength as the excitation one was used,⁸ which is resonant with Pr³⁺ ESA (¹G₄ → ³P₁) and simultaneously far from resonance with the GSA of these ions, as is required for PA. Measurements were performed at room temperature (RT). At the beginning, singly Pr³⁺-doped samples were investigated. The range of ion concentrations was deliberately chosen starting with a very low content, namely 0.1% of Pr³⁺ ions, ending at 3%. It can be expected that a high concentration of ions would lead to the concentration quenching of luminescence intensity. However, even nanocrystals with low Pr³⁺ ion concentrations showed no emission under these excitation conditions (Fig. 2a). The peak at around 725 nm is an artifact resulting from a laser diode and no emission was observed from singly Pr³⁺-doped nanocrystals. Therefore, nanocrystals co-doped with Pr³⁺ and Yb³⁺ ions were investigated. The 852 nm wavelength can excite Yb³⁺ ions, and then, by energy transfer (ET) processes, the absorbed energy can be further transferred to Pr³⁺ ions. The core and core–shell samples co-doped with 15%Yb³⁺ ions and 0.1%, 0.3%, 0.5% or 0.7% of Pr³⁺ ions successfully showed multicolor emission in the range from 450 nm to 700 nm (Fig. 2b and c). Comparing the emission spectra of the core and core–shell samples, the core–shell nanocrystals feature significantly higher luminescence intensity, confirming the role of the inert passive shell, which protects optically active ions from environment impacts. Furthermore, for the sample singly doped with 0.5%Pr³⁺ ions, no avalanche emission was observed, and the sample co-doped with 0.5%Pr³⁺ and 15%Yb³⁺ ions exhibited multicolor emission, which proves that Yb³⁺ ions play a key role as a sensitizer of PA emission.

Fig. 2 Anti-Stokes emission spectra of β-NaYF₄ nanocrystals: (a) core–shells singly doped with Pr³⁺ ions; (b) cores co-doped with x%Pr³⁺, 15%Yb³⁺; (c) x%Pr³⁺, 15%Yb³⁺ co-doped cores with a passive shell with assigned transitions. The presented emission spectra were recorded under 852 nm excitation (1.5 MW cm⁻²) at room temperature.

The emission bands of the samples co-doped with Pr³⁺ and Yb³⁺ ions were assigned to the following transitions: ³P₁ → ³H₄ (468 nm), ³P₀ → ³H₄ (482 nm), ³P₁ → ³H₅ (522 nm), ³P₀ → ³H₅ (538 nm), ³P₁ → ³H₆ (583 nm), ³P₀ → ³H₆ (607 nm), ³P₀ → ³F₂ (641 nm), ³P₁ → ³F₃ (672 nm), ³P₁ → ³F₄ (693 nm), and ³P₀ → ³F₄ (720 nm) (Fig. 2c). In the literature, the assignment of emission bands in the 580–620 nm range is ambiguous, as in some reports, they are interpreted as emission from the ³P₁ level, while others are from the ¹D₂ level.^8,12–17 Here, as also earlier,¹⁰ we assigned the emission at 607 nm to the ³P₀ → ³H₆ transition. Considering the emission at 607 nm (16 475 cm⁻¹), the energy mismatch is lower for the ³P₀ → ³H₆ transition (ΔE = 16 400 cm⁻¹) than for the ¹D₂ → ³H₄ transition (ΔE = 16 800 cm⁻¹). Additionally, to prove that the emission at 607 nm comes from the same level as 482 nm, fluorescence lifetime decay curves for these wavelengths were measured in a wide range of temperatures. Luminescence lifetime curves measured at RT are presented in Fig. S3.† The values of the short and long components of the decay time curves and as a function of temperature are shown in Fig. S4a–d.† The average values of the lifetimes were calculated using the formula:

giving 13.6 μs and 17.6 μs for emissions at 607 and 482 nm (both at RT), respectively (Fig. S4a and c†). Measurements of the mentioned luminescence lifetimes helped to confirm that the 607 nm and 482 nm emissions originate from the same level and assign them to 482 nm (³P₀ → ³H₄) and 607 nm (³P₀ → ³H₆) transitions. These emission bands, as one of the most intense, were additionally chosen to measure the emission intensity as a function of pump power density. The power density of excitation wavelength increases from 10⁴ W cm⁻² to around 10⁷ W cm⁻² with small (between 1000 and 2000 W cm⁻²) steps. As a result, characteristic s-shaped curves were obtained (Fig. 3a and b). The pump power dependence measurements were carried out at least two times and the average maximum S and average PA_TH were calculated (Fig. 3c–f and Table S2†). By comparing the core samples with their core–shell counterparts, it can be noted that the slope values (Fig. 3c and d) are higher for the core–shell samples, while thresholds (Fig. 3e and f) are lower in their case. Secondly, the inert shell plays an important role in protecting the optically active core from environment impacts, which might result in luminescence quenching. Analyzing Fig. 3c and d, which show the results obtained for various Pr³⁺ doping concentrations, it can be observed that average slope values increase with the amount of Pr³⁺ ions up to 0.5% and then decrease a bit for 0.7%Pr³⁺ ions. For both PA emissions, i.e. 482 nm and 607 nm, the biggest average slopes were detected for the core–shell β-NaYF₄ nanocrystals co-doped with 15%Yb³⁺, 0.5%Pr³⁺ ions, which amount to similar values of 9.0 and 8.6, respectively (Table S2†). In the case of thresholds (Fig. 3e and f), for PA emissions at 482 nm and 607 nm, the smallest PA_TH values were observed for the core–shell β-NaYF₄:15%Yb³⁺,0.5%Pr³⁺@NaYF₄ nanocrystals (281 kW cm⁻² and 287 kW cm⁻², respectively) (Table S2†). Overall, besides Stokes emission lifetime similarity (Fig. S3†), the similarity of the PA behavior at 482 and 607 nm further indicates the same origin of both emission bands, i.e.³P₀, and further suggests that no or negligible emission originates from the ¹D₂ level. This simplifies the description of the sensitized photon avalanche mechanism and has serious implications for further, more detailed analysis and simulations that were carried out. Based on the PA performance, the core–shell 15%Yb³⁺ and 0.5%Pr³⁺ co-doped β-NaYF₄ nanocrystals were found to be optimal for PA emission in this investigated set of samples. It is compliant with earlier research.⁸ We expect that PA emissions are extremely sensitive to even minor factors responsible for excitation, and therefore the resulting slopes of the pump power dependence curves are not as high as reported from confocal setups. Moreover, one may expect that in core–shell nanoparticle architectures, ion intermixing between the core and shell may occur,^19,20 and thus the actual dopant ion concentration may be slightly diluted there. In consequence, optimal conditions for PA emission may become diminished, leading to reduced nonlinearity and an increased PA threshold. However, regardless of the exact values of S nonlinearity, we draw general conclusions and observe and discuss trends in order to instruct further development of PA nanomaterials as well as clarify the mechanism of PA sensitization.

Fig. 3 Characteristic of PA emission in the core and core–shell β-NaYF₄ nanocrystals co-doped with 15%Yb³⁺ and 0.1%, 0.3%, 0.5% or 0.7% of Pr³⁺ ions. Subsequent columns: (a and b) pump-power dependence of PA luminescence; (c and d) maximum values of S, which are a measure of nonlinearity of the PA process (eqn (1)); (e and f) PA_TH pump powers as a function of the concentration of Pr³⁺ ions. (g and h) Mean values of D_AV parameters indicating the efficiency of the PA process. These properties were recorded for core (●) and core–shell (○) nanoparticles at 482 nm (top row) and at 607 nm (bottom row). Average values and standard deviations for (c–h) were calculated based on a few (2–4) measurements.

There were calculated D_AV values (Fig. 3g and h), which, according to eqn (2), describe how much the luminescence intensity increases upon doubling the power density. The core–shell nanocrystals turned out to be more sensitive to the changes and the most sensitive was the β-NaYF₄:15%Yb³⁺,0.7%Pr³⁺@NaYF₄ sample. Another characteristic PA feature is the slowing down of the emission rise time for power densities close to the threshold. Here, parameters τ_50% and τ_80% were adopted to describe the time needed to obtain 50% and 80% of maximum luminescence intensity (at an infinitely long excitation time), respectively. The average values of τ_50% and τ_80% are presented in Fig. S5.† The highest values of rise times were observed for β-NaYF₄:15%Yb³⁺,0.1%Pr³⁺@NaYF₄ nanocrystals.

To understand the PA mechanism, besides experimental investigations, simulations of the PA process in the Pr³⁺, Yb³⁺ co-doped system and in singly Pr³⁺-doped nanocrystals were performed. Compared with earlier studies,⁸ here a significantly simpler energy diagram of the Pr³⁺, Yb³⁺ system was adopted for modeling. The supposed PA mechanism in the Pr³⁺, Yb³⁺ ion co-doped system is demonstrated in Fig. 4a and, excluding gray elements, was used to write DREs (eqn (S1)–(S7)†). The correspondence of the properties measured at 607 and 482 nm indicates that the emissions come from the same level and therefore the ¹D₂ level can be excluded from analysis. This feature allowed us to rationalize the rate equation model as compared to earlier studies, perform the next computational studies in a more effective way and draw general and more clear conclusions about the origin of PA in the PrYb system studied here. As expected, the 852 nm wavelength, thanks to its resonance with the ESA [¹G₄ → ³P₁] transition in Pr³⁺ ions as well as its incompatibility with the GSA of Pr³⁺ (Fig. 4a), turned out to be appropriate for PA. Absorption cross sections from the ground and excited states in Pr³⁺ were indicated in modeling as σ_ESA and σ_GSA, respectively. As was experimentally evidenced, the Yb³⁺ ions are necessary for the existence of PA. Thus, the absorption cross section in Yb³⁺ was also taken into consideration (σ_Yb).

Fig. 4 The mechanism and susceptibility of Yb³⁺-sensitized Pr³⁺ photon avalanching to the processes involved in the SPA phenomenon: (a) energy diagram and the PA mechanism of the Pr³⁺, Yb³⁺ co-doped system. This graph, excluding the gray elements, has been used to define the DREs. Panels from (b) to (k) contain simulated s-shaped dependences of luminescence intensity as a function of pump power density depending on the values of particular parameters involved in the SPA, namely nonradiative transitions (w₆₅, w₇₁, w₇₄, w₆₄, NR₂₁), radiative lifetimes (τ_Yb, τ₄, τ₅) and absorption cross sections (σ_ESA, σ_Yb), used in the developed model. Individual panels show how the variation of these parameters (one at a time) affects the s-shape profiles. Saturated and desaturated colors correspond to the increase or decrease of a given parameter value, respectively, while s-shapes obtained for pristine (literature) values are black. (l) Pump power-dependent Pr³⁺ PA luminescence intensity profiles obtained from theoretical modeling (navy blue for Yb³⁺, Pr³⁺ and pink for pure Pr³⁺-doped systems). The impact of the Yb³⁺ sensitizer on PA emission characteristics is evaluated by modeling, i.e. the Yb³⁺, Pr³⁺ co-doped system exhibits PA emission, while the singly Pr³⁺-doped system shows conventional upconversion.

The metastable level (starting with ESA) should be initially populated to provide efficient ESA. Here, three energy transfer processes populating the metastable ¹G₄ level in Pr³⁺ ions were considered. First, CR between Pr³⁺ ions [³P₁, ³H₄] → [¹G₄, ¹G₄] (W_CR1), second, w₆₅ indicating energy transfer upconversion (ETU) from Pr³⁺ to Yb³⁺ ions [Yb³⁺: ²F_7/2, Pr³⁺: ³P₁] → [Yb³⁺: ²F_5/2, Pr³⁺: ¹G₄] and finally, w₇₁ energy transfer (ET) from Yb³⁺ to Pr³⁺ ions [Yb³⁺: ²F_5/2, Pr³⁺: ³H₄] → [Yb³⁺: ²F_7/2, Pr³⁺: ¹G₄]. These ETU and ET were also considered as energy transfer processes going in opposite directions and indicated as w₇₄ and w₆₄, respectively. Additionally, CR populating the ³H₅ level in Pr³⁺ ions (W_CR2) was considered. This process was the basis for obtaining PA in Pr³⁺ for the first time by Jay S. Chivian,¹ therefore, it should not be overlooked in the first approximation, despite the fact that the ESA occurs from ¹G₄ and not ³H₅. Consequently, radiative and nonradiative emission processes from ³H₆ and ³H₅ levels were taken into consideration, namely τ₃, τ₂, NR₃₂ and NR₂₁. Other two nonradiative processes, i.e. NR₅₄ and NR₄₃, were also considered. Because both PA emissions (482 nm and 607 nm) originate from the same ³P₀ level, we could disregard the ¹D₂ level from further analysis and thus simplify the description of the phenomenological energy transfer DRE model and speed up the data simulation presented in Fig. 4. Simultaneously, emissions from ³P₁ are weaker compared with emissions from ³P₀ (Fig. 2c). This is because of a small energy break between levels ³P₁ and ³P₀ (<600 cm⁻¹), which can be easily bridged by matrix phonons (energy of one phonon in NaYF₄ ∼350 cm⁻¹). Thus, in the modeling, the ³P₁ level has been disregarded. Emission bands from ³P₀, ¹G₄ and Yb³⁺ ions were characterized by lifetimes at these levels: τ₅, τ₄ and τ_Yb, respectively. Based on the supposed PA mechanism and analyzing the emission spectra of PA nanocrystals, the processes illustrated as gray in Fig. 4a have been omitted from the modeling. Finally, the DRE (eqn (S1)–(S7)†) model was designed to evaluate and qualitatively understand the PA phenomenon and its susceptibility to other material properties. The parameters utilized to solve the DREs were mostly adopted from the literature on LiYF₄ matrix-based luminescent materials and are summarized in Table S3.† The rates of nonradiative transition were estimated on the basis of the chart of probabilities of nonradiative transitions of Ln³⁺ ions in a tetragonal LiYF₄ laser crystal as a function of energy gap at T = 0.²¹ The probability of nonradiative transitions from the starting level, J, to the end level, J′, is described as W_JJ′ (eqn (S9)†). These calculations are presented in the ESI.†

The prepared DREs were resolved and the characteristic s-shape curve illustrating luminescence intensity as a function of pump power density was obtained, which is presented in Fig. 4 in all small panels (b–k) as a black curve. All the other color s-shaped curves indicate how the increase (more intense color) or decrease (paler shade) of individual parameters influences the features of PA, what will be discussed later. It is important to notice that the prepared DREs were resolved also for the singly Pr³⁺-doped system (i.e. by zeroing all processes describing Yb³⁺ and Pr³⁺ interactions, namely w₇₄, w₆₅, w₇₁, w₆₄ and σ_Yb). The model excluding Yb³⁺ ions (Fig. 4l) clearly showed that singly Pr³⁺ ions under 852 nm excitation are not sufficient to obtain PA emission, and energy transfer processes between the two selected ions play a key role.

The original DREs (eqn (S1)–(S7))† were then used to experimentally evaluate how the individual parameters influence the features of PA. To understand their roles, the values of these individual parameters of DREs were artificially and purposefully increased and decreased around the literature values. The changes were not uniform for each of the parameters, but particular parameters were tuned individually, and specific values are presented in Table S4.† The results obtained from these calculations are presented in Fig. 4(b–k) and Fig. S6.† Not all parameters affect PA in a significant way, thus, only those parameters that substantially influence the shape of the pump power dependence or slope are presented here. Although W_CR1 seems to be the key process for PA performance, it does not influence PA features, because there exist other more intense processes that populate the metastable ¹G₄ level. The Pr³⁺ concentration is low, thus CR processes, including W_CR2, are not efficient enough to power the avalanching. Here, w₆₅ plays a role of CR and populates the ¹G₄ metastable level, which is necessary for ESA.²² Moreover, the value of w₆₅ is higher than that of W_CR1, which supports the hypothesis that Yb³⁺, not W_CR1, is responsible for PA. Yb³⁺ ions not only support avalanching, but are also essential components of the system, which enables PA emission in this configuration. To obtain PA in a single Pr³⁺-doped material, there will be necessary excitation conditions used by Chivian.¹ An increase of w₆₅ enables more efficient ESA, which causes a steeper slope and shifts the PA_TH to higher values (Fig. 4b and Fig. S6a†). Similarly to w₆₅, w₇₁ populates the ¹G₄ level of Pr³⁺ ions and for more intense w₇₁, the slopes are higher (Fig. S6b†). The w₇₄ process could be assigned to ESA and its higher values increase slopes (Fig. S6c†). The w₆₄ transfers energy from Pr³⁺ to Yb³⁺ ions, picking up the energy from the metastable state. The greater the share of w₆₄, the more difficult it is to achieve PA (Fig. 4e) and the slope decreases (Fig. S6d†). However, w₆₄ pumps Yb³⁺, which is necessary for PA, but Yb³⁺ can get entangled in energy migration to the surface. It is widely known that Yb³⁺ ions are very susceptible to surface quenching by the overtones of water vibrations (3300 cm⁻¹); therefore, surface effects also have a vital impact. Thus, the role of Yb³⁺ is twofold. First, Yb³⁺ enables PA, but facilitates quenching on the surface, which may not be prevented by a passive shell even for the thicknesses studied here. Decreasing the radiative lifetime of Yb³⁺ ions (τ_Yb), for example, due to surface quenching, increases PA_TH by up to one order of magnitude (Fig. 4f). Also, the saturation of the curve is smoother. The shorter lifetime of Yb³⁺ ions may indicate that w₆₄ and w₆₅ processes are more intense and consequently PA_TH also increases. Let us consider the other radiative lifetimes. First, the τ₄, which is a radiative lifetime of the ¹G₄ metastable level, is considered. A longer τ₄ facilitates the more efficient occurrence of ESA. Consequently, PA_TH is lower (Fig. 4g) but also slopes decrease (Fig. S6f†). For τ₅, only minute variation was observed (Fig. S6g†). In the cases of τ₂ and τ₃, insignificant changes have been observed. These lifetimes determine the population of levels (2) and (3), which are necessary for cross-relaxation described by W_CR2. These processes will be critical for excitation based on the ³H₅ → ³P₁ transition applied by Jay S. Chivian, but here, under 852 nm photoexcitation wavelength, are not significant. If these lifetimes were very long, respective levels could store electrons and reduce the ground state population, thus hindering the CR processes. But here, their impact is negligible, which in turn suggests that the ET between Yb³⁺ and Pr³⁺ is more important than the CR within the Pr³⁺ ion couples. The lifetimes considered here are radiative, thus nonradiative transitions were additionally taken into account in the model. Here, to make the model easier, some Pr³⁺ levels were overlooked and NR₅₄ and NR₄₃ were adopted to describe nonradiative transitions between levels ³P₀ → ¹G₄ (10 800 cm⁻¹) and ¹G₄ → ³H₄ (9850 cm⁻¹), respectively. The energy gaps of these transitions are wide; therefore the probabilities of the processes are not high. However, it is important to remember that under experimental conditions, these transitions could be more efficient than how they appear in the numerical considerations. Calculations for increased and decreased NR₅₄ and NR₄₃ were performed, but no clear differences in the s-shape were observed for values up to three orders of magnitude smaller or larger than the original values. The other two nonradiative processes are closely connected with W_CR2 and here do not change too much; however the impact of NR₂₁ was observed (Fig. 4i and Fig. S6h†), indicating the importance of having Pr³⁺ ions in the ground state to initiate PA emission.

Ending the description of modeling, let's come back to the key conditions necessary to obtain PA emission. One of these conditions is a significantly high value of cross section for absorption from the excited state and a simultaneously negligible cross section for absorption from the ground state. The 852 nm excitation wavelength fulfills these conditions. As expected, σ_GSA has not influenced the features of PA, while σ_ESA has the biggest impact on the PA_TH values. Higher values of σ_ESA facilitate PA, shifting the PA_TH to smaller values, and conversely, lower values of σ_ESA increase the threshold value (Fig. 4j). Moreover, the slopes increase for bigger σ_ESA (Fig. S6f†). It confirms that ESA plays a key role in being necessary in the PA process and higher values of σ_ESA positively influence PA. In the case of σ_Yb, the changes are negligible. A decrease of the parameter impedes PA by moving PA_TH to higher power densities (Fig. 4k); however, the slopes increase (Fig. S6j†). Yb³⁺ ions take part in two processes populating ¹G₄. For w₆₅, Yb³⁺ ions in the ground state are required, while for w₇₁, Yb³⁺ ions in the excited state are necessary. From these two processes, w₆₅ is found to be more intense; therefore, in this case Yb³⁺ ions in the ground state could help to increase a slope.

Besides PA measurements, basic spectroscopy studies of the synthesized samples has been carried out, including measurements of the absorption, emission and excitation spectra as well as luminescence lifetimes. These results are presented in the ESI.† In all cases, the amounts of the investigated samples were the same to enable a quantitative comparison. Considering the samples co-doped with Pr³⁺ and Yb³⁺ ions, where the amount of Yb³⁺ ions was constant (15%), one can see that it is reflected in the absorption spectra (Fig. S7a†) because the absorption values coming from Yb³⁺ ions are at the same level. Simultaneously, the absorption of Pr³⁺ ions increases with an increase of Pr³⁺ amount, as illustrated in the inset of Fig. S7a†. A similar situation is observed in the case of singly Pr³⁺-doped NaYF₄ nanocrystals, where absorption also increases with Pr³⁺ ion concentration (inset in Fig. S7b†). Insets were prepared based on the integral area of the peak at 444 nm corresponding to the ³H₄ → ³P₂ transition. Excitation spectra (Fig. S8a and S8c†) were recorded for the samples co-doped with Pr³⁺, Yb³⁺ ions and singly Pr³⁺ doped, by monitoring emission at 980 nm and 607 nm, respectively. In all cases the most intensive peaks were obtained at 444 nm. For the core–shell nanocrystals co-doped with Pr³⁺ and Yb³⁺ ions, the integral area under the peak at 444 nm decreases with an increasing concentration of Pr³⁺ ions (Fig. S8a†). It means that the absorbed energy is less effectively transferred to Yb³⁺ ions when the Pr³⁺ concentration is higher, which is due to the increased concentration quenching between neighboring Pr³⁺ ions (i.e. increasing Pr³⁺ concentration from 0.1 to 0.7% quenched the emission intensity by ca. 2 fold). It is astonishing to see that the concentration quenching is not that strong in singly Pr³⁺-doped samples, as the integral area under the peak at the same wavelength (444 nm, monitoring Pr³⁺ emission at 607 nm) initially increases with Pr³⁺ ion concentration up to 1%, and then decreases slowly by ca.30% for samples doped with 1 and 8% Pr³⁺. This can only be explained by the larger amounts of lanthanides (>10%), the cooping of large²³ and small ions, i.e. 1.179 Å and 1.042 Å Pr³⁺ and Yb³⁺ ions, respectively, for 9-fold coordination positions and/or the role of Yb³⁺ ions, which most probably cluster dopants more effectively than for dopant levels of 1%. It is well known that low concentrations of optically active ions minimize luminescence quenching relative to high dopants. However, the arrangement of atoms in the lattice is also important. It is worth mentioning here studies on nanocrystals with orthorhombic crystallographic structures, in which lanthanide ions were arranged in tetrahedral arrays.²⁴ Such an arrangement of ions resulted in minimizing the migration of excitation energy to defects by preserving the excitation energy within the sublattice domain. The lack of control over the organization of nanomaterials at the atomic scale is one of the reasons limiting their optical performance. Providing control over the mixing of cations at the core–shell interface can provide a dramatic improvement in quantum efficiency.¹⁸

Considering that the most intense peak in the absorption spectra was obtained at 444 nm, emission spectra (Fig. S8b and d†) were recorded under excitation at 444 nm and multicolor emission was observed. Insets for emission spectra were prepared on the basis of the integral area under the peak at 482 nm assigned to the ³P₀ → ³H₄ emission. For small amounts of Pr³⁺ ions, namely 0.1%, 0.3%, 0.5%, and 0.7%, the emission intensity increased with the concentration of Pr³⁺ ions (inset in Fig. S8b†). Moreover, the core–shell samples, where the inert shell protects optically active ions from the environment, have shown more intense luminescence in comparison with the cores, as was expected. In the case of singly Pr³⁺-doped samples, where the range of concentrations of Pr³⁺ ions was wider (from 0.1% to 8%), initially the luminescence intensity increases with the concentration of Pr³⁺ ions up to the sample doped with 3% of Pr³⁺. Subsequently, for samples doped with 5% and 8% of Pr³⁺ ions, the luminescence intensity decreases. It is connected with the concentration quenching effect, which occurs for highly doped nanocrystals.

Experimental

Materials and methods

β-NaYF₄ nanocrystals doped with praseodymium as well as co-doped with praseodymium and ytterbium ions were synthesised. The selected crystal matrix is appropriate for the PA phenomenon due to its low phonon energy. Additionally, compared with a cubic α-NaYF₄ structure, a hexagonal one is brighter. All nanocrystals were prepared in a core and core-undoped shell architecture.

Synthesis methods

Reagents. All chemical reagents were used without any further purification. Yttrium oxide Y₂O₃ (99.99%), praseodymium oxide Pr₆O₁₁ (99.99%), ytterbium oxide Yb₂O₃ (99.99%) and trifluoroacetic acid (99%) were purchased from Alfa Aesar. Acetic acid (99.5%), sodium hydroxide NaOH (≥98.0%), methanol (99.8%), ethanol (96%), n-hexane (czda) and chloroform (98.5%) were purchased from POCH S.A. Ammonium fluoride NH₄F (≥98.0%), sodium trifluoroacetate (98%), octadecene (90%) and oleic acid (90%) were purchased from Sigma-Aldrich.

Preparation of the precursor in the form of lanthanide acetates. Stoichiometric amounts of lanthanide oxides, namely Y₂O₃, Yb₂O₃ and Pr₆O₁₁ (2.5 mmol), were mixed in acetic acid and water (volume ratio 1 : 1) in a Teflon vessel. The prepared mixture was kept at 200 °C for 60 minutes to form acetates. The final precursor was obtained by evaporation of residual acetic acid and water using a rotary evaporator in a pre-vacuum (<8 hPa) and drying at 140 °C for 12 hours.

Preparation of the precursor in the form of lanthanide trifluoroacetates. A stoichiometric amount of Y₂O₃ (2.5 mmol) was dissolved in a solution of water (10 mL) and trifluoroacetic acid (10 mL) and kept at 90 °C for one hour to form yttrium trifluoroacetates. The final precursor was prepared by evaporation of residual trifluoroacetic acid and water using a rotary evaporator and drying at 140 °C for 12 hours.

Synthesis of core β-NaYF₄:x%Pr³⁺ and β-NaYF₄:x%Pr³⁺,15%Yb³⁺ nanoparticles. The core nanoparticles were synthesized by the thermal decomposition of oleate salts in high boiling solvents, based on the procedure described earlier by Abel et al.²⁵ First, a precursor in the form of lanthanide acetates was prepared as described above. In a typical synthesis, oleic acid (30 mL) and 1-octadecene (75 mL) were added to a flask with the precursor. The solution was stirred and heated to 140 °C under vacuum in order to remove residual oxygen and water. Under such conditions, the mixture was kept for 30 minutes. Subsequently, the solution was cooled down to around 70 °C under a nitrogen atmosphere and a mixture of NaOH (12 mmol) and NH₄F (20 mmol) in 13 mL of methanol was added. Then, the solution was stirred for 10 minutes at 60 °C. Subsequently, the temperature was increased to 80 °C and the solution was maintained under these conditions for 45 minutes to slowly evaporate methanol. Then, the mixture was heated up to 110 °C and kept under vacuum for 15 minutes. After removing methanol, the reaction temperature was quickly increased to 300 °C and the synthesis spanned for one hour under a nitrogen atmosphere. Then, the solution was cooled down to around 80 °C and the nanoparticles were precipitated by the addition of ethanol and centrifugation at 10 000 rpm for 10 minutes. After that, the prepared nanocrystals were dispersed in a minimal amount of n-hexane and again precipitated with ethanol at 14 000 rpm for 10 minutes. The resulting nanoparticles were dispersed in 10 mL of chloroform.

Ellipsoidal shell deposition. Nanocrystals with an inert ellipsoidal shell were prepared by the thermal decomposition of oleate salts. At the beginning a precursor in the form of lanthanide acetates from Y₂O₃ (2.5 mmol) was prepared as described earlier. The next steps were analogous to the core synthesis. Oleic acid (30 mL) and octadecene (75 mL) were added to the round-bottomed flask containing the precursor. The solution was heated up to 140 °C and kept under vacuum for half an hour. Subsequently, the mixture was cooled down to around 60 °C under a nitrogen atmosphere and a solution of NH₄F (12 mmol) and NaOH (20 mmol) in 13 mL of methanol was added to the flask. The dilution was heated up to 80 °C and the flask was opened for 45 minutes to slowly evaporate methanol. The temperature was then increased to 110 °C and the solution was kept under vacuum for 15 minutes. Then, the solution was cooled down to 60 °C and the earlier prepared core nanocrystals dispersed in chloroform were added to the flask under a nitrogen atmosphere. Then, chloroform was evaporated. Afterwards, the mixture was heated up to 300 °C and kept under a nitrogen atmosphere for one hour. After synthesis, the solution was cooled down to about 80 °C and ethanol was added to precipitate the nanoparticles. The nanocrystals were separated by centrifugation at 10 000 rpm for 10 minutes. Afterwards, the nanoparticles were washed with n-hexane, once more precipitated with ethanol, centrifuged at 14 000 rpm for 10 minutes and finally dispersed in chloroform (10 mL).

Spherical shell deposition. Nanoparticles with a spherical shell were prepared by the thermal decomposition of trifluoroacetates. At the beginning a precursor in the form of trifluoroacetates was prepared, as described above. Afterwards, sodium trifluoroacetate (5 mmol), oleic acid (40 mL) and 1-octadecene (40 mL) were placed in a round-bottomed flask containing the precursor. Subsequently, the solution was maintained under vacuum at 120 °C for half one hour to remove residual oxygen and water. Afterwards, the mixture was cooled down to 60 °C under a nitrogen atmosphere, the core nanocrystals (5 mL) were added to the flask and the mixture was stirred for 10 minutes. Then, chloroform was evaporated. At the beginning the temperature was increased up to 80 °C and the flask was opened for 30 minutes, and subsequently, the temperature was increased up to 110 °C and the solution was kept under vacuum for 15 minutes. Afterwards, the solution was quickly heated up to 300 °C and maintained under a nitrogen atmosphere for one hour. After that, the solution was cooled down to 80 °C and the nanocrystals were precipitated by the addition of ethanol and centrifuged at 10 000 rpm for 10 minutes. Then, the precipitated nanoparticles were dispersed in n-hexane and once more precipitated with ethanol and then centrifuged at 14 000 rpm for 10 minutes. Finally, the nanocrystals were dispersed in chloroform (5 mL).

Setup description and the methodology of PA measurements. The photoexcitation power density dependence of luminescence intensity was investigated and measurements of the PA rise times were performed on a custom-built optical microscopy system. The range in which the pump power density laser can be measured is 10²–10⁷ W cm⁻². The sample was excited with a single mode laser diode (FPL852S – 852 nm, 350 mW), producing an 852 nm beam. The outgoing laser beam passes through a set of gray filters of different densities that attenuate the light, so the power density of the excitation light can be precisely controlled without spectral tuning. Then, a dichroic mirror (DMSP805R) reflects the excitation beam to the side (left) port of an inverted microscope (Nikon Ti-2 Eclipse). Furthermore, the laser beam was focused by the objective lens (Nikon Plan Apo 40×, NA = 0.95, W.D = 0.21 mm) on a sample placed on a microscope cover slip (Menzel-Glaser, 24 × 60 mm). The sample's emission was collected by the objective lens and it exited through the left port of the microscope and then passed through the dichroic mirror (DMSP805R), where it could be simultaneously recorded using two PMT detectors (1001M and 2101M from Thorlabs) connected to a four-channel photon counter (quTAG). The luminescent light of the sample was separated by a dichroic mirror (DMSP505R), and then selected in front of the detectors by bandpass filters (F01-475 – Semrock and 600BP – Omega Optical). Thus, it is possible to record luminescence intensities by two channels at 480 nm and 610 nm, simultaneously. The obtained experimental data were analyzed using our home-developed Matlab code, which is described in the following point. Using an Andor Shamrock i500 spectrograph equipped with a Newton CCD camera (DU920P-BEX2-DD), emission spectra were recorded. The absorption spectra shown in Fig. S7† were recorded using a Cary 5000 UV-Vis-NIR spectrophotometer (Agilent Technologies). Excitation and emission spectra as well as lifetimes were collected with a FLS1000 (Edinburgh Instruments).

Calculations of S and PA_TH values. S and PA_TH values were determined using our home-developed Matlab code, which is available in our web repository (https://github.com/LuNASIanalysis/Photon-Avalanche-PA-). Our algorithm was based on the fitting of the linear function to the logarithm from the raw data (which is analogous to fitting the exponential function to raw data) point by point. For each of the points, the fit covers the maximal range of the surrounding data, which fulfils the given certainty. Such an approach allowed us to reduce the impact of the noise on the determined S values. The highest S values which were obtained for the analyzed dataset were selected for presentation. Similarly, the PA_TH values were determined as the power corresponding to the cross-section point of the lines tangent to the logarithm of the raw data for the pre-avalanche region and tangent to the logarithm of the raw data in the range of maximum S values. The s-shaped curves were measured few times (2–4) and based on them, the average S, PA_TH and D_AV values together with the standard deviation were determined.

Conclusions

In summary, SPA emission at 482 nm (³P₀ → ³H₄) and 607 nm (³P₀ → ³H₆) under excitation at 852 nm in β-NaYF₄ colloidal nanocrystals co-doped with 15%Yb³⁺ and a small (0.1 to 0.7%) amount of Pr³⁺ ions was successfully observed and studied. We have clearly demonstrated the critical role of undoped shell passivation in the PA emission characteristics, which is obviously beneficial and diminishes the impact of the local chemical environment (quenchers, solvent) on both the emitting (Pr³⁺) and sensitizing–migrating (Yb³⁺) ions. Under the same experimental conditions no emission from singly Pr³⁺ ion-doped β-NaYF₄ nanocrystals was observed under 852 nm photoexcitation. It emphasizes the key role of Yb³⁺ ions, which are necessary to obtain PA emission under chosen experimental conditions. During experimental investigations, the highest S of 8.6 and 9.0 as well as the smallest PA_TH of 286 kW cm⁻² and 281 kW cm⁻² were observed for the core–shell sample co-doped with 15%Yb³⁺ and 0.5%Pr³⁺ ions for emissions at 607 nm and 482 nm, respectively. Therefore, the optimal sample composition for SPA NaYF₄:0.5%Pr³⁺,15%Yb³⁺@NaYF₄ was confirmed.⁸ Both Stokes emission and photon avalanche emission let us clearly confirm that the ¹D₂ level does not take part in energy transfer, and thus can be excluded, unlike in some other existing reports, from the analysis. In consequence, elimination of processes related to ¹D₂ level this let us simplify the phenomenological set of rate equations and ultimately enablws to simulate the behavior, mechanism and performance of the sensitized photon avalanche emission, indicating the critical role of the Yb³⁺ sensitizer in this system under 852 nm photoexcitation. Moreover, the simulations of SPA allowed us to understand the impact of energy transfer parameters on PA features. These calculations confirmed that PA emissions from Pr³⁺ ions under excitation at 852 nm are possible only with the assistance of Yb³⁺ ions; thus, crystals singly doped with Pr³⁺ ions do not exhibit PA features at such a photoexcitation wavelength. The obtained experimental and theoretical results have shown and allowed us to understand the important role of Yb³⁺ ions, which are necessary to obtain PA emission from Pr³⁺ ions under excitation at 852 nm.

Author contributions

MD: conceptualization, synthesis, data curation, data analysis, investigation, visualization, and writing the original draft; ZK: data analysis, investigation, visualization, and writing – review & editing; KP: data analysis and writing – review & editing; OB: investigation and data analysis; LS: writing – review & editing; MS: investigation, methodology, software, and writing – review & editing; AB: conceptualization, funding acquisition, methodology, software, project administration, resources, supervision, validation, writing the original draft, review and editing.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This research was funded in whole or in part by the following projects 2018/31/B/ST5/01827 (M. D., Z. K., M. S., A. B.) and 2021/43/B/ST5/01244 (A. B.). K. P. acknowledges the financial support from NCN, Poland, grant number 2018/31/D/ST5/01328. Calculations have been carried out using the resources provided by the Wroclaw Centre for Networking and Supercomputing (https://wcss.pl), grant no. 529.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3nr04409b

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