Optical and Dielectric Properties of BaF₂:(Er,Yb) Co-Doped Crystal

Abstract

A BaF₂ single crystal co-doped with Er³⁺ and Yb³⁺ was grown by the vertical Bridgman technique and investigated for its optical and dielectric properties. Judd–Ofelt analysis yielded intensity parameters Ω₂ = 0.59, Ω₄ = 0.38, and Ω₆ = 0.27 (×10⁻²⁰ cm²), with a quality factor χ = 1.41, indicating strong radiative transitions. Under UV and near-UV excitation, emissions at 321, 405, 518, and 536 nm were observed, with radiative lifetimes ranging from 1.1 to 3.4 ms. A single dielectric relaxation process was identified, with activation energy of 0.58 eV and associated with trigonal NNN dipoles. The NNN dipole concentration was estimated at ~2.5 × 10¹⁸ cm⁻³. These results support the suitability of Er³⁺,Yb³⁺ co-doped BaF₂ crystals for luminescent and dielectric applications in advanced photonic materials.

1. Introduction

Luminescent materials utilizing rare-earth (RE) ions continue to captivate the scientific community due to their intriguing properties and hold significant relevance for a wide array of emerging applications. Rare-earth-doped fluorite materials are well known as active media for solid-state lasers from ultraviolet up to middle infrared spectral region [1,2,3,4]. Due to the up-conversion phenomenon of fluorites doped or co-doped with rare earth ions, they are being studied for their applications as optical temperature sensors [5,6,7,8,9,10], solar radiation converters [11], or scintillators [12,13]. Although many types of materials co-doped with Er and Yb ions have been studied [14,15,16,17,18,19,20,21,22], there are few studies concerning the optical and dielectric properties investigation of Er-doped and Yb-codoped BaF₂ crystal. Recently, N. Li et al. [23] reported a color-tuned up-conversion luminescence of BaF₂:Yb³⁺, Er³⁺ 1D nanostructures containing nanofibers, nanobelts, and hollow nanofibers under excitation on 980 nm. S. Balabhadra et al. have reported an up-conversion fluorescence upon selective excitation of Yb³⁺ ions at 980 nm, resulting in a strong visible emission from ²H_11/2 (blue), ⁴S_3/2 (green) and ⁴F_9/2 (red) multiplets [24]. However, there is limited information regarding the luminescent properties of double-doped BaF₂:(Er,Yb) crystal [25], but for up-conversion luminescence only. BaF₂ crystal have a cubic structure in the Fm-3m space group, with Ba²⁺ ions suitable for rare-earth doping. For a low dopant concentration, the charge compensation takes place through interstitial F⁻ ions in the next-nearest neighbor (NNN) lattice positions, resulting in trigonal site symmetries [3]. Higher dopant concentrations lead to the formation of complex defects and clusters in the crystal. The presence of local charge compensation results in the formation of electric dipoles, which exhibit relaxations detected as dielectric absorption. Dielectric relaxation measurements of Er³⁺ ions in BaF₂ crystals were recently reported [26]. A single dielectric relaxation has been observed, characterized by activation energy of 0.54 eV which is attributed to the trigonal (C_3v) centers. Similarly, the dielectric spectra of Yb³⁺ doped BaF₂ crystals have shown a relaxation peak around 0.54 eV also assigned to the NNN dipoles (namely Yb³⁺–F(i) compensating defect with C_3v site symmetry) [27]. It is important to mention that in both cases, the dopant concentration is less than or equal to 0.5 mol%. With the increase in dopant concentration, the number of isolated centers decreases, giving rise to new aggregates or clusters. Using the method described by Nicoara et al. [26], the number of NNN dipoles can be calculated. It is known that certain trivalent rare earth ions (such as Eu, Sm, Ho, Tm, and Yb) can dissolve in the BaF₂ crystal in both trivalent and divalent state. The presence of Yb²⁺ ions is revealed in the UV absorption spectrum and the presence of Yb³⁺ ions is in the near-IR. The Yb²⁺ ions replace Ba²⁺ ions, require no charge compensation, exhibit cubic site symmetry, and do not contribute to the dielectric spectra [27]. Compared to CaF₂ or SrF₂, BaF₂ allows efficient charge compensation through interstitial fluorine ions located in next-nearest-neighbor positions (NNN), leading to well-characterized trigonal (C_3v) dipoles. This makes it a model system for studying dielectric relaxation associated with NNN dipoles [27]. Moreover, BaF₂’s low phonon cutoff energy and high transparency in the UV–MIR range enhance its suitability for investigating the luminescence phenomena in RE³⁺/RE²⁺ co-doped systems [11,19]. Despite its advantages, detailed studies on the dielectric and photoluminescent behavior of Er³⁺,Yb³⁺ co-doped BaF₂ crystals remain limited, particularly in relation to defect–dipole dynamics, which we aim to address in this work. Both dielectric and absorption spectra can provide information about the nature and site symmetry of the defects. Spectroscopic properties of Yb²⁺ and Yb³⁺ ions and, separately, Er³⁺ ions doped in BaF₂, crystals have been investigated, particularly for applications as up-conversion luminescent materials in the UV–VIS–NIR spectral range [3,12,13,28,29,30]. There are few papers related to the luminescence properties of Er/Yb double-doped crystals under excitation in the visible range (downshifting emission), but no study reported for BaF₂:(Er,Yb) crystals. In 2019, Liu et al. [31] reported that PbF₂:(Er,Yb) phosphors show a green emission, around 550 nm, and red emission at 660 nm, under excitation at 376 nm, which correspond to ⁴S_3/2→⁴I_15/2 and ⁴F_9/2→⁴I_15/2 transitions of Er³⁺ ions, respectively. Red, green, and blue up-conversion emissions were also observed in Er/Yb double-doped Ba₂LaF₇ nanocrystals [32]. Kaczmarek et al. [33] reported a near infrared emission, at room temperature, around 1540 and 978 nm of (Er³⁺,Yb³⁺):LaF₃ nanoparticles under excitation at 378 nm (direct excitation of ⁴G_11/2 level). On the other hand, by direct excitation at 380 nm, a broad and intensive emission, with maximum around 410 nm, was observed in Bi₂ZnOB₂O₆:Yb³⁺/Er³⁺ single crystal which correspond to to the ³P₁ → ¹S₀ transition of Bi³⁺ ions from the host crystal [34].

The focus of this paper is to investigate the spectroscopic and dielectric properties of double-doped BaF₂:(Er,Yb) crystal. To achieve this objective, optical absorption and photoluminescence (PL) measurements were obtained, and the Judd–Ofelt (JO) model was used to obtain information about the luminescence properties of the crystal. The obtained theoretical values were compared with those reported by other authors. The dielectric measurements were utilized to investigate the nature of charge compensating defects arranged in a dipolar configuration. Within this analysis, the dielectric absorption unveiled the relaxation process of these dipoles. The study identified a singular type of charge compensating defect, specifically, the isolated C_3v site symmetry. The examination of these physical properties of the low concentration double-doped BaF₂:(Er,Yb) crystal holds significance not just from a scientific perspective but also for practical applications.

2. Materials and Methods

Double-doped BaF₂:(0.05 mol% YbF₃, 0.2 mol% ErF₃) crystal was grown in our Crystal Growth Laboratory by vertical Bridgman method using a shaped graphite furnace [35]. The doping concentrations mentioned pertain to the inclusion of ErF₃ and YbF₃ into the molten substance. The crystal was grown under a vacuum of approximately 10⁻¹ Pa in a crucible made of spectrally pure graphite. We utilized crushed BaF₂ optical UV–VIS windows sourced from Crystran Ltd., Dorset, UK (derived from 99.99% BaF₂ powder) as our initial material. To this raw material, the specified quantity of ErF₃ and YbF₃ (purchased from Merck, Dramstadt, Germany, 99.99% purity) was added. Figure 1a shows the preparation stage of the Bridgman setup for reaching the rated electrical power of 4.3 kW on the graphite heater. This electrical power is required to achieve the melting temperature of the raw material (1375 °C). A period of three hours at the 4.3 kW is required in order to stabilize the melt. Figure 1b illustrates the typical temperature distribution measured at the bottom of the crucible using an S-type thermocouple. The crucible, with the melt, goes down in the heater with a pulling rate of 4 mm/h. In the first hour of the growth process, the temperature gradient was 36 °C/h. The grown crystal has approximately 6 cm long and 10 mm in diameter. The crystal was cleaved along the (111) crystallographic direction into several slices. For this study, a 2.33 mm thick slice was selected from the top of the crystal (slice 12). It exhibits transparency and is devoid of any visible inclusions or cracks, as depicted in Figure 1c.

The room temperature optical absorption spectra in the UV–VIS–NIR spectral range were recorded using a Shimadzu 1650PC, Schimadzu Corporation, Kyoto, Japan and Nexus 470 FTIR spectrophotometer, Thermo Fisher Scientific, Waltham, MA, USA The spectrophotometers use an automatic correction for baseline correction. The correction takes into account the effect of instrument noise and the light scattering due to the possible undesired particles in the sample. In order to measure the room temperature luminescence spectra, at room temperature, in the UV–VIS domain, the PerkinElmer LS55 spectrofluorometer, Perkin Elmer Inc., Waltham, MA, USA, was used. The room temperature luminescence spectra were obtained by excitation at two wavelengths, 290 and 378 nm, respectively, which correspond to the ⁴I_15/2→⁴G_7/2 and ⁴I_15/2→⁴G_11/2 transition of Er³⁺ ions [13]. The local charge compensations by pairing the trivalent rare-earth (RE³⁺) cations with interstitial F⁻(i) anions create electric dipoles, (RE³⁺-F⁻). These dipoles can reorient under a variable electric field. The reorientation process can be studied by various methods as anelastic relaxation, thermally stimulated depolarization, or dielectric relaxation measurements. The temperature and frequency dependence of the complex dielectric constant, ε* = ε₁ − iε₂ constitutes the dielectric spectra. The dielectric spectra, spanning ten audio frequencies, were obtained using the RLC Meter ZM2355 NF Corporation, Yokohama, Japan, across temperatures ranging from 150 K to 320 K. The real part of the dielectric constant, ε₁, was derived from the measured capacitance C, while the imaginary part, ε₂, was subsequently calculated through relation D = ε₂/ε₁, where D = tan θ is the dielectric loss. The measurements were conducted employing linear heating rates of 1 K/min. A sample cleaved from single crystal, polished to ~0.6 mm thick disk, was used, with Ag (Leitsilber) contacts facilitating the dielectric property assessments.

3. Experimental Results and Discussion

3.1. Optical Absorption Spectra

Figure 2a,b shows the absorption spectra of the selected sample. The absorption spectra show both the characteristic absorption bands of trivalent Er ions and those of Yb³⁺/Yb²⁺ ions, the latter resulting from the Yb³⁺→Yb²⁺ electric charge conversion, similar to that observed in CaF₂:(Er,Yb) crystals [19]. The absorption peaks, denoting the transitions from the ground state ⁴I_15/2 to the excited states of Er³⁺ ions, are specifically highlighted in Figure 2a,b. Also, in Figure 2a, the absorption bands of Er³⁺ ions around 290 and 378 nm, used for excitation during emission monitoring, are highlighted. Due to the charge compensation process, involving both rare earth ions, the energy levels of the RE ions split causing the formation of broad and structured absorption bands.

3.2. Judd–Ofelt Analysis

The application of the standard Judd–Ofelt analysis (J-O analysis) [36,37] facilitated the calculation of spectroscopic parameters based on optical absorption spectra. By utilizing a set of four absorption bands corresponding to specific Er³⁺ transitions from the ground level ⁴I₁₅ to excited levels (⁴I_13/2, ⁴F_9/2, ²H(2)_11/2, and ⁴F_7/2), the J–O intensity parameters, namely Ω₂, Ω₄, and Ω₆, were computed. The 980 nm absorption band was excluded due to overlapping transitions of Er³⁺ ions and Yb³⁺ ions (specifically, ⁴I_15/2→⁴I_11/2 for Er³⁺ ions and ²F_7/2→²F_5/2 for Yb³⁺ ions). Additionally, the absorption band around 378 nm (⁴I_15/2→⁴G_11/2 transition), characteristic of Er³⁺ ions, was excluded from the J–O analysis due to overlapping with the Yb²⁺ ion absorption band related to the 4f¹⁴(¹S₀)→4f¹³5d (²F_3/2) transition.

In order to obtain the J-O parameters Ω_t (t = 2, 4, 6) and the measured line strength (see

Table 1. The mean wavelength, the wavelength range, and the integrated absorption cross-section of the measured and calculated absorption line strengths of the selected absorption peaks of BaF₂: (0.05 mol% YbF₃, 0.2 mol% ErF₃) crystal.

Trasition, 4I15/2 →	λmean (nm)	Wavelength Range (nm)	$\mathbf{\sum }=\int \mathit{\alpha }$(λ)$\mathbf{d}\mathit{\lambda }$ (×10−20 cm2·nm)	${\mathit{S}}_{\mathit{D}\mathit{E}}^{\mathbf{m}\mathbf{e}\mathbf{a}\mathbf{s}}$ (×10−20 cm2)	${\mathit{S}}_{\mathit{D}\mathit{E}}^{\mathbf{c}\mathbf{a}\mathbf{l}\mathbf{c}}$ (×10−20 cm2)
4I13/2	1530	1450–1600	14.353	0.405	0.444
4F9/2	648	620–695	1.576	0.309	0.330
2H(2)11/2	518	502–533	2.471	0.604	0.603
4F7/2	486	476–498	1.271	0.331	0.226

), we solved a set of four equations, using the Levenberg–Marquardt algorithm, corresponding to the four transitions. The experimental line strength has been determined using the following expression:

$$S_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}={\displaystyle \frac{3\hslash c\left(2J+1\right)n\left(\lambda \right)}{8{\pi }^3{N}_0{\lambda }_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}{e}^2}}\left[{\displaystyle \frac{9}{{\left({n\left(\lambda \right)}^2+2\right)}^2}}\int \alpha \left(\lambda \right)\mathrm{d}\lambda \right]$$

(1)

where J is the total angular momentum quantum number of the initial state, n(λ) is the refractive index, N₀ is the Er³⁺ ions concentration (added to the melt), λ_mean is the mean wavelength of the specific absorption bands, $\mathsf{\Sigma }=\int \alpha \left(\lambda \right)\mathrm{d}\lambda$ is the integrated absorption coefficient as a function of λ, α(λ) is absorption coefficient, c is the vacuum speed of light, and ħ is Planck’s constant. The refractive index, n, was determined from the Sellmeier dispersion equation [38].

Deriving the J–O parameters involved solving a set of four equations for transitions between J and J′ manifolds using the experimental line strengths. This simultaneous calculation, is based on Expression (1) and the electric dipole line strength:

$$S_{J{J}^{\prime }}^{\mathrm{e}\mathrm{d}}=\sum _{t=\mathrm{2,4},6}{\mathsf{\Omega }}_{\mathrm{t}}{\left|⟨\left(\mathrm{S},\mathrm{L}\right)\mathrm{J}|\left|{\mathrm{U}}^{\left(\mathrm{t}\right)}\right||\left({\mathrm{S}}^{\prime },{\mathrm{L}}^{\prime }\right)\mathrm{J}\prime ⟩\right|}^2$$

(2)

and the magnetic dipole (md) line strength:

$$S_{J{J}^{\prime }}^{\mathrm{m}\mathrm{d}}={\left|⟨\left(\mathrm{S},\mathrm{L}\right)\mathrm{J}|\left|\mathrm{L}+2\mathrm{S}\right||\left({\mathrm{S}}^{\prime },{\mathrm{L}}^{\prime }\right)\mathrm{J}\prime ⟩\right|}^2$$

(3)

where $⟨ |\left|U^{\left(t\right)}\right|| ⟩$ are the reduced matrix elements of rank t (t = 2, 4, and 6) of tensor operators between states characterized by the quantum numbers (S, L and J) and (S′, L′ and J′). We have used the values of the reduced matrix elements for the chosen Er³⁺ transitions from those tabulated in the work of Kaminskii [39]. Only the ⁴I_15/2→⁴I_13/2 transition has a magnetic dipole contribution corresponding to the absorption band around 1530 nm. The calculated line strength in this case is the sum of electric and magnetic dipole line strength. The calculated J-O parameters, by applying a least-squares fitting of S_meas and S_calc, and the spectroscopic quality factor, χ, are given in

Table 2. The Judd–Ofelt parameters, Ω_t (expressed in 10⁻²⁰ cm²) and the spectroscopic quality factors, χ.

Ωt (10−20 cm2)	BaF2: 0.2 mol% ErF3 + 0.05 mol% YbF3 (This Paper)	BaF2: 0.15 mol% ErF3 [13]	BaF2: 0.2 mol% ErF3 [12]	BaF2:Er3+ [1]	BaF2: 1.96 mol% ErF3, 2.91 mol% YbF3 [25]	YSAG: Er3+ [40]
Ω2	0.58983 ± 0.08026	0.6749	0.932	1.98	0.95	0.4681
Ω4	0.38250 ± 0.05751	0.1118	0.153	1.18	0.49	0.8378
Ω6	0.27115 ± 0.12981	0.5525	1.074	1.20	1.45	0.6741
χ = Ω4/Ω6	1.41	0.20	0.14	0.98	0.34	1.24

. A comparison of the obtained J–O parameters with those obtained for BaF₂:ErF₃ [1,12,13,20,21], for BaF₂:(Er,Yb) reported by [25] and for Er:YAG [40], is also provided. The observed variations between the Judd–Ofelt parameters obtained in this work and those reported in previous studies on BaF₂:Er³⁺ or BaF₂:(Er³⁺,Yb³⁺) systems can be attributed to several factors. Firstly, differences in dopant concentrations affect the local field environment and may lead to the formation of clusters rather than isolated centers, thereby altering the local symmetry and affecting the oscillator strengths [13]. Secondly, in co-doped systems such as the present one, additional energy transfer pathways and local charge compensation mechanisms are introduced due to the presence of both Er³⁺ and Yb³⁺ ions. This can modify the effective transition probabilities compared to a singly doped host. Thirdly, spectral overlap, especially in the near-UV region, may introduce uncertainties in the evaluation of certain bands; for instance, the ⁴I_15/2 → ⁴G_11/2 transition of Er³⁺ at ~378 nm overlaps with the 4f¹⁴→4f¹³5d transition of Yb²⁺, complicating the spectral deconvolution. Such overlapping transitions were excluded from the fit to ensure reliability of the extracted parameters.

The root mean square deviation, defined by $\mathsf{\Delta }{S}_{\mathrm{r}\mathrm{m}\mathrm{s}}={[{(q-p)}^{-1}\sum {(S_{\mathrm{calc}}-S_{\mathrm{meas}})}^2]}^{1/2}$ is a measure of the accuracy of the fitting procedure; q = 4 is the number of analyzed spectral bands and p = 3 is the number of sought parameters. In this case, ΔS_rms = 0.08·10⁻²⁰ cm², which is comparable with those obtained in other papers [1,12,13,28,40]. In order to calculate the radiative lifetime (τ_rad) for an excited state, J′ can be used the relationship τ_rad = 1/∑A_JJ′, where:

$$A_{J{J}^{\prime }}={\displaystyle \frac{64{\pi }^2{e}^2}{3\hslash (2J+1){\lambda }_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}^3}}\left[{\displaystyle \frac{n{(n^2+2)}^2}{9}}{S}_{J{J}^{\prime }}^{\mathrm{e}\mathrm{d}}+n^2{S}_{J{J}^{\prime }}^{\mathrm{m}\mathrm{d}}\right]$$

(4)

is the sponaneous emission probability and the sum is taken over all final lower-lying states, $S_{J{J}^{\prime }}^{\mathrm{e}\mathrm{d}}$ and $S_{J{J}^{\prime }}^{\mathrm{m}\mathrm{d}}$ are defined by Equations (2) and (3). The calculation of fluorescence branching ratios, B_JJ^′ involves utilizing the expression B_JJ^′ = A_JJ^′·τ_rad.

Table 3. The radiative emission probabilities, the branching ratios, and the calculated and measured radiative lifetimes for transitions where luminescence was observed.

Transition	λem. [nm]	AJJ′ (s−1)	β	τrad (ms) This Work	τexp (ms)
4S3/2→4I15/2	536	191.8	0.66	3.4	1.9 [13]
					0.96 [12]
					0.38 [19]
					0.56 [28]
2H(2)11/2→4I15/2	518	733.4	0.88	1.9	1.351 [13]
					0.96 [12]
					0.39 [19]
2G(1)9/2→4I15/2	405	209.1	0.42	2.0	-
2P3/2→4I15/2	321	79.2	0.09	1.1	0.906 [13]

shows the values of the radiative emission probabilities, the branching ratios, and the radiative lifetimes for transitions where luminescence was observed. A comparison of the calculated radiative lifetimes and those measured by other authors is also given in Table 3. The observed difference between calculated and measured lifetimes may imply the presence of energy migration, intense emission reabsorption or thermal coupling across manifolds, factors not taken into account in the standard J-O model.

3.3. Emission Spectra

In order to obtain the room temperature emission spectra, two absorption bands were used for excitation, namely λ_exc. = 290 nm (⁴I_15/2→⁴G_7/2 transition) and λ_exc. = 378 nm (⁴I_15/2→⁴G_11/2 transition). The emission spectrum of the studied sample under 290 nm excitation and the energy level diagram are shown in Figure 3a,b. By excitation at 290 nm, the emission spectrum is characterized by an emission band centered around 321 nm (Figure 3a). This UV emission was reported in our previous work for various concentrations of simple-doped ErF₃ doped BaF₂ crystals [13] but it was not reported before in double-doped BaF₂:(Er,Yb).

This emission can be attributed to the self-trapped exciton (STE). When the energy level ⁴G_7/2 is pumped, the Er-bond exciton emission took place from the ²P_3/2 manifold, which is placed in the middle of the STE emission band, therefore the energy transfer from STE to the Er³⁺ ion becomes effective [13].

By excitation at 378 nm, we obtained three broad emission bands (Figure 4a). The green band has two peaks at 536 nm and 518 nm (weak) which correspond to the transition ⁴S_3/2→⁴I_15/2 and ²H(2)_11/2→⁴I_15/2 of Er³⁺ ions (Figure 4b). The blue band, around 405 nm, corresponds to the ²G(1)_9/2→⁴I_15/2 transition. The intensities of the blue and green emissions are over two times higher than the UV emission band. The International Commission on Illumination (CIE) chart corresponding to the visible emissions is shown in Figure 4c. The CIE coordinates were obtained using Gocie V2 software [41]. For the excitation at 378 nm, the color coordinates are X = 0.28 and Y = 0.66. These values are in a good agreement with those obtained for BaF₂:0.15 mol% ErF₃ (X = 0.30 and Y = 0.67) [13].

3.4. Dielectric Relaxation

Figure 5a,b show the temperature and frequency variations of both the real, ε₁, and imaginary, ε₂, components of the complex dielectric constant, ε*, specifically for the BaF₂:(0.05 mol% YbF₃, 0.2 mol% ErF₃) crystal. In the studied temperature range, the imaginary part (ε₂) of the complex dielectric constant exhibits a singular peak at a temperature denoted as T_relax, consistent across all frequencies. The T_relax temperature demonstrates a tendency to shift towards higher temperatures with increasing frequency, which indicates a dielectric relaxation phenomenon. The fundamentals of dielectric relaxation and the interpretation of relaxation phenomena in complex systems are well detailed in recent works [42,43,44]. The relaxation process is characterized by the activation energy for re-orientation, E, and by a relaxation time, τ, given by the Arrhenius relation τ = τ₀ exp (E/kT), where τ₀ is [45] the reciprocal frequency factor. The imaginary part, ε₂, has a maximum for ωτ = 1 [45]. Therefore, the plot of 1/T_relax versus ln ω permits to determine the activation energy, E, and the τ₀ (Figure 5c). The values of the relaxation parameters for the observed dipoles are E = 0.58 eV and τ₀ = 4.1 × 10⁻¹⁵ s. These values correspond to the NNN dipoles (trigonal, C_3v) sites (inset of Figure 5b). The RE³⁺ ions (erbium or ytterbium) substitute for Ba²⁺ ions in BaF₂ crystals. Charge compensation is achieved by a fluorine ion in trigonal site symmetry. This result is in good agreement with our recent study regarding the analysis of site symmetries of Er³⁺ doped CaF₂ and BaF₂ crystals by high resolution photoluminescence spectroscopy [13]. For comparison, the relaxation parameters of Yb/Er/Gd doped BaF₂ crystals are given in Table 4. Although the activation energy of 0.58 eV is consistent with values attributed in previous works to NNN dipoles of trigonal (C_3v) symmetry in rare-earth doped BaF₂ crystals, the potential contribution of other defect configurations cannot be entirely ruled out. Defect clusters or residual oxygen-related complexes may lead to similar relaxation signatures, especially at higher doping levels or under different growth atmospheres. Nevertheless, the low dopant concentrations employed here, and the close match to known C_3v-related values, support our current assignment [13,27]. Further studies using EPR or impurity profiling would help to clarify the role of such alternative configurations.

The number of dipoles, N_D, that contribute to the dielectric relaxation peak can be calculated from the dielectric spectra using the Campos method [46] that consists of plotting (T tan δ) versus 1000/T, where T is the temperature and tan δ is the dielectric loss. The maximum of this curve occurs when ωτ = 1, which corresponds to

$$N_D={\displaystyle \frac{6{\epsilon }_{0}k}{{\mu }^2}}{\left(T\mathrm{t}\mathrm{a}\mathrm{n}\delta \right)}_{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(5)

where μ = 8.596 × 10⁻²⁹ C·m is the dipole moment of the NNN dipole, k is the Boltzmann constant, ε₀ is the free space dielectric constant, and N_D is the number of dipoles contributing to the relaxation.

Table 4. Relaxation parameters of NNN dipoles in Yb/Er/Gd doped BaF₂ crystals.

Sample	E (eV)	τ0 (10−15 s)	ND (1017 cm−3)
BaF2:0.05 mol% YbF3, 0.2 mol% ErF3	0.58	4.1	25.2 This work
BaF2:0.2 mol% ErF3	0.56	6.7	22.6 [26]
BaF2:0.05 mol% YbF3	0.53	16	20.95 [27]
BaF2:0.05 mol% GdF3	0.53–0.61	7 [47]	-

Table 4. Relaxation parameters of NNN dipoles in Yb/Er/Gd doped BaF₂ crystals.

Sample	E (eV)	τ0 (10−15 s)	ND (1017 cm−3)
BaF2:0.05 mol% YbF3, 0.2 mol% ErF3	0.58	4.1	25.2 This work
BaF2:0.2 mol% ErF3	0.56	6.7	22.6 [26]
BaF2:0.05 mol% YbF3	0.53	16	20.95 [27]
BaF2:0.05 mol% GdF3	0.53–0.61	7 [47]	-

After the Gaussian multi-peak fit of the plot (T tan δ) versus 1000/T for a frequency f = 1 kHz, the values of the N_D for both RE ions can be estimated (Figure 6). In Table 4, the values for the concentration of NNN dipoles are shown. The NNN dipole concentration of Er³⁺ ions is 15.9·10¹⁷ cm⁻³ being slightly lower than for simply doped BaF₂:Er crystals (see Table 4) [26]. Similarly, a lower NNN dipole concentration value of 9.3·10¹⁷ cm⁻³ was obtained for Yb³⁺ ions [27]. This can be explained by the clustering process in the double-doped crystal. No detailed study on the dielectric spectra of double-doped BaF₂:(ErF₃, YbF₃) crystals has been previously reported.

4. Conclusions

A double-doped BaF₂:(ErF₃, YbF₃) crystal was grown by using the conventional Bridgman technique. The optical and dielectric properties of the crystal were investigated. The significant results found in the present work are as follows: (1) The optical absorption spectrum shows both the characteristic bands of trivalent Er and Yb ions and the characteristic bands of Yb²⁺ ions (in the UV spectral range). (2) Using the J-O formalism, the radiative emission probabilities, the branching ratios, and the radiative liftimes were calculated. (3) Room temperature emission spectrum of BaF₂:(0.05 mol% YbF₃, 0.2 mol% ErF₃) crystal under excitation at 290 and 378 nm was measured. (4) In the investigated temperature range, only one type of dielectric relaxation has been observed. This relaxation, with activation energy of 0.58 eV is associated with trigonal NNN centers. (5) The number of dipoles corresponding to NNN centers was determined. The values obtained are in agreement with those observed in singly doped crystals. The study of the dielectric behavior and optical properties of double-doped BaF₂ crystal with low concentrations of YbF₃ and ErF₃ has not been reported so far.

In summary, the co-doped BaF₂:Er³⁺,Yb³⁺ single crystal investigated in this work demonstrates a combination of favorable optical and dielectric properties, including well-resolved emission bands, radiative lifetimes in the sub-millisecond range, and a clearly defined dielectric relaxation process The Judd–Ofelt analysis reveals a relatively high Ω₂ parameter and quality factor (χ = 1.41), indicating good radiative efficiency. These results highlight the suitability of such crystals for UV–visible photonic applications, particularly in compact solid-state luminescent sources. The findings contribute to the growing interest in rare-earth doped fluorides as multifunctional materials combining optical and dielectric functionalities.