Luminescence Intensity Ratio Thermometry with Er3+: Performance Overview

The figures of merit of luminescence intensity ratio (LIR) thermometry for Er3+ in 40 different crystals and glasses have been calculated and compared. For calculations, the relevant data has been collected from the literature while the missing data were derived from available absorption and emission spectra. The calculated parameters include Judd–Ofelt parameters, refractive indexes, Slater integrals, spin–orbit coupling parameters, reduced matrix elements (RMEs), energy differences between emitting levels used for LIR, absolute, and relative sensitivities. We found a slight variation of RMEs between hosts because of variations in values of Slater integrals and spin–orbit coupling parameters, and we calculated their average values over 40 hosts. The calculations showed that crystals perform better than glasses in Er3+-based thermometry, and we identified hosts that have large values of both absolute and relative sensitivity.


Introduction
The measurements of temperature, one of seven fundamental physical quantities, can be classified according to the nature of contact between the measurement object and instrument to invasive (where there is direct contact, e.g., thermocouples, thermistors), semiinvasive (where measuring object is altered in a way to enable contactless measurements), and non-invasive (where the temperature is estimated remotely, e.g., optical pyrometers) [1]. The first type necessarily perturbs the temperature of measurement objects which limits its use in microscopic objects. In addition, such approaches are difficult to implement on moving objects or in harsh environments, for example, in high-intensity electromagnetic fields, radioactive, or chemically challenging surroundings. Thus, the current market of thermometers, accounting for more than 80% of all sensors [2], demands methods that allow for remote or microscopic measurements. Among many perspective optical semiinvasive techniques, luminescence thermometry which uses thermographic phosphors has drawn the largest attention [3,4]. The thermographic phosphor probe can be incorporated within the measured object or on its surface, on macroscale to nanoscale sizes, or can be mounted on the surface of the fiber-optic cables and bring to proximity of measuring objects. Luminescent thermometry has found a range of valuable applications, from engineering to biomedical [5], and, currently, it is a widely researched topic with an exponentially increasing number of published research papers [6].
Presently, many types of materials are used for the construction of thermometry probes. These include rare-earth and transition metal activated phosphors, semiconductor quantum dots, organic dyes, and metal-organic complexes, carbon dots, and luminescent polymers. Among the rare-earth crystals are by far the most exploited type [5], usually exploited in the so-called luminescence intensity ratio (LIR, sometimes called fluorescence intensity ratio (FIR) or labeled as ∆) temperature read-out scheme that is based on the LIR(T) = I H (T) where B is the temperature invariant parameter that depends only on the host. The thermometer's performance is estimated by the absolute (S a ) and relative (S r ) sensitivities, and temperature resolution (∆T) given by [10]: where σ a and σ r are the absolute and relative uncertainties in measurement of LIR, presented as standard deviations.
For the temperature dependence of LIR given by Equation (1), sensitivities have the following form: The absolute sensitivity reaches maximum at T = ∆E/2k with the value of [8]: where e = 2.718 is a number (the natural logarithm base). The ideal situation for LIR is the Boltzmann luminescence thermometer since it is easily calibrated with the well-known and simple theory. According to Equation (3), the relative sensitivity only depends on the value of energy difference between thermalized energy levels. The choice of levels with the energy gap larger than 2000 cm −1 may result in the thermalization loss at low temperatures, and even around room temperatures, while the small energy gap gives small relative sensitivities. One should consider that for achieving the Boltzmann's thermal equilibrium some other conditions must be fulfilled besides the suitable energy difference between the levels, as recently demonstrated by Geitenbeek et al. [11] and Suta et al. [12]. Furthermore, considering the adjacent energy levels of trivalent rare-earth used for thermometry, the largest energy gap is in the Eu 3+ (between 5 D 1 and 5 D 0 levels) and is approximately 1750 cm −1 .
Thus, the current research of LIR of a single emission center is aimed at increasing the relative sensitivity without the loss of thermalization (and deviation from the Boltzmann distribution). One recently demonstrated solution is the inclusion of the third, non-adjacent level, with higher energy, which is thermalized with the second level. If the first and second levels are thermalized, and second and third levels are thermalized, then the ratios of emission intensities of the first and third levels will follow the Boltzmann distribution, even if their separation is greater than stated above, see Figure 1 for the case of Er 3+ . The conventional LIR of Er 3+ is equal to the ratio of emissions from 2 H 11/2 (~523 nm) and 4 S 3/2 (~542 nm) levels, which are separated by~700 cm −1 , thus giving the relative sensitivity of~1.1% K −1 . By observing intensities to the ground level from 4 F 7/2 (~485 nm) with the 4 S 3/2 , it is evident that their relative change is much larger than with 2 H 11/2 . This larger energy difference, according to Equation (3), ultimately results in more than a three-fold increase in relative sensitivity.

Judd-Ofelt Theory and Its Relevance for Luminescence Thermometry
The electronic configuration of trivalent erbium is that of Xenon plus the 11 electrons in the 4f shell, i.e., Er 3+ has [Xe]4f 11 electronic configuration. With only 3 electrons missing from the completely filled 4f shell, Er 3+ shares the same LS terms and LSJ levels as the Nd 3+ who has 3 electrons in the 4f shell. The transitions from one to another level are followed by the reception or release of energy. The probability for such phenomena for a given set of initial and final levels is given by the wavefunctions and the appropriate moment operator. The exchange of energy in the intra-configurational 4f transitions with the highest intensity is of induced electric dipole and magnetic dipole types [13]. What was puzzling only half a century ago was the origin of these "electric dipole" interactions, as they were clearly and strictly forbidden by the Parity selection rule, also known as the Laporte rule. The solution to this problem came in 1962 in the papers simultaneously published by Judd [14] and Ofelt [15], to what is latter known as the Judd-Ofelt theory (JO). For the sake of brevity, it will not be explained here, and the reader is instead referred to the excellent references [16][17][18]; however, we will touch on several basics that are the most relevant for the present research.

Judd-Ofelt Theory and Its Relevance for Luminescence Thermometry
The electronic configuration of trivalent erbium is that of Xenon plus the 11 electrons in the 4f shell, i.e., Er 3+ has [Xe]4f 11 electronic configuration. With only 3 electrons missing from the completely filled 4f shell, Er 3+ shares the same LS terms and LSJ levels as the Nd 3+ who has 3 electrons in the 4f shell. The transitions from one to another level are followed by the reception or release of energy. The probability for such phenomena for a given set of initial and final levels is given by the wavefunctions and the appropriate moment operator. The exchange of energy in the intra-configurational 4f transitions with the highest intensity is of induced electric dipole and magnetic dipole types [13]. What was puzzling only half a century ago was the origin of these "electric dipole" interactions, as they were clearly and strictly forbidden by the Parity selection rule, also known as the Laporte rule. The solution to this problem came in 1962 in the papers simultaneously published by Judd [14] and Ofelt [15], to what is latter known as the Judd-Ofelt theory (JO). For the sake of brevity, it will not be explained here, and the reader is instead referred to the excellent references [16][17][18]; however, we will touch on several basics that are the most relevant for the present research.
In RE 3+ (trivalent rare-earth) ions in general, the electrostatic (He) and s-o (Hso) interactions between 4f electrons are dominant and are of the approximately same magnitude, thus the Hamiltonian can be given approximately as H = He + Hso [17].
The electrostatic Hamiltonian can be reduced to the electron-electron repulsion form [17], which can be split into its radial (F k ) and angular (fk) parts: The radial parameters are the Slater integrals given by [19]:  In RE 3+ (trivalent rare-earth) ions in general, the electrostatic (H e ) and s-o (H so ) interactions between 4f electrons are dominant and are of the approximately same magnitude, thus the Hamiltonian can be given approximately as H = H e + H so [17].
The electrostatic Hamiltonian can be reduced to the electron-electron repulsion form [17], which can be split into its radial (F k ) and angular (f k ) parts: The radial parameters are the Slater integrals given by [19]: where r > is greater and r < is smaller than r i and r j , and R are the radial parts of the wavefunction. The Slater integrals can be evaluated by the Hartree-Fock method; however, it does not provide accurate results, and it is best to obtain them semi-empirically by adjusting them to the experimentally observed energies of 4f levels [20]. H so mixes all states that have the same J quantum number, and it is proportional to the s-o coupling parameter, ζ, which is further proportional to the number of electrons within the 4f shell. Er 3+ has a relatively high value of ζ in comparison with other trivalent lanthanide ions, providing a large mixing of states [21]. In this intermediate coupling approximation scheme, the wavefunctions are expressed as a linear combination of all other states in the configuration with the same J quantum number [17,18,22,23]: As the 4f electrons are shielded by the outer higher-energy electrons the crystal field (CF) introduces only a perturbation to the Hamiltonian [16]. Nevertheless, that perturbation weakens the already mentioned Laporte (parity) selection rule that forbids the ED transitions within the configuration. The 4f-4f transitions of electric dipole (ED) type become allowed and are known as the induced ED [24]. The radiative transition probability for such spontaneous emission is then equal to [25]: or for the purely induced ED emission (MD is an abbreviation for the magnetic dipole): where h = 6.626 × 10 −27 erg·s is the Planck's constant. X is the local field correction, ν SLJ→S L J is the emission barycenter energy, and D is the dipole strength given in esu 2 cm 2 units. The emission barycenter is [26]: where i is the intensity at a given energy. The local field correction for ED emission is given by [27]: where n is the refractive index that should be given at the wavelength of the barycenter of the emission. It can be calculated from the Sellmeier's equation for a given material, which is given in the form [28,29]: In the JO scheme, the ED strength is given by [26]: where e = 4.803 × 10 10 esu, U λ SLJ→S L J is the abbreviation for squared RMEs 4 f 11 SLJ U λ 4 f 11 S L J 2 , which in turn can be calculated from the Slater integrals and the s-o coupling parameter. Ω λ are the JO intensity parameters, obtained semi-empirically or by the ab initio calculations (from the crystal-field parameters). The integrated emission intensity for the transition SLJ→S L J is given by [30,31]: or without the hν if the spectrum is recorded in counts instead of power units [32]. LIR of two emissions from the thermally coupled levels is then given by: where I H/L are the integrated intensities from the higher and lower level, respectively (without ν H /ν L if recorded in counts). According to the Boltzmann distribution, the optical center population is given by: where g = 2J + 1 are the degeneracies of the selected levels.
Equation (15) can be rewritten as Equation (1) where B is the temperature invariant parameter that is given by: or if the intensities are recorded in counts instead of power units, without the ν H /ν L . As we have demonstrated in our previous article [8], by inserting Equation (8) into Equation (17), the LIR, the absolute sensitivity (and everything related to it) and temperature resolution can be predicted by JO parameters, as the B parameter can be obtained from: (18) or in the case of the pure ED transitions: For the case of spectra recorded in counts, ν H /ν L should be to the power of 3. The shielding of 4f electrons by electrons from outer orbitals ensures that the RE 3+ spectra are featured by sharp peaks whose energies are almost host-independent. This is reflected in the almost host invariant reduced matrix elements. However, as the Slater parameters deviate significantly in Er 3+ , using such approximation may introduce significant errors. For the analysis of this type, it is more accurate to use the reduced matrix elements that are calculated from Slater integrals and s-o coupling parameters, which are calculated semi-empirically from the positions of the energy levels. Analogously, the small variations in energy level positions may provide significant variations in energy level differences, and thus large deviations in absolute and relative sensitivities. Finally, the small differences in refractive index become enormous when they propagate in the local field correction coefficient and, thus, it is of utmost importance to use accurate values. In this study, the observed levels are energetically very close, thus, it is a good approximation to consider the refractive index as wavelength-independent; however, the exact method is always preferred.

Calculations of Er 3+ Radiative Properties in Different Hosts
For the study, we have selected 40 different hosts doped with Er 3+ (Table 1), from the literature that contained the most complete set of data needed for the analysis presented in this paper. As the JO parametrization is traditionally performed semi-empirically from the absorption spectrum, powders and non-transparent materials are not included in this analysis. In the 3rd column, Table 2 gives the energies of the 4 S 3/2 , 2 H 11/2 , and 4 F 7/2 levels used for the two LIRs that will be theoretically investigated. As stated in the introduction, this is important in the estimation of the thermometric figures of merit, and it is linked to the Slater integrals and s-o parameters. The table also includes the Slater integrals and s-o coupling parameters of the 16 out of the 40 hosts, and the JO intensity parameters for all the hosts, taken from references Table 1.  (15) in Ref. [17]. b Slater integrals calculated from Racah parameters by Equation (17) in Ref. [17]. c Slater integrals and spin-orbit coupling parameter not provided, the RME values the authors used are by Carnall in Ref. [66], or by d Weber in Ref. [35]. e Energy levels are not given in the literature, values in the table are provided approximately. Figure 2 presents the variation of Slater integrals and s-o coupling parameters in those 16 hosts. Although there are no large differences in parameters between the crystals and glasses, there are certain trends that may be observed by the type of compound. Deviations in parameters' values from host to host can be large, so the use of Carnall or Weber tables [22,35] for Er 3+ RMEs can introduce large errors in the later calculations. Figure 3 presents the JO parameters as given in Table 2. Glass hosts have smaller values of JO parameters than crystals, on average. When crystals are analyzed, the largest values of Ω 2 parameter are found in tungstates and molybdates, while the smallest values are in garnets, phosphates, silicates, and oxysulfides. Ω 6 are expectedly higher in fluorides,  [17]. b Slater integrals calculated from Racah parameters by Equation 17 in Ref. [17]. c Slater integrals and spin-orbit coupling parameter not provided, the RME values the authors used are by Carnall in Ref. [66], or by d Weber in Ref. [35]. e Energy levels are not given in the literature, values in the table are provided approximately. Figure 2 presents the variation of Slater integrals and s-o coupling parameters in those 16 hosts. Although there are no large differences in parameters between the crystals and glasses, there are certain trends that may be observed by the type of compound. Deviations in parameters' values from host to host can be large, so the use of Carnall or Weber tables [22,35] for Er 3+ RMEs can introduce large errors in the later calculations.  Table 2. Figure 3 presents the JO parameters as given in Table 2. Glass hosts have smaller values of JO parameters than crystals, on average. When crystals are analyzed, the largest values of Ω2 parameter are found in tungstates and molybdates, while the smallest values are in garnets, phosphates, silicates, and oxysulfides. Ω6 are expectedly higher in fluorides, phosphates, and silicates. In glasses, borate glasses have lower Ω2, while phosphate glasses have higher Ω2. Ω6 is on average higher in phosphate glasses. No clear correlation could be given for the Ω4 parameter in crystals or glasses.  Table 2.  Table 2.
The squared RMEs for each transition investigated for LIR are given in Table 3. This list can be used beyond the scope of this paper for accurate calculations of JO parameters. The deviations from the average RME values are given in Figure 4, and they are large for the 2 H11/2→ 4 I15/2 transition. Thus, the use of Carnall's or Weber's values [22,35] might introduce significant errors in the JO parameters estimation, as the RMEs were calculated for the LaF3 and YAlO3, respectively. The average RMEs values calculated from Table 3 Table 2.
The squared RMEs for each transition investigated for LIR are given in Table 3. This list can be used beyond the scope of this paper for accurate calculations of JO parameters.
The deviations from the average RME values are given in Figure 4, and they are large for the 2 H 11/2 → 4 I 15/2 transition. Thus, the use of Carnall's or Weber's values [22,35] might introduce significant errors in the JO parameters estimation, as the RMEs were calculated for the LaF 3 and YAlO 3 , respectively. The average RMEs values calculated from Table 3 are given in Table 4, together with the deviations from the values by Carnall and Weber. The refractive index values taken from the corresponding references are also listed in Table 3. If Sellmeier's equation is given, the refractive index is calculated at the wavelength of the emission. From the refractive index value, the local field correction is calculated according to Equation (11). The induced ED strengths (the last column of Table 3) are calculated for each transition and for each using JO parameters from Table 2 and local field corrections and RMEs from Table 3. Table 3. Squared RMEs for hosts in Table 1, recalculated by RELIC software [17] from Slater integrals and s-o coupling parameters in Table 2, refractive index values, local corrections for emission, and induced electric dipole strengths. Note: if Slater integrals were not provided in Table 2, the squared RMEs will be given from the tables by Carnall [66], unless indicated that the authors used tables by Weber [35].

No.
Initial   a RME values not calculated by RELIC software, but given in the corresponding reference. b Refractive Index values approx. wavelength-independent. c RME values from Carnall [66], d from Weber [35].  Table 3 from their average values. Initial level U 2 U 4 U 6 U 2 (C) U 4 (C) U 6 (C) U 2 (W) U 4 (W) U 6 (W) 4 Table 3 from their average values. Table 4. Average RME values estimated from squared RMEs listed in Table 3. Deviations of average values from squared RMEs reported by Carnall (C) and Weber (W), in percentage.

Calculations of LIR Parameters
For this theoretical analysis, two Er 3+ -based LIRs are considered, the traditional LIR that uses the temperature-dependent ratio of emissions from 4 S 3/2 and 2 H 11 levels, and the relatively novel concept that uses the temperature-dependent ratio emissions from 4 S 3/2 and 4 F 7/2 levels. Table 5 provides the energy differences between 4 S 3/2 and 2 H 11 and 4 S 3/2 and 4 F 7/2 that are used to calculate the room-temperature-relative sensitivities for each host using Equation (2). The temperature invariant B parameters are calculated from the data in Table 3 using Equation (19) (version for spectra recorded in counts). Then, using Equations (2)-(4) and calculated B values, it was possible to derive the LIR's absolute sensitivity, the maximal absolute sensitivity value, and the temperature at which maximal absolute sensitivity occurs.
The relation between relative and absolute sensitivities of traditional LIR (that uses Er 3+ emissions from 2 H 11/2 and 4 S 3/2 levels) for different hosts is presented in Figure 5a-c. As a rule of thumb, the higher the sensitivity value the better is the performance of thermometry. From Figure 5a, one can see that glasses tend to perform slightly weaker than crystals, on average. Figure 5b compares the LIR performance of different crystals. Fluorides', garnets', phosphates', and silicates' performances are worse than for other hosts. The best results are obtained with simple oxides, vanadates, niobates, molybdates, and tungstates. Figure 5c illustrates the performances of only glass hosts. Even the number of hosts in this set is rather small, it is possible to observe that Er 3+ activated borate glasses perform worse than other glasses. Fluorophosphate glasses show high relative sensitivities, but somewhat small absolute sensitivities. The best combination of sensitivities is achieved in PbO-PbF 2 glass. Similar conclusions can be drawn for the novel LIR type (that uses Er 3+ emissions from 4 F 7/2 and 4 S 3/2 levels), Figure 5d-f. Among different glasses, telluritefluoride glasses show the best performance. For crystals, the situation is almost equivalent to that of traditional LIR. Figure 6a-c gives the relation between relative sensitivity and absolute sensitivity at the temperature at which the absolute sensitivity has its maximum for the traditional LIR, while Figure 6d-f show the same relationship for the novel type LIR. Analogous conclusions can be drawn as in the previous analysis ( Figure 5). Among glass hosts, tellurite-fluoride, tungstate, and molybdate glasses show the best performances. Among crystals, the performance trend is almost the same, but the NaY(MoO 4 ) 2 shows the worst performance at elevated temperatures. The best overall performer is LiLa(WO 4 ) 2 .
As a limit of the study, we must note that the values of the energy levels, Slater integrals and s-o parameters, refractive index values, and JO parameters are taken from literature, so one cannot estimate the level of their accuracy. The extreme outliers are to be taken with caution. Table 5. Calculated luminescence thermometry parameters: energy gaps (∆E) from Er 3+ 4 S 3/2 level to 2 H 11/2 and 4 F 7/2 , relative temperature sensitivities (S r ) for LIRs between selected levels, B LIR parameters, absolute sensitivities at room temperature (S a ), maximum sensitivity value (S amax ), temperatures at which maximum absolute sensitivity occurs (T(S amax )), and relative sensitivities at T(S amax ) (S r (T(S amax )).

No.
Higher    Figures 6a-c gives the relation between relative sensitivity and absolute sensitivity at the temperature at which the absolute sensitivity has its maximum for the traditional LIR, while Figures 6d-f show the same relationship for the novel type LIR. Analogous conclusions can be drawn as in the previous analysis ( Figure 5). Among glass hosts, tellurite-  As a limit of the study, we must note that the values of the energy levels, Slater integrals and s-o parameters, refractive index values, and JO parameters are taken from literature, so one cannot estimate the level of their accuracy. The extreme outliers are to be taken with caution.

Conclusions
The conventional thermometric characterizations are lengthy, complicated, and expensive. Given that there is an infinite number of possible hosts and doping concentrations of luminescent activators, the guidelines in selecting the appropriate material are important, and they can be provided by the Judd-Ofelt thermometric model which predicts thermometric figures of merit from its 3 intensity parameters.
Er 3+ deserves special attention in luminescence thermometry. It features LIR between 2 H11/2 and 4 S3/2 levels with energy separation of ~700 cm −1 , and a recently introduced LIR between 4 F7/2 and 4 S3/2 levels, whose higher energy separation allows for up to 3x larger relative sensitivity. The performances of 40 various crystals and glasses were predicted by the Judd-Ofelt thermometric model, and guidelines were set to aid the search for the best phosphor for LIR thermometry.
It was demonstrated that the Slater integrals and s-o coupling parameters significantly vary from host to host so that their values should not be adopted from other hosts. Consequently, for Er 3+ , the squared reduced matrix elements also significantly vary between hosts (especially for the 2 H11/2→ 4 I15/2 transition). Therefore, RMEs from frequently used Carnall or Weber tables should be replaced by the average RMEs for the three transitions that are used in these LIR read-out schemes, if the exact RMEs cannot be obtained. This will allow for the improved precision in the prediction of thermometric sensor performances, as well as for the improved Judd-Ofelt parametrization of Er 3+ doped compounds.

Conclusions
The conventional thermometric characterizations are lengthy, complicated, and expensive. Given that there is an infinite number of possible hosts and doping concentrations of luminescent activators, the guidelines in selecting the appropriate material are important, and they can be provided by the Judd-Ofelt thermometric model which predicts thermometric figures of merit from its 3 intensity parameters.
Er 3+ deserves special attention in luminescence thermometry. It features LIR between 2 H 11/2 and 4 S 3/2 levels with energy separation of~700 cm −1 , and a recently introduced LIR between 4 F 7/2 and 4 S 3/2 levels, whose higher energy separation allows for up to 3× larger relative sensitivity. The performances of 40 various crystals and glasses were predicted by the Judd-Ofelt thermometric model, and guidelines were set to aid the search for the best phosphor for LIR thermometry.
It was demonstrated that the Slater integrals and s-o coupling parameters significantly vary from host to host so that their values should not be adopted from other hosts. Consequently, for Er 3+ , the squared reduced matrix elements also significantly vary between hosts (especially for the 2 H 11/2 → 4 I 15/2 transition). Therefore, RMEs from frequently used Carnall or Weber tables should be replaced by the average RMEs for the three transitions that are used in these LIR read-out schemes, if the exact RMEs cannot be obtained. This will allow for the improved precision in the prediction of thermometric sensor performances, as well as for the improved Judd-Ofelt parametrization of Er 3+ doped compounds.