Energy Transfer and Cross-Relaxation Induced Efﬁcient 2.78 µ m Emission in Er 3+ /Tm 3+ : PbF 2 mid-Infrared Laser Crystal

: An efﬁcient enhancement of 2.78 µ m emission from the transition of Er 3+ : 4 I 11/2 → 4 I 13/2 by Tm 3+ introduction in the Er/Tm: PbF 2 crystal was grown by the Bridgman technique for the ﬁrst time. The spectroscopic properties, energy transfer mechanism, and ﬁrst-principles calculations of as-grown crystals were investigated in detail. The co-doped Tm 3+ ion can offer an appropriate sensitization and deactivation effect for Er 3+ ion at the same time in PbF 2 crystal under the pump of conventional 800 nm laser diodes (LDs). With the introduction of Tm 3+ ion into the Er 3+ : PbF 2 crystal, the Er/Tm: PbF 2 crystal exhibited an enhancing 2.78 µ m mid-infrared (MIR) emission. Furthermore, the cyclic energy transfer mechanism that contains several energy transfer processes and cross-relaxation processes was proposed, which would well achieve the population inversion between the Er 3+ : 4 I 11/2 and Er 3+ : 4 I 13/2 levels. First-principles calculations were performed to ﬁnd that good performance originates from the uniform distribution of Er 3+ and Tm 3+ ions in PbF 2 crystal. This work will provide an avenue to design MIR laser materials with good performance.

In this paper, Er: PbF 2 , Tm: PbF 2 , Er/Tm: PbF 2 crystals were successfully prepared by the Bridgman technique. The spectroscopic properties of prepared crystals were analyzed based on absorption spectra, emission spectra, and fluorescence decay curves. Compared with the Er: PbF 2 crystal, the Er/Tm co-doped PbF 2 crystal presents a larger 2.78 µm fluorescence emission intensity and higher fluorescence branching ratio. Moreover, theoretical calculations were performed to discover that the co-doping of the Tm 3+ ion can make the Er 3+ and Tm 3+ ions more evenly distributed in PbF 2 crystals, which can effectively break the local clusters of the Er 3+ in Er: PbF 2 crystal, thus ensuring efficient energy transfer between Er 3+ and Tm 3+ ions, and resulting in the enhancement of 2.78 µm MIR fluorescence emission.

Experimental Section
The 1.0 at.% Er: PbF 2 , 0.5 at.% Tm: PbF 2 , and 1.0 at.% Er/0.5 at.% Tm: PbF 2 crystals were grown by the conventional Bridgman method in an atmosphere of N 2 with intermediate molybdenum heating. The fluoride powders of the PbF 2 (99.999%), ErF 3 (99.999%), and TmF 3 (99.999%) were all raw materials. The raw materials were weighed and thoroughly mixed. The process of crystal growth was similar to our previous work [37]. The melt was homogenized in a covered graphite crucible in a high-temperature zone at 1000 • C for 8 h, and the crystal growth process was driven by lowering the graphite crucible at a speed of 0.5 mm/h. After the growth process was completed, the cooling rate of the crystal was 30 • C/cm-40 • C/h. The actual concentration of Er 3+ and Tm 3+ ions in the grown samples were measured utilizing inductively coupled plasma atomic emission Crystals 2021, 11, 1024 3 of 13 spectrometry (ICP-AES). The concentrations of Er 3+ and Tm 3+ ions in dual-doped Er/Tm: PbF 2 crystal were 1.15 at.%, and 0.58 at.%, respectively. The concentration of Er 3+ ion in the Er: PbF 2 crystal was 1.15 at.%, and the concentration of Tm 3+ ion in the Tm: PbF 2 crystal was 0.59 at.%.
The crystalline structure of as-grown samples was observed utilizing D/max2550 Xray diffraction (XRD) with Cu K α radiation. The Perkin-Elmer UV-VIS-NIR spectrometer (Lambda 900) with a resolution of 1 nm was used to detect the absorption spectra of prepared samples in the range of 400-2200 nm. The emission spectra, up-conversion fluorescence spectra, and fluorescence decay curves were detected and recorded using the Edinburgh Instruments FLS920 and FSP920 spectrophotometers. The repetition frequency of the excitation pulse for measuring the fluorescence decay curves was set to 20 Hz, and the duration of the excitation pulses was 30s. All the measurements were performed at room temperature.

Calculation Method
In the framework of density functional theory, VASP codes and the plane-wave basis set method were used for calculation [38,39]. The mutual interactions were described by the projector augmented-wave pseudopotential with an exchange-correlation function (Perdew-Burke-Ernzerhof form) [40,41]. The cut-off was set at 550 eV and a 1 × 1 × 1 Gamma k-grid was used to guarantee the relaxation accuracy of 10 −5 eV and 0.01 eVÅ −1 within a 2 × 2 × 2 supercell, respectively. The spin polarization was included in the calculations. According to the method reported previously [42], the formation energy (∆E) and cluster symbols were obtained. It is pointed that the energy correction of the PbF 2 crystal was different from that of CaF 2 , SrF 2 , and BaF 2 crystals. For a 2 × 2 × 2 supercell with a net charge, the calculated value in PbF 2 crystal was 0.069 eV. Figure 1 shows the XRD patterns and refined XRD patterns of the Er: PbF 2 , Tm: PbF 2 , Er/Tm: PbF 2 crystals, and the JCPDS standard card of the PbF 2 crystal (nos. 06-2051) [37]. The residuals of refinements (fit profiles shown in Figure 1) of Er: PbF 2 , Tm: PbF 2 , Er/Tm: PbF 2 crystals were 9.61%, 7.71%, 10.17%, respectively. It is obvious that no clear shift in the phase diffraction peaks was observed and all XRD curves were well matched with the standard card of the β-PbF 2 crystalline phase (nos. 06-2051). The results demonstrate successful co-doping of Er 3+ and Tm 3+ ions in PbF 2 crystal without phase transitions. The crystalline structure of as-grown samples was observed utilizing D/max2550 Xray diffraction (XRD) with Cu Kα radiation. The Perkin-Elmer UV-VIS-NIR spectrometer (Lambda 900) with a resolution of 1 nm was used to detect the absorption spectra of prepared samples in the range of 400-2200 nm. The emission spectra, up-conversion fluorescence spectra, and fluorescence decay curves were detected and recorded using the Edinburgh Instruments FLS920 and FSP920 spectrophotometers. The repetition frequency of the excitation pulse for measuring the fluorescence decay curves was set to 20 Hz, and the duration of the excitation pulses was 30s. All the measurements were performed at room temperature.

Calculation Method
In the framework of density functional theory, VASP codes and the plane-wave basis set method were used for calculation [38,39]. The mutual interactions were described by the projector augmented-wave pseudopotential with an exchange-correlation function (Perdew-Burke-Ernzerhof form) [40,41]. The cut-off was set at 550 eV and a 1 × 1 × 1 Gamma k-grid was used to guarantee the relaxation accuracy of 10 −5 eV and 0.01 eVÅ −1 within a 2 × 2 × 2 supercell, respectively. The spin polarization was included in the calculations. According to the method reported previously [42], the formation energy (∆E) and cluster symbols were obtained. It is pointed that the energy correction of the PbF2 crystal was different from that of CaF2, SrF2, and BaF2 crystals. For a 2 × 2 × 2 supercell with a net charge, the calculated value in PbF2 crystal was 0.069 eV. Figure 1 shows the XRD patterns and refined XRD patterns of the Er: PbF2, Tm: PbF2, Er/Tm: PbF2 crystals, and the JCPDS standard card of the PbF2 crystal (nos. 06-2051) [37]. The residuals of refinements (fit profiles shown in Figure 1) of Er: PbF2, Tm: PbF2, Er/Tm: PbF2 crystals were 9.61%, 7.71%, 10.17%, respectively. It is obvious that no clear shift in the phase diffraction peaks was observed and all XRD curves were well matched with the standard card of the β-PbF2 crystalline phase (nos. 06-2051). The results demonstrate successful co-doping of Er 3+ and Tm 3+ ions in PbF2 crystal without phase transitions.

First-Principles Calculations
Based on the first-principal calculations, the cluster structure of Tm 3+ and Er 3+ ions were simulated to research the change of local structures of doping ions in PbF 2 crystal. The possible thermodynamically stable Er 3+ and Tm 3+ centers in PbF 2 crystals are shown in Figure 2a,b. It is clear to see that there are 9 different types of centers in each Tm: PbF 2 and Er: PbF 2 crystals. In particular, only the 3 1 |0|8|4 1 -C center in the Tm: PbF 2 crystal varies from the 2 1 |0|6|3 1 center in the Er: PbF 2 crystal, and the other eight different types of centers in the Tm: PbF 2 crystal are the same as the Er: PbF 2 crystal. Moreover, Figure 2c shows the formation energy of Er 3+ and Tm 3+ versus the number of Er 3+ and Tm 3+ ions within a cluster, respectively. It can be seen that the slope of Er 3+ clusters in PbF 2 crystal is −0.988 eV, which is almost the same as the slope of Tm 3+ clusters in the PbF 2 crystal (−1.003 eV). These results indicate that the clustering characteristics of Er 3+ and Tm 3+ ions in PbF 2 crystal are almost consistent. This phenomenon agreed well with the approximately equal segregator coefficients of Er 3+ (1.15) and Tm 3+ (1.16) in the Er/Tm: PbF 2 crystal mentioned above, which may be owing to the slightly different ion radii between Er 3+ (88.1 pm) and Tm 3+ (86.9 pm) ions. That is to say, it can be considered that the Er 3+ and Tm 3+ ions replace Pb 2+ ions with equal probability when they are co-doped in PbF 2 crystal, which makes the Er 3+ and Tm 3+ ions more evenly distributed in the PbF 2 crystal. The results suggest that the efficient energy transfer between Er 3+ and Tm 3+ ions can be guaranteed due to the uniform distribution of Er 3+ and Tm 3+ ions, and result in the enhancing of 2.78 µm MIR fluorescence emission in the ensuing discussion.

First-Principles Calculations
Based on the first-principal calculations, the cluster structure of Tm 3+ and Er 3+ ions were simulated to research the change of local structures of doping ions in PbF2 crystal. The possible thermodynamically stable Er 3+ and Tm 3+ centers in PbF2 crystals are shown in Figure 2a,b. It is clear to see that there are 9 different types of centers in each Tm: PbF2 and Er: PbF2 crystals. In particular, only the 31|0|8|41-C center in the Tm: PbF2 crystal varies from the 21|0|6|31 center in the Er: PbF2 crystal, and the other eight different types of centers in the Tm: PbF2 crystal are the same as the Er: PbF2 crystal. Moreover, Figure 2c shows the formation energy of Er 3+ and Tm 3+ versus the number of Er 3+ and Tm 3+ ions within a cluster, respectively. It can be seen that the slope of Er 3+ clusters in PbF2 crystal is −0.988 eV, which is almost the same as the slope of Tm 3+ clusters in the PbF2 crystal (−1.003 eV). These results indicate that the clustering characteristics of Er 3+ and Tm 3+ ions in PbF2 crystal are almost consistent. This phenomenon agreed well with the approximately equal segregator coefficients of Er 3+ (1.15) and Tm 3+ (1.16) in the Er/Tm: PbF2 crystal mentioned above, which may be owing to the slightly different ion radii between Er 3+ (88.1 pm) and Tm 3+ (86.9 pm) ions. That is to say, it can be considered that the Er 3+ and Tm 3+ ions replace Pb 2+ ions with equal probability when they are co-doped in PbF2 crystal, which makes the Er 3+ and Tm 3+ ions more evenly distributed in the PbF2 crystal. The results suggest that the efficient energy transfer between Er 3+ and Tm 3+ ions can be guaranteed due to the uniform distribution of Er 3+ and Tm 3+ ions, and result in the enhancing of 2.78 μm MIR fluorescence emission in the ensuing discussion.

Absorption Spectroscopy
The illustrations in Figure 3 shows the photos of Er/Tm: PbF 2 , Er: PbF 2 , Tm: PbF 2 crystals and their cut and polished crystal pieces; their sizes are also marked, respectively. It can be seen that all the crystal pieces are transparent and have no inclusions. Figure 3 illustrates the room temperature absorption spectra of Er: PbF 2 , Tm: PbF 2 , and Er/Tm: PbF 2 crystals ranging from 400 nm to 2200 nm. Clearly, the typical absorption bands centered at approximately 417, 451, 486, 521, 541, 650, 802, 975, and 1509 nm in the Er: PbF 2 crystal originated from the transitions from the ground state 4 I 15/2 level to upperlying 2 H 9/2 , 4 F 5/2 , 3/2 , 4 F 7/2 , 2 H 11/2 , 4 S 3/2 , 4 F 9/2 , 4 I 9/2 , 4 I 11/2 and 4 I 13/2 levels of Er 3+ ion, respectively [37]. While in the Tm: PbF 2 crystal mainly five absorption bands of Tm 3+ ion are labeled, the absorption peaks centered at round 464, 680, 792, 1211, and 1618 nm are in accord with the transitions from ground state 3 H 6 level to upper-lying 1 G 4 , 3 F 2 , 3 , 3 H 4 , 3 H 5 and 3 F 4 levels, respectively. Obviously, the huge absorption band centered at around 792 nm in the range of 750-830 nm corresponding to Tm 3+ : 3 H 6 → 3 H 4 transition well coincides with the wavelength of 808 nm AlGaAs LD pumping. The absorption bands in the Er/Tm: PbF 2 crystal are altogether composed of the transitions of Er 3+ and Tm 3+ ions discussed above, indicating the successful introduction of both Er 3+ and Tm 3+ ions. Strong overlap between the Tm 3+ : 3 H 6 → 3 H 4 absorption transition and the Er 3+ : 4 I 15/2 → 4 I 9/2 absorption transition can be seen in the Er/Tm: PbF 2 crystal. The absorption overlap indicates that a possible nonradiative energy transfer process Tm 3+ : 3 H 4 → Er 3+ : 4 I 9/2 would effectively occur for enhancing the absorption efficiency of Er 3+ ion~800 nm. Therefore, benefiting from the broad absorption band of Tm 3+ ion centered at around LD pump wavelength and the possibility for energy transfer, the Tm 3+ ion can act as a suitable sensitizer for Er 3+ ion in the Er/Tm dual-doped PbF 2 crystal.
AlGaAs LD pumping. The absorption bands in the Er/Tm: PbF2 crystal are altogether composed of the transitions of Er 3+ and Tm 3+ ions discussed above, indicating the successful introduction of both Er 3+ and Tm 3+ ions. Strong overlap between the Tm 3+ : 3 H6 → 3 H4 absorption transition and the Er 3+ : 4 I15/2 → 4 I9/2 absorption transition can be seen in the Er/Tm: PbF2 crystal. The absorption overlap indicates that a possible nonradiative energy transfer process Tm 3+ : 3 H4 → Er 3+ : 4 I9/2 would effectively occur for enhancing the absorption efficiency of Er 3+ ion ~800 nm. Therefore, benefiting from the broad absorption band of Tm 3+ ion centered at around LD pump wavelength and the possibility for energy transfer, the Tm 3+ ion can act as a suitable sensitizer for Er 3+ ion in the Er/Tm dual-doped PbF2 crystal. For demonstrating the sensitization effect of Tm 3+ ion for Er 3+ ion via the Tm 3+ : 3 H4 → Er 3+ : 4 I9/2 energy transfer transition, the lifetimes of Tm 3+ : 3 H4 level in the Tm 3+ single-doped and Er/Tm dual-doped PbF2 crystals were measured and shown in Figure 4a,b, respectively. The decay curves were measured under the condition of 1.47 μm emission (Tm 3+ : 3 H4 → 3 F4) and 800 nm excitation (Tm 3+ : 3 H6 → 3 H4) and were all well fitted by single-exponential behavior. As shown in Figure 4a, the measured lifetime of the Tm 3+ : 3 H4 manifold is 1.67 ms in the Tm: PbF2 crystal, while the lifetime is 0.54 ms in the Er/Tm: PbF2 crystal shown in Figure 4b. The remarkable decreasing lifetime in the Er/Tm: PbF2 crystal indicates the effective sensitization effect of the Tm 3+ ion. The energy transfer efficiency from Tm 3+ : 3 H4 to Er 3+ : 4 I9/2 level can be calculated by the following equation: ηET1 = 1 − τEr/Tm/τTm, where τEr/Tm and τTm are the lifetimes of Tm 3+ : 3 H4 level in Tm: PbF2, Er/Tm: PbF2 crystals, respectively. The high value of ηET1 (67.66%) confirms that the Tm 3+ ion has a significant influence on Er 3+ : 4 I9/2 level in PbF2 crystal, and can effectively act as a sensitizer for Er 3+ ion for enhancing ~2.7 μm MIR emission. For demonstrating the sensitization effect of Tm 3+ ion for Er 3+ ion via the Tm 3+ : 3 H 4 → Er 3+ : 4 I 9/2 energy transfer transition, the lifetimes of Tm 3+ : 3 H 4 level in the Tm 3+ singledoped and Er/Tm dual-doped PbF 2 crystals were measured and shown in Figure 4a,b, respectively. The decay curves were measured under the condition of 1.47 µm emission (Tm 3+ : 3 H 4 → 3 F 4 ) and 800 nm excitation (Tm 3+ : 3 H 6 → 3 H 4 ) and were all well fitted by single-exponential behavior. As shown in Figure 4a, the measured lifetime of the Tm 3+ : 3 H 4 manifold is 1.67 ms in the Tm: PbF 2 crystal, while the lifetime is 0.54 ms in the Er/Tm: PbF 2 crystal shown in Figure 4b. The remarkable decreasing lifetime in the Er/Tm: PbF 2 crystal indicates the effective sensitization effect of the Tm 3+ ion. The energy transfer efficiency from Tm 3+ : 3 H 4 to Er 3+ : 4 I 9/2 level can be calculated by the following equation: η ET1 = 1 − τ Er/Tm /τ Tm , where τ Er/Tm and τ Tm are the lifetimes of Tm 3+ : 3 H 4 level in Tm: PbF 2 , Er/Tm: PbF 2 crystals, respectively. The high value of η ET1 (67.66%) confirms that the Tm 3+ ion has a significant influence on Er 3+ : 4 I 9/2 level in PbF 2 crystal, and can effectively act as a sensitizer for Er 3+ ion for enhancing~2.7 µm MIR emission.

Emission Spectra and Emission Cross-Sections
For further clarifying the energy transfer mechanism between Tm 3+ and Er 3+ ions, the emission spectra of Er/Tm: PbF2, Er: PbF2 samples in the range of 1400-1700 nm, and Er/Tm: PbF2, Tm: PbF2 samples in the 1700-2200 nm region are shown in Figure 5a,b, respectively. The test parameters of the luminescence performance of the prepared samples, such as pump power and slits, are uniformed. As shown in Figure 5a, compared with the Er: PbF2 crystal the emission intensity centered at around 1.55 μm corresponding to the Er 3+ : 4 I13/2 → 4 I15/2 transition in the Er/Tm: PbF2 crystal weakened sharply, at almost ten times lower. The result shows that the introduction of Tm 3+ ion would significantly reduce the population of the Er 3+ : 4 I13/2 energy level, thereby enhancing the ~2.7 μm mid-infrared emission and reversely weakening the 1.55 μm infrared emission. This depopulation of Er 3+ : 4 I13/2 energy level is mainly attributed to the deactivation effect of Tm 3+ ions via energy transfer process: Er 3+ : 4 I13/2 → Tm 3+ : 3 F4 in Er/Tm: PbF2 crystal. As the deactivation energy transfer process occurs, the population on the Tm 3+ : 3 F4 level would increase, thereby enhancing the 1.91 μm emission (Tm 3+ : 3 F4 → 3 H6 transition) in the Er/Tm: PbF2 crystal, but it is actually weakened (shown in Figure 5b). The 1.91 μm emission intensity of the Tm 3+ ion in Er/Tm: PbF2 crystal is nearly three times lower than that in the Tm 3+ single doped PbF2 crystal. This result is mainly assigned to the cross-relaxation (CR) process between Tm 3+ and Er 3+ ions (Tm 3+ : 3 F4 + Er 3+ : 4 I13/2 → Tm 3+ : 3 H4 + Er 3+ : 4 I15/2), bringing about the depopulation of the Tm 3+ : 3 F4 level and Er 3+ : 4 I13/2 level. Therefore, the reduced emission intensity of 1.55 μm of Er 3+ ion and 1.91 μm of Tm 3+ ion both would depopulate the ions on the Er 3+ : 4 I13/2 level, which is beneficial to enhance ~2.7 μm MIR emission. More importantly, as shown in Figure 6, the emission intensity of the Er/Tm: PbF2 crystal centered at around 2.7 μm in the 2500-3100 nm region is remarkably larger than that of the Er: PbF2 crystal, confirming that the efficient enhanced ~2.7 μm emission is achieved in the Er/Tm: PbF2 designed crystal. To further confirm the prospects of Er: PbF2, Er/Tm: PbF2 crystals as the mid-infrared luminescent material in laser applications, the 2.78 μm emission cross-sections are subsequently calculated according to the Fuchtbauere-Ladenburg theory [43]:

Emission Spectra and Emission Cross-Sections
For further clarifying the energy transfer mechanism between Tm 3+ and Er 3+ ions, the emission spectra of Er/Tm: PbF 2 , Er: PbF 2 samples in the range of 1400-1700 nm, and Er/Tm: PbF 2 , Tm: PbF 2 samples in the 1700-2200 nm region are shown in Figure 5a,b, respectively. The test parameters of the luminescence performance of the prepared samples, such as pump power and slits, are uniformed. As shown in Figure 5a, compared with the Er: PbF 2 crystal the emission intensity centered at around 1.55 µm corresponding to the Er 3+ : 4 I 13/2 → 4 I 15/2 transition in the Er/Tm: PbF 2 crystal weakened sharply, at almost ten times lower. The result shows that the introduction of Tm 3+ ion would significantly reduce the population of the Er 3+ : 4 I 13/2 energy level, thereby enhancing the~2.7 µm mid-infrared emission and reversely weakening the 1.55 µm infrared emission. This depopulation of Er 3+ : 4 I 13/2 energy level is mainly attributed to the deactivation effect of Tm 3+ ions via energy transfer process: Er 3+ : 4 I 13/2 → Tm 3+ : 3 F 4 in Er/Tm: PbF 2 crystal. As the deactivation energy transfer process occurs, the population on the Tm 3+ : 3 F 4 level would increase, thereby enhancing the 1.91 µm emission (Tm 3+ : 3 F 4 → 3 H 6 transition) in the Er/Tm: PbF 2 crystal, but it is actually weakened (shown in Figure 5b). The 1.91 µm emission intensity of the Tm 3+ ion in Er/Tm: PbF 2 crystal is nearly three times lower than that in the Tm 3+ single doped PbF 2 crystal. This result is mainly assigned to the crossrelaxation (CR) process between Tm 3+ and Er 3+ ions (Tm 3+ : 3 F 4 + Er 3+ : 4 I 13/2 → Tm 3+ : 3 H 4 + Er 3+ : 4 I 15/2 ), bringing about the depopulation of the Tm 3+ : 3 F 4 level and Er 3+ : 4 I 13/2 level. Therefore, the reduced emission intensity of 1.55 µm of Er 3+ ion and 1.91 µm of Tm 3+ ion both would depopulate the ions on the Er 3+ : 4 I 13/2 level, which is beneficial to enhance~2.7 µm MIR emission. More importantly, as shown in Figure 6, the emission intensity of the Er/Tm: PbF 2 crystal centered at around 2.7 µm in the 2500-3100 nm region is remarkably larger than that of the Er: PbF 2 crystal, confirming that the efficient enhanced 2.7 µm emission is achieved in the Er/Tm: PbF 2 designed crystal. To further confirm the prospects of Er: PbF 2 , Er/Tm: PbF 2 crystals as the mid-infrared luminescent material in laser applications, the 2.78 µm emission cross-sections are subsequently calculated according to the Fuchtbauere-Ladenburg theory [43]: where λ denotes the wavelength of fluorescence spectrum, I (λ) is the intensity of emission spectrum at λ, I(λ)/ λI(λ)dλ is the normalized line shape function of the emission spectrum of prepared crystal, n is the refractive index of PbF 2 crystal, c is the speed of light in a vacuum, β is the fluorescence branching ratio of 4 I 11/2 → 4 I 13/2 transition, and A is

Energy Transfer Mechanism between Tm 3+ and Er 3+ Ions
Based on spectroscopic results discussed above, the simplified energy level scheme and electron transitions of the Er 3+ /Tm 3+ co-doped PbF 2 crystal are presented in Figure 7. The cyclic related processes of the Tm 3+ and Er 3+ ions in the crystal under optical excitation are as follows: cross-relaxation, energy transfer between Tm 3+ and Er 3+ ions, and multiphonon relaxation. The main two ET (namely ET1, ET2) and three CR (namely CR1, CR2, CR3) processes are listed as follows:

Energy Transfer Mechanism between Tm 3+ and Er 3+ Ions
Based on spectroscopic results discussed above, the simplified energy level scheme and electron transitions of the Er 3+ /Tm 3+ co-doped PbF2 crystal are presented in Figure 7. The cyclic related processes of the Tm 3+ and Er 3+ ions in the crystal under optical excitation are as follows: cross-relaxation, energy transfer between Tm 3+ and Er 3+ ions, and multiphonon relaxation. The main two ET (namely ET1, ET2) and three CR (namely CR1, CR2, CR3) processes are listed as follows: ET 1:  As discussed, the Tm 3+ : 3 H4 → 3 H6 transition is resonant with the Er 3+ : 4 I15/2 → 4 I9/2 transition in the Er/Tm: PbF2 crystal. Therefore, after the crystal is excited to the Tm 3+ : 3 H4 level by a pump of 800 nm LD, ET1 process Tm 3+ : 3 H4 → Er 3+ : 4 I9/2 would occur. Ions in the Er 3+ : 4 I9/2 level decay non-radiatively to the lower Er 3+ : 4 I11/2 level, and then decay radiatively to the Er 3+ : 4 I13/2 level and emit 2.78 μm mid-infrared light. Ions in the Er 3+ : 4 I13/2 level continue to decay radiatively to the ground state Er 3+ : 4 I15/2 level and emit 1.55 μm infrared light. Similarly, the Er 3+ : 4 I13/2 → 4 I15/2 transition is resonant with the Tm 3+ : 3 H6 → 3 F4 transition, and the ET2 process from Er 3+ : 4 I13/2 to Tm 3+ : 3 F4 level takes place. The ET2 process would reduce the population of the lower level of Er 3+ : 4 I13/2, thereby enhancing the 2.78 μm emis- As discussed, the Tm 3+ : 3 H 4 → 3 H 6 transition is resonant with the Er 3+ : 4 I 15/2 → 4 I 9/2 transition in the Er/Tm: PbF 2 crystal. Therefore, after the crystal is excited to the Tm 3+ : 3 H 4 level by a pump of 800 nm LD, ET1 process Tm 3+ : 3 H 4 → Er 3+ : 4 I 9/2 would occur. Ions in the Er 3+ : 4 I 9/2 level decay non-radiatively to the lower Er 3+ : 4 I 11/2 level, and then decay radiatively to the Er 3+ : 4 I 13/2 level and emit 2.78 µm mid-infrared light. Ions in the Er 3+ : 4 I 13/2 level continue to decay radiatively to the ground state Er 3+ : 4 I 15/2 level and emit 1.55 µm infrared light. Similarly, the Er 3+ : 4 I 13/2 → 4 I 15/2 transition is resonant with the Tm 3+ : 3 H 6 → 3 F 4 transition, and the ET2 process from Er 3+ : 4 I 13/2 to Tm 3+ : 3 F 4 level takes place. The ET2 process would reduce the population of the lower level of Er 3+ : 4 I 13/2 , thereby enhancing the 2.78 µm emission and weakening the 1.55 µm emission, as shown in Figures 5a and 6. Meantime, the energy transfer up-conversion (UC) CR3 process (Er 3+ : 2 4 I 13/2 → 4 I 15/2 + 4 I 9/2 ) in the crystal can also populate the Er 3+ : 4 I 11/2 level and depopulate the Er 3+ : 4 I 13/2 level. Additionally, ions in the Tm 3+ : 3 F 4 level decay radiatively to the 3 H 6 level and emit 1.91 µm emission. The subsequent CR1 populates the Er 3+ : 4 I 9/2 level, and then the Er 3+ : 4 I 11/2 level is populated through the nonradiative decay from the 4 I 9/2 level to the 4 I 11/2 level, increasing the population ratio of 4 I 11/2 / 4 I 13/2 levels. Moreover, the ions in the Tm 3+ : 3 F 4 energy level will also absorb energy and jump to the upper Tm 3+ : 3 H 4 energy level due to Stark level splitting, and then the CR2 process described above occurs . The CR2 process can simultaneously reduce the population Er 3+ : 4 I 13/2 , Tm 3+ : 3 F 4 levels, to achieve 2.78 µm emission enhancement and 1.91 µm emission reduction, as shown in Figures 5b and 6. The CR2 process also brings about the increasing population of the Tm 3+ : 3 H 4 level. Besides emitting 1.47 µm light via the Tm 3+ : 3 H 4 → 3 F 4 transition , ions in the Tm 3+ : 3 H 4 level can populate the Er 3+ : 4 I 9/2 level via ET1 process, resulting in further enhancement of the sensitization effect. To prove the CR2 process, the UC emission spectra of Er: PbF 2 and Er/Tm: PbF 2 crystals are shown in Figure 8 under 980 nm excitation. Clearly, as shown in Figure 7, under 980 nm NIR light excitation, the electrons in the ground level 4 I 15/2 can be excited to the intermediate level 4 I 11/2 , and the electrons in the 4 I 11/2 level sequentially populate the 4 F 7/2 level ( 4 I 15/2 → 4 I 11/2 → 4 F 7/2 ). Additionally, then, the multiple nonradiative multi-phonon relaxation in the 4 F 7/2 state in turn populate the lower 2 H 11/2 , 4 S 3/2 , 4 F 9/2 , and 4 I 9/2 levels, which would produce 800 nm light via the process: 4 I 9/2 → 4 I 15/2 . It is clear to see that the UC emission intensity of the Er/Tm: PbF 2 crystal is at least two times larger than that of the Er: PbF 2 crystal at around 800 nm. Obviously, Tm 3+ ions have no absorption band matching the 980 nm excitation (shown in Figure 3). This enhancing UC emissions phenomenon is possibly assigned to the CR2 and ET1 mechanism processes illustrated in Figure 7. To summarize, the ET1, ET2, CR1, CR2, CR3 processes all have significant effects on narrowing the lifetime gap of upper-lying Er 3+ : 4 I 11/2 and lower-lying Er 3+ : 4 I 13/2 levels or even achieving population conversion of these two levels, thereby obtaining efficient enhanced 2.78 µm emission. Additionally, then, the multiple nonradiative multi-phonon relaxation in the 4 F7/2 state in turn populate the lower 2 H11/2, 4 S3/2, 4 F9/2, and 4 I9/2 levels, which would produce 800 nm light via the process: 4 I9/2 → 4 I15/2. It is clear to see that the UC emission intensity of the Er/Tm: PbF2 crystal is at least two times larger than that of the Er: PbF2 crystal at around 800 nm. Obviously, Tm 3+ ions have no absorption band matching the 980 nm excitation (shown in Figure 3). This enhancing UC emissions phenomenon is possibly assigned to the CR2 and ET1 mechanism processes illustrated in Figure 7. To summarize, the ET1, ET2, CR1, CR2, CR3 processes all have significant effects on narrowing the lifetime gap of upper-lying Er 3+ : 4 I11/2 and lower-lying Er 3+ : 4 I13/2 levels or even achieving population conversion of these two levels, thereby obtaining efficient enhanced 2.78 μm emission.

Conclusions
In summary, Er 3+ : PbF 2 , Tm 3+ : PbF 2 , and Er 3+ /Tm 3+ : PbF 2 crystals were prepared successfully by the Bridgman technique. An efficient enhanced 2.78 µm emission was obtained in the Er/Tm: PbF 2 crystal for the first time, and the proposed energy transfer mechanism of the Er/Tm: PbF 2 crystal was systematically investigated. The theoretical calculations were performed to discover that the co-doping of Tm 3+ ions can make the Er 3+ and Tm 3+ ions more evenly distributed in PbF 2 crystals, which can effectively break the local clusters of Er 3+ in Er: PbF 2 crystal, thus ensuring efficient energy transfer between Er 3+ and Tm 3+ ions, and resulting in the enhancing of 2.78 µm MIR fluorescence emission. The cyclic energy transfer mechanism contains several energy transfer processes and crossrelaxation processes, which all have significant effects on narrowing the lifetime gap of upper-lying Er 3+ : 4 I 11/2 and lower-lying Er 3+ : 4 I 13/2 levels or even achieving population conversion of these two levels. As proved, the Tm 3+ ion can simultaneously act as an appropriate sensitized and deactivated ion for the Er 3+ ion in the PbF 2 crystal. Compared with the Er 3+ single-doped crystal, the Er 3+ /Tm 3+ co-doped PbF 2 crystal has the larger 2.78 µm mid-infrared fluorescence emission intensity, higher fluorescence branching ratio (20.24%), and higher stimulated emission cross-section (0.63 ×10 −20 cm 2 ), corresponding to Er 3+ : 4 I 11/2 → 4 I 13/2 transition. Therefore, the introduction of Tm 3+ ions is favorable for achieving efficient enhanced 2.78 µm emission in the Er/Tm: PbF 2 crystal, which can become a promising material for low threshold, and high-efficiency mid-infrared laser applications under the pump of a conventional 800 nm LD.