Investigation of crystal field effects for the spectral broadening of Yb3+-doped LuxY2−xO3 sesquioxide crystals

The investigation of crystal field effects is significant for elucidating the spectral characteristics of Yb3+-doped sesquioxide crystals for ultrafast laser generation. The narrow spectra of Yb3+-doped single sesquioxide crystals limit the generation of ultrafast lasers; in this study, the Y3+ ions were introduced into Lu2O3 single crystals by the employment of ion replacement to broaden the spectra. To analyze the spectral broadening, the responsible crystal field parameters (CFPs) were calculated. The conversion of the host dominant ion and the distortion of the ligand affected the values and signs of the CFPs, and further determined the energy level splitting and fluorescence spectra. A linear relationship expressed by the semi-empirical equations for Yb3+-doped sesquioxide crystals was produced, which could be used for high throughput spectral prediction. Opposite variations of high- and low-frequency vibrational energies and the influence of the electron–phonon coupling on the spectra were also achieved. The redshift from the crystal field and the blueshift from the electron–phonon coupling make the optimal spectral broadening appear when x = 1.19 in the Yb:LuxY2−xO3 crystals. The results of these analyses could provide some key clues for the development of Yb3+-doped crystals for the generation and amplification of ultrafast lasers.


Introduction
The crystal eld theory, which describes how the crystal eld inuences active ions, has been used to analyze the relationship between the spectroscopic characteristics and the crystallographic structure of laser gain materials. [1][2][3][4] The investigation of crystal eld effects is demanded for Yb 3+ -based laser gain materials for a 1 mm emission, owing to the fact that energy level distributions are sensitive to crystal eld interactions. The weaker shielding effect of outer 5s 2 5p 6 for 4f 13 shell electrons in Yb 3+ ions enhances the effect of the crystal eld compared with other lanthanide ions. 5 A main feature of Yb 3+ -based lasers is their quasi-three-level operating scheme, in which the fundamental and terminal laser levels belong to the same groundstate manifold. In order to limit the thermal population of the terminal laser level for highly efficient laser generation, a large energy level splitting is required. 6,7 It can be realized in Yb 3+doped cubic sesquioxide (Re 2 O 3 , Re ¼ Lu, Y, and Sc) through providing a strong crystal eld in comparison with various Yb 3+doped materials. 8,9 The adequate spectroscopic and thermal capabilities possessed by Yb 3+ -doped cubic sesquioxides facilitate their application in ultrafast laser operations. [10][11][12] The Yb:Lu 2 O 3 thindisk laser (TDL) generated a pulse width of 96 fs with the highest average output power of 21.1 W in the sub-100 fs regime. 13 However, the insufficient spectral width of Yb 3+ -doped pure sesquioxide (e.g. Yb:Lu 2 O 3 : $13 nm, Yb:Sc 2 O 3 : $12 nm, Yb:Y 2 O 3 : $14 nm) 14 limits the development of ultrashort pulse generation. The Yb 3+ -doped sesquioxide solid solutions formed by host mixing are effective for broadening the emission spectra and obtaining short pulse widths. [15][16][17][18][19][20] C. J. Saraceno et al. used Yb:Lu 2 O 3 with a full width at half maximum (FWHM) emission spectrum of 13 nm and the mixed sesquioxide Yb:LuScO 3 with an FWHM of 22 nm to generate ultrashort pulses, and achieved shortened pulse durations from 142 fs to 96 fs. 14,21,22 Compared with the Sc 3+ ions, the presence of Y 3+ ions in Yb:Lu 2 O 3 largely retains the thermal characteristic, which is the best among these Yb 3+ -doped sesquioxide solid solutions. 23,24 In our previous study, a ligand engineering strategy was used to disorder coordination in the Yb:Lu x Y 2Àx O 3 crystal system and ultimately realized spectrum broadening. 25 Furthermore, crystal eld computations and analyses of Yb 3+ -doped sesquioxides are required for investigating the varying regularity of CFPs in the solid solutions and the mechanism of spectral broadening.
In this study, further analysis was conducted based on the results of the previous experimental research in the Yb:Lu x -Y 2Àx O 3 crystal system. 25 The energy level sequence of Yb 3+ ions in the coordinate environment was obtained from the decomposed uorescence and absorption spectra. The CFPs were tted with reference to the gained energy levels, and the crystal eld strength and its inuence on the energy level splitting and overall spectral broadening were analyzed. Meanwhile, the semi-empirical equations were obtained to predict the energy level splitting from the component of the solid solution. Furthermore, the vibrational modes obtained from Raman spectroscopy and calculated by rst principles were used to analyze their inuences on electron-phonon coupling and the spectral broadening mechanism. This study can serve as a basis for a deeper understanding of the crystal eld effects as well as for further research on ultrafast laser gain crystal materials.

Experimental section
The 1 at% Yb:Lu x Y 2Àx O 3 (x ¼ 0, 0.79, 0.99, 1.19, 1.39, 2) series crystals had been grown using the optical oating zone (OFZ) method. 25 The uorescence and absorption spectra of the crystals were measured at room temperature ($25 C) using a uorescence spectrometer (FLS920, Edinburgh Instruments) excited at 900 nm and an ultraviolet-visible/near-infrared region spectrophotometer (UH4150, Hitachi), respectively. The observed energy levels of Yb 3+ ions in Lu x Y 2Àx O 3 crystals were obtained from the decomposed uorescence and absorption spectra. The Raman spectra of the crystals were collected by a Raman spectrometer system (LABRAM HR-800, Horiba) excited with a 532 nm solid-state laser. Programs for superposition model tting (SMFit) and crystal eld parameter tting (CFPFit) 26,27 were executed to obtain the phenomenological CFPs in the Yb 3+ -doped sesquioxide crystal system. Two tting steps were performed: rst, the SMFit program 28 was run to obtain the intrinsic CFPs B k and initial CFPs B k q (0) using the lattice parameters and observed energy levels. Second, the obtained B k q (0) were substituted into the CFPFit program as the initial values, and the nal CFPs B k q were tted based on the observed energy levels using a numerical iteration method. The phonon state densities of the Lu 2 O 3 and Y 2 O 3 crystals were calculated using rst principles based on the density functional theory, 29 and the generalized gradient approximation with Perdew-Burke-Ernzerhof 30 was used to describe the exchange-correlation function. The linear response method was used in the phonon calculations, and the norm-conserving method was selected as the pseudopotential.

Results and discussion
Based on the X-ray diffraction results of the grown crystals, all components of the 1 at% Yb:Lu x Y 2Àx O 3 (x ¼ 0, 0.79, 0.99, 1.19, 1.39, and 2) crystals exhibited the same crystallographic structure, and their point and space groups were m 3 and Ia 3. 25 Their lattice parameters are list in Table S1. † As shown in Fig. 1(a), two cationic sites were found in the crystallographic structure, namely the centrosymmetric C 3i site denoted as Re1, and the non-centrosymmetric C 2 site denoted as Re2. The Yb 3+ ions occupied both cationic sites when incorporated into the crystal. Because the uorescence was mainly derived from the electric dipole transition of Yb 3+ ions at the non-centrosymmetric Re2site, 31 and the number of Re2 sites was larger than that of Re1, the cations at the Re2-site were the focus of this study. As shown in Fig. 1(b), distorted octahedron coordination was formed by the Re2-site central cation and its surrounding six oxygen ions. These oxygen ions were classied as three pairs according to the different Re-O bond lengths.
The 4f 13 shell electrons of Yb 3+ ion split into two manifolds under the crystal eld effect when the Yb 3+ ion is doped into the sesquioxide crystals: the ground state 2 F 7/2 with four Stark Table 1 Intrinsic crystal field parameters, residuals, and s obtained by SM fitting  splitting energy levels, and the excited state 2 F 5/2 with three Stark splitting energy levels. 32 The seven energy levels of each component of the Yb:Lu x Y 2Àx O 3 (x ¼ 0, 0.79, 0.99, 1.19, 1.39, and 2) crystals were derived according to the peak positions of the experimental uorescence and absorption spectra. Additionally, these energy levels were used as reference values for the subsequent energy-level tting.
In the Yb 3+ -doped sesquioxide system, the Hamiltonian is expressed as follows: 33 where E ave is the average energy level, which indicates the inuence of the spherically symmetric central force eld; A SO and z are respectively the angular and radial parts of the spinorbit coupling parameter. The last term indicates the inuence of the parameterized crystal eld, where B k q and C k q represent the radial and angular parts of the CFPs, respectively. The radial parts of CFPs are impossible to calculate directly and are generally obtained by tting. Furthermore, the residual R and root-mean-square deviation s are calculated to measure the difference between the calculated and experimental energy levels.
The SMFit program was developed based on the superposition model, 28 which reduced the number of initial parameters and simplied the tting calculation. For the sites occupied by Yb 3+ ions with C 2 symmetry, three intrinsic CFPs B k (k ¼ 2, 4, and 6) were considered. Additionally, there were een mutually independent parameters in the plural B k q , where k ¼ 2, 4, and 6, and q was an even integer in the range of Àk # q # k. The intrinsic CFPs B k obtained using SMFit are listed in Table 1. The CFPs B k q (0), which were used as the initial parameters of CFPFit, are listed in Table S2. † For each B k q (0) with specic k and q values, the signs were the same for different x values, indicating that the inuence of the coordination structure on the parameter signs was consistent. The CFPs B k q obtained from the CFPFit are listed in Table 2, the corresponding experimental and calculated energy levels are listed in Table S3, † and the trends of  the B k q values with respect to the host mixing contents x are shown in Fig. 2. The tting precision indices R and s improved signicantly aer CFPFit, indicating the effectiveness of a twostep calculation. The parameter values of B 4 2 and B 4 4 were the largest, which suggested that they played a major role among the CFPs. The parameter values uctuated with respect to x. As shown in Fig. 2(a), the local minimum values of each parameter were obtained at x ¼ 0.99, which were the opposite of the trends shown in Fig. 2(b). The dominant ion converted at the point of x ¼ 0.99 between Lu 3+ and Y 3+ ions in the matrix crystals, which induced the singularity of the CFPs at this component, and resulted in two opposite trends of crystal eld effects.
It was imperative to understand the variation in the parameter signs. Based on a comparison of the B k q signs of all x values before and aer the CFPFit calculation, as shown in Tables S2 † and 2, the real and imaginary parts of B 4 4 remained negative, and the imaginary parts of B 2 2 and B 4 2 remained positive. Several signs signicantly varied, such as that of B 2 0 , which changed from negative to positive, whereas the signs of the real parts of B 4 2 and B 6 2 changed from positive to negative. The point charge electrostatic model (PCEM) was considered as a potential method for parsing the relationship between structures and signs. The expression for the B k q parameters in the PCEM contains a summation over discrete point charges situated at the positions of the ligands: 33 where hr k i is the radial integral, Z L e 2 represents the multiplication of the charges of the L-th ligand and electron, R L is the Fig. 3 Sign variation of the CFPs with respect to the coordination ion angle q L and 4 L (red: positive, green: negative).
distance between the L-th ligand and central ion, and the subscript L indicates the L-th coordination ion. The only part that determines the sign is the angular part expanding with spherical harmonics, and the angles q L and 4 L of the coordination ions are the decisive factors. The spherical coordinate system was established with the central cation as the coordinate origin and the polar axis along the crystallographic c axis. The polar angle q L and azimuth angle 4 L of the coordinate oxygen ions are shown in Fig. 1(b). Assuming that each pair of O 2À ions has a corresponding structural distortion state, one ion from each pair of O 2À ions was selected as the observation object and denoted as O1, O2, and O3, respectively. According to the structural analysis results, the q L values of O1, O2, and O3 were approximately 136 , 57 , and 69 , respectively; and their 4 L values were approximately 42 , 50 , and 139 , respectively. Under the inuence of lattice vibration, the positions of the O 2À ions relative to the central cation changed in accordance with the ligand distortion. Assuming that the angle deviation is within 10 , we simulated the sign part of eqn (2) and obtained the results shown in Fig. 3. As is evident from the gure, with a gradual decrease in q 2 and gradual increase in q 1 , 4 1 , q 3 , and 4 3 , the real parts of B 4 2 and B 6 2 tend to be negative, whereas B 2 0 and the imaginary parts of B 6 4 tend to be positive.
The crystal eld strength parameter N n is a simplied description of the crystal eld and is expressed as follows: 33 The calculated crystal eld strengths of the Yb:Lu x Y 2Àx O 3 crystal series are shown in Fig. 4(a) 34 and 35) were different, and the total cationic radius (r ion ) varied with respect to the mixing content x of the host crystal. Given that r(Y 3+ ) > r(Lu 3+ ), r ion gradually increased as x gradually decreased, as shown in Fig. 4(a); N n was negatively correlated with r ion . The dashed line in Fig. 4(a) is the crystal eld strength line tted by the Yb:Lu x Sc 2Àx O 3 crystal series, 36 which is expressed as follows.
The relationship between the two crystal series was consistent, and the semi-empirical formula is suitable for various Yb 3+ -doped cubic sesquioxide crystal systems. Stark splitting occurs at the Yb 3+ ion energy levels under the inuence of the crystal eld. 5 The differences in the split energy levels of the ground state (DE 7/2 ) and the excited state (DE 5/2 ) are shown in Fig. 4(b). With an increase in x and the crystal eld strength, the extent of splitting increased. Overall, DE 7/2 > DE 5/2 , given that 2 F 7/2 exhibits one more splitting energy level than 2 F 5/ 2 , thus resulting in a greater difference. The average energy level E ave of the Yb 3+ ions with respect to x is shown in Fig. 4(b). Because of the energy-level expansion caused by splitting, E ave increased with x. Moreover, E ave acted as a spherically symmetric central force eld in the Hamiltonian.
The crystal eld strength N n is related to the energy level splitting, which has been deduced by F Auzel et al. using relation equations. 37 For the ground state 2 F 7/2 of Yb 3+ ions, the relation is expressed as DE( 2 F 7/2 ) ¼ 0.246N( 2 F 7/2 ); however, the splitting of Yb 3+ ion in sesquioxide was beyond the predictive value. 6 Here, by combining the trends from Yb:Lu x Sc 2Àx O 3 and Yb:Lu x Y 2Àx O 3 crystals, two semi-empirical equations were proposed for dening the relation between the crystal eld strength N n and the maxima splitting DE J of ground (J ¼ 7/2) and excited (J ¼ 5/2) states of Yb 3+ ion in sesquioxide, which are expressed as: These corresponding straight lines are also represented in Fig. 4(c), and most components of the Yb 3+ doped sesquioxide t the trend lines well. The relations described by the semiempirical eqn (4)-(6) can be used to predict the energy level splitting from the determined mixed content and ionic radii.
The variation in the spin-orbit coupling parameters z with respect to x is shown in Fig. 4(d). Theoretically, the spin-orbit coupling parameter is a free ion parameter that increases rapidly with an increase in the atomic number Z, and z is approximately 2900 cm À1 for the Yb 3+ ion. Moreover, z varied slightly with respect to x and reached its maximum value at x ¼ 0.99, which may be due to the inuence of the central force eld. Compared with the Yb 3+ -doped Lu x Sc 2Àx O 3 crystal, the variation range of z was reduced for Lu x Y 2Àx O 3 owing to the smaller radius difference and weaker disturbance of the central force eld.
Under the crystal eld effect, the splitting degree of the ground state 2 F 7/2 energy levels increased with increases in x; this caused a decrease in the spacing between the excited and terminal levels of the transition, resulting in a redshi of emission wavelength and an extension of the uorescence spectra framework. Meanwhile, lattice vibrations and electroncoupling broadened the uorescence spectra. Therefore, the vibrational states of the lattice were experimentally and numerically investigated.
The Raman spectra of the Yb:Lu x Y 2Àx O 3 (x ¼ 0, 0.79, 0.99, 1.19, 1.39, and 2) crystals are shown in Fig. 5(a). The density of phonon states of the Y 2 O 3 crystal calculated using the rstprinciples method is shown in Fig. 5(b). The calculated values were higher than the experimental values, perhaps because of the different conditions and the inherent defects of the computational models. 38 As is evident from the gure, oxygen ions exhibited high-frequency (>300 cm À1 ) vibrations, which could be attributed to the stretching and bending of the Re-O bonds in the lattice. The vibrational energy was negatively correlated with the bond length that was directly related to the cationic radius. Typically, the frequency of the most intense Raman peak shied from 376.7 cm À1 (Yb:Y 2 O 3 ) 39 to 391.0 cm À1 (Yb:Lu 2 O 3 ) with x increasing and the cationic radius decreasing. On the other hand, the vibrations with less than 300 cm À1 energy were primarily from the cationic vibrations, and the cationic contribution of the Re2 site (Y2) was more signicant than that of the Re1 site (Y1). It was a negative relationship between the vibrational energy and the cationic mass. As shown in Fig. 5(a), the vibration at approximately 100 cm À1 corresponding to the heavier Lu 3+ ion increased with increased x, whereas the vibration at approximately 150 cm À1 corresponding to the lighter Y 3+ ion decreased with increased x. The experimental uorescence spectra showed that the maximum broadening spectrum was 24.55 nm in the Yb:Lu x -Y 2Àx O 3 crystals at x ¼ 1.19, 25 which was expected to obtain pulse continuation of 45 fs. 40 In order to distinguish the electronic and vibrational transitions, the uorescence spectra were decomposed as shown in Fig. 6. The main transition peaks with the same positions as the spectra obtained in low temperature (77 K), researchers generally identied them as the transitions between the upper and lower energy levels which resulting from the Stark splitting. 8,41,42 The other peaks were assigned as vibrational transitions since the enhanced electron-phonon coupling strength with the increased temperature. 43,44 The electronic transitions were represented using Gauss line shapes because of the inhomogeneous broadening. Meanwhile, the vibrational transitions were represented using Lorentz line shapes because of the homogeneous broadening. As mentioned above, the redshi appeared on the wavelength from electronic transition with increased x, and the broadening from electronic transition reached the maximum at x ¼ 0.99. 25 Additionally, the lattice vibrations involved in the vibrational transition were identied. As for high-frequency vibrational transitions, the excited energy levels increased owing to the stronger crystal eld, and the terminal energy levels increased owing to the higher vibrational frequency, with increased x. This eventually stabilized the transition peaks at approximately 1006.5, 1012.5, and 1036 nm. With respect to the low-frequency vibrational transitions, the energy level spacing was increased from the opposite changes in the excited and terminal energy levels. The blueshi was observed from approximately 990 nm to 980 nm on low-frequency transition peaks with increased x. The content of x ¼ 1.19 for optimal spectral broadening was achieved considering the inuence of the redshi from the crystal eld and the blueshi from the electron-phonon coupling.

Conclusions
In this study, the crystal eld effects in Yb:Lu x Y 2Àx O 3 crystals were systematically analyzed, and the intrinsic CFPs B k and CFPs B k q were obtained by two-step tting. The CFPs B 4 q were discovered to play a major role because of their large values among all the B k q . The parameter signs were affected by ligand distortion and the changes of polar and azimuth angles were discovered. Subsequently, the semi-empirical equations describing the linear relationship between the energy level splitting, crystal eld strength, and the cation radius were deduced, which could be taken as a reference in Yb 3+ -doped sesquioxides research. The energy-level splitting increased along with the x value; consequently, the spectral framework  7) to the four 2 F 7/2 energy levels (1)-(4). The vibrational transitions (i) corresponding to the phonon sidebands of (5)-(i), (i ¼ 1, 2, 3, and 4) energy levels transitions, respectively. broadened depending on crystal eld effects. Based on vibrational mode experiments and calculations, the vibrations were found to participate in spectral broadening through the electron-phonon coupling process. Specically, the high-frequency vibrations raised the intensities of the spectra and the lowfrequency vibrations brought the blueshi of the transition peaks of the spectra. The ndings of this study can therefore serve as a basis for the development of ultrafast laser applications based on other rare-earth ion-doped crystal materials.

Conflicts of interest
There are no conicts to declare.